NASA Conference Publication 3034 

Part 1 

Computational 

Methods for 

Structural 

Mechanics and 

Dynamics 

Edited by 

W. Jefferson Stroud 

Jerrold M. Housner 

John A. Tanner 

and Robert J. Hayduk 

NASA Langley Research Center 

Hampton, Virginia 

Proceedings of a workshop sponsored by 

the National Aeronautics and Space 

Administration and held at 

NASA Langley Research Center 

Hampton, Virginia 

June 19-21, 1985 

National Aeronautics and 

Space Administration 

Office of Management 

Scientific and Technical 

Information Division 

1989 
PREFACE 

This document contains the proceedings of the Workshop on Computational 

Methods for Structural Mechanics and Dynamics held at NASA Langley Research 

Center, June 19-21, 1985. The workshop was sponsored by NASA Langley 

Research Center. 

The workshop had two objectives. The first objective was to introduce to the 

structural analysis technical community a new Langley research activity in 

structural analysis called Computational Structural Mechanics, or CSM. The 

second objective was to hear experts discuss important structural analysis 

problems and methods for solving those problems. 

The workshop was organized into the following four sessions: 

1. Local/Global Nonlinear Stress Analysis - Full day - June 19 

2. Tire Modeling Half day - June 20 

3. Transient Dynamics 

4. Multi-Body Dynamics 

Half day - June 20 

Full day - June 21 

Each session closed with a panel discussion. 

Papers in these proceedings are grouped by session and identified in the contents. 

The order of the papers is the order of the presentations at the workshop. The 

proceedings also include any transcription of questions and answers that followed 

each paper and panel discussions that followed each session. 

The use of trade names or names of manufacturers in this publication does not 

constitute an official endorsement of such products or manufacturers, either 

expressed or implied, by the National Aeronautics and Space Administration. 

W. Jefferson Stroud 

Jerrold M. Housner 

John A. Tanner 

Robert J. Hayduk 

Workshop Co-Chairmen 

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iii 
CONTENTS 

PREFACE ......................................................................... iii 

CONTENTS ....................................................................... v 

INTRODUCTION: Computational Structural Mechanics Activity ...................... 1 

PART 1 

LOCAL/GLOBAL NONLINEAR STRESS ANALYSIS 

SOME ISSUES IN NUMERICAL SIMULATION OF NONLINEAR STRUCTURAL RESPONSE ............ 7 

H. D. Hibbitt 

COMPUTERIZED STRUCTURAL MECHANICS FOR 1990's: ADVANCED AIRCRAFT NEEDS .......... 25 

A. V. Viswanathan and B. F. Backman 

NONLINEAR RESPONSE OF A BLADE-STIFFENED GRAPHITE-EPOXY PANEL WITH A 

DISCONTINUOUS STIFFENER: WORK IN PROGRESS .................................... 51 

Norman F. Knight, Jr., William H. Greene, and W. Jefferson Stroud 

COMPUTATIONAL PROCEDURES FOR POSTBUCKLING OF COMPOSITE SHELLS ................... 67 

G. M. Stanley and C. A. Felippa 

A REVIEW OF SOME PROBLEMS IN GLOBAL-LOCAL STRESS ANALYSIS ....................... 89 

Richard B. Nelson 

SOME COMMENTS ON GLOBAL LOCAL ANALYSES .......................................... 103 

Satya N. Atluri 

ON COMPUTATIONAL SCHEMES FOR GLOBAL-LOCAL STRESS ANALYSIS ....................... 123 

J. N. Reddy 

GLOBAL FUNCTIONS IN GLOBAL-LOCAL FINITE-ELEMENT ANALYSIS OF LOCALIZED 

STRESSES IN PRISMATIC STRUCTURES .............................................. 135 

Stanley B. Dong 

GLOBAL-LOCAL METHODOLOGIES AND THEIR APPLICATION TO NONLINEAR ANALYSIS .......... 151 

Ahmed K. Noor 

APPLICATION OF THE P-VERSION OF THE FINITE-ELEMENT METHOD TO 

GLOBAL-LOCAL PROBLEMS .......................................................... 169 

Barna A. Szabo' 

QUESTION AND ANSWER SESSION ..................................................... 179 

PANEL DISCUSSION ................................................................. 190 

pRECEDfNG PAGE BLAFiK NOT FLMfD 

V 
TIRE MODELING 

FEATURES AND CHARACTERIZATION NEEDS OF RUBBER COMPOSITE STRUCTURES .............. 211 

Farhad Tabaddor 

NONLINEAR HIERARCHICAL SUBSTRUCTURAL PARALLELISM AND COMPUTER ARCHITECTURE ...... 249 

Joe Padovan 

ADAPTIVE METHODS, ROLLING CONTACT, AND NONCLASSICAL FRICTION LAWS ............... 269 

J. T. Oden 

CONTACT SOLUTION ALGORITHMS ..................................................... 291 

John T. Tielking 

EXPLOITING SYMMETRIES IN THE MODELING AND ANALYSIS OF TIRES ..................... 317 

Ahmed K. Noor, C. M. Andersen and John A. Tanner 

PANEL DISCUSSION ................................................................ 331 

PART 2* 

TRANSIENT DYNAMICS 

SUMMARY: TRANSIENT DYNAMICS .................................................... 341 

Robert J. Hayduk 

IMPROVING TRANSIENT ANALYSIS TECHNOLOGY FOR AIRCRAFT STRUCTURES ................. 343 

R. J. Melosh and Mladen Chargin 

EXPLICIT, IMPLICIT, AND HYBRID METHODS .......................................... 355 

T. Belytschko 

UNCONDITIONALLY STABLE CONCURRENT PROCEDURES FOR TRANSIENT 

FINITE-ELEMENT ANALYSIS ....................................................... 369 

Michael Ortiz and Bahram Nour-Omid 

TRANSIENT ANALYSIS TECHNIQUES IN PERFORMING IMPACT AND CRASH DYNAMIC STUDIES .... 383 

A. B. Pifko and R. Winter 

APPLICATION AND IMPLEMENTATION OF TRANSIENT ALGORITHMS IN COMPUTER PROGRAMS ..... 407 

David J. Benson 

QUESTION AND ANSWER SESSION ..................................................... 419 

PANEL DISCUSSION ................................................................ 425 

*Presented under separate cover. 

vi 
MULTI-BODY DYNAMICS 

RECENT DEVELOPMENTS IN DEPLOYMENT ANALYSIS SIMULATION USING A MULTI-BODY 

COMPUTER CODE ................................................................. 435 

Jerrold M. Housner 

DYNAMIC ANALYSIS OF SPACE STRUCTURES INCLUDING ELASTIC, MULTIBODY, AND 

CONTROL BEHAVIOR .............................................................. 443 

Larry Pinson and Keto Soosaar 

Part 1: DYNAMICS OF FLEXIBLE MULTI-BODY MECHANISMS AND MANIPULATORS: 

AN OVERVIEW ................................................................... 455 

Steven Dubowsky 

Part 2: APPLICATION OF FINITE-ELEMENT METHODS TO DYNAMIC ANALYSIS OF 

FLEXIBLE SPATIAL AND CO-PLANAR LINKAGE SYSTEMS ................................ 459 

Steven Dubowsky 

DYNAMICS OF ARTICULATED STRUCTURES .............................................. 491 

Edward J. Haug 

MODAL REDUCTION STRATEGIES FOR INTERCONNECTED FLEXIBLE BODIES SIMULATION ........ 517 

F. 0. Eke and G. K. Man 

COMPUTATIONAL ASPECTS OF MULTI-BODY DYNAMICS .................................... 527 

K. C. Park 

CONSTRAINT ELIMINATION IN DYNAMICAL SYSTEMS ..................................... 537 

R. P. Singh and P. W. Likins 

NONLINEAR CHARACTERISTICS OF JOINTS AS ELEMENTS OF MULTI-BODY DYNAMIC 

SYSTEMS ....................................................................... 543 

Edward F. Crawley 

QUESTION AND ANSWER SESSION ..................................................... 571 

PANEL DISCUSSION ................................................................ 589 

vi i 
PART 1 

INTRODUCTION 

Computational Structural Mechanics Activity 

NASA Langley has initiated a new research activity in structural analysis called 

Computational Structural Mechanics (CSM). The broad objective of the CSM activity 

is to develop advanced structural analysis technology that will exploit modern and 

emerging computers, such as computers with vector and parallel processing capabilities. There are three main research activities under way in CSM: (1) parallel processing, (2) software test bed, and (3) structural analysis methods. The following 

three paragraphs provide a brief description of the three main CSM research 

activities. 

Parallel processing.- Within the next 5 years, all new high-performance computers 

will have multiple processors. CSM researchers are developing software approaches 

and structural analysis algorithms that will exploit this new capability. This 

research activity has been under way for several years and has used an experimental, 

in-house-developed multiprocessor computer known as the finite-element machine 

(FEM). At this writing, FEM has 16 processors. A new commercial multiprocessor 

computer will be delivered to Langley for parallel processing research. The 

parallel processing research activity involves several in-house researchers and 

universities. 

Software test bed.- The test bed is to be a modern, modular system that handles data 

efficiently, that contains a command language which is powerful and easy to learn 

and use, and that has an architecture which allows users to add software with minimal 

difficulty. The test bed will be used to study the ingredients of modern software 

and how those ingredients should fit together. More importantly, the test bed will 

be used to study and evaluate structural analysis methods on practical applications 

problems. The test bed research is just now getting under way. It involves in-house 

researchers and contractors. 

Structural analysis methods.- The structural analysis methods research has several 

goals. One goal is to develop analysis methods that are general. This goal of 

generality leads naturally to finite-element methods, but the research will also 

include other structural analysis methods. Another goal is that the methods be 

amenable to error analysis; that is, given a physical problem and a mathematical 

model of that problem, an analyst would like to know the probable error in predicting 

a given response quantity. The ultimate objective is to specify the error tolerances 

and to use automated logic to adjust the mathematical model or solution strategy to 

obtain that accuracy. A third goal is to develop structural analysis methods that 

can exploit parallel processing computers. Our structural analysis methods research 

will focus initially on three types of problems: local/global nonlinear stress 

analysis, nonlinear transient dynamics, and tire modeling. These three types of 

problems are the topics of the workshop and are discussed in detail subsequently. 

Workshop Description 

This workshop had several objectives, among which were 

1. To introduce CSM to the structural analysis technical community, 

particularly the university community 

1 
2. To hear experts discuss important structural analysis problems and 

methods for solving those problems 

3. TO help make decisions regarding research thrusts in structural 

analysis methods development 

Three types of problems were addressed at the workshop: iocal/global nonlinear 

stress analysis, nonlinear transient dynamics, and tire modeling. The following 

is a description of these three types of problems. 

Local/global nonlinear stress analysis.- A local/global stress analysis is a local, 

detailed stress analysis within a larger, less-refined analysis model. Both the 

application and the solution procedure are taken to be arbitrary. The definition 

of local/global is not precise. Because of this ambiguity, two focus problems are 

used to define (for the workshop) what is meant by the words local and global. The 

first focus problem is a stiffened flat composite panel with a circular hole that 

causes a stiffener to be discontinuous. The panel is subjected to an in-plane compressive load in the direction of the stiffeners. The second focus problem is 

similar, except that it is curved and unstiffened. In each case, the overall 

response of the panel is the global problem; the response near the hole is the 

local problem. In their talks, speakers might not have presented solutions to the 

workshop focus problems, but their comments regarding analysis methods referred to 

the focus problems. 

Structural problems requiring a local/global stress analysis generally involve discontinuities which cause rapid changes in stress. The high stress gradients are a 

local phenomenon. Analysis procedures that must be used to predict these stress 

gradients are not required away from the discontinuity. In regions in which stresses 

vary slowly, less-refined analyses are adequate. Practical considerations demand a 

multilevel approach. It is these various levels of analysis that cause difficulties 

in the local/global problem. In the vicinity of the discontinuity, an analyst may 

refine a finite-element grid, introduce a three-dimensional grid, use more powerful 

elements, apply a classical solution, and use other analysis tools and approaches. 

Usually, the analyst will take additional steps to insure that his analysis is 

adequate. He may rework the problem with a modified model, or he may make comparisons with the results from other analysis procedures. 

To help the structural analyst overcome the difficulties just described, a systematic analysis procedure is needed that is amenable to error analysis. The analyst 

would like to prescribe the error tolerances in predicting the stress at specified 

locations and use automated logic to adjust the mathematical model or solution 

strategy to obtain that accuracy. In addition, an analyst needs the capability to 

calculate derivatives of response quantities with respect to parameters that define 

the problem. Such a sensitivity analysis is necessary for design purposes and is 

helpful in determining the validity of an analysis. 

analysis procedure that includes error analysis, adaptive solution refinement, and 

sensitivity analysis should be easy to use. The goal is for solutions to local/ 

global problems, as well as other structural analysis problems, to become routine. 

Finally, this systematic 

Tire modeling.- One of the most challenging problems in the field of Computational 

Structural Mechanics is predicting the response of automotive and aircraft tires 

during ground handling operations. The pneumatic tire exhibits a complex geometry 

which can generally be described as a noncircular, incomplete torus. Under normal 

operational conditions, these tires are subjected to large deformations and 

2 
structural rotations. The heavy weight loading requirements imposed on many aircraft and off-road vehicle tires require their carcasses to be so thick that a significant portion of the resulting strain energy is attributed to transverse shear 

deformation. Furthermore, the most interesting loading conditions associated with 

braking and steering operations are nonsymmetric. 

I 

The tire is a laminated composite structure consisting of rubber and textile constituents. The material properties of the constituents may differ by as much as 

2 or 3 orders of magnitude, thereby pushing the limits of current composite theory. 

The rubber exhibits nearly incompressible characteristics and is subject to large 

hysteresis effects. Under dynamic loading conditions, these hysteresis effects can 

lead to large thermal gradients in the tire carcass and, hence, significantly affect 

the material properties of the constituents. Furthermore, the material properties 

of the tire carcass can be highly anisotropic and nonhomogeneous. 

A central feature of any tire model must be a sophisticated contact algorithm which 

is capable of handling contact friction forces. The tire designer will also require 

the model to predict stress distributions in critical regions of the tire carcass. 

This requirement might necessitate the local integration of three-dimensional finite 

elements into an otherwise two-dimensional model. Finally, the model must demonstrate its cost effectiveness before it is generally accepted by the United States 

tire industry. 

Nonlinear transient dynamics.- During the last 2 decades there has been a growing 

interest in the numerical solution of time varying equations of structural and 

mechanical systems. Such interest comes from such diverse fields of application as 

impact mechanisms, robotic manipulation, aircraft structures and space structures. 

These applications often involve the response of systems composed of many structural 

or mechanical components. 

Dynamic analyses of these systems involve both the formulation of the governing 

equations and their solution. Usually, the governing equations are discretized 

spatially and then integrated temporally using one of a wide variety of time integration procedures. It is therefore reasonable to consider the dynamic analysis as 

consisting of two activities: a formulation and spatial modeling activity and a 

temporal solution activity. 

The formulation and spatial modeling activity involves treatment of constraints, 

controls, possible large deformations of components and large relative angular motion 

between components. It is not unusual that the formulation and modeling results in 

large sets of nonlinear governing equations. Consequently, the temporal solution of 

such systems can be computationally intensive, and it becomes the aim of the formulation and modeling activity to improve computational efficiency and computer 

implementation. 

The solution of the governing equations involves the temporal integration of equations of motion subject to initial conditions and constraints. The integration is 

usually carried out using either some type of modal superposition procedure or a 

direct time integration procedure. The modal superposition procedure, though very 

powerful for the solution of linear equations, loses much of its effectiveness in 

dealing with nonlinear equations and for this reason is not included in this volume. 

Generally, direct time integration procedures are used for dynamic analysis of 

mechanical and structural systems governed by nonlinear equations. 

3 
Direct time integration procedures are usually referred to as algorithms since it is 

the computation of the response quantities which are sought by these procedures 

rather than theorems relating to solution existence, uniqueness, etc. A very large 

number of these algorithms exist. Their popularity is attested to by the inclusion 

of such algorithms in nearly all general-purpose computer programs for analyzing 

structures, mechanisms, and satellites. Many such programs offer the user a selection of algorithms. However, for the solution of problems involving the temporal 

integration of a large number of equations, the use of some of these algorithms 

leads to very intensive computer usage due to a lack of computational efficiency. 

Not only is solution efficiency in need of improvement, but so is solution reliability. The user desires a certain level of confidence that the computed results 

are reasonably accurate. Unlike problems of statics, solution satisfaction of the 

governing equations does not provide a guarantee that the solution is correct, for 

the solution must come from a prescribed set of initial conditions. Conservation of 

energy affords a level of confidence in the solution, if such is applicable, but in 

many problems it is not applicable. Thus, both efficiency and reliability are items 

which need improvement. 

Research into developing improved formulations and modeling and their solution is 

presently being performed. However, much remains to be done. Furthermore, it takes 

considerable time for such developments to find their way into computer programs 

which are used in routine practice. Yet the application needs to continue to grow. 

Two sessions in this research area were held. One half-day session dealt exclusively 

with solution procedures, while one full-day session dealt with development and 

treatment of items critical to proper formulation and modeling of multi-body systems. 

Thirty papers dealing with the three classes of problems just discussed were presented at the workshop. In these papers, solution concepts are reviewed, computational algorithms are outlined, an assessment of various methodologies is made, and 

problem areas are identified. In addition, directions for future research are 

recommended . 

4 
LOCAL/GLOBAL NONLINEAR 

STRESS ANALYSIS 

5 
Some Issues in Numerical Simulation of Nonlinear Structural Response 

H. D. Hibbitt . Hibbitt, Karlsson and Sorensen 

Providence, R.I. 

1. Introduction 

For some considerable time the author has been involved in the development of commercial finite element software, and this paper is written from that perspective. One of the 

remarkable features of the finite element method is its generality, and there is no better 

reflection of this than the observation that our company and several of our competitors 

each market one product only-a single, “general purpose” finite element code (our particular code is called Abaqus). These codes provide practical tools that are used in an 

astonishingly wide range of engineering applications that include critical aspects of the 

safety evaluation of nuclear power plants or of heavily loaded offshore structures in the 

hostile environments of the Korth Sea or the Arctic, major design activities associated with 

the development of airframes for high strength and minimum weight, thermal analysis of 

electronic components, and the design of sports equipment. For various reasons, the code 

that my company develops and markets is generally used in the area of more advanced 

applications. These applications almost always involve nonlinear effects. There is little 

doubt in my mind that the need for such analysis will continue to grow-it is very easy to 

identify problems which should be reachable in the next generation or two of computers 

and software and which have substantial economic importance. 

The development, maintenance and support of production software involves many activities, but-at least in the more advanced application areas where we try to contributethe effectiveness of our product depends critically on the quality of the mechanics and 

mechanics related algorithms that we implement. It is generally true that the end users 

are not sophisticated with respect to what is now being called “computational mechanics.” 

They have other interests and motivations. not the least of which is the need to complete 

work successfully and to a schedule. Thus. ”algorithmic robustness” is of primary concern to us: we should choose those methods that we believe will maximize reliability with 

minimal understanding on the part of the user. Computational efficiencj- is important 

because there are always limited resources. and hence problems that we would like to do 

but which are too time consuming or costly. In reality. we compromise: for example. we 

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7 
knowingly commit what Strang and Fix (1973) call “variational crimes’‘ because we get 

awa\ Mith therri often enough for it to be worthwhile. But robustness implies a need for 

thorough understanding of the algorithms: we should at least know where the limitations 

of an algorithm are likely to be. This is not a simple task: for example, we still do not 

have practical ways or assessing local error, even in a linear numerical solution. 

It is easy to identify important practical problems that. presently, are computationally 

rather difficult. but which will become relatively routine in the not too distance future. 

Two that are taking up much of our time just now are the problem of simulating vehicle 

(especially automobile) crashes. and the simulation of rather complicated contact situations, such as the analysis of threaded connectors in drill pipe or casing which is subjected 

to very large axial forces, causing possible thread jump and quite substantial strains in the 

pipe. Both problems are modeled today on current generation computers (the Cray-1 and 

X-MP) with Abaqus and other codes. They challenge the limits of the algorithms in our 

code. and are computationally intensive-typical run times are several hours per case. The 

observations made in this paper are based on our experience to date with such problems. 

Large scale general purpose codes have a rather long life span: two codes that are 

widely used at this time (Ansys and Kastran) were begun in the mid 1960’s. Thus, in 

designing such a code, it is important to try to anticipate what sort of computers will be 

in general use for such applications in 10-15 years. Based on past history, this is a difficult 

extrapolation-computers are still developing at a rapid rate. We can make some guesses. 

For example, at Cray Research’s Science and Engineering Symposium held in Minneapolis 

in 1985 Seymore Cray discussed the specification of the Cray-3. which he expects will be 

available in three years. The most relevant parts of that specification from our point of 

view are the size of the high-speed, directly addressable, memory (10” words), and the 

machine‘s parallelism. Consider a problem of order 50000 unknowns: many important 

cases of the types mentioned above are of this size or smaller. The rms half-bandwidth in 

such a case would be about 5000, so that the assembled symmetric part of the Jacobian 

(stiffness) matrix will occupy 250 x loG words-a quarter of the available memory; the 

complete matrix will occupy half the memory. (More and more problems are likely to need 

the full Jacobian matrix in the context of Newton methods, because of the use of more 

realistic constitutive models, such as damage mechanics models for brittle or composite 

materials, which have a non-symmetric Jacobian). Element matrices, state variables for 

constitutive calculations, etc. are unlikely to occupy more than 5 x 10‘ words in such 

a case. Thus, it would seem that, for such applications. we need not worry about “I/O” 

problems-we can assume the model can run “in-core.” This means that, for our purposes, 

the practical computational limitation will be the time taken to do the arithmetic. Cray 

Research and others seem to be moving rapidly toward parallelism to provide arithmetic 

speed. This should fit well with that part of our finite element codes that perform element 

and constitutive calculations: there we process a large number of such calculation points 

(perhaps 4000 elements, 30000 integration points in the 50000 degree-of-freedom case) so 

that, provided the code is designed to allow inner loops to spread over the processors, load 

balancing should not be much of a problem. It is interesting to note that. in this part 

of the code. typical vectors are of order 10-100, so that any vector processor with a long 

start-up time (like the Cyber 205) is not very suitable. The solution of the linear equations 

7 

8 
is not such a natural fit into parallel architecture. but the problem is such an important 

one that it is likely that very effective modules will become available (such as Floating 

Point Systems now provide for their processors). Since we in fact are solving a nonlinear 

system, it may be that non-fiewton methods. not requiring the resolution of a Jacobian 

matrix, may be preferable in such an environment. I assume that load balancing is the 

critical issue in a multi-processor machine. and that the most effective approach to that 

issue is at the inner loop level, rather than the macro (finite element) level. 

The remainder of the paper discusses three principal areas. First, the paper discusses 

our experiences with the approaches we use to the two initial value problems of primary 

interest here-static and dynamic nonlinear response of structures. It then has a brief 

section on shell modeling. Then it discusses our current approach to the constitutive 

integration problem, in the context of conventional plasticity models. Finally, the paper 

lists some areas where we hope that research work will provide us with new methods, or 

with improvements to our present approaches. 

2. Experience with initial value problems 

2.1 Structural dynamics 

The nonlinear dynamic problems that we most often see involve globally nonlinear 

response-most of the structure yields and undergoes finite rotations and, possibly. finite 

strains. However. locally nonlinear dynamic problems are not uncommon: in fact they 

represent a sufficiently important application area that we have found it worthwhile to 

provide methods to model them with some efficiency. A good example of such a case is 

the flow induced vibration of a steam generator tube in a nuclear plant. The tube rattles 

in its support plates due to vibration excited by instabilities in the coolant flow, and this 

intermittent contact causes wear and, eventually, may lead to leakage. The tube itself may 

be ”sprung”-that is, the initial stress in the tube may be large enough that it forms a 

significant contribution to the tube’s stiffness-but the vibration amplitude is not large 

enough that the stiffness of the tube changes significantly during the motion. The nonlinearities are therefore confined to the support interaction, where they involve intermittent, 

“chattering” contact and friction. Our approach to such problems has been through conventional Guyan reduction. retaining the transverse displacements at the support points 

and enough other degrees of freedom in the tube to model its response accurately in the 

frequency range of interest (up to about 150 Hz in this case). The Guyan method has 

two drawbacks-it is based on guesswork, and the reduced model may be much bigger 

than necessary. Since the majority of the applications that have come to our attention 

involve relatively small models (typically less than 1000 degrees of freedom, with 30-80 

degrees of freedom directly involved in local nonlinearities). these limitations are not very 

critical: it is not much effort to extract the modes of interest on the full model and then 

to experiment with reduced models until a reasonably small one is found that provides 

adequate matching. 

Nonlinear dynamic problems are integrated explicitly or implicitly. The explicit approach brings with it several advantages because it is always used with a lumped mass 

9 
matrix and so no equations need be solved: this leads to great simplification in the computer code. But conditional stability is a serious limitation, especially in structural models 

where the thickness of the shell is usually the determining factor with respect to the time 

step. Many of the problems that we see are ”event and response” cases, in which an initial 

input of energy is mostly dissipated as inelastic response in the structure. In these cases 

the response usually damps out fairly quickly into a pattern of plastic hinges, until some 

later disturbance-perhaps a secondary impact, or a sudden effect associated with geometry changes-causes this pattern to undergo a redistribution. Indeed, this phenomenon of 

plastic hinge formation is so much a part of such problems that it has given rise to the successful mode form solution approach of P. S. Symonds, J. B. Martin, and others (Symonds, 

1967). It is difficult to accept that a conditionally stable integration method which can 

never expand the time step beyond some fraction of the shortest period exhibited by the 

model provides the optimal approach for such an application. My own experience with 

explicit methods is limited, so that I do not know how one deals with constraints (such as 

arise in mixed element formulations) within these methods. But if the methods are only 

useful when we do not solve equations, it seems that the generality of their application is 

limited. 

Implicit methods are chosen for their numerical stability. Stability is usually discussed 

in the context of linear systems, and 1 am not sure of what stability proofs exist for 

the nonlinear problems we are trying to model. My own practical experience has been 

that numerical stability of the operator has never been a limitation: except for accuracy 

considerations, the time step is most usually limited by our ability to solve the equations. 

I will return to the equation solving problem later, because it is. in our experience, a 

key issue. lmplicit methods have the great advantage of generality-the time step can be 

chosen for modeling reasons. And since we have to solve equations anyway, we are at liberty 

to introduce whatever additional equations we care to, including, for example, Lagrange 

multipliers to impose constraints. In Abaqus we have for some time been using the HilberHughes-Taylor (1978) operator (an extension of the trapezoidal rule that provides the 

ability to introduce some numerical damping) with a simple “automatic” time step selection 

scheme, and this approach has been of great practical benefit. Again, consider the car 

crashworthiness problem. A front end collision test case for a car design usually involves 

about 10-15 milliseconds of response. With the shell elements in Abaqus and a mesh 

that is adequate to model the response usefully, the stability limit of the central difference 

operator is in the microsecond range. Using the implicit method, the analysis is usually 

completed in 150-300 time increments, which range from a few microseconds just after a 

major impact up t,o a significant fraction of a millisecond during the fixed plastic hinge 

rkgime. The utility of the method then depends on whether we can solve the nonlinear 

equations 150-300 times in an acceptable amount of computer time. 

2.2 Statics 

Smoot.hly nonlinear static problems are not uncommon, but it seems that many important applications involve abrupt changes in the response. which is often unstable as well. 

The car crash problem again serves to provide an example. During a front end crash, most 

conventional designs of front rails in cars are responding unstably during about 80% of 
their usable deformation, and the switch into this collapsing response occurs quite sharply, 

as a buckling of the structure (which is a shell), after it has undergone considerable plastic 

deformation. Most shell buckling cases involve relatively thin shells, so that the equilibrium equations are not well conditioned. Elastic-plastic buckling often involves sudden 

“localization” of the plastic deformation into narrow regions, while major parts of the 

structure, which were yielding in the pre-collapse phase, unload elastically. Such problems 

make demands on the solution strategy-it must be able to detect, and to switch into, 

the alternate equilibrium path, and the instability of the collapse phase of the response 

must be handled. For the latter purpose we have found that our version of a “Riks” algorithm (Riks, 1979) has been most valuable. We use it with an L‘automati”’ incrementation 

scheme, and find that it can often march right through into the collapse as far as we want 

to go, without too much stuttering. The most common complaint that we receive is that 

the solution sometimes turns back on itself. When this occurs it is at critical points where, 

presumably, the equilibrium path has very high curvature. For the present we ascribe this 

to a weakness in our implementation only: the method finds an equilibrium path, but it 

is not the path of interest. We have nothing in our code that can detect the possibility of 

alternative equilibrium paths: for example, we know we cannot obtain sensible results for 

a round bar in compression (Euler buckling) or tension (necking), without knowing that 

the switch may occur and seeding the problem definition with a suitable imperfection. A 

robust algorithm would not need this. 

Rate form constitutive models (such as conventional plasticity or visco-plasticity theories) still must be integrated, even though the overall problem is quasi-static. There seems 

to be relatively little motivation for choosing explicit methods in general for this aspect of 

the problem, although there are some particular cases where an explicit approach makes 

sense: for example, many high temperature creep problems associated with metal structures can be treated efficiently with explicit integration of the creep model, because in these 

cases the response times of interest are usually not very long compared to characteristic 

relaxation times for the material subjected to the stresses that arise in the structureotherwise the design would be unacceptable anyway. In Abaqus, except for this particular 

case, we use implicit integration for rate plasticity models, as discussed below. 

3. Equation solving 

Almost all of the procedures we use in Abaqus are based on a “full” Newton method. 

We have tried a few alternatives (modified Newton and quasi-Newton methods), but so 

many problems of interest are not very well conditioned and exhibit such knotty response 

that we have not looked at many of our standard test cases before we have rejected the 

alternatives that we have tried, and returned to the quadratic convergence of the full 

Newton method. However. our work is hardly rigorous or complete. 

A significant part of the usage of our code involves systems for which the Jacobian 

matrix of the Newton method is not symmetric. Examples are the non-associated flow 

plasticity models that are often used in soil mechanics applications, and loading cases with 

“follower forces”, like Morison drag on offshore pipes and risers. In these cases we usually 

form and solve the non-symmetric system. There are some problems where we offer the 

11 
possibility of approximating the Jacobian with its symmetric part because our experience 

has suggested that many typical cases of that class are analysed with less computational 

expense in that way. An example is mentioned later in this paper in connection with 

integration of plasticity models. But we always provide the possibility of invoking the 

non-symmetric capability if needed, typically by restarting the simulation at soTle point, 

because it often happens that the symmetric approximation becomes less effective as the 

solution develops. 

Newton’s method introduces difficulties of its own. The most expensive is the need to 

form and solve a system of linear equations at each iteration, and the most awkward is the 

need to define the Jacobian matrix. The algebraic manipulations involved in defining this 

matrix can be formidable-and there are obviously cases when it cannot be defined, except 

perhaps numerically. For example, a recent paper on numerical methods in plasticity (Simo 

and Ortiz, 1984) contains the remark: 

“In general, however, the task of evaluating the consistent tangent 

moduli in closed form may prove exceedingly laborious. It would appear, 

therefore, that a general purpose implementation of the physically more 

compelling algorithm.. . may require the use of quasi-Newton or secantNewton methods . . .” 

This difficulty should not be underestimated. 

The straightforward approach we have been using for conventional plasticity models is 

discussed in Section 5. One further comment concerning that method is appropriate 

here: even in cases when the rate plasticity model exhibits the usual symmetry property 

obtained from assuming associated flow and a smooth yield surface, the exact Jacobian of 

the integrated model-using the integration operator that we have chosen-is not always 

symmetric. 

The full Newton method is expensive computationally. We can provide some numbers 

to quantify this on computers that are available today. Abaqus is a general purpose code 

and is used on many different computer systems, presently ranging from the Apollo 300 

to the Cray X-MP. The code is not particularly “tuned” to any system, and therefore 

should be representative of typical straightforward finite element codes (we know from 

benchmarks that this is generally true). We use some standard benchmark problems to 

estimate performance of our code on various computers: experience has shown that these 

benchmarks are reliable. One of these is a shell model. with 1000 eight-node elements. 

It has about 17000 degrees of freedom and an rms wavefront of 400 degrees of freedom. 

Such a mesh would be adequate for the typical front-end collision analysis that I used 

as an example above. On a CrayX-MP this problem takes 48 cp seconds per iteration: 

60% of that time is in the linear equation solver. and most of the rest is associated with 

element and constitutive calculations. A typical dynamic analysis of a front end collision 

needs 150-300 time increments with the implicit operator that we use, and between three 

and four iterations per increment with the “almost full Newton’‘ implementation of the 

shell in Abaqus (the initial stress terms associated with bending are not defined exactly 

in the Jacobian--there is no basic problem, we have simply not completed some lengthy 

manipulations. I do not think the terms we omit are very significant). Thus, such a crash 

12 
simulation can be expected to pass through the basic loop about 500-1200 times if our 

algorithms for time stepping, impact, etc. all are working well. This implies about 6; to 

16 hours on the Cray for the job. 

4. Shells 

A substantial part of the modeling for which Abaqus is used involves shells, much of it 

in cases where geometric and material nonlinearities dominate the response. We have 

tried to provide useful capabilities for shell modeling, but we know that there are serious 

deficiencies in what we offer. Irons wrote often about shells: his definitive account of 

SemiLoof (Irons, 1976) and his textbook (Irons and Ahmad, 1980) both contain succinct 

statements of the difficulties. My impression is that things are improving rapidly. I hope 

so: I think we are all aware of the need. 

In Abaqus we always treat shells as shells-we do not have any numerically degenerated solid elements acting as shells. The elements are formulated in terms of forces and 

moments at integration points. The behavior of the cross section is defined in closed form, 

or by numerical integration through the thickness. In most cases we use shear deformable 

elements, but the transverse shears are not considered in the constitutive calculations-we 

assume the shell is thin enough that transverse shear is not very important: it is simply 

a numerical technique that allows us to manage with low order interpolation. Low order 

elements are often desirable in practical cases. 

Abaqus includes three types of shell model: axi-symmetric shells with axi-symmetric 

deformations. general shells. and pipes and elbows with deforming sections (ovalization and 

warping). These last elements are a rather special case which turns out to be important 

in certain piping applications that arise in the nuclear power industry and in the offshore 

oil industry. We often see problems where a mixture of shell and solid modeling is needed 

(li-joints in offshore platforms; Tees in pressure vessels). We have a standard technique for 

joining the shells to the solids, based on kinematic assumptions introduced as constraints. 

The approach appears to be satisfactory for the small strain cases where we have seen 

results. 

The purely axi-symmetric case is rather simple because the problem is one dimensional. For this case we use a linear interpolation element with one integration point and a 

quadratic interpolation element with two. Our impression is that these elements are quite 

effective. Our implementation of these elements is based on a simplified finite membrane 

strain theory. The theory is similar in concept to Rodal’s thesis (Rodal and Witmer, 1979), 

except that we use somewhat different strain measures because our applications involve 

material models for which logarithmic strain seems to be appropriate, and, in a case like 

this where the principal directions do not rotate. it is easy to work directly in terms of 

this strain. The main simplifying assumptions in the theory are that the thinning of the 

shell is uniform through the thickness and is defined by an incompressibility assumption 

on the reference surface of the shell. and that the thinning occurs smoothly, so that we 

can neglect gradients of thickness change with respect to position. 

There are many applications for which a finite membrane strain model should be 

useful-obvious examples arise in sheet metal forming processes. The axi-symmetric case 

13 
14 

is particularly simple and our version of a finite membrane strain theory is quite basic. 

Nevertheless. it takes two pages of the Abaqus theory manual just to write out the definition 

of the initial stress matrix. and the manipulations involved in reaching that definition are 

lengthy. The extension of the same formulation to general shell deformations will involve 

substantial algebraic manipulation. 

The general shell elements that we use are the Batoz triangle, and the four and eight 

node “Ahmad quadrilaterals” (shear flexible elements using reduced integration). Our 

implementation of the Batoz triangle uses Batoz’ interpolation functions in a co-rotational 

frame for bending, together with constant membrane strain. The element is in Abaqus 

because it is essential for us to offer a three-node triangular element for shell problems, 

and we have been advised that this element is among the better elements of this category. 

The element appears to behave well in bending-it converges rapidly and is relatively 

insensitive to distortion. The shortcomings of our implementation of the element are the 

constant membrane strain assumption, the faceted geometric approximation, and the need 

to use three integration points, this last because it makes the element more expensive than 

it should be, given the constant membrane strain assumption. 

The Ahmad quadrilaterals are very attractive for cases involving material nonlinearity 

because the use of reduced integration minimizes the constitutive evaluations needed to 

form the element. The basis functions are also very simple. The four node element is 

of limited value without hourglass control: we use the hourglass controls proposed by 

Belytschko (Belytschko and Tsay. 1983) for this element. This improves the situation, 

but it is still not entirely satisfactory-we still find it necessary to change the hourglass 

stiffnesses from time to time. without being too sure of the reasons for the values we choose. 

Abaqus includes a print option which summarizes the energy content of the solution. It 

is not uncommon to see relatively significant “artificial strain energy’‘ associated with the 

hourglass control: in typical high energy dynamic events this artificial strain energy may 

even exceed the residual “physical” energy (strain energy plus kinetic energy) at the end 

of the event. Yet the overall predictions of the significant aspects of the response are 

generally usable. 

As Irons pointed out in his last paper (Irons and Loikkanen, 1983), the eight node 

Ahmad element does not pass the patch test except as a rectangle. We offer this element 

for historical reasons-it seemed to be a good choice at the time. In fact it does respectably 

well on shells that are not too thin, even if it is somewhat distorted. But there is another 

practical objection to the eight node interpolation scheme: contact problems are awkward 

with such functions. Contact over the entire element does not present serious difficulties, 

but. with the contact algorithms that we use (which have Lagrange multipliers to represent 

the contact pressure), we do not know how to deal with partial contact over an elementthat is. with the possibility that contact or separation may occur over part of the surface 

represented b> an element. We plan to add the nine-node version of the same element to 

overcome this last objection, even though Irons has told us that this form of the element 

onl> passes the patch test when it is geometrically bilinear (which would soon not be the 

case in a large displacement analysis. even if it were true to start with). 

Both of the A4hmad elements, as we have implemented them, perform poorly if they 

are distorted and the shell is thin. This is a serious objection, but it would appear that 
the element of K. C. Park (Park and Stanley, 1~84) alleviates this problem. I think it 

is important. for practical cases, that we retain the simplicity of the low order elements 

as well as the cost effectiveness of reduced integration, especially when relatively severe 

nonlinearities (requiring, for example, modeling of finite strain effects) are present. 

5. Integration of Conventional Plasticity Models 

Conventional plasticity models present an interesting integration problem: the material behavior changes drastically at yield; the plastic strain is defined only as a rate, and 

the stress measures usually used are defined on the current configuration, so that the rotation of the material must be accounted for in some way. The problem has been receiving 

attention recently (for example Ortiz and Popov, 1984), and new approaches have been 

suggested. I would like to describe the approach that we have been using in Abaqus: overall: we are reasonably satisfied with it, but it raises some questions that we have not been 

able to answer, and which may also be relevant to the more recently proposed approaches. 

Abaqus is primarily an implicit code, so that it is desirable to use an algorithm that 

is unconditionally stable. This is not strictly essential. In typical cases involving metals 

under ordinary conditions, our experience has been that we are unlikely to succeed in 

using increments in which the rotations exceed 10-15 degrees, or the strains exceed a 

few percent over important parts of the mesh (these limitations are associated with our 

Newton approach to solving the equations). Thus, any method that is stable for this size of 

increment should be satisfactory. However, I would be surprised to see conditionally stable 

algorithms that can handle such increments: typical yield strains in metals are about 

so that the increment size is about ten times the size of the yield surface in strain space, 

and I assume that the stability limit of a conditionally stable operator would not be larger 

than the yield strain. Because we have found the full Newton method to be effective, we 

want an algorithm that is sufficiently simple so that the Jacobian matrix, (dT/dE)t+nt, 

where T is the stress , E is the strain, and t i At is the time at the end of the increment 

(where we write the equilibrium equations in an implicit code), can be obtained exactly 

for common plasticity models. It is desirable that this matrix have an additional property: 

several plasticity models of practical importance are derived from a locally smooth potential 

which is also the yield surface, so that the rate equations of the model give a symmetric 

form dT/aE. We would like the integration operator to preserve this symmetry-this is of 

practical significance with respect to computational cost. The operator we use in Abaqus 

does not do so, except in a restricted class of plasticity models (which does not include 

some of those that are available the code and which have a symmetric rate form). 

We first integrate the kinematic aspect of the model by using the algorithm of Hughes 

and Winget (1980) to define an increment of rotation. which we apply to all vector and 

tensor valued functions at the material point; we also follow the Hughes-Winget suggestion 

and define the strain increment by the central difference approximation. Various authors 

have commented on the inadequacies of this class of method in the context of theories that 

involve tensor valued hardening variables, such as classical kinematic hardening theory. I 

do not see much utility in simple kinematic hardening with respect to finite strain applications. and the approach does not suffer the same obvious deficiency when it is used with 

15 
the isotropic yield models that we generally use for metals and soils at finite strains. The 

algorithm leads to a symmetric initial stress matrix. which is tedious to derive. 

We are left with the integration of the change of state associated with deformation. The following discussion considers only isothermal, rate independent behavior with 

isotropic hardening. In Abaqus we use the same approach for rate dependent models (except where we integrate explicitly. as mentioned in Section 2.2 above), for non-isothermal 

cases (including fully coupled temperature-displacement calculations) and for more complicated hardening models. such as kinematic hardening. 

The isotropic hardening plasticity models in Abaqus all have the following structure. 

We assume a strain rate decomposition, 

dE = dE"' + dEp' 

where dE is a differential change in total mechanical strain, dE"' is a differential change 

in the elastic strain. and dE'" is a differential change in the inelastic (plastic) strain. We 

have introduced the Hughes- Winget approach to account, in an approximate way, for the 

rigid-body rotation of the material during the increment, and to define a finite increment of 

total strain during the increment. This allows us to write this decomposition in integrated 

form as 

(1) E ~ E" - E?'' 

where E. E"' and ET'' are defined as summations of the corresponding rotated values at 

the start of each increment and the incremental values associated with that increment. 

During the constitutive calculations E is known, except in the case of plane stress. 

We assume an elastic strain energy potential so that the stress. T. is defined as 

where II' = W(E"'.O) is the strain energy potential, and 8 is the temperature. 

Some of the plasticity models that we use in Abaqus assume linear elasticity. while 

others (soils models) use a nonlinear elasticity in which the logarithm of the pressure 

stress is proportional to the volumetric strain. None of these elasticity models has internal 

constraints such as incompressibility, so that (2) defines the stress completely. It can be 

differentiated to give 

which we write as 

dT 'D " : dE" 

A simple isotropic hardening model has the yield function 

f=O 

(3) 

(4) 

I 16 
where f = f(T.O,H"), with Ha being a set of hardening parameters; f is defined such 

that. whenever f < 0, the response is purely elastic. The models we use in Abaqus all 

have a smooth yield function. so that aj I aT. a2j aT 3T. and d f /dH" are well defined 

everywhere on f. 

The flow rule is written 

dE"' = dX- dT (5) 

where dX is a positive scalar and g(T, 8, H") is the flow potential. Since the material 

models being discussed are rate independent, dX is determined only by the kinematic 

solution at the material point being considered. 

The hardening parameters evolve with plastic strain: 

dH" = h"(dEP', T, 8, HP) 

where h" defines the hardening: h" must be homogeneous of degree one in dEP" for the 

model t>o be rate independent. Therefore 

The plasticity model is now defined. except for choosing particular forms for the elastic 

strain energy potential, K', the yield function, f, the flow potential. g, and the hardening 

rules, h". 

The only rate equations in the formal definition of the material model are the evolutionary rule for the hardening and the flow rule. The simplest operator which may at 

least fulfill the requirement of unconditional stability mentioned at the beginning of this 

section is the backward Euler method, which can be introduced into (5) to give 

while (6) is writt,en 

ag 

aT 

AH" = AX h"( -, T, 8, H') 

In these equations and in all of t,he following. all quantities are evaluated at the end 

of the time increment. 

Remark: Ortiz and Popov (1984) propose the midpoint rule as a more accurate operator. based on an error analysis with small strain increments. Their analysis 

appears to assume that the material is yielding during the entire increment. 

This is often not the case-in most calculations there are always material 

points where yielding begins part way through the increment. At such points 

the mid-point rule requires solution for the initial intersection with the yield 

surface during the increment. and application of the rule only within the 

17 
yielding part of the increment. This creates some formidable algebra. especially if one wishes to derive the Jacobian. Our analysis of this problem leads 

us to the conclusion that, even for an associated flow material, the Jacobian 

will, in this case. not be symmetric, and that its non-symmetric part may 

not be very small. We choose the backward Euler method partly because it 

does not introduce this difficulty, and so retains sufficient tractability that 

we can complete the algebra rather easily. It is interesting to note that Ortiz 

and Popov confirm the conclusion of Schreyer et al. (1979): that, for strain 

increments that are not small compared to the size of the yield surface in 

strain space, the backward Euler method exhibits the highest accuracy of 

any of the simple methods that they tested. 

From a computational viewpoint the problem is now algebraic: we must solve the 

nonlinear equations (l), (2), (4), (7) and (8), and thus define the state of the material at 

the end of the increment. The "material stiffness matrix", 

must also be defined for the overall Newton scheme that we use for the equilibrium equations. Simple manipulations lead to the definition 

ID] = [II] + AA[H] : [L]]-'[H] 

where 11; is the fourth order identity tensor, and 

[HI = IDJe' : IZ] 

1- 

d iZ; = [I] - -NM : 

18 
and 

For a non-associated flow material, af/dT and N are not related, so that [D] cannot 

be symmetric. For an associated flow material, 13fjdT and N (and hence N) will be 

co-linear. Therefore ID] will be symmetric if 

which is the case, for example, for the simple Prandtl-Reuss model that is commonly used 

for metals, but is not true for the modified Cam-clay model that is sometimes used for 

clays. However, in associated flow cases when it is not true, the non-symmetric contributions to ID] appear only in the terms multiplied by (AA)2, so that they are of the 

order of (the plastic strain increment)2 compared to unity. This suggests that, for practical purposes, the lack of symmetry should not degrade the convergence of the Newton 

iterations for equilibrium if we approximate ID] with its symmetric part, and our experience to date confirms this-at least, the performance of the algorithm with the symmetric 

approximation has been satisfactory, by our standards. 

There remains the problem of solving the algebraic equations for the state at the end 

of the increment. In the general case, this is not a simple matter: whatever method is used 

should not be expensive on the computer. but it must work for the tightly curved yield 

functions that appear in some models of practical interest, even when the strain increment 

is many times the typical size of the yield surface in strain space. The problem is made 

more awkward in cases such as Cam-clay, where exponential terms appear in the elasticity 

and in the hardening. Our approach has been as follows. A Newton scheme should work, 

provided we choose suitable variables as a basis, and provided we start with a good guess. 

The plastic strain increments, AEp', should be a suitable set of variables. Then Kewton's 

method for (I), (Z), (4), (7) and (8) is the linear equations 

[!I] t AA[Z] : [L] : [D]"'] : Cp'= [Z] : (AAN - -1 

d AE"') t N- f 

which are solved for CP' , the improvement to the solution for GEP': 

The elastic strain is then obtained from (l), the stress from (2). LA from the projection 

of (7) onto N: 

and AH" from (8). 

19 
This loop is repeated until the components of LE"' converge. A tight convergence 

criterion is necessar), so that the solution is accurately defined: this is essential for the 

overall Newton scheme for the equilibrium equations to converge quadratically. 

The Newton method requires solution of the linear system at each iteration. Although 

the system of equations is not large (at most six), these computations are done at each 

integration point and at each iteration of the overall equilibrium solution, so that it is 

desirable to solve the problem more efficiently than by direct application of the Newton 

method. In addition, when the strain increment is large, a reasonable starting guess is 

necessary for the method to converge. For these reasons we have been using a projection 

of the problem onto a smaller number of variables to start the solution. If this subspace 

is chosen appropriately, we should be able to develop a useful estimate of the solution at 

low cost. The general idea is as follows. 

Let KO, (Y = I,. . . . n, be a set of tensors that are orthonormal: 

and choose n to be less than the number of stress components in the actual problem. 

guess. 

The Ku are chosen for a particular guess and are fixed during the solution for that 

Assume that the plastic strain increment is 

,%-E"' = ne='Ka (9) 

The elasticity, (2), and the strain rate decomposition. (1). then define the stress, 

which we require to satisfy yield, (4). The integrated flow equation, (7), is imposed in the 

sub-space: 

where 

Since the Ka are orthonormal, (9) and (10) then define 

Aep$ = AxNa 

where 

The Newton scheme described above for the full stress space projects directly: 

20 
where 

and 

These linear equations provide the ca, the improvements to the Aeg': 

At each iteration the solution is updated as described for the full stress-space solution, 

except that Ax is calculated by projecting (11) onto Na: 

The utility of this technique depends on the choice of the Ka, and on n. Clearly it is 

unlikely that it would be worthwhile to use n > 2. Most of the plasticity models in Abaqus 

are isotropic. in the sense t,hat the yield function, f, and the flow potential, g, are defined 

in terms of the stress invariants. An obvious choice for the KO in these cases is n = 2, 

with 

(here I is the identity matrix), 

and 

where S" is the deviatoric part of To, the stress that would occur at the end of the 

increment if there were no plasticity occurring in the increment. and 

If the only st,ress dependencies in f and g are the effective pressure stress, 

1 

p = --I : T 

3 

and the deviatoric stress magnitude, q; and the elasticity is isotropic, (and the hardening 

is isotropic, as has been assumed here), the subspace solution is the exact answer to the 

21 
problem, except for the plane stress case. For some simple yield surfaces and hardening 

definitions the subspace problem can be solved in cloyed form, without the need for iteration. For the simplest case of a Mises material. where q is the only stress term in f and 

g. the one-dimensional sub-space defined by K2 above provides the exact solution. For 

perfect plasticity the method is then precisely Wilkins‘ “radial return” algorithm. 

Our experience with more general yield functions and flow potentials, where the third 

stress invariant is also used, is that the two-dimensional sub-space provides a satisfactory 

guess, from which the solution can be completed usually with no more than two iterations 

of the full stress-space problem. 

6. Closure 

The paper has presented a brief review of our experience in the recent past in the 

general area of nonlinear structural analysis of metal shells. It is not difficult to identify 

the most severe limitations in the methods we have used so far, to write down a “wish 

list” of areas where we would like to be using better methods than those we have used to 

date. The list is as follows. 

0 We do not like the rapid deterioration in the accuracy of the isoparametric shell 

elements that we currently use when they are not rectangles and the shell is thin. 

0 Reduced integration elements are attractive because they minimize constitutive calculations, which are often a significant part of the computational cost when the material 

model is not simple. Numerical artifices, such as hourglass control, are not objectionable, when they are effective and are well understood. Our own understanding so far 

is lacking. 

0 The characteristic hinging of thin metal shells under compressive loads raises difficulties and opportunities. The difficulties are associated with concerns about capturing 

this behavior with smooth discretizations. The opportunities are indicated by the remarkable success of the rigid-plastic models of Wierzbicki. Perhaps it is true that, in 

practical cases, load bearing members are too thick for this to be important, although 

movies of front-end collision experiments on cars shown very pronounced hinging. 

0 A straightforward finite membrane strain shell formulation would have wide application in several problem areas that we often encounter. The axi-symmetric formulation 

we have been using is based substantially on the assumptions introduced by Rodal in 

his thesis, although we have taken a rather different approach in detail. We do not 

have a good appreciation of the limitations of the formulation. 

0 We expect to continue to work with implicit methods in many problem areas, including 

a large part of the analysis work associated with structural design. If this is a correct 

assessment. it would be highly desirable to have equation solution methods that are 

more effective and less difficult to use than Kewton’s method. This is, from our point 

of view. one of the most severe limitat,ions that we face in practical applications. 

0 So much shell behavior involves branching on the solution path that it would be very 

satisfying to have methods that can handle this unassisted. 

22 
0 Finally, we have not mentioned rezoning, or the treatment of localization, but these 

are. becoming important issues in practical cases. 

7. References 

T. Belytschko and C.-S. Tsay, “A Stabilization Procedure for the Quadrilateral Plate 

Element with One-Point Quadrature”, Int. J. Num. Meth. Eng., Vol. 19, pp. 405-419, 

1983. 

H.M. Hilber, T.J.R. Hughes and R.L. Taylor. ‘bCollocation, Dissipation and Overshoot 

for Time Integration Schemes in Structural Dynamics”, Earthquake Eng. and Struc. Dyn., 

T. J. R. Hughes and J. Winget, “Finite Rotation Effects in Numerical Integration of 

Rate Constitutive Equations Arising in Large Deformation Analysis”, Int. J. Num. Meth. 

Eng., Vol. 15. pp. 1862-1867, 1980. 

B. M. Irons, “The Semi-Loof Shell Elements”, Chapter 11 of Finite Elements for Thin 

Shells & Curved Members, edited by D. G. Ashwell and R. H. Gallagher, John Wiley & 

Sons, London, 1976. 

B. M. Irons and S. Ahmad. Techniques of Finite Elements, Ellis Horwood Ltd., Chichester, England, 1980. 

B. M. Irons and M. Loikkanen, “An Engineers’ Defense of the Patch Test”, Int. J. 

Kum. Meth. Eng., Vol. 19, pp. 1391-1401, 1983. 

M. Ortiz and E. P. Popov, “Accuracy and Stability of Integration Algorithms for 

Elastoplastic Constitutive Relations”, Brown University. Division of Engineering Report, 

1984. 

K. C. Park and G. M. Stanley, “A Curved C* Shell Element Based On Assumed 

Natural-Coordinate Strains”, Applied Mechanics Lab.. Lockheed Palo Alto Research Lab., 

Palo Alto, Cal., 1984. 

E. Riks, “An Incremental Approach to the Solution of Snapping and Buckling Problems”, Int. J. Solids and Struc.. Vol. 15, pp. 329-551. 1979. 

J . J . -4. Rodal and E. A. Witmer, “Finite-Strain Large-Deflection Elastic-Viscoplastic 

Finite-Element Transient Response Analysis of Structures“. contractor report 159874, 

NASA Lewis, 1979. 

J. C. Simo and M. Oritz, “A Unified Approach to Finite Deformation Elastoplastic 

Analysis Based on the Use of Hyperelastic Constitutive Equations”, Brown University 

Division of Engineering Report, 1984. 

G. Strang, and G. J. Fix, An Analysis of the Finite Element hlethod, Prentice-Hall, 

New Jersey, 1973. 

H. L. Schreyer. R. F. Kulak and J. M. Kramer, “.4ccurate Numerical Solutions for 

Elastic-Plastic Models”, Trans. ASME, J. Pressure Vessel Technology. Vol. 101, 1979. 

P. S. Symonds, “Survey of Methods of Analysis for Plastic Deformation of Structures 

under Dynamic Loading“, Brown University Division of Engineering Report. 1967. 

VO~. 16, pp. 99-117, 1978. 

23 
COMPUTER I ZED STRU TU AL ME iANICS FOR 1990's: ADV N 

N89-24640 

ED AIRCRAFT NEEDS 

A. V. Viswanathan and B. F. Backman 

Boei ng Mi 1 i tary Ai rpl ane Company 

Seattle, Washington 

Abstract 

This paper describes the needs for computerized structural mechanics (CSM) as 

seen from the standpoint of the aircraft industry. It projects these needs 

into the 1990's with special focus on the new advanced materials. It identifies the major areas: 

0 preliminary design/analysis 

research 

0 detail design/analysis 

and elaborates on the role of local/global analyses in these different areas. 

The lessons learned in the past are used as a basis for the design of a CSM 

framework that could modi fy and consol i date exi sti ng techno1 ogy and i ncl ude 

future developments in a rational and useful way. 

A philosophy is stated, and a set of analyses needs driven by the emerging 

advanced composites is enumerated. The roles of NASA, the universities, and 

the industry are identified. 

Finally, a set of rational research targets is recommended based on both the 

new types of computers and the increased complexity the "industry" faces. 

Computerized structural mechanics should be more than new methods in 

structural mechanics and numerical analyses. It should be a set of engineering applications software products that combines innovations in structural 

mechanics, numerical analysis, data processing, search and display features, 

and recent hardware advances and is organized in a framework that directly 

supports the desi gn process. 

BRECEDiNG PAGE BLMK NOT FlLMED 

25 
There are two aspects to the development of engi neeri ng appl i cations software: 

(1) i nnovati ons and (2) improvements in the producti vi ty of engineers . While this paper concentrates on the first aspect, it is still important to 

consider the producti vi ty aspect in the development of new methods because the 

software ultimately will be part of the set of tools used in the design of 

ai rcraft structures. 

Structural Analysis Needs 

for the Aircraft Industry 

0 New capabilities 

Concepts 

Materials 

Details 

0 Improved productivity 

Engineering workstations 

Tutorial software 

CAD/CAM interfaces 

I 26 
New non-ductile materials have entered the scene and will be part of the 

picture for a long time to come. These materials require more detailed analyses 

not only because of micro-considerations and interlaminar effects but also 

because of their unforgiving nature which entails a significant participation 

of "secondary" effects in the failure modes. The complex definition of these 

materials and their responses required to identify behavior has resulted in a 

large increase in the information volumes for processing of a typical aircraft 

problem in the structures field. 

At the same time, we have seen tightening up of requirements not only in terms 

of adverse environment but also in terms of improved quality. In both cases 

this tightening results in a need for more sophisticated analyses. 

The general areas of computing technology and information science have seen 

dramatic changes in both hardware and software. There have been hardware 

changes that could be parlayed into: (1) optimization of structures with 

practical constraints, (2) nonlinear analyses at reasonable cost, and 

(3) mi cro-analyses at acceptable storage requi rements. Developments have 

occurred in the field of data processing that make data base management a 

natural cornerstone in the future of CSM. 

ments in the fields of graphics and CAD/CAM which would make the establishment 

of standardized user interfaces extraordinarily useful both from the standpoi nt of techno1 ogy and management. 

Finally, there have been developMotivation for Increased Development 

Advanced composites 

Higher complexity 

More unforgiving 

Larger information volumes 

More stringent requirements 

Lighter weights 

Higher temperatures 

Longer lives 

Improved computing technology 

Numerical methods 

Data processing/search and display 

CAD/CAM 

27 
The only result of significance in the aircraft industry is to produce a new 

"better" aircraft. Any other result is intermediate and will only be acceptable if it contributes to this improvement. The success of CSM is therefore 

conti ngent on three abi 1 i ti es : 

e To promote and support new technology in the fields of structures, 

numerical methods, and data processing 

To include and improve methods in the preliminary design and analysis of 

new aircrafts (the need to develop and refine conceptual methods for 

the selection and comparisons of sophisticated candidates with no 

experience basis should be a strong driver in goal setting) 

e 

To introduce new rnechods and integrate these methods in a framework that 

can produce the visibility, data reduction, and sophistications that 

are necessary to support a production effort in a new era 

What Constitutes Success? 

e Better aerospace vehicles 

I) Therefore "CSM" must 

Support technology development 

Provide NEW preliminary design and analysis methods 

Automate detail design and analysis methods 

28 
The successful development of new methods in computerized structural mechanics 

requires an understanding of the psychology of the situation. 

1. An efficient method that supports the needs of the industry must be 

established. 

2. A practical input and output language must be used. 

3. A set of useful user interfaces must be available. 

4. An appropriate amount of visibility and data reduction features should 

be provided. 

5. A realistic marketing effort should be launched. 

6. A developer and user interaction is required. 

Yardstick for Success for CSM 

Computerized methods development 

will not be successful unless 

the new “product” is to tally acceptable 

to the users 

29 
The CSM framework should be designed and implemented with common design questions in mind. Of these, the question of material choice and mix is very 

complex. An optimum distribution of materials in the primary structure of an 

airplane is dependent on the critical failure modes. Is it strength (in 

tension), stability, fatigue, damage tolerance, or stiffness requirements that 

size a local detail? Again, it can be seen how a global/local/global cycle 

must be used to come to grips with this design problem. All the design 

drivers are closely related, and the question of environment involves the 

determination of temperature, moisture, presence of chemical, corrosion, and 

risks of FOD*. 

The performance requirements, such as speed, load factors, roll rates, and 

landing speeds, all are very basic and need to be addressed again in a local/ 

global fashion. 

process and evolve in a cyclic manner as the local/global analyses of the 

structure mature. 

These effects need to be assessed at least on a parallel track. 

Finally, design criteria are llsoft" early in the design 

What are the Structural Design Drivers? 

Material choice and mix 

Strength 

Stiffness 

Damage tolerance 

Fatigue performance 

Environment 

Performance requirements 

Design criteria 

*foreign object damage 

30 
There are three fields of structures that have to be included in order to give 

attention to all the aspects of engineering design in the aircraft industry. 

The first of these fields is the research activity that leads to a new technology and a better understanding of the ingredients in the design process. 

The second field is the preliminary design and analysis that leads to the 

selection of the appropriate family of vehicles or products. Both syntheses 

and analyses on one side and parameters and variables on the other are involved in this design and analysis. 

Research 

Preliminary Design (Pilots) 

The third field is the detail design and analysis, of which perhaps the locall 

global concept is more applicable than anywhere else. Here the objective is 

to learn as much as possible about a specific design candidate. The demands 

are especially stringent on CSM to include features that facilitate data 

reduction and display as well as sophisticated local analyses methods. 

Production 

(Detail Design) 

CSM, What Should be Included? 

c 

Optimization Trade-Studies Synthesis Detail Design 

31 

Local-Global Sensitivities Analysis Optimization 

Parameter 

Evaluation Local-G lobal 

LocaCGlobal Error Estimation 
The overwhelming requi rement in the preliminary design and analysis phase 

deals with synthesis methods that discriminate between parameters and vari - 

ables in the design process and makes it possible to study alternative 

formul ati ons. 

Preliminary Design and Analysis 

.Access to expert system data base (trade study 

results) 

Parametric results from local-global analyses as 

baseline for optimization 

Determination of sensitivities (data base) 

Algorithms for parameter determinations and 

variable selections 

Strategy algorithms, history function 

32 
The role of research in the CSM arena is central and its primary purpose is to 

promote better understanding of the materials and structures technologies 

required to produce better, vehicles. 

significant development in (1) structural and continuum mechanics, 

(2) numerical methods, and (3) failure prediction and test evaluation. This 

must be done in a framework that allows for empiricism that is sensitive to 

user needs, aims at synthesis, and produces pilot capabilities with welldefined interfaces. 

To that end, it is essential to have 

Research 

Phenomena evaluation for better understanding 

Special purpose detail analyses 

New numerical methods 

“Pilot ” development 

Test data evaluation methods 

Empirical corrections methods 

33 
The detail design and analysis fields have always been characterized by large 

volumes of data. This is becoming more and more the case as the new advanced 

composite materials are introduced into the production environment. 

consider the simple problem of calculating margins of safety.) This together 

with more complex requirements and the less forgiving nature of the materials 

has resulted in order-of-magnitude increases in data volumes in addition to a 

myriad of local analyses methods demands. 

(Just 

Production (Detail Design and Analysis) 

Postprocessing, search and display (graphics) 

Strength checking (software for detail analyses) 

Management visibility and access 

Data base access (parametric representation ... allowables) 

Automated resize. (optimization), baseline updates 

Strategy for local-global analyses 

Error estimation 

34 
The CSM framework has to have an information center that supports the local/ 

global analyses and makes it possible to combine all aspects of theory, 

empiricism, test data, and criteria. 

This data base should be such that it supports the preliminary design, detail 

design, and research. 

Information Center (Data Base) 

Self educating expert system (better modeling, improved 

parameter evaluation, trade projections) 

Test data evaluation and empirical correction 

Sensitivities 

History writing 

Search and display 

35 
The building blocks for CSM are naturally many and of very different nature, 

but the central core in a local/global system will have to be a set of special 

purpose methods that in a very flexible way can be included, updated, and 

rep1 aced. 

What are the Building Blocks? 

interfaces 

36 
Without the proper discipline in the structuring and targeting for CSM, there 

is a significant risk that spur-of-the-moment CSM designs will prevail. Such 

designs will be avoided by a central NASA leadership. 

Should the Philosophy be Uncontrolled Growth? 

THE ANSWER IS NO! 

+ 

37 
From the standpoint 

increased re1 i ance 

(1 oca1 ) analyses. 

should provide the requi rements necessary to curtail and reverse 

enaineerina trend. At the same time a number of qeneral systems 

budget 

/g1 oba 

of the aircraft industry, the last 20 years have seen an 

on highly special ized experts to perform special purpose 

This is an undesirable development, and the CSM framework 

this software 

have emerged, 

needs, and 

communi careiched mahrity, expanded to a high level of maintenance 

then stagnated. Finally, the framework must provide loca 

tions and user interfaces. 

Need from Users 

Present situation 

Increasing dependence on highly specialized 

experts 

General purpose systems in use require more 

user driven long term plans 

*As the present trend shows high maintenance burden and early stagnation 

Proper framework for global to multi-local 

communications is missing 

38 
The st rateyy for the 1 ong-term CSM development could i ncl ude: (1 ) i dent i f i cation of existing software that can be modified for a new environment, (2) a 

plan that is based on an evaluation of industry needs and experience with new 

operating systems, compilers, hardware, and methods, (3) the new hardware potential that must be part of the picture when targeting special fields; optimization is still a cost/hardware constrained activity, and (4) emphasis that 

should be based on the priorities that come out of industry needs. NASA has a 

key role to play in pulling all these together. 

Strategy 

Plan that modifies and uses existing software where feasible 

Plans driven by what the industry presently knows 

Plan that recognizes new hardware potentials 

Plan that encourages more “applied research” into fields 

driven by industry needs 

?9 
The 

issues that are essential to CSM development follow-through. First, the size 

of the software demand is such that a national commitment is necessary if the 

technology benefits are to be realized. Second, a number of general-purpose 

sci enti f i c systems and an assortment of speci a1 -purpose software have demonstrated the need for a framework designed to draw upon the benefits of both. 

Finally, user acceptance will depend on good visibility and easy access to 

solutions and results. 

requirements mentioned previously translate into a set of organizational 

Fundamental Organizational Issues 

The successful development of 

CSM requires central leadership 

CSM should provide the framework 

for unification of general/global and 

special/local methods 

Search and display algorithms and 

standardized user interfaces are as 

important as solutions 

40 
The backbone of the CSM design approach is the data management, but the foundation for success can be found in the quality of the special-purpose methods 

and the associated software. 

tance of these methods are very dependent on availability of required data, 

ease of interpretation of results, and visibility of the steps leading to the 

final solution. 

Experience has shown that the success and accepThe approach should also deal with the present trends in structures, in which 

on one hand we are moving into fields with very limited experience bases and 

on the other hand we produce innovation through point design testing. We can 

address both situations in a framework that not only accepts new methods with 

ease but also uses these methods for experience development and empirical 

evolution. 

Could This be a “Better” Approach? 

Data communication 

Problem definition 

(analyseddesign strategy) 

Structural Preliminary Production 

analysis design (detail design) Research 

I I I I 

Search and 

display algorithms 

General Ad hoc purpose 

structural analysis software 

system 

data bases 

Requested 

results Learning module 

Parameter selection 

Empirical rules 

History and 

management 

visibility 

History and management visibility 

41 
The typical situation in the structures field involves a number of analyses 

that are directed at different levels of resolution. These levels will have 

to be revisited a number of times during the design process. 

the desi gn, therefore, obviously requi res a number of global -to-1 oca1 -toglobal transitions involving huge data volumes. The efficiency of these 

transitions requires as much attention as the methods development. 

The evolution of 

Lo cal-Glo bal Analyses 

Internal loads/global, overall FE analyses, stability, 

aeroelastic effects, flutter, ... 

Detail stress analysis (linear/nonlinear), “local” 

buckling, all0 wables processing, postbuckling 

analyses, interlaminar analyses, residual strength, 

damage tolerance analyses, fatigue analysis 

Local optimization, multilevel optimization, 

automated remodeling, ... 

Error estimation 

Microdetail-macro strategy (transitions) 

42 
An understanding of the design drivers leads to the identification of a number 

of fields in which work should be done in order to efficiently produce preliminary and final designs. Here the preliminary and final design processes 

are different in focus, but similar principles could serve both enterprises 

and emphasis should be given to synthesis-1 i ke features. 

Development Support 

Determination of material development targets 

Selection of “best” structural materials (based on most important 

design drivers, and determine failure progression) 

Establishing environmental effects 

Determining an optimum set of performance requirements 

(temperature/speed, weight, load factor, ...) 

Exploration of sensitivities to variation in design criteria 

Multilevel strength, stress/stability analyses 

Multilevel optimization 

Pre- and postprocessing with search and display features 

43 
The framework for CSM must be such that both categories of users/developers 

(typical engineers and experts) can be allowed their proper influence in a 

manner that promotes natural roles and a1 1 ows for organized communications of 

both needs and results. 

What are We Aiming For? 

.A system for technology development and improved 

scientific understanding for experts 

.A system for preliminary design, detail design and 

analyses for typical engineers in production environments 

44 
The local/global analysis development is seen as a three-pronged effort that 

i ncl udes : ( 1) advanced methods, (2) typi cal engi neeri ng analyses, and 

(3) data base and associated methods for experience development. The advanced 

methods would primarily be intended for expert evaluations, but could, if 

properly packaged, be included among the typical engi neeri ng analyses. The 

expert evaluations would produce direct input to the overall design effort, 

but would also feed the data base and indirectly support both parametric 

evaluations and preliminary design approaches. The example problem represents 

a number of analyses that belong on the local level and should be considered 

from the desi gn standpoi nt. 

Local-Global Analysis 

data 

base I 

system 

Visibility 

I Test-empiricism-theory - 

U 

quality design 

45 
The example problem in the context shown in the previous figure represents a 

number of analyses and syntheses. All of these belong to the local or micro 

level when seen from the standpoint of the design process. All of these, and 

a number of others, are natural development (research) targets that after 

implementation should either directly feed results into the "typical" 

engineering activity or produce results to the local data base for future use 

or paramet ri c access. A1 1 these devel opments should result in pi 1 ot capabi 1 i - 

ties or products that in an ad hoc fashion would satisfy one of the above two 

requirements. 

What are the Analyses for the Example Problem? 

0 Local failure modes and progression 

based on test, for correlation and 

empirical inputs 

0 Multilevel analyses: interlaminar, 

local damage growth, local influence 

of postbuckling 

Stability analyses 

Postbuckling 

0 Nonlinear material 

analyses 

response,. . . 

damages 

0 Flaw-growth for a number of 

0 Allowables determination 

Local optin 

5" 

nization 

Parametric representation and 

empirical input to data base 

0 Automated "remodeling" 

0 Screening for criticalities, 

0 "Gross" FE properties 

0 Residual stresses 

transition regions 

requirements 

' and 

critical 

46 
The example problem is one of many analogous local problems that must be 

solved as part of the design process. Each one of these problems is solved a 

number of times during the evolution of the design. 

to minimize the number of micro evaluations and expert involvement in order to 

produce an efficient design process. This can be done by including the 

results on a communal experience basis in a way that supports the practical 

engineering tasks. 

minimum involvement by experts. In many types of problems, it will be 

necessary to identify important parameters and produce solutions in advance in 

a format that can be accessed in a data base management environment. Very 

similar considerations apply to the allowables question, whether‘it is processing of material structural allowables. Finally, in the failure prediction 

development, one can see elements of the other types and here one can hardly 

expect practical support of the design process without a two-level software 

development that ul i tmately depends on empi ri cal modifications. 

We are therefore forced 

Many of the local analysis packages must be designed for 

The Example Problem 

Local with “micro” requirements 

Local analyses with minimum expert requirements 

Basis for parameter selection 

Panel for allowables generation 

Candidate for failure prediction development 

47 
In conclusion, we find that there are a number of research targets waiting for 

initiative from the engineering community. It has become clear, however, that 

the majority of these targets will be missed if not approached from the standpoint of their overall role in the design process. It is also felt that the 

objectives will not be met without a proper technology transfer to the users. 

This naturally involves both the software packaging and the promotional 

act i vi ty necessary for di ssemi nat i on. 

Conclusions 

Research should include 

Analysis methods (nonlinear, advanced materials,... 

structures) 

Numerical methods 

Artificial intelligence 

Search and display/data processing 

Emphasis on technology transfer to industry 

48 
NASA has a key role in the development of CSM for the 1990's and beyond. 

national effort is required if new materials (composites), new computers, new 

methods, and new requirements are to be addressed in a manner that establishes 

economic advances' and preserves the superior safety record established by the 

aircraft industry. NASA has a role not only in leadership but also as communicator assuring technology transfer and promoting user acceptance. 

A 

Recommendations 

Clear interfaces 

0 Standardization 

.Standing committee 

49 
NONLINEAR RESPONSE OF A BLADE-STIFFENED GRAPHITE-EPOXY PANEL 

WITH A DISCONTINUOUS STIFFENER: WORK IN PROGRESS 

Norman F. Knight, Jr., William H. Greene, 

and W. Jefferson Stroud 

NASA Langley Research Center 

Hampton, Virginia 

INTRODUCTION 

Discontinuities and eccentricities are usually present in practical 

structures. In addition, potential damage of otherwise perfect structures is often 

an important design consideration. Predicting the structural response in the 

presence of discontinuities, eccentricities, and damage is particularly difficult 

when the component is built from graphite-epoxy materials or is loaded into the 

nonlinear range. Recent interest in applying graphite-epoxy materials to aircraft 

primary structures has led to several studies of postbuckling behavior and failure 

characteristics of graphite-epoxy structural components (e.g., refs. 1-31. 

However, these studies concentrated on two topics: prediction of the overall 

response of composite structural components in the postbuckling range or failure 

mechanisms and analytical failure prediction techniques for fibrous composite 

materials. The problem of calculating detailed stress distributions around 

discontinuities in buckled, composite structural components for use with the 

various analytical failure prediction techniques has not been thoroughly explored. 

The purpose of this paper is the application of computational methods to the 

detailed stress analysis problem which is the focus of this session of the 

workshop. One approach to uncovering the difficulties of this type of analysis and 

to providing specific directions for future research in this area is a direct 

attack on the problem using currently available analysis tools. A candidate 

problem has been selected and the remainder of the paper describes experiences from 

calculating its structural response. 

PRECEDW PAGE RAMK NOT FILMED 
BLADE-STIFFENED GRAPHITE-EPOXY PANEL WITH A DISCONTINUOUS STIFFENER: 

FOCUS PROBLEM 

The focus problem for the local/global stress analysis session of this 

workshop is to determine the nonlinear response of a flat blade-stiffened graphiteepoxy panel with a discontinuous stiffener. 

T300/5208 graphite-epoxy with a nominal ply thickness of 0.0055 in. Typical lamina 

properties for this graphite-epoxy system are 19,000 ksi for the longitudinal 

Young's modulus, 1,890 ksi for the transverse Young's modulus, 930 ksi for the 

shear modulus, and 0.38 for the major Poisson's ratio. The panel skin has 25 plies 

[445/0, /545/0 /?45/0, /T45/0 , /+45/0,/545 1) and the blade stiffeners have 24 plies [ [: f 45 10 , IT45 1 . The overall length of the panel is 30 in., the overall width is 

11.5 in., stiffener spacing is 4.5 in., stiffener height is 1.4 in., and the hole 

diameter is 2 in. The loading is uniform axial compression. The loaded ends of 

the panel are clamped and the sides are free. 

The material system for the panel is 

This problem was selected as the focus problem because experimental results 

are available and because it has characteristics which often require a local/global 

analysis. These characteristics include a discontinuity, eccentric loading, large 

displacements, large stress gradients, high inplane loading, and a brittle material 

system. This problem represents a generic class of laminated composite structures 

with discontinuities in which the interlaminar stress state becomes important. 

* 

* 

e 

0 

0 

0 

0 

* 

I! 

G~aphjt~-ep~xy (T300f5208) 

Flat panel with t 

30 in. long 

11.5 in. wide 

Stiffener spacing of 4.5 in. 

Stiffener height of 1.4 in. 

2,0--in,-diameter hole 

25-ply panel skin 

24-ply blade stiffeners 

0 Axia~ly loaded with loaded ends clamped and sides free 

52 
-- 

LOCAL/GLOBAL TERMINOLOGY 

The definition of a local/global structural analysis is not precise. For 

example, several levels of detail are considered in the analysis of an aircraft 

structure, and the concepts of local and global can change with every change in 

analysis level. To some, the entire aircraft is the global model, and a fuselage 

section is the local model. To others, lamination theory represents the global 

model, and micromechanics is used for the local model. For this workshop, a 

local/global analysis is a local, detailed stress analysis within a larger, lessrefined global analysis model. The overall response of the panel is the global 

problem; the response near the hole is the local problem. 

Current areas of research associated with local/global methodologies are 

described in the literature (e.g., refs. 4 and 5). One research area 

(discretization procedures) includes finite-element methods, finite-difference 

methods, and boundary element or boundary integral methods. Adaptive mesh 

refinement, h- and p-convergence, and error analysis are current research topics in 

this area which address discretization effects in the presence of a large stress 

gradient. A second area (refined theories) includes research in transverse shear 

formulations and three-dimensional elasticity solutions. These topics focus on the 

mathematical representation of the mechanics of the problem. A third area includes 

classical and closed-form solutions'which are often restricted to simple 

geometries, specific boundary conditions and ,material systems, and often to a 

linear response prediction. A fourth area is hybrid techniques in which two or 

more methods are used simultaneously but in different domains of the structure. 

All four of thes'e areas of research are addressed in the local/global session of 

this workshop. 

0 Concept of IocaVglobal changes with analysis level 

0 Definitions 

--- Global means overall panel response 

Local means response near the hole --- 

0 LocaVglobal methodologies 

--- Discretization procedures 

Refined theories --- 

--- Classical and closed-form solutions 

Hybrid techniques --- 

53 
APPLICATION OF THE FINITE-ELEMENT METHOD 

TO LOCAL/GLOBAL STRESS ANALYSIS 

The local/global methodology adopted for this paper is the finite-element 

method because of its generality. The first step in applying the method was to 

develop and verify a finite-element model of the focus problem. Model verification 

involved solving simpler example cases and comparing the results with other 

analytical results. 

of a flexible mesh generation capability which allowed various finite-element 

discretizations to be evaluated rapidly and systematically. The mesh generation 

capability also provided an easy way to construct and study several idealized 

example cases. 

This model verification process was aided by the development 

Once an adequate finite-element model for the global response was verified, 

the nonlinear structural response was calculated. To identify the local modeling 

detail required to predict accurately the stress distribution near the hole, linear 

stress analyses were performed on a rectangular plate with a circular hole using 

several refined 2-D models near the hole. 

0 Finite-element model development 

0 Finite-element model verification 

0 Global nonlinear response prediction 

0 Local linear stress analysis 

54 
FINITE-ELEMENT MODEL DEVELOPMENT 

The model development strategy is to predict the global nonlinear response 

using the complete model and then to construct a refined, local 2-D model for a 

small distance away from the hole to predict accurately the large stress gradient. 

Displacements and rotations from the global nonlinear solution obtained using the 

complete model will be applied to the refined model and the state of stress will be 

determined. This strategy will be referred to as a multi-level or ffzoom-inff 

approach. 

The automated mesh generation capability allowed versatile modeling of the 

complete problem as well as local regions near the hole. The analyst could specify 

the number of elements across the stiffener depth and down the length of the panel, 

the number of rings of elements around the hole,and the number of elements around the 

hole, and could control the element spacing in the vicinity of the hole. Models could 

also be generated with the hole and discontinuous stiffener filled-in or with no 

stiffeners. 

Complete global model 

Global model 

of panel skin 

model 

55 
FINITE-ELEMENT MODEL VERIFICATION: 

BLADE STIFFENED PANEL 

In the first step of the verification process, a simplified version of the 

focus problem was studied. This simpler problem was identical to the focus problem 

except that the hole and discontinuous stiffener are filled-in and the end boundary 

conditions are now simple support conditions. For this prismatic panel, an exact 

solution was obtained using the PASCO computer code (Panel Analysis and Sizing 

Code, ref. 6). The finite-element analysis system EAL (Engineering Analysis 

Language;ref. 7) was used for the finite-element analysis. The finite-element 

model used in the verification was developed from that of the focus problem to 

determine if any problems related to element distortion or aspect ratio were 

present. 

uniform stress state is present in the skin and the blade stiffeners. The three 

lowest buckling eigenvalues obtained using EAL are very close to the PASCO 

solutions. 

The prebuckling boundary conditions and end loading are such that a 

Buckling mode 

P 

---- 

Eigenvalues 

1 2 3 

PASCO 44536 51063 61601 

EAL 44652 51182 60975 

Difference +0.26% +0.23% - 1 .O2% 

56 
FINITE-ELEMENT MODEL VERIFICATION: 

FOCUS PROBLEM 

The next step of the verification process was to define an adequate finiteelement model for the global response of the focus problem. Finite-element model 

verification for the focus problem started with a "reasonable grid" of 376 4-node 

assumed-stress quadrilateral elements and 422 nodes. 

referred to as Mesh 1. A second, refined grid of 1088 4-node quadrilateral 

elements and 1168 nodes (Mesh 2) was generated for model verification. Linear 

bifurcation buckling solutions for Mesh 1 and Mesh 2 were compared to establish the 

adequacy of the models. 

agree within approximately one percent. 

This discretization is 

The three lowest eigenvalues from both discretizations 

Mesh 1 Mesh 2 

376 elements 1088 elements 

422 nodes 

P 

Eigenvalues 

1 2 3 4 

Mesh 1 41378 52754 54288 55344 

Mesh 2 41829 52533 54259 56895 

Change -1.08% +0.42% +0.05% -2.73% 

57 
LINEAR BIFURCATION BUCKLING SOLUTION 

Oblique views of the prebuckling deformation pattern and the eigenmodes 

corresponding to the four lowest eigenvalues are shown in this figure. 

out-of-plane displacements to develop near the hole from the onset of loading. 

Because of this coupling between inplane and out-of-plane displacements, no linear 

traditional meaning. 

selecting the initial load for the nonlinear analysis and in choosing a load step 

size. However, their main use is in studying the effects of spatial 

I discretization. 

The I discontinuous stiffener leads to an eccentric loading condition which causes large 

equilibrium path exists and the linear bifurcation buckling results do not have the I 

The linear buckling solutions may be used as a guide in 

Linear prebuckling deformation pattern 

Mode 3 

Mode 2 

58 
GLOBAL NONLINEAR RESPONSE PREDICTION 

The global nonlinear response predicted for the focus problem was obtained 

using a new release of EAL. This new release has a nonlinear analysis capability 

using a corotational formulation with linear strain-displacement relations within 

the elements. For this problem, the loading was applied in increments with a full 

Newton-Raphson algorithm. Convergence was based on the maximum error in the 

residual force vector. 

An oblique view of the deformed geometry for the last calculated solution is 

similar to the linear solution shown previously, indicating that the primary 

equilibrium path is being followed. A global response quantity, end shortening, is 

nearly a linear function of the applied load. 

plane displacement at the edge of the hole and blade stiffener, indicates large 

displacements from the onset of loading. Longitudinal inplane stress-resultant 

d i s t r i b u t i o n s f o r two values of the applied load, as a function of distance from 

the hole, indicate high inplane stresses and a high stress gradient near the hole. 

A local response quantity, out-ofThese high inplane stresses and stress gradients coupled with the large outof-plane displacements and the free edge of the hole may cause material 

nonlinearities, local failures, and/or delaminations to develop in order to provide 

local stress relief mechanisms (like plasticity in metal structures) near the hole 

and blade stiffener. However, an accurate prediction of the effects of these 

mechanisms on the global nonlinear response is beyond the current analysis 

capabilities. Stress fracture criteria are developed in reference 8 for an inplane 

loading condition in which the influence of these stress relief mechanisms can be 

accounted for in failure studies without knowing exactly what is happening locally 

near the hole. The point stress failure criterion developed in reference 8 and 

applied in reference 9 to a broad class of laminated composite plates with holes 

will be used in this study as a guide for establishing an adequate finite-element 

model for predicting the stress distributions near the hole. 

50 

25 

P 

(KIPS) 

0 

End shortening 

--- Extension of 

linear solution 

Linear buckling solution 

.02 .04 -06 

U (in.) 

" 5c 

25 

P 

(KIPS) 

Out-of-plane 

displacement 

0 .1 .2 .3 

W (in.) 

59 
GLOBAL NONLINEAR RESPONSE PREDICTION 

4000 

NX Deformed geometry (lb/in.) 

lnplane stress resultant 

at panel midlength I 

I 

I FEdge of hole I 

I 

I \ I 

\\ Load 6 1 

I 

.\ Load A I I 

I 

I 

I I I ll 

1.0 2.0 3.0 4.0 

Y (in.) 

60 
POINT STRESS FAILURE CRITERION 

The point stress failure criterion assumes that failure occurs when the stress 

at a distance do away from the edge of the hole reaches ultimate. The distance do 

is a characteristic dimension which takes the form of a material property. A 

consequence of using this criterion is that an accurate prediction of the state of 

stress precisely at the free edge of the hole is not required. It is only 

necessary to have an accurate stress prediction at a distance do from the edge of 

the hole. Based on results of reference 9 with a similar graphite-epoxy material 

system, a value of do = 0.05 inches was assumed. Model A corresponds to the global 

model near the hole for Mesh 1. Model B is a refined model which more accurately 

predicts the stress gradient near the hole. 

Assumed value of 

do is 0.05 in. 

Model A Model B 

61 
~~ 

LOCAL LINEAR STRESS N LYSIS 

To understand how the spatial discretization near the hole affects the 

prediction of the stress at do, a simpler structural configuration was considered 

which did not include the stiffeners. Using this planar structure, an adequate 2-D 

finite-element model was identified for the local stress analysis. This approach 

provided the necessary insight required for a multi-level model of the focus 

problem. An alternate approach would have been to use an adaptive mesh refinement 

procedure. However, no such procedure was available. The longitudinal inplane 

stress resultant distributions as a function of distance away from the edge of the 

hole are shown for two finite-element models. The results from both models 

approach one another away from the hole. However, at a distance do from the edge 

of the hole, the solutions for Models A and B differ by 12.5 percent. The finiteelement model in the vicinity of the hole was refined by doubling the number of 

elements. The inplane stress resultant at do changed by only 2.2 percent between 

Model B and a model with half as many elements. 

Unstiffened Flat Panel 

(Iblin.1 I j \\ NX I i\\ V 

Y 

0 1.0 2.0 

62 
STATUS AND ADDITIONAL TASKS 

The overall strategy for this study is to predict the global nonlinear 

response using the complete global model and then to construct a refined, local 2-D 

model for a small distance away from the hole. The global nonlinear response has 

been predicted for the focus problem and the local modeling detail required for an 

accurate local stress analysis near the hole of an unstiffened panel has been 

identified. The tasks that remain to be completed for the focus problem include 

performing the multi-level analysis and applying a failure criterion. The multilevel analysis will involve applying the displacements and rotations from the 

global nonlinear solution on the refined local 2-D model and determining the state 

of stress at do. In addition, a three-dimensional model near the discontinuity 

will be required for an accurate determination of the through-the-thickness state 

of stress (i.e., normal and transverse shearing stress distributions). The use of 

3-D elements within a 2-D model will also require a strategy for the transition or 

blending of the two models. 

Status 

0 Finite element model developed and verified 

0 Global nonlinear response predicted 

0 Required modeling detail identified for stress gradient 

near the hole for an unstiffened panel 

Additional tasks for focus problem 

0 Perform multi-level 2-D analysis (refined, local 2-D model) 

0 Apply point stress failure criterion 

0 Perform multi-level 3-0 analysis (refined, local 3-D model) 

63 
SUMMARY 

The local/global nonlinear stress analysis of a blade-stiffened graphite-epoxy 

panel with a discontinuous stiffener is indeed a computational challenge. 

Substantial engineering effort is required in modeling the structure, in verifying 

that the physics of the problem are modeled, and in interpreting the predicted 

nonlinear solutions. Approximately fifty percent of the analysis effort to date was 

devoted to model development and verification. 

generation capability was essential for model verification. 

similar but simpler structures were required and easily generated using the 

automated mesh generator. 

The development of a flexible mesh 

Several models of 

To complete the analysis effort for the focus problem several issues need to be 

addressed. The transition or interface between the various levels of the multilevel model needs to be defined. An adaptive mesh refinement procedure is needed 

to automate the definition of the finite-element models at each stage of the multilevel approach. To obtain a detailed through-the-thickness stress distribution, a 

three-dimensional analysis will be required and the number of three-D elements throughthe-thickness of the laminate needs to be determined. In addition, to predict the 

response of the structure up to overall structural failure, a progressive failure 

analysis capability would be required in which various failure mechanism and 

failure criteria are incorporated. 

0 Substantial engineering effort required in modeling, 

model verification, and response interpretation. 

0 Flexible mesh generation capability essential to model 

verification. 

0 Definition of transitionlinterface region between 

multi-level models required. 

0 Required number of 3-D elements through-the-thickness 

to be determined. 

0 Nonlinear analysis procedure with progressive failure 

analysis capability needed. 

64 
REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

Starnes, James H.b Jr., Dickson, John N. and Rouse, Marshall, "Postbuckling of 

Graphite-Epoxy Panels;" ACEE Composite Structures Technology: Review of 

Selected NASA Research on Composite Materials and Structures, NASA CP-2321, 

1984, pp. 137-160. 

- 

Starnes, James H., Jr. and Williams, Jerry C., "Failure Characteristics of 

Graphite-Epoxy Structural Components Loaded in Compression," NASA TM-84552, 

September 1982. Also, in Mechanics of Composite Materials; Recent Advances, 

Proceedings of the IUTAM Symposium on Mechanics of Composite Materials, Z. 

Hashin and C. T. Herakovich (editors), Pergamon Press, Inc., 1983, pp. 283- 

306. 

Noor, A. K., Shuart, M. J., Starnes, J. H., Jr., and Williams, J. G. 

(Compilers), Failure Analysis and Mechanisms of Failure of Fibrous Composite 

Structures, NASA CP-2278, 1983. 

Zienkiewicz, 0. C., "The Generalized Finite Element Method---State of the Art 

and Future Directions," Journal of Applied Mechanics, ASME, Vol. 50, December 

1983, pp. 1210-1217. 

Noor, A. K. and Pilkey, W. D. (editors), State-of-the-Art Surveys on Finite 

Element Technology, ASME, New York, 1983. 

Anderson, Melvin S. and Stroud, W. Jefferson, "A General Panel Sizing Computer 

Code and Its Application to Composite Structural Panels," AIAA Journal, Vol. 

17, No. 8, August 1979, pp. 892-897. 

Whetstone, W. D., EISI-EAL Engineering Analysis Language Reference Manual - 

EISI-EAL System'Level 2091, Engineering Information Systems, Inc., Sm ;os€, 

Ck , July 1983. 

Whitney, J. M. and Nuismer, R. J., "Stress Fracture Criteria for Laminated 

Composites Containing Stress ConCentrations," Journal of Composite Materials, 

Vol. 8, July 1974, pp. 253-265. 

Mikulas, Martin M., Jr., "Failure Prediction Techniques for Compression Loaded 

Composite Laminates with Holes," Selected NASA Research in Composite Materials - 

and Structures, NASA CP-2142, 1980, pp. 1-34. 

65 
N89-24642 ~ 

COM P UTATI 0 N A L P R 0 C E D U R E S FOR 

POSTBUCKLING OF COMPOSITE SHELLS 

G. M. Stanley and C. A. Felippa 

Lockheed Palo Alto Research Laboratory 

Palo Alto, California 

SUMMARY 

A recently developed finite-element capability for general nonlinear shell analysis. featuring the use of three-dimensional constitutive equations within an efficient resultantoriented framework, is employed to simulate the postbuckling response of an axially 

compressed composite cylindrical panel with a circular cutout. The problem is a 

generic example of modern composite aircraft components for which postbuckling 

strength (Le., fail-safety) is desired in the presence of local discontinuities such as 

holes and cracked stiffeners. While the computational software does a reasonable 

job of predicting both the buckling load and the qualitative aspects of postbuckling 

(compared both with experiment and another code) there are some discrepancies due 

to (1) uncertainties in the nominal layer material properties. (2) structural sensitivity 

to initial imperfections, and (3) the neglect of dynamic and local material delamination effects in the numerical model. Corresponding refinements are suggested for the 

realistic continuation of this type of analysis. 

31. INTRODUCTION 

Advanced composite materials, due to their superior strength-to-weight ratios 

and stiffness tailorability, have become key ingredients in the design of modern 

aerospace vehicles. However, the complex structural response associated with such 

materials coupled with the intricacy of their fabrication creates harsh requirements 

for numerical simulation. 

Specifically, a problem that is of current interest to NASA/LaRC is the determination of the postbuckling strength of thin laminated composite shells comprising 

the “skin” of stiffened air-transport fuselages [I]. These shells are required to 

maintain safe load-carrying capability substantially beyond the point at which 

skin buckling (Le., wrinkling) occurs. To complicate matters, aircraft fuselages 

typically feature local discontinuites, such as fasteners, stiffeners and cutouts, 

which can induce high local stress gradients that tend to delaminate the composite material. In the presence of buckling, these local delaminations can propagate 

throughout - and hence fail - a composite structure. 

It is NASA’s ultimate goal to be able to predict such phenomena analytically. 

To this end, they have asked us to match some experimental data obtained for a 

. --- 

PRECEDING PAGE BLANK NOT FILMED 

67 
representative test specimen [l].* We have completed the first phase of the global 

analysis, which is described in the following sections. It employs a recently developed finite-element shell analysis capability, featuring the use of three-dimensional 

constitutive equations within an efficient resultant-oriented framework, to determine the buckling load and explore the postbuckling regime of an elastically in-tact 

composite model. The more ambitious global/local analysis, which involves the 

prediction of local material delaminat,ion and it,s interaction with global structural 

response to determine postbuckling strength, is still in the planning phase. 

$2. PR,OBLEM DESCRIPTION 

$2.1 Setup 

The focal problem mentioned above is depicted in Figure 1. Shown is a moderately deep (55.6 deg) cylindrical panel with a circular cutout. The panel has a 

14 in. square planform, a 15 in. radius of curvature and a radius-to-thickness ratio 

of 150. The hole is centrally located and measures 2 in. in diameter. The composite shell wall consists of 16 layers of unidirectional graphite fibers in epoxy-resin. 

Each layer is .0056 in. thick (for a total of .086 in.) and the layers are arranged 

in the symmetric, quasi-isotropic *’ stacking sequence: { *45/90/0/0/90/ 45) 

degrees - repeated twice. The orthotropic-elastic material properties for each 

layer are listed in the figure. Note that these properties represent nominal values 

and will require some adjustment in the sequel (see $5). 

Surrounded by a metallic test frame, the appropriate boundary conditions for the 

cylindrical panel are (i) fully clamped on the bottom edge, (ii) clamped except for 

axial motion on the top edge and (iii) simply supported on the vertical edges. 

§ 2.2 Experimental Results 

The test conducted by NASA consisted of statically imposing a uniform endshortening, 6, to determine the load-carrying capability of the panel beyond the 

initial buckling load. Experimental results are shown in Figure 2. 

Figure 2a represents a normalized “load versus end-shortening” curve. Note that 

buckling occurs abruptly, followed by a rapid drop in the axial load. Somewhere 

* 

** 

This problem was first suggested by Dr. Norman F. Knight, Jr. at NASA Langley 

Research Center. We amusingly refer to it as “Knight’s problem” both as an acknowledgment to the originator and as a reminder of the many pitfalls obstructing 

its numerical solution. 

The term quasi-istropic refers to the fact that the resultant constitutive matrix is essentially isotropic, due to a balanced sequence of fiber angles through the thickness. 

However, in contrast to truly isotropic materials, there is some additional coupling 

between bending and twisting deformations. 

68 
between the top and bottom of this vertical branch, which spans a period of rnilliseconds in the experiment, delamination occurs near the hole (Fig. 2b). Although 

the delamination gets progressively worse and eventually distorts the hole, there 

is a secondary (postbuckling) stiffening branch in the load/displacement curve. 

The test was stopped at the point on this secondary branch labeled bbco11apse77, at 

which point extensive delamination is evident (see [ 11 for details). 

$3. COMPUTATIONAL APPROACH 

$3.1 Formulation 

The formulation of the governing equations and associated solution algorithms 

used to analyze the above problem is outlined in Figure 3. A detailed description 

of this approach may be found in [a]. Briefly, it employs continuurn-based (CB) 

shell elements, similar to the Ahmad element 131, but extended to the nonlinear 

regime and reduced to an economical resultant form in which element stiffness and 

force operators are pre-integrated through the thickness. 

Shell Equations 

Thickness pre-inbegration of the CB shell equations is achieved by augmenting the 

standard, Mindlin-type [4] hypotheses of straight normals and zero normal stress 

with two additional hypotheses, namely: small trunsverse-shear strains and mild 

taper. As argued in [2], these additional hypotheses do not significantly alter the 

range of applicability of the original formulation. The resulting theory, which is 

expressed in terms of stress-resultants rather than pointwise (continuum) stresses, 

is referred to as a continuum-bused resultant (CBR) shell formulation. 

Note that unlike earlier efforts to pre-integrate the CB shell equations, which typically assume a constant through-thickness variation of the surface metric (e.g., [5]), 

the present CBR formulation bypasses this assumption and hence is not restricted 

to very thin shells. 

Shell Elements 

To spatially discretize the CBR shell equations, a variety of shell finite elements 

have been implemented within the above framework. However, on the basis of the 

numerical evaluation conducted in [2], only the following two shell elements were 

considered for the present analysis: (i) the nine-node Heterosis (HET) element [6], 

and (ii) a new nine-node assumed natural strain (ANS) shell-element [7]. While 

both elements are parabolically curved (Fig. 3) and use standard isoparametric 

interpolation as a starting point, each departs from the basic recipe in order to 

properly represent inextensional bending deformation for thin shells. 

69 
In particular, the 9 HET element selectively under-integrates all stiffness and force 

terms involving membrane strains to avoid membrane “locking”, and uses a mixture of Lagrange shape functions (for rotations) and Serendipity shape functions 

(for translations) to avoid “spurious modes” otherwise evoked by reduced integration. 

In contrast, the 9-ANS element assumes an appropriate (inextensionally accurate) 

strain field frorn the outset, using a modified set of Lagrange shape functions and 

employing full numerical integration throughout. Due to the fact that the strains 

are assumed in the generally non-orthogonal isoparametric coordinate basis, an 

apparent advantage of the ANS approach is its decreased sensitivity to element 

mesh distortion. 

Nonlinear Solution Procedures 

After performing an updated-lagrange linearization of the equilibrium equations, 

a modified Newton-Raphson (NR) version of the Riks-Crisfield (RC) arc-length 

control algorithm IS] is used to trace incrementally the load-displacement curve. 

The RC procedure was adopted as a convenient means of statically traversing the 

bifurcation points that arise in shell postbuckling analysis. However, such methods 

are not foolproof, and special attention by the analyst (e.g., in the selection of 

imperfections, step-sizes and error tolerances) is often required in the vicinity of 

closely spaced or multiple bifurcation points. 

Two kinds of update procedure are required to advance the solution of the discrete 

nonlinear shell equations from one load-step to the next: (i) a kinematic update 

that “accumulates” incremental nodal displacement - both translational and rotational components; and (ii) a constitutive update that generates new element 

stresses from the corresponding displacement field. 

Only the rotational part, of the kinematic update is non-trivial.* In the present 

approach, the rotation increments are used to update orthogonal triads defined at 

each shell node. The triad-update algorithm involves no trigonometric functions, 

maintains orthogonality at each step and provides a shell-oriented coordinate system in which the normal rotational degree of freedom may be eliminated at all shell 

nodes (except at junctures). Furthermore, once the nodal triads and referencesurface coordinates have been so updated, the current element configuration is 

completely and uniquely defined (see (21 and [9] for details). 

The stress update is handled via an incrementally objective algorithm 1101 that 

* Since the translational components of displacement are vectorial, translation increments are simply added to obtain total displacements and hence update the nodal 

coordinates. 

70 
features a midpoint-rule numerical integration of rate-type constitutive equations. 

For finite-strain analysis, the constitutive algorithm additionally involves shell 

thickness updates that account for large Poisson effects. These are computed (as 

in [ 11)) by recovering the normal strain increments from the constitutive equations 

via the zero normal stress (ZNS) hypothesis. 

Summary of Computational Features 

The computational features of the above approach may be summarized as follows: 

0 Applicable to Both Thin and Moderately Thick Shells 

0 Rotations May Be Arbitrarily Large 

0 Strains May Be Large ( Except Transverse Shears ) 

0 Resultant Format Yields I Cost Savings c( Number of Layers I 

0 Shell Elements Are Fairly Robust 

- No Locking 

- No Spurious Mechanisms 

- Low Sensitivity to Mesh Distortion (Especially ANS Elements) 

Finally, note that while the formulation allows for finite strains and inelastic material behavior, the present analysis simply employs orthotropic linear-elasticity 

within each layer and does not account for the material delamination (z.e., damage) observed in the NASA experiment. 

53.2 Implement,ation (NICE Software Architecture) 

The shell-element capabilities mentioned above have been implemented in a modular fashion to facilitate research and transferral to other finite element codes. 

Some of the specific shell-element functions that are available via independent 

FORTRAN77 subroutine calls are shown in Figure 4a. A complete description of 

the shell-element software (including listings) is given in Appendix S of reference 

PI. 

The host finite-element program used by the authors is actually a network of independently executable programs (or processors) which are coordinated via highlevel procedures written in a mathematically oriented command-language (Fig. 

4b). In such an environment, the shell-element software is embedded in a single processor and the global solution algorithms are implemented as procedures; 

examples are given in [12]. 

71 
The software architecture (i.e., utilities) used to construct this particular analysis 

system is known as NICE (for Network of Interactive Computational Elements, 

11 2,131). Due to the flexibility provided by the NICE archit,ecture and its suitability 

for nonlinear and coupled-field problems, it is currently being explored by NASA 

as the basis for a standard generic testbed system for Computational Structural 

Mechanics (see [14] and other presentations therein). One of the motivating factors 

for developing such a system is the implementational complexity associated with 

a comprehensive global/local analysis of the present composite-shell problem. 

54. FINITE-ELEMENT MODELS 

Due to the physics of the problem, a full numerical model of the test specimen is 

required (Fig. 5). The slight anisotropy emanating from the composite material 

stacking sequence is only partly responsible for the lack of available symmetry." 

As will become apparent, the nonlinear postbuckling response is inherently nonsymmetric due t,o the presence of the hole and the participation of many diverse 

mode shapes. 

Several combinations of shell-element type and mesh density were employed during 

the course of the linear (pre-buckling), stability (buckling eigenvalue) and nonlinear (postbuckling) analyses. Figure 5 shows three representative grids, involving 

300, 1500 and 5000 degrees-of-freedom, respectively. These grids correspond to 

16, 80 and 256 nine-node elements (or alternatively to 64, 320 and 1024 four-node 

elements), respectively. Note that there is intrinsic element mesh-distortion in 

these models - bot,h in-plane and out,-of-pla.ne - due to t,he focus on the hole 

and the curvature of the shell. However, the elernents nearest the hole (where it 

counts most) have the most regular shapes. 

The coarsest grid (1) was used to verify the modeling procedure, the finest grid 

(3) was used exclusively to check convergence of the linear and eigen solutions, 

and the intermediate grid (2) became the workhorse for nonlinear analysis. Furthermore, as little difference was observed between the 9-HET and 9-ANS elements 

(53.1) during the early stages of analysis, the 9-HET element (which is slightly less 

expensive) is fea.tured in the analytical results that follow. 

Boundary coiiditions were imposed as described in Section 2 and illustrated in 

Figure 5. To simulate end-shortening, an axial force was applied in conjunction 

with a degree-of-freedom equivalence among all axial displacements on the loaded 

edge. This was done to avoid the use of specified displacements, which tend to 

complicate the adaptive, arc-length-based nonlinear solution algorithm. 

* We have confirmed this via numerical experiments with an isotropic model. 

72 
55. LINEAR (PRE-BUCKLING) ANALYSIS 

Results for the linear pre-buckling analysis are shown in Figure 6. The defornia.tions due to an applied axial compressive load of 22480 Ibs (1 kN) are shown 

rnagriified by a factor of 10 in the top half of the figure ~~ for grid 2. The corresponding distribution of the axial stress resultant, N,, along the panel circumference at mid-span is shown in the bottom half of the figure - for both grids 2 and 

3. 

R,egarding the displacement solution, convergence of the axial end-shortening, 

6, was achieved with grid 2. (Grid 3 yielded less than a 1% increase in endshortening.) However, the converged end-shortening solution, 6 = .0316, is approximately 15% lowcr than thc cxpcrimcntal valuc; as deduced from the linear 

portion of the experimental load-displacement curve (Fig. 2). It is presumed that 

this over-estimation of the axial stiffness is due to the uncertainty in the nominal 

material propert,ies, a conclusion that is reinforced in 58. Thus, to compensate 

for the rriisrnatcli, the lamina principle elastic modulus (E,) is reduced by a corresponding fact,or in the subsequent nonlinear analysis ($7). 

From a qualitative perspective, the solution shows substantial bending deformation 

in the vicinity of the hole (Fig. 6a). This suggests that geometric nonlinearity 

may be important even at relatively low load levels and diminishes the credibility 

of linear response and buckling-eigenvalue analyses. 

Regarding the linear stress solution, note that the compressive axial resultant, 

N,, is distributed evenly along most of the panel circumference except for a very 

localized region near the hole. While grid 2 was adequate for the displacement 

solution, grid 3 provides much more accurate resolution of this stress concentration. 

In particular, grid 2 yielded a peak stress concentration factor (SCF) of 2.8; about 

14% less than the SCF obtained with grid 3. 

As experimental data were not available for verification of the computed stresses, 

the convergence of the grid 3 stress solution was inferred by comparison with 

a closed-form (asymptotic) solution due to C.R. Steele (private communication, 

Stanford University, 1985). While the closed-form solution pertained to a purely 

isotropic panel, the linear finite-element stress solutions for isotropic and quasiisotropic panels were found to be quite similar. It is also interesting that the SCFs 

for both the isotropic model (3.1) and the quasi-isotropic model (3.25) are not 

very different from the classical SCF for a flat plate with a circular hole (3.0). 

Finally, note that by strongly biasing the mesh towards the hole, it was possible 

to obtain grid-3 accuracy with grid 2 for the local stress gradients. Such biasing, 

however, was found to be unnecessary in the subsequent, globally oriented buckling 

and postbuckling analyses. 

73 
$6. STABILITY (BUCKLING) ANALYSIS 

Figure 7 shows the first 5 buckling eigenmodes for grid 2. These results represent 

perturbations about the linear pre-buckling solution described. The eigenvalues, 

X ::~ 1.084,1.10ci,1.181,2.432,1.582, are the ratios of the corresponding buckling 

loads to the axial load applied in the linear pre-buckling analysis. As before, grid 

2 seems to provide adequate resolution, with grid 3 giving only a 2% reduction in 

the first two eigenvalues, and a 4% reduction in the remaining three. 

The following observations are important for subsequent computational purposes: 

(i) the eigenvalues are closely spaced; (ii) t,he eigenmodes are vastly different in 

character; (iii) therc is no single form of symmetry to be exploited computationally; (iv) the first, buckling mode is symmetric and bears the most resemblance 

to the linear pre-buckling solution; (iv) the second and third modes possess skew 

symmetries; (v) the fourth and fifth modes are symmetric; the latter mode featuring pra.ctically no distortion of the circular hole; and (vi) higher modes (not 

shown) look much like those for a cylindrical panel without a hole, though the 

values remain closely spaced. 

Finally, it, was found that the first (z.e., critical) buckling load is approximately 

25% lower than that of an identical cylindrical panel without a hole. Hence, while 

the influence of the hole on the buckling loud is only moderate (i.e., relative to the 

stress concentration fact>or), its influence on the buckling modes is profound. As 

we shall see, the hole has an even stronger influence on the postbuckling response. 

$7. NONLINEAR (POSTB‘IJCKLINC,) ANALYSIS 

In practice), more than just a linearly converged model and an adaptive solution 

strategy were necessary to obtain a reasonable nonlinear solution. One additional 

pre-requisite was a 15% reduction in the nominal elastic modulus, El (from 19.6 x 

IOG to 17.1 x lo”), to match the linear branch of the load versus end-shortening 

curve (as explained in $5). Another important ingredient was the specification of 

initial imperfections. In this regard, three versions of the analysis were run: (1) 

one involving no imperfections; (2) one with an imperfection amplitude of 1% of 

t,he thickness applied to each of the first 4 buckling modes (see Fig. 7); and (3) 

one with an imperfection arnp1it)ude of 10% of the thickness applied to each of the 

first 4 buckling modes. A discussion of the results for these three cases follows. 

0 7.1 No Imperfect ions 

One would think that the out-of-plane “imperfections” introduced by the linear 

pre-buckling solution (Fig. 6) would be sufficient to trigger a realistic buckling 

response. However, this was not the case. With no imperfections, the computed 

solution path resembled the experiment only up to the descending part of the 

74 
load-displacement curve (Fig. sa). The computed curve then rolled back onto 

itself with the stiffening branch of the postbuckling curve practically aligned with 

the pre-buckling curve. 

To gain further insight, it is useful to look at the deformation and stress histories 

portrayed in Figure 8b. Shown is a sequence of computational “snapshots” taken 

at various points (i.e., load steps) on the nonlinear load-displacement curve.* The 

variation of the axial stress resultant along the mid-span circumference is plotted 

below each frame. 

Note that the deformation starts out (at load step 10) much like the first linear 

buckling mode (Fig. 7), with inward dimples both fore and aft of the hole, then 

articulates through the second and third modes during the initial postbuckling 

phase (steps 15-20). This rotation of the two dimples about the hole is probably 

triggered by the bending/twisting coupling inherent in the composite stacking 

sequence. The dimples continue to rotate and broaden until, at step 40, the 

pattern begins to resemble the fifth linear buckling mode. Evidently, it is this 

“locking” into mode 5 that is responsible for the excessive secondary stiffening 

in the load-displacement curve (Fig. Sa). Clearly, mode 5 is an unrealistically 

stifl one, resembling what might occur if a ring sliflener had been placed around 

the hole. This is also evident in the axial stress distribution, where the stress 

concentration has practically been shed by step 40. 

57.2 Large Imperfections 

To avoid the unrealistic mode-5 locking observed in the preceding analysis, a fairly 

large initial imperfection was introduced in the numerical model. This was accomplished by adding an equal measure of each of the first 4 buckling modes to the 

initial geometry, such that the maximum radial (i.e., shell-normal) displacement 

in each mode was equal to 10% of the panel thickness. Thus, the magnitude of 

the combined radial imperfection approached 40% of the thickness at some points. 

It is emphasized that this rather arbitrary choice of imperfections was designed 

primarily to minimize the influence of mode 5. 

The computed nonlinear response for the imperfect panel is shown in Figure 9. 

Note that the secondary stiffening branch bears more resemblance to the experiment than did the imperfection-less analysis. Unfortunately, it is also true that 

the buckling (i.e., peak) load has dropped by about 15%, and now underestimates 

the experimental buckling load by more than 20%. A heuristic explanation is 

* Note that the pre-buckling phase of the analysis appears linear with respect to the 

axial displacement, 6. However, due to the rapid growth in radial displacements 

(not shown), the analysis is actually quite nonlinear from the outset; which explains 

the relatively large number of load-steps required on the ‘‘linear” branch of Fig. 8. 

75 
provided by considering the deformation and stress histories. 

In Figure 9b, we see that with the lO%-h imperfections, the buckling pattern leaves 

mode 1 almost immediately and develops an intensified inward dimple on one side 

of the panel. Consequently, the full axial load is re-distributed to the other side 

of the panel (see N, plots in Fig. 9b), accounting for the reduction in both the 

buckling load and the postbuckling stiffness (fig. sa). 

57.3 Small Imperfections (the &Bottom Line”) 

Finally, the “best of both worlds” was obtained with a 1%-h imperfection in each 

of the first 4 modes. Compared with the previous analysis, the computed loaddisplacement curve (Fig. 10) shows both an increase in the maximum load and 

a decrease in the minimum load, thus bringing the solution more in line with 

experiment. 

The improved performance, obtained by reducing the imperfections, may again 

be related to the deformation history (Fig. lob). Here, as in the case without 

imperfections, two inward dimples develop and proceed to rotate about the hole. 

Just after buckling, however, one dimple tends to deepen while the other diminishes, and eventually there is a double snap-through. This accounts for the double 

dip in the load-displacement curve (Fig. loa) and seems to explain why a lower 

minimum load is obtained with the smaller imperfection. 

Still, there are some serious descrepancies between analysis and experiment, 

namely: (i) a 7% under-estimation of the buckling load, (ii) a 25% over-estimation 

of the minimum load, (iii) a 30% under-estimation of the postbuckling endshortening, and (iv) a 10% over-estimation in the postbuckling stiffness. These 

will be addressed in $9. 

$8. CORROBORATION WITH ANOTHER CODE 

To support the above results, obtained via the computational procedures described 

in Section 3, parallel analyses were performed with another finite-element computer code. For this purpose, we employed the STAGS code [15], which has been 

used for more than a decade by varous government agencies and industrial firms 

(e.g., NASA and Lockheed) to analyze difficult nonlinear shell problems. Another 

reason for using STAGS is that it features finite-element computational procedures that are substantially different from those used in the present approach, 

thus adding strength to an analytical comparison. 

For the linear (pre-buckling and buckling) analysis, excellent agreement was obtained between the STAGS and NICE-based solutions. For completeness, the 

STAGS runs were performed with two radically different shell-element types, which 

both converged to the same solution as the NICE/g-HET element, albeit at slower 

76 
rates and from below in stiffness. (Note that while the axial pre-buckling stiffness 

of the STAGS elements converged from below, the buckling loads converged from 

above). 

In particular, the comparison included: (i) the commonly used STAGS/410 element ~~ a flat quadrilateral plate element based on Kirchhoff-Love theory and 

cubic membrane/bending displacement interpolation; and (ii) the less frequently 

used STAGS/422 element - a quadrilateral composed of two Kirchhoff-based triangular plate sub-elements with cubic bending interpolation and quadratic membrane interpolation. It is believed that the relatively slow convergence “from 

below” of the STAGS/410 element is due to warping sensitivity, while that of the 

STAGS/422 element is probably due to the incornputibility between membrane 

and bending displacement fields for non-flat quadrilateral element shapes.* 

For the nonlinear comparisons with NICE/S-HET, the STAGS/410 element was 

used exclusively. The resulting load-displacement curves (for 1%-h and 10%-h 

imperfections) are summarized in Figure 11. The dashed curves represent the 

NICE/9_HET solutions and the dotted curves represent the STAGS/410 solutions; 

both were obtained with Grid 2 (Fig. 5). 

Note that although the STAGS and NICE-based solutions use different finiteelement types, large-rotation update procedures and nonlinear solution strategies, 

the correlation is remarkable especially during the postbuckling phase. Even 

the STAGS zero-imperfection analysis (not shown) resulted in the same excessive postbuckling stiffness as displayed in Figure 8a. One other point: While 

the STAGS/410 element consistently shows about a 5% higher buckling load 

than the NICE/9_HET element, thus coming closer to the experimental peak; 

STAGS/410 is actually less accurate - with respect to discretization errors 

- than NICE/g-HET. This follows from thc fact that both STAGS/410 and 

NICE/g-HET converge from above in the buckling load. This was confirmed by 

running the NlCE/9_HET element with a coarser grid, for which it too showed a 

5% higher peak. That non-converged solut,ions compare better with experiment 

than converged ones suggests that spatial discretization is not the only source of 

error here (e.g., see $9). 

* It is interesting to note that the STAGS/422 element was used in the related study 

conducted in [I], where it yielded a 17% more flexible linear solution, and thus agreed 

better with the linear portion of the experiment. Nevertheless, it has since been 

found that the boundary conditions were not consistently applied to the element’s 

mid-side freedoms in that analysis. By correcting this implementation error, the 

17% discrepancy with the other elements has been completely eliminated. Thus, it 

appears that the “accuracy” obtained in [l] with the STAGS/422 element is due to 

compensating errors in the nominal material properties. 

77 
$9. CONCLUSIONS 

59.1 Summary 

The present study may be summarized as follows: 

0 PURPOSE: 

- Validate continuum-based resultant (CBR) shell formulation 

- Evaluate new shell elements 

- Gain experience with composite-shell postbuckling analysis 

0 RESULTS: 

- Good agreement in pre-buckling/buckling range 

- Good qualitutive agreement in postbuckling range 

- Discrepancies due to: 

-_ Material properties 

-_ Imperfection sensitivity 

__ Dynamic effects 

-_ Delamination 

The “good” agreemerit obtained between the present shell-element formulation 

and experiment in the pre-buckling and buckling range was possible only after 

adjusting the nominal material properties so that the linear axial stiffnesses coalesced. The material-property modification was further justified via corroboration 

with the STAGS finite-element code, which features a substantially different computational approach. 

The “best” solution for the nonlinear response was obtained by introducing small 

(1 %-thickness) imperfections corresponding to each of the first four buckling 

modes. The computed load-displacement curve (Fig. loa), which again compared 

well with STAGS (Fig. ll), still showed major discrepancies with the experiment: The discrepancy in the buckling load (which is relatively small) may be 

due to the inadequacy of adjusting only the principal layer elastic modulus, El, 

rather than the complete set of orthotropic rriaterial constants. The discrepancy in 

the unloading phase of postbuckling (i. e., the computed end-shortening reversal) 

is attributed to the quasi-static approximation of what is, in reality, a dynamic 

phenomenon. Finally, the discrepancy in the stiflening phase of postbuckling is 

clearly dominated by damage, ie., delamination observed in the experiment but 

not represented in the model. 

78 
59.2 Recommendations 

The goal is to eliminate the discrepancies listed with a minimum of computational 

cost and complexity. To this end, the following steps are recommended: 

1) PERFORM ADDITIONAL EXPERIMENTS. First, more experimental data 

are required to verify existing computational capabilities for composite shell postbuckling. For example, panel imperfections should be carefully measured and 

selected strains and overall deformation patterns should be monitored at frequent 

intervals. Additionally, an isotropic panel should be tested in order to eliminate 

the material identification problem and also to provide a standard benchmark for 

shell-element evaluation. The isotropic problem would be valuable for screening 

out geometrically sensitive elements. 

2) REFINE NONLINEAR GRID. The nonlinear analysis should be repeated 

with a finer grid (e.g., Grid 3), as convergence in the linear regime is no guarantee 

of convergence in the nonlinear regime. Moreover, a study of modal participation both in the imperfections and in the nonlinear response may help establish 

modeling guidelines for future analysis. 

3) INCLUDE DYNAMIC EFFECTS. Dynamic effects, which are relatively 

straightforward to incorporate, should be assessed at the first opportunity. The 

analysis could be started in a quasi-static mode, switched to an explicit transient 

response algorithm during the unstable phase, and switched again to an implicit 

algorithm during the stable postbuckling phase. 

4) REPRESENT LOCAL FAILURE: (DELAMINATION). To account for the 

composite delamination mechanism, appropriate failure criteria and progressivefailure modeling procedures need to be developed, implemented and evaluated. 

For shell-based failure criteria, accurate stress recovery is essential. Improved 

stress resolution is required both along the surface (e.g., via stress/displacement 

iterat,ions and adapt,ive refinement,) and thrOiigh-th,e-thl.ckncs.s, as both normal and 

transverse-shear stresses can play a dominant role in delamination. Progressive 

failure may then be simulatcd by methods ranging from a simplified shell model 

that selectively degrades luyer yroperlies, to a full 3l) uridysis riear the hole with ail 

evolving 2D/3D transition. An intermediate approach is to “split” shell elements 

along delaminating boundaries. (See Fig. 12.) The simplest approach, however, 

has obvious implementational advantages. 

5) DEVELOP EFFICIENT GLOBAL/LOCAL ALGORITHMS. Finally, there is 

a need to reduce the cost of nonlinear analysis for such problems. The cost of the 

global analysis is dominated by the large number of iterations/steps in the linearto-postbuckling transition regime. Possible approaches include the reduced-basis 

79 
technique 116,171, Thurston’s method 1181, and improved extrapolators. For the 

combined global/local problem, where material “properties’) are changing rapidly 

during postbuckling. additional feat,ures such as line-searches [ 191, quasi-Newton 

stiffness updates (201 and nonlinear substructuring may greatly improve effic ienc y. 

We arc presently acting on rccomrnendations (2)-(3) and will report the outcome 

in a forthcoming paper. 

ACKNOWLEDGMENTS 

The preparation of this paper was supported by the Independent Research Program of 

Lockheed Missiles and Space Company, lnc. Ongoing research is being supported by 

NASA/Langley Research Center as part of their newly launched program in Computational Structural Mechanics. 

REFERENCES 

Knight, N.F, Jr. and Starnes, J.H., Jr. (1984), “Postbuckling Behavior of Axially 

Compressed Graphitc-Epoxy Cylindrical I’ancls with Circular Holes”, prescnted at 

t,hc 1984 ASME Joint Pressrirc Vcsscls and Piping / Applied Mechanics Converence, 

San Antonio, Texas. 

Stanley, G.M. (1985), Continuum-Based Shell Elements, Stanford University Ph.D. 

thesis, also Lockheed report’ LMSC-F035839. 

Ahrriad, S., Irons, B.M. and Zienkiewiecz, O.C. (1970), “Analysis of Thick and Thin 

Shell St,ruct,ures by Curved Finit,e Elerrients” , International Journal for Numerical 

Melhods in Enyineer1rig, 2, 419-451. 

Mindlin, R.D. (1951), "influence of Rotat,ory lnertia and Shear on Flexural Motions 

of Isotropic, Elastic Plates”, Journal of Applied Mechanics, 18, 31-38. 

Parisch, H. (1981), “Nonlinear Analysis of Shells [Jsing lsoparametric Elements”, 

pp. 47- 63 iri Nonlinear Finite Element Analysis of Plates and Shells, (eds. T.J.R. 

Hughes et al.), ASME, New York. 

Hughes, T.J.R., and Cohen! M. (1979), “The ‘Heterosis’ Finite Element for Plate 

Bendirig” , Computers and Structures, 9, 445-450. 

Park, K.C. and Stanley, G.M. (1986)) “A Curved Co Shell Element based on Assumed Natural-Coordinate Strains”, in Journal of Applied ,l/lechanics, June 

1986. 

Crisfield, M.A. (1983), “A Fast Incremental/lterative Solution Procedure that Handles Snap-Through” , Computers and Structures, 13, 55-62. 

Rankin, C.C. and Brogan, F.A. (1984), “An Element-Independent Corotational 

Procedure for the Treatment of Large Rotations”, pp. 85-100 in Collapse Analysis 

of Structures (eds. L.H. Sobel and K. Thomas), ASME, New York. 

80 
[lo] Hughes, T.J.R. and Winget, 3. (1980), “Finite Rotation Effects in Numerical lntegration of Rate Constitutive Equations Arising in Large-Deformation Analysis”, 

International Journal for Numerical Methods in Engineering, 15, 1862-1867. 

[I l] Hughes, T.J.R. and Carnoy, E. (1981)) “Nonlinear Finite Element Shell Formulation 

Accounting for Large Membrane Strains” , pp. 193-208 in Nonlinear Finite ElerneriL 

Analysis of Plates and Shells (eds. T.J.R. Hughes el al.), ASME, New York. 

1121 Felippa, C.A. and Stanley, G.M. (1985), “NICE: A Utility Architecture for Computational Mechanics”, proceedings of US-Europe Symposium on Finite-Element 

Methods for Nonlinear Problems, Univ. of Trondheim, Norway (ed. by P. Bergan). 

1131 Felippa, C.A. (1981), “Architecture of a Distributed Analysis Network for Computational Mechanics”, Computers and Structures, 13, 405-413. 

[ 141 Geers, T.L. (1984)) “Comments on Proposed Research Program in Computational 

Structural Mcchanics” ) proceedings of Workshop on Computational Structural Mechanics, held 5-7 Sep 1984 at NASA/LaRC, edited by W.J. Stroud. 

1151 Almroth, B.O., Brogan, F.A. and Stanley, G.M. (1979), “Structural Analysis of 

General Slrells, Vol. IT: User Instructions for the STAGS(C-I) Computer Code”, 

Lockheed Report LMSC-D33873. 

116) Noor, A.K. and Peters, J.M. (1983), “Recent Advances in Reduction Methods for 

Instability Analysis of Structures”, Computers and Structures, 16, 67-80. 

1171 Stehlin, P. and Brogan, F.A. (1984), “Analysis of Stjructjural Collapse by the Reduced Basis Technique”, pp. 69-84 in Collapse Analysis of Structures (eds. L.H. 

Sobel and I(. Thomas), ASME, New York. 

(181 Thurston, G.A., Brogan, F.A. and Stehlin, P. (1985), “Postbuckling Analysis Using 

a General Purpose Code”, presented at AIAA 26th Structures, Structural Dynamics 

and Materials Conference) Orlando, Florida, AIAA Paper No. 85-07 19-CP. 

1191 Crisfield, M.A. (1982), “An Arc-Length Method Including Line Searches and Acceleration”, International Journal for Numerical Methods in Engineering, 19, 1269- 

1289. 

120) Dennis, J .E. and More, J.J. (1977)) “Quasi-Newton Methods, Motivation and Theory”, SIA M Review, 19, 46-89. 

81 
COMPOSITE CYLINDRICAL PANEL WITH CIRCULAR HOLE 

R = 15 in. 

Metallic 

Fixture 

Test 7 

I 

v 

c 

1 = 14 in. 

I 

NOMINAL COMPOS1 TE 

MATERIAL PROPERTIES: 

16 Layers 

C ra ph i te- Epoxy 

hlayer = 0.0056 in. 

htotal = 0.09 in. 

6 E = 19.6 x 10 psi 

E2 = 1.89 x 10 6 psi 

vz1 = 0.38, v12 = 0.037 

CtZ = 0.93 x 10 6 psi 

{t45/90/0/0/90r45}o syrn 

--. 

1 

.... 

------I 

Figure 1. Knight’s problem, NASA test specimen. 

0 2 4 6 x lo-= 

END-SHORTEN ING, d/L 

Figure 2. Experimental results. 

82 
0 Continuum-based (CB) shell equations 

- 3-D Continuum equations (equilibrium /con st i tut ive) 

- Embed sheil hypotheses (straight normals, zero normal stress) 

0 Reduce to "resultant" (CBR) form 

- Assume: small transverse-shear strains, mild taper 

- Preintegrate through-the thickness 

0 Discretize via curved "isoparametrlc" elements 

- Selective/reduced integration (SRI) 

- Assumed natural strain (ANSI 

a __--- 

-1 <\ 

I 

1 

0 Solve nonlinear matrix equations via: 

- Linearization w.r.t. currrent configuration (UL) 

- Modified NR algorithm with adaptive (RC) strategy 

- Nodal triad updates for large rotations 

0 Solve rate constitutive equations via: 

- "Midpoint rule" incremental algorithm 

- ZNS recovery of normal strains (thickness updates) 

I 

Figure 3. Computational approach. 

USER 

... I I I I 

GLOBAL DATABASE 

a. NICE (Network of Interactive Computational Elements) 

Figure 4. Implementation. 

83 
I x 16 

01 

NODES 300 DOF 

a. Grid I IColrrcI 

I.* 

C. Grid 3 (Refined) 

Figure 5. Finite-element models. 

a. Linear Displacement Solution 

(Magnified by 10) 

Figure 6. Linear (prebuckling) analysis. 

84 
3 

x10- 

2. h 

P 

0 

< 

1. 

s 

0. 

 BUCKLING MODE 21 

/BUCKLING MODE 41 

[BUCKLING MODE 51 

Figure 7. Stability (buckling eigenvalue) analysis. 

I I 

1.0 2.0 3.0 4 

/, I 

STEP 10 STEP 30 

X' I 

STEP 35 

STEP 20 

END-SHORTEN1 NC, 6/L x1~-3 

- EXPERIMENTAL 

Nx 1 -...-- FIN IT€-ELEMENT 0 LOAD-STEP NUMBER 

a. Load versus End-Shortening b. Deformations and Stresses 

Figure 8. Nonlinear (postbuckling) analysis; no imperfections. 

85 
3.0 

x10 - 

2.0 

a 

w 

a 

0 

a 

s 

1.0 

0.0 

1 o-~ 

F.E. 

LOAD 

2.0 - 

5 

d 

s 

. 

n. 

4 

1.0 - 

EXPERIMENTAL 

0 LOAD-STEP NUMBER 

+-t* FINITE-ELEMENT 

0.0 I I 1 I 

0.0 1.0 2.0 3.0 4.0 x10-3 

END-SHORTENING, d/L 

a. Load Versus End Shortening 

Figure 9. Nonlinear (postbuckling) ana 

.INEAR BUCKLING LOAD 

DAMAGE 

Nx 1 &-+----I 

b. Deformations and Stresses 

lysis; 10%-h imperfections. 

STEP 10 STEP 30 

STEP 20 

STEP 40 

a. Load versus End-Shortening b. Deformations and Stresses 

Figure 10. Nonlinear (postbuckling) analysis; 1%-h imperfections. 

86 
3.0 

2.0 4 

d 

\ 

n 

< 

0 

J 1.0 

0.0 

EXPERIMENT 

--- STAGS / 41 0 ELEMENT 

NICE/S-HET ELEMENT --- 

/ I 1 I I 

0.0 1.0 2.0 3.0 4.0 5. 

END SHORTENING, 6/L 

a. 10%-h Imperfections 

EXPERIMENT 

STAGS/4lO ELEMENT 

NICE/g-HET ELEMENT 

--- 

--- 

Figure 11. Corroboration with the STAGS code, postbuckline analyses. 

G = G, 

= - G2 

SHELL ELEMENT SHELL ELEMENT 

a. Simple: Shell-Element Material-Property Degradation 

SHELL ELEMENTS SHELL ELEMENT 

b. Intermediate: Shell-Element "Splitting" 

SOLID ELEMENTS t SHELL ELEMENT 

CONSTRAINED 

INTERFACE 

C. Complex: 30/20 Transition 

Figure 12. Composite failure-progression models. 

87 
N89-24643 * 

A REVIEW OF SOME PROBLEMS IN GLOBAL-LOCAL 

STRESS ANALYSIS 

Richard B. Nelson 

UCLA 

Los Angeles, California 

INTRODUCTION 

The continually increasing power and economy of computers provides the 

structural engineering and mechanics community with an opportunity and challenge to make major advances in computer intensive areas of analysis, design 

and nondestructive evaluation of complex structural systems. Certainly the 

availability of modern computers is making it possible to consider increasingly larger and more complex structural analyses. State-of-the-art commercial grade software is generally available to use for analyzing a variety of 

linear and nonlinear problems on large mini or mainframe computers. At the 

same time, structural analysis programs are being "down sized" for use on personal computers. 

Yet, given all of these advances, some important and perhaps even critical problems are developing which must be resolved if the remarkable improvements in computer-based analysis and design over the past 10 years are to 

continue and the structural engineering community is to take full advantage of 

the new computing power. Several problems should be briefly mentioned. 

First, modern structural analysis software is generally a proprietary 

product of an active and very competitive commercial software industry. As 

such the software is beyond the control of the engineers who are almost completely dependent on it €or performing structural analysis and design. mus, 

they do not directly control the computer analysis and therefore are not able 

to fully understand the results they obtain. As a consequence engineers must 

rely on faith and earlier experience with given software to justify their 

analyses and subsequent designs. 

Second, the software packages available to the engineering community are 

enormously complex, so that even if source listing of the programs were available, few engineers would be able, much less willing, to learn how the program 

works. Thus, the shear size and complexity of the software encourages both 

the user and the software vendor to let well enough alone. Incremental 

changes in software are surprisingly difficult and most software requires 

almost continuous support by a technical staff. 

As a result little incentive exists for either engineers or software 

firms to push for major modifications in existing software or to develop fundamentally new and more powerful software. Unfortunately, in addition to 

being hampered by size and complexity, today's software is a product developed 

for yesterday's computers. For example, most programs are written using logic 

designed around one-dimensional arrays to store compacted stiffnesses and 

column solution techniques to solve equations in order to minimize storage 

requirements. However, modern computers have almost unlimited (virtual) 

PRECEDING PAGE BLANK NOT FlLMED 
memory and array processors which would be more effectively used if the software had a different program architecture. Thus, much of today's structural 

analysis software is unable to take Eull advantage of the most advanced computing machinery. A major revision of an existing program is a tremendous 

undertaking and, if developed, may have significant short-term costs to both 

the software developer and the user. 

Finally, it should be recognized that a number of "research" areas in 

structural mechanics have reached the development stage and are starting to be 

used by the structural engineering profession. Examples are structural life 

predictions using fracture mechanics, structural system identification, structural design optimization, semi-automated, adaptive computer modeling, and the 

analysis of infinite domain problems. The development of these areas has 

been and will continue to be slowed until the time when the necessary analysis 

techniques become generally available in commercial grade programs. 

If new generation software is to have maximum impact on the structural 

engineering profession it must be configured so that basic analysis can be 

used as a tool in a more general comprehensive engineering program. The software also should be as adaptable as possible to the evolution of computing 

machinery. 

It is the purpose of this presentation to provide some areas of structural engineering which are not well served by today's software and which 

should be given serious attention by developers of future structural analysis 

programs. In keeping with the theme of this workshop session, several aspects 

of glohal-local stress analysis will be discussed, with attention drawn to 

both the nature of the problem and the type of computational software which 

should be developed to investigate the problem. 

BASELINE MODELING CONCEPTS 

Perhaps the most difficult decision the engineer analyst must make is the 

choice of the proper mathematical model for use in investigating structural 

behavior (fig. 1). Generally the analyst inust choose first the dimensional 

level of the model, e.g., l-D (truss/beam) technical theory, 2-D technical 

theory, 2-D continuum modeling, and 3-D continuum modeling or a model composed 

of a mixture of some or all of the above. A major consideration in this 

choice is the available computer element library (which, for most programs is 

reasonably complete) and the performance of given finite elements. Also, 

elements and/or procedures for interfacing different types of element types 

are very important for problems where mixed element types are to be used. 

The analyst must also specify the level of physics to be treated by the 

computer program, especially the material constituency, the effect of initial 

stress, large motion (stability), dynamical response and nonlinear effects. 

The analyst should be free to investigate all these effects if necessary and 

not be constrained by limits on the features of a given software package 

(e.g., lack of a geometric stiffness matrix for a 3-D continuum finite element). As may be shown with a number of systems, structural behavior may 

initially be linear, elastic and even quasistatic in nature under given loads, 

but given "snall" changes in configuration, such as a hole or notch in a 

critical section, the problem may be fundamentally different involving 

90 
nonlinear elastoplastic response, large deformation and leading to large-scale 

failure. 

Thus to investigate a system thoroughly the analyst needs to have access 

to a very complete analysis package involving comprehensive physics and numerical modeling. 

The effective use of modern finite-element software depends on the basic 

skill of the analyst/engineer, who may be tempted to replace insight and basic 

knowledge of structural analysis with degrees of freedom (DOF'S). Such analyses first of all cost more than is necessary. To make matters worse large 

DOF models with excellent graphical characteristics may contain low resolution 

physics. (However, three dimensional models may be much easier to present to 

clients or company executives!) 

Consider the cantilever beam shown in figure 2. The simple 2 DOF model 

gives excellent beam deflections from which the trained engineer can obtain a 

wealth of accurate stress data. The planar 96 DOF model is in many respects 

less accurate. For example, the (exact) cubic lateral displacement is only 

approximately modeled by a series of parabolas. The stresses near the tip and 

near the left end are captured with little, if any, additional precision. The 

192 DOF planar model is, if anything, less precise than the 96 DOF model since 

lower precision 4 node elements are being used. Finally, the complex 3-D 

model with 768 DOF is still less accurate since the plane stress assumption is 

no longer built in to the model and now must be obtained (approximately) 

through the solution of a large system of equations. 

This example shows the efficiency that can be achieved using a wellthought-out finite-element model but it also shows some difficulties that can 

result. The deceptive ease with which software can be used invites abuse by 

unwary or poorly trained users. It is the author's opinion that some 

(optional) diagnostics should be available to warn or guide users in the 

generation of finite-element models. 

one of the most common uses of finite-element analysis is to investigate 

the behavior of structures in the vicinity of sudden variations in structural 

geometry or configuration in which sinqularities in the stress field or stress 

concentrations may occur. This area of analysis will be termed near-field 

modeling (fig. 3). Two different approaches for investigating the behavior of 

structures in the vicinity of stress concentrations reflect basic concepts 

global-local stress analysis. In the first approach, the finite-element model 

of the general structure away from the area of stress concentration (a hole in 

fig. 3) is coupled with an analytical solution [ll. The finite-element model 

may or may not extend into the region 

solution. For the problem shown the analytic stress field is in the form of 

Kirsh's solution [21 with an unknown stress factor u # uo. The interaction 

between the exterior finite-element model and the interior analytical model is 

used to find u as well as the nodal displacements and internal stresses in 

the exterior grid. 

R < RL influenced by the analytical 

- 

- 

The approach requires an analytical solution simple enough to be effectively used in a computer analysis, This approach appears to be limited to 

primarily isotropic homogeneous elastic structures with very simple stress 

concentrations, and for simple loadings. 

91 
Current software packages generally do not contain a library of analytical solutions and, if the numerical and analytical regions overlap, may not 

have the capability for generating and solving the necessary equations of 

equilibrium. 

A different approach to solving the same problem is to simply model the 

entire structure with a finite-element model using a mesh which is sufficiently finely zoned near the hole to capture the stress concentration 

(fig. 4). The generation of such a mesh requires a basic understanding of the 

physics of the stress concentration, especially the characteristic lengths of 

the decay of the stress concentration, and also a good understanding of the 

capabilities of the finite elements chosen to model the structure. 

An assessment of the quality of the finite-element model may be difficult 

if the analyst has a limited understanding of the finite elements being used 

to represent the physics, i.e., a problem if the analyst is using a program as 

a "black box. 'I 

In order to check for a valid solution the engineer often uses a finer 

mesh to re-analyze the problem especially in the vicinity of the apparent 

singularity indicated from the previous solution. This "field" application of 

the patch test is commonly used to check the convergence characteristics of a 

computer simulation. It would be helpful if it were necessary to generate 

only the data in the revised part of the structure and the data generated for 

the unchanged part could be reused. This is a simple task and yet one which 

is not commonly available in commercial computer programs. This feature may 

become quite important if the physics is more complicated than first believed, 

such as might be the case if a central stiffener were present, which would 

force the analyst to consider the stress concentration due to the hole and the 

nearby shear lag problem due to the interrupted stiffener. 

The problem of analyzing systems in which a part of the system involves 

an infinite or semi-infinite continuum is a perplexing and difficult problem, 

since an unlimited number of finite elements may be needed to model the complete system. The idea of using an analytical solution for far-field behavior 

together with a finite-element model of the near-field structure is a compelling one and has been used by a number of authors [3,41. 

denotes the outer radius of the finite-element model. 

In figure 5, RM 

As in the near-field problem, an analytical solution that can be effectively utilized in the context of a finite-element analysis is required. The 

need €or using an analytical solution has restricted the method to problems 

involving elastic isotropic homogeneous media and to a relatively small class 

of static or forced vibration problems. Recently efforts have been undertaken 

to generalize the method to problems where the medium was orthotropic or 

layered using finite-element solutions €or the far-field response of layered 

media in place oE analytical functions. 

In this approach, analytical solutions may have a number of terms, each 

with some characteristic factor Fi 

load. As in the near-field case, the region of the analytical solution may or 

may not extend into the region of the finite-element model. 

which must be related to the applied 

92 
The technique is not easily applied using conventional finite-element 

analysis packages since neither the definition of the far-field solution nor 

the techniques for matching the Einite-element grid with the far-f ield analytical solutions are contained in the programs. 

The far-field analysis problem is often investigated by using a so-called 

media island to treat structure media interaction (fig. 6). The finiteelement model typically extends as far as is economically practical from the 

site of interest. On the outer boundary, some special procedure is used to 

make the boundary a transmitting one, i.e., to permit outgoing waves to pass out 

of the computational grid and eliminate spurious reflections that might 

contaminate the solution. Typical boundary treatments are to use dampers, or 

special paraxial boundary elements [51, or recently, to use a so-called boundary zone superposition zone [6,71 to trap and cancel spurious waves. 

The latter method appears quite promising and requires only some basic 

knowledge of wave speeds in the boundary zone. The region in the interior may 

behave in a linear or nonlinear fashion. An important characteristic of the 

boundary zone superposition method is that it is very simple to program and 

may be used in principle with any finite-element software package. Unfortunately, in practice this is not the case since the analyst may be using 

"black box" software over which he has no control. 

The development of a practical tool €or near-field/€ar-field analysis has 

major implications €or such problems as, using ultrasonics for nondestructive 

testing, developing sensor/control systems on very large spacecraft, and 

studying impact of large bodies, such as spacecraft and Shuttle-type transport 

vehicles. 

The behavior of connections in structures is a persistent problem for 

structural engineers, especially on structures such as a space station which 

may have hundreds or even thousands of connections. Unlike terrestrial 

systems the connections on spacecraft may be very lightly loaded and therefore 

play a very important role in determining structural response of the overall 

system. The connection is a physical stress concentration (or substructure) in 

which structural characteristics may be quite complex (fig. 7). Yet for 

purposes of analyzing a very large system the highly resolved behavior of a 

connection must be consistently and appropriately reduced to a level usable 

€or the analysis of the large-scale system. 

The development of a simple substructure model which gives the essential 

behavior of the substructure in a global system is a non-trivial task. Certainly "downsizing" a highly resolved model of a connection to a much simpler 

model suitable for use in a global analysis should be based on a consistent 

formulation in which overall internal energy under specific deformation patterns is maintained, and the model should account for appropriate rigid-body 

behavior. 

The use of connection elements has been incorporated in a number of 

finite-element programs, especially €or use in piping analyses in the nuclear 

power industry. However, the software for an analysis such as described for 

the connection will have to have the capability of giving the user control 

over the definition of the element (as compared to having access to one of 

several basic connection elements in a general element library). 

93 
A different type of substructure problem which is encountered in globallocal finite-element analysis is the case where the important physics is in a 

small region of a large structure (fig. 8). This is the typical case in a 

problem involving the analysis of crack growth using fracture mechanics. In 

order to determine the rate at which the crack shown on the structure will 

grow given applied cyclic loadings, the stress intensity factor at the front 

of the crack must be determined for the crack as it grows during loading, or 

alternatively, the strain energy release rate. This is accomplished by 

analyzing the structure with a given length of crack, and then releasing the 

connection between elements at the tip of the crack (allowing it to advance 

for one element) and reanalyzing the structure. 

This procedure amounts to a model revision; thus, the entire structural 

analysis problem must be reestablished and resolved. This is only practical 

on a large system involving many thousands of DOF's if the surrounding structure is treated as a substructure and the crack growth region as the primary 

structure (which may be repeatedly modified to perform the strain energy 

release rate calculation). This type of analysis can be done using available 

commercial software, but only in a one-solution-at-a-time mode. It would be 

very helpful if the procedure could be carried forward in a semiautomatic 

manner that would require substantial software development. 

This analysis is very important in making safe-life predictions for 

critical components in aircraft and spacecraft. Of course, the problem is 

much more complicated if the direction of crack growth is unknown, since the 

finite-element models of the substructure and structure could not be determined prior to analysis. In short the finite-element model would have to be 

adaptive. 

Based on the comments in figure 8 it is evident that finite-element 

modeling must be adaptive in order to make safe-life predictions, a process 

which now involves the engineer analyst directly. In fact, considerable 

research has been done to develop semi-automatic, adaptive finite-element mesh 

generators [8-101. These procedures operate in basically one of two ways, 

refinement of the mesh itself, using similar finite elements the same order of 

approximation within each element (H-convergence), or leaving the grid fixed 

but refining the physics within each element (P-convergence). Different 

strategies are used to assess the quality of a solution for a given finiteelement grid. The same information is then used to revise the model and 

improve the solution. 

In response to a test problem proposed by NASA as a vehicle for discussion at the workshop (fig. 9) a simple highly idealized model of the structure 

was prepared by Dr. Paolo Roberti and analyzed using his algorithm [lo]. This 

algorithm uses triangle constant strain finite elements and H-convergence. 

The results are remarkable, giving almost a map of the stress concentration in 

the vicinity of the hole in the stiffened panel, figure 10a-f. Of course, 

this analysis was conducted only for a linear static solution. In reality, 

the presence of the hole in the panel may lead to instability or even failure. 

Nevertheless, the analysis is important in developing a finite-element model 

with a specified precision. 

Approaches such as this only hint at the tremendous problem solving power 

that can be brought to bear on structural engineering problems if the software 

94 
available can be designed to be flexible enough to adopt new and different 

concepts in analysis. 

CONCLUDING REMARKS 

The various types of local-global f inite-element problems point out the 

need to develop a new generation of software. First, this new software needs 

to have a complete analysis capability, encompassing linear and nonlinear 

analysis of 1-, 2-, and 3-dimensional finite-element models, as well as mixed 

dimensional models. The software must be capable of treating static and 

dynamic (vibration and transient response) problems, including the stability 

effects of initial stress, and the software should be able to treat both 

elastic and elasto-plastic materials. 

The software should carry a set of optional diagnostics to assist the 

program user during model generation in order to help avoid obvious structural 

modeling errors. In addition, the program software should be well documented 

so the user has a complete technical reference for each type of element contained in the program library, including information on such topics as the 

type of numerical integration, use of underintegration, and inclusion of 

incompatible modes, etc. Some packaged information should also be available 

to assist the user in building mixed-dimensional models. 

An important advancement in finite-element software should be in the 

development of program modularity, so that the user can select from a menu 

various basic operations in matrix structural analysis, including matrix formulation and storage, assembly (by row or column), solution (by row, column or 

wave front), and method of time integration. Most important, the software 

should permit the user/analyst to link to the computer program his own specialized software. User programs might include formulation of (substructure) 

stiffness matrices, specialized solution packages (matrix inversion, partial 

inversion), time integration and, for nonlinear problems, input of different 

types of materials. 

The next generation of finite-element software also should be developed 

with the idea of analysis serving as a basic tool in design, system identification and optimization. 

The implementation of adaptive finite-element modeling techniques in commercial grade software will have a major impact on the structural engineering 

community which now invests a significant effort on basic analysis, especially 

in the modeling, solution, and remodeling cycle. A number of problems in nonlinear structural analysis will also benefit from adaptive computer modeling, 

such as making safe-life predictions €or structures using fracture mechanics 

concepts. 

Hopefully, a new generation of software can be developed with many, if 

not all, of the features described. If it is possible to do so, then structural analysis software will become a much more complete, versatile and 

reliable tool for the structural engineer. 

95 
REFERENCES 

1. Bradford, L. G., Dong, S. B., Nicol, D. A. C., and Westmann, R. A., 

"Application of Global-Local Finite Element Method to Fracture Mechanics," 

Electric Power Research Institute Report #EPRI NP-239 (Research Project 

299-1 1 Technical Report 1 , September 1976. 

2. Timoshenko, S. e., and Goodier, J. N., TheoryofElasticity, Third Edition, McGraw Hill, New York, 1970. 

3. Muki, R., and Dong, S. B., "Some Remarks on the Use of Asymptotic Solutions in Global Local Finite Element Analysis €or an Elastic Half Space," 

Recent Research on Mechanical Behavior of Solids, University of Tokyo 

Press, 1979, pp. 55-78. 

- - 

4. Medina, F., and Taylor, R. L., "Finite Element Techniques for Problems of 

Unbounded Domains," International Journal for Numerical Methods in Engineering, Vol. 19, No. 8, August 1983, pp. 1209-1226. -~ 

5. Cohen, M., and Jennings, P. C., "Silent Boundary Methods €or Analysis," 

Computational Methods for Infinite Domain Media-Structure Interaction, 

ASME, 1982, pp. 183-204. 

6. Kumar, R. R., and Marti, J., "A Non-Reflecting Boundary for Explicit 

Calculations," Computational Methods for Infinite Domain Media - Structure 

Interaction, ASME, 1981 , pp. 183-204. 

7. Muki, Y., "A Radiating Boundary Method for Linear and Nonlinear Transient 

Finite Element Analysis," M.S. Thesis, UCLA School of Engineering and 

Applied Science, 1985. 

8. de S. R. Gago, Kelly, D. W., Zienkiewicz, 0. C., and Babuska, I., "A 

Posteriori Error Analysis and Adaptive Processes in the Finite Element 

Method: Part I - Error Analysis," International Journal for Numerical 

Methods - in Engineering, Vol. 19, No. 8, August 1983, pp. 1593-1620. 

9. de S. R. Gago, Kelly, D. W., Zienkiewicz, 0. C., and Babuska, I., "A 

Posteriori Error Analysis and Adaptive Processes in the Finite Element 

Method: Part I1 - Adaptive Mesh Refinement," International Journal for 

Numerical --- Methods in Engineering, Vol. 19, No. 8, August 1983, 

pp. 1621-1656. 

10. Roberti, P. , "Self Adaptive Mesh in Stress Analysis," Ph.D. Dissertation, 

IJCLA School of Engineering and Applied Science, 1984. 

96 
BASELINE MODELING CONCEPTS 

I 

Dimensional Level of Model 

- Technical Theory 

1-'D Truss/Beam 

21- D Panel/PIate/Shell 

- Continuum Representation 

2- D Plane Stress, Plane Strain 

nth order Symmetry 

31- D 

Physical Requirements of Model 

- Material (Elastic, Plastic, Anisotropic) 

- Initial Stress (Stability) 

- Dynamics (Vibration, Transient Analysis) 

- Nonlinear (Large Deformation, Separation) 

FINITE-ELEMENT MODELING 

- Which Model is Better? 

dp BEAM PLANAR 2-D 1, * 2DOF 

rITIiIiIIIiIT-11 96D0F 

4P 

PLANAR 2-D 

P 

P 

192 DOF 

768 DOF 

Figure 1 Figure 2 

97 
NEAR-FIELD MODELING 

Singularities/Stress Cmcentrations 

- Finite-Element Model with Near-Field 

Analytical Solution 

- Analytical Solution(s1 Required 

Capable of Being Evaluated and Utilized 

- Restricts Approach to 

Isotropic, Homogeneous Elastic Systems, 

Very Simple Regions, Simple Loadings, 

Statical Problems, or Forced Vibrations 

- Finite-Element Modeling with Mesh 

Refinement 

Understanding of Characteristic Lengths of 

Physical Processes Required 

Evaluation of Model, Solution Difficult 

Model Refinement Capability Important 

Figure 3 Figure 4 

98 
FAR-FIELD MODELING 

Finite Structure Imbedded in Infinite Continuum 

- Finite Element Model with Far Field 

Analytical Solution 

Analytical Solution(s) Required 

Capable of Being Evaluated and Utilized 

Restricted to Isotropic Homogeneous Elastic 

Far Field, Static or Forced Vibration 

Figure 5 

-Finite-Element Model In Continuum Island 

\ 

J 

RADIATION 

B. C. 

Formulation Requires Special Boundary 

Conditions to Insure Radiation 

Dampers 

Reflecting Zones 

Knowledge of Wave Transmitting 

Characteristics of Medium Required 

Figure 6 

99 
SUBSTRUCTURE CONCEPTS 

Use of Highly Resolved Models 

- Imbedded Substructures 

I '* 

- Surrounding Substructure 

CRACKGROWTH 

REGION 

Figure 7 Figure 8 

100 
ADAPTIVE FINITE ELEMENT MODELING 

BLADE-STIFFENED PANEL WITH DISCONTINUOUS STIFFENER 

IP UNIFORM END SHORTENING 

\ 

RESULTS OF ROBERTI'S ALGORITHM FOR 

SUCCESSIVELY REFINED MESHES TO 

CAPTURE STRESS CONCENTRATIONS 

IN NASA TEST PROBLEM 

b 

Figure 9 

C 

d 

e 

f 

Figure 10 

101 
N89- 24644 

SOME COMMENTS ON GLOBAL-LOCAL ANALYSES 

Satya N. Atluri 

Georgia Institute of Technology 

Atlanta, Georgia 

ABSTRACT 

The maln theme of this paper concerns methods that may be classified as global 

(approximate) and local (exact). Some specific applications of these methods are 

found in: 

(1) Fracture and fatigue analysis of structures with 3-D surface flaws 

(2) Large-deformation, post-buckling analysis of large space trusses and space 

frames, and their control 

(3) Stresses around holes in composite laminates 

PRECEDING PAGE UNK NOT FKMED 103 
A typical engineering problem is illustrated in figure 1, which shows a corner 

flaw at the intersection of a nozzle and a pressure vessel. The shape of the 

surface flaw may often be approximated mathematically as quarter-elliptical or 

quarter-circular. For the problem shown in figure 1, wherein the crack is located 

in the longitudinal plane of symmetry of the structure, only the so-called Mode I 

conditions prevail. In figure 1, the presence of a traction-free crack, in an 

otherwise unflawed solid, alters the stress-state only locally. From a viewpoint of 

fracture mechanics, however, the main quantities of interest are only the stressintensity factors (strengths of asymptotic stress singularities) near the crack 

front. For analyzing fatigue crack growth and crack instability under thermal shock 

various flaw sizes and shapes need to be considered. The primary objective of 

analysis is to determine the variation of the Mode I stress-intensity factor along 

the border on the surface flaw. 

Figure 1. Corner surface-flaw at the pressure-vessel-nozzle intersection. 

104 
I Figure 2 shows the schematic of a 12-bay space frame. The equations of dynamic 

motion of the frame, assuming large deformations and plasticity, may be written as: 

t where M, is the mass matrix, 

ness matrix (which includes the effect of large defgrmation and plasticity), f4 the 

control-actuator force, gE the external load, (N+l)X_ the acceleration vector at 

time tN+1, (N+l)? N the velocity vector at tN+1, the incremental displacement between tN and tN+1, and “1% the internal-force vector at tN. In order to implement 

the control algorithms in an efficient manner, the order of the above system of 

equations must be as small as possible (i.e., each frame member must be modeled by 

no more than one finite element). Further, the control must be implemented for 

pulse-type loading of high intensity, such that the above system of equations must 

be integrated directly rather than using a modal-decomposition. Also requirements 

of on-line control may necessitate that tK_, C,, and M, be known explicia (in closed 

form) for arbitrary values of deformation, without the need for introducing approximate shape functions for deformation of each element and without the need €or any 

numerical integrations over each element. In figure 2, the object of inquiry is 

what effect does local (member) instability have on global (system) stability? How 

can we control the dynamic deformations locally to improve global behavior? Each 

member may be treated as a truss member, or a 3-D beam-type member, depending on 

joint design. How can local effects be accounted for simply and efficiently, so 

that algorithms for control of dynamic motion may be implemented, on line, using 

on-board computers in a large space structure? 

the matrix of passive damping, K, the tangent stiff4 4 

72 8 3 

Figure 2. Schematic of a 12-bay space frame. 

105 
, An appraisal of the computational mechanics methods is given in figure 3. 

These methods include the finite-element, boundary-element, and edge function 

methods (fig. 3). 

F: I N I TE ELEMENTS : 

TRIAL AND TEST 

TRIAL AND TEST 

APPROACH 

FUNCTIONS ARE BOTH APPROXIMATE 

FUNCTIONS ARE, IN GENERAL, ALIKE - GALERKIN 

IN SOME INSTANCES IT IS BEST TO HAVE TEST FUNCTIONS DIFFERENT 

FROM TRIAL FUNCTIONS - PETROV-GALERKIN APPROACH 

THE SOLUTION IS BOTH -GLOBALLY AND LOCALLY APPROXIMATE 

VERSATILE OR ARBITRARY GEOMETFIY, BOUNDARY CONDITIONS, SUITED 

FOR GLOBALLY APPROXIMATE NCINLINEAR SOLUTIONS 

BOUNDARY ELEMENTS :

TEST FUNCTIONS ARE GLOBALLY EXACT FOR THE GIVEN LINEAR PROBLEM, 

OR AT LEAST FOR THE HIGHEST-ORDER DIFFERENTIAL OPERATOR OF 

THE PROBLEM 

TRIAL FUNCTIONS ARE APPROXIMATE (AT BOUNDARY ONLY FOR LINEAR 

THE SOLUTION IS BOTH LOCALLY AND GLOBALLY APPROXIMATE 

NOT AS VERSATILE AS THE FINITE-ELEMENT METHOD, BUT EXCELLENT 

PROBLEMS, AND IN INTERIOR ALSO FOR NONLINEAR PROBLEMS) 

FOR SOME SPECIFIC PROBLEMS 

EDGE FUNCTION METHOD: 

TRIAL FUNCTIQNS ARE GLOBALLY EXACT 

TEST FUNCTIONS ASSUMED ONLY AT BOUNDARY 

LIMITED TO LINEAR PROBLEMS POSED BY CLASSICAL DIFFERENTIAL 

EQUATIONS 

Figure 3. Appraisal of computational mechanics methods. 

106 
In the most commonly used Galerkin finite-element approach in computational 

solid mechanics, the trial and test function spaces are identical and consist of 

simple piecewise continuous algebraic polynomials over each finite element, such 

that these functions and their appropriate-order derivatives (as dictated by the 

problem on hand) are continuous at the interelement boundaries. For problems of 

fourth or higher order, such as those of plates and shells, the development of 

1 finite elements has long been, and continues to be, plagued by the need for C 

(or higher order) continuity at the interelement boundaries. However, the success 

of the finite-element method in structural mechanics is unparalleled and is mainly 

due to the intuitive and 'geometric' interpretation of the method. The method is 

versatile in its ability to deal with complicated structural assemblies, such as of 

beams, plates, and shells, of the type used in aerospace applications. The solutions obtained through the finite-element method may be classified, in general, as 

being both globally as well as locally approximate. 

On the other hand, in linear and nonlinear solid mechanics, it is often possible to derive certain integral representations for displacements. A key ingredient which makes such derivations possible is the singular solution, in an infinite 

space, of the corresponding differential equation (in certain linear problems) or of 

the highest-order differential operator (in the nonlinear case, or even in the 

linear case when the full linear equation cannot be conveniently solved), for a 

'unit' load applied at a generic point in the infinite space. When the problem is 

linear and the singular solution can be established for the complete linear differential equation of the problem, the aforementioned integral representations for 

displacements involve only boundary integrals of unknown trial functions and their 

appropriate derivatives. Such an integral representation, when discretized, leads 

to the so-called boundary-element method. Such pure boundary-element methods are 

possible in linear, isotropic, elastostatics, and in problems of static bending of 

linear elastic isotropic plates. On the other hand, as in such cases as (i) linear 

problems wherein the singular solutions cannot be established for the entire differential equations, (ii) anisotropic materials, and (iii) problems of large 

deformation and material inelasticity, the integral representations (if any) for 

displacements would involve not only boundary integrals but also interior-domain 

integrals of the trial functions and/or their derivatives. A discretization of such 

integral equations would lead not only to a simple boundary-element method but also 

to a sort of hybrid boundary/interior element method. 

When asymptotic solutions to the governing differential equations of the problem are used as assumed trial functions, the interior residual error is zero, and 

only the boundary conditions need to be satisfied in a weighted residual method. 

Such an approach is called the edge-function method, but is limited mostly to linear 

problems. For further details, see references 1 through 4. 

107 
The surface-flow problems for current methods are noted in figure 4. Problems 

for the proposed method are also shown. 

A1 1 Present Methods: Singular stress-state near the flaw 

border is model ed by 1 oca1 ly approximate methods 

(I) Finite-Element Methods (singular elements) 

Atluri & Kathiresan (refs. 5-10) 

(Hybrid crack elements) 3-6,000 d.0.f. 

Tracy, Barsoum, Newman & Raju (refs. 11-13) 

(Distorted i soparametric elements and 

singular shape fn.) 5-10,000 d.0.f. 

These are very expensive, but accommodate 

arbitrary geometries of structure and flaw. 

( I I - Boundary-El ement Methods (for local ly approximate stress analysis and K-estimation from 

stress extrapol ition) 

S ti 11 very expensive. 

Cruse (ref. 141, Heliot et al. (ref. 15). 

Not suitable for 'thin' shells with flaws. 

( I I I 1 Li ne-Spri ng Method 

Emited to simple geometries -of structure and 

flaw. 

Proposed Method: --------- 

It is a GLOBALLY APPROXIMATE, but -- LOCALLY EXACT 

Singular stress-state near the flaw is NOT MODELED 

It is about 30 times cheaper than the singular 

' Details (Atluri & Nishioka (refs. 16-21) - several 

METHOD 

NUMERICALLY 

finite-element method 

papers with varied examples) 

Figure 4. Surface-flaw problems. 

108 
solution for 

uncracked body 

w 

Locally exact solution 

for pressurized crack 

Remarks: 1. Solution D: A rather complicated analytical 

solution (Atluri & Vi jayakumar, Journal of 

Appl ied Mechanics, 1981 1 (ref. 22) 

is (i.e., the Solution C) the source of singularity. The stresses due to this local solution 

2. Local solution due to crack-face traction alone 

decay- very rapidly. 

are sufficient to obtain K-solutions with 1% 

Only one or two 1 terations -

accuracy. 

About 30 times cheaper than the usual finiteelement method for a typical problem such as 

flawed BWR nozzle. 

Figure 5. Global (approximate) and local (exact) analyses of embedded flaws. 

109 
Some comments concerning the solution of surface flaws in finite bodies using 

the present procedure are in order (fig. 4). 

elliptical flaw embedded in an infinite solid is used as solution D, it is necessary 

to define the residual stresses over the entire crack-plane including the fictitious 

portion of the crack which lies outside of the finite body containing only a surface 

flaw (fig. 5). 

type funct€on-interpolation inside the interpolated region can be increased with the 

number of polynomial terms; however, the interpolating curve may change drastically 

outside of the region of interpolation. The optimum variation of pressure on the 

crack surface extended into the fictitious region should be as shown in figure 6. 

For in-depth discussions of a variety of surface problems and their solutions, see 

refs. 19-21, 23-28. 

Since the analytical solution OF an 

Moreover it is well known that the accuracy of the ‘least-squares’ 

/ ACTUAL \ ASSUMED 

ICAL 

H 

Figure 6. Postulated residual stress distributions on fictitious portions of an 

elliptical flaw, in the case of semi-elliptical or quarter-elliptical 

surf ace flaws. 

110 
For both 3-dimensional truss and frame members, explicit (locally exact) 

tangent stiffness matrices have been derived (fig. 7). Some effects of local 

(member) buckling on global (structural) behavior are illustrated in figures 8 

and 9. 

Truss Member: 

Member material is nonlinear 

Each member may buckle and become curved (what 

tach member undergoes large displacement and large 

rigid rotation 

effect does it have on global stability?) 

Frame Member: 

'-Concepts of 3-dimensional semi-tangential rotations 

Each member undergoes arbitrarily large rigid rotaBending-stretching coupling incorporated in each 

Plastic-hinge method used to account for plasticity 

Member forces: axi a1 , shear, and bending- twi s ti ng 

employed 

ti ons and ri gi d di spl acements 

member 

in each member 

moments 

Kondoh and Atluri (refs. 29-30) and Kondoh et a1 . (refs. 31-32) 

Figure 7. Space trusses and space frames. 

111 
L = 66.04(cm) 

(a) Thompson's Strut 

system imperfection considered 

no system imperfection 

50 

Both member 

buckling and 

considered (two different 

30:: ; / magnitudes of imperfections ) 

~ oj$ 

1, 

00 1 

Dlsplocement of Node No 10 In the X Direction 

03 20 $0 60 60 '00 '20 '40 I60 

(CU) 

(b) Effect of local (member) buckling on global (structural 1 behavior. 

Tangent stiffness of each member is exact in both the pre-buckled 

and post-buckled states of member (ref. 29) 

Figure 8. Thompson's strut and effect of local buckling on global behavior. 

112 
10-5 

I t x3 i 

I I 

(a) Load System: (i) PI: Vertical 

point loads as all nodes; 

(ii) Pz: Verticrpoint loads at 

nodes in quadrants XI, x2 > 0 

9 -9, \ 

N 

n 

I1 

-e 

n 

-7 

-6 

-5 

-4 

-7 

Member EA = 4.59Kg; Buckling E1 = 250 Kgm2 

EA = 4.59Kq; EI= 123 

Kgm2 

I 

-5 - 10 -I5 - 20 -25 I 104 

pI - PI /EA 

(b) Stabi 1 i ty boundary under 1 oads 

PI and P2 

Figure 9. Study of the effect of member buckling on global (system) stability. 

113 
Examples of the efficiency of the global/local approach in analyzing frames are 

illustrated in figures 10 and 11. In figure 10, the classical problem of a two-bar 

frame is schematically illustrated. In the present approach, the tangent stiffness 

matrix of each member (respresented by a single finite element) is derived from 

exact solutions of governing differential equations which account for the bendingstretching coupling. Thus, no "shape functions" are assumed in each element, and no 

numerical integrations are performed in forming the tangent stiffness matrix. The 

present numerical integrations are performed in forming the tangent stiffness 

matrix. The present numerical results are shown to agree excellently with those of 

Wood and Zienkiewicz (ref. 34), as well as the experimental results of Williams 

(ref. 35). However, Wood and Zienkiewicz use five finite elements to model each 

member of the frame. 

In figure 11, the problem of plastic collapse of a frame is €llustrated. Here 

again, the tangent stiffness matrix of each member (represented by a single finite 

element) is derived in closed form, accounting for large deformations, large rotations, and plasticity. A plastic-hinge method is used, and the progressive development of plastic hinges, at various load levels, Is indicated in figure 11. 

E = I 03r107~1b/~n~ 

y 1000 2000 2000 4000 5COO 6000 1000 9"h) 

00 01 02 03 04 05 06 07 Slnl 

00 - 

Figure 10. Variation of load-point displacement and support reaction with applied 

load in a two-bar frame. Tangent stiffness of each 3-D beam member 

undergoing large deformation, large rotation, and plasticity is exact. 

Locally exact solution (ref. 30). 

114 
PL ~b 

3~1 1 2.667 (Limit Analysis, U. B 1 

2.51 ~$f--~-d= 2.500 (Llmlt Analyslsi L. 8.1 

I.Oi i I I Iii / 

1 

M; 0 Fully Plostic Bendin Moment 

of Sectlon 6 

0.5ll/ / 1'. Moment of lnertla of Sectlon @ 

05 10 15 2.0 

2.3962 

2.5454 

12 

2.2198 &I4 

2 6186 

R 0 PIos11c Hinge 

' TANGENT STIFFNESS OF EACH MEMBER UNDERGOING LARGE DEFORMATION, LARGE ROTATION AND 

PLASTICITY IS EXPLICIT AND EXACT, PLASTIC-HINGE METHOD USED, 

EXTENSION TO CRASH ANALYSIS OF FRAMES BEING STUDIED, 

Figure 11. Plastic collapse of frame. 

115 
Figure 12 shows a problem of current interest in the analysis of stiffened 

composite plates. Issues involve the following: (1) stress concentrations near the 

hole in a composite laminate, (2) local buckling of stiffeners, (3) effect of geometric imperfections, (4) effect of discontinuities, and (5) three-dimensional 

effects and delaminations near the hole. An efficient globally approximate and 

locally exact approach could possibly include: (1) use of locally exact, laminated 

hole elements with embedded three-dimensional stress state (refs. 36 and 371, 

(2) use of locally exact stiffener elements as described earlier (ref. 32), 

(3) techniques for proper interacting of various elements, and (4) hole elements 

that can be improved by incorporating possible free-edge singularities in u 3i' 

Focus PROBLEM 

(GLOBALLY 8 LOCALLY APPROXIMATE) FINITE ELEMENTS] 

Figure 12. A stiffened laminated-composite panel with a hole. 

116 
Another example of the advantages of using a global/local approach is illustrated here in the problem of analysis of stresses near a hole in a laminated 

composite [two cases of (-45/+45), and (90/0), laminates are discussed]. 

shows a typical finite-element model with "special-hole elements" in which a 3-D 

asymptotic hole solution is embedded. Figures 13b and 13c illustrate the excellent 

accuracy obtained from the present approach, in comparison with a fully 3-D finiteelement solution of Rybicki and Hopper (ref. 38). 

an order of magnitude less expensive. 

and 37. 

I 

I Figure 13a 

The present solution is, however, 

Further details are given in references 36 

TYPICAL FUI MODEL OF A LAMINATE 

WITH HOLE, 3-0 ASYMPTOTIC "HOLE- IN (-45/+45), LAMINATE IN (90/0), LAMINATE 

SOLUTIONS" EMBEDDED IN ELEMENTS 

NEAR THE HOLD(Ref. 36) 

STRESS AROUND A HOLE STRESS AROUND A HOLE 

Figure 13. Analysis of stress state near a hole in laminated composites. 

117 
The following conclusions and recommendations are given. 

. 

. Hybrid analytical/numerical methodologies should be explored 

Simplified analysis procedures for elasto-plastic should be considered 

(Dynamic response calculations should be studied (some benchmark problems 

essential)) 

Constitutive models badly need improvement 

Methods of coupling of problem-specific methodologies for use in general 

purpose programs should be explored 

Trends to treat structural mechanics problems as continuum mechanics 

problems should be critically reviewed; the knowledge base in structural 

mechanics should be fruitfully utilized 

Attempts to bridge the gap between micromechanics and macromechanics of 

heterogeneous (composite) media through computational mechanics should be 

explored 

Computational stochastic structural analysis methods should be developed 

Algorithms for new computing systems (MIMD) should be explored 

Expert systems, . . . . (?) 

NASA’s role should be to provide: 

Increased research support to universities 

Predoctoral NASA fellowships (up to 20K per year, tax-free) that could be 

awarded to attract the best students 

Long-range funding to properly plan and sustain high-quality research 

efforts 

Increased access to supercomputers 

Frequent visits to NASAfacilitiesby graduate students to participate in 

laboratory testing. University facilities in this area are scarce; 

students in computational mechanics should get some first-hand experience 

in experimental mechanics 

118 
REFERENCES 

1. Atluri, S.N. and Kathiresan, K., "Influence of Flaw Shapes on Stress Intensity 

Factors for Pressure Vessel Surface Flaws and Nozzle Corner Cracks", Journal of 

Pressure Vessel Technology, Vol. 102, August 1980, pp. 78-286. 

2. Atluri, S.N. (Editor and Contributor to Part 11, Finite Element Methods), Finite 

Element Handbook, McGraw-Hill Book Co., 1200 pp. (In Press). 

3. Atluri, S.N., "Computational Solid Mechanics (Finite Elements and Boundary 

Elements): 

International Conference on Applied Numerical Modeling (Eds: H.M. Hsia, et al.) , pp. 19-37, 1984- 

Present Status and Future Directions", (Keynote Paper) in Zoc. 4th 

-- 

4. O'Donoghue, P.E. and Atluri, S.N., "Control of Dynamic Response of a Continuum 

Model of a Large Space Structure", AIAA 26th Structures, Structural Dynamics & 

Materials Conference, Orlando, Florida, AGi1 15-17, 1985. 

5. Atluri, S.N., Kathiresan, K., and Kobayashi, A.S., "Three-Dimensional Linear 

Fracture Mechanics Analysis by a Displacement-Hybrid Finite-Element Model", 

Trans. of the 3rd International Conference on Structural Mechanics in Reactor 

- Technology, London, Vol. 5, Part L, Paper No. L-7/3, September 1975, 13 pp. I_ -- 

6. Atluri, S.N. and Kathiresan, K., "On a 3-D 'Singularity Element' for Computation 

of Combined Mode Stress Intensities", Advances in Engineering Science, Vol. 1, 

November 1976, pp. 267-274. 

7. Atluri, S.N., Kathiresan, K., Kobayashi, A.S., and Nakagaki, M., "Inner 

Surface Cracks in an Internally Pressurized Cylinder Analyzed by a ThreeDimens i onal Di s p 1 acemen t -Hy b r i d F i ni t e E 1 ement Me t hod", Pres sur e Ves s e 1 

Technology, Part I1 - Materials and Fabrication, ASME, 1977, pp. 527-533. 

8. Atluri, S.N. and Kathiresan, K., "Outer and Inner Surface Flaws in Thick-Walled 

Pressure Vessels", Trans. of 4th International Conference on Structural 

Mechanics in Reactor Technology, Paper G 5/4, 1977, 13 pp. 

9. Atluri, S.N. and Kathiresan, K., "Stress Analysis of Typical Flaws in Aerospace 

Structural Components Using 3-D Hybrid Displacement Finite Element Method", 

Proceedings of 19th AIAA/ASME/SAE Structures, Structural Dynamics and Materials 

Conference, Bethesda, Maryland, 1978, pp. 340-351. 

10. Atluri, S.N. and Kathiresan, K., "3-D Analysis of Surface Flaws in Thick-Walled 

Reactor Pressure Vessels Using Displacement-Hybrid Finite Element Method", 

Nuclear Engineering & Design,"Vol. 51, No. 2, January 1979, pp. 163-176. 

11. Tracy, D.M., "Finite Elements for Three-Dimensional Elastic Crack Analysis", 

- Nuclear Engineering 6 Design, Vol. 16, 1977, pp. 282-290. 

12. Barsoum, R.S., "On the Use of Isoparametric Finite Elements in Linear Fracture 

Mechanics", International Journal- for Numerical Methods - in Engineering, Vol. 10, 

NO. 1, 1976, pp. 25-37. 

119 
13. Newman, J.C., Jr. and Raju, I.S., "Stress-Intensity Factor Equations for Cracks 

in Three-Dimensional Finite Bodies Subjected to Tension and Bending Loads'' 9 

(S.N. Atluri, Ed.) Computational Method in the Mechanics of Fracture, NorthHolland Publishing Co. (In Press). 

14. Cruse, T.A., "Boundary-Integral Equation Method for Three-Dimensional Elastic 

Fracture Mechanics", AFOSR-TR-75-0813, U.S. Air Force, 1975, 52 pp. 

15. Heliot, J., Labbens, R.C., and Pellissier-Tannon, A., "Semi-Elliptical Cracks in 

a Cylinder Subjected to Stress Gradients", in Fracture Mechanics, ASTM STP 677, 

1979, pp. 341-364. 

16. Atluri, S.N. and Nishioka, T., "Analytical Solution for Embedded Elliptical 

Cracks, and Finite Alternating Method for Elliptical Surface Cracks, Subjected 

to Arbitrary Loadings", Engineering Fracture Mechanics, -- Vol. 17, No. 3, 1982, 

pp. 247-268. 

17. Atluri, S.N. and Nishioka, T., "Integrity Analyses of Surface-Flawed Aircraft 

Attachment Lugs: A New, Inexpensive, 3-D Alternating Method", Proceedings of 

22nd AIAA/ASME/ASCE/AHS Structures and Material Conference, New Orleans, 

Louisiana, 1982, pp. 287-301. 

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a New 3-Dimensional Alternating Method", Aspects of Fracture Mechanics in 

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Also Journal of Pressure Vessel Technology, ASME Val. 104, 1982, pp. 299-307. 

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and Crack Depth for Long Flaws of Semi-Elliptical Shape", Advances in Aerospace 

Structures, AD Vol. 2, ASME, 1982, pp. 47-50. 

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Method €or the Analysis of Surface Flawed Aircraft Structural Components", AIkA 

-- Journal, Vol. 21, No. 5, 1983, pp. 749-758. 

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Pressure Vessels", Journal of PressureVessel Technology, Trans. ASME, Vol. 54, 

1982, pp. 612-619. 

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Solid, . Subject to A;bi trary Crack-Face Tractions", Journal of Applied Mechanics, 

Vol. 103, No. 1, 1981, pp. 88-96. 

23. O'Donoghue, P.E., Nishioka, T., and Atluri, S.N., '$Multiple, Coplanar, Embedded 

Elliptical Cracks, in an Infinite Solid Subject to Arbitrary Crack Face 

Tractions", International Journal of Numerical Methods in Engineering, Vol. 21, 

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Pressure Vessels", Engineering Fracture Mechanics, Vol. 20, No. 3, 1984, 

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120 
I 

26. Atluri, S.N., "Computational and Theoretical Studies on Dynamic Fracture Mechanics and Three-Dimensional Crack Problems", Keynote Lecture, Proceedings of ICF 

International Symposium on Fracture Mechanics, Science Press, Beijing, 1983, 

pp. 1-17. 

I27. O'Donoghue, P.E., Nishioka, T., and Atluri, S.N., "Analysis of Interaction 

Behavior of Surface Flaws in Pressure Vessels", Journal of Pressure Vessels 

Technology, Trans. ASME, 1984. 

28. Punch, E.F. and Atluri, S.N., Development and Testing of Stable, Invariant, 

Isoparametric Curvilinear 2- and 3-D Hybrid Stress Elements", Computer Methods 

in Applied Mechanics and Engineering, Vol. 47, 1984, pp. 331-356. 

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Deformation, Post-Buckling Analyses of Large Frame Structures, Using Explicitly 

Derived Tangent StifEness Matrices", International Journal of Numerical Methods 

- in Engineering, Vol. 23, No. 1, 1986, pp. 69-90. 

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Nonlinear Analysis of Large Space-Trusses and Frames, Using Explicity Derived 

Tangent Stiffnesses, and Accounting for Local Buckling", Report to Air Force 

Wright Aeronautical Laborato.ries, 1985, 171 pp, USAF, WPAFB. 

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Using Exact Tangent-Stiffness Matrices" in Finite Element Analysis & Design, 

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-- 

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Mechanics, Vol. 16, No. 4, 1982, pp. 573-583. 

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121 
ON COMPUTATIONAL SCHEMES FOR GLOBAL-LOCAL 

STRESS ANALYSIS 

J. N. Reddy 

Virginia Polytechnic Institute and State University B1 acksburg, Vi rgi ni a 

1. INTRODUCTION 

This paper primarily deals with an overview of global-local stress analysis 

methods and associated difficulties and recommendations for future research. The 

phrase global-local analysis is understood to be an analysis in which some parts of 

the domain or structure are identified, for reasons of accurate determination of 

stresses and displacements or for more refined analysis than in the remaining parts. 

The parts of refined analysis are termed local and the remaining parts are called 

global. Typically local regions are smallsize compared to global regions, while 

themputational effort can be larger in local regions than in global regions. 

2. CONTENTS 

This paper is divided into the following parts: 

Mo ti va ti on-P rob1 ems ( p rob 

Common Features 

Focus Problem 

Analysis Methods 

G1 obal -Local Approaches 

Example Problem 

ems that motivated global -1 ocal ana 

Conclusions and Recommendations 

3. MOTIVATION PROBLEMS 

ysis) 

The following stress analysis problems, among many others, motivate us to use 

gl obal-1 ocal aproaches: 

Free-Edge Stress Concentration in Laminates 

Contact Stress Problems 

Impact 

Fracture Mechanics 

Unbounded-Domai n Problems 

123 

PRECEDING PAGE BUNK NOT FILMED 
Almost all laminated composite structural elements have free edges at the 

boundaries (including holes) of the elements. It is well known that the transverse 

normal and shear stresses are very large at the edges (more precisely, the stresses 

are large within a distance of the order of thickness of the laminate from the free 

edge). While the classical laminate theory is adequate to describe the behavior of 

the laminate everywhere except in the "boundary layer" in which the transverse normal 

and shear stresses are large, a refined theory, often the 3-D elasticity theory, is 

needed to describe the state of stress near the edges. 

Contact stress problems (for example, bolted and bonded joints, tire contact, 

and metal-forming problems) require the use of a special theory that accounts for 

appropriate constitutive laws and friction and allows for slip, slide, and separation 

of the mating parts in the contact regions. Elsewhere, appropriate elasticity theory 

can be used. 

Impact of two solid bodies can be modelled by the use of one theory in the 

vicinity of contact and by another theory elsewhere. 

larger in the contact region than elsewhere, more refined theory and analysis are 

required in contact regions. Of course, the theory and analysis used depend on the 

type of structure , loadi ng and deformation. 

Since the stresses are much 

Structures containing cracks, whether formed during manufacturing or service, 

require special treatment of stress fields around cracks, often using 3-D stress 

analyses and/or nonlinear fracture mechanics theories, while the linear elastic 

fracture mechanics theory is adequate away from the cracks. 

Problems involving unbounded regions (for example, soil mechanics and earthquake 

engineering) are by their nature divided into local and global regions. Global 

regions, in theory, can be infinite but in practice they are finitely large, and less 

refined theory and/or analysis is used to determine the stress field and other 

pertinent information. 

4. COMMON FEATURES 

The motivating problems 1 isted previously share certain common physical features 

that are significant from the modelling and analysis points of view. 

list provides some of these features: 

The following 

Stress Concentration ( 1 arge 1 oca1 gradients) 

Three-Dimensional State of Stress 

Large Rotations/Strains 

. Local Di sconti nui ties (holes , di sconti nuous fibers , etc. 

. Material Nonlinearities (nonlinear elasticity, plasticity, etc.) 

A global-local stress analysis should account for all features that are present 

in the problem. Of course, some of those features are not to be included in the 

global model. 

124 
5. FOCUS PROBLEM 

The focus problem identified by NASA Langley Research Center is a bladestiffened panel with a discontinuous stiffener. 

p hy s i ca 1 f ea tu res : 

The problem has the following 

Geometric Discontinuities 

Local Stress Gradients 

Eccentric Loading 

Large Di spl acements 

Free Edges 

We shall return to this problem later to discuss the global-local analysis approach. 

6. ANALYSIS METHODS 

6.1 COMMON APPROACHES 

The commonly used analysis methods for structural problems are 

Classical (or Analytical) Methods 

C1 assical Variational Methods 

Fi ni te-Di fference Method 

Fi ni te-El ement Method 

Boundary Element Method 

Some noted advantages and disadvantages of these methods are outlined next. 

6.2 CLASSICAL AND VARIATIONAL METHODS 

The classical method of solving problems exactly is the best there is. However, 

most practical problems (which have irregular geometries, anisotropic materials, discontinuities, geometric and/or material nonlinearities, etc.) do not admit exact 

solutions by the classical approach. 

The classical variational (e.g., Ritz, Galerkin and weighted-residual ) methods 

yield continuous solutions throughout the domain, giving high resolution of displacements and stresses. They are computationally efficient for a given problem. For a 

given order of approximation, previously computed (for lower order approximation) 

matrix coefficients can be used. These methods, however, have two major shortcomings: (i) the approximation functions are not easy and are often impossible to 

125 
construct for most practical structural problems; and (ii) the variational methods 

cannot be implemented on a computer for the analysis of a class of problems because 

the resulting a1 gebraic equations depend on the approximat5%-?h3ionsy which in 

turn depend on a specific problem. 

6.3 FINITE-DIFFERENCE METHOD (FDM) 

The finite difference method is simple in formulation (based on the representa- 

. tion of derivatives of a function in terms of a finite Taylor's series expansion) and 

easy to implement on the computer. 

methods until the sixties, when the finite-element method gained popularity, especially in solid and structural mechanics. The disadvantages of the finite-difference 

method include: difficulty in representing complex geometries, inexact representation 

of boundary conditions on non-straight boundaries, and difficulty in developing 

higher-order approximations. 

itself for general-purpose code development. 

approximations in structural mechanics problems. 

The method dominated the field of numerical 

Because of these difficulties the method does not lend 

The method is seldom used for spatial 

6.4 FINITE-ELEMENT METHOD 

The f i ni te-el ement method overcomes the shortcomings of the cl assical vari ational methods. This approach is systematic (modular) and natural and allows an 

accurate representation of complex geometries. 

to use without changing the modular structure of the approach. 

suited for general purpose and computer program development. The disadvantages, 

compared to other computing methods, are the large formulative and computational 

efforts. The finite-element method is the most frequently used numerical method in 

structural analysis. It is now a key component of any mechanical CAD/CAM system. 

Higher-order approximations are easy 

The method is ideally 

6.5 BOUNDARY ELEMENT METHOD (BEM) 

The finite-difference and finite-element methods can be classified as domain 

methods because they involve approximations of the entire domain. 

element method, also known as the boundary integral method (BIE), seeks approximations only on the boundary of the domain by converting the governing differential 

equations to integrals over the boundary of the domain. The dimensionality of the 

problem is thereby effectively reduced by one. 

1 not approximated, the computational time involved is less (BEM/FEM PI - where nxn is n' 

the finite-element mesh). The BEM offers continuous interior modelling within the 

sol uti on domain, gi vi ng high resol ution of di spl acements and stresses. The method is 

unsui tab1 e for probl ems requi ri ng information at a 1 arge number of internal points. 

Application of BEM to nonlinear problems and problems with discontinuities is not 

fully established. 

The boundary 

Because the interior of the domain is 

126 
6.6 CONCLUSIONS 

In conclusion, the general application of the finite-e ,?merit metl,oc to structural problems is unmatched to date. Structural problems, because of the modular 

nature of FEM, are better modelled by FEM. 

be advantageous in some problems (e.g. , unbounded-domain problems). 

A suitable combination of FEM and REM can 

7. GLOBAL-LOCAL APPROACHES 

7.1 MODELS AND METHODS 

In formulating a given problem, either the same theory is used throughout or a 

refined theory is used locally and a less refined theory globally. 

The global-local analysis methods can be FEM throughout, FEM and classical 

solution, or BEM and FEM. When the same theory and FEM are used throughout, it is 

understood that locally special elements are used (e.g., friction element, interface 

element with sliding, or elements which allow opening and closing along element 

interfaces). Classical solutions are available, for example, for infinite plates 

with holes. The solution is not valid far away from the hole if the plate is long 

but not infinite. In such cases, the FEM can be used globally and the classical 

solution can be used locally. For soil mechanics and earthquake engineering problems, a combination of BEM and FEM proves computationally efficient. 

tions experimental methods globally and computational methods local ly are 

recommended. 

In some situa7.2 SOME EXAMPLES 

Some exampl e probl ems that requi re g1 obal-1 oca1 analysis are 1 i sted here. 

Free-edge stress analysis of laminates 

Contact stress problems 

Stress analysis of structures with discontinuities 

A blade-stiffened panel with a discontinuous stiffener - the focus 

p rob1 em 

As mentioned earlier, free-edge stress problem requires a refined theory near 

free edges. For example, the classical laminate theory globally and .either quasi-30 

or full 3-0 theory locally (depending on the lamination scheme, geometry and loading) 

can be used to analyze the problem. The problem will be discussed in more detail 

1 ater . 

i nterferometry can be used to determi ne the surface displacements (and hence strains 

and stresses) and the finite-element method can be used to determine the interior 

di spl acement and stress fi el ds. 

In bolted joint problems, an experimental technique such as the Moir6 

127 
In the stress analysis of plates with holes, one can use a laminate theory away 

from holes and 3-D elasticity theory around the hole, and use the finite-element 

method to model the entire problem. 

In the case of the focus problem, which has all features that are present in the 

examples discussed above, the global theory should be a shear deformation plate theory (2D) with the Von Karmen geometric non-linearity and the local theory can be 

the fully 3D laminate theory. The finite-element method should be used throughout. 

When FEM is used locally, the fully 3D elements or 3D degenerate elements can be 

used. 

7.3 DIFFICULTIES 

In using global-local approaches, we face some difficulties. Some of these are 

Interfacing between regions 

Interfacing between methods 

Selection of regions 

Changing regions and interfaces 

When the finite-element method is used, the elements used globally and locally 

can be different. Then it is important to have compatibility of the nodal degrees of 

freedom at the interface of the elements. 

needed in some situations. When two different methods are used, the unknowns in the 

two methods should be the same. Selection of the local and global regions depends on 

the physical features and accuracy desired. In some cases, the regions might have to 

be determined only after a preliminary analysis. The global and local regions can 

change during the history of deformation/ loading. analysis, the plastic zones are unknown a priori and they change with loading. 

A special interface element might be 

For example, in elastic-plastic 

8. EXAMPLE PROBLEM 

Here we briefly discuss the free-edge stress problem in symmetric laminates. 

Figure 1 shows the laminate geometry, loading, and the domain modelled. Because of 

the assumed symmetry of the lamination about the midplane and the constant straining 

along the x-axis, the displacement field can be approximated (ref. 1) as 

I u = uox + u(y,z) 

v = V(y,z) 

w = W(y,z) 

where Uo is a constant, K. 

128 
The displacement field is three dimensional but it leads, when substituted into 

the Navier equations of equilibrium, to three partial differential equations in two 

independent variables, y and z. 

It is well known that the transverse normal stress, uZ, is very large (unbounded) near the free edge. To reduce its magnitude a cap is used on the free 

edge. The effect of the cap on the stress distribution uZ is investigated. The 

f i ni te-el ement method with the four-node bi 1 i near rectangul ar el ement i s used to 

model the computational domain. A refined mesh is used near the free edge and in the 

cap. 

Figures 2 and 3 show the distribution of the transverse normal stress uZ along 

the width of the laminate for [Oo/9OoIs and [45"/-45"Is laminates, respectively, 

(E1 = 137.89 GPa, E2 = E3 = 14.48 GPa, 612 = 613 = 623 = 5.86 GPa, 

~12 = vi3 = 93 = 0.21). Results for both capped and uncapped laminates are 

presented (for k = 0.001, b = 25.4 cm, h = 2.54 cm and thickness of the cap, 

t = 0.08 cm). 

has quite a large magnitude within a distance of y/b = 0.1 (one-tenth of the width) 

from the free edge. Hence a laminate theory is sufficient to model the interior, 

while the quasi-3D can be used to model the free-edge stress field. The effect of 

the free-edge reinforcement (i .e. , cap) on the stress magnitude is significant; the 

magnitude is reduced to less than one-third of that without cap. 

We observe that the stress is essentially zero inside the laminate but 

For a more detailed and complete stress distribution near the free edge of a 

more general laminate (e.g., without symmetry about the midplane), a threedimensional model is needed. 

9. CONCLUSIONS AND RECOMMENDATIONS 

9.1 AREAS NEEDING SUPPORT 

A review of the literature shows there are very few cases of global-local 

analyses of structural problems involving the "physical features'' discussed earl ier. 

It is recommended that the following areas of global-local approaches be 

investigated: 

G1 obal-1 oca1 analysis of problems with "common features" out1 i ned 

earl i er 

Investigation and development of i nterface elements 

Feasibility of BEM as a computational tool for nonlinear problems and 

its interface with FEM 

Development of adaptive mesh refinements and time-stepping algorithms 

Exploitation of the vector and parallel processor computers for 

efficient structural analysis 

129 
0 Finite-element calculations 

Solution of equations 

Ei genval ue computations 

The use of parallel processors can dictate the solution procedures, for 

example, iterative methods over one-step methods. 

9.2 NASA's INVOLVEMENT 

NASA (CSM) shoul d be i nvol ved in the gl obal-1 ocal analysis devel opment because 

of the tremendous impact this field has on computational mechanics applied to space 

structures. 

1 ocal analysis area: 

In particul ar, NASA shoul d undertake the foll owi ng tasks in the global - 

Support individual grants (as opposed to large group grants) 

Collaborate with university faculty and graduate students by identifying 

specific problem areas and providing computational time and scientific 

advice 

Give graduate student residentships, during which students spend a few 

weeks (perhaps the summers) at NASA 

Conduct workshops (say, once in two years) to bring the latest 

developments for critical evaluation and to set future directions. 

ACKNOWLEDGMENTS 

This paper was prepared via the support of the Structural Mechanics Branch of 

NASA Langley Research Center (NAG-1-459 1. 

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Symmetric Laminates,"Virginia Polytechnic Institute and State University, 

Blacksburg, VA, CCMS Report 85-01, January 1985. 

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130 
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Processor System," IEEE Trans. Computers, Vol. 26, pp. 838-47, 1977. 

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pp. 618-23, 1976. 

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pp. 637-647, 1984. 

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Engineering Stress Analysis," J. Strain Analysis, Vol. 18, No. 4, 1983. 

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Structures, Vol. 19, No. 5/6, pp. 865-877, 1984. 

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Vol. 20, pp. 740-77, 1978. 

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Stiffened Graphite-Epoxy Panels Loaded in Axial Compression," 26th 

AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materiamonference, AIAA 

Paper No. 85-0168 - CP , Orlando, Fl., 1985. 

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131 
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I Zienkiewicz, 0. C., Gago, J. P., De, S. R., and Kelly, D. W., "The Hierarchical 

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132 
t Domain modelled 

b" 

v) 8 1.0- 

E 

c L 

v) 

- 

(d 

0 

Q, 

Q, 

> 

v) 

c 

!- 

2 0.0 

2 

2h Y 

? 2b - -4 

Y -bcapped 

- 

uncapped 

Figure 1. Laminate geometry and loading (cap is not shown). 

\" I""""' 

0,8 0.85 0.9 0,s 1 

YIb 

Figure 2. Distribution of the transverse normal 

stress along the width of a cross-ply laminate 

co 0/90° 1 s. 

133 
1.04 

I'"'I'' I'Z" 

0,8 0.85 r3,9 0,95 1,o 

Y/b 

Figure 3. Distribution of the transverse normal stress (uz) 

along the width (y/b) of the laminate C45O/-45"Is. 

134 
GLOBAL FUNCTIONS IN GLOBAL-LOCAL FINITE-ELEMENT ANALYSIS 

OF 

LOCALIZED STRESSES IN PRISMATIC STRUCTURES 

Stanley B. Dong 

University of California 

Los Angeles, California 

Abstract 

An important consideration in the global local fini te-el ement method 

(GLFEM) is the availability of global functions for the given problem. 

The rol e and mathematical requi rements of these gl obal functions in a GLFEM 

analysi s of 1 oca1 ized stress states in pri snatic structures are discussed. 

A method is described for determining these global functions. Underlying 

this method are theorems due to Toupin and Knowles on strain energy decay 

rates, which are re1 ated to a quantitative expression of Sai nt-Venantf s 

prlnciple. It is mentioned that a mathematically complete set of global 

functions can be generated, so that any arbitrary interface condition 

between the finite element and global subregions can be represented. 

Convergence to the true behavior can be achieved with increasing global 

functions and f ini te-el ement degrees of freedom. Speci f ic attenti on is 

devoted to mathematically two-dimensional and three-dimensional prismatic 

structures. Comments are offered on the GLFEM analysis of NASA flat panel 

with a discontinuous stiffener. Methods for determining global functions 

for other effects are also indicated, such as steady-state dynamics and 

bodies under initial stress. 

135 
Introduction 

136 

The f ini te-el ement method (FEM) has revol uti onized structural and 

stress analyses in the last quarter of this century. Its impact has been 

widespread, even extending beyond the preserve of structural engineers to 

other fields. 

model ing technique, the analysis community with its collective experience 

will admit that it is not the quintessential technique. There are problems 

not well suited to FEM that result in clunsy, ineffective and costly 

mathematical models. Examples can be cited fran problems involving stress 

si ngul ari ti es and i nf ini te domai ns. 

modifications to FEM have been explored. One alternate approach which 

bodes considerable pranise is the so-called Global-Local Finite-Element 

Method (GLFEM). 

Although EM is acknowledged as an extremely powerful 

To obvi ate the di f f icul ti es, 

GLFEM util izes both conventi onal finite el ements and cl assi cal Ritz 

functions in the modeling process. Their respective roles are readily 

apparent; finite elements work well in regions where complicated geanetry 

and inhomogeneous material characterizations prevail, and Ritz functi ons, 

hereinafter referred to as global functions in GLFEM, enable the behavior 

other regions to be represented accurately and efficiently. At this stage 

of development, GLFEM can be assessed to be in its maturing phase. It is 

of good lineage, has already exhibited an enhanced capability above EM in 

certain problems, and pranises effectiveness in other classes of problems 

upon its full development. 

Herein, GLFEM as appl ied to the analysis of local ized stresses in 

prismatic structures is discussed. 

various GLFEM modeling layouts are sunmarized. A brief review of some 

problems that have been successfully analyzed by GLFEM is given. Then, 

the main theme re1 ating to GLFEM analysis of local ized stress states is 

addressed. Prismatic structures that can be described mathematically 

by two spatial variables are discussed first. Attention is devoted to 

the g1 obal functionsr thei r devel opment and the1 r roles in the present 

setting. Then, three-dimensional structures are considered, wfth 

reference to the NASA example problem, where an outline of a method of 

attack is given. Last, comments on the analysis of localized stresses 

involving steady-state dynamic effects as well as other conditions are 

given. 

Fi rst, the essence of GLFEM and 

Basi c Concepts of GLFEM and the Various Mesh Conf igurati ons 

Hamil ton's pri nci ple, or a1 ternatively the theorem of minimun 

potential energy when no inertial effects are present, may be considered 

as the basis for generating GLFEM equations. The theory and variational 

derivation of these equations may be found in Ref. Cl,pp.451-4741. Also 

included therein is a survey of GLFEM contributions to the literature up 

to 1982. 

As noted earlier, the technique util izes finite-element model ing with 

classical Ritz approximations simultaneously. It enjoys the advantages of 
more versatile model Ing capabil Ities with substantially fewer degrees of 

freedom. Vari ous gl obal /local model ing conf igurati ons are 11 1 ustrated 1 n 

Fig. 1. Figs. la and If represent, respectively, the classical Rltz and 

f in1 te-el ement conf Igurati ons. The others are possi bl e GLFEM mesh 1 ayouts. 

In a given problem, the model ing may take the form of any one of these configurations or a combination of two or more of them for various subregions. 

An important key is the enforcement of ki nemati c 1 nter-reg1 onal conti nul ty 

between various global and 1 oca1 subregions by means of constrai nt equations. In problems on local ized stress states, only the Fig. IC conf iguration will be usedr where finite elements exclusively are used in one subregion and global functions in the other. MOrewer, the global subregion 

may be infinite in extent. 

The governing matrix equations in a GLFEM analysis have the form: 

where 161 denotes the finite-element degrees of freedom and {SI contains 

the array of general ized coordi nates associ ated with the g1 obal functi ons. 

In Eq. (11, CKgg3 CMg 18 CK,I and CMgalrefer to the global and loc 1 

= CMgg IT represent gl obal -local coup1 ing f ran imposi ng ki nemati c 

Detail s on the formati on 

stiffness and mass matr 9 ces of the system. The matrices CKagl = CKggl ? 

at 1 nterf ace (s 1 between subregi on(s 1. 

of these matrices may be found in Ref. C11. 

It is mentioned that GLFEM variants are possible, which do not lead 

to the same set of governing equations as Eq. (1). These variants contain 

the spi rit of GLFEM and employ the model ing conf igurati ons shown in Fi g. 1; 

however, the method of enforcing inter-regional continuity may differ. An 

appl icati on concerned with el astic wave scatter1 ng will I1 1 ustrate one such 

vari ant. 

Another key point In GLFEM is the availability of an appropriate set 

of global functions for a given problem or a class of problems. 

accuracy and effectiveness of the method are dependent upon the quality 

of the global functions. The cfioice of these global functions for the 

analysis of lccalized stresses in primatic structures and their method 

of derivation will be discussed in what follows. It will become.apparent 

why these global functions, together witti the finlte-elenent model of 

the subregion that contains the localized stresses, will lead to a 

super1 or model . 

The 

Sane Examples of Global Functions for GLFEM 

Two areas ideally suited to GLFEM are fracture mechanics and Infinite 

137 
and/or semi-infinite domain problems. Much has been published on various 

aspects of fracture mechanics problems. Many nunerical methods have been 

used, many falling within a GLEM classification or its variants. The 

global subregion model usually takes the form of special crack tip 

elements, where the singular stress field is incorporated into the 

stiffness matrix. These elments are well known. No further elaboration 

on this subject will be given here. 

problems, wherein the far field behavior is represented by global functions 

(for example, Boussinesq or Cerruti solutions), have also met with 

considerable success. A nunber of references on both of these subjects 

may be found in Ref. C11. 

Elastostatic analyses of half-space 

Herei n, two recent GLFEM appl ications are mentioned to emphasize the 

roles of the global functions and their mathmatical suitabil ity. They are 

concerned with (1) steady-state el astic wave scattering by axisymmetric 

objects embedded in an infinite isotropic medium and (2) steady-state 

soil-structure interaction involving an axisymmetric structure occupying 

some locale in a semi-infinite mediun. The feature of note is that these 

global functions constitute a complete set of eigenfunctions and have the 

capabil ity of mathematically representi ng an arbitrary scattered f iel d to 

any given accuracy. Hence, the true behavior in the far field can be 

ach i wed. 

In Fig. 2 is shown an el astic, axisymmetric incl uslon embedded in an 

elastic, isotropic mediun. 

it may have i nhanogeneous, orthotropi c properties. The f ini te-el ement 

subregion includes this object and a portion of the surrounding mediun. 

For convenience in the analysis, the interface is taken to be spherical. 

Outside of the finite-element subregion is the outer field, where a 

complete set of outgoing spherfcal harmonics is used to model the scattered 

fie1 d. Each component satisfies the equations of motion and the Sanmerfel d 

radi ati on conditions. 

di spl acement di stri buti ons at the interface, and thei r undetermined 

strengths are the global function coefficients or the terms in {SI. A 

given incident wave illuninates this object. 

determined by solving the finite-element equations and requiring that the 

sun of incident and scattered wave fields based on the global functions 

have both traction and di spl acement conti nul ty with the fini te-el ement 

data at the interface. Details of this analysis may be found in Ref. C21. 

Here, attenti on is called to the mathematical flexi bil ity of the global 

functions for accommodating interface continuity to any precision with a 

sufficient nunber of terms. 

Because finite elements are used for the object 

The gl obal functions have speci f ic stress and 

The scattered field is 

The dynamic soil-structure i nterati on problem under steady-state 

conditions is shown in Fig. 3. The approach used here is similar to that 

for elastic wave scattering by an object embedded in the entire space. 

In fact, the same set of spherical harmonics for the entire space may be 

applied to this half-space problem. However, traction-free surface 

conditions are not satisfied by the spherical harmonics. Thus, in 

addition to traction and di spl acement conti nui ty at the hemispherical 

interface, it is necessary to enforce the traction-free surface in the 

gl obal subregion. In Refs. C3,41, detail s concerni ng an integral 
constraint condition to meet this traction-free surface condition are 

given. 

functions is used (that is, a set capable of modeling any arbitrary traction 

and di spl acement condi ti ons between various subregi ons i n a GLFEM 1 ayout ), 

the analysis procedure enjoys the opportunity of converging onto the 

true behavior with increasing FEM and global degrees of freedom. 

Again, it is noted that because a complete set of global 

Mathematical ly Two-Dimensi onal Structures 

The choice of gl obal functi ons for mathematical ly two-dimensional 

structures will now be discussed. As illustrations of this class of 

problems and their GLFEM layouts, refer to Fig. 4, where examples of a 

laminated composite plate and cylinders are given. 

may be considered as a plane strain problem herein. The scarf joint 

joining two cylinders may be taken as an axisymmetric structure under 

axisymmetric or asymmetric loads. 

in these joi nts. 

The double lap joint 

The purpose i s t o study the stresses 

Uniform stress states exist at points well away fran these local ized 

stress regions. 

woul d result. In GLFEM, two-dimensional fi ni te el ements (pl anar or 

axi symnetri c toroi dal el ements 1 are used for the subregi on contai ni ng the 

1 ocal ized stresses. If the 1 ocal ized stress state contai ns a si ngul arity, 

a global subregion within the finite-element subregion may be added. The 

i nterf ace 1 ocati on is dependent on the gl obal functi ons' mathemati cal 

capabil ity for capturing the transitional stress and di spl acement f le1 ds 

accurately. For gl obal functions capable of representi ng the true 

behavior, the finite-element subregion can be quite small with the 

interf acek 1 near to the local ized stress area. An independent set of 

global functions must be adopted at each interface. For the lap joint 

in Fig. 4, two or three distinct systems of global functions may be 

needed depending on the thickness and material properties of the plate 

components. Each set of global functions is associated with its own 

set of general ized coordinates or global coefficients. For the 

cylindrical scarf joint, two independent sets are needed. 

a quantitative expression of St. Venant's principle. Toupin C51 and 

Knowles C61 presented upper bound estimates of strain energy decay rates 

in terms of distance f ran a sel f-qui1 ibrated stress state. Thei r 

results can be stated in the form of a strain energy inequality: 

If FEM were used, it is obvious that an awkward model 

The global functions in these casesare based on theorems relating to 

d '2YX V(0) e 

where Y is the inverse of the characteristic decay length, V(0) is the 

total strain energy and V(x) is that portion of V(0) in the body beyond 

x. Since the strain energy is quadratic, the mechanical variables such 

as stress, strain and displacement are of the forms: 

139 
where Ki (i=lr2,3) are constants. 

Based on these theorems, a boundary-Val ue problem can be formulated 

for a prismatic structure. Using the solution form in the prismatic 

direction as e-Yx, the analysis leads to an efgenvalue problem. 

eigenvalues Y'S are the characteristic inverse decay lengths and the 

ei genf uncti ons are di stri buti ons of sel f-qui 1 i brated stress states. 

These eigendata comprise a complete set fran which any arbitrary sel fequilibrated stress state may be represented. These eigendata may be 

used as global functions for describtng the far-field behavior in a 

prismatic structure. Horgan and his colleagues have solved a nunber 

of problems on hanogeneous and sandwich plates under plane strain using 

the Airy stress function as the primary dependent variable (see, for 

exampl es, Refs. C7,81). 

The 

For a laninated composite structure, it is more convenient to 

determine the eigendata nunerically. Dong and Goetschel C91 developed 

a one-dimenslonal finite-element analysis for extracting eigendata for 

a laminated composite plate with an arbitrary nunber of bonded, elastic 

1 aminates. Fi nite-el ement discretization occurs in the thickness 

direction, see Fig. 5. Applying the theorem of minimun potential energy, 

a system of second-order ordinar differential equations is obtained. 

By invoki ng exponenti a1 decay e-;', the foll owing second-order a1 gebrai c 

ei genval ue probl em results: 

where (0) is an ordered set of the plate's nodal displacements. This 

equation is reducible to first order with a non-symmetric matrix. If a 

1 arge nunber of degrees of freedom are involved, a Block-Stodol a iterati on 

technique E101 can be used to extract the eigendata efficiently. The 

sol ution consists of a complete set of eigenval ues and corresponding 

eigenvectors, which are the sel f-qui1 ibrated displacement states for the 

given composite plate. Stresses can be computed fran these displacements. 

Laminated cylinders may also be solved using the same finite-element 

scheme, see Ref. C113. The mechanical variables have ci rcunferenti a1 

dependence, which may be expressed analytically by Fourier series. As 

a circmferential mode nunber m occurs in this case, the counterpart to 

Eq. (4) for each circunferential mode has the form: 

The solution to Eq. (4) or (5) prwides the global function data 

base for the nunerical evaluation of the global stiffness matrix and the 

140 
gl obal-local coup1 ing as a prel ude to a mathematically two-dimensional 

GLFEM analysis of the local ized stress zones. 

of this type have been obtained, which are contained in Refs. C12,133. 

These 1 imited scope studies indicate an overall feasi bil ity for this 

approach. 

Sae prel iminary results 

The accuracy of the global functions depends on the fineness of the 

one-dimensi onal finite-el ement model adopted for the ei genproblem. Since 

only one-dimensional finite elements are used, a large model does not incur 

an inordinate computational effort because of a very small bandwidth. 

The nunber of global functions required in a GLFEM analysis depends 

on both the nature of the localized stress and the location of the 

interface. Having an interface near the localized stress zone will 

require a larger nunber of global functions, but with a decrease in the 

f i ni te-el ement coordi nates. 

the local ized zone needs fewer global functionst but is counteracted by 

a greater nunber of finite-element degrees of freedom. 

Converse1 y, an i nterf ace far removed f ran 

Three-Dimensi onal Structures and the NASA Probl em 

A schematic of a three-dimensional prismatic structure and the NASA 

problem of a flat stiffened composite panel with a discontinuous stiffener 

are shown in Fig. 6. In this class of problems, three-dimensional finite 

elements must be employed in the localized stress region. Global functions 

must be used at the interface. They can be obtained fran a two-dimensional 

finite-element analysis of the inverse characteristic decay lengths. 

The analysis to determine the global functions follows the same 

methodology as that for mathematical ly two-dimensi onal structures. The 

pri smati c cross-secti on i s model ed by two-dimensi onal finite el ements. 

With the dependence in the prismatic di rection taken as e-yx, an 

eigenproblem emerges for the extraction of eigendata that form the 

global function data base for the given cross section. 

aspects are the same as that described in the previous section. 

obvious thatsin this case, the computational effort is greater. 

The other 

It is 

Sane comments can be given on a GLFEM analysis of the NASA flat panel. 

The set of two-dimensional global functions constitutes a complete system 

of el genf uncti ons, with the non-zero ei genval ues associ ated with inverse 

characteristic decay lengths of sel f-qui1 ibrated stress states. There 

are two zero eigenvalues for two stress distributions exhibiting no decay. 

They are the uniform axial deformation and pure bending states. These two 

global functions are needed in a GLFEM analysis of the NASA flat panel# 

since the discontinuous stiffener may produce bending in addition to its 

uniform end shortening. A set of global functions with all of these 

members present should permit a three-dimensional finite-element model 

to be concentrated on the details of the discontinuous stiffener region. 

Parametric studies wherein the hole and the gap length in the 

disconti nuous stiffener are varied may be conducted. Each configuration 

will require a change of the three-dimensional finite-element mesh, but 

the same set of global functions may be used in all cases. 

141 
Appl icati ons to Steady-State Probl ems 

The di scussion of global functions in the two earl ier sections 

pertained to elastostatic analysis of the localized stress zones. 

some ranarks on steady-state dynamic effects are made. The one-dimensional 

finite-element method for generating global function data bases can be 

modified for steady-state inertial effects by including kinetic energy in 

the problem formulation. Instead of Eqs. (4) and (51, those equations 

become, respectively: 

Here 

where w is the steady-state forcing frequency. 

equations are given in Refs. C11,143. 

The derivations of these 

With these global functions, it is possible to study elastic wave 

scatter1 ng 1 n pri mati c structures by di sconti nut tles durl ng vi brati on 

or by some other steady-state dynamic input. GLEM analysis of this 

type of prismatic structures will be similar to problems of elastic wave 

scattering by an object embedded in an inflnlte mediun or soil-structure 

1 nteracti on. 

Effects of Initial Stress 

Us1 ng the same methodol ogy, pri smati c structures under i ni ti a1 

stress may also be analyzed. In this the global functions must 

1 ncl ude the prestress1 ng effect. One-dimensi onal finite-element 

analysis of wave propagation in laninated composite plates and cylinders 

under initial stress have been explored, see Refs. C15,161. It is a 

straightforward task to adapt these formulations to generate an 

eigenproblan for the global functions for a prismatic structure under 

initial stress. Also, no conceptual difficulties are seen in an 

extensi on to three-dimensi onal prismatic structures under prestress. 

Concl udi ng Remarks 

Considerable discussion has been devoted to the strategies of GLFEM 

analyses of prismatic structures with localized stress regions and other 

discontinuities. The role of the global functions has been clearly 

out1 ined and the1 r mathematical requi rments Indicated. The method 

for derlving these global functions for prismatic structures, whose 

cross-sect1 onal geunetries are compl icated by laninated construct1 on, 

has been discussed. Frun the discussion of GLFEM analysis strategy, 

it should be clear that GLEM is feasible and effective. Considerable 

economy of computational efforts over a strictly FEM approach should 

be real ized. 

142 
References 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10 0 

11. 

12. 

Dong, S. B. , "G1 obal -Local Fi ni te El ement Methods, State-of-the-Art 

Surveys on Finite Element Techno1 ogy, edited by A.K. Noor and W.D. 

Pi1 key, Chap. 14, pp.451-474, herican Society of Mechanical 

Engi neers, 1983. 

Goetschel, D.B.8 Dong, S.B. and Muki, R.8 "A Global-Local Finite 

Element Analysis of Axisymmetric Scattering of Elastic Waves," 

Journal of Appl ied Mechani cs, heri can Society of Mechanical 

Engineers, Vol . 49, pp. 816-820, 1982. 

Avanessian, V., Muki, R. and Dong, S.B.8 *'Forced Oscillation of an 

Axisymmetric Structure in Contact with an Elastic Half-space by a 

Version of Global Local Finite Element,11 Journal of Sound and 

Vibration, Vol . 104, pp. 449-463, 1986. 

Avanessi an, V. , Muki , R. and Dong, S. 8. , "Axi symmetric Sol 1 -Structure 

Interaction by G1 obal -Local Fi ni te Elements, Internati onal Journal 

of Earthquake Engineering and Structural Dynamics, in press. 

Toupi n, R. A. "Sai nt-Venant' s Pri nci pl e, Archive for Rati onal 

Mechanics and Analysis, Vol. 18, pp. 83-96, 1965. 

Knowles, J.K., "On Saint-Venant's Principle in Two-Dimensional Linear 

Theory of Elasticityrtl Archive for Rational Mechanics and Analysis, 

Vol. 21, pp. 1-22, 1966. 

Choir I. and Horgan, C.O., "Saint-Venant's Principle and End Effects 

in Anisotropic Elasticity," Journal of Appl led Mechanics, Anerican 

Soci ety of Mechanical Engi neem Vol . 44, pp. 424-430, 1977. 

Horgan, C.O., "Sane Remarks on Saint-Venant's Principle for 

Transversely Isotropic Composites, 1) Journal of El astici ty, 

Vole 2, pp. 335-339, 1972. 

Dong, S.B. and Goetschel, D.B., !'Edge Effects in Laminated Composite 

Journal of Appl led Mechanics, Anerican Society of Mechanical 

Engineersr Vol. 49, pp. 129-135, 1982. 

Dong, S.B.8 "A Block-Stodol a Eigensol &ion Technique for Large 

A1 gebraic Systems with Non-Symmetrical Matricesrtl International 

Journal for Nunerical Methods in Engineering, Vol. 11, pp. 247-267, 

1977. 

Huang, K.H. and Dong, S.B., "Propagating Waves and Standing Vibrations 

in a Composite Journal of Sound and Vibration, Vol. 96, 

pp. 363-379, 1984. 

Chen, W.T. , 9tudi es Re1 ati ng to Appl icati on of the Global-Local Fi ni te 

Element Method to Elastostatic Analysis of End Effects in Cylindersrff 

M.S. Research Project Report, Dept. of Civil Engineering, University 

of California, Los Angeles, March 1985. 

143 
13. Nystran, D.M., TJunerical Evaluation of Global Stiffness Matrix for 

Analysis of Sel f-Equil i brated Stress States, M. S. Thesi s, Graduate 

Divisi on, University of Cal iforni a, Los Angel es, June 1985. 

14. Dong, S.B. and Huang, K.H., '!Edge Vibrations in Laminated Composite 

P1 ates,", Journal of Appl ied Mechanics, herican Society of 

Mechanical Engineers, Vol . 52, pp. 433-438, 1985. 

15. Bradford, L.G. and Dong, S.B., Vatural Vibrations of Orthotropic 

P1 ates under Ini ti a1 Stress, International Journal of Sol ids and 

Structures, Vol. 11, pp. 213-230, 1975. 

16. Bradford, L.G. and Dong, S.B., "Natural Vibrations of Orthotropic 

Cy1 inders under Ini ti a1 Stress," Journal of Sound and Vi brati on, 

Vol. 60, pp. 157-175, 1978. 

144 
(a) GLOBAL FUNCTIONS 

ONLY OVER ENTIRE 

DOMAIN 

(b) GLOBAL FUNCTIONS 

OVER (VOLjg 

GLOBAL & 

LOCAL FUNCTIONS 

OVER (VOL)eg 

(c) GLOBAL FUNCTIONS (d) GLOBAL & LOCAL 

OVER (VOL)g FUNCTIONS OVER 

LOCAL FUNCTIONS 

OVER (VOL)i 

ENTIRE DOMAIN 

(e) GLOBAL & LOCAL 

FUNCTIONS OVER 

LOCAL FUNCTIONS 

(VOL) e, 

OVER (V0L)i 

(f) LOCAL FUNCTIONS 

OVER ENTIRE DOMAIN 

1- GLOBAL REGION {g}= [Bg]{S} 

FffR LOCAL (FINITE ELEMENT) REGION 

{E}= [Nil{&} 

(.} = [$, I {4} + [Ngl{ s} 

COMBINED GLOBAL-LOCAL REGION 

Figure 1. Basic global-local mesh configurations. 

145 
S C ATT E RE D 

finite elements 

MESH BOUNDARY 

INCIDENT WAVES 

global functions 

SCATTERED WAVE FIELD - SPHERICAL HARMONICS 

J n=o 

Figure 2. Elastic wave scattering by axisymmetric inclusion embedded in 

an infinite isotropic medium. 

146 
TRRCTION-FREE 

Figure 3. Soil-structure interaction problem. 

FINITE ELEMENT SUBREGION 

(TWO-OIMENSIONAL PLANAR ELEMENTS) 

ALUM I NUM 

- I’ COMPOS I TE 

DOUBLE LAP JOINT OF ALUMINUM AND COMPOSITE PLATES 

UNDER PLANE STRAIN 

SCARF 

I 

JOINT OF ALUM I NUM AN0 COMPOS I TE CYL I NDERS 

Figure 4. Two-dimensional prismatic structures. 

147 
STRAlN ENERGY WCAY RATE 

surface & 

I self-equilibrated 

tractions on this 

-.-. --. 

FORMULATION OF TWO-DIMENSIONAL PROBLEM FOR 

PLATE WITH PLANE ANISOTROPIC flATERlALS 

t 

' [TYPICAL LAMINATE 

f= (Y -Yb 1 (Y, -Yb 1 

SOLUTION FORM IN X-DIRECTION SECOND ORDER ALGEBRAIC EIGENVALUE PROB. 

Figure 5. Finite-element analysis of self-equil ibrated edge effects 

in a composite plate. 

148 
EXAMPLE OF A 

THREE-DIMENSIONAL PRISMATIC STRUCTURE 

UNIFORM END SHORTENING- - 

I 

I 

I 

I 

THE NASA FLAT PANEL WITH 

DISCONTINUOUS STIFFENER 

FINITE ELEMENT SUBREGION 

(SOL ID, PLATE AND SHELL ELEMENTS) 

P 

I -j-L/2- - 

'tee 

Figure 6. Three-dimensional prismatic structures. 

149 
GLOBAL-LOCAL METHODOLOGIES AND THEIR APPLICATION 

TO NONLINEAR ANALYSIS 

Ahmed K. Noor 

George Washington University 

NASA Langley Research Center 

Hampton, Virginia 

ABSTRACT 

An assessment is made of the potential of different global-local analysis 

strategies for predicting the nonlinear and postbuckling responses of 

structures. Two postbuckling problems of composite panels are used as benchmarks and the application of different global-local methodologies to these 

benchmarks is outlined. The key elements of each of the global-local 

strategies are discussed and future research areas needed to realize the full 

potential of global-local methodologies are identified. 

NOMENCLATURE 

Elastic moduli of the individual layers in the 

direction of fibers and normal to it, respectively 

Shear moduli in plane of fibers and normal to it, 

respectively 

Side lengths of panel 

Total axial force acting on the edge of the panel 

ELJ ET 

GLTJ GTT 

L1' L2 

h Thickness of panel 

N 

9 Edge displacement 

R Radius of curvature of the panel middle surface 

r Number of global approximation vectors 

U Total strain energy of the panel 

uw Displacement components in the coordinate directions 

Orthogonal curvi 1 inear coordinate sys tem 

Major Poisson's ratio of the individual layers 

Axial strain 

a' 

a' 3 

LT 

1 

xx 

V 

E 

The range of the subscript a is 1.2. 

1. INTRODUCTION 

Considerable progress has recently been made in computational mechanics 

which is manifested by the development of versatile and powerful finiteelement discretization methods, improved numerical algorithms and programming 

techniques (see, for example, BELYTSCHKO & HUGHES C19831; NOOR & PILKEY 

C19831; NOOR C19831; LIU, BELYTSCHKO & PARK C19841; and KARDESTUNCER 

C19851). Also, an explosive growth has taken place in computer technology. 

In particular, the introduction of large expensive computer systems, usually 

referred to as supersystems, such as CRAY X-MP, CDC CYBER 205 and Denelcor 

HEP1 has made possible new levels of sophistication in the modeling of complex 

structures which were not possible before (NOOR, STORAASLI & FULTON C19841). 

In spite of these advances the detailed stress analysis of complex structures 

is very time consuming and, therefore, is not economically feasible. To date 

the only realistic structural response simulations that have been obtained 

PRECEDING PAGE BLANK NOT FlLMED 

151 
involve either simple structural configurations or components of complicated 

configurations. Prediction of the response of future complex structures such 

as those of flight vehicles is likely to require more sophisticated analysis 

models than has heretcfore been done. This is because of the requirements of 

high performance, light weight and economy and the associated stringent design 

criteria. Also, analysis may of necessity replace tests in some missioncritical areas. 

the accurate prediction of detailed stress distribution in structures without 

overtaxing the available computational resources are the global-local 

methodologies which are basically hybrid modeling and/or analysis techniques. 

In order to put these methodologies in proper perspective, a brief summary is 

given of the different approaches for reducing the cost and/or time for 

solving nonlinear problems. The efforts devoted to this activity can be 

grouped into three different levels. 

The first level is modeling. Reductions in cost of analysis can be 

achieved by using simple models that capture major effects in the responses 

and by exploiting all the symmetries and quasi-symmetries in the problem (see 

NOOR & PETERS C19851; and NOOR, ANDERSEN & TANNER C19851). 

ductions in time can be achieved by incorporating the known physical behavior 

into the computational model of the structure and by using global-local 

methodologies in which different analysis methods and/or models are coupled 

for predicting the nonlinear response of the structure. 

algorithms for solution of equations (e.g., multigrid methods, operator 

splitting techniques, dynamic relaxation, and element-by-element techniques 

-HACKBUSCH & TROTTENBERG C19821; UNDERWOOD C19831; and HUGHES, RAEFSKY, 

MULLER, WINGET & LEVIT C19841); as well as the vectorized and parallel 

numerical algorithms for use on pipeline and parallel processors (SCHENDEL 

C19841; MIKLOSKO & KOTOV C19841; and PADDON C19841). 

the second category. Specifically, the objectives of this paper are: 

local analysis strategies for predicting the nonlinear and postbuckling 

responses of structures 

2) To identify the future directions for research required to realize 

their full potential 

Discussion of global-local methodologies is primarily focused on 

nonlinear analysis of composite panels with discontinuities (e.g., stiffeners 

and cutouts). Two benchmark problems of composite cylindrical panels with 

cutouts typical of those used in modern aircraft structures are selected to 

provide a focus for the discussion. However, many of the conclusions apply to 

other complex structural configurations. 

benchmark problems, identifies their major characteristics, and lists the 

difficulties encountered in analyzing them using conventional finite-element 

methods. In the second part of the paper four global-local analysis 

strategies are reviewed and the potential for using these strategies in 

analyzing the benchmark problems is assessed. 

of global-local methodologies and their application to nonlinear analysis. 

These are the recommended future directions of research. 

Among the different analysis methodologies that have high potential for 

The second level is that of computational strategies. Significant reThe third level is that of numerical --- algorithms. These include fast 

The present study deals with global-local methodologies which belong to 

1) To review and assess the potential of a number of different globalThe paper is divided into three parts. The first part describes the two 

The third part of the paper identifies the items that pace the progress 

152 
2. BENCHMARK PROBLEMS AND THEIR CHARACTERISTICS 

The two benchmark problems selected for this study are shown in Fig. 1. 

They are postbuckling problems of laminated composite cylindrical panels with 

central circular cutouts. The loading consists of applied axial end 

displacements. One of the panels is unstiffened and the other has discrete 

blade stiffeners. The two panels are generic examples of modern composite 

aircraft components for which postbuckling strength is desired in the presence 

of local discontinuities such as holes and cracked stiffeners. In the 

conventional finite-element approach the panels are modeled using twodimensional shell elements and the stiffeners are modeled using twodimensional plate elements. The unstiffened panel was analyzed using an inhouse research program. Shear-flexible mixed finite elements were used in the 

modeling (see NOOR & ANDERSEN C19821). Also, extensive numerical solutions to 

this problem using continuum-based shell elements are presented in STANLEY 

C19851 in which the imperfection sensitivity of the panel is assessed. The 

stiffened panel was analyzed using the EISI-EAL program and the analysis 

results are given in KNIGHT, GREENE & STROUD C19851. The characteristics of 

the finite-element models used in the present study and those used in KNIGHT, 

GREENE & STROUD C19851 are summarized in Table 1. 

exhibits inversion symmetry, and therefore, only one half of each panel needs 

to be analyzed (see NOOR, MATHERS & ANDERSON C19771). The numbers between 

parentheses in Table 1 refer to the number of elements and degrees of freedom 

required for analyzing the full panel. 

postbuckling response of the panel exhibits a sudden drop in the loading due 

to delamination in the neighborhood of the cutout (see KNIGHT & STARNES 

C19841). Figure 2 shows plots of the total axial load versus axial and normal 

displacements up to the maximum load reached. For the range of loading 

considered, the results shown in Fig. 2 agree reasonably well with the 

experimental and numerical results presented in KNIGHT & STARNES C19841. 

ments as well as the axial strains on the top and bottom surfaces. Note the 

high strain concentration at the cutout. A finer model near the cutout is 

required for the accurate prediction of the strain in that region. On the 

basis of the studies made, the following characteristics of the two benchmark 

problems can be identified: 

large numbers of degrees of freedom in the finite-element models. 

stresses) near the cutout requires either a higher order two-dimensional 

theory or a three-dimensional theory. Failure analysis (including prediction 

of delamination) requires even more sophistication in the modeling and 

analysis. 

3. The postbuckling response exhibits large rotations in certain 

regions. 

4. The postbuckling response of the unstiffened panel is highly 

sensitive to initial imperfections. Tracing the postbuckling response past 

the maximum load point requires the inclusion of initial imperfections in the 

model (see STANLEY C19851). 

The response of both panels 

The response of the unstiffened panel is shown in Figs. 2 and 3. The 

Figure 3 shows normalized contour plots of the axial and normal displace1. The presence of discontinuities (cutouts and stiffeners) results in 

2. The detailed stress analysis (including determination of interlaminar 

3. GLOBAL-LOCAL ANALYSIS STRATEGIES 

In this section, the application of four different global-local analysis 

153 
strategies to the prediction of the nonlinear response of benchmark composite 

panels is discussed. In each case, the key features of the global-local 

approach and the requirements for its effective implementation are identified. 

3.1 Zooming Technique 

The first global-local approach considered is the zooming technique in 

which a global solution is obtained using a coarse grid. Then the detailed 

stress distribution near the cutout (local solution) is obtained by zooming on 

that area, refining the model, and using the displacements from the coarser 

model as input for the refined model (see Fig. 4). Single or multiple levels 

of zooming can be made corresponding to multiple passes from coarse to fine 

subdivisions (WILKINS C19831, and HIRAI, UCHIYAMA, MIZUTA & PILKEY C19851). 

Interelement compatibility can be maintained by either embedding the region of 

the fine grid into a superelement or by using transition elements at the 

interface between the coarse and the fine grids (ARMEN, Grumman Aircraft 

Systems, Bethpage, NY, Private Communication, 1985). 

A number of questions remain in connection with this technique, namely: 

1. Limitations of the technique for nonlinear problems 

2. Criteria for selecting the extent of the region for refining the 

3. Treatment of the interfaces between the coarse and the fine grids and 

model and selection of the refined model 

the effect of the error on the boundary data for the refined model on the 

accuracy of the stresses in that model 

approaches for selecting the fine model. More recent work is based on the 

distribution of the strain energy density function. Currently, a variety of 

adaptive refinement techniques are available. Some of these techniques will 

be discussed in subsequent sections. 

The early work on zooming techniques was based on heuristic and intuitive 

3.2 - Simultaneous Application of Two Discretization Techniques 

The early applications of this approach consisted of the simultaneous use 

of the finite-element method and the global (classical) variational technique 

(see MOTE C19711). A variety of options are available depending on the extent 

of using each of the two techniques in the model (DONG C19831). The concept 

has been later generalized to cover other techniques for the global 

(approximate) solution and the local (detailed) solution. A list of the 

common techniques used for generating these solutions is shown in Table 2. 

The global solution can be obtained by using classical variational 

methods (e.g., Rayleigh-Ritz or weighted-residual approaches), or any of the 

discrete element methods (conventional finite elements, global elements, 

boundary element method) or their combinations. In the global (or macro) 

element method the structure is divided into a small number of elements and a 

suitable (usually nonpolynomial) approximation is made within each element. 

The continuity of the field variables across the interfaces is imposed 

implicitly in the variational functional (DELVES & HALL C19791, and DELVES & 

PHILLIPS C19801). The global element method is a compromise between classical 

variational techniques and finite-element method. It combines the rapid 

convergence of global variational methods with the ability to handle 

complicated geometries. 

The boundary element method is an effective technique for solution of 

linear and materially nonlinear structural problems with high stress gradients 

(or singularities). However, it is not competitive with either the finite154 
element or the global element method for geometrically nonlinear problems (see 

KAMIYA & SAWAKI C19821; and KAMIYA, SAWAKI & NAKAMURA C19841). 

ment methods and analytical solutions (e.g., polar and/or edge functions 

PATTIBIRAMAN, RAMAMURTI & REDDY [1974]). Figure 5 shows three different 

combinations of discrete element methods for analyzing !he unstiffened 

composite panel. 

a) criteria for selecting the proper global and local analysis techniques and 

b) problem-adaptive strategies for generating global solutions and treatment 

of interfaces. 

Among the different techniques for local analysis are the discrete eleThe effective implementation of this global-local strategy requires: 

3.3 Reduction Methods - 

These are hybrid two-step techniques which are based on the successive 

application of a discrete element method (finite elements, boundary elements 

or combination of finite elements and boundary elements) and classical 

variational techniques (see, for example, NOOR C19821; and NOOR & PETERS 

L19831). The discrete element method is used to generate - few global 

approximation vectors (or modes). The classical variational technique is then 

used to compute the amplitudes of these modes. The primary objective of using 

reduction methods is to reduce considerably the number of degrees of freedom 

in the initial discretization, and hence, reduce the computational effort 

involved in the solution of the nonlinear problem. 

problem is depicted in Figs. 6 and 7. Figure 6 shows the accuracy of the 

normal displacements and total strain energy obtained by using four global 

approximation vectors (generated at zero loading). Figure 7 shows contours of 

the normal displacement w of the first three global approximation vectors. 

Two recent applications of reduction methods deserve further examination. 

In the first, only partial reduction is made. The degrees of freedom in the 

region of strong nonlinearity (e.g., near the cutout) are retained; the other 

degrees of freedom are reduced. In the second application, the response of a 

complex structure (e.g., stiffened panel) is generated using small (or large) 

perturbations from the response of a simpler system (e.g., unstiffened 

panel). It is also possible to use a hierarchy of simpler structural systems 

in generating the response of the original complex structure. This is 

accomplished by choosing a number of perturbation parameters, and successively 

applying a single-parameter reduction method with each of the parameters. 

Application of this strategy to the nonlinear analysis of anisotropic panels 

is described in NOOR c19851. 

commercial programs requires: a> the selection of a simple set of global 

approximation vectors and b) the development of a problem-adaptive.strategy 

for error sensing and control. 

The application of reduction methods to the unstiffened composite panel 

The wide acceptance of reduction methods and their incorporation into 

3.4 

Techniques -._I 

Hierarchy of Mathematical Models and/or Numerical Approximation - 

The last global-local approach considered is that based on a hierarchy of 

mathematical models for different parts of the structure. The application of 

this approach to the stiffened composite panel is depicted in Fig. 8 where a 

heuristic choice is made of the mathematical models. For the panel a 

boundary-layer (or a higher order two-dimensional) theory is used near the 

155 
cutout, followed by a first-order shear deformation theory, and then a 

classical shell theory. For the stiffeners, a plate theory is used near the 

cutout, followed by a thin-walled beam theory and then a shear deformation or 

a classical beam theory. The effective implementation of this approach 

requires the following: 

models (e.g., the method of initial functions of V. Z. Vlasov - VLASOV & 

LEONTEV C19661; and IYENGAR, CHANDRASHEKHARAN & SEBASTIAN C19741); or the 

asymptotic integration technique - GOLDENVEIZER C19761) 

1. Systematic procedure for generating the hierarchy of mathematical 

2. Criteria for the adaptive refinement of the mathematical model 

3. Treatment of the interfaces between the different regions 

4. TREATMENT OF INTERFACES 

The treatment of interfaces is one of the key elements of the globallocal analysis. The two commonly used approaches for maintaining displacement 

compatibility and traction reciprocity at the interfaces are: 1) Lagrange 

multiplier method; and 2) penalty function method. The second approach has 

the advantage that it does not lead to any extra unknowns or equations (DELVES 

& HALL C19791). The numerical problem associated with increasing the penalty 

weight, to meet constraint satisfaction tolerances can be overcome by using 

the iterative procedure described in FELIPPA C19781. 

5. QUALITY CONTROL OF NUMERICAL SOLUTIONS 

One of the most difficult aspects of numerical modeling is the validation 

of the results and ensuring that a given model is adequate for the particular 

problem at hand. In general, there are three types of errors in the numerical 

solution. These errors are (see UTKU & MELOSH C19841): 

made in abstracting the mathematical model from the real structure. 

zation of the continuous mathematical model. 

of the computers (limitation in representing real numbers due to the finiteness of the computer word length); and b) the errors resulting in the process 

of solving the equations of the discrete model (e.g., using iterative 

methods). 

In this paper only the second type, namely: d i s c r e t i z a t i o n e r r o r s , is 

considered. There are two classical approaches for estimating these errors 

(KELLY, GAG0 & ZIENKIEWICZ C19831). 

1. Extension methods - based on reanalysis of the structure on a 

sequence of meshes of increasing refinements (h extension) ; with a hierarchic 

set of interpolation polynomials (p extension); or using a combination of the 

two (h-p extension). 

2. Dual (or complementary) -- procedure - based on obtaining two solutions 

with two different computer programs to provide bounds on global response 

characteristics. 

1. Mathematical modeling errors, which result from the simplifications 

2. Discretization errors, which are caused by the numerical discreti3. Mza>ipulation-errors, which are caused by: a) the finite precision 

--I_--- 

Both of these approaches are too expensive for practical implementation. 

In recent years, considerable effort has been devoted to the development 

of a posteriori error estimates that are based on information obtained during 

the solution process itself (KELLY, GAG0 & ZIENKIEWICZ C19831; BABUSKA & CUI 

C19851; SZABO C19841; and SPECHT C19841). For structural mechanics problems, 

all these error estimates were developed for compatible displacement models. 

- 

156 
Among the error estimators developed to date are the following two: 

1. Local energy norm error. This is the square root of the strain 

energy ofThe error. 

measures a global response characteristic, locally (within an individual 

element). 

around a nonlinear solution and evaluating the energy norm of the linearized 

problem. 

2. Interior and boundary residuals. These represent the equilibrium 

defects in the interior on the portion of the boundary where tractions are 

prescribed as well as the jumps in the tractions at interelement boundaries. 

For uniform grids with linear, bilinear and trilinear shape functions the 

contributions of the jumps dominate the residual and, therefore, the residual 

can be approximated by the traction jumps. A simple approximate method of 

evaluating these residuals for elements with hierarchic shape functions was 

given in KELLY, GAG0 & ZIENKIEWICZ C19831. 

The error estimators, in addition to providing information about the 

quality of solutions, form the basis for adaptive improvement of the finiteelement solution. This can be accomplished by enriching or improving the 

approximation using one of the following approaches (or possibly, their 

combinations). 

This is a local-global measure in the sense that it 

In nonlinear problems, the measure can be used by linearization 

1. Refining the mesh 

2. Moving the nodes (node relocation) 

3. Increasing the local order of the approximation 

4. Using the iterated defect correction method 

The third approach has the advantages over the first two of being easy to 

implement and of providing a simple formula for the error estimator. The 

fourth approach is based on using the numerical solution obtained to construct a pseudo or neighboring problem whose exact solution is known (e.g., 

polynomial or spline interpolation of the discrete numerical solution). The 

pseudo problem is then solved using the same finite-element model as that used 

for the original problem. The error in the pseudo problem is assumed to be a 

close approximation of the error in the original problem and is used as a 

correction to that solution. The technique has been successfully applied to 

the numerical solution of stiff systems of ordinary differential equations and 

appears to have high potential for application to finite-element boundary 

value-problems (ZADUNAISKY C19761; FRANK, HERTLING & MONNET [ 19831 ; and BOHMER 

& STETTER [I 9841). 

6. POSTPROCESSING AND STRESS CALCULATION 

In displacement finite-element models, the strain energy of the structure 

is the highest quality information that can be extracted from the finiteelement solution. The accuracy and rate of convergence of stresses depend on 

how (and where) they are computed. Several approaches have been suggested for 

improving the accuracy of stress calculations (see, for example, HINTON & 

CAMPBELL [ 19743 ; CAREY [ 19821 ; ZIENKIEWICZ, XI-KUI & NAKAZAWA [19851; and 

BABUSKA & MILLER C19841). Among these are: 

1. Evaluating the stresses at numerical quadrature points and determining their values at the nodes by extrapolation 

2. Computing the stresses using the discarded structural equations 

(corresponding to prescribed displacement boundary conditions) 

3. Averaging or smoothing based on projection techniques 

4. Using influence function methods 

157 
The first approach is by far the most commonly used. Superconvergence 

(increased accuracy and improved rates of convergence) has been observed for 

stresses evaluated at quadrature points. 

faster convergence than the strain energy. The second approach is particularly useful for evaluating the stresses at the boundaries. The third and 

fourth approaches improve the accuracy of stress predictions through the 

filtering of spurious oscillations. A systematic assessment of the latter two 

approaches is needed. 

Note that the stresses ~- cannot have a 

7. FUTURE DIRECTIONS FOR RESEARCH 

Global-local analysis strategies have high potential for the reliable and 

efficient prediction of the nonlinear response of complex structures subjected 

to different loadings. To realize this potential the global-local strategy 

must include the following seven key elements: 

1. Rational selection of a hierarchy of mathematical models for 

different parts of the structure and a strategy for the adaptive refinement of 

these models 

2. Use of global (or -- macro) elements for discretization whenever appropriate with interface conditions satisfied via exterior penalty method 

3. Use of reliable failure criteria and discrete elements that account 

for the progressive failure mechanisms 

4. Application of operator splitting -- in conjunction with reduction 

method for generating the response of the complex structure by using large 

perturbations from the response of a simpler structure 

-- 

-- 

5. Postprocessing to increase the accuracy of stress calculations 

6. 

7. 

- Quality control of numerical solutions 

---I Exploiting the computational power of new multEocessor -- machines 

through parallelization of the problem formulation, computational strategy as 

well as the numerical algorithms. 

reach the level of maturity needed for routine inclusion in the global-local 

strategy. To this end, there are pacing items that must be addressed by the 

research community. Among the items that pace the progress of global-local 

methodologies are the development of: 

as well as adaptive strategies for refining the model whenever needed. Also, 

strategies for blending regions of different structural behavior (e.g., 

boundary layer, two/three dimensional models of the structure). 

composite delamination mechanisms. 

the same accuracy as that of the strain energy. 

area which requires more attention by researchers. In particular, error 

estimators that satisfy the following four criteria need to be developed for 

nonlinear analysis: 

global estimation 

Each of the aforementioned key elements requires major development to 

1. Criteria and control parameters for selecting the mathematical model, 

2. Reliable failure criteria and shell elements that account for the 

3. Simple and accurate techniques for stress calculations which provide 

4. Error estimation and adaptive improvement strategies. This is an 

a) provide reliable local assessment of the error with extrapolation to 

b) computationally inexpensive to evaluate 

c) applicable to a wide class of discrete elements 

d) easy to use in conjunction with adaptive improvement 

5. Parallel computational strategies for multiprocessor computers. 

158 
These strategies include the use of: a) primitive variables (e.g., threefield mixed formulation); b) domain decomposition (with minimization of 

interfaces); and c) operator splitting to uncouple the algebraic equations. 

In addition, the intense research effort currently under way on parallel 

numerical algorithms (see, for example, NOOR C19831, and PADDON C19841) should 

be brought to bear on global-local methodologies. Due to the wide variety of 

new parallel computers, the idea of developing macro algorithms which are 

efficient on different parallel machines should be investigated. The 

numerical tasks in these algorithms are performed by different programs which 

are optimized for each of the individual machines. 

CONCLUDING REMARKS 

A review and an assessment were made of global-local strategies for the 

nonlinear analysis of structures. To provide a focus for the discussion two 

benchmark problems of postbuckling of laminated composite cylindrical panels 

were selected. The major characteristics of these problems were identified. 

A number of global-local analysis strategies were reviewed, their 

potential for solving the benchmark problems discussed and their shortcomings 

delineated. Also, error estimation and postprocessing techniques were 

reviewed. 

identified and are, therefore, recommended as future directions for 

research. These include the coupling of different global-local methodologies; 

postprocessing and stress calculation methods; quality control and adaptive 

improvement of numerical algorithms; and effective computational strategies 

for new computing systems. 

The items that pace the progress of global-local methodologies are 

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162 
TABLE 1. CHARACTERISTICS OF THE FINITE-ELEMENT MODELS USED FOR THE BENCHMARK 

PROBLEMS 

Type of Element 

Number of 

Elements 

Blade-Stiffened Panel 

(Knight, Greene 6 Stroud [1985]) Unstiffened Panel 

Mesh 1 Mesh 2 

Mixed, 9-Noded + Hybrid 4-Noded 

188 544 

(376) (1088) 66 

Number of Displacement Degrees of 

Freedom' 

1338 1266 

(2432) 

3504 

(7008) 

TABLE 2. PARTIAL LIST OF THE COMMONLY USED TECHNIQUES FOR GLOBAL AND LOCAL 

ANALYSES 

o Global variational methods 

o Discrete element methods 

o Conventional finite elements 

o Global element method 

o Boundary element method 

o Analytic solutions 

o Discrete element methods 

o Conventional finite elements 

o Special elements 

o Superelements 

o Boundary element method 

o Analytic solutions 

163 
16-PLY UNSTIFRNEO PANEL BLADE STIFFENED PANEL 

- L2 

ETh3 

L = 1 = 0,3556 m 

R = 0.381 m. 

h = 2.276 x m. 

E = 1.31 x 10l1 Pa 

E = 1.303 x lolo Pa 

G = 6.412~ 10 Pa 

C = 5.102 x 10 Pa 

uLT = 0.392 

Fiber orientation: 

[+ 45/90/02/90/ 7 451 

12 

L 

T 

LT 

n 

9 

9 

f 

1 

I 

I1 

25-PLY SKIN 

Boundary conditions 1 ~45/02/~45/O~~45/Oj/~45/O~~45/02/~451, 

Atxl=*L /2 

1 24-PLY BLADES u = &9/2, u =w= d = d =o 

1 2 12 I I 45/O2d; 45 IT 

Atx,= tl /2 -2 

w= 0 30 1 

Figure 1. Benchmark problems considered in present study. 

LINEAR 

SOLUTION ' 

- 128 

- 64 

LINEAR 

SOLUTION ' 

- 128 

- 64 

- 32 

1 

0 17.5 35.0 52.5 70.0 0 0.15 0.30 0.45 0.60 

Figure 2. Nonlinear response of unstiffened panel subjected to end 

shortening (see fig. 1). 

164 

ORfGlNAL PAGE fS 

OF POOR QUALITY 
=i) fL. 

Figure 3. Normalized contour pilots for displacements 

and strains in untiffened panel at - = 126.6 

(see fig. 1). 

NL2 

ETh 

3 

Figure 4. Application Application of zooming zooming technique technique to unstif€ened unstif€ened panel. 

165 
GLOBAL GLOBAL ELEMENTS CONVENTIONAL FINITE ELEMENTS 

ANALY S I S 

BOUNDARY ELEMENTS LOCAL CONVENTIONAL (OR SPECIAL) 

ANALYSIS FINITE ELEMENTS 

Figure 5. Simultaneous application of two discrete element methods to the 

analysis of unstiffened panel. 

GLOBAL ELEMENTS 

BOUNDARY ELEMENTS 

& 

ETh3 

-64 

-32 p Fi (4 VECTORS) 

0 0.15 0.30 0.45 0.60 

wJh 

0 15 30 45 60 

5 

ETh4 

Figure 6. Accuracy of normal displacement w and total strain energy U 

obtained by reduction method. 

shortening (see fig. 1). 

Unstiffened panel subjected to axial end 

166 
I 1 

aw aw -I - I ''1 max. 

Figure 7. Normalized contour plots for the global approximation 

vectors - unstiffened panel subjected to axial end shortening 

(see fig. 1). 

CLASS I CAL 

BEAM THEORY - 

TH I N-IIVA LLED 

BEAM THEORY 

PLATE THEORY -. 

- 

7 

CLASS I CA L THEORY 

SHEAR - DEFORMAT I ON 

THEORY 

BOUNDARY 

7 LAYER THEORY 

Figure 8. Heuristic approach for selecting a hierarchy of 

mathematical models for blade stiffened panel. 

167 
N89- 24648 

APPLICATION OF THE P-VERSION OF THE FINITE-ELEMENT METHOD 

TO GLOBAL-LOCAL PROBLEMS 

Barna A. Szab6 

Washington University 

St. Louis, MO 

1. INTRODUCTION 

The following is a brief survey of some recent developments in finite-element 

analysis technology which bear upon the three main research areas under consideration in this workshop: (1) analysis methods; (2) software testing and quality 

assurance; and (3) parallel processing. 

The variational principle incorporated in a finite-element computer program, 

together with a particular set of input data, determines the exact solution corresponding to that input data. Most finite-element analysis computer programs are 

based on the principle of virtual work. In the following we consider only programs 

based on the principle of virtual work and denote the exact displacement vector 

field corresponding to some specific set of input data by iiEx. The exact solution 

iiEx is independent of the design of the mesh or the choice of elements. Except for 

very simple problems, or specially constructed test problems, iiEx is not known. 

We perform a finite-element analysis (or any other numerical analysis) because 

we wish to make conclusions concerning the response of a physical system to 

certain imposed conditions, as if iiEx were known. We know the finite-element 

solution only which we denote by iiFE. The solution iiFE depends not only on the 

variational principle and the input data but (also on the finite-element mesh and 

the choice of elements. We will assume that the finite-elements are exactly and 

minimally conforming and therefore the elements are completely characterized by 

their polynomial degree. We therefore control iiFE by mesh design and the choice 

of the polynomial degree of elements. 

We wish to compute iiFE so that iiFE is close to iiEx in some sense. For example, 

if we are interested in determining a stress intensity factor then we wish to have 

the stress intensity factor computed from iiFE to be close to the stress intensity 

factor computed from iiEx within some prespecified level of tolerance 7. In general, 

we wish to determine functionals Q,(~FE) (i = 1, 2...,n) so that: 

The question naturally arises: how can we tell whether Q;(iiFE) is close to Q;(iiEX) 

if we do not know zEX? The answer is: by performing extensions. ,Both the 

estimation and control of error are based on extensions. 

PRECEDING PAGE BLANK NOT FILMED 169 
Extensions are systematic increases in the number of degrees of freedom either 

by mesh refinement, increase in the polynomial degree of elements or a combination 

of both. If the extension is by mesh refinement then the process is called hextension*. If the extension is by increase in the polynomial degree of elements 

then the process is called p-extension**. If the extension is by a combination 

of proper mesh refinement and concurrent increase in the polynomial degree of 

elements then it is called h-p extension. Having performed an extension, we may 

draw conclusions concerning the overall quality of the approximate solution and 

the quality of any functional computed from CFE. 

2. OVERALL QUALITY 

The overall quality of approximation can be judged in terms of the estimated 

error in energy norm and errors in equilibrium. Estimation of error in energy 

norm is outlined in some detail and an example is presented. Procedures for 

assessment of the quality of approximation in terms of errors in equilibrium are 

briefly discussed. 

2.1. Estimation of error in energy norm. 

We know that the strain energy of the error U(ilEx - CFE) must decrease monotonically as we systematically refine the mesh or increase the polynomial degree 

of elements and a well developed, elaborate theoretical basis exists for the estimation of error in energy norm for the h-, p- and h-p extension processes. (See, for 

example, [1,2,3,4,5].) The error in energy norm is defined as: 

where n represents the solution domain and U represents the strain energy. 11 UEX - 

iiFEIIE(n) is closely related to the root- mean-square of error in stresses [6]. 

In the case of h- and p-extensions the estimate is of the form: 

where k and p are positive constants, N is the number of degrees of freedom. In 

the case of h-p extensions the estimate is of the form: 

where k, 7 and e are positive constants. These estimators are 'sharp' for large N 

values hence the 'less than or equal' (I) can be replaced by 'approximately equal' 

(w) in (3), (4) when N is large. Therefore from (3) for large N values we have: 

(5) 

If we plot log JJuEx -uFEJJE(n) versus log N we see a downward sloping straight line. 

The absolute value of the slope is p, called the asymptotic rate of convergence. 

When p is large then the error decreases rapidly as N is increased. When p is 

small then the error decreases slowly. Of course, the error also depends on k which 

log IldEX - ~FE IIE(n) log k - p log N 

* h represents the size of elements. h-Eztension involves letting h,,, --t 0. ** p represents the polynomial degree of elements. p-Eztension involves letting pmin -+ 00. 

170 
is generally not known a priori, but can be estimated from data obtained from 

properly performed extensions. This will be discussed later. When the estimate 

is of the form (3) the rate of convergence is said to be algebraic. 

When the estimate is of the form (4) and we plot 1ogI)iiEx - ZFEIIE(n) versus 

logN then for large N values we see a downward curving line [2,3,4,5,7]. In this 

case the rate of convergence is exponential: 

log IliiEX - ZFEIIE(n) W logk- 7 (loge) Ne (6) 

where e is the base of the natural logarithm. If we plot logJJiiEx - iiFEIIE(n, versus 

Ne (not logN as before) then we see a downward sloping straight line. It is known 

that under conditions which are generally satisfied in practice e 2 1/3 [5]. 

All error estimation techniques are based on extension. Because in general 

the exact solution iiEx is not known, the only information available to us is how 

the finite-element solution iiFE behaves when the number of degrees of freedom is 

increased either through mesh refinement or increase in the polynomial degree of 

elements. Such information, together with an estimate or hypothesis concerning 

the magnitude of the error, or its rate of change with respect to N, is essential to 

all error estimation. Of course, the estimate or hypothesis must be asymptotically 

correct: as N + 00 the estimated error must approach zero at the same rate as the 

true error does. Therefore the quality of error estimators should increase with N. 

P-extension makes it convenient and inexpensive to obtain information concerning the rate of change of U(iiFE) with respect to N. In the p-version hierarchic 

basis functions are used. Therefore the stiffness matrices and load vectors corresponding to polynomial degree p are embedded in the stiffness matrices and load 

vectors of polynomial degree p + I. Once a solution is available for polynomial 

degree pmas, all solutions corresponding to p = I, 2,. . . ,pmaz - 1 can be readily and 

inexpensively obtained. Specifically, we write: 

Let us assume for the moment that U(CEX) > U(iiFE). In that case: 

We have three unknowns: U(iiEx), k and p. If we have three values of U(iiFE) and 

N corresponding to three different values of p, then we have three equations for 

computing the unknowns. Let us denote these three values by Up, up-l, Up-2 and 

Np, Np-l, Np--2 and U(iiEx) by U. Then from (8) we have: 

u - up 

log u - u,-1 W 

Np-1 log - 

NP 

Np-2 log - 

Np-1 

Denoting the right hand side of (9) by Q, we have: 

8 u - up 

u - up-l 

(9) 

To obtain an estimate of the exact strain energy U, we need to solve (10). The 

solution is expected in the neighborhood of Up. Because convergence of the strain 

171 
energy is monotonic, we know that U 2 Up when Up > Up-l. Conversely, U 5 Up 

when Up < Up-l. Eq.(lO) would not be different if u(iiEx) < U(iiFE); therefore, the 

restriction that U(CEX) > U(CFE) is not essential. Computational experience has 

shown this estimate to be reliable and generally accurate, with the accuracy of the 

estimate increasing with the accuracy of Up. 

2.2. Example. 

The following test problem is representative of plate and shell intersections 

and reentrant corner problems in general. An L-shaped plane elastic body of 

thickness t is loaded by tractions. The tractions are computed from a stress field 

which satisfies the equilibrium and compatibility equations and the stress free 

conditions along the reentrant edges. Specifically, the stress field corresponds to 

the first (symmetric or 'Mode 1') term of the asymptotic expansion of iiEx about 

the reentrant corner. Therefore the exact solution is 

known. Specifically, the components of iiEx in the coordinate system shown in 

Fig. 1 are: 

(See, for example, [8].) 

(1la) 

(W 

A 

2G 

A 

2G 

u, = - rx [(K - Q(X + 1)) COS X 6' - X COS(X - 2) 6'1 

uy = -rA [(K+ Q(X + 1))sinX 0 + Xsin(X - 2) 6'1 

where A is a generalized stress intensity factor; X = 0.544483737; Q = 0.543075579; G is 

the modulus of rigidity and K depends on Poisson's ratio Y only. For plane strain: 

K. = 3 - 4v. We assume plane strain conditions and v = 0.3, therefore in this case 

K = 1.8. 

1 / 

Fig. 1. L-shaped plane elastic body. 

The stress tensor components are: 

[T, = AX [p - Q(X + 1)) cos(^ - 1) e - (X - 1) cos(^ - 3) e] 

cy = AX rx-l [ (2 + Q(X + 1)) cos(X - 1) 0 + (A - 1) cos(X - 3) 8) 

T~~ = AX rx-' [(A - 1) sin(X - 3) B + Q(X + 1) sh(X - 1) 6'1. 

172 
Because we know the exact displacement and stress fields we can compute the 

strain energy of the exact solution: 

A2azX t ~(CEX) = 4.15454423- E 

where E is the modulus of elasticity. The relative error in energy norm is defined 

as follows: 

Using the mesh shown in Fig. 2 finite-element solutions were obtained for 

p=l to 8. The computations were performed by a new computer program, called 

PROBE [9]. The number of degrees of freedom, the computed strain energy, the 

estimated and true relative errors in energy norm, computed from eq. (14), are 

shown in Table 1. 

22 

25a 

Fig. 2. Mesh design. 

The results presented in Table 1 are typical of the quality of the error estimate 

we can obtain by means of the procedure described in Section 2.1. When the mesh 

is strongly graded toward the point of singularity then the convergence path the 

the rate of convergence is nearly exponential and the downward slope increases 

with N. In this segment the estimated error is conservative. Near the inflection 

log(e,)E versus logN curve) looks like an inverted S [3,4,5,10]. For low N vaues I 

173 
Table 1. Estimated and true relative error in energy norm. 

1 41 

2 119 

3 209 

4 335 

5 497 

6 695 

7 929 

8 1199 

00 00 

_____ 

3.8860880 

4.1248326 

4.1481150 

4.1526504 

4.1536354 

4.1539746 

4.1541390 

4.1542378 

4.1545442 

-__-- 

- 

- 

1.93 

2.76 

3.05 

2.45 

1.83 

1.39 

1.09 

__ 

-- -- 

- 25.42 

- 8.46 

5.34 3.93 

2.02 2.14 

1.01 1.48 

0.80 1.17 

0.75 0.99 

0.75 0.86 

0 

__ -- 

- 

point, (i.e. where the curvature of the convergence path changes from negative to 

positive) the estimate is the least accurate and not conservative, nevertheless as 

we see in this example, it remains close. The estimate then becomes progressively 

more accurate as the asymptotic range of the p-extension is entered. In this case 

the correct asymptotic rate of convergence is p = X = 0.5445. At p = 8 the computed 

value of p is approximately 0.7 with B decreasing. 

2.3. Equilibrium tests. 

Smallness of error in energy norm is a necessary but not sufficient condition 

for ensuring that the overall quality of the finite-element solution is good. It is 

possible to produce examples where the estimated error in energy norm is small 

(under 1 percent) yet the error in overall equilibrium is large (well over 10 percent). 

Although we do not know ZEX we know that ZEx satisfies the equations of 

equilibrium and the law of action and reaction. We can, therefore, assess the 

quality of the finite-element solution by examining to what degree zFE satisfies 

equilibrium and the law of action and reaction. Specifically, we can perform: (I) 

overall equilibrium tests; (2) element by element equilibrium tests and (3) actionreaction tests. 

In the overall equilibrium test we 'cut' the structure from its supports and 

integrate the tractions, computed from the ZFE, to obtain the reactions. In this 

way a free body diagram is produced. The error in equilibrium must be small in 

relation to the magnitude of applied forces. For example, we can define: 

where r; are the (global) load vector components. The term F is a suitable measure 

of the magnitude of the applied load. 

In the element-by-element equilibrium test individual elements (or any group 

of elements) are separated from the model and tested for equilibrium. Specifically, 

we denote the element domain by Re and its boundary by dR,. We compute: 

174 
where uij are the stress components computed from the finite element solution; Xi 

represents the applied body force components and n, represents the unit normal 

to an,. The summation convention is used. We should have 4- as 

well as the absolute values of each component of both expressions on the right 

hand side of (16) small in relation to the magnitude of the applied loads. If lqje)l is 

small but the absolute values of the integral expressions are not small then there 

is a local error but, according to Saint-Venant's principle, the effect of the local 

error will not be substantial at some distance from the element in question. If 

Iqje)l is large, even after p-extension was performed, then element e, and possibly 

its neighbors, should be subdivided. Thus the element by element equilibrium 

test provides information about the quality of mesh design. In many cases minor 

local refinement (for example, dividing one element into two elements) can have a 

highly beneficial effect on the overall quality of approximation when p-extension 

is used. 

In the action-reaction test we compute the stress result ants along interelement 

boundaries and external element boundaries where tractions are applied. Along 

interelement boundaries the stress resultants computed for neighboring elements 

should have nearly the same absolute value and opposite sense. Along external 

boundaries the resultants of the applied tractions and the tractions computed from 

the finite element solution should be nearly the same. 

Examples of equilibrium tests are presented in [ll]. 

3. LOCAL QUALITY 

Having ascertained that the overall solution quality is acceptable, we are ready 

to compute the quantities which are of principal interest, i.e. gi(cFE). Smallness of 

error in energy and equilibrium does not guarantee that all functionals computed 

from cFE are accurate. It is advisable to perform convergence tests on at least 

the more difficult functionals. We demonstrate the procedure by computing the 

direction and magnitude of the principal stresses at a point close to the reentrant 

corner. We selected the point r = 0.025a; 6 = 30' in the coordinate system shown in 

Fig. 1. The stress components computed from the exact solution are: 

u2 = 3.67198AaX-l uy = 7.68696AaXd1 rzy = 0.698375AaX-l (17) 

Therefore the principal stresses u1 and c2 and the direction of the first principal 

stress u1 from the positive x-axis, denoted by dl, are: 

In general functionals, other than the strain energy, do not converge monotonically, nevertheless the fact that convergence has occurred should be obvious. 

Here u1 and c2 happen to converge monotonically but el does not. We see that the 

state of stress is known with sufficient accuracy for engineering purposes at p=4 

(335 degrees of freedom, see Table 1). Extension beyond p=4 merely confirms 

that convergence has occurred to within the range of precision normally expected 

in engineering computations and thereby establishes reliability of the data. 

This test problem demonstrates that accurate stress data can be obtained in 

the very close proximity of stress singularities. Other examples and additional 

discussion of this point are presented in [ 10,121. 

175 
Table 2. Principal stresses at r = 0.025a; e = 30". 

P 

1 

2 

3 

4 

5 

6 

7 

8 

-- 

__ 

00 

c1 

AaX-l 

7.096 

7.441 

7.532 

7.731 

7.754 

7.773 

7.786 

7.791 

7.804 

-__ 

___ 

AaX-l 

2.854 

3.047 

3.294 

3.483 

3.525 

3.545 

3.551 

3.553 

3.554 

___ 

___ 

61 

(degrees) 

92.1 

84.1 

81.6 

80.9 

80.6 

80.5 

80.5 

80.4 

80.4 

___ 

--- 

It is possible also to compute various functionals from ciFE using advanced 

methods of extraction [13,14,15] .For example,we may wishtodetermine the value 

of the generalized stress intensity factor A [see eq.'s (lla,b), (12a,b,c)]. Such 

procedures based on [ 13,14,15] have been implemented in PROBE [9]. 

4. CONCLUSIONS AND RECOMMENDATIONS 

(1) Extensions are essential for both the estimation and control of error in finiteelement computations. 

(2) We are in a much better position today than we were, even just one year 

ago, from the point of view of understanding how an advanced finite element 

software system should be designed so that (a) the solution is obtained at very 

nearly the theoretically optimal efficiency and (b) the user is provided with the 

capability to estimate and control the quality of engineering data computed 

from the finite-element solution at a small marginal cost. This is because now 

we understand the interplay between mesh design and the polynomial degree 

of elements. 

(3) P-extension, coupled with properly graded meshes, is the most efficient method 

for controlling error in finite-element computations. 

(4) The proper mesh design is such that points of singularity (and areas where the 

solution changes rapidly over short distances are isolated by one or more layers 

the points of singularity. In this way both the global and local behavior of the 

solution can be represented without compromising the accuracy of either. 

(5) Implementation of advanced extraction methods for the the computation of 

certain engineering data, such as stress intensity factors, will further increase 

the efficiency and reliability of computations. 

(6) The p-version is well suited for implementation on parallel processors because 

the data are organized in relatively few, large units. This logical organization 

reduces the overhead associated with parallel processing. 

(7) The substantial increase of efficiency in finite-element computations through 

the use of h-p extension and the availability of parallel and vector processing 

technology make it possible and desirable to model plate and shell problems 

of small elements, with the elements grade d in geometric progression toward 

176 
(9) 

using hierarchic sequences of plate and shell theories in conjunction with fully 

three-dimensional representation. The various plate and shell theories are 

nothing more than specializations of the three-dimensional theory of elasticity through restrictions imposed on the variation of the displacement field in 

the direction of the normal. Such restrictions generally do not hold near supports, stiffeners, cut-outs, plate and shell intersections, etc. which are the 

areas where cracking and delaminations originiate and therefore of the greatest concern to analysts and designers. These areas can be properly modeled 

by three-dimensional representation only. The use of hierarchic extensions toward higher order plate and shell theories will permit us to assess and control 

the quality of approximation in relation to three-dimensional theory. 

Although linear theory is properly the first and most generally used approach 

to structural modeling, it should be possible to ascertain by a posteriori analysis whether engineering conclusions drawn from a numerical model would be 

different if geometric and material nonlinearities were considered. We can view 

linear theory as the simplest of a hierarchic system of theories. Much the same 

way as we estimate error by the use of extension processes within the framework of linear theory, we should be able to estimate error by extension within 

the hierarchic system of theories. This important area has not received much 

attention in the past. Because it bears on the reliability of computed data, and 

the engineering conclusions based on them, it deserves serious consideration. 

In some areas our ability to compute data is already greater than the material 

scientists’ ability to tell us what data should be computed. For example, it 

is not fully understood what parameters govern crack initiation. The reason, 

at least in part, is that the conventional finite-element method tends to yield 

’fuzzy’ data in areas where stresses change substantially over short distances. 

Proper use of h-p extension, coupled with advanced extraction methods, permits us to compute any stress field parameter with arbitrary precision. This 

removes an uncertainty from the phenomenological characterization of material response to various stress fields. Of course, such characterization can be 

developed only through joint experimental-analytical investigations. 

5. REFERENCES 

[l] BabuSka, I., Szab6, B. and Katz, I. N., ”The p-Version of the Finite-element 

Method”, SIAM J. Numer. Anal., Vol. 18, pp. 515-545, (1981). 

[2] Babugka, I. and Szab6, B., ”On the Rates of Convergence of the Finite-element 

Method”, Int. J. Num. Meth. Engng., Vol. 18, pp. 323-341 (1982). 

[3] BabuSka, I., Gui, W. and Szab6, B. A., ”Performance of the h, p and h-p Versions of the Finite-element Method”, Institute for Physical Science and Technology, Laboratory for Numerical Analysis, Technical Note BN-1027 (September, 1984). 

[4] Gui, W., ”The h-p Version of the Finite-element Method for One Dimensional 

Problem”, Doctoral Dissertation, University of Maryland, (1985). 

[5] Guo, B., ”The h-p Version of the Finite-element Method in Two Dimensions”, - 

Mathematical Theory and Computational Experience”, Doctoral Dissertation, 

University of Maryland (1985). 

[6] Szab6, B. A. and BabuSka, I., ”Stress Approximations by the h- and p-Versions 

of the Finite-element Method” Proc., 6th Invitational Symposium on the Uni177 
fication of Finite-elements, Finite Differences and Calculus of Variations”, 

Edited by H. Kardestuncer, The University of Connecticut, May 1982. 

[7] Szab6, B., ”Estimation and Control of Error Based on P-Convergence” Proc. 

Int. Conf. on Accuracy Estimates and Adaptive Refinements in Finite-element 

Computations (ARFEC), Lisbon, Portugal (1984). 

[8] Williams, M. L., ” Stress Singularities Resulting from Various Boundary Conditions in Angular Corners of Plates in Extension”, J. Appl. Mech. ASME, 

[9] Szab6, B. A., ”PROBE: Theoretical Manual”, Noetic Technologies Corp., St. 

Louis, Missouri (1985). 

[lo] Szab6, B. A., ”Mesh Design for the p-Version of the Finite-element Method”, 

Report WU/CCM-85/2, Center for Computational Mechanics, Washington 

University (1985). Presented at the Symposium on Recent Advances in Computational Mechanics, Joint ASME,’ ASCE Mechanics Conference, Albuquerque, NM (June 24-26, 1985). 

[ll] Szab6, B. A., ”Implementation of a Finite-element Software System with hand p-Extension Capabilities”, Proc., 8th Invitational UFEM Symposium: 

Unification of Finite Element Software Systems, Edited by H. Kardestuncer, 

The University of Connecticut, May 1985. 

[12] Szab6, B. A., ”Computation of Stress Field Parameters in Areas of Steep 

Stress Gradients”, Communications in Applied Numerical Methods, Vol. 2, 

pp. 133-137 (1986). 

[13] BabuSka, I. and Miller, A., ”The Post-Processing Approach in the Finiteelement Method” - Part 1: Calculation of Displacements, Stresses and Other 

Higher Derivatives of the Displacements”, Int. J. Num. Meth. Engng., Vol. 

20, pp. 1085-1109 (1984). 

[14] BabuSka, I. and Miller, A., ”The Post-Processing Approach in the Finiteelement Method” - Part 2: The Calculation of Stress Intensity Factors”, Int. 

J. Num. Meth. Engng., Vol. 20, pp. 1111-1129 (1984). 

[15] BabuSka, I. and Miller, A., ”The Post-Processing Approach in the Finiteelement Method” - Part 3: A-Posteriori Error Estimates and Adaptive Mesh 

Selection”, Int. J. Num. Meth. Engng., Vol. 20, pp. 2311-2324 (1984). 

pp. 526-528 (1952). 

178 
Yednesday , June 19, 1985 

LOCAL/GLOBAL NONLINEAR STRESS ANALYSIS 

Questions and Answers following: 

Nonl i near Structural Response" by H. D. Hibbi tt. 

"Some Issues in Numerical Simulation of 

Nelson Bauld, Jr., Clemson University: 

method, for computing the limit point on the buckling load. 

Crisfield method? 

You used a linear constraint, a Riks 

Have you tried the 

Hibbitt: 

done quite a lot of work on Riks' algorithms. 

he's mentioned. 

on the Riks concept. 

exactly the' same as Issac Fried published. 

cases. It certainly has done as well as the alternates we've tried. The linear 

constraint makes it very easy to code and deal with in a program like ABAQUS. 

It's critical to use some kind of automatic incrementation scheme along with 

it. By and large, it seems to work well. In the problems where it fails, I've 

certainly had fail ures with some of Cri sfiel d' s a1 ternate a1 gori thms as we1 1. 

We played around with a number of these methods, and Crisfield has 

We've tried some of the things 

I think everybody has his own little perturbation or variation 

We're pretty happy with it, in most 

We're using one that I found out a few months ago is 

Questions and Answers following: 

1990's" by B. F. Backman. 

'Canputerized Structural Mechanics for the 

W. J. Stroud, NASA Langley Research Center: Bjorn, in your talk, you mentioned 

computer program development and the usefulness, most often the lack of usefulness, of these computer programs to industry people. That is really a problem. 

I'll give you my personal opinion about some of these things. 

cult for NASA to develop computer programs for industry. 

sane very special cases throughout our history, we really have not had that as 

our goal. 

contractors develop computer programs because that i s how we cal cul ate numbers. 

Ue try to make those codes and the supporting technology available to industry. 

But it's very hard, as you well know, to develop computer programs for someone 

else's use. 

It is very diffiAs a result, except in 

On the other hand, as a part of our research, we and our grantees and 

Would you like to comment on that? 

179 
Backman: I think NASA has a key role in this area. When I referred to second 

generation programs I meant that there is a strong need for an organized interface between the developers and the users, and I see NASA as having a key role 

here. Maybe it' s a standing commission--maybe it' s an establ ished relation for 

review--I don't know. But where the needs can be established and where the 

first generation software could be used and evaluated by the industry and recommendations coming back to the developers, I think there exists an interest in 

this. 

mutually beneficial. 

The only question is how you stimulate this to the point where it becomes 

Questions and Answers following: 

Graphite-Epoxy Panel with a Discontinuous Stiffener: 

F. Knight. 

"Nonlinear Response of a Blade-Stiffened 

Work in Progress'' by N. 

K. C. Park, Lockheed Palo Alto Research Laboratory: Norm, is that specimen 

based on an optimized design for which the panel buckling load is about the same 

as the skin buckling load? 

Knight: This panel was one that's been around for a number of years. It was 

first built as part of one of the early studies in our composite design program 

and, I believe, it was designed without the hole--as part of a wing-type 

structure near the tip, so it is primarily a strength-type panel. Now with the 

hole, the characteristics of the panel change. 

Yung J. Chiang, Uniroyal , Inc.: 

You assume that the failure occurs when the stress at a distant do away from 

the edge of the hole reaches ultimate. 

referring to? 

By using the point-stress failure criterion, 

What is the stress that you are 

Knight: It was just an 

average longitudinal stress-type measure. We were using stress resul tants not 

necessarily to predict failure but to guide us in defining the mesh refinement 

needed near the hole in order to predict the stress distribution. 

that we were using do not have a transverse shear formulation, so we can't 

really recover the shear stress distribution other than from classical plate 

theory by taking derivatives of the moments and things. 

We applied just the average inplane stress resultant. 

The elements 

180 
Chiang: 

the failure? 

Can the energy calculation be better than stress for the prediction of 

Knight: We didn't calcu 

late a local energy norm 

that calculation. I can 

ate the energy so I don't really know. 

and use that to predict failure, but we did not make 

t answer that. 

You can calcuQuestions and Answers fol lowing: 

Composite Shells" by G. M. Stanley. 

"Computational Perspectives on Postbuckl ing of 

Gerald Goudreau, Lawrence Livermore National Laboratory: That fifth buck1 ing 

node that you said was nonphysical, did it vary with the different element types 

that you' ve studied? 

Stanley: No, not really, the eigenvalues moved around by a few percent here or 

there. There was actually more of a discrepancy in the linear solution than in 

the buckling solutions. 

Goudreau: Then to what do you attribute the nonphysical basis for that mode? 

Stanley: 

would not occur in practice, because any imperfection would prevent it from 

occurring. 

I'm not saying that that mode is wrong. I'm saying that that mode 

Jerrold M. Housner, NASA Langley Research Center: 

pretty rough problem once you get to the buckling load. 

experimental results, it looks like there was a distinct softening of the panel 

just prior to its first fail ure--col lapse. 

Gary, I noticed that it's a 

In the slide showing 

Stanley: 

picture than it was in practice. 

Before you continue, that distinction was a little sharper in this 

That was handdrawn. 

Housner: But, in any case, the analysis doesn't seem to pick that up. The 

softening is in the last 10 percent of the buckling load as you're coming up the 

load deflection curve. Could that be-- 

Stanley: The analysis doesn't pick up the peak; the analysis does roll over. 

181 
Housner: 

is that the softening that takes place--could that be what triggers the failure 

and is not being duplicated in the analysis? 

It does roll over for a short way I noticed, but what I'm driving at 

Stanley: The slide which shows stress contours in color would help us here. 

As we move up the load-shortening curve--near, but before, the peak--the analysis predicts very high (near ultimate) axial stresses in the vicinity of the 

hole. So, it's very possible that delamination is already occurring up there. 

This softening is due to material instability. (See figure, p. 207).* These 

are the compressive axial stresses in the, I think, fourth layer--in which the 

fibers are oriented along the axis--and this color scale is such that the maximum is white and the minimum is black. Everything in white here, though, is 

actually beyond the nominal ul timate stress for this particular composite. 

at the buckling load we're already at ultimate stress on both sides of the hole. 

This shows that as we roll around, the white region begins to rotate a little 

bit,then is unloading and drops off. When we're back on the secondary path, all 

the load is picked up here and it's again way beyond ultimate. You can't read 

it from the scale, but as we go up the curve, it's maybe 50 percent or more 

higher than ultimate. So there are a lot of things that need to be explained, 

and I think that we need to try one thing at a time. I don't think we should 

jump into a 3-D analysis until we have tried some failure criteria--just degradi ng the materi a1 properti es. 

So 

Housner: 

cri teri a? 

That's what your next step is going to be--putting in failure 

Stanley: I think so. I plan to use this workshop, the feedback and opinions 

expressed here, to help formulate the next phase of the research, as a matter of 

fact. But, that's one possible direction that we might begin next. 

Questions and Answers following: 

Stress Analysis'' by R. B. Nelson. 

"A Review of Some Problems in Global/Local 

Stroud: In the model that Paolo Roberti generated, what was the logic behind 

the model generation and ref i nment? 

Nelson: In words, the approach is to say: I'll take a specific grid and then 

I'll refine it one tine to see if the stresses change much. And I compare 

182 
stresses across neighboring elements. If the stress change is more than a set 

tolerance (and the stress change was, by the way, a root mean square type Of 

stress change), then I will refine the model further in these particular areas. 

There are a number of issues regarding the convergence that should be used and 

how one should go about this. 

that topic. 

think merits serious research consideration. 

There are a number of authors who have written on 

I'm only bringing it forward to show that it is an area which I 

Stroud: 

are nice to have. 

things that might even suggest having an expert system built into it because 

that's considered by many to be a diversion. 

energy and funds to put into something, you can't go into these diversions. 

I agree with you. I think that many of the things that you alluded to 

Yet, today, it is difficult for us to get support to do some 

When you have a limited amount of 

Nelson: It is true, but I think that the development of an automated technique 

for node1 regeneration, or adaption, or refinement, something that the engineer 

really shouldn't have to do, - isappropriate. 

tend the screens. 

putting their time to exploring whether or not we need to have a large deformation option being turned on, or whether or not we are worried about there being 

a problem with the material model used to model a composite. Is it a mean 

square of stress or a delamination stress that we should be looking at? 

kinds of issues are engineering issues I think should be pursued. 

The engineer should not have to 

So many people tend the screens, and I think they ought to be 

Those 

Questions and Answers following: 

S. N. Atluri. 

"Some Comments on Global/Local Analysis" by 

James C. Robinson, NASA Langley Research Center: I think when we start talking 

about either globally or locally exact solutions, as structual engineers, we 

should show proper respect for a negative force displacement curve. 

very nice when itJs plotted up in a rigid test machine, but when it is in an 

actual structure that is surrounded by other equally sensitive elements and you 

dOnJt do a dynamic solution, you have probably missed failure at the ultimate 

load point. 

tual ly use1 ess. 

It looks 

To look at it from there on is a fine academic exercise, but strucAtluri: I agree with you totally. These are illustrative examples to test the 

theory that is presented. I n these examples , a di spl acement-control 1 ed 1 oadi ng 

is assumed. 

183 
K. N. Shivakumar, AS&M, Inc., Hampton. 

some theories would allow you to replace a hole by a crack. 

theories? To me, a crack has an entirely different model from a hole, and you 

may be missing some physics in the problem. 

You mentioned that in a composite panel, 

What are the 

Atluri: If you consider an isotropic plate with a hole in it, the stress 

concentration is always three, if the size of the plate is roughly five times 

bigger than the hole size. 

ence. Even if you consider a plate five times bigger than the hole diameter, 

the stress concentation still depends upon the size of the hole. And in that 

sense, if you were to plot the failure stress of the laminated plate with the 

hole versus the hole radius, it would vary as ra. And in fracture mechanics, 

that variation is--if you were to consider r to be the length of the crack-- 

the failure stress for a cracked plate would vary as r-.5, or something like 

that. There is no great theory behind it--just an empirical relation. Mar at 

MIT has taken that, and there is a theory called the Mar-Lin theory wherein the 

fracture stress in a laminated hole varies as the diameter of the hole to some 

power alpha. So there is some evidence that a hole in a laminate behaves like a 

crack. 

But in a composite laminate, there is a size dependThe physics of that needs to be explored much further. 

Stanley: I agree with your advice to incorporate locally exact solutions in 

global analysis, but in this particular case of the composite panel with a hole, 

what about the subsequent propagation of failure in which the hole begins to 

distort--it's no longer a circular hole. Do you think it's actually feasible to 

continue to follow the complete analysis through with locally exact solutions? 

Atluri: 

better handle on the three-dimensional stress state--1 ike the transverse shear 

and so forth. 

would not be locally exact, but they would be much better than the traditional 

finite element basis functions you use. 

I think so. At least these locally exact solutions could give you a 

And when large geometric changes take place, those solutions 

Stanley: 

top of the normal shape functions. 

Perhaps they could be incorported as deviational basis functions on 

184 
Questions and Answers following: 

Stress Analysis" by J. N. Reddy. 

"On Computational Schemes for Global/Local 

Glenn Sahrmann, PRC Kentron: 

were the stresses becoming si ngul ar--unbounded? 

In your analysis of the 2-D free edge problem, 

Reddy: 

as you refine the mesh, oZ 

But we know that uZ must have some finite value. I did not address that 

problem here. 

a cap. 

modeled the 'cap as part of the region. 

Without free edge cap, yes, the analyses done by many people show that 

still keeps getting larger. So, it is unbounded. 

My objective was is to reduce the stress concentration by adding 

We accounted for the cap in the finite element analysis. We also 

Questions and Answers foll owing: 

Localized Stresses in Prismatic Structures' by S. B. Dong. 

"G1 obal /Local Finite El ement Analysis of 

Housner: After you compute what you feel is the position of the interface 

between the local and the far field solutions, have you gone back to examine the 

sensitivity of the solution to the location of that boundary--that interface? 

Donz: Yes, in fact, I had a student that took the interface at different 

regions. What happens is that when you have self-equil ibrated stresses, those 

which you subtract out from what the Bernoulli-Euler theory or Kirchoff theory 

would give you in terms of a plate analysis in the far field, the selfequilibrated stresses will decay. 

close to the localized area, you would obviously need more global functions. 

But that's really not a severe penalty because when you include 10 global functions, you can probably eliminate one or two layers of finite elements that 

would involve many more degress of freedom. 

finite element region very close to the localized area is that you will determine on a mathematical basis how the self-equilibrated stesses diffuse as they 

go into the interior rather than by letting the finite element solution--which 

depends on the model ing--indicate how the sel f-equil i brated stresses are 

diffused. 

difficul ty in going as close as you please to the local ized region. 

sively taking more and more global functions, you have a basis for evaluating 

the error thatJs incurred in the analysis. 

If you take the finite element region very 

A second advantage of taking the 

If you have a complete set of global functions, then there's no 

By succes185 
Housner: 

really wanted to. Years ago, I dabbled a little in wave propagation and if YOU 

have the complete set of waves, including those that attenuate, you can 

actually, theoretically, at least, go all the way back to the inclusion. 

Although, in those days years ago, we didn't have the computers to actually do 

that. 

I guess you could go all the way back to the inclusion itself if you 

Dong: I think if you go back to the inclusion, Eli Sternberg would be very 

happy. Then you would cut out all the finite elements, and, you would be where 

he was before all of us messed up his conquest of all the problems we have. 

Questions and Answers following: 

tions to Nonlinear Problems'' by A. K. Noor. 

"Global /Local Methodologies and Their Appl ica- 

- Park: 

have found that a symmetric boundary condition does not yield the physically 

correct buckling solution. 

What was the motivation for choosing a symmetric boundary condition? We 

- Noor: 

reflection symmetry or mirror symmetry, then I would agree with you. What we 

found in these problems is that you have inversion symmetry which is typical of 

the response of many anisotropic panels. 

presented by Norm Knight and Jim Starnes and we also looked at our results. 

inversion symmetry was exhibited by the response of the panel up to the maximum 

load given by the experiments. 

Well , I think you have to qualify the type of symmetry. If you say 

I have looked at the results that were 

The 

- Park: 

and a1 so postbuckl i ng analysis? 

Do you foresee that one may choose this boundary condition to do collapse 

- Noor: If you start from the beginning conducting a 

nonlinear analysis, the inversion symmetry would a1 ways be preserved until and 

unless you have a branching point. 

point from your equations. 

possibility that you might go on a branch where that symmetry is lost. 

Let me put it this way. 

And you can always detect the branching 

If you get that branching point, then there is a I 

However, 

to synthesize the solution of the 

symmetric solutions, even for the 

as you will hear 

unsymmetric prob 

tomorrow, there are ways then 

em from the symmetric and ant 

186 
nonlinear case. 

ning with a nonlinear solution, the inversion symmetry properties of that 

nonlinear solution will not change unless you have a branching point. 

Summarizing what I said, I think if you start from the begin- 

- Park: How far have you carried along your solution in your example? 

- Noor: 

experiments. 

In the unstiffened panel, up to the maximum load that was given in the 

- Park: 

a bifurcation that occurs about 10 or 15 percent below the collapse load, and 

that mode happens to be antisymmetric. 

We happen to disagree with that because just before the collapse there is 

- Noor: 

but if you have bifurcation, you can loose the symmetry. But even then, as I 

said, you can synthesize the unsymmetric solution from symmetric and 

antisymmetric components. And you'll hear about this tomorrow. 

We did not detect bifurcation up to the level of the loading that we had, 

Stanley: I think it would be good for us to compare quantitative results, but 

just qualitatively, are the path derivatives more accurate than the modes in 

this particular problem or has this been your general experience? 

- Noor: 

them in several other probl ems--pl ate probl ems , shear-1 oaded plates, axially1 oaded pl ates--and without any exception, you get considerably higher accuracy 

with the path derivatives than with equal number of buckling modes. 

mention also that the path derivatives are considerably less expensive 

to generate than the buckling moies 

loading or zero end displacement, you need to decompose only the linear global 

stiffness matrix. 

Ne did not try the buckling modes in this particular problem but I tried 

Not to 

because if you generate them at zero 

Questions and Answers following: 

Element Method to Global/Local Problems' by 8. A. Szabo. 

'Application of the p-version of the Finite 

Carlos A. Felippa, Lockheed Palo Alto Research Laboratory: 

depends essential ly on the assumption that the elements are conforming. 

I think your theory 

- Szabo: Yes. 

187 
Fel ippa: 

And if you use nonconforming elements, a funny thing can happen. 

the mesh and get a worse solution 

deteriorate. 

el ements? 

Now, most production computer programs do not use conforming elements. 

You can refine 

because the aspect ratio of the elements may 

How would you extend the theory to programs that use nonconforming 

Szabo: 

that the solution will converge in a certain norm. There are several methods, 

such as methods involving nonconforming elements, which do not meet with the 

Babuska-Brezzi criterion which guarantees the desired convergence properties. 

This criterion is difficult for engineers to understand because it is based on 

mathematical considerations. There are several methods in existence (methods 

that do not meet the Babuska-Brezzi criterion) for which one could select a set 

of input data and the methods would work well. 

the mesh, select the input data, etc., and the method would work. And somebody 

else could select a different set of input data, and because the rnethod does not 

satisfy the Babuska-Brezzi condition , it may "blow up". Now it frequently 

happens that a new method is shown to work on smooth problems, typically textbook problems. And the researcher says, "Look, my method works because it can 

solve some specific problem like the simply supported square plate." 

numerical method should be able to do this. The real difficulty is when these 

methods are applied to nonsmooth problems which are the important problems in 

engineering. Then they may not work; they may give a completely nonsensical 

result. For this reason, I choose to stay away from those methods. Instead, I 

use a method that is based on the principle of virtual work because I know this 

method satisfies the Babuska-Brezzi condition. Nonconforming methods, which do 

not satisfy those conditions, may give you unpleasant surprises. 

you may be doing more work and getting worse results. On the other hand, if you 

use a method that satisfies the Babuska-Brezzi condition and if you do more 

work, (for example, use finer meshes) the analysis will give you better results. 

To me, a very important consideration is that there is an existing proof 

In other words, one could design 

Any 

For example, 

Stroud: Because of Carlos' comments and some comments that you just made, do 

you recommend that conforming elements be used exclusively? 

Szabo: Yes, until proofs are developed that the other methods are robust--in 

the sense that no matter what (admissible) data you select, the method should 

work. And this proof is not available for the nonconforming elements, as far 

188 
as I know. 

something like that), but I have talked with mathematicians about this and they 

have the view that it is dangerous to use methods which are not meeting that 

criterion. 

Now it could be that somebody has just worked it out yesterday (or 

Reddy: 

a sufficient condition. And I think that there are some elements which are 

proven to work even if they do not satisfy the LBB condition. 

should not just throw out everything that will not pass the LBB conditon 

because it is only a sufficient condition. 

I think that the LBB condition is not really a necessary condition, itJs 

I think that we 

Not necessary. 

Szabo: IJve heard that argument. IJm not a mathematician; I've only picked up 

infomation on this topic by osmosis. But I believe that it is a necessary 

condition. Basically, the argument is the following. If your method does not 

meet the Babuska-Brezzi condition, then one can always select input data (maybe 

not the kind of data that you are interested in) for which the method will not 

work, and for which you were not able to say ahead of time, "Please do not make 

my input data like this or that," because it is very difficult to predict which 

set of input data will cause the method to fail. 

issues that relate to finer mathematical points; I'm not a mathematician. 

But please donJt press me on 

Reddy: 

LBB condition and if it fails then they should not give up on the method. 

I'm only trying to let the other people here know that if they check the 

The 

same th 

the o rem 

may not 

general 

Szabo: 

method , 

ng about Lax-Milgram theorem for existence. 

It can be shown that a problem has a solution but 

meet the same conditions as stated in the Lax-Milgram theorem or Babuska 

zed theorem. 

It is a sufficiency 

but not necessary. 

I just would like to invert your question. 

for example the method based on the principle of virtual work, why would 

If you know you have a good 

YOU wish to select a different method which may give you unpleasant surprises? 

In other words, since we do have good methods that we know are working, what 

WOUl d be the reason for using methods--nonconfomi ng methods, for exampl e-- that 

have not given satisfactory solutions at various times? 

Reddy: 

work principle. 

problems where we think stress is an important quantity and that should be known 

Well , very rarely do we use methods other than those based on virtual 

For example, mixed-type finite elements are used in certain 

189 
on the boundary of the element, like hybrid methods and 

are the cases where you need to use mixed methods which 

mixed methods. Those 

are not based on the 

virtual work principles--I should say traditional virtual work principles 

because the mixed principles are also some kind of virtual work principles. 

Szabo: I'm not against any particular method--or for it--all I am saying is 

that I would like to see a set of objective performance measures, convergence 

characteristics in certain norms, proven and demonstrated, in order for me to be 

convinced I should use that method. Until that is available, I don't want to 

use it. And I have not seen really detailed performance studies on the methods 

that you have mentioned. 

haven't seen them and I'm quite satisfied with the performance of virtual work 

fonnul ations. 

It could be that they exist. All I'm saying is that I 

Reddy: I think that history shows it works the other way around. We, as 

engineers, design methods which we think indeed do work. Then mathematicians, 

or people interested in mathematical aspects, explore the question of whether it 

is really a right method. 

Szabo: I agree. If we.waited for the mathematicians, I think we would still be 

living in caves. But, on the other hand, a large part of this meeting is about 

concerns of the performance of the various methods, and that's what I wanted to 

address. 

other methods as well. I haven't so far seen those. 

I would love to see similar performance studies presented for the 

Panel Discus si on 

Bjorn F. Backman, Boeing Military Airplane Company: 

making a controversial statement. I think any product of engineering development in the methods field is an engineering tool. 

distinction here, initially, between scientific development and engineering 

development. 

engineering is involved in the rational process of making decisions in an 

environment of limited knowledge. 

of the two things. 

structual mechanics to either be a tool that directly applied to the design 

process or a tool that comes with a recipe that says this is how you use it in 

order to prepare the set of data to be used for the design process. 

190 

I would like to start by 

And I would 1 ike to make the 

I think science is involved in the pursuit of the truth, while 

To that extent, I would have liked to see one 

I would have liked the end product of computational 

That's from 
one side. On the other side I see if we want to--in an academic environment--to 

pursue the scientific side of engineering to the exclusion of design 

considerations, then I think NASA has an even more important role in making the 

translation into the tools that can be applied to the design process. 

you. 

Thank 

W. Jefferson Stroud, NASA Langley Research Center: 

sial statement. I want to make sure I understand what you said, Bjorn. YOU 

feel that the CSM activity should produce products. I will even go further and 

say software products. 

Yes, that was a controverBackman: Yes. 

Stroud: 

ware would be a shortfall? 

Would you say that merely developing the know-how to produce the softBackman: 

principle and solving problems. If you look at any solution technique that's 

going to require you 3 months to come up with a number. If you want to apply 

that technique to the design process where the question is not what is the 

response, but the question is, what is a good design--(For instance, for this 

curved panel that we have been studying, where is the optimum location of the 

hole, or what is be best pad-up around the hole, or what is the sensitivity of 

the design to realistic boundary conditions?)--then you see 3 months very 

quickly translates into at least a hundred times that long in order to solve the 

design problem. So I'm implying that if you don't ask yourself the question, 

what am I going to use this technique for, or you say this is the way you can 

use it in the design process, you have only solved the principle not the 

probl em. 

We1 1 , it reminds me of the difference between solving problerns in 

Gary M. Stanley, Lockheed Palo Alto Research Laboratory: 

comment. 

methods development is nontrivial and that we should be pursuing methods 

imp1 ementation issues simultaneously with methods development--rather than 

waiting till later. 

I'd just like to 

I think that what he's saying is that the implementational aspect Of 

David Herting, MacNeal-Schwendler Corporation: 

left field here. 

Wed re sort of coming in from 

Let me cut in, because Universities and NASA have been 

19 1 
dancing this nice little dance, and we are involved in this 

We (at MSC) are involved in this research ourselves, and we 

developing similar code. We are now out in the business of 

development ourselves, in effect, also doing what NASAJs do 

very interesting to us. However, IJd probably like to make 

comments. 

in several aspects. 

are also involved in 

paying for code 

ng. So this is all 

a couple of 

One thing IJm glad to see is NASA is getting out of the role of trying to 

develop massive computer codes. 

the research mode itJs going to be much more valuable to us, and much less 

duplication. Also, I'm glad to see that you have practical applications. I'm 

glad to see people looking at the realistic problems and developing techniques 

to solve them because we are also looking at the same problems, particularly 

tires and cracks. 

I think that's a good trend. If you go into 

We are also in the development mode. 

However, I'm disturbed about a few things. 

morning, and maybe I should bring up a couple of points that should stimulate 

some discussion. 

commercial codes. I speak for Swanson, I speak for Hibbit. A lot of the 

complaints that were coming up about the codes are not true. 

capabil i ties exist. 

implicitly in NASTRAN, or MSC NASTRAN anyway. 

capabil i ties. 

I've kept quiet here during the 

One of the things is the lack of knowledge about the 

A lot of these 

The capability to do the local ized nonlinear analysis is 

I'm sure ABAQUS has similar 

This brings up the previous point of the ivory tower aspect. 

you are in an ivory tower if you're not aware of whatJs in the codes. 

it should be a part of the academic world to look at commercial codes. We 

certainly look at the academic world as far as looking at what the research is 

and what the current applications and hot methods are. 

more equally. 

It appears that 

I think 

Let's do it a little 

A couple of other comments: 

develop new methods, new codes. 

done? 

away and hides. We can't seem to find it. 

COSMIC says, "NO, that's our property," and we can't use it. 

devel op a1 1 these nice el ement routi nes--sol vers, nonl inear methods--how do we 

Some of these research projects are going to 

What's going to happen to these after they' re 

Sometimes we find it at COSMIC and 

There seems to be a lot of development at NASA that just sort of goes 

If you're going to 

192 
get those out to the public? 

some restrictions; we'd be glad to work around them. 

WeJ d certainly 1 ike to use them. There may be 

Stroud: You said a couple of things that I didn't quite understand. Although 

COSMIC charges for the software, I thought you could get any software that you 

wanted. 

Herting: 

distribute it out to everybody in the world. 

Well , the question occurs because we'd 1 ike to put it in NASTRAN and 

Stroud: Worldwide distribution of COSfIIC software is different. If software 

is developed with government funds, and if it's considered to the advantage of 

the United States to have the software available only to United States companies, then there is a restriction. But as far as COSMIC restricting software 

from anybody in the United States, I don't believe that's the case. 

Herting: I think the problem is, do we go out and require every customer to pay for a COSMIC code that is 

incorporated in MSC/NASTRAN? Take a program like CONMIEJ that takes $200 or $300 

to purchase from COSMIC, our legal staff tells us that we have to go collect 

that $200 or $300 from every delivery we make of CONMIN and ship it off to 

COSMIC. 

to work out. 

We've had some troubles just for the United States. 

And it's really questions--details--like this that we haven't been able 

Stroud: 

at this workshop. 

these things." 

lot of things. 

existing commercial code could do all of the local/global stuff that we've done. 

But I think you agree that it would not be easy or routine. We want to begin 

now to look ahead at what we ought to be able to do 5 to 10 years from now. 

I can't address that question. Your other question was something that 

I was really concerned about when we were considering having software developers 

Would the software developers say, "We can already do all 

I think that a sharp person with a good existing code can do a 

I would not question that one bit. A sharp person with a good 

H. David Hibbitt, Hibbitt, Karlsson, and Sorenson, Inc.: I'd like to address 

the comment that the academics should look at commercial codes. I didn't come 

here to be looked at, I came here to look. 

a lot from academic work. 

for the guys out there who have real problems. 

We commercial code vendors can learn 

Our job is to try and package it to make it sensible 

I don't really think it's 

193 
terribly important that the academics look at us. I came here, in fact, to look 

at an issue that is becoming important and that I haven't heard discussed very 

much except in very general terms. And this is the issue of parallel machines. 

Langley, I think, is the place where you have the finite element machine. 

the other extreme, people at Cray tell me that the way to do parallel processing 

is to get the data down into a microlevel and do a lot of local parallelism. 

hope our code is going to have a long life. 

parallel architectures and we have to decide what to do. 

anybody here discussing the finite element machine; I haven't heard anybody 

addressing detailed issues of writing software on para1 1 el machi nes. Is there 

somebody here who can help me with that, and where is he, and could he address 

the question? 

On 

1 

So we really have to look at these 

Now I haven't heard 

Stroud: You are right, Dave. We are doing work in parallel processing. And 

there is no paper here today that is oriented specifically towards parallel 

processing. Tomorrow, Joe Padovan will talk about it. He's into tire modelling. We are hoping that within the next year or so we will be able to have a 

conference that will focus on parallel processing in structural analysis. 

trying to learn. 

replace the finite element machine for parallel processing research. 

called a FLEX/32 Multicomputer. 

VAX 11/750. And I hope that we will be able to address some questions that YOU 

raised. 

and audience right now. 

We're 

We're buying a new multiple processer computer that will 

It's 

Each one of the 20 processers is about like a 

Why don't you throw the question of parallel processing to the panel 

Hibbitt: I think that the point that I was getting to, Jeff, if I could just 

continue for a minute is it seems that the philosophy you've take.n here at 

Langley is to go into parallelism at the macro level , putting one finite element 

on one processor, while if you talk to people, say at Cray Research, the 

impression I get is that they want to do it all on a micro level. There are 

some very successful finite element applications that have taken advantage of 

this approach. 

this. 

right and a wrong way, or are both ways successful? 

I think Jerry Goudreau is here, and I think he knows a lot about 

And I was hoping that he would give us some good advice here. Is there a 

Stroud: Let me comment on what our group has been doing in parallel processing. 

First of all, the Finite Element Machine started out, in the mid-to-late 1970's, 

with the concept of a microprocessor for each finite element. By the beginning 

i.9L 
of the ~O'S, we decided that one processor for each finite element was not the 

way to go. 

about a nonth ago. 

clear that we here at Langley could not develop a computer having several 

hundred processors. 

algorithms that could exploit multiple processors. 

at 1 inear simultaneous equation sol vers--both iterative and direct solution 

techniques. 

eigenvalue shifts. 

processors and let each processor get the eigenvalues near its shift parameter. 

you could use some logic to insure that you've gotten all the eigenvalues. A 

third example is substructuring, in which you divide the structure into pieces, 

assign those pieces to different processors, let each processor calculate the 

response of the substructure assigned to it, and then work the interface 

problem. In each case, computational speedups can occur if several processors 

are working on their assigned tasks simultaneously. But, the approach that 

we've totally abandoned is the idea of a microprocessor per finite element. 

We just got 16 processors operational on the Finite Element Machine 

Granted it was not a high priority research item, but it was 

So we and several university grantees have been looking at 

For example, we are looking 

Another example is eigenvalue solution techniques in which you have 

You could assign a value of the shift to each of the 

Gerald Goudreau, Lawrence Livermore National Laboratory: 

moment, not to preempt the panel, but to respond to Dave a little bit about our 

experience in a Cray environment with primarily the pipeline concept of the 

Cray. We also have implicit codes that concern themselves with optimization Of 

profile equation solvers and eigen packages, but a large portion of our finite 

element applications is in dynamic impact using explicit finite element hydra 

codes which don't have any matrices to deal with whatsoever. And so without 

linear algebra to exploit, what are you going to do to take advantage of the 

Cray environment? And the thing that we hit on, and this is John Hallquist, and 

primarily in the DYMA code applications, is that a big chunk of the effort, in 

fact 95 percent of the effort, is on what you would call the right hand side 

evaluation, you know, f of x and t, which is really f of the stress stateIt's the element stress divergence in solid mechanics, and there are comparable 

things (B-transpose sigma) in any of the structual elements as well. And rather 

than try to vectorize the element, meaning that the loop over whatever degrees 

of freedom are within an element, we used element chunking. 

chunking, first you scalerize every aspect of tne algorithm within the element 

and then, in the innermost do-loop, you tightly code a loop over a large number 

of elements. Idow some of the vector machines require very large elements, and 

if you turn all the scaler variables that are in the element logic into vectors 

Let me just take a 

In element 

195 
Over a set of elements, you may have a horrendous amount of storage required to 

get long enough vectors. 

memory that the Crays have provided, we've been able to chunk at that level very 

very efficiently. 

strain computation--they're just about at the limit of what we think we can get 

out of it--the Cray environment that we live in. This does present a problem 

with constitutive models, or equations of state because every time a new one 

comes along it immediately slows down the code by a factor of 2 to 10. 

only after it's proven its worth that we then get in and vectorize or optimize 

But with both 64 and 128 vector lengths and with the 

And I think the B-transpose sigma part that--and also the 

It's 

that element 

Now let 

ing the pipe 

ing us more, 

been itching 

the XMP is n 

me say a word about parallel machines which is sort of concatenatine to do more. The Cray XMP and future Cray machines are promisand we have John Hallquist's protege here, Dave Benson, who has 

to multitask the XMP. A problem we have at Livermore is that when 

t in the hands of one research group, but in the hands of a laboratory with 500 to 1000 equally important researchers, what you find is that the 

multitasking priority is not for individual code, but for the operating system 

as a whole. And so at Livermore the operating system is trying to multitask all 

these users trying to simultaneously do their work. 

we have access to is a single processor and we haven't gotten advantage of that. 

Now I was up at White Oak, the Naval Weapons Center, yesterday and found out 

that ABAQUS is being explored by CSPI, a small array processing company up in 

Massachusetts, and we're getting one too I guess, not in my group, but another 

group at the Laboratory. So I would be interested in talking further with YOU 

And so, at the moment, all 

-- 

about--maybe at a smaller level than at a Cray-looking at 

processing ideas and how we could proceed and then maybe 

machines down the road. 

Moktar Salama, Jet Propulsion Laboratory: I was also hop 

speakers more about how their problems fit into the broad 

some of these array 

nfluence the bigger 

ng to hear from the 

objective of the CSM 

activity, which as stated here, is to develop advanced structural analysis technology that will exploit modern and emerging computers. I was hoping that the 

speakers would give us an idea of the amount of computing that their problems 

take. Because if the problems we're talking about here require little effort, 

little computing, then today's computers are adequate. 

deal of computational intensity, a large amount of data, or unless the kind of 

problem that's being solved requires a real time solution then these problems 

Unless there is a great 

196 
don't fit into the category of problems that need modern computing beyond what 

we have today. 

Barna A. Szabo, Washington University: I had occasion to speak to many aerospace engineers in different companies. Their general opinion is that as the 

1990's come along they foresee the use of 3-dimensional finite element technology in airframe design. The wing skins are getting thicker and more complicated, and new materials are being used. There is no doubt in their minds that 

the aerospace industry will be directed toward very heavy computing, very large 

computational loads, very complicated geometries, many el ements, new material S, 

singularities, damage tolerance requirements, and the like. As we look to the 

1990'~~ I think it is clear that parallel processing is going to be increasingly 

important. 

addressi ng--where we can obtain high accuracy with much fewer degrees of freedom 

by properly designing the finite element spaces. There is a real need out there 

for us to look at how the airframes for the 1990's will be designed. 

such systems are not accessible to us at the university. We're very happy to 

find time on the VAX 11/750. Even there I have to twist the dean's arm. I 

would certainly be very interested in exploring different computer systems to 

find out exactly how we should design algorithms for those systems. 

have access to parallel systems to experiment with them--I don't think we can 

really answer your questions and Dr. Hibbitt's questions, which are very important questions. It's not that we're not interested in parallel processing. 

It's just that we first have to get access to them. 

And I believe they are very interested in the technoloqy I was 

On the other hand, I cannot really speak about parallel systems because 

Unless we 

Sal ama: We1 1 , I 'm real ly not saying that the problems you discussed do not 

require heavy computing. I'm not saying that. 

brought out, that's all. 

I'm saying that this fact was not 

Stroud: 

going to make our Flex mu1 ti processor computer avail ab1 e to the universities 

that we are working with. 

able to have a handle on several structural analysis methods that look good on 

mu1 tiple processor computers. 

The CSM group is going to be having grants with universities. We are 

We hope that 12 to 18 months from now we ought to be 

Richard B. Nelson, U. C. L. A.: I want to second the comment that Professor 

Szabo made, and that is that UCLA recently got some new computing equipment 

197 
which did involve an array processor. And now we are all looking around, saying 

to ourselves, now that we have it what are we going to do. 

mother's milk of research and for us to go out and explore array processing in a 

big way we need to have support for a relatively intensive program. 

students, one of the things we do is invest a great deal of time in having a 

student generally write, almost from scratch, a finite element code, or pick UP 

SAP 4, which is by now quite dog-eared, or possibly NONSAP, one of the codes 

which we more or less have inherited from Berkeley more than 10 years ago. I'm 

speaking as a researcher in need of a kind of test bed that we across the 

country could pretty much rely on and plug our research efforts into so that 

people across the country could make use of our research product on a general 

basis. 

codes that are specifical ly designed for certain industrial appl ications. But 

rather that it gives people both in industry and in the academic cornunity the 

opportunity to communicate across technical lines, and let's face it, the 

language of comunication today is through computers and software. And we have 

to develop some kind of a liaison through channels of software and that's going 

to require significant support to develop. 

Ploney is the 

With Ph.0. 

I don't consider that to be particularly competitive with optimized 

E. Thomas Moyer, Jr., The George Washington University: 

comments, one on the hardware aspect of parallel processing. As the technology 

gets cheap enough, rather than making time available at one central location, 

it's becoming feasible for universities to buy parallel processing computers. 

For instance, with the new INTEL machines, you can pick up a parallel system 

with 32 processors in it for under $100,000. 

should buy the computers either through a NASA program or an NSF program in a 

wide way so that a lot of researchers will have access to them. As far as the 

comment that was made with regard to university professors not being aware Of 

finite element codes capabilities and that type of thing, I think that the 

universities should have a lot stronger access to the codes at either the code 

level or, at least, at the executable level at a price, if not nothing, then 

extremely negligible. 

commercial endeavor and to the research environment as a whole. And I think 

that that ought to be explored in more depth. 

I'd like to make two 

As the prices fall , universities 

And that I think would be productive both to the 

Stroud: 

universities at quite reduced rates. 

My bet is that commercial software companies do provide their codes to 

(Agreement from audience. 1 

198 
Backman: 

industry. I think it's quite clear that for the 199O!s we have seen a number of 

problems, many of which perhaps have not been very dominant before, that require 

both hardware and software improvements of orders of magnitude. You look at the 

optimization side, you look at the emerging technology in the damage tolerance 

fie1 ds that--in composites--requi re del ami nation growth under compression , and 

postbuckle states, and I see substantial development requirements. There is 

very little doubt that there are a number of applications producing problem 

sizes such that present software and hardware is not adequate. 

get into the large optimization problems that we need to be doing. 

need to gain additional first-hand hardware experience in order to truly 

understand and develop the requirements. 

I wanted to address the question about the needs of the aerospace 

We still can't 

We really 

Stroud: 

that if we are going to have good structural analysis in the future, we need to 

have better software architecture in order to exploit the new computers. 

We've touched on multiple processor computers. What about the idea 

Leonard A. Lopez, University of Illinois: 

serious look at the architecture of computer systems for finite element 

analysis. Or. Nelson talked about testbed systems. I am a strong believer in 

that approach. 

Illinois since 1975. The system uses a virtual machine concept and emphasizes 

flexibility, machine independence, easy addition of elements, and easy addition 

of nonlinear material models. 

individuals in our department in the early 1970!s, and this capability has 

permitted these researchers to try new ideas without needing to deal with all of 

the other problems associated with finite element techno1 ogy. 

The design of the POLO-FINITE testbed is based on 1970 software and hardware technology. The biggest problem we are facing today is that our expectations have grown significantly, and that the kinds of problems being envisioned 

by our faculty far exceed the capacity of the combined hardware/software testbed. 

parallel (mu1 ti-processor) computers in order to solve the next generation of 

probl ems. 

There would be many objectives for a multi-processor testbed system. 

Clearly one of them must be that the testbed be flexible enough to take into 

account existing multi-processing architectures, as well as architectures which 

have not yet been conceived. 

I think we do need to take a very 

We have been running a testbed system at the University of 

Basically these were areas of interest to 

I think we now must proceed to investigate and exploit the architecture Of 

It will take years to develop such a system and 

199 
put it into production. The corresponding hardware technology is changing very 

rapidly, and it will be necessary to be able to move the testbed to various new 

architectures as they are designed and implemented. 

idea about how fast things are changing, I would like to again point to our 

experience at the University of Illinois. Six months ago we were confined 

primarily to uti1 izing a Cyberll75 computer. Today we have two national centers 

for supercomputers--one based on mu1 tiple Cray XMP!s and the other based on 

the CEDAR machine. The CEDAR machine is a research machine based on ALLIANT 

computers (today). 

can be used as a multi-processor network; it is also viable for finite element 

calculations. All of these additions in hardware took place in six months. 

Additional changes will occur within the next one to two years. 

We, as individuals working in finite element software technology, must be 

careful not to box ourselves in, or exclude the kinds of things which will be 

happening in the near future. I think we can accomplish this objective by 

seriously studying and redesigning the architecture of our computer systems, and 

developing new testbeds that are flexible enough to meet the challenges of the 

future. 

In order to give you an 

We a1 so have a network of Apol lo super-micro-computers which 

K. C. Park, Lockheed Palo Alto Research Laboratory: You are opening a box of 

worms. Dr. Bahram Nour-Omid and I have been exploring the Cal . Tech Hypercube 

for the last 6 months. It took us 2-1/2 months to successfully run a 

one-dimensional wave propagation problem. And then the word I just got from him 

on the plane was, after struggling about 2 to 2-1/2 months, we just have been 

able to run a simple 2-D plate bending problem. What this means is that it is 

very difficult to implement software on parallel machines, at the moment, mainly 

because the supporting system utilities aren't there. 

about 3 to 5 years before we engineers can--in a routine manner--use a parallel 

machine, either a FLEX machine or a Hypercube, or the like. Right now, for 

example, the FLEX people insist that they have a FORTRAN compiler. But when we 

looked at it, it really doesn't have a good solid FORTRAN cross compiler and 

neither does the Cal . Tech Hypercube. 

over again using the C language. 

rewriting sophisticated finite element programs into the C language. So I think 

we have 3 to 5 years before we can start to see the routine uses of the parallel 

machines. 

And it's going to take 

What this means is we have to start all 

And imagine the conversion labor involved in 

200 
The other comment that I would like to make is that parallel machines are not 

cheap. What that means is that only wealthy private companies, such as some of 

the profi t-maki ng finite element analysis companies, 1 i ke Marc , or 

MacNeal-Schwendler, may be able to afford parallel machines. But in the 

meantime we want to minimize the time it takes to move from university research 

products to practical applications--say on a parallel machine. Then the cost of 

the machine has got to go down, or somebody should provide the funds. Otherwise 

you will not see a wide dissemination of parallel machines. 

Stroud: 

it's the lack of software. 

compiler--and that's a problem. 

before we are routinely using the Flex computer. I think that we will be able 

to begin talking about structural analysis on parallel machines in 18 months. 

Today, it's not so much the cost of the multiple processor machines, 

That's right, FLEX might not even have a FORTRAN 

But, I certainly hope it won't be 5 years 

Vincent Godino, Bolt, Beranek, and Newman: I would like to say a few words 

about our experience in parallel processing. Under DARPA sponsorship, BBN has 

developed a para1 1 el processor call ed the Butterfly. 

currently operational . The Structural Mechanics department, which I head, has 

been developing structural mechanics software on this machine including matrix 

operations, Gauss e l i m i n a t i o n , etc. 

particular machine, I must take issue with some of K. C. Park's comments. In 

terms of ease of implementation, our scientific programmers, with no previous 

experience on any parallel processors, were able to program several extremely 

effective algorithms in a short time and with minimum help from people 

experienced on the machine. 

through the ARPA net. 

thousands of nodes are currently under development. 

agree with Jeff Stroud in that I think that we!re a lot closer than five years 

before these machines are going to be real tools being used to solve. structural 

mechanics problems. 

We have a 128-node machi ne 

Based upon our experience on this 

The Butterfly is available to DARPA contractors 

I guess I would like to 

Next generation p a r a l l e l processors consisting of 

Jimmy Y. L. Ho, Lockheed, Sunnyvale: 

properties and various other properties, such as joint behavior, are statistical 

quantities. They do not have a precise value. 

We need to realize that the material 

201 
- Szabo: 

equations. The loading, the material properties, the thicknesses, and so on are 

really stochastic variables. 

just beginning to address this fact. 

question because of the size of practical problems. We cannot perform so many 

different analyses with different input data. 

certain bounding characteristics of the data and ref1 ect those characteristics 

in our answers in an honest way. 

very important topic for NASA to take up as a research study--how to be more 

realistic in the way we treat computed data. 

rather, it is a stochastic variable. Understanding the bounds within which the 

answer is valid is very important. Delivering a number doesn't mean very much 

unless I'm able to state the confidence level I can associate with that number. 

The realities are that we do indeed deal with stochastic differential 

I'm aware of some research in this area that's 

Monte Carlo methods are out of the 

But it is possible to study 

And the point raised here could really be a 

The number is not 10.75, but, 

Robert Melosh, Duke University: 

that's probably the basis for the conflict. 

to be responsive to the demands of their employers and turn out good analyses, 

and right now, as I see it, the analyst has a very heroic job to turn out 

analyses that he can depend on. There is another group here represented by 

program developers who are responsive to the needs of the analysts. 

to supply programs to help the analyst do his job, but they don't have the final 

responsibility for the analysis. 

group you represent, Jeff, the Langley group who want to do relevant research to 

help the first two groups. The fourth group incl udes the university people who 

want to do relevant research to help the first three groups. 

want to center on what you should do as a group, you have to decide who you want 

to satisfy in tens of setting your priorities. 

analysts at that end of the extreme or the researcher at the other end of the 

extreme? You have to settle on that in order to settle on what your objectives 

will be for research. 

I see four different groups of people here, and 

I see a group of analysts who want 

They want 

The engineer does. The third group is the 

I think if you 

Do you want to satisfy the 

Bahram Nour-Omid, Lockheed Palo Alto Laboratory: There are a couple of issues 

that have come up in my work on the parallel processor--in particular, the one 

down in Cal Tech. 

sors. 

205 computers. 

First of all I'd like to distinguish between parallel procesThe one based on single instruction multi-data, which are Cray and Cyber 

And the next generation computers which are going to be multi202 
instruction, mu1 ti-data-stream computers. 

of different architectures, some of them are based on message passing between 

processors, and others are based on a shared memory environment. It is quite 

possible to write programs--code--and get them to work on these computers. 

whether those codes are going to be efficient is a completely different story. 

One can write a Gauss elimination procedure, probably in a matter of months. 

However, it turns out that on concurrent processors, which are the multiinstruction, multi-data computers, the time taken for a message to go from one 

processor to another is equivalent to between 10 and 100 multiplies. 

one has to look at algorithms that utilize the given architecture of the computer. It is pointless to look at algorithms that require a lot of interprocessor communication. 

Particular, the most promising ones that we have seen are iterative methods and 

the type which are based on partitioning of a structure into substructures. 

Now these computers come in a variety 

But 

Therefore, 

One has to look at algorithms which don't do that. In 

That brings up another problem which I think is quite unsolved and that is the 

mapping of a finite element mesh onto a set of processors. People have done a 

lot of work on partitioning meshes, regular meshes--and mapping them on the 

concurrent processors. But once the mesh gets quite general, such as the ones 

we have seen--finite element mesh of an aircraft--then I think it's very hard to 

say how to automate the process of mapping those meshes on the computer. 

Joop Nagtegaal, MARC Analysis Research Corp.: I would like to make two brief 

comments on two very different subjects. One is the parallel processing. I 

believe there are a lot of algorithms already around which lend themselves to 

parallel izing computations. I know of Gaussian elimination equation solvers 

which can be para1 1 el ized by sel fcontinuous substructuring technique. Certainly 

you can do all element calculations in parallel. What I see is missing at the 

moment is operating software on these multiprocessor machines which we can 

program such that they work. 

But I see that things are changing. 

things going on. 

uncertainty business, stochastic analysis. I'd like to add that NASA is sponsoring such a project. It's a different branch of NASA, but I'd like to think 

of NASA as one big organization. And there are certinly some promising things 

going on. 

That's what I see currently as the bottleneck. 

The second comment on a whole different subject is on this 

So, a l i t t l e b i t of time and I see a lot of 

And I would be happy to talk about that with anyone. 

203 
Stroud: NASA Lewis is doing that work. 

Nagtegaal: And there are several people from Lewis at this workshop. 

Stroud: We're a1 1 one good team. 

Nagtegaal: That's what I like to think, yes, thank you. 

Stanley: 

from both the panel and the audience--that people seem to view this idea of 

global/local analysis as an interim measure pending the full-scale uti1 ization 

of parallel processing--that as soon as computers become fast enough, there 

won't be a need to use global functions and exact solutions. We'll just go in 

there with brute force with 3-D finite element analysis. 

that's the case, because if that is the feeling then perhaps we should retrain 

in midstream here. Maybe that's a rhetorical question. On the other hand, if 

we do continue with these global/lccal methodologies, I would like to see them 

become more problem oriented, I'd like to see them continue in this manner, I'd 

1 i ke these benchmark probl ems-- the NASA problems-- to be di stri buted among people 

that are collaborating in this area. And I would like to then compare the 

methods used to solve these problems based on a number of criteria, such as 

(1) reliability; (2) computational effort, which would include the analyst's 

effort as well ; (3) implementational effort, that's important because retreading 

software can be a huge cost and, finally, (4) generality. If worse comes to 

worst, we will have to build special purpose codes as they do in the fluids 

business. But, optimally, it would be nice to hang on to the general purpose 

benefits we get from the finite element method by using such techniques as 

reduced basis global functions, based on finite element basis functions, in the 

first place, so that we can automate this global/local ization. 

I'm getting the impression now from listening to all the comments-- 

I'm wondering if 

Ahmed K. Noor, George Washington University: 

regard to the question about uncertainty and confidence in the results. 

of doing a stochastic analysis I think we can get some confidence in the results 

by carrying out a sensitivity analysis. 

respect to the input parameters, for example. 

dence in the numerical results. The second comment deals with parallel processing. Parallel processing is a system issue. Unfortunately, much of the work 

that is being done has addressed certain aspects of the analysis process. 

Just two comments. One with 

Short 

Sensitivity of the response with 

This would give us some confiIn 

204 
particular, the numerical a1 gorithms for hand1 ing the a1 gebraic equations. 

is very important, of course, and computer time spent in algorithms is dominant 

in present-day finite element analysis, but it might not be enough to address 

that alone if we want a very efficient solution procedure. 

address the question from a higher level starting with the formulation, looking 

at the different phases of the analyses, and, as I said in'my presentation, that 

we might have to move away from some of the traditional finite element analysis 

processes 1 ike, for example, bypassing the assembly process. 

difficult part of the solution process for vectorization or parallel ization. 

So, I think we have to look more into the overall computational strategy rather 

than just the numeri cal a1 gori thm. 

This 

I think we have to 

This is the most 

Nelson: 

although we saw a lot of pictures. 

of an old Tinius Olsen testing motto which says that one test is worth a thouThis comment has to do with something we haven't talked a lot about, 

That was the idea of testing. I'm reminded 

sand expert opinions. 

tool. Which means that the finite element software has to be configured to be 

flexible. 

panel, in which we begin to see failure such as delamination and things of this 

nature, we want to be able to incorporate in the software new material models or 

new failure laws for composites. So then we can have tight feedback between 

experiment and analysis. We will all benefit from having codes that are not 

just bigger--the codes are just a little bit smarter, because we are. 

We want finite element analysis to be a true engineering 

If, for example, we start looking at such things as a NASA test 

J. N. Reddy, Virginia Polytechnic Institute: 

response to the first comment made by Dr. Backman. Most of my colleagues from 

universities, I hope, share at least the spirit of what I'm going to say. 

think most of the students that are coming out of the universities, I am disappointed to say, are taught the recipe-type approach, formula plugging and 

checking compared to my older colleagues who have a lot of ingenuity and physics 

background. 

current students. And I don't want to use the same kind of approach as proposed 

Or suggested by Dr. Backman. 

number crunching and formula plugging. I think if they have an understanding Of 

the basic physics and if they have certain tools, when they get out of school I 

think they can be trained on the job. I'll give an example of my own. After 1 

graduated, I took a job with Lockheed Huntsville Space Company. 

that job, I only worked with Hilbert spaces and Sobolev spaces, and also, of 

I want to make one comment in 

I 

My older colleagues are better at modeling than, I think, the 

I want the students to know more physics than 

Before taking 

205 
course, I learned a great deal from Professor Oden and other people in terms of 

physics and mathematical modeling. At my new job, I was asked to do a 3-D 

analysis of impact, hydrodynamic approximation. One week after I got into the 

job, I had a job offer to go to the University of Oklahoma, but I decided to 

Stay and complete the project given to me. And I did complete it in 6 months. 

The point is, if you have the right background in terms of'the mathematics and 

physics, I think you can learn number crunching very easily. But if you train 

yourself with only number crunching, you may not get the physical understanding. 

So I think we should be very careful in training our students. 

S troud: 

particu 

ware. 

Backman 

My bet is that Bjorn wasn't suggesting that students be trained a 

ar way. Rather, he wanted the product of our CSM efforts to be softYes. I'm not sure what I said that caused this comment. It's possible that I hinted that the level of so-called blackbox engineering has to be 

increased. 

experts has reached a level such that we simply can't replace them fast enough 

in the production field. And what I'm trying to say is, of all these students 

that are coming up with the proper background in science and structural 

mechanics and so on, how many do you expect to be proficient in the operating 

systems that exist on the machines they are using? How many do you expect to 

know the details of the compilers we use on the system? How many of them do you 

expect to understand, in detail , the equation solvers? And how many do you 

expect to understand the inner workings of the optimizers? Somewhere in here we 

are going to have to stop asking these people to become renaissance 

personalities. 

like finite element systems ultimately fit into the same category as operating 

systens. 

design problems. 

sciences. I'd rather see the industry taking on some additional educational 

responsibility in this area. 

to require 10 years or more of higher education or you have to identify how 

deeply into the problem you want to go. 

that Dr. Szabo has been advocating is probably a first step towards the ability 

to heuristically understand the result of your analysis without knowing the guts 

of each step of the solution. 

I think that we are slowly reaching the point where proliferation of 

I'm just trying to achieve some balance here where maybe things 

And maybe the engineering community will be dealing with the actual 

I'm, by all means, in favor of a solid background in the 

The point is that sooner or later you either have 

And, I think, a method like this one 

And I'll stop with that. 

206 
ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

I Stroud: Thank you. 

have not resolved them. And opening them up is certainly one of the objectives 

of this workshop. 

attempt to resolve some of these questions. I want to thank you all for coming 

and participating. In particular, I want to thank the speakers. 

It's clear that we have opened up a lot of topics and we 

There will be many more workshops and conferences that will 

*Original color slide shown here in black and white. 

207 
TIRE MODELING 

PRECEDING PAGE BLANK NOT FILMED 

209 
N89-24649 

FEATURES AND CHARACTERIZATION NEEDS OF 

RUBBER COMPOSITE STRUCTURES 

Farhad Tabaddor 

Goodrich Company 

Akron, Ohio 

Abstract 

This paper outlines some of the major unique features of rubber 

composite structures. The features covered are those related to 

the material properties, but the analytical features are also 

briefly discussed. It is essential to recognize these features 

at the planning stage of any long-range analytical, experimental, 

or application program. 

The development of a general and comprehensive program which fully 

accounts for all the important characteristics of tires, under 

all the relevant modes of operation, may present a prohibitively 

expensive and impractical task at the near future. There is therefore 

a need to develop application methodologies which can utilize 

the less general models, beyond their theoretical limitations and 

yet with reasonable reliability, by proper mix of analytical, 

experimental, and testing activities. 

OUTLINE: 

1. CORDS 

2. RUBBER 

3. RUBBER COMPOSITES 

Single Ply 

Laminate 

4. ANALYTICAL AND 

COMPUTATIONAL ASPECTS 

PRECEDING PAGE BLANK NOT FILMED 

211 
INTRODUCTION 

The textile-cord-reinforced-rubber composites used in various 

industrial products differ in many respects from the classical 

rigid composites. There are a great number of papers, books, and 

professional journals devoted to rigid composites. 

on rubber composites, however, is very limited as compared to 

classical composites. 

it is essential to develop an in-depth understanding of the way in 

which the internal variables of the constituents participate and 

interact in responding to external agents, i.e., mechanical, 

thermal or other environmental forces. It is, however, important 

to first study the properties of each constituent before dealing 

with the material properties of rubber composites. 

The literature 

To understand the mechanics of such composites, 

CORDS 

The textile cord reinforcements are structural members which may be 

viewed as one dimensional in a microscopic sense. The cords 

generally consist of several yarns twisted together. The yarns 

also consist of numerous filament components organized in a 

geometrical array with the view towards enhancing certain target 

properties. 

of the filaments and the geometrical organization as well as the 

interfacial characteristics. The filament itself has also been known 

to possess an internal structure at the microscopic level. 

The cord properties therefore depend on the properties 

In view of the above consideration, the cord itself is a complex 

structure which may be studied at the microscopic level (figs. 1 

and 2). The way in which the cord properties are related to 

filament properties, geometries (ref. 1) and other variables is 

the subject of textile mechanics (fig. 3). Properties such as 

strength, stiffness, and fatigue characteristics can be controlled by internal variables when their relationships are well 

understood. 

From a higher scale of continuum mechanics we are, however, 

concerned only with the phenomenological properties of the cords as 

experimentally obtained without any attempt for relating such 

212 
measured values to the micro-structure in the sense of textile 

mechanics. The properties of interest at this continuum level are 

therefore the effective cord properties. 

Such an approach enables us to bypass the complexity of the textile 

mechanics in our formulation of composite properties. The 

limitation is, however, that such measured properties serve as 

llaveragesll only and therefore the continuum elements should be at 

least in the order of cord diameters. 

Figure 1. Typical tire cord. 

213 
BLACK 

ORlC I N AL 

AND WHITE 

Figure 2. Cross section of tire cord (2). 

0 5 IO 15 20 25 

STRAIN 

(1) RAYON (5) NYLON-66 

(2) NOMEX (6) STEEL 

(7) GLASS (3) POLYESTER 

(4) NYLON-6 

TESTED AT 70°F.,65% R.H. 

INSTRON - 100% STRAIN RATE 

Figure 3. Stress-strain characteristics of typical tire cords (2). 

2 14 
The cord is the major load-bearing member of rubber composite 

structures and as such should provide strength and many other 

characteristics of interest. 

characteristics of,the tire, and for that matter any other 

structure, can be directly related to cord properties. 

consider some of the tire performance characteristics which are 

affected by cord properties. 

Some of the expected performance 

Let us 

A partial list is given in Table 1. 

Table 1 

TIRE PERFORMANCE AFFECTED BY CORDS 

o Burst Strength 

o Bruise Resistance 

o Tire Endurance (Separations) 

o Power Loss 

o Tread Wear 

o High-speed Endurance 

o Tire Size and Shape 

o Groove Cracking 

o Flat Spotting and Non-Uniformity 

o Tire Cornering Force 

o Tire Spring Rate 

o Noise 

215 
Tables 2 to 4 of reference (3) provide a list of tire 

requirements and the related cord requirements. 

should be viewed with caution and qualifications. 

Such relations 

Table 2 

(REF. 3) 

RADIAL PASSENGER 

Vehicle Trends 

High Performance 

Downs i z ing 

Front Wheel Drive 

Fuel Economy 

Tire Requirements 

Low Aspect Ratio 

(Increased Cornering' 

Forces 1 

Downsizing - Monoply 

Improved Tread Wear 

Retreading 

Rolling Resistance 

Cord Requirements 

Modulus 

Lateral Stiffness 

Dimensional Stability 

Fatigue 

Modulus 

Tenacity 

Dimensional Stability 

Lateral Stiffness 

Aged Adhesion 

Thermal Stability 

Fatigue 

Tenacity 

Modulus 

Hysteresis 

216 
Vehicle Trends 

Increased Drive 

Position Loads 

Improved Fuel 

Economy/New 

Transportation 

Act 

Tire Requirements 

High Deflection 

Reduced Weight 

Retread 

Table 3 

(REF. 3) 

RADIAL TRUCK 

Tire Requirements 

Increased Durability 

Lighter Weight 

Low Profile 

Retreading 

Rolling Resistance 

Table 4 

(REF. 3) 

RADIAL AIRCRAFT 

Cord Requirements 

Tenacity 

Strength/Area Ratio 

Tenacity 

Fatigue 

Aged Adhesion 

Fatigue 

Thermal Stability 

Tenacity 

Modulus 

Hysteresis 

Cord Requirements 

Tenacity 

Modulus 

Thermal Stability 

Tenacity 

Strength/Area Ratio 

Aged Adhesion 

Dimensional/Thermal Stability 

Fatigue 

217 
CORD VISCOELASTICITY 

The textile cords used in rubber-reinforced composites are often 

nonlinear viscoelastic. The viscoelastic deformations are associated 

with the loss of energy. 

leads to temperature rise which in turn affects the material properties. 

The cord is therefore an important contributor to the energy loss in 

rubber-composite structures such as tires (4). Figures 4 and 5 show 

some of the viscoelastic properties of polyester cords. 

The dissipated energy appears as heat and 

Figure 4. Carpet plot of 1000/2 polyester real modulus. 

I 218 
.! e 

.I 6 

.I 4 

.I 2 

TAN 6 

.IO 

.O 8 

.06 

.04 

7-1 

I 

I bo ” I 1000/2 POLYESTER 

STRAIN x 15 MAXIMUM 

.9 

Figure 5. Carpet plot of 1000/2 polyester loss modulus. 

219 
SUMMARY ON CORDS DISCUSSION 

o Cords are structures and as such their effective properties are 

geometry and boundary condition dependent. 

o The effective cord properties are different when cords are 

embedded in the rubber matrix. 

o The calculation of the three-dimensional effective properties 

by analytical homogenization is very complex and impractical. 

o The properties of some cords such as nylon are highly temperature sensitive, particularly around the glass transition 

temperature. 

o The properties are different in tension and compression. 

o Cords are often nonlinear and viscoelastic. 

220 
RUBBER 

The physical properties of rubber compounds depend on various 

processing parameters and components. Here again, we are not 

concerned as how these properties are related to the molecular 

structure of the rubber and the physics of rubber vulcanization. 

The focus is on determination of the relevant properties by proper 

experiment and within the framework of continuum physics. The most 

important property is the elasticity of the rubber, which is 

distinctly different from other conventional materials. The most 

distinctive features of rubber elasticity are the deformability and 

the rapid recovery of the deformations when loads are removed. 

Rubber remains elastic at extension ratios of several hundred 

percent. Such elastic characteristics make the rubber unique in 

this respect. In fact, the major developments in the theory of 

large elastic deformations evolved around application to rubber 

elasticity. The rubber elasticity is an important subject in 

understanding finite-element analysis of such products. 

RUBBER ELASTICITY 

To understand rubber elasticity, we may first examine the 

thermodynamics of reversible processes. The first and the second 

laws of thermodynamics state that (5) 

dE = Tds + dw 

where E is the internal energy, T is the absolute temperature, S 

is the entropy and w is the work done on the system. Experimental 

work has shown that the rubber elasticity resides basically in the 

entropy term. The rubber elasticity therefore has an entirely 

different molecular origin than other elastic materials whose 

elasticities are primarily associated with the increase in internal 

energy through changes in molecular or atomic spaces. 

221 
Much work has been carried out to formulate rubber elasticity. 

One example of the molecular approach is one which considers the 

molecular chain length having Gaussian distribution. 

parameters are calculated from such quantities as finite molecular 

length and molecular weight between crosslinks. 

resulting from Gaussian theory leads to 

The elasticity 

The entropy change 

in which N is t e number of network chains per unit volume 

andX1 12 and '3 are the three prjncipal extension ratios 

along the three mutually perpendicular axes of strain for pure 

homogeneous strains. 

The above equation provides a first order approximation to rubber 

elasticity but is not adequate over a large range of deformations. 

i 222 
PHENOMENOLOGICAL APPROACH 

An entirely different approach (6) is the phenomenological approach 

of continuum mechanics. In this approach, the existence of a strain 

energy function is postulated. It has been shown that such a strain 

energy function should depend on deformation gradients. The equations for isotropic incompressible elastic media are as follows: 

w = w (I1, 12, 13) 

I1 = x, 2 + x2 2 + 13 2

I2 = x 22 x + AllQ 22 + x2x3 22 12 

For the incompressible case 

13 = 1 

In series form 

223 
An alternative form, suggested in (7), is as follows 

w = t (A1) + t (A2> + t (A3> 

The stress-strain relations in the principal directions (the 

stresses are force per deformed area) are 

2 24 
The one-dimensional stress-strain relation can be obtained from 

the three-dimensional relations as shown in figure 6. 

0.4 

0.2 

Figure 6. One-dimensional stress-strain relation. 

225 
It can be seen that the tangent compressive moduli increase 

significantly as the compressive strain increases. 

realistically describes the rubber response in compression. 

finite-element treatment of rubber materials, the rubber is often 

modeled as linear, on the ground that the strains are small and 

that the linear constitutive law should then provide a reasonable 

approximation. This view is very disputable. It suffices to say 

that the compressive strains remain small due to stiffening of 

rubber in compression. This feature can only be handled properly 

by nonlinear constitutive laws. 

material stiffness remains unchanged and therefore moderate 

compressive forces produce high compressive strains that are not 

seen in the actual structure. Such exceedingly large strains result 

in distortion of elements which quickly leads to severe numerical 

problems. 

This feature 

In the 

In the linear model of rubber, the 

A difficult problem, in dealing with rubber elasticity, arises from 

the incompressibility constraint condition. 

condition leads to certain simplifications in the exact analysis of 

the problem, basically because of reduction in the number of 

unknown parameters. 

element approach. In the variational formulation, for example, the 

Lagrange multiplier introduces an additional unknown scalar 

function into the finite-element formulation. This unknown should 

be accommodated at the element level. The procedure results in a 

significant increase in the number of total unknowns and also may result in an ill-conditioned stiffness matrix. A great deal of 

research has been carried out to find the most suitable element for 

handling incompressibility. The subject is still open but the fact 

remains that the incompressibility imposes an additional burden in 

numerical analysis. The incompressibility condition, aside from 

being inconvenient in the finite-element analysis, is an 

approximation for rubber-like materials. 

becomes increasingly less accurate as the percentage of carbon 

black increases in the rubber compound. The exact enforcement 

of incompressibility is therefore not actually needed. 

The incompressibility 

Such is not, however, the case with the finiteSuch an approximation 

226 
NEARLY INCOMPRESSIBLE MATERIALS 

The strain energy function for nearly incompressible materials can 

be obtained by a series expansion of the strain energy function 

about (13-1), and retaining the leading terms of up to second order 

in (13-1) as follows 

w = cc cij (I1 - 3) i (I2 - 39 + w2 - (I3 - 1) + wg (I3 - 1)* + . . . ij 

One special case of the above equations is when the function W2 

and W3 are considered constants. The constants are described as 

follows: 

227 
The near incompressibility can now be enforced by assigning large 

values to H. In fact, as H approaches infinity, (13-L) approaches 

zero so that the strain energy remains finite. The higher the 

value of H, the closer the incompressibility would be satisfied. 

H is referred to as the penalty number. In finite-element analysis, 

however, the large values of H can lead to overriding stiffness 

which results in numerical problems. The penalty method nevertheless 

permits the satisfaction of near incompressibility without increasing 

the number of unknowns in finite-element formulation. Figure 7 shows 

a set of typical properties for the rubber. 

-.2 .. 

I 

/ 

8 

I 

I 

I 

I 

I 

I 

I 

C,,= 0472 (kg Imrn’) 

C0l = 0211 

C = - 00414 

C,,= - 000262 

c,o= 00112 

HI= -(c,o+zcol) 

H = 11.138 

Figure 7. Stress-strain relation, (8). 

228 
FRACTURE AND FATIGUE PROPERTIES 

The fatigue of the rubber has been the subject of many investigations 

in the past. 

of rubber fracture and cut growth. The fracture mechanics approach 

for rubber was first adopted by Rivlin and Thomas (9) and Thomas (10) 

who promoted the concept of the tear energy in describing the cut 

growth mechanism. The tear energy approach has been applied to the 

study of the crack growth problem and to the description of fatigue 

behavior of rubber. 

and this therefore excludes the fatigue caused by or resulting from 

non-mechanical sources such as aggressive environment. Some of the 

non-mechanical sources, such as ozone cracking in elastomers, may, 

however, be more damaging than mechanical sources. 

The'rubber fatigue is intimately related to the nature 

We only consider the mechanically induced fatigue 

Bussels early results (11) on NR compounds, shown in Table 5, clearly 

demonstrate the unique feature of rubber fatigue. 

Table 5 

EFFECT OF STRAIN CYCLE 

STRAIN CYCLE LIFE, MINUTE 

5 - 60% Extension 10 

25 II 10 - 60% 

15 - 60% II 90 

150+ II 17 - 60% 

20 - 60% 00 00 

Some experimental fatigue data on various compounds are shown in 

Figures 8 to 11 taken from References (12-14). 

229 
( 100) 70 LMIN - Lo 

%MINIMUM STRAIN = LO 

Figure 8. Fatigue life of NR as a function of minimum 

range (from ref. 12). 

strain 

0 9 

I o3 lo4 lo5 IO6 10' IOD IO 

NUMBER OF CYCLES TO BREAK 

Figure 9. Imposed extension versus number of cycles to break 

(from ref. 12). 

230 
DUMEELLS OF REVOLUTION 

75% TOTAL STRAIN CYCLE 

1 LIFE V5 % MINIMUM STRAIN 

- - NR - SER 

IO4 t 

I 1 

7. MINIUUN STRAIN 

Figure 10. Cycle of failure versus minimum strain 

for NR and SBR (from refs. 13 and 14). 

75HR 25EE 

IC0 NR 

25 h?-75SE 

100 SBR 

t 

I 

IO5' -:5 -3: 1/2 6 + 50 

% MINMUM STRAIN 

Figure 11. Cycle of failure versus minimum strain for 

various compositions (from refs. 13 and 14). 

231 
SUMMARY ON RUBBER 

o Rubber remains elastic at extension ratios of several hundred. 

o Rubber is almost incompressible. 

o The rubber elasticity can be most conveniently expressed 

through the strain energy function. 

o Rubber has unique fatigue properties. Extensive experimental 

work needs to be carried out for fatigue characterization of 

various rubber compounds. 

o Appropriate failure theories are needed for interpretation of 

stress data from finite-element analysis. 

o Properties are temperature dependent. 

232 
CORD REINFORCED RUBBER COMPOSITE - SINGLE PLY 

In the macro-mechanics approach to composites, the actual 

heterogeneous medium, Figure 12, is replaced by an "equivalent" 

homogeneous medium. The equivalent medium has the same geometry as 

the heterogeneous medium but different material properties. The 

vveffectivevf properties of the latter may belong to a different 

class of symmetry than those of the constituent materials. For 

instance, for isotropic constituents, the effective properties may 

be isotropic, orthotropic or anisotropic depending on the internal 

geometry of the composite. 

Let us consider a two-dimensional composite made of an isotropic 

matrix and reinforced by doubly periodic arrays of isotropic 

fibers as shown in Figure 13. A compo-site representation of the 

problem is a homogeneous medium with the same external geometry and 

boundary conditions as those of the heterogeneous case, but with 

orthotropic properties. The effective orthotropic properties are 

then defined for a vvrepresentativevv element. It is very important 

to first understand what is meant by that. Hashin (15) defined and 

discussed this question and has drawn some analogies to the case of 

homogeneous continua. In such continua, the hypothesis is that the 

continua retain their properties even for infinitesimal elements. 

Due to the discrete nature and microstructure of homogeneous media, 

the infinitesimal element of continuum mechanics should be large 

compared to the scales of the microstructure. The infinitesimal 

element therefore exhibits properties which are some statistical 

averages of the microstructure properties. On the other hand, the 

infinitesimal elements should be very small compared to the 

dimensions of the continua. It then follows that the physical 

quantities at a point, such as moduli, stress and strain components, 

are in reality associated with averages over infinitesimal elements 

and not with a geometrical point. Due to the complex nature of the 

microstructure, the properties of the continuum infinitesimal 

elements are determined experimentally as opposed to calculating 

the averages from a microstructural theory. The same comments also 

apply to the lvrepresentativev' element of composite theories, that 

is to say that a Ivrepresentativevv element is the infinitesimal 

element of composite materials. It therefore should be large 

compared to the dimensions of its material phases. A 

Ivrepresentativevv element, as defined, would retain and represent 

the properties of the composite continua and furthermore these 

properties would be insensitive to boundary conditions. 

233 
Figure 12. Representative element. 

2 34 
A distinction must be made between a "representative" element and 

unit elements or unit cells. 

blocks, so that the continua can be constructed by repeated use of 

such units. For example, consider the composite of Figure 13. 

There are a number of unit cell configurations which can equally 

serve as building blocks. 

in Figure 14. Figure 15 shows the boundary effects on shear 

deformations. The average properties of these units, unlike 

those of a llrepresentativell eement, are highly boundary condition 

dependent (16) . A llrepresentativell eement, however, consists of a 

large number of unit cells. 

The latter is defined as building 

Some of these possible choices are shown 

From the definition of a Ilrepresentativell element, it is apparent that the properties calculated or experimentally measured would be 

rather insensitive to boundary conditions. The problem, however, 

is that such a calculation would be a formidable task. Most 

published works dealing with a calculation of effective properties 

use a unit cell as the basis of their computations. Even for unit 

cells, the exact solutions can be obtained for only some simple 

geometries and simple boundary conditions. 

adopted is to determine the upper and the lower bounds for these 

properties through approximate solutions of energy formulations. 

These bounds are, however, far apart for composites with high 

cord-to-matrix stiffness ratios, such as rubber composites, and are 

therefore of little practical use. 

Another approach 

The immediate question is, therefore, how sensitive these properties 

are to boundary conditions specified over the surface of the 

unit cell, and, furthermore, how the element boundary conditions are 

influenced by global boundary conditions. 

been fully investigated in the literature. It can, however, be 

stated that such sensitivities should depend on the relative 

stiffnesses of the cord and the matrix. As the stiffness of the 

matrix approaches that of the reinforcing materials, the 

effective properties should become less sensitive to boundary 

conditions and obviously independent of the boundary conditions in 

the limiting case of identical constituents1 properties. These 

sensitivities are thus of greater concern in rubber composites than 

conventional rigid composites. 

stiffness can exceed 30,000 in some composites which is far greater 

than those of rigid composites. 

These questions have not 

The ratio of cord to rubber 

235 
Figure 13. Componsite with doubly periodic fiber arrays. 

(11) (111) 

Figure 14. Several unit cells. 

236 
(a) Undeformed (b) Deformed 

Figure 15. miform shear strain boundary condition. 

237 
SUMMARY ON BASICS OF RUBBER COMPOSITES 

o The calculation of effective properties of rubber composites 

by homogenization is subject to more limitations than rigid 

composites. These limitations need to be established on a 

sound theoretical basis. 

o In experimental determination of the effective properties, 

the effects of size and the boundary conditions should be 

investigated. 

o The homogenized properties of a single ply cannot be easily 

calculated since all the 3-D homogenized cord properties are 

not generally available, with sufficient accuracy. Some of 

the effective composite properties, however, may prove to be 

somewhat insensitive to inaccuracies in cord properties. 

o A necessary characteristic of a composite material is 

statistical homogeneity. A representative element, used for 

calculation of the effective properties, should be large 

compared to typical phase regions. A representative element 

of a single ply does not have such a characteristic in the 

thickness direction. The out of plane properties of a single 

ply, therefore, are subject to question. 

o The effective properties are different in tension and 

compression. 

238 
CHARACTERIZATION OF LAMINATED COMPOSITES 

We now consider a laminated structure, composed of N layers of 

cord-reinforced composite materials, as shown in Figure 16. 

Each layer of heterogeneous composite may then be modeled as 

homogeneous but orthotropic with respect to the proper local 

coordinate of each layer. The constitutive equations of the 

laminated composite, however, must be obtained with respect to 

a global coordinate system XYZ, as shown in Figure 16. The 

transformation relations may then be used for the appropriate 

layers to carry out the required transformation. The layers 

are numbered from top to bottom and no symmetry is assumed with 

respect to any axis. The stress and moment resultants for the 

laminated structure in terms of stresses are defined as follows: 

Y 

Figure 16. Geometry of typical laminated composites. 

239 
~~ ~ 

The displacement components 

are 

and the resulting constitutive relations 

240 
SUMMARY ON LAMINATED COMPOSITES 

o Each ply may be homogenized, subject to the same limitations 

as those of a single ply. 

o Complete homogenization is not possible since the coupling 

between the forces and the moments can only be accounted for 

by preserving nonhomogeneity in the thickness direction. 

o Classical kinematics constraints, such as the Kirchhoff 

hypothesis, do not apply. 

o Plate/shell type stiffnesses are extremely hard to measure. 

o Constitutive relations are nonlinear due to angle change 

between the cords of adjacent plies. 

241 
ANALYTICAL AND COMPUTATIONAL FEATURES 

KINEMATICS 

The rubber composite structures undergo large strains as well as 

large rotations. The major kinematic features are listed in table 6. 

Table 6 

KINEMATIC FEATURES 

LARGE RUBBER STRAINS 

LARGE ROTATIONS 

CORD ANGLE CHANGES 

NEAR INCOMPRESSIBILITY CONDITION 

CONSTITUTIVE RELATIONS FOR COMPOSITES 

Single ply 

The single-ply composites can be considered as linear orthotropic 

when referred to axes of symmetry. For large displacements but 

moderately small strains, the rubber and the cords can be considered 

linear, but the strain-displacement gradient relations are nonlinear. 

The most appropriate form of constitutive equations, in such case, is 

tS = tC t 

o ij o ijkl o"k1 

where tC ijkl are orthotropic properties referred to coordinates 

of initial material symmetry. 

the second Piolla-Kirchhoff stress tensor and 

The term kSij are the components of 

are the compot 

o~kl 

nents of the Lagrangian strain tensor. This form is invariant under 

a rigid body motion and therefore needs to be updated due to the cord 

angle change. This does not hold for constitutive equations 

242 
which utilize other stress and strain measures. The major drawback 

of modeling each ply separately is the increased size of the 

finite-element problem. 

Several plies 

It is often more convenient to lump several plies together in one 

element and therefore reduce the size of the problem. The preceding 

equation can be utilized but not longer remains unchanged due to the 

change in the angle between the minus and the plus ply. In such 

models the orthotropic properties of the combined plies continually 

change as functions of the cord angle. 

Composites with nonlinear constituents 

It is often necessary to account for the rubber nonlinearities in 

the finite-element analysis, even for small strains. The reason 

for such a need is that the typical compressive forces can produce 

large rubber strains if the rubber stiffening in not properly 

accounted for in the material modeling. These unrealistic large 

strains may lead to element distortion and eventual loss of 

numerical stability in the finite-element model. No rigorous 

nonlinear composite constitutive equations have appeared in the 

literature for rubber composites. This awaits further research 

in this field. 

Various modelinglevelsmay be required depending on the nature 

of the problem. These models and their respective features are 

listed in Table 7. 

243 
ELEMENT -- TYPE FOR COMPOSITES 

MEMBRANE ELEMENTS 

TWO- AND THREE-DIMENSIONAL 

ELEMENTS REPRESENTING: 

SINGLE PLY 

SEVERAL PLIES 

SHELL-TYPE ELEMENTS 

Table 7 

MODE LING FEATURES 

FEATURES 

REQUIRES 2-D MATERIAL PROPERTIES, 

INTERPLIES MODELED INDEPENDENTLY 

ONE ELEMENT PER PLY AND ONE ELEMENT 

PER INTERPLY IN THE THICKNESS 

DIRECTION, NO NEED FOR MATERIAL 

UPDATE DUE TO THE CORD ANGLE CHANGE 

REQUIRES 3-D MATERIAL PROPERTIES, 

ONE ELEMENT PER PLY IN THICKNESS 

DIRECTION, NO NEED FOR MATERIAL 

UPDATE DUE TO THE CORD ANGLE 

CHANGE 

REQUIRES 3-D MATERIAL PROPERTIES, 

LESS ELEMENTS, MATERIAL PROPERTIES 

MUST BE UPDATED DUE TO THE CORD 

ANGLE CHANGE, CANNOT MODEL THE 

COUPLING BETWEEN THE BENDING 

AND IN-PLANE FORCES 

INCREASED MATERIAL PROPERTY INPUT, 

LEAST NUMBER OF ELEMENTS, 

CAN MODEL THE COUPLING BETWEEN THE 

BENDING AND IN-PLANE FORCES, 

KINEMATICS CONSTRAINTS OF THE 

CORRESPONDING SHELL THEORY, MATERIAL 

PROPERTIES MUST BE UPDATED DUE TO 

THE CORD ANGLE CHANGE 

I 

~ 

I 

244 
SUMMARY OF MAJOR ANALYTICAL AND COMPUTATIONAL NEEDS AND FEATURES 

o Large elastic strains. 

o Large rotations. 

o Incompressibility condition, natural rubber can be considered 

as incompressible, but filled rubber exhibits compressibility. 

o Nonconservative loading. 

o Element type and aspect ratio. 

o Self-adoptive schemes for load increments and step size. 

o Contact algorithms for frictional loadings. 

o Finite-element formulation in rotating coordinate system. 

o Substructuring for localized analysis. 

o Stresses are very erratic at regions of sudden change in 

stiffness, such as cord-rubber interface. When calculated 

from the finite-element displacement method, proper smoothing 

algorithms must be developed for nonlinear problems. 

o Sensitivity analysis for uncertain input, material properties 

or other variables. 

245 
I REFERENCES 

I 1. Riding G. "Study of the Geometrical Structure of Multi-Ply 

Yarns," Journal of Textile Institute, vol. 52, pp. 366-381, 

1961. 

2. Clark, S.K., "Mechanics of Pneumatic Tires," National Bureau of 

Standards Monograph 122, First Edition, 1971. 

3. Oblath, R.M., "Toward 21st Century - The Challenges Ahead of 

Tire Reinforcements," Akron Rubber Group, Akron, Ohio, 

Jan. 26, 1984. 

4. Tabaddor, F. and Clark, S.K., "Linear and Nonlinear Viscoelastic 

Constitutive Relations of Cord Reinforced Elastomers," Proceedings of the International Symposium on Constitutive Equations 

for Non-Linear Materials, University of Arizona, Tuscon, 

Jan. 1983. 

I 5. Treloar, L.R.G., "The Elasticity of a Network of Long Chain 

Molecules (I and II)," Trans. Faraday Society, vol. 40, p. 59, I 1944. 

I 6. Rivlin, R.S. and Saunders, D.W., "Large Elastic Deformations of 

Isotropic Materials - VII. Experiments on the Deformation of 

Rubber," Phil. Trans. Royal Society, vol. A243, p. 251, 1948. 

I 7. Valanis, K.C. and Landel, R.F., "The Strain Energy Function of a 

Hyper-Elastic Material in Terms of the Extension Ratios," 

Journal of Applied Physics, vol. 38, no. 7, pp. 2997-3002, 1967. 

8. Tavazza, G. and Rimondi, G., "Stress and Strain Calculations of 

Rubber Goods," Proceedings of the International Rubber Conference, Kiev USSR, 1978. 

9. Rivlin, R.S. and Thomas, A.G., J Polymer Science, vol. 10, 

p. 291, 1953. 

I 10. Thomas, A.G., J. Polymer Science, vol. 31, p. 467, 1958. 

11. Busses, W.F., Ind. Eng. Chem., vol. 26, p. 1194, 1934. 

12. Cadwell, S.M., Merril, R.A., Sloman, C.M. and Yost, F.L., Ind. 

Eng. Chem. Anal., vol. 12(1), p. 19, 1940. 

13. Beatty, J.R., Rubber Chemistry and Technology, no. 5, 

pp. 1341-1346, Dec. 1964. 

14. Beatty, J.R., J. of Elastomers and Plastics, vol. 11, p. 147, 

April 1974. 

I 15. Hashin, Z., "Analysis of Composite Materials--A Survey," Journal 

of Applied Mechanics, vol. 50, September 1983. 

246 
16. Tabaddor, F., "Mechanical Properties of Cord-Rubber Composites," 

Composite Structures, vol. 3, pp. 33-53, 1985. 

247 
N89=24650 

NONLINEAR HIERARCHICAL SUBSTIIUCTURAL PARALLELISM 

AND COMPUTER ARCHITECTURE 

Joe Padovan 

The University of Akron 

Akron, Ohio 

Abstract 

This paper investigates computer architecture in conjunction with the 

algorithmic structures of nonlinear finite-element analysis. To help set the stage 

for this goal, the development is undertaken by considering the wide-ranging needs 

associated with the analysis of rolling tires which possess the full range of 

kinematic, material and boundary condition induced nonlinearity in addition to gross 

and local cord-matrix material properties. 

1. Introduction - 

With the advent of the finite-element method (FEM), the analysis of large-scale 

structure is finally possible. While large-scale linear finite-element simulations 

are relatively economical, such is not the case for nonlinear situations involving 

geometric, material and boundary induced nonl inearit~l-~. 

aerospace and commercial structures which require full-scale nonlinear analysis to 

enable their improved design. This includes such structural systems as gas turbines, 

space structures, aircraft structure, autos, etc. Perhaps the most commonplace of 

such structures is the tire, which serves as a component to a wide variety of 

aerospace and auto systems. 

There are numerous 

To bypass the difficulties associated with nonlinear FE analysis, significant 

work has been channeled into two main areas, namely: 

i) The development of algorithmic improvements, element-element ,5 constrained 

7 Newton/Raphson (NR) , and hierarchical least squares, 

ii) The design of new computer architecture enabling hardware speedup, i.e., as in 

vector processors (Cray, Cyber 205 and true parallel machines8s9) 

In the context of such thrusts, not enough e€fort has been undertaken to 

consider how algorithmic structures might effect machine architecture or vice versa. 

Based on the foregoing comments, this paper will investigate machine architecture in conjunction with algorithmic structure. To achieve this goal, the development will be undertaken by considering the wide-ranging needs associated with the 

analysis of tires. This approach was taken since, as will be seen in later sections, 

the needs of tire modeling embody essentially all the requirements of nonlinear 

continuum mechanics, namely 10 

i) Material nonlinearity 

ii) Inelastic behavior 

PRECEDING PAGE BLANK NOT FILMED 249 
iii) Large deformation/strain kinematics 

iv) Complex inertial fields 

v) Nonlinear boundary conditions 

vi) Microstructure 

I vii) Thermomechanical response 

1 viii) Solid fluid interaction 

All this leads to the development of what is called hierarchical substructural 

parallelism which enables bottom-up/top-down modeling. l1 

multilevel substructuring scheme is overviewed which enables the simplification of 

the data based management (DBM) of parallel-type operators while still yielding 

enhanced computational speeds as well as reducing core requirements. 

Overall a nonlinear 

In the sections that follow, detailed tire modeling discussions embody the 

diversity of needs of nonlinear simulations, various types of current machine 

architectures, and potentials of hierarchical substructural parallelism. Examples 

that define enhanced properties will also be given. 

2. -- Shortcominp ---- of FEM Vis-;-vis ---I-- Tire Structural -- Analysis 

Noting Figure 1, the tire possesses a very regionalized/substructural form of 

12 construction. Overall it consists of: 

I i) Carcass plies, steel/glass/Kevlar cord-rubber composites 

ii) Belt plies (same as above) 

I 

iii) Bead, bundled steel cords 

iv) Thread conFiguration 

v) Regionalized rubber types 

vi) Belt edges, turnup plies 

I 250 
The operating environment consists of: 

i) The tire-road interface which involves varying pavement textures, 

flexibilities and resulting frictional characteristics 13,14 

ii) The tire-rim interface 

iii) The tire-rim-suspension behavior 

iv) Cornering, braking and accelerating maneuvers 

v) Standing, steady/transient rolling 10,13-15 

vi) Obstacle/hole envelopment roll over events 13,14 

16,17 vii) Pressurization 

As seen from Figures 2 and 3, the pressurization and subsequent loading into 

standing contact can lead to large deformations and associated rotations. For 

instance Table 1 illustrates comparisons of the deflection fields generated from 

linear and nonlinear FE simulations. 

In this context, it follows that there are several sources of response 

nonlinearity, namely 

i) Large deformation kinematics 

ii) The road-tire-rim interfaces 

iii) Bimodular behavior of cord-rubber composites in transitions from 

tension to compression 

iv) Thermomechanical interactions 

v) Material nonlinearity 

vi) Local large strain levels in various regions of the tire; 

belt edges, bead region, and tread 

vii) Dynamic impact interactions 

Each of the foregoing sources of nonlinearity initiates different forms of response 

behavior. 

For instance, from a kinematics point of view, the pressurization process causes 

rotations and deflections which lead to an overall stiffening of the tire. 

Similarly, as with Hertizian contact problems, the tire-road interface also exhibits 

hardening-type properties, namely, the hub force-deflection response is stiffening in 

character as noted in Figure 4. 

251 
In addition to the foregoing modeling difficulties, in general the tire response 

needs to be handled in several levels, namely 

i) Cord-matrix and regionalized rubber interfaces 

i i )Who le co rd-r ubbe r p 1 i e s / 1 ami nae 

iii) Full laminate structures, several plies as in belt and 

carcass laminates 

iv) Full (global) structure 

As one preceeds from (i)-(iv), a "bottom-up" modelingll approach is required wherein 

fine detail is handled at the lowest level while the upper level models are increasingly coarser so as to reduce overall degrees of freedom in a global model. 

Once the global-level model is solved what is needed is a "top-down" scheme'' to 

provide proper mechanics information at the constituent level. Such an approach is 

necessary if proper stress and strain €ields are to be captured hence enabling proper 

description of internal. fields. 

Current FE models of tires start from level (iii) and proceed to (iv). In this 

way, a true local-level description of mechanical fields is not possible. 

3. Types of Parallelism ----- 

Multiprocessor computers fall basically into two main categories, namely 

i) Vector processors (Cray, Cyber 205) 

ii) True parallel processors (Flex, Goodyear) 

Compared with single processor units (IBM 3084, CDC7600), vector processors 

enable quicker more eEficient handling of matrix manipulations. This is achieved 

through the use of multiple processors which operate simultaneously on a succession 

oE matrix elements. Data transfer for such operations is typically from a single 

common core storage. 

In true parallel processors, different functions/operations are performed in 

separate processors. In such machines data transfer usually involves both a common 

core as well as individual. local processor cores. For such machines very high speeds 

can be realized. 

In the context of programming languages, vector processors typically can be 

programmed in enhanced versions of FORTRAN or the like. For true parallel processors, overall programming is generally achieved at two levels. At the local processor level, languages such as FORTRAN can be employed. At the total system level, 

machine control language MCL is usually employed. 

252 
4. -- Classical Solution Algorithm 

The solution of large-scale FE simulations typically involves either some 

variant of the NewtonjRaphson scheme NR, or an explicitlimplicit time integration 

procedure. For the current demonstration purposes, the presentation will concentrate 

on static equation solvers. The most recent improvements €or such problems fall into 

several categories, namely 

i) Element-by-element preconditioners (Hughes et a~~) 

6 ii) Constrained NR procedures of Padovan and Arechaga 

iii) Constrained hierarchical least-squares algorithms of 

7 Padovan and Lackney 

Assuming large deformation kinematics along with potential material 

nonlinearity, the governing FE formulation takes the form 697 

where S_ is the second Piola Kirchoff stress tensor, F, is the nodal force vector 

and G is the vector of body forces. Typically (1) is nonlinear and must be solved 

via N< schemes. 

following NR algorithm 

After expansion into truncated Taylor series, (1) yields the 

;k T 

AG + [Ki]AYi+l = F+AF - .f[Bi] Sidv - -- - 

R 

where [K] defines the tangent stiffness matrix, that is697 

such that [Si] is the prestress matrix and [DTi] is the tangent material stiEEness. 

As noted earlier, the solution involves either the use of constrained procedures6 for 

appropriate load increment control or a direct Gaussian-type inversion scheme. 1 

To date such methodologies have been employed either in single processor or 

vector processor machines. The shortcomings of the FEM outlined in the previous 

section are essentially a direct outgrowth of the limitations of the architecture of 

single and vector-processor-type machines. In the next section, the intrinsic 

structure of the INR algorithm will be explored to define new computer architectures 

to bypass such difficulties. 

253 
5. Hierarchical Substructurint? 

From a conceptual point of view, the INR scheme defined by (2) does not confine 

the FEM scheme to a particular type of computer configuration. Rather the problems 

of speed and storage are essentially hardware based. Specifically the main questions 

and problems evolve out of the need to define architectures which enable the use of 

multiple processors so as to enhance overall machine speed as well as memory size. 

While the CRAY and CYBER systems are certainly a step in the right direction, they 

fall short of the ultimate requirements. Currently very large-scale FE models can 

easily outstrip the available core storage and machine CPU speeds. 

In seeking to develop new computer architectures one is faced with the fact that 

i) Vector processors require extensive cores as well as complex logic flows 

ii) True parallel processors still await the fruition of properly organized DBM 

Based on the foregoing, this paper seeks to develop what is called a hierarch€cal 

form of substructural parallelism. Following the pioneering efforts of the NASA 

Langley groupgy9 specifically, a nonlinear FE simulation, say of the tire, can be 

logically divided into a hierarchy of substructural groups defined by a variety of 

attributes, namely 

i) Material group 

ii) Geometric conEiguration 

iii) Kinematic behavior 

I iv) Boundary conditions 

At the lowest rung of the hierarchy, items (i)-(iv) are employed to define the 

specific local level substructural groups. The choice of the number of Eirst-order 

groups is contingent on: 

i) Minimizing core requirements of local level processors 

ii) Minimizing number of perimeter nodes so that higher order substructural 

groups also have reduced core requirements for associated processors. 

As can be seen, the main thrust is to maintain in core solutions for each local 

substructural processor. 

Noting Figure 5, a given FE simulation can be broken up into a number of substructural levels. At each level internal nodes are eliminated to enable assembly 

through perimeter nodes. In terms of (21, the NR algorithm and its constrained 

counterpart can be substructured to yield the following first-level algorithms, that 

is : 

254 
k = 1,2,....Number of first-level substructure such that 

where ( )('sk) d notes the first level kth substructure, ( li+l the (i+llth 

iteration, @-i+l (' 'ke the nodal load increment, [Ki] the substructural tangent 

stiffness, A:i+l (lSk) the nodal deflection increment and E!:ik) the body force 

increment . 

To enable assembly into second-order substructural groups, (4) is partitioned 

into internal and perimeter nodes yielding 

Employing (7)-(10) we obtain the following relationships for the inner and perimeter 

nodes 

I 
, where 

I Assembling (11) yields the second-level substructural relationships, namely 

k = 1,2,.. . Number of second level substructure 

By partitioning (16) into inner and perimeter degrees of freedom we yield the thirdorder substructural relations after the appropriate assembly process. Continuing the 

partitioning and assembly process yields the various higher order substructural 

relations specifically 

wherein the associated inner and perimeter partitions take the form 

256 
such that 

Based on (11)-(24), we see that the overall nonlinear hierarchical substructuring 

requires a forward calculation phase as well as a backward stage. The forward phase 

involves the use of (111, (131, (141, (19), (21) and (22). In contrast the backward 

phase, which involves the definition of inner nodes, incorporates the use of (12), 

(15), (16), (20), (23) and (24). In terms of the forward iterative algorithms, the 

overall required machine architecture takes the form defined in Figure 6. Note the 

common data buses linking successive substructural levels need only provide access to 

perimeter data. In this way, significantly less data need to be accessed by the 

global-level DBM. This applies throughout the forward phase of the iteration 

process. Overall the steps handled by each of the succeeding levels involve 

assembly, inner/perimeter partitioning, and setting up effective stiffnesses €or the 

forward and backward phases. In terms of (21)-(24), the stiffnesses associated wi h 

the perimeter and inner nodes involve an inverse of the inner partition of the 

substructural stiffness. All such manipulations must be performed by processors 

dedicated to each of the k individual substructures associated with the various 

hierarchical levels. 

F; kt 

Once the forward loop of calculations is complete, the perimeter data must be 

back tracked to the inner nodes of each of the various substructures at the different 

substructural levels. The overall flow of control/calculation is depicted in 

Figure 7. As can be seen, the perimeter data are used to determine the inner nodal 

incremental excursions. This is achieved through the use of the family of expressions defined by (20). Once the back substitutions to the succeeding levels up to 

and including the first are completed, the standard norm type convergence checks must 

be implemented to ascertain the quality of convergence. Contingent on the convergence check, the iteration process can be cycled through the forward and backward 

phases of the substructural hierarchy. 

257 
6. Discussion 

To illustrate the hierarchical substructural scheme, consider the three-level 

simulation defined in Figure 8. The number of nodes and substructure associated with 

the example are given in Figure 9. 

variables depicted, the expressions defining the number of respective nodes are given 

by : 

Based on the number of inner and perimeter 

i) Level 1 

Perimeter Nodes = 2(R1 + $2 + 23) 

Inner Nodes = (21 - 2)(R2 - 2) 

ii) Level 2 

iii) Level 3 

Perimeter Nodes = 2mlnl(Rl - 1) + 2m2n2(R2 - 1) 

+mn +mn - m1m2 + 1 (30) 11 22 

Employing (25-30) we see that the storage effectiveness of each of the various levels 

is expressed by the relations 

Per i me t er 

Perimeter + Inner 

&') = -~.--~-- 

where k denotes the level number. In the context oE (31), it follows that 

2(t n 

n II (n 

p + t2n2 - n1 - n,) = 11 

+ 1) + E2n2(nl + 1) - 3n1n2 - 4(nl + n2) + 1 11 2 

(33) 

2mlnl(Ql - 1)' t 2m2n2(I12 - 1) 

L m n,(m2 + 1) + II m n (m &3) = (34) + 1) -mlm2(n1 + n2 + 1) - mlnl - m2n2 + 1 11, 222 1 

258 
Consider the case wherein 

El = 100, L2 = 50 

n =5, n2=4 1 

m =3, m =4 1 2 

In terms of the foregoing, Table 2 gives the total number of 

0 degrees of freedom 

0 processors required at each level 

perimeter/inner nodes 

as well as the storage effectiveness of each of the substructural leve s. Noting 

that a straight solution of the given problem would require a 1.2 x 10 order 

stiffness matrix, it follows from Table 2 that very significant storage savings as 

well as speed enhancements can be achieved. 

t! 

In the context of the foregong development, it follows that hierarchical substructural parallelism has decided advantages over vector-type processors, namely: 

i) Global common core is reduced in size 

ii) Substructures are handled in smaller local cores which could employ 

vector processors and which are controlled by local DBM 

iii) Data transfer between succeeding levels of substructural hierarchy 

are reduced thereby reducing load on DBM 

iv) Various substructures are updated, inverted, and assembled 

simultaneously hence emhancing the overall speed 

v) The overall addressing requirements are reduced since the size of 

individual substructural zones is much smaller 

iv) Extensive use of cash memory (Ram Disk) can be made at the local 

level thereby reducing disk 1/0 

vii) Backward and forward steps follow natural formulational lines 

viii) Element-to-element or hierarchical least-squares algorithms can 

be employed at the local substructural level 

ix) Linear/nonlinear problem partitioning can be more logically handled 

x) Overall control of the machine is more logical and less difficult 

since local processors are essentially autonomous within updating 

and inverting phases of the operation 

259 
xi) The MCL can be patterned about well-defined substructuring methodology; the 

transfer of control from level to level is contingent on the 

monitoring status of stiffness/inversion calculations 

xii) The data base manager needs only to deal with data residing on perimeters 

of the substructure; as noted earlier, this significantly reduces the amount 

of data transferred between levels. 

As discussed earlier, the modeling of tires in their use environment represents 

perhaps one of the most comprehensive single component nonlinear structural response 

problems currently available. This follows from the fact that geometric-, material-, 

and boundary-induced nonlinearity all simultaneously act to define the global 

response behavior. Due to their regionalized/ substructural form of construction, 

tires represent a good modeling problem to help define the architecture of high-level 

multiprocessor machines. In this context, a hierarchical form of substructural 

parallelism has decided advantages over other forms of multiprocessors. As has been 

seen such a procedure has several theoretical advantages for nonlinear problems. 

These evolve about the simplified DBM structure, reduced data flow, smaller global 

core, and reduced addressing requirements. 

Overall Euture work in this area should 

0 Place qreater emphasis on algorithmic architecture and its possible effects on 

machine structure 

0 Establish proper control configuration for hierarchical DBM 

Extend scheme to constrained incremental Newton/Raphson (INR) least-squares 

algorithms as well as transient schemes 

0 Apply concept to available parallel processors 

0 Structure procedure so as to enable either direct or iterative solutions at 

substructural level 

0 Establish criteria to enable determination of quality of convergence at 

local substructural level 

260 
6. References 

~ 

I 1. Zienkiewicz, O.C., The Finite Element Method, McGraw Hill Company (1983). 

2. Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice Hall 

(1983). 

3. Oden, J.T., Finite Elements of Nonlinear Continuum,-McGraw Hill Co. (1972). 

4. Cook, R.D., Concepts and Applications of Finite Element Analysis, Wiley (1983). 

5. Hughes, T.J.R., Winget, J. and Levit, I., Comp. Meth. Appl. Mech. Engineering, 

36, 241 (1983). 

6. Padovan, J. and Arechaga, T., Int. Jr. Engineering Science, 20, 1077 (1982). 

7. Padovan, J. and Lackney, .J., Computers and Structures, 19, 535 (1984). 

8. Noor, A.K., Storaasli, 0.0. and Fulton, R.E., ASME Special Publ., H00275, 1 

(1983). 

9. Bostic, S.W. and Fulton, R.D., AIAA Paper No. 85-0783-CP (1985). 

10. Padovan, J. and Tovichakchaikul, S., Computers and Structures, 18, 191 (1984). 

11. Padovan, J., Global-Local Finite Element Modeling, 126th Meeting of the ACS 

Rubber Division (1984). 

12. Clark, S., Mechanics of Pneumatic Tires, U.S. Department of Transportation 

(1980). 

13. Padovan, J. and Nakajima, Y., Numerical Simulation of Tire Skidding Events 

Involving Impacts with Holes and Bumps, 4th Annual Tire Society Meeting (1985). 

14. Padovan, J. and Nakajima, Y., Numerical Simulation of Tire Transiently 

Sliding/Rolling Over Holes and Bumps, ACS Meeting (April 1985). 

15. Padovan, J. and Paramadilok, O., Transient and Steady State Viscoelastic Rolling 

Contact, Computers and Structures, 20, 545-553 (1985). 

16. Trinko, J.J., Ply and Rubber Stresses and Contact Forces for a Loaded Radial 

Tire, First Annual Meeting of Tire Society, Akron (1982). 

17. Tabaddor, F. and Stafford, J., Computers and Structures, 13, 737 (1981). 

261 
TABLE 1 COMPARISON OF LINEAR AND NONLINEAR FE SIMULATION OF PRESSURIZED TIRE 

VONLINEAR (IN) 

MAXIMUM DEFLECT ION 

LINEAR (IN) 

MAX IYUY DEFLECT I ON 

PRESSURE 

(PSI 1 

5. 

10. 

15. 

30. 

CROWN SIDEWALL CROWN SIDEWALL 

.003 ,036 003 

,006 ,073 006 ,047 

,009 . in9 009 064 

. 01 8 21 8 01 8 ,107 

TABLE 2 COMPARISON OF HIERARCHICAL SUBSTRUCTURAL PARALLEL AND SINGLE 

PROCESSOR SYSTEMS 

LEVEL 

1 

2 

3 

GLOBAL 

*D of F/P - deqrees of freedom per processor 

**Total nuniher of perimeter and inner 0 of F 

262 
TREAD STOCK 

/ 

I BE~.D *BE40 FILLER 

b 

Figure 1 Tire geometry and cross section. 

INITIAL SHAPE 

EX PER1 MEN TAL 

FINITE ELEMENT 

--- 

-.-. 

Figure 2 Pressure tire profile. 

263 
n 

rl 500 

v 

Figure 3 Loading into standing contact. 

spring rate -2Opsi 

-- 

Predicted 

Experimental 

-4 -8 

Radial Deflection (in-) 

Figure 4 Force deflection characteristics into standing contact. 

2 64 
@ Steel belt 

@ Textile carcass 

@ Inner liner 

@ Belt edge cushion 

@ Side wall 

@ Bead filler 

@) Textile finishing 

@ Bead 

Figure 5 Substructural zones of tire. 

COMMON DATA BUS 1 

LEVEL 1 1 k ... 

COMMON DATA BUS 

COMMON DATA BUS I . 

. 

LEVEL j 1 COMMON DATA BUS 

MAIN 1/0 BUS 

----------- 

LOCAL STIFFNESS UPDATE 

SUBSTRUCTURE ASSEMBLED 

PERIMETER-INNER PARTITIONING 

-- ---_-___ - PERIMETER DATA 1/0 

-_---_----- 

SUBSTRUCTURE ASSEMBLED 

PERIMETER-INNER PARTITIONING 

_---------- 

PERIMETER DATA 1/0 

__--------- 

GLOBAL SUBSTRUCTURE 

ASSEMBLED AND SOLVED 

Figure 6 Flow of control: forward loop. 

265 
COMMON DATA BlJS 

LEVEL 

L COMMON DATA BUS 

COMMON DATA BUS J . I I e 

i COMMON DATA BUS 

I3 LEVEL j 

- -------____ 

MAIN 1/0 BUS -------____ 

INNER VARIABLES CALC. 

ALL FIELD VARIABLES 

UPDATED 

-- -------_ - PERIMETER DATA 1/0 

__-_------- 

INNER VARIABLES CALC. 

----_------ 

PERIMETER DATA I/O 

---_---___- INNER VARIABLES CALC. 

Figure 7 Flow of control: backward loop. 

- LEVEL 1 (LOCAL) 

- LEVEL 2 <INTERMEDIATE) 

’/ 

Figure 8 Example oE three-level hierarchical substructural system. 

266 
LEVEL 

LEVEL 

LEVEL 

1 

2 

Figure 9 Number of perimeter substructures at each level. 

267 
ADAPTIVE METHODS, ROLLING CONTACT, 

AND NONCLASSICAL FRICTION LAWS 

J. T. Oden 

The University of Texas at Austin 

Austin, Texas 

ABSTRACT 

Results and methods on three dif€erent areas oE contemporary research are 

outlined. These include adaptive methods, the rolling contact problem for 

finite deformation of a hyperelastic or viscoelastic cylinder, and nonclassical friction laws for modeling dynamic friction phenomena. 

PRECEDING PAGE RANK NOT FILMED 269 
1. INTRODUCTION 

This paper addresses three subjects that impact on the computer simulation of nonlinear tire behavior: adaptive methods, which represent schemes 

for assessing numerical error and automatically adapting the mesh so as to 

improve accuracy; the rolling contact problem, which is at the heart of tire 

analysis; and new friction laws, which are essential in developing realistic 

models of frictional contact. Space limitations preclude a detailed discussion of these issues; but further details can be found in recent papers and 

reports by the author and his colleagues [l-171. 

2. ROLLING CONTACT 

The general rolling contact problem as a basis for nonlinear tire analysis involves some of the most challenging and difficult problems in structural mechanics. Among the complicating features are the presence of 

unilateral contact, friction, inertia effects, multi-parameter bifurcations, 

the emergence of standing waves, viscoelastic and thermal effects, large 

deformations, the necessity of modeling of complex materials such as fiberreinforced rubbers, and the presence of non-conservative pressurization 

loadings. A first step toward resolving such issues is the formulation of 

correct kinematics and variational principles for a special case: the steadystate rolling of a hyperelastic or viscoelastic cylinder in contact with a 

rigid foundation and in a state of plane strain. 

The kinematical situation is shown in Fig. 1 where the geometry of the 

reference configuration (a rigid spinning cylinder with no contact) is compared to the geometry of a deformed cylinder in steady-state rolling contact 

with a rough (frictional) roadbed (foundation). 

Key features of the kinematic description are listed as follows: 

1) Time appears only implicitly in the formulation; if (R,@, Z,) are 

material coordinates, the referential coordinates are 

r = R, 8 =@+ ut, z = Z 

where w is the angular velocity of the rigid, reference cylinder. 

2) If the motion is defined by the map 

xi = xi (r, e, z) i = 1, 2, 3 

where 

guration, then velocity and acceleration components are 

xi = spatial Cartesian coordinates of particles in the current confi2 

2 a Xi a =u-- axi 

2 v =wae i ae i 

270 
3) The unilateral contact constraint can be expressed in the form 

where 

hub center to the foundation. This condition can also be written rC is the exterior contact surface and @ is the distance Erom the 

where (.I+ denotes the positive part OF (.). 

4) The time history of deformation can be expressed in terms of strains 

of particles located on the same circular arc in the reference configuration. 

For example, if is the Green-Saint Venant strain tensor, its time history 

over an internal 0 T t satisfies: - - 

This property makes it possible to incorporate viscoelastic effects into the 

rolling contact problem in a straightforward way. 

For illustration purposes, we consider a class of rolling contact problems in which the following constitutive properties hold: 

a) The material is either an isotropic hyperelastic material characterized by a strain energy function 

(or W = W(11].12) if I3 = 1 - an incompressible material) in which Il,12,T3 

are the principal invariants of the deformation tensor 

material is a viscoelastic material characterized by a linear viscoelastic 

perturbation of a hyperelastic material; e.g., 

C, = zT F,, or the 

with y a viscosity parameter, k a material kernel, and S, the second 

Piola-Kirchhoff stress tensor. 

271 
___- 

S) The normal stiffness of the contact interface obeys aconstitutive 

law of the type 

m 

tn = cn(x2 - I{)+ on rc n 

where t, is the normal stress and cn and m, are material constants. 

If m, = 1 and cn = 1/~, where E is a positive constant, this relation 

coincides with the normal contact stress associated with an exterior penalty 

approximation of the unilateral constraint condition. 

c) If the cylinder is given a prescribed velocity vo relative to 

the roadway, the slip velocity on the contact surface is 

wT - vo - i, = v - wagX 0 

5) The friction law is (with 

t I < llltnl = > WT = 0 I -T 

t I = pJtnJ = > WT = -XI I -T 

Variational principles for var 

the frictional stress) LT 

ous rolling contact problems are summarized in Figs. 2-8, beginning with the pure spinning of a cylinder without 

contact and progressing to the general variational inequality for finite deformation rolling contact with friction. Various spaces of admissible functions are defined in these figures as well as several nonlinear forms. In 

particular, 

A(x, n) = the internal virtual work produced by the Piola-Kirchhoff NN stresses To 

B(x, n) = the virtual work produced by inertia (radial acceleration) 

effects per unit of angular velocity velocity w NN 

C(x, 0) = work term due to normal compliance of the interface 

I(x, 5, n) = a virtual work term representing the work done by the 

hydrostatic pressure p (present when the material is 

incompressible) 

NN 

NNN 

j(x, rl) = the virtual work term due to frictional forces 

f(n> = the virtual work due to external forces 

NN 

N 

272 
A finite-element code has been developed based on this general variational principle which has the following €eatures: 

1. Biquadratic (Q2) elements are used to approximate the displacement 

field and, for incompressible materials, el, discontinuous linear elements are 

used to approximate hydrostatic pressures 

I 2. The frictional functions are regularized in a standard way 

~ 

3. A Riks-Crisfield method with Newton-Raphson corrections is used to 

solve the nonlinear systems of equations characterizing the discrete problem 

To date, an extensive set of numerical solutions has been obtained using 

these concepts and methods. Here only one example is cited, which is interesting because of the slow emergence of standing waves as the angular velocity 

is increased for a fixed penetration H of a hollow rubber cylinder into a 

rough roadway with coefficient of frictLon u = 0.03. Computed deformed 

shapes and stress contours are shown in Figs. 9-13 for various values of w. 

We notice the steady development of more-or-less periodic wavelets on the exterior surface which meet at points at which singularities appear to be 

forming. The presence of Eriction on the contact surface destroys the symmetry of this wavelet pattern. Mild viscoelastic effects, such as those in 

rubbers at moderate temperatures, do not appreciably alter the structure of 

these deformed geometries. 

The generality of the Formulation and of the methods employed here makes 

it possible to study numerically a wide range of rolling contact problems. 

Further work shall involvs generalizations of these results to threedimensional rolling contact problems which include the effects o€ turning, 

steering forces, and tilting relative to the roadway plane. 

3. ADAPTIVE METHODS 

We shall now turn to the important subject of adaptive finite-element 

methods. Adaptive methods should have a significant impact on not only tire 

analysis but also on general computational structural mechanics in the relatively near future. 

In general, adaptive methods seek to change the structure of an approximate method to improve the quality of the solution. By structure, we mean the 

mesh density, locations of nodes, and order of the local polynomials. By 

quality of an approximation, we generally mean the error in approximation in 

some appropriate norm. There are, thus, two primary aspects of any adaptive 

E inite-e lement method : 

1) The estimation of the error 

2) The adaptive strategy 

In the first o€ these, it is assumed that an initial approximation of the 

solution is known, perhaps from a computation on a coarse mesh, and that this 

rough solution can be used to obtain an a posteriori estimate of the local 

273 
error over each finite element. Once an estimate oE the local error is known, 

one must call upon some technique to reduce the local error and thereby improve the quality of the solution. 

There are two general types of methods we have studied for a posteriori 

error estimation of the local error over each finite element. Once an estimate of the local error is known, one must call upon some technique to reduce 

the local error and thereby improve the quality of the solution. 

There are two general types of methods we have studied for a posteriori 

error estimation: 

1. Residual methods 

2. Interpolation (or a priori) methods 

As the name implies, residual methods make use of element residuals - the 

residual or unbalance left over when the approximate solution is substituted 

into the governing equations and edge conditions on each element. 

The residual itself (e.g., the equilibrium unbalance in element forces) 

is not necessarily a good indication of local error. Indeed, the local 

residual can be nearly zero while the error can be quite large. For this 

reason, it is generally necessary to calculate certain local error indicators 

9, The calculation of error indicators generally requires the solution of special local 

problems over each element in which the element residuals enter as data. For 

example, in the model elliptic problem, 

which bound the error above and below in appropriate norms. 

2 -Au = f in Q CIR 

u = on an 

(with A the Laplacian and f given), the finite-element solution Uh 

satisfies 

I Vuh Vvh dxdy = fvhdxdy 

n n 

for arbitrary test €unction Vh, and over each element K of a mesh, the 

residual is 

rh=- Au,, - f 

274 
Over element K, an error indicator #Ik is computed which satisEies 

vd s 

auh rh vdxdy + #I -- 

an ,f V#k Vv dxdy = I 

K K aK 

for v in H1(K). One can show that the error eh = u - Uh in the energy norm 

(1 lehl Il,Q = {I lVehl2 dxdy 1 1'2) satis€ies the bound 

n 

Various residual methods differ in the way these error indicators are 

defined and calculated. There are some residual techniques which can produce 

sharp local error estimates in virtually any norm for certain classes of problems. (See Demkowicz and Oden 14, 51). These methods are not restricted to 

linear problems and have been used to produce error estimates in highly nonlinear problems (see [7, 161). 

The interpolation methods make use of the fact that the interpolation 

error over an element K of diameter hK over an element on which polynomials of degree p are used is 

where u is a given smooth function, I$,u is its interpolant, 

+ u2 + u2 )dxdy 

and C is a constant independen 8 of hK and u. The idea behind interpolation methods is to estimate 

tion (e.g., using finite-difference methods or extraction methods [6]). The 

constant C cannot, in general, be determined, so such interpolation methods 

can only be used to assess relative error in various finite elements. 

lu12,K using results of a coarse-mesh approximaOnce an estimate of the error is obtained, the local error is reduced 

adaptively using one of the following techniques: 
1. h-methods: the mesh size h is reduced and the number of elements 

is increased in regions of high error. 

2. p-methods: the mesh is fixed, but the local order p of the 

polynomial shape functions is increased. 

3. Moving mesh methods: the nodes in a finite-element mesh are moved 

and concentrated in areas of high error. 

4. Combined methods: these involve combinations of the above three 

techniques. 

We have developed test codes which employ all of these methods. The 

results of some tests are given in Figs. 14-20, and specific comments follow. 

1. In Fig. 14 we see a distorted mesh obtained using a moving mesh 

strategy on the driven cavity problem for an incompressible viscous fluid (see 

[71). 

2. Figures 15, 16, and 17 contain computed results in adaptive schemes 

based on residual methods devised by Demkowicz and Oden [4, 51. The results 

shown here are for transient heat conduction problems with dominate convection 

effects and for nonlinear Burgers' equation vector-valued solutions which 

simulate the Navier-Stokes equations. A special h-method is used here which 

employs a fine grid and a coarse-grid approximation to estimate error. 

3. Results oE a new p-method for Navier-Stokes equations in twodimensions are illustrated in Figs. 18, 19 (see [l, 161). Here a rather 

coarse mesh is used and errors are reduced by increasing local polynomial 

degrees from 1 to 2 to 3 Different shading in these figures indicates different levels of local L error, with black cells indicating an error of less 

than 5 percent, grey an error of less than 10 percent, and white an error of 

over 20 percent. Such large local errors are reduced before the solution is 

allowed to advance in time. Remarkably, the so-called effectivity index 0 

for this proljlem (which represent the ratio of the estimated local error to 

the actual local error) for an L' - norm varied from around 1.001 to 0.860 

for the time ranges considered in a test example. This suggests that 

residual-type error estimates based on p-type strategies can be very accurate, 

even for transient nonlinear problems on coarse meshes. 

i 

Figure 20 shows refined mesh patterns for a class of linear elliptic 

problems in which a very fast vectorizable h-method is used in conjunction 

with an interpolation-type error estimator (see [61). One interesting aspect 

of this work, indicated by different shadings of elements in the figure, is 

that the distinction between "optimal" meshes determined using a very sophisticated error estimator (see [171) and very crude estimates (C61) is negligible whenever strong singularities are present. 

4. NON-CLASSICAL FRICTION LAWS 

In our most recent calculations of rolling contact, we have employed 

special interface constitutive equations for the normal compliance OF the 

276 
interface and the tangential .frictional forces. Some of these laws are mentioned in Section 2 of this paper (see also Fig. 6). The physical interpretation and the motivation of such models are discussed in references [14, 15, and 

181. 

ACKNOWLEDGMENT 

The work reported here was supported by several agencies: the work 

on adaptive methods by The Office of Naval Research under Contract 

N00014-84-K-0409; the work on rolling contact by Contract NAS1-17359 from the 

NASA Langley Research Center, and work on non-classical friction by the Air 

Force Office of Scientific Research under Contract F49620-84-0024. 

277 
REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

DEMKOIJICZ, L., ODEN, J.T., STROUBOULIS, T. and CAMPOS L., Variational 

Principles and Adaptive Schemes for a Class of Flow Problems with Moving 

Boundaries, Finite Elements in Flow Problems, Vol. VI, John Wiley and 

Sons, Ltd., London, 1985. ----I--- ------- 

ODEN, J.T., and DEMKOWICZ, L., A Posteriori Error Estimates and 

Adaptivity in Finite Element Methods, Integral Methods _____l-_

in Science and 

Engineering, Hemisphere Publishers, Washingtion, 1986. 

ODEN, J.T., and DEMKOWICZ, L., Adaptive Finite Element Methods €or Complex Problems in Solid and Fluid Mechanics, International Conference on 

Finite Elements in Computational Mechanics, 2-6 December, 1985, Bombay 

India. 

DEMKOWICZ, J., and ODEN, J.T., An Adaptive Characteristic Petrov-Galerkin 

Finite Element Method €or Convection-Dominated Linear and Nonlinear 

Parabolic Problems in One Space Variables, Texas Institute For Computational Mechnics, TICOM Report - 85-3, Austin, TX, June, 1985. 

DEMKOWICZ, L., and ODEN, J.T., An Adaptive Characteristic Petrov-Galerkin 

Finite Element Method for Convection-Dominated Linear and Nonlinear 

Parabolic Problems in Two Space Variables, TICOM Report 85-7, 1985. 

DEMKOWICZ, L., DEVLOO, P.Y., and ODEN, J.T., On an H-Type Mesh Refinement 

Strategy Based on Minimization of Interpolation Errors, Computer Methods 

in Applied Mechanics and Engineering. Vol. 53, pp. 67-89, 1985. 

DEMKOWICZ, L., and ODEN J.T., On a Mesh Optimization Method Based on 

Minimization of Interpolation Error, Lnt. J. Eng. Sci., Vol. 24, No. 1, 

pp. 55-68, 1986s. 

ODEN J.T., DEMKOWICZ, L., and STROUBOULIS, T., Adaptive Finite Element 

For Flow Problems With Moving Boundaries. I. Variational Principles and 

A Posteriori Estimates, Computer Methods in Applied Mechanics and 

Engineerix, Vol. 46, pp. 217-251, 1984. 

DEMKOWICZ, L., and ODEN J.T., An Adaptive Characteristic Petrov-Galerkin 

Finite Element Method For Convection-Dominated Linear and Nonlinear 

Parabolic Problems in Two Space Variable, Computational Methods - in 

Applied -- Mechanics, (to appear). 

ODEN J.T., DEMKOWICZ, L., and STROUBOULIS, T., Adaptive Methods €or 

Incompressible Viscous Flow Problems with Moving Boundaries, - Accuracy 

Estimates and Adaptive Refinements in Finite Element Computations 

Wiley and Sons Ltd., London, (to appear), (Proceedings, Lisbon Protugal, 

John 

June 19-22, 1984). 

STROUBOULIS, T., ODEN J.T., and DEMKOWICZ, L., A Posteriori Error 

Estimates for Some Galerkin Space-Time Approximations of the Unsteady 

Navier-Stokes Equations, =OM REPORT 85-5, JUNE 1985. 

278 
12. ODEN J.T., RECKER, E.B., LIN, T.L., and DEMKOWICZ, L., Formulation and 

Finite Element Analysis of a General Class of Rolling Contact Problems 

with Finite Elastic Deformation, Mathematics of Finite Elements with 

Applications, Academic Press Ltd., London, (to appear). 

13. ODEN J.T., BECKER, E.R., LIN, T.L., and HSIEH, K.T., Numerical Analysis 

of Some Problems Related to the Mechanics of Pneumatic Tires: Finite 

Deformation of Cord-Reinforced Rubber Composites, Symposium on Advances 

and Trends in Structures and Dynamics, NASA CN, NASA Langley Research 

Center, (to appear). NASA CP 2335, Oct. 1984, pp 297-307. 

--- --- 

14. ODEN J.T., and LIN, T.L., On the General Rolling Contact Problem for 

Finite Deformations of a Viscoelastic Cylinder, TICOM REPORT 85-, Austin, 

TX, 1985. 

- -- 

15. ODEN J.T., and MARTINS, J.A.C., Models and Computational Methods for 

Dynamic Friction Phenomena, Computer Methods in Applied Mechanics and 

Engineering, Vol. 52, pp. 527-634, 1985. ------- -- 

16. ODEN J.T., and DEMKOWICZ, L., Extraction Methods for Second-Derivatives 

in Finite Element Approximations of Linear Elasticity Problems, 

Communictions in Applied Numerical ---- Methods, Vol. 1, pp. 137-139, 1985. .__-____---- 

17. MARTINS, J.A.C., and ODEN J.T., Interface Models, Variational Principles 

and Numerical Solutions for Dynamic Friction Problems, Mechanics OE -- Material Interfaces, (Proceedings Session on Mechanics of Material 

Interfaces aid Mechanics of Contact Problems, at the ASCE/ASME Mechanics 

Conference, Albuquerque, NM, June 24-26, 1985) 

279 
Figure 1 

M: W = POLYNOMIAL OF IC, IIc, Illc 

V = SOBOLEV SPACE OF ORDER (1 $1 

= (w'.P(Q~))~ for some p. 2 i p < m 

280 

Figure 2 
EM = BU 

ROL I ING CONTACT WITHOUT 

FRICTION 

3( = UNILATERAL CONSTRAINT SET 

=(qfvl q,iH onr,l 

INCOMPRESSIBLF M ATERIAL 

Iq E v I q2 I H on Tc , detVq = 1 a.e. in no) 

Figure 4 

281 
ROI I ING CONTACT WITH FRICTION 

'u = SPACE OF VELOCITY-MOTIONS 

5 [ q 6 v I (waeq1,q2) 6 v I 

YARlATlONALITY B.V.P. 

Figure 6 

I 282 
- 

Q = SPACE OF MULTIPLIER 

detVq E Lp(Qo) ---> 9 = LP’(Q0) 

(l/p) + (l+p’) = 1 

i(p,)(,~) = TRILINEAR FORM 

= p adjV)( : Vq dvo 

Figure 7 

Figure 8 

283 
C1=80 psi C2=20 psi VIS=O FRIC=.3 DISP=R, - H = .1 in. 

300 rpm 

C1=80 p 

Figure 9 

lsi C2=20 psi VIS=O FRIC =.3 DISP=R, - H = .1 in. 

300 rpm 

Figure 10 

2 84 
C1=80 psi C2 = 20 psi VIS=O FRIC=.3 DISP=R, - H = .1 in. 

'00 rpm 

Figure 11 

C1=80 psi C2=20 psi VIS=O FRIC=.3 DISP=R, - H = 

Figure 12 

.1 in. 

300 rpm 

285 
C1=80 psi C2=20 psi VIS=O FRIC=.3 OISP=R, - H = .1 in. 

286 

.6 rpm 

Figure 13 

Driven cavity problem. Optimal mesh after 8 FE recalculations. a=4, ~~4, Y=O 

Figure 14 

c- 4 
ORIGlNAL PAGE 

AND WHITE PHOTOGRAPH 

Heat equation with a dominating convection - 

solution after 1 time step, t =0.02 

Figure 15 

Heat equation with a dominating convection - 

solution after 25 time steps, t = 0.5 

Figure 16 

287 
ORlG INAL 

BLACK AND WHITE 

,, 

. c 

PAGE 

PH07 

Burger's equations. First component of the 

solution after 1 time step, t = 0.02 

Figure 17 

I TIME=@. 1888E-i-88 

-0GRAPH 

Problem with exponential solution. Deviations of the 

from unity. Parameters: h = 2, K,1 local effectivity ratios e 

At = 0.1, 6 = 0.25, M = 4 

Figure 18 

288 
T" 

ORIGINAL PAGE 

BLACK AND WHITE PHOT'OG'R~H 

Problem with exDonentia1 solution. Deviations of the local 

from unity. Parameters h = KY5 effectivity ratios e 

At = 0.1, 6 = 0.5, M = 4 

Figure 1.9 

m 

m 

m 

W 

E 

- 

m 

m 

E 

W 

-I 

W 

z 

m 

rtz 

z H 

LL 

W 

w 

Ill 

z 

w 

W 

w 

W 

H Z 

\ 

Icn 

W 

: 

0 

W 

a 

W 

LL 

\ 

Icn 

W 

$ 

2, 

% 

H 

R 

I 

W 

3 

I- I1c 

w 

3 

J 

m 

I 

IH Z 

> 

E 

W 

-I 

R 

a 

3 

R 

W 

U 

\ 

IZ 

2 

H 

1. Preconditioned Jacobi Conjugate Gradient Scheme 

2. 4-El ements/Ref i nement 

3. Rates = Uniform/Const. = (CI1i2 

Figure 20 

289 
CONTACT SOLUTION ALGORITHMS 

John T. Tielking 

Texas A and M University 

College Station, Texas 

EXPANDED ABSTRACT 

Two algorithms for obtaining static contact solutions are described in this 

presentation. Although they were derived for contact problems involving specific 

structures (a tire and a solid rubber cylinder), they are sufficiently general to 

be applied to other shell-of-revolution and solid-body contact problems. 

The shell-of-revolution contact algorithm is a method of obtaining a point 

load influence coefficient matrix for the portion of shell surface that is expected to carry a contact load. If the shell is sufficiently linear with respect to 

contact loading, a single influence coefficient matrix can be used to obtain a good 

approximation of the contact pressure distribution. Otherwise, the matrix will be 

updated to reflect nonlinear load-deflection behavior. 

The solid-body contact algorithm utilizes a Lagrange multiplier to include the 

contact constraint in a potential energy functional. The solution is found by 

applying the principle of minimum potential energy. The Lagrange multiplier is 

identified as the contact load resultant for a specific deflection. 

At present, only frictionless contact solutions have been obtained with these 

algorithms. A sliding tread element has been developed to calculate friction shear 

force in the contact region of the rolling shell-of-revolution tire model. This 

element allows a relatively general, non-Coulomb, friction law to be specified for 

the contact interface. It has the added advantage of allowing friction to be calculated in the continuous interface and, when coupled with the solid-body contact 

algorithm, will permit analytic investigation of various continuum friction theories 

that have been proposed. 

The outline of future directions for the development of contact solution 

algorithms is: 

I. SHELL-OF-REVOLUTJON CONTACT ALGORITHM 

11. SOLID-BODY CONTACT ALGORJTHM 

111. STATTC AND ROLLJNG CONTACT FRICTTON 

IV. FUTURE DIRECTIONS FOR CONTACT SOLUTION ALGORTTHMS 

291 
I. SHELL-OF-REVOLUTION CONTACT ALGORITHM 

A shell whose geometry and material properties are axisymmetric can be economically modeled by shell-of-revolution finite elements. The SAMMSOR/SNASOR programs (refs. 1,2), for example, permit nonlinear behavior of orthotropic shells of 

revolution to be calculated, including response to nonaxisymmetric loads. The 

algorithm described here was developed to calculate the shell deflection in response 

to a nodal point load, utilizing the calculated response to a sequence of harmonically varying ring loads on the node. The point load solution is then used to construct an influence coefficient matrix, from which the shell contact solution is 

obtained. 

292 
FINITE-ELEMENT TIRE MODEL 

The tire is modeled here by an assembly of axisymmetric shell elements connected to form a meridian of arbitrary curvature and following the carcass midsurface. The elements are homogeneous orthotropic, with moduli determined by the 

ply structure of a particular tire. Details of this model are given in reference 3. 

If the deformation is symmetric about the wheel plane only one-half of the meridian 

is modeled, as shown in figure 1. The finite elements are joined at nodal circles, 

referred to here as nodes. Node 12 in figure 1 is located at the tire bead and is 

given in built-in end condition. 

Figure 1 

293 
SINGLE HARMONIC RING LOADS 

The finite-element tire model will respond to single harmonic ring loads on 

the nodal circles in addition to a uniform inflation pressure load. An approximately linear ring load-deflection response is obtained when an individual ring load 

is applied to any node of the pressurized tire model. 

deflection calculation for a passenger tire model is shown in figure 2. A harmonic 

sequence of stiffness matrices is obtained by applying a sequence of single harmonic 

ring loads to each of the nodes that may be in the tire-pavement contact region. 

An example ring loadTOTAL 3000 - 

RADIAL 

RING 

LOAD (Ib) 

2000 - 

PRESSURE 

I 

0 0.5 I .o 1.5 

RADIAL DEFLECTION (in) 

CROWN LOAD-DEFLECTION DATA CALCULATED WITH A UNIFORM RING LOAD APPLIED TO THE CROWN NODE 

SINGLE HAWNIC RING LOADS APPLIED TO A FINITE-ELEMENT NODE 

Figure 2 

I 2 94 
TRANSFER FUNCTION DEFINITION 

As a consequence of the linearity of the ring load-deflectionresponse, the 

application of a single harmonic ring load produces a displacement field that 

varies circumferentially in the same harmonic as the applied ring load. The definition of the transfer function Tn as the ratio oftheoutput and input amplitudes 

is given below (ref. 4). Since each node responds differently, a transfer function 

matrix Tikln is used to store the stiffness information generated by the ring loads. 

The partitions of this matrix are determined by the direction of the ring load.(Fig.3). 

Single Harmonic Ring Load 

Single Harmonic Displacement 

An cos ne (input) 

Bn cos ne (output) 

Bn 

n 

TRANSFER FIJNCTTON T, = A 

= nth harmonic transfer function relating displacement of node i to an 

nth harmonic ring load on node k Tikln 

LOAD TYPE 

Circumfe rential Radial Axial 

r- I 1 

I I I 

z 

W 

a 

n=0,1,. . . ., N/2 

Figure 3 

295 
POINT LOAD VECTOR {p) AND THE DISCRETE FOURIER TRANSFORM (DFT) 

This application of the discrete Fourier transform uses an even number of 

points (N), equally spaced around the circumference. The example shown in figure 

4 uses N=8 points. A unit load is applied at any point, say point 0. The DFT 

of the load vector yields a set of N coefficients, 

of the coefficients of the conventional Fourier series deftned on the continuous 

interval 0 < 8 < 27~ and representing the unit point load. The point load is 

applied, sequentially, in the radial, axial, and circumferential directions. 

which are approximate values Gj 9 

INFLUENCE COEFFICIENT GENERAT ION 

4 

I1 

{PI = (1, 0, 0, 0, 0, 0, 0, 0) load Lector 

1 N-1 -iPn/N DFT G. = - gk$k We = e 

k=O 

9k = {PI ’ 1 G. = - 

JN j = 0, 1, . . ., N-1 

Figure 4 

296 
INVERSE DISCRETE FOURIER TRANSFORM (IDFT) AND THE INFLUENCE COEFFICIENTS 

Having the unit point load represented by a conventional Fourier series, whose 

of the 

coefficients an are approximately given by the DFT coefficients, the transfer 

functions Tikln are applied, on each harmonic, to obtain the coefficients b 

Fourier series representing the response of the nodal circle to the unit point 

load. The inverse discrete Fourier transform is then used to evaluate the displacements, u , at the N points. These displacements are the elements of the 

influence coeyficienf matrix [AijkL]. 

P 

(Fig. 5). 

- 1 INPUT SERIES COEFFICIENTS a 2 Gn - 

n 

OUTPUT SERIES COEFFICIENTS b = anTikln = -T 1 ikln 

n 

n 

DFT OF DISPLACEMENT VECTOR Gn b 

I!-1 

m = 0, 1, . . .) N-1 ik = G WWmn 

ne 

IDFT u 

n=O 

j = 1, 2, . . .) N ik 

=u Ai j kl j -1 INFLUENCE COEFFICIENTS 

j = l, &+1, . . ., N ik 

j -1 SHIFT: Aijke =u 

- 

SYMMETRY: Akeij - Aijkl 

Figure 5 

297 
INFLUENCE COEFFICIENT MATRIX 

The influence coefficient matrix relates the radial, axial, and circumferential components of the displacement of points on the tire surface to the radial, 

axial, and circumferential components of load at these points. The radial response 

partition, shown in figure 6, is used to obtain a solution for frictionless contact, in which the axial and circumferential force components are known to be zero. 

The matrix here covers 3 points on each of 5 nodes. The point separation with this 

matrix is 11.25 degrees. 

-. 

dl 1 

d21 

d31 

d4 1 

d51 

dl 2 

d22 

d32 

d42 

d52 

.d 

- 

All11 

A2111 

A4111 

A1211 

12211 

4211 

5211 

A3111 

A5111 

\32 11 

- 

A2121 

A4121 

A5121 

A122 1 

A2221 

A422 1 

A5221 

A3121 

A322 1 

A3131 

h131 

k131 

h231 

A2231 

A3231 

4231 

A5231 

Pke = load at point e on node k 

d.. = deflection of point j on node i 

1J 

4141 

A5141 A5151 

A1241 A1251 A1212 

A2241 A2251 A2212 A2222 

A3241 A3251 A3212 A3222 A3232 

4241 4251 4212 4222 4232 h242 

4241 k251 A5252 A5222 k232 4242 h252 

- 

-- 

p1 1 

p21 

'31 

'4 1 

'51 

'1 2 

p22 

'32 

'42 

'52 

-- 

Figure 6 

298 
TOROIDAL SHELL CONTACT SCHEMATIC 

A cylindrical coordinate system is used to locate points on the toroidal 

surface. 

axial directions, respectively. 

plane) and a tire meridian is in an r-z plane. 

The coordinates r, 0, and z indicate the radial, circumferential, and 

The tire equator lies in the r-0 plane (wheel 

After the inflation solution has been obtained, the tire model is deflected 

against a frictionless, flat surface. The contacting surface is perpendicular 

to the wheel plane and positioned at the specified loaded radius Rl, as shown 

in figure 7. 

a priori. 

The vertical load and the contact pressure distribution are unknown, 

Figure 7 

299 
DEFLECTED MERIDIAN 

The deflected shape of the meridian passing through the center of contact is 

shown in figure 8. This shape is calculated by the finite-element tire model for 

the specified tire deflection of one inch. The tire load that will produce a one 

inch tire deflection is calculated to be 10,590 lb. Figure 8 also shows the 

meridian prior to inflation and the calculated shape of the meridian of the inflated 

tire, prior to contact loading. These finite-element meridians follow the carcass 

midsurface, as indicated in Eigure 1. Geometric and material property data on the 

Space Shuttle nose gear tirewereused for the calculated results shown in figures 

8, 9, and 10. 

-- --_ \ 6 - 1.0 in (specified) I 

.,I --__ Uninflated Shape 

1 I I I I 1 

0 1 2 3 4 5 

a (in) 

Figure 8 

300 
CONTACT PRESSURE DISTRIBUTIONS 

The static contact pressure values (psi) calculated for two different loads 

on the Shuttle nose gear tire are shown in figure 9. The number of finite-element 

points in the contact region increases as the tire load increases. A rough estimate of the contact boundary is obtained by extrapolation of the pressure distribution, Integration of the pressure distribution gives the tire load. 

NOSE GEAR TIRE CONTACT PRESSURE DISTRIBUTIONS 

32~8.8 20 PR TYPE VI1 

Inflation Pressure = 300 psi 

\ -- -- . ________- ---------- - - 

703.1 I \ 

A- - / 

/ 

\ i 191.2 327.2 191.2 ' 

\ 

I 

I 117.5 

I I I 364.2 482.6 psi 364.2 I 

I I 

I 117.5 I 

\ I 

\ 191.2 327.2 191.2 I 

703.1 

/ 

/ 

/ 

/ 

--- - -________ -- ------- -_ -- 

-- '. 

(a) 6 = 0.75 in. F, = 5.708 lb 

---_ c______ ------- -- -- 

-- .- 

--- / _--- 527.9 I 

/ 

I 388.6 

285.3 I 

I 

I 48.9 

401.3 I 

I 

I 48.9 

285.3 I 

\ 

\ 388.6 

822.3 

318.5 

173.0 

529.7 psi 

173.0 

318.5 

822.3 

\ 

I 

I 

48.9 I 

I 

388.6 

285.3 I 

401.3 1 

I 

I 48.9 

285.3 I 

I 

388.6 / \ I 

'-- ~ 527.9 _--- #MM 

---- -_____ ___- ----- 

--- -- 

(b) 6 = 1.00 in, F, = 10,590 lb 

Figure 9 

301 
TIRE LOAD VERSUS TIRE DEFLECTION 

An important test of a tire model is its ability to calculate a static 

load-deflection curve. Figure 10 compares the load-deflection curve calculated 

for the Shuttle nose gear tire with measured data for a similiar aircraft tire. 

Although these are both 32 x 8.8 Type VI1 tires, constructional details can 

alter the load-deflection curve (and many other aspects of tire behavior). The 

cord used in the test tire is unknown and may be quite different from the nylon 

cord in the Space Shuttle tire. 

Load-Deflection Curves For 32 x 8.8 Type VI1 Aircraft Tires 

30000 

25000 

20000 

15000 

10000 

5000 

0 

Tire Calculated Load 

Fz (lb) 

/* 

// 24 Pr 

/ 285 Psi '/ Measured Data 

/ 

/ 

(Similiar Tire) 

1 1 I I I I I I I 

0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 

Tire Deflect ion 6 (in) 

0 

Tire Calculated Load 

Fz (lb) 

/ 

/ 

/ 

1 1 I I I I I 

0.25 0.50 0.75 1.00 1.25 1.50 1.75 

Tire Deflection 6 (in) 

/* 

/ 24 Pr 

285 Psi Measured Data 

(Similiar Tire) 

/ 

/ 

- 

2.00 2.25 

Figure 10 

302 

I 
11. SOLID-BODY CONTACT ALGORITHM 

A solid body with a cylindrical surface is often loaded in contact against a 

rigid surface. The contact load may revolve around the body, as in the case of 

a roller or a solid tire, or may remain stationary if the cylinder is used as a 

support cushion. 

Interfacial friction, present in all contact problems, is currently an active 

field of research. The study of frictional behavior is facilitated if the contact 

region is relatively large, as is produced when the body is highly deformable. 

This makes it easier to calculate distributions of normal pressure and tangential 

motion (slip) in the interface. In the case of rubber contact, the behavior 

deviates sufficiently from the Coulomb friction law that other, more physically 

realistic laws, can be easily tested. Since friction is a microscopic phenomenon, 

a contact solution giving continuous distributions of interfacial pressure and 

slip is desirable for analytic purposes. The contact algorithm described here 

provides a continuum solution for frictionless contact, the first' step toward 

analysis of friction in the continuous contact interface. 

303 
A PLANE STRAIN CONTACT PROBLEM 

An elastic semicylinder of radius R is bonded to a fixed surface, figure 

ll(a). 

as shown in figure ll(b). 

A contact load is applied by a rigid plate that deflects the semicylinder 

The problem is formulated in terms of cylindrical material coordinates 

(r,0,z) which identify points in the undeformed body, Bo. 

located by Cartesian coordinates x1 and xg axes shown in figure ll(a) and 

A point Po in Bo is 

x = r cos 0 x2 = r sin 0 x=z 3 

1 

The contact load is assumed to produce a plane strain deformation. Point Po 

moves to position P in the deformed body, B. Point P is located by the Cartesian 

coordinates y.. 

where X3 = 1 2s a specified constant extension ratio. 

With plane strain, y, = yl(r,O), y, = y2(r,0), and y, = X,z 

t x2 

t y2 

Figure 11 

304 
GEOMETRIC DESCRIPTION 

The metric tensors g and Gij, given below, completely describe the elastic iB 

semicylinder before and after deformation. 

problem is solved by finding the functions y,(r,8) and y2(r,e) which determine 

Gij. 

course, be found when x and y are known. (Fig. 12). i i 

Since y3 is known a priori, the 

The displacement field is not utilized in this formulation but it can, of 

(implied sum 

with a = 1,2) 

The Green/Saint-Venant strain tensor components are defined as 

yij = 3i(Gij - 

Und e f ormed 

Metric [g. .I 

1J 

Deformed 

Metric [G..] 1J 

Figure 12 

305 
MATERIAL DESCRIPTION 

The material is assumed to be hyperelastic so that its constitutive properties 

are contained in a strain energy density, W. Isotropy is also assumed. For 

plane strain of an isotropic material, the strain energy is known to be a function 

of only the first and third strain invarients, I, and I3 (ref. 5) 

For general deformation, the strain invarients are given by 

ij 11= g G.. 

1J I,= G/g 

2 

where g = det [g. .] = r for the semicylinder and G = det [G. .]. 1J 1J 

When the material is also assumed to be incompressible (I3 = 1)the constitutive 

behavior is not completely determined by the strain energy density. 

pressure becomes an additional unknown, which can be determined as a Lagrange 

multiplier (ref. 6). This difficulty is avoided if a compressible material model 

is used. 

Hydrostatic 

The material description selected for the contact problem solved here is the 

compressible neo-Hookean model developed for continuum rubber by Blatz and KO 

(ref. 7). The Blatz-Ko model may be expressed as 

where p is the classical shear modulus, K is the bulk modulus, and k is a parameter 

related to atomic repulsion. 

neo-Hookean model for incompressible material. For small strains it reduces to 

the energy density giving Hooke's law for compressible isotropic material. 

When I, = 1, the Blatz-Ko model reduces to the 

306 
CONTACT CONSTRAINT 

The assumption of frictionless contact with a rigid surface, positioned perpendicular to the y, axis (see fig. ll(b)), implies a geometric constraint on1.y on 

the solution function y2(r,0). 

known a priori, the deformed surface is flat and it is known that y2= Rl 

where Rp. is the specified location of the contact surface. 

region, and in the interior of B, the solution must satisfy y, < RL. 

equality constrainton y is converted to an equality constraint by introducing a 

new function s(r,f3), defined by the following equation 

Within the contact region, whose extent is not 

Outside the contact 

This in2 

y, + s = Rl (constraint equation) 

which is valid everywhere on the boundary and in the interior of B. 

s(r,0), called a slack variable, has been used previously in optimization problems 

with an inequality constraint. 

slack variables. 

The function 

Reference 8 gives several examples of the use of 

The contact problem is solved by minimizing the strain energy in B, subject 

to the constraint equation given above. Since plane strain is assumed, the energy 

is uniform along the axis of the semicylinder. Using symmetry, integration of the 

energy density is taken over one-half of the r-8 plane contained in Bo. 

means of a Lagrange multiplier function A(r,O). 

governed by the following functional 

The constraint equation is brought into the energy density functional by 

The contact problem is then 

where 

Although rW is positive definite, F is not positive definite due to the addition of the constraint. Therefore, I may only be regarded as being made stationary instead of minimized by equilibrium solution functions. The integral 

of W, however, is minimized by the equilibrium solution and this is used as a 

check during the solution finding process. 

Through additional analysis (J. T. Tielking, Texas A and M University, 

unpublished data) the Lagrange multiplier function is shown to be an unknown 

constant, identified as the resultant load in the contact region. The constraint 

condition may then be removed from the integral, the slack variable is no longer 

needed, and the contact problem is now governed by 

The solution is obtained by finding ya(r,8) and the constant X which make 

I(ya, A) stationary. (Fig. 13). 

307 
I Exact 

- 

/ I \ 

I 

I I I 

/ / / / / / / -/ R / 

Example: M= N = 2 

Figure 13 

308 
SOLUTION FUNCTIONS 

The numerical solution is obtained by application of the principle of 

stationary potential energy (ref. 6), using the functional I(ya, A)in which X is 

an unknown constant. The solution functions y are taken as two-dimensional 

finite series a 

M N 

i-1 j=l 

M N 

where a 

conditigjis evideat in figure 11 are met by 

and bi. are unknown coefficients. The symmetry and geometric boundary 

i (r,e) = r sin (2je) ij 

yl 

i ij 

y, (rye) = r sin [(2j -1.)0] 

The above functions allow the energy density to be integrated, thereby reducing the functional I to an algebraic function of the 2 x MX N + 1 unknown 

constants a ijy bij. and A 

The functional I is made stationary by the constants obtained from the following 

set of simultaneous nonlinear equations: 

aaij =O and---- - 0 for i= 1,2,...,M and j = 1,2, ..., N 31 81 

abij 

This system is solved in an iterative manner by the Newton-Raphson method. Using 

the starting values a = b = X = 0, five or six iterations (which give successive 

corrections to these canstakis) are usually sufficient. 

tinued until the corrections appear to have negligible effect on the solution 

functions y9 (rye). 

check for minimization. 

i' The iterations are conThe energy density is evaluated after each iteration to 

309 
DEFORMATION SOLUTION 

Numerical results have been obtained using material constants 1-1 = 75 psi 

(shear modulus), K = 475,000 psi (bulk modulus), and k = 13.3 in the Blatz-Ko 

model. The values of K and k are taken from reference 7 where they are shown to 

give a good fit to hydrostatic compression data on Butyl tread rubber (polyisobutylene). The shear modulus is believed to be a realistic estimate, based on 

Treloar's statement (ref. 9) that the shear modulus of rubber is lower than the 

bulk modulus by a factor of about 10 . 4 

The computer-generated drawing below shows coordinate circles and radii 

before deformation (xl and x2, dashed lines) and the deformed configuration 

(yl and y2, solid lines) of these circles and radii. 

duced by a 10-percent de€lection of the contacting surface (shown dashed). The 

deformation solution, y1 (rye) and y2 (rye), is obtained in a 16-term series 

for each function; the strain energy is minimized by the coefficients a and 

bij for i=1,2,3,4 and j=1,2,3,4, found after six Newton-Raphson iterati%. 

This computation took 30 seconds of CPU time on a mainframe computer (Amdahl 

470/V8). 

preted as a 93.1 lb load needed for a 10-percent deflection if the semicylinder 

extends 1 inch in the z-direction. (Fig. 14). 

The deformation is proThe Lagrange multiplier obtained in this solution is X=93.1, inter- 

--- Undeflected 

10% Deflection 

Figure 14 

310 
111. STATIC AND ROLLING CONTACT FRICTION 

Static Contact Friction. A body is brought into static contact by motion 

perpendicular to the contact plane. During this motion, the contact boundary 

expands until the resultant of the normal contact pressure reaches equilibrium 

with the external load applied to the body. As the contact region is formed, shear 

forces are generated by tangential motion of contacting surface points. These 

shear forces are frictional and transient, reaching equilibrium levels when the 

body itself comes into equilibrium. Although the body may be assumed elastic, and 

thus conservative, the frictional shear forces are not conservative. 

The formidable problem of calculating a static contact solution including the 

effect of friction is alleviated somewhat by assuming Coulomb's law of friction 

is valid in the contact region. An algorithm for including Coulomb friction in 

a static contact problem has been developed by Rothert et al. (ref. lo). Although 

Coulomb friction may be taken for an approximate analysis of frictional contact, 

mathematical and physical uncertainties arise when it is assumed. Nonclassical 

friction laws have been proposed by Oden and his coworkers (e.g., ref. 11). Although these appear to have been developed mainly for metals, they may be applicable to more deformable material such as rubber. 

Rolling Contact Friction. This discussion is limited to steady rolling 

under a constant load. Neglecting hysteretic effects in the body, the power input to maintain steady rolling is balanced by the work rate of the friction forces 

in the contact region. In steady rolling, a contact region, whose boundary is 

fixed by the load, is continuously generated. The normal pressure and sliding 

velocity at a given location within the contact boundary do not change with time 

so steady-state frictional behavior is maintained. 

with friction is therefore much easier to analyze and provides a mechanism for 

the study of nonclassical friction theories. 

in the contact region of a tire rolling at constant velocity will be described 

next . 

The rolling contact problem 

An algorithm for calculating friction 

311 
SLIDING TREAD MODEL 

This is a tread element model developed to convert a frictionless rolling 

contact solution into a solution for rolling contact with friction. 

Passes through the contact region with the velocity found for frictionless contact. 

This is termed the carcass velocity, V,, whose distribution is symmetric about the 

center of the footprint as sketched below 

The element 

V 

entry4 V exit 

C 

// / / 1 1 1 

For a free-rolling tire, the amplitude of the footprint sliding velocity is 

very small. 

be about 50 ips in a frictionless footprint. 

At 60 mph (1056 ips), the peak V, is calculated (by the author) to 

In free-rolling, the normal contact pressure distribution, p, is essentially 

unchanged by friction. The sliding velocity distribution, however, is significally altered in an interactive manner. 

force, F,, on the actual sliding velocity, Vs, and normal pressure, p, at a 

A hysteretic theory of tire-pavement 

I friction proposed by Schapery (ref. 12) gives the dependence of the friction 

I point in the contact region. This is expressed as 

(hysteretic theory) 

where a, b, and B are material friction properties. The sliding tread element 

model, shown in figure 15(a), is viscoelastic with stiffness and damping parameters 

K and c. 

the sliding velocity Vs. 

integrated to calculate Vs. 

Sliding friction, F,, causes the element to deform, thereby influencing 

The following nonlinear differential equation is 

dVS C K(Vc-Vs) + CV 

I In this equation, dF,/dVs is the rate of change of sliding friction with sliding 

velocity. 

above or measured experimentally. Footprint transit time, t, is taken as the 

independent variable. 

steady rolling. 

This can be obtained by differentiating the hysteretic theory given 

The time is equivalent to location in the footprint for 

312 
Figure 15(b) shows a schematic diagram of the sliding tread model and its 

function in converting a frictionless sliding velocity distribution into sliding 

velocity influenced by friction. 

SL I DING 

TREAD 

MODEL 

C 

I+ 

vs I BDa6_T-L”y”” 

I I 

I I I 

I I 

I I 

(b) 

Figure 15 

313 
IV. FUTURE DIRECTIONS FOR CONTACT. SOLUTION ALGORITHMS 

As in other areas of solid mechanics, future research on contact problems 

will be directed towards obtaining solutions valid for large deformations. The 

finite-element method seems particularly well suited for application to contact 

problems and special elements have already been developed for this purpose. Continuum mechanics research on contact problems should not be neglected, however. 

A large-deformation contact solution in terms of continuous functions will prove 

valuable in the analysis of contact with friction and the assessment of friction 

laws now being proposed for deformable bodies. 

A true contact problem is one in which the contact boundary and interfacial 

pressure distributions are unknown a priori. 

problems will be displacement prescribed: 

contact surface is specified and integration of the calculated normal component 

of the contact pressure gives the resultant load. Some effort should be directed 

towards a load-specified contact problem, perhaps utilizing the principle of 

stationary complementary energy to calculate the interfacial pressure distribution subject to the prescribed load constraint. Validated solutions for frictionless contact are essential prior to including the effect of friction on the 

contact solution. 

At present, it appears that such 

Deflection of the body toward the 

In the analysis of friction, it seems that the study of rolling contact as 

a steady-state problem has much to offer. As friction is an interactive phenomenon, at least in regard to sliding velocity, sophisticated algorithms are 

needed to generate the frictional contact solution from the solution for frictionless contact (which will undoubtably be the starting point). 

The following schematic, figure 16, outlines a progression of research on 

contact problems. Linear sliding contact is excluded from the outline as this 

is usually a transient situation leading to accelerated wear and abrasion. 

Much more will be gained by research focused on rolling contact which will, in 

any case, include sliding. 

3 14 
PROGRESSION OF RESEARCH ON FRICTIONAL CONTACT PROBLEMS 

Large Deformation 

Contact Problems 

Influence Coefficient 

Finite-Element Methods 

Algorithms 

Lagrange Multiplier 

Algorithms 

Continuum Methods 

+ Frictionless Contact 

Surface 

Characterization Friction Laws 

I I J 

I 

Frictional Contact 

I 

r’ I 

Static Contact with Friction Rolling Contact with Friction 

Steady-State ProbZem 

I 

I I I 

Assessment of Nonclassical 

Friction Laws 

Analysis of Wear 

Figure 16 

315 
REFERENCES 

1. Tillerson, J.R.: and Haisler, W.E.: SAMMSOR I1 - A Finite Element Program to 

Determine Stiffness and Mass Matrices of Shells of Revolution. TEES-RPT-70-18, 

Texas A&M University, Oct. 1970. 

2. Haisler, W.E.; and Stricklin, J.A.: SNASOR I1 - A Finite Element Program for 

the Statis Nonlinear Analysis of Shells of Revolution. TEES-RPT-70-20, Texas 

A&M University, Oct. 1970. 

3. Schapery, R.A.; and Tielking, J.T.: Investigation of Tire-Pavement Interaction During Maneuvering: Theory and Results. Report No. FHWA-RD-78-72, 

Federal Highway Administration, June 1977. 

4. Tielking, J.T.; and Schapery, R.A.: A Method for Shell Contact Analysis. 

Computer Methods in Applied Mechanics and Engineering, Vol. 26, No. 2, 

pp. 181-195, May 1981. 

5. Green, A.E.; and Adkins, J.E.: Large Elastic Deformations. Oxford University Press, London, 1960. 

6. Levinson, M.: The Application of the Principle of Stationary Potential 

Energy to Some Problems in Finite Elasticity. Journal of Applied Mechanics, 

Vol. 32, NO. 3, 1965, pp. 656-660. 

7. Blatz, P.J.; and KO, W.L.: Application of Finite Elastic Theory to the 

Deformation of Rubbery Materials. Transactions of the Society of Rheology, 

Vol. 6, 1962, pp. 223-261. 

8. Beveridge, G.S-G.; and Schecter, R.S.: Optimization: Theory and Practice. 

McGraw-Hill, 1970, p. 264. 

9. Treloar, L.R.G.: The Physics of Rubber Elasticity, 2nd edition. Oxford 

University Press, London, 1958, p. 74. 

10. Rothert, H.; Idelberger, H.; Jacobi, W.; and Niemann, L.: On Geometrically 

Nonlinear Contact Problems with Friction. Computer Yethods in Applied 

Mechanics and Engineering, Vol. 51, 1985, pp. 139-155. 

11. Oden, J.T.; and Pires, E.B.: Nonlocal and Nonlinear Friction Laws and Variational Principles for Contact Problems in Elasticity. Journal of Applied 

Mechanics, Vol. 50, No. 1, March 1983, pp. 67-76. 

12. Schapery, R.A.: Analytical Models for the Deformation and Adhesion Components of Rubber Friction. Tire Science and Technology, Vol. 6, No. 1, 

February 1978, pp. 3-47. 

316 
N89=24653 

EXPLOITING SYMMETRIES IN THE MODELING 

AND ANALYSIS OF TIRES 

Ahmed K. Noor, C. M. Andersen, and John A. Tanner 

NASA Langley Research Center 

Hampton, Virginia 23665 

INTRODUCTION 

In recent years nonlinear analysis of static and dynamic problems has become 

the focus of intense research efforts. This endeavor has prompted the development 

of versatile and powerful finite-element discretization methods as well as of improved numerical methods and software systems for nonlinear static and dynamic 

analysis of structures and solids. One of the most challenging applications of 

computational structural mechanics is the numerical simulation of the response of 

aircraft tires during taxi, takeoff and landing operations. The commonly used 

models for predicting the tire response are reviewed in Refs. 1 to 3. Figure 1 

lists some of the difficulties encountered in the modeling and analysis of tires and 

their implications. 

fht, the tire is a composite structure composed of rubber and textile constituents which exhibit anisotropic nonhomogeneous material properties. The laminated carcass of the aircraft tire is thick enough to allow significant transverse 

shear deformation. 

of unavoidable imperfections, the cross section is unsymmetric; and tkirtd, the tire 

is subjected to inflation pressure and to a variety of unsymmetric mechanical and 

thermal loads which can result in large structural rotations and deformation, as 

well as to a variation in the characteristics of the tire constituents. Moreover, 

the detailed stress and temperature distribution in the tire may require the use of 

three-dimensional finite elements in certain regions of the tire. 

Second, the tire geometry is complicated and due to the presence 

The aforementioned difficulties make the computational expense of the numerical 

simulation of the tire response prohibitive. Hence, the need for the development of 

modeling techniques and analysis methods to reduce this expense. Among the inodeling 

strategies which show promise in reducing the cost is the exploitation of symmetries 

and qu~i-~ymm~~ exhibited by the tire response. 

D I FF I CULTY i 

o TIRE SHAPE 

- COMPLEX GEOMETRY 

- IMPERFECTIONS 

o TIRE MATERIAL AND CONSTRUCTION 

- ANISOTROPY 

- NONHOMOGENEITY 

o TIRE LOADS 

- UNSYMMETRY 

IMPLICATION 

o LARGE MODEL SIZE 

o COMPUTATIONALLY EXPEN I VE 

Figure 1 

317 
OBJECTIVES AND SCOPE 

The objectives of this paper are listed in Figure 2. They are 

1) To review the different types of symmetry exhibited by the tire response 

2) To present simple and efficient computational procedures for reducing the 

size of the analysis model of tires 

3) To discuss the potential of the proposed techniques and their application 

to practical, quasi-symmetric tire problems 

To sharpen the focus of the study, discussion is limited to two-dimensional 

shell models of the tire, with elliptic cross-section and linear material response. 

The analytical formulation is based on a Sanders-Budiansky-type shell theory with 

transverse shear deformation, anisotropic material behavior, and geometric nonlinearities (moderate rotations) included (Refs. 4 and 5). Displacement finiteelement models are used for the discretization. However, the procedure presented 

herein is expected to be particularly useful for the analysis of three-dimensional 

tire models. 

0 B JECTl VES 

0 REVIEW SYMMETRIES PRESENT IN TIRES 

PRESENT TECHNIQUES FOR MODEL-S I ZE REDUCTIONS I N QUASI -SYMMETRI C 

PRO B LEM S 

0 DISCUSS POTENTIAL APPLICATIONS OF TECHNIQUES 

SCOPE 

TWO-DIMENSIONAL SHELL MODELS 

0 LINEAR MATERIAL RESPONSE 

0 ELLIPTIC CROSS SECTION 

0 DISPLACEMENT F. E. MODELS 

Figure 2 

318 
I SYMMETRIES EXHIBITED BY TIRE RESPONSE 

The three types of symmetry commonly exhibited by the tire response are shown 

in Figs. 3 and 4. Also, the difference between the symmetries of orthotropic and 

anisotropic tires are illustrated. 

The first type of symmetry is the aXh! hyt??mw exhibited by tires whose 

geometric, material characteristics, and loading are independent of the circumferential coordinate, i.e., axisymmetric. The response of these tires will also be axisymmetric. An example of this situation is that of a tire subjected to uniform 

inflation pressure. For an orthotropic tire the generalized displacements u, w and 

$1 are axisymmetric and v and $2 are zero. 

alized displacements are nonzero for an anisotropic tire. 

In contradistinction, the five generRe&tection (O/t m&tOhl hymm~ with respect to coordinate planes is exhibited 

by the response of orthotropic tires when subjected to loadings that exhibit the 

same type of symmetry. 

hVUAi0n) hynnntfky with respect to the center of symmetry shown in Fig. 4. 

these symmetries are exploited in the finite-element analysis, the size of analysis 

model for an anisotropic tire is twice that of the corresponding orthotropic tire 

(see, for example, Refs. 6 and 7). 

Anisotropic tires, on the other hand, exhibit /tatationat (OIL 

If 

When the external loading exhibits phodic (on tAanA~ationcd) aymmd&.j, the 

tire response also exhibits periodic symmetry. In orthotropic tires this is demonstrated by the presence of more than two planes of reflection symmetry. In anisotropic tires periodic symmetry is demonstrated by the presence of more than two 

centers of rotational (inversion) symmetry. Again, the size of the analysis model 

for an anisotropic tire with periodic symmetry is twice that of the corresponding 

orthotropic tire (see Fiq. 4). 

AXIAL SYMMETRY - e. g. INFLATION PRESSURE 

W 

ANI SOTROP I C 

U u, v, w, A* 

V 

ORTHOTROPIC 

u, W#+I 

V v,+2 =o 

ANALY S I S MODEL 

TYP CAL MER DlAN 

Figure 3 

319 
REFLECT1 ON AND 

ROTAT I ONAL SYMMETRIES --_ . . 

wt 0 RTHOTRO P I C 

RCITAT13NAL SYMMETRIES) 

(REFLECT1 ON AND 

s+ 

ANISOTROPIC 

(ROTAT IO NAL SYMMETRY 

ONLY) 

PER I OD I C SYMMETRY I I I 1 

Note: Shading shows analysis region. 

Figure 4 

320 
QUASI-SYMMETRIC PROBLEMS 

LOAD I NG 

SYMMETR I C 

Figure 5 lists a number of quasi-symmetric problems. The three basic quasisymmetric problems are the ones for which either the tire material, loading, or geometry are not symmetric, but the other two are symmetric. Here material anisotropy 

is considered to be a source of reflection unsymmetry. The unsynmetry in geometry 

can be caused by the presence of unsymmetric imperfections in the tire. In a practical situation combination of the three basic types of unsymmetry can exist. In the 

present study, a computational procedure is presented for reducing the size of the 

analysis models for quasi-symmetric problems of tires to those of the corresponding 

symmetric problems. 

GEOMETRY 

AXIALLY SYMMETRIC 

CASE 

- - 

I1 

111 

MATER I AL 

Figure 5 

321 
BASIC IDEA OF MODEL-REDUCTION TECHNIQUE 

FOR QUASI-SYMMETRIC PROBLEMS 

Figure 6 lists the two key elements of the model reduction technique when applied to the finite-element analysis of anisotropic tires with symmetric geometry 

subjected to symmetric loading. The two elements are: a) decomposition of the 

stiffness matrix into the sum of an orthotropic and nonorthotropic (anisotropic) 

parts; and b) successive application of the finite-element method and the classical 

Rayleigh-Ritz technique. The finite-element method is first used to generate few 

global approximation vectors (or modes). Then the amplitudes of these modes are 

computed by using the Rayleigh-Ritz technique. 

ANISOTROPIC MATERIALS 

0 DECOMPOSITION OF MATRICES IN GOVERNING FIKITE-ELEMENT EQUATIONS 

INTO ORTHOTROPIC AND NONORTHOTROPIC PARTS 

0 SUCCESSIVE APPLl CATIO\! OF 

- FINITE-ELEMENT ANALYSIS TO GENERATE A FEW GLOBAL APPROXIMATION 

VECTORS USING SAME SIZE MODEL AS FOR THE ORTHOTROPIC CASE 

- CLASSICAL RAYLEIGH-RITZ TECHNIQUE TO COMPUTE AMPLITUDES OF 

AP PROX I MAT ION VECTORS 

Figure 6 

322 
MATHEMATICAL FORMULATION 

Figure 7 outlines the mathematical formulation for the proposed model-reduction I 

technique when applied to linear problems of anisotropic tires. The global stiffness matrix [K] is decomposed into orthotropic and nonorthotropic matrices [KIo 

and [K],, respectively. The nonorthotropic matrix [Kla is multiplied by a tracing 

parameter X which identifies & the nonorthotropic material coefficients. The 

original finite-element equations correspond to the case A=l. 

The global approximation vectors are selected to be the solution corresponding 

to X=O (zero nonorthotropic matrix) and its various-order derivatives with respect 

to X (path derivatives). The path derivatives are obtained by successive differentiation of the governing finite-element equations with respect to A. Note that the 

coefficient matrix appearing on the left-hand sides of the recursion formulas is 

[KIo, and that the size of the analysis region used in evaluating each of the global 

approximation vectors is the same as that for the orthotropic case (X=O). 

The vector {XIo and its path derivatives are now chosen as approximation vectors, and the vector of nodal displacements for the anisotropic tire, {XI, is expressed as a linear combination of these vectors. A Rayleigh-Ritz technique is used 

to replace the original finite-element equations by a reduced system of equations in 

the unknown parameters, {$I, which represent the amplitudes of the global approximation vectors. 

GOVERNING FINITE-ELEMENT EQUATIONS 

[K] IX; = jP; 

LET [K] = [K], + h[K], 

h = TRACING PARAMETER 

BAS1 S REDUCTION 

WHERE 

It= 1Vj VECTOR OF AMPLITUDES OF MODES 

GLOBAL APPROXIMATION VECTORS 

I [K], ;xio = IPI OBTAINED FROM SAME SIZE MODEL 

AS FOR THE ORTHOTROPIC CASE 

REDUCED EQUATIONS (ERAYLEIGH-R ITZ TECHNIQUE) 

Figure 7 

323 
APPLICATION TO ANISOTROPIC TIRES 

As a first application of the proposed model-reduction technique consider the 

anisotropic tire subjected to the symmetric localized loading shown in Figure 8. As 

shown in Figure 4, the tire response exhibits rotational (inversion) symmetry. It 

does not exhibit reflection symmetry, and therefore, the analysis model consists of 

half the tire. As can be seen from the contour plots of Figure 8, all the global 

approximation vectors exhibit reflection symmetry (and antisymmetry) and therefore, 

they can be obtained by analyzing only one quadrant of the tire (same size model as 

that used for analyzing the corresponding orthotropic tire). 

GLOBAL APPROXIMATION 

DOUBLY SYMMETR IC AND 

PATTERNS ARE KNOWN 

VECTORS ARE ALL 

THE SYMMETRY 

Figure 8 

324 
ACCURACY AND CONVERGENCE OF SOLUTIONS OBTAINED 

BY PROPOSED TECHNIQUE 

An indication of the accuracy and convergence of the solutions obtained by the 

proposed model reduction technique is given in Fig. 9. The standard of comparison 

is taken to be the direct finite-,element solution of the anisotropic tire. As can 

be seen from Fig. 9, the solutions obtained by the proposed technique are highly 

accurate even when a small number of approximation vectors are used. Numerical experiments have shown that for highly anisotropic tires no more than five approximation vectors are needed. 

1.0 }- 

lo 

U = TOTAL STRAIN ENERGY 

W 

W Full 1 

1 I 1 I 1 I I 

U 

FULL 

- Oa5 t 

1234567 

NUMBER OF APPROX. VECTORS 

Figure 9 

325 
APPLICATION OF MODEL-REDUCTION TECHNIQUE TO PREDICTING NONLINEAR 

RESPONSE OF ORTHOTROPIC TIRES SUBJECTED TO ASYMMETRIC LOADING 

As another application of the proposed model-reduction technique, consider the 

orthotropic tire subjected to the localized loading shown in Fig. 10. The analysis 

model consists of one quadrant of the tire. For linear problems, the decomposition 

of the loading into symmetric and antisymmetric components and the consequent reduction of the size of the analysis model to one-octant of the tire are well known. 

However, it is generally assumed that such a decomposition is not useful for nonlinear problems in which the principle of superposition is not applicable. 

The foregoing model-reduction technique can be used to reduce the size of the 

analysis model to one octant of the tire. This is accomplished by: 1) decomposition of the given loading into symmetric and antisymmetric components with load 

parameters p1 and p2, respectively; and 2) use of the multiple-parameter reduction 

technique described in Refs. 8 and 9. The vector of nodal displacements of the tire 

is approximated by a linear combination of few global vectors or modes. These vectors are selected to be the various-order derivatives of the displacment vector with 

respect to load parameters p1 and p2. 

p1=p2=0. 

symmetries (and/or antisymmetries), and therefore, each vector can be evaluated using 

only one octant of the tire. 

Each of these vectors is evaluated at 

Figure 10 shows that the global approximation vectors exhibit periodic 

p1 p2 

A S A S A 

W 

ASYMMETRIC LOAD I NG 

S and A refer to lines 

of reflection symmetry 

A S A S A 

U S S S S S 

and antisymmetry, - '0 

re s pect i ve I y. 

Figure 10 

326 
POTENTIAL OF PROPOSED MODEL-REDUCTION TECHNIQUE 

The proposed model reduction technique appears to have high potential for analysis of practical tire problems. In particular, in the presence of combinations of 

unsymmetries in the material (viz. anisotropy), the geometry or loading on the tire, 

the same size model can be used as for symmetric material (viz., orthotropic), geometry and loading (Fig. 11). This is accomplished by the introduction of a tracing 

parameter for each of these unsymmetric effects; and the successive application of a 

reduction method with each of these parameters. The global approximation vectors 

are selected to be the various-order derivatives with respect to the tracing parameters. 

0 ASYMMETRIC LOAD1 NG AND ANI SOTROPY 

0 ANISOTROPY AND INITIAL UNSYMMETRIC IMPERFECTIONS 

0 ASYMMETRIC LOADING AND INITIAL UNSYMMETRIC IMPERFECTIONS 

THREE-DIMENSIONAL MODELS 

MIXED FI NITE-ELEMENT MODELS 

Figure 11 

327 
SUMMARY 

In summary, a computational procedure is presented for reducing the size of the 

analysis models of tires having unsymmetric material, geometry and/or loading. The 

two key elements of the procedure when applied to anisotropic tires are: a) decomposition of the stiffness matrix into the sum of an orthotropic and nonorthotropic 

parts; and b) successive application of the finite-element method and the classical 

Rayleigh-Ritz technique. 

global approximation vectors (or modes). Then the amplitudes of these modes are 

computed by using the Rayleigh-Ritz technique. 

The finite-element method is first used to generate ~QLU 

The proposed technique has high potential for handling practical tire problems 

with anisotropic materials, unsymmetric imperfections and asymmetric loading. It is 

also particularly useful for use with three-dimensional finite-element models of 

tires. 

MODEL-SIZE REDUCTION PROCEDURE PRESENTED FOR ANALYSIS OF TIRES 

BASED ON 

- DECOMPOSITION OF MATRICES 

- SUCCESSIVE APPLICATIONS OF FINITE ELEMENTS AND 

RAYLEIGH-RITZ TECHNIQUES 

0 PROPOSED TECHNIQUE HAS HIGH POTENTIAL FOR HANDLING 

- ASYMMETR I C LOAD I NG 

- ANISOTROPIC MATERIALS 

- UNSYMMETRIC IMPERFECTIONS 

- THREE DIMENSIONAL MODELS 

Figure 12 

328 
REFERENCES 

1. Clark, S. K. (ea.), "Mechanics of Pneumatic Tires," U.S. Dept. of Transportation, National Highway Traffic Safety Administration, Washington, D.C., 1981. 

2. Tanner, J. A. (compiler), "Tire Modeling," NASA CP-2264, 1983. 

3. Noor, A. K. and Tanner, J. A., "Advances and Trends in the Development of Computational Models for Tires," Computers and Structures, 1985. 

4. Sanders, J. L., "Nonlinear Theories for Thin Shells," Quarterly of Applied 

Mathematics, Vol. 21, April 1963, pp. 21-36. 

5. Budiansky, B., "Notes on Nonlinear Shell Theory," Journal of Applied Mechanics, 

Vol. 35, June 1968, pp. 393-401. 

6. Noor, A. K. and Camin, R. A., "Symmetry Considerations for Anisotropic Shells," 

Computer Methods in Applied Mechanics and Engineering, Vol. 9, 1976, pp. 317- 

335. 

7. Noor, A. K., Mathers, M. D. and Anderson, M. S., "Exploiting Symmetries for 

Efficient Postbuckling Analysis of Composite Plates," AIAA Journal, Vol. 15, 

January 1977, pp. 24-32. 

8. Noor, A. K. and Peters, J. M., "Recent Advances in Reduction Methods for Instability Analysis of Structures," Computers and Structures, Vol. 16, 1983, pp. 67- 

80. 

9. Noor, A. K., Andersen, C. M. and Tanner, J. A., "Mixed Models and Reduction 

Techniques for Large Rotation, Nonlinear Analysis of Shells of Revolution with 

Application to Tires," NASA TP-2343, October 1984. 

329 
PANEL DISCUSSION, Wednesday, June 19 - TIRE HODEL-ING 

J. A. Tanner, NASA Langley Research Center: 

ing in the general area of tire modeling. However, I believe that the content 

of the papers is actually broader than tire modeling and that we are really 

dealing with the more general issues associated with computational structural 

mechanics. I was rushing some of the authors this morning, trying to rush them 

through their presentations, and I am concerned that they may not have made all 

the points they wanted to make. 

ask each author to take about 2 or 3 minutes to summarize any points that he 

would like to emphasize and to address points not made in his presentation. 

After the summary statements, the floor will be opened for additional questions. 

I am going to start immediately to my right with Dr. Tabaddor who gave the first 

paper this morni ng. 

You've heard five papers this mornIn an effort to make amends, I would like to 

Farhad Tabbador, 9. F. Goodrich Company: 

point on the material properties of composites. There is a great need for 

efforts in characterization of composites, and I think that theories developed 

over the past decade or so for rigid composites do not quite apply to soft 

composites. 

initiated by NASA to do this basic work. 

I would just like to reiterate the 

I would recommend that maybe some sort of grant effort might be 

Tanner: 

characteristics of the new computers that will be coming on line. 

maybe you would have some more things you wanted to say on that subject. 

Dr. Padovan, in your presentation you dealt with the architectural 

I thought 

Joe Padovan, University of Akron: 

of some real number crunching power so that we could really push models to the 

limit. Secondly, I think that we are all too interested in the local detail 

information in the model and I think we are not really trying to capture the 

true essence of what a tire does, and that is to provide good ride characteristics. 

I think more effort should be put towards characterizing the overall features of 

the true structure and leave most of the stress analysis to the tire industry 

where it belongs. I think the dynamic characteristics of the tire are most important. More effort has to be put into that kind 

of modeling. 

you really want major changes in the technologies that have been in place for 40 

years then you have to spend a great deal of money. 

I think we would like to see the availability 

Now I am speaking for myself. 

Thirdly, more money has to be put into the effort in general. If 

331 

PRECEDING PAGE UNM ROT FlLMED 
Tanner: 

about error analysis. 

their solution accuracies are verified. 

cal to the verification of any modeling strategy. 

further on this subject? 

Dr. Oden, in your presentation this morning you spent some time talking 

I imagine that error analysis is critiThe tire industry will not use any analysis tools unless 

Would you care to comment 

J. Tinsley Oden, The Computational Mechanics Company, Inc.: 

see the word "error" used more and more in meetings such as this because I 

believe the users of finite element methods have finally reached a point where 

they are beginning to demand some assurance as to the re1 iabil ity and qual ity of 

solutions. 

now machines to handle the algorithms, but how good are the answers? 

are evolving that will someday allow us to answer that question with some degree 

of confidence. However, there are methods at our disposal now that are based on 

estimations of errors that will allow us to incorporate adaptive strategies into 

very general finite element structural calculations. I heartily support the 

activity in this area. I think we are going to see some error estimation and 

adaptivity become an integral part of finite element calculations in the near 

future. 

It is gratifying to 

We have spent 10 years in developing elements, then algorithms, and 

Methods 

Tanner: 

and you presented results from static contact, both with and without friction. 

I was particularly interested in your last comment, however, when you indicated 

that future studies ought to be concerned with rolling contact. 

might want to expand a little on that thought. 

Dr. Tielking, the central thane of your paper was contact algorithms 

I thought YOU 

J. T. Tielking, Texas A&M University: 

more about the study of contact with friction. Of course, all contact involves 

friction. 

of a rigid body--we really don't have such structures. I feel that the finite 

elanent models certainly do have a place in the study of friction, but friction 

forces vary continuously through the interface, they depend, of course, on the 

variations of several variables, typically interfacial pressure, normal pressure, and sliding velocity. It is now recognized that there is sliding velocity 

everywhere in the tire contact region. The assumption of regions of adhesion 

and sliding are good for approximations, but not for more accurate results. 

feel that the continuous sliding models should not be neglected. One of my 

motivations for doing that plain strain cylinder problem was to get a large 

Yes, I would like to say a little bit 

The assumption of frictionless contact is analogous to the assumption 

I 

332 
deformation contact solution which could be used in the study of friction. 

Therefore, I am very pleased to see that you are not restricting the CSM 

activity to finite element methods and that you will entertain alternative 

methods of solution. Finally, I appreciate your concern for university funding. 

Foreign governments have been sending their best students to our universities 

for a long time and we are very happy to have them, but we would like to be able 

to fund our own students. I think organizations such as NASA could certainly do 

a great deal to see that students from our own country can be funded as well as 

the top students fran foreign countries. 

Tanner: Now I will let the first two authors on the last paper have their say. 

A. K. Noor, George Washington University: I would like to make a few comments 

about the paper. I think first of all this was really a study that aims at 

developing a general computational strategy. In very general terms, when we are 

confronted with a complex problem, we generate the solution to that problem 

starting from a simpler problem or perhaps even from a heirachy of simpler problems. I think this is very much an engineering-type approach. These simpler 

problems will each be associated with a control parameter. What we are doing is 

essentially using a perturbation technique. 

presentation was an extension of the classical perturbation technique which is 

limited to small values of the perturbation parameter. The technique that we 

have developed is not limited to small values of the perturbation parameter; it 

works well even if you have very large values for the perturbation parameter. 

Our example was for a highly anisotropic tire. The perturbation parameter was 

indicative of the degree of anisotropy. The technique, I think, can be essentially summed up as a combination of operator splitting, where we did the splitting on the matrices, and a multiple parameter extended perturbation. The other 

comment which I would like to make is that you get the sensitivity information 

free because all that information is needed in the daily use of the equations. 

In other words, the sensitivity information is a by product of the solution 

algorithm. The final comment is that the CPLJ time was reduced by a factor Of 

eight or more for the problems that you have seen but that reduction factor can 

be much higher for dynamic problems. 

What we have heard in the last 

333 
C. M. Anderson, College of William and Mary: Well , to paraphrase our paper just 

one other way, if the geometry, material properties, and loading in a structure 

exhibit a kind of symmetry, then you may well expect the solution will exhibit 

that same symmetry. It is very easy to take that symmetry into account and 

reduce your model size. Nhat we are saying is that one can now introduce a 

higher symmetry group into the problem. 

have that full symmetry, one can find basis functions which represent the higher 

symmetry group. 

For nonlinear problems which do not 

The solution that we are looking for will be a linear combinathose basis functions. Therefore, we can still achieve the reduced 

ze even though the full symmetry is not present in the final solution 

are 1 ooki ng for. 

tion of 

model s 

that we 

Tanner: 

tire industry have been involved in a joint program called the National Tire 

Modeling Program. 

out infringing on the proprietary rights of the individual tire companies. 

objective of the program is to develop tire analysis tools to help streamline 

the ti re design process. 

in this workshop is an indication of the strong industry support for the 

p r og ram. 

I have a few closing comments. For the past 2 years, NASA and the U.S. 

The program provides a forum for technical discussions withThe 

The number of ti re industry representatives involved 

The last point that I want to make involves the relationship between tire modeling studies and computational structural mechanics in general. The paper that 

Dr. Tabaddor presented this morning gives an indication of the material property 

concerns that exist in the tire industry. A1 though some of the concerns expressed by Dr. Tabaddor are unique to the rubber-cord composites, I believe that 

many of his material property concerns are common to both soft and rigid composites. 

modeling efforts are common to each of the disciplines represented here today. 

This CSM workshop is a great way of getting a lot of people together to find out 

what we have in common in our technical pursuits. We should do this more often. 

I think that many of the technical challenges that we face in our tire 

Any questions from the audience? 

Barna A. Szabo, Washinaton University: 

Oden. 

YOU presented. 

the energy norm of each element to the total error of approximation. 

I'd like to address my question to Prof. 

I understand that you are computing the contribution of error in 

You mentioned the error estimation which uses the error indicators that 

You would 

334 
like to see that each element contributes about the same error if the mesh is 

properly designed. 

the energy norm, something of less than 1 percent and, yet, have more than 20 

percent error in equilibrium. 

supports and recompute the reactions from the finite element solution using the 

direct method, we still can have a fairly large error in equilibrium. 

important to look at the error in energy norm but it is also important to check 

other quantities of interest. Have you observed the same thing? I am particularly fond of the equil ibrium check because it can be appl ied to nonlinear problems as well as to linear problems and also it is meaningful to engineers. One 

last comment, if an element or group of elements is found to be out of equilibrium then the possibility of pollution of that error arises, whereas if the group 

Of elements is in equilibrium then according to St. Venant's principle you would 

expect the error to be localized. This is a qualitative view, but, nevertheless, it has a certain convincing flavor to it that engineers will appreciate. 

We found that in some cases we can have very small errors in 

For example, if you cut the structure off the 

It is 

- Oden: 

ual in a solution when you are talking about linear or nonlinear elliptic problems. The calcuation of the error indicator does compute a local residual. It 

computes a local residual that enters the right-hand side of the equation. The 

problem is that the residual by itself is not an indication of error. 

residual is large, chances are the error is large, but the converse is not true. 

You can have a system of forces in equilibrium. and have an enormous error. That 

is the complicating feature of these kinds of error estimation. One must select 

an indication of the error that has the right asymtotic behavior such that as 

the indicator goes to zero, so also does the error go to zero. 

exhibits this characteristic. The points you raise are subtle but very important issues in calculating error estimators. 

feature in this particular kind of strategy because one has to design an error 

indicator with the proper asymtotic properties. 

The presence or absence of equilibrium globally has to do with the residIf your 

The energy norm 

These points are the complicating 

There are many methods for 

estimating local error, but the one I am speaking of now, wh 

the best in that it overcomes the very problems that you are 

produces a robust and accurate estimation of the local error 

fact, vanish at a rate equal to the vanishing of the residua 

refi ned. 

le expensive 

referring to 

which does, 

as the mesh 

is 

It 

n 

is 

335 
Gerald Goudreau, Lawrence Livermore National Laboratory: I 'm new to tire modeling and the one thing I didn't get out of this morning was a feeling for how YOU 

get your composite material models of a tire. I get a feeling that you go out 

and measure rubber properties, even get finite deformation or finite strain 

models of rubber, you do load deflection tests on cords and, yet, somehow this 

is all put together into some kind of a model which may be orthotropic, maybe 

anisotropic, but who's doing what? Anyone can respond. 

Joe Padovan: 

been going on for many years. The industry does not measure rubber properties 

and cord properties separately but rather the aggregate cord/rubber composite. 

Typically, the rule that has been used for mixtures is Halpin-Tsai, although 

this does not fully apply in a nonlinear setting like the tire. 

Measurements of the material properties of tire constituents have 

Goudreau: Are these small strain models then? 

Padovan: There are not really any 1 arge deformation constitutive models except 

for some very isolated materials. In the case of rubber/cord composites, there 

certainly are none yet. Basically what you do is assume Halgin-Tsia or whatever 

linear model you are using works and then assume large strains and repeal 

Kirchhoff's assumptions on the other side of the equation. 

Goudreau: How are you characterizi ng this anisotropy? 

Padovan: The ani sotropy i s hand1 ed through standard 1 aminate theory. 

Goudreau: You're testing things lamina by lamina then? 

Padovan: Yes, lamina by lamina. 

Marion G. Pottinger, B. F. Goodrich Company: 

comments. Structural mechanics love to calculate the ti re stresses and strains, 

but tire users are concerned with performance characteristics which are very 

different things. Since the cords are pretty big and you are treating the tire 

as a continuum, there are some definitive limitations on what you can do. 

are worrying about being exact, but how does this exactness fit in with the 

tried and proven design rules of thumb. 

I would like to make some 

We 

The old design rules of thumb are very 

336 
important to design people and so are the methods of qualitative characterization. Finally, I would like to mention the question of combining the analytical 

and the experimental and the question of experimental verification. Not only do 

you have to satisfy yourself that things really work, you also have to satisfy 

1 awyers. 

Richard B. Nelson, U.C.L.A.: I am a user of finite elements, but I also use 

cars and tires. 

kill tires. 

need to develop models which are capable of predicting heat build-up and what 

that does in the way of generating premature tire delaminations. 

My experience is that heat, abuse, and, delamination are what 

I wonder if there has been attention given in the industry to the 

Tanner: Of course, we are all extremely interested in the thermal characteristics of the tire constituents. As Dr. Tabaddor pointed out this morning, the 

constituents are highly sensitive to temperature. 

ti res, we are very concerned about the transient temperatures that are generated 

during normal taxi operations. Truck and automobile tire people have similar 

concerns except that they deal with rolling for many many hundreds of miles at a 

time at stabi 1 i zed temperatures. 

In the area of aircraft 

Dr. Tabaddor: 

there is much concern about the effect of the temperature on material properties, especial ly fatigue and fracture properties. 

dation of these qualities with increasing temperature. The analytical model 

will also be extremely complex because you have a mechanical model and rolling 

contact while viscoelasticity generates heat and that heat in turn affects the 

tire material properties. 

With respect to Dr. Ne1 son's question, experimentally, of course, 

There is a significant degraDr. Tielking: 

friction and visoelasticity is to get the energy dissipation due to tire flexing 

into a heat conduction analysis. 

way in the National Tire Modeling Program that's involved in transient heat conduction analysis for rolling tires. 

great extent both experimentally and computationally over the 1 ast couple of 

years. 

One objective of the rolling contact analysis which includes 

There is some preliminary work already under 

Sam Clark has been involved in that to a 

Joop Nagtegaal, MARC Analysis Research Corp.: 

ence with the tire industry. 

I have had quite a bit of experiI would like to make a comment on the remarks from 

337 
the gentleman of Goodrich. In the late 70's when I was dealing with several 

tire industries on another continent, they would pose a simple problem that 

could be solved, but then say that that is not the whole problem. 

to account first for one effect and then another until, finally, the problem 

would become unsolvable. 

stand that, but I think it would be extremely useful if this National Tire 

Modeling Program could come up with a clear set of problems to solve. 

this is already being done 

which are very different from many years ago. 

I would have 

Now tire problems are extremely complicated, I underMaybe 

because I see a lot of positive contributions here 

338 
Nalwl kmulrs and l- Snace Mrn~ruslralm 

1. Report No. 

NASA CP- 3034, Part 1 

Report Documentation Page 

2. Government Accession No. 

17. Key Words (Suggested by Author(s)) 

Local/Global Parallel Processing 

Global/Local 

Tire Modeling 

7. Authods) 

W. Jefferson Stroud, Jerrold M. Housner, 

John A. Tanner, and Robert J. Hayduk, Editors 

9. Performing Organization Name and Address 

NASA Langley Research Center 

Hampton, VA 23665-5225 

18. Distribution Statement 

Unclassified - Unlimited 

12. Sponsoring Agency Name and Address 

19. Security Classif. (of this report) 20. Security Classif. (of this pagel 

Unclassified unclassified 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

21. No. of pages 22. Price 

342 A1 5 

I 15. Supplementary Notes 

3. Recipient’s Catalog No. 

5. Report Date 

May 1989 

6. Performing Organization Code 

8. Performing Organization Report No. 

L-16560 

10. Work Unit No. 

505-63-01-10 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Conference Publication 

14. Sponsoring Agency Code 

16. Abstract 

The conference publication contains the proceedings of the Workshop on Computatione 

Methods for Structural Mechanics and Dynamics held at NASA Langley Research Center, 

June 19-21, 1985. The Workshop was organized into the following four sessions: 

(1) Local/Global Nonlinear Stress Analysis 

(2) Tire Modeling 

(3) Transient Dynamics 

(4) Multibody Dynamics 

Transcriptions of discussions are also included. 

Transient Dynamics 1 Multibody Dynamics I Subject Category - 39