THEOFIETICAL AND SOFTWARE: 

CONSIDE:RATIONS FOR NONLINE:AR 

DYNAMIC ANALYSIS 

By 

NASA-CR-174504 

1984000752] 

Richard J. Schmidt 

Robert H. Dodds Jr. 

liBRARY (~Opy 

I ., '7 1Q';:>' 1 IJC L i \.,\)," 

LANGLEY RESEAR(' . 

LIBRARY, NASA 

HAMPTON, VIHG\N1A 

, .R 

An I ntE~ri m Report on Flesearch Sponsored by the 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 

Research Grant 

NAG 3-32 

Structural Engineering and Engineering Materials 

SM Report NO.8 

February 1983 

II -------.. --4-----------.--------------------- THE UNIVERSITY OF KANSAS CENTER FOR RESEARCH, INC. 

2291 I rving Hill Drive-Campus West Lawrence, Kansas 66045 
THEORETICAL AND SOFTWARE 

CONSIDERATIONS FOR NONl.INEAR 

DYNAMIC ANALYSIS 

by 

RIchard J. SchmIdt 

Robert H. Dodds Jr. 

An InterIm Report on Research Sponsored by the 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 

Research Grant NAG 3··32 

UnIversIty of Kansas 

Lawrence, Kansas 

February, 1983 
3. Recipient's Accession No. 

PAGiE 

REPORT DOCUMENTATION 1_1,-REPORT NO. 1 2• 

---------~------------------------~~-----------+--------------------~ 4. Title and Subtitle 

Theoretical and Software Considerations for 

Nonl inear Dynamic Analysis ~----------------.. -----------_._------_._------------- 7. Author(s) 

R. J. Schmidt and R. H. Cbdds, Jr. 

9. Performing Organization Name and Address 

University of Kansas Center for Research, Inc. 

2291 Irving Hill Drive, West Campus 

Cleveland, Ohio 

12. Sponsoring Organization Name and Address 

5. Report Date 

6. 

8. Performing Organization Rept. No. 

SM Report No. 8 

10. Project/Task/Work Unit No. 

11. Contract(C) or Grant(G) No. 

(C) 

(G) 

13. Type of Report & Period Covered 

~------.---------.----I 

14. 

~---------.-----.---------------------------------------------~----------------------; 15. Supplementary Notes 

1----------·_----_·_---------------- ---------------.-- . --' ------------------1 -16. Abstract (Limit: 200 words.) 

In the finite element method for structural analysis, it is generally necessary 

to d i screti ze the structura I model into a very I arge number of el ements in ol-der to 

accurately evaluate displacements, strains, and stresses. As the complexity of the 

model increases, the number of degrees of freedmm can eas i I Y exceed the capac i ty of 

present-day softwa re system. 

A solution to this problem lies in improvements of structural analysis software 

including more efficient use of existing hardware and improved structural model ing 

techniques. One model ing technique that is used successfully in static I inear and 

nonl inear analysis is multi level substructuring. 

This reseal-ch extends the use o-f multi'/tevel substructure model ing to include 

dynamic analysis and defines the requirements fora general purpose software system 

capable of efficient nonl inear dynamic analysis. 

This report contains a presentation of the multi level substructuring technique, 

a review of the analytical formulations and computational procedures for dynamic 

analysis and nonl inear mechanics, and concludes with a presentation of an approach 

to the dE3sign and implementation of a general purpose structural software system. 

~-----------------------------------------.------------------------------------------__l 17. Document Analysis a. Descriptors 

dynamics, modal synthesis, finite elements, nonl inear analysis, 

multilevel substructures, transient analysis 

b. Identlfiers/Open·Ended 'rerms 

c. COSATI Field/Group 

18. Availability Stalement 19. Security Class (This Report) 21. No. of Pages 

Unl imited 292 ~---~--~~-------~--~~~----------

20. Security Class (This Page) 

Release Unl imited 

22. Price 

_Un I. Lm_LtRd 
This Page Intentionally Left Blank 
ACKNOWLEDGEMENT 

Portions of this report were extracted from a special project submitted by Richard J. Schmidt to the Clvl I Engineering Department, 

University of Kansas, In partial fulfl I Iment of the requIrements for the 

degree of Master of Science In Civil Engineering. The study was conducted under the direction of Dr. Robert H. Dodds Jr. 

The research was supported by the National Aeronautics and Space 

Administration under Grant No. 3-32. Special thanks go to Dr. Murray S. 

HI rschbEll n, NASA Techn I ca I Mon I tor. 

Numerical computations were performed on the HarrIs 500 computer at 

i·he Computer Aided Engineering Facility, School of Engineering, University of Kansas. 

I I 
This Page Intentionally Left Blank 
TABLE OF CONTENTS 

a-IAPTER PAGE 

1. INTRODUCTION 

1. 'I 

1 .2 

1.:3 

1.4 

Genaral •....••. 

Objectives and Scope •. 

Notation ••••••. 

References. • • • • • . . . .'. . 

1 

3 

5 

7 

2. MULTILEVEL SUBSTRUCTURE STATIC ANALYSIS. 9 

2.2 

2 • ~5 

2.4 

2.:; 

2.6 

2.7 

General ••••••... 9 

Substructured vs Standard Models. • . • • • •• 12 

Literature Survey. • • . . • • • . . . • . • • • . . •• 13 

User-Software Interface. • • • • • • •• •••••• 18 

Computational and Software Issues. • • • • • • • 32 

2.5.1 Substructure Reduction. • • • •• 33 

2.5.2 Logical Control and Data Structures. .•••• 41 

2.5.3 Linear Equation Solving. • • • • • • • • •• 53 

Examples of Substructured Analysis. . • • • • • • 67 

2.6.1 Linear Example.. • ••••••••••••• 69 

2.6.2 Non I Inear Example. . . • • • . • • • • • • • • • • 79 

References. • • • • • • • • • • • • • • • 91 

3. DYNAMIC REDUCTION OF STIFFNESS AND MASS MATRICES. 93 

4. 

3. 'I .Genera I • • • • • • • • • .• • • • • • • • • • • 93 

3.2 Guyan Reduction • • • • • • • • • • • • • • . • 96 

3.2.1 Basic Formulation. •• • • • • • •• 96 

3.2.2 Automat I c Sa I act i on of tvlaster DOF. . • • • • • • . 102 

3.2.3 Improved Displacement Recovery. • • • • 103 

3.2.4 Evaluation of Guyan Reduction Techniques. •• 105 

3.3 Modal Synthesis. 0 • • • • • • • • • • • • • •• 107 

3.3.1 Introduction............ • •• 107 

3.3.2 Fixed-Interface Method. • • • • • • • 108 

3.3.3 Free-Interface Method with Interface Loading ••• 114 

3.3.4 Branch Mode Analysis. • • • •••••• 118 

3.3.5 Dynamic Stiffness Matrix •••••••.•.••• 119 

3.3.6 Attachment Modes and Interface Mode Sets ••••• 121 

3.3.7 Improved Displacement Recovery in Modal Synthesis. 124 

3.3.8 Residual Mass and Residual Flexibil ity • • •• 125 

3.3.9 Low-Order Polynomial Transformations. •• •• 126 

3.3.10 Evaluation of Modal Synthesis Techniques ••••• 129 

3.4 Selection of fl.lethocls for Dynamic Reduction. • ••• 134 

3.5 References. • • • • • • • • • • • • • • • •• •• 135 

COMPUTATIONAL ALGORITHMS FOR DYNAMIC ANALYSIS. • • • 137 
CHAPTER PAGE 

4.2.2.1 Simultaneous Iteration .....••... 142 

4.2.2.2 Transformation Methods. .. . .••• 144 

4.2.2.3 Polynomial/Vector Iteration. . .•. 147 

4.2.3 Evaluation of Eigenproblem Solution t·1ethods •••• 150 

4.2.4 Choice of Eigenproblem Solution Methods.. .• 155 

4.3 Solution of the Equations of Motion •..•...•••• 157 

4.3.1 Introduction........... . . 157 

4.3.2 Mode Superposition. • • . • • . . • • • • 158 

4.3.3 Time-History Integration. • . • .160 

4.4 Additional Computational Considerations. • • ••• 163 

4.5 References. • • . • . • • • • . • • • • • . 166 

5. MATR I X FORfvl OF NONL I NEAR CONT I NUUM MECHAN I CS • • . . • • 169 

5.1 Introduction.................. • 169 

5.2 Coordinate Systems and Transformations. • . . •• 172 

5.3 Strain-Displacement Relations. • • • • • • • • . 179 

5.4 stress Measures and Rates. • • • • • ••• • • • 184 

5.5 Principle of Virtual Displacements ••••.••••••• 189 

5.6 Constitutive Models • .; • • • • • • • • ••••••• 194 

5.7 Summary and Comparisons. • ••••••••••••• 201 

5.8 References. • • • • • • • • • • • • • 205 

5.9 Appendix -- Notation. • • • • • • • • • 207 

6. NONLINEAR FINITE ELEMENT EQUATIONS • • • 213 

7. 

6.1 

6.2 

6.3 

6.4 

6.5 

6.6 

6.7 

General ••..••••••• 

Nonl inear Equations of Motion •••••• 

Transient Analysis with Substructuring •• 

Total Lagrangian Stiffness ••• 

Updated Lagrangian Stiffness. 

Comparison of Formulations •••••• 

References. • • • • • • . • 

• 213 . 

• 215 

• • • • • 221 

• • • • • • 229 

• • 231 

• 238 

• 244 

USER INTERFACE -- INPUT DESIGN . • • • 247 

7. 1 Genera I • • • • • • • • • • • • • • • • 247 

7.2 Description of the POL. • • • • • •• 249 

7.3 POL Structure. • • • • . • . • • • • . . ••• 258 

7.3.1 Symbol ic Conventions of the Syntax. • .258 

7.3.2 Command Syntax. • • • • • • • • • • • • • • 262 

7.4 Sample Input. • • • • • . • • • • • • • • • 269 

7.4.1 Standard Linear Structure - Vibration Analysis •• 269 

7.4.2 Standard Linear Structure - Spectrum Analysis ••• 273 

7.4.3 Standard Nonl inear Structure. • • • 275 

7.4.4 Substructured Linear Analysis ••••••••••• 278 

7.4.5 Substructured Nonlinear Analysis. • • • • . 282 

8 SUMMARY AND TOPICS FOR FURTHER STUDY. • 289 

8.1 

8.2 

Summary •• • • • 

Topics for Further Study. 

v 

• 289 

• 292 
FIGURE 

2.1 

2.2 

LIST OF FIGURES 

Bridge Structure to I I lustrate Substructurlng. 

Substructured rvlodel for Bridge Example •• 

2.3 Structural Hierarchy for Bridge Example. 

2.4 POLO-FINITE Input Data for Structure SPAN •• · . . . . . 

PAGE 

22 

23 

24 

26 

2.5 POLO-FINITE Input Data for Structure SPAN_CON. 27 

2.6 POLO-FINITE Input Data for Structure BRIDGE. • • • ••• 29 

2.7 Topologlc Sort to Order Substructures for Assembly. 44 

2.8 Multilevel Substructure Assembly Procedure. . . . . . 46 

2.9 Data Structures for Stiffness MatrIces. • . 48 

2.10 Data Structures for Stor-age of Substructure Resu I ts. • • 51 

.2.11 storage Scheme for Blocked, VarIable Bandwidth Procedure 56 

2.12 Effects of Element Numbering on Frontal Condensation . . · · 61 

2.13 Hypermatrlx Storage of Sparse Matrices. . . . . · 63 

2.14 Schematic of Hypermatrl>< Choleski Triangulation. · · · 66 

2.15 Jet Engine Exhaust Duct for Example Linear Analysis. 70 

2.16 Element Types for Exhaust Duct Model . . . . . . · • · · 72 

2.17 Substructured 90 Deg. Model for Exhaust Duct • 74 

2.18 90 Deg. Section of Exhaust Duct Without Substructurlng . 75 

2.19 Structura I Mode I Hierarchy for Exhaust Duct. • • • 

2.20 CPU Time Distribution for Exhaust Duct Analysis •• 

2.21 Shal low Shel I Subjected to Project I Ie Impact. 

2.22 Substructures for Nonlinear Impact Example · . . . . . 2.23 CPU Times Comparison for Example Nonlinear Analysis. 

vi 

77 

80 

81 

82 

86 
FIGURE PAGE 

2.24 CPU Time Distribution for the Example Nonl inear Analysis 88 

2.25 I/O Comparison for the Example Nonl inear Analysis. . . . • • 89 

2.26 CPU Time Distribution for Substructured Nonl inear Example. • 90 

3.1 Internal and External Boundaries for Substructure FIN. • 97 

3.2 Statically Determinate and Redundant Constraints. • • 109 

3.3 DOF Sets for Interface Loading. • • • • • • • • . • • • 117 

3.4 Mode Classes for Interface Mode Sets. 

7.1 Three Element, Plane Frame (Example Input 1-3) •• 

7.2 Two Span Bridge (Example Input 4) •••••••• 

7.3 Two Span Nonl inear Bridge (Example Input 5). 

vi i 

123 

.272 

• • • • 281 

• • • 287 
TABLE 

:2 • 1 

:2.2 

-4.1 

6. 1 

6.2 

6.3 

LIST OF TABLES 

CPU Times for the Exhaust Duct Analysis. 

Computation Summary for Nonl inear Example. . . . . · 

Opl3ratlon Counts for Eigenproblem Solution. · 

Procedure for Transient Analysis with Substructurlng. 

Summary of Element Matrices for Total Lagrangian. . · 

Summary of Element ~latr Ices for Updated Lagrangian. · 

vi i i 

PAGE 

78 

. . . · 85 

. . . · 1 ~54 

• 226 

• 2:52 

• 2:56 
This Page Intentionally Left Blank 
1 • 1 Ba.n.er:.aJ.. 

CHAPTER 1 

INTRODUCTION 

Complex structural systems are most often modeled for analysis as 

assemblies of discrete structural components. The most generally appi Icable discretization approach is the finite element method (FEM). In 

this method It Is often necessary to divide the structural model Into a 

very large number of elements In order to accurately evaluate displacements, strains, and stresses. As the number of elements Increases, the 

number of degrees of freedom (DOF) In the model can easily exceed the 

capacity of many present-day computer facll itles (both hardware and 

so'ftware) or can make the solution of the large order matrix equations 

prohibitively expensive [l.lJ. This problem becomes particularly acute 

In nonlinear analysis. The Iterative nature of nonlinear analysis requires that the matrix equations be solved repetitively, thus compounding the computational expense. 

The appl icatlon of time-dependent (dynamic) loading to complex 

structural models Imposes additional difficulty on current structural 

so'ftware systems. A linear dynamic analysis can be orders of magnitude 

more expensive than a static analysis of the same model. When nonlinear 

response I s a I so cons I dered, i-he computat I ona I effort req u I red for 

analysis can quickly make the solution Infeasible. 
2 

Recent advances in computer architecture, primarily in brute computational speed, have somewhat rei ieved the problems of excessively 

large and costly analyses. It is not expected, however, that new 

developments in hardware wi I I keep pace with the growing demands, In 

terms of model size and complexity, made by 

'wi II the very expensive super-computers 

structural analysts. Nor 

become widely avai lable. 

Moreover, It appears unlikely that orders of magnitude increases In 

high-speed memory capacity and data transmission rates (which analysts 

have come to expect every few years) can be extracted from current technology. If any short term rei lef Is to come, It must derive from Improvements In the structural analysis software. Such Improvements I Ie 

In more efficient use of existing hardware and In Improved structural 

modeling techniques. The focus of this work Is on the Improvement of 

structural modeling techniques. 

One procedure that is used successfully in static analysis Is mUltilevel substructured model ing [1.2J. This approach al lows the various 

major structural units, or substructures, to be treated Independently 

prior to final assembly. With the use of condensation techniques, a 

reduction In total model size can be achieved whl Ie exactly retaining 

the original system characteristics for static analysis. Substructurlng 

techniques also find broad appl Icabi I ity to the various types of computer hardware used by engineers today. Efficient use of both mainframes and virtual memory minicomputers with either serial or plpel Ina 

processors has bean demonstrated [1.3J. In view of Independent substructures, the possible adaptation of the software to a system of Independently operating processors (or computers) under the logical conC " 
3 

trol ot a single machine becomes quite attractive. 

Many investigators have presented extensions of the substructuring 

approach from static analysis to dynamic analysis of finite element 

models. However, these efforts are I imlted to one level of substructures. No attempt has yet been made to formulate and implement mMl-

. j:J I eye I substructur I ng for dynam I c ana I ys I sin a genera I purpose FEM 

system. Thus, the techniques are not proven effective from the practIcal viewpoint of large scale structural analysis. The need exists for 

a comprehensive dynamic analysis system capable of processing mUlti level 

substructured models and incorporating nonlinear response Into the solutl'on. 

1 .. 2 ~tl ves .§1ld Scope 

The objectives of this work are fourfold: 

1. To review the state-'of-the-art of the multilevel substructure 

methodology and modeling procedures for the static analysis of 

complex structures by the FEM. Included is a presentation of 

the design and Implementation of the required software and an 

Illustration of the modeling technique by way of example problems. 

2. To review the analytical formulations and computational procedures available for the analysis of complex structural 

systems subjected to time-dependent loads and capable of 

I inear or nonlinear response. EmphasIs Is placed on methods 

for reducing the sIze of the finite element model for dynamiC analysis. Also studied are elgenproblem solution procedures, 

solution of the equations of motIon, and formulations for 

tracking nonlinear response. 

3. To Identify the reduction ,and computational procedures most 

suitable for incorporation into a general FEM software system 

with multilevel substructurlng capabl Iities. 

4. To discuss the Implementation considerations of the above 

relative to a multi level substructured environment. 
4 

The discussion of dynamic analysis methods is based on a review and 

interpretation of the open literature. Whi Ie the authors, along with 

several other researchers, have experience in large scale linear and 

nonl inear static analysis with substructurlng, large scale dynamic 

analysis has been performed using only the simplest of techniques. 

The remainder of this report Is divided Into chapters which Identify the major topics covered. Chapter 2 Is a pre~entation of the 

methodology and implementation procedures of multi level substructuring 

In a general purpose software system. Methods for reducing the order of 

the coefficient matrices In the differential equations of motion are 

reviewed and evaluated In Chapter 3. Chapter 4 considers the varfous 

computational algorithms required In dynamic analysis. Included are 

elgenproblem solution methods, procedures for solving the differential 

equations of motton, and selected minor topics. The nonlinear continuum 

mechanics equations for finite deformation, cast In matrix form, are 

presented In Chapter 5. Exact forms for both the Total Lagrangian and 

the Updated Lagrangian approaches are described In addition to the 

finite deformation theories of elasto-plastlclty. Chapter 6 presents 

specific forms of the fInite element matrIces for the Total and Updated 

Lagrangian approaches. Details of the transient solution procedure 

based on an Implicit Integration operator with the effects of substructuring are discussed. Chapter 7 describes the Input language designed 

to provide a convenient user Interface wfth the application software. A 

summary of work performed thus far is presented In Chapter 8 along with 

proposed future activities necessary for successful dynamic analysis of 

multi level substructured models. 
5 

1 .2> Notat lQn 

Most of the notation used In the fol lowing chapters Is defined as 

It is Introduced. However, the fol lowing I ist may prove to be a useful 

re1:erence .. Consistency Is maintained whenever possible and exceptions 

. arEl noted I n the text. The comp I ex! ty of the notation used I n Chapter's 

5 and 6 warrants special discussion In those chapters. 

{ } a vector 

[ :1 a matrix 

pCU characteristic polynomial 

x - yg 9 global coordinate system 

x - I YI local coordinate system 

{q} vector of generalized displacement coordinates 

{u}, {~}, {~} displacement, velocity, and acceleration vectors In 

geometriC coordinates 

S.E. 

[A] 

[8], [G] 

[C(t)] 

[D] 

[K], [M] 

displacement vectors for master and slave DOF 

matrix of trial mode shapes for Iteration "j" 

k I netl c energy fc)r a structure 

stra I n energy for' a structure 

a tridiagonal matrix 

an Interaction matrix used In simultaneous Iteration 

methods 

time dependent damping matrix 

substructure dynamic stiffness matrix 

stiffness and mass matrices or submatrices 
[KG]' [MG] 

[KF], [MF] 

6 

Guyan reduced stiffness and mass matrices 

fixed-interface substructure reduced stiffness and mass 

matrices 

[K~k], [M~k] submatrices of generalized stiffness and mass for subI I 

* * [K], [M] 

* * [Kmm], [Mmm] 

[L] 

[M(tu) ] 

[pJ, [Q] 

{pet)} 

[R] 

[s] 

c [TG], [<I> ] 

[Tw] 

2 

Wj 

[w2] 

2 r:l 

A, [)J 

structure "i" 

stiffness and mass matrices assembled from lower level 

substructures 

stiffness and mass submatrices for the system master OOF 

within * * [K] and [M] 

lower triangular matrix of Choleskl factors of a stiffness 

matrix 

dynamic mass matrix 

orthogonal transformation matrices 

time dependent load vector 

an upper triangular matrix 

symmetric coefficient matrix for an elgenproblem in standard form 

Guyan transformation matrix (static constraint modes) 

frequency dependent transformation matrix 

square of the substructure natural frequency for mode "I" 

diagonal matrix of substructure natural frequencies 

squared 

unknown system frequency 

an eigenvalue, diagonal matrix of eigenvalues 

matrix of substructure normal modes of vibration 

set of retained substructure normal modes 
7 

1.4 Refer-ences 

1.1 Craig, Roy R. Jr. and Bampton, Mervyn C. C., "Coupl ing of 

Substructures for Dynamic Analyses," AJlili. Journal, Vol. 6, 

No.7, pp. 1313-1319 (1968) 

1.2 Dodds, Robert H. Jr. and Lopez, L. A., "Substructuring in 

Linear and Nonlinear Analysis," International Journal 1~ 

Numerical Methods In Engineering, Vol. 15, pp. 583-597 (1980) 

1.3 Noor, Ahmed K., Kamel, Hussein A., and Fulton. Robert E., 

"Substructuring Techniques -- Status and ProJections," .QQ.mputersSUl.d.Structur:..e...s., Vol. 8, pp. 621-632 (1978) 
8 
CHAPTER 2 

MULTILEVEL SUBSTRUCTURE STATIC ANALYSIS 

2 • 1 Ge neral. 

Complex structures frequently consist of repeated, Identical components. This may be dictated by economic, construction, or symmetry constraints. The boundaries between components (either real or artificial) 

partition a complex structure Into a natural system of simpler substructures. E':lch substructure may I n turn be parti tl oned to exp I 01 t aadf'tfonal repetition. The associated finite element model generation 

prc:>cess Is cons I derab I y simp I I fled through the repeated use of 

pnwlously defined components. In many cases, the computational costs 

of analysis are reduced accordingly. This concept of substructured 

modeling has been termed the Vlsuperelement" technique In the literature 

due:! to th4~ similarity of the substructure merging process with that uSE~d 

to connect finite elements. The term "user-defined" has also been employed to distinguish analyst specified substructurlng from automatic 

partitioning of the equilibrium equations. 

The structural frame of an aircraft provides the Classic example to 

Illustrate the concepts and advantages of a substructured analysis. Ir~ 

delPendent des I gn groups deve I op the I nd I v I dua I substructures, for e,<-

amlPle: -the wing assembly, fusealage sections, and vertical stabilizer". 

Th(~ substructures Interface through relatively small boundaries (in 

terms of the number of nodes). Even with such "first-level" substruc.... 

turing, the number of nodes and elements may be too large for efficient 

pr4:>cessl n!~ of the substructures. The same substructurl ng process, In 

9 
10 

theory, can be repeated within each first-level substructure to Introduce second, third, fourth, ••• level substructures. The engine 

nacelles, shrouds, flaps, ribs, and skin panel sections within a wing 

assembly may comprise substructures five or six levels removed from the 

"highest" level structure that represents the complete airframe. The 

. conceptual organization for this type of structural model paral leis that 

of a tree. The tree has a single root node (the highest level structure). Any number .of substructure levels may be defined below the root 

node. No theoretical limit exists on the number of branches that enter 

a node (substructure) at level "I" from level "1-1". AI I terminal 

nodes of the tree are Individual finite elements. 

Substructure techniques have been utilized extensively by the Norwelgan ship building Industry [2.3J to construct and analyse finite element models of ofl tankerso Repeated bulkheads and common stiffener arrangements In ships are weI I suited for substructurtng. WIthout multilevel substructures, typical analyses would Involve 100-150 thousand 

degrees of freedom. Problems of this size remain Impractical to solve 

despite the availabIlity of super-computers. 80th the aIrframe and ship 

building examples clearly demonstrate the usefulness of multi-level substructuring to support practical analyses. 

In the context of linear, static analysis, a substructured model 

yields the same solution (to wIthin round-off errors) as a "standard" 

model that considers the structure as a single collection of nodes and 

elements. The solution equivalence remains valid for static, nonlinear 

analysts provided the substructures experiencing nonlinear behavior are 

Included In the Iterative solution. Normally, regions of a structure 
II 

that remain linear are substructured and eliminated via static condensation fran the iterative solution. For example, It Is a simple matter to 

anticipate the plastic deformation that occurs near a stress concentration. Portions of the structure removed from the stress concentration 

are defined as linear substructures and condensed. Nonlinear finite 

'el ements and reduced II near" substructures comprl se the highest I eve I 

s·t·ructun~ for the Iterative solution. Linear substructures Simply 

provide elastic restraint on the nonlinear region. In such cases, the 

standard and substructured models again yield Identical solutions. 

However" the substructured model generally requires much less computational' effort as a consequences of Its reduced size. 

This chapter provides detailed background Information on substructured m()del Ing and solution procedures for static analysis. The advantages of substructured analysis relative to the standard procedure are 

first described. The literature concerned with static substructurlng Is 

rEW I ewed.. Various techn Iques adopted to address the user-software I ntE~rface are descr I bed and a simp I e examp I e I s presented to I I I ustrate 

the techniques used In the POLO-FINITE system. Computational and 

software Issues that arise In programs for general substructured 

analysis are surveyed. The chapter concludes with a study of two example pr"oblems that Illustrc~te the computational savings possible with 

substructured models for linear and nonlinear static analysis. 
12 

2.2 Substructured Ya Standard Models 

Compared with a standard modeling and solution procedure, a multllevel substructured approach offers a number of advantages. These Include: 

1. Input data reqUirements are reduced. GeometrIc and topologlc 

descriptions of a substructure are specified only once. When 

the same substructure appears repeatedly at higher levels, Input data that must be provided by the user Is significantly 

reduced. 

2. Tne Impact of design changes on reanalysis costs can be 

reduced. Stiffness matrices and loading vectors for only the 

modified substructures are computed during reanalysis. 

3. Models may be generated Independently. Because substructures 

have clearly defined Interfaces, the design and modeling 

groups may work almost independently. Element numbering, node 

numbering, and load case naming schemes are usually Independent across substructures which slmpl iffes model generation. 

4. Isolated substructures may be Independently verified. Each 

substructure may be constrained, loaded, and analyzed to study 

the response of the Isolated model. Checkout analy~es are 

usually Inexpensive relatIve to the cost of a complete structure analysIs. The higher execution priority assigned to jobs 

that request fewer machine resources also decreases the total 

analysis time. A complete analysts performed In smaller segments frequently costs less, and requires less residency tIme,' 

than a comparable standard analysis performed In a single execution. 

5. Identical substructures may be used repeatedly. This Is the 

most often cited advantage of a substructured model. Substructure quantities, for example the reduced stiffness 

matrix, are computed but once and used repeatedly to form 

other structure stIffness matrices. The degree of computational savIngs Is highly problem dependent with cost 

reductIons In the range of 2 to 100 having been reported 

[2.2J. The level of savings In nonlinear analysis depends on 

the degree of size reductIon and the frequency of tangent 

stiffness updates. 

6. Numerical conditioning problems are often reduced. Certain 

types of numerical problems may be remedied through the use of 

a substructured modeling and solution procedure. A common 
13 

situation involves the joining of a very stiff substructure to 

a comparatively flexible one which results In a loss of precision during stiffness assembly (a very large number overwhelms 

the much smaller term). Condensatlol1of a very stiff substructure frequently reduces the magnitude of stiffness coefficients for the remaining nodes to a level comparable with 

those of the adjacent, more flexible component. The loss of 

precision in the Important stiffness assembly process can thus 

be minimized without extended precision arithmetic. 

7. Exposure to machine fal lure Is reduced. The solution ot a 

structure with a very large numbE~r of DOF may require long 

residency times on multipurpose computer systems. During 

this time, the execut'ing program Is exposed to the possibility 

of a machine failure that would require restarting the 

analysis. Procedures have been developed, termed checkpoint 

restart, that save snapshots of the program status on disk 

files at specified Intervals. This process often requires extensive machine dependent coding which restricts the software 

portability_ Substructured models provide a more natural 

solution to the failure protection problem. Substructures are 

processed In a logically Independent, sequential manner during 

execution. Natural breakpoints occur" between each substructure at which the the execution may be terminated and the 

databases saved on tape. If a fail ure occurs while the next 

substructure Is being processed, the databases on tape are 

simply restored to disk and the analysis resumed. 

Thelse advantages of a substructur'ed approach are equally applicable In 

a I I types of ana lyses -- statl c, dynam I c, II near, 'and non I I near. 

2.3, LI tera~ Survey 

Although the concepts and equations of substructured analysts are 

generally well known for static analysis, relatively few papers on the 

subject hc:lve appeared in the literature. A contributing factor to this 

apparent lack of Interest Is the enormous software complexity required 

to support substructurlng, coupled with the limited avallabt I Ity OT 

software to researchers. For static linear and nonlinear analysis, the 

governing equations are straightforward and simple to derive. The few 

researchers who have examined substructurlng have focused on Improving 
14 

the computational efficiency and on the computer ImplementatIon 

problems. Because only approximate substructured solutIons are feasible 

In dynamic analysis, the literature concerned with Improving the algorithms continues to expand and Is reviewed In the next chapter. This 

section reviews previous studies that addressed static, linear and non-

. linear substructured analyses. 

In the early 1960s, Przemlenleckl [2.15J presented the first comprehensive formulation for substructured analysis using the conventional 

displacement method. Talg [2.17J described an attempt to Implement 

these procedures for general analysis. These two early studies viewed 

substructurlng as automatic partitionIng of the equIlibrium equations to 

overcome computer sIze I Imitations. Interest In substructured analysts 

declined during the late sixties when commercial programs using sparse 

matrix technIques became ope~ational on third generation computerso The 

non-computational aspects of analysis, Including model generation and 

technical coordination, replaced computer I Imitations as the major 

problem areas. Substructure techniques regained appeal as an approach 

to solve these problems but In the form of analyst defined, rather than 

automatic partitioning. Thus the term "user-defined" substructurlng was 

coined. In thIs same perIod, WillIams [2.18J showed through operation 

counts that equation solving with sparse matrix techniques can never be 

more computationally efficient than a substructured solution, when the 

substructure arrangement Is suitably defIned. 
15 

Furuike [2.5J described a software system capable of processing 

three levels of substructures. The root and the next two levels of the 

tn:~e may contain only substructures. Fourth level structures consist 

entirely of finite elements. The analyst supplies, through Input data, 

the;) order of substructure process I ng and the proper sequencl ng of noda I 

'DOF across substructure boundar~ I es. Substructure stl ffness condensation 

Is performed with the Inefficient Inversion algorithm described In Se(~ 

tlon 2.5.1. The author presents several example solutions with and 

without substructurlng that demonstrate the numerical equivalence of the 

results. Unfortunately, no comparisons of computer resources ar-e 

prov I ded. 

Egelan and Araldsen [2.3J briefly surveyed the substructurlng 

capabilitIes of the SESAM-69 program. No details of the user-Interfac:e 

fol" descr~lblng the substructure connectivity and orientation are 

provided. SESAM supports multilevel substructured models. Finite elements and substructures cannot be mixed at the same level In SESAM. 

This restr~lctlon Is Imposed by the adoption of separate assembly processorbs for structures composed 01: on lye I ements and those composed of on I y 

substructures. The classical Inversion algorithm for stiffness condensation Is Impl led In the survey. 

Descr~lptlons of substructurlng capabilities In NASTRAN are given by 

MacNeal and McCormick [2.11J clOd In ASKA by Schrem [2.16J. Each system 

Initially supported only modest substructurlng; the major emphasis being 

directed toward efficient sparse matrix techniques. Proprietary ver'-

slons of NASTRAN have been expanded to support more comprehensive substructuring, Including some techniques for dynamic analysis beyond Guyan 
16 

reduction. Software details are not publicly available. To perform a 

substructured analysis, users must write "driver" programs In a special 

language to manipulate' disk files and to control the execution of 

processing modules. Completely automated solution procedures for substructured analysis are not available. 

More recently, Peterson and Popov [2.14J addressed the computational penalty that occurs with stiffness matrix rearrangement prior 

to condensation. They propose a scheme to eliminate nodal freedoms at 

the element level before assembly of the condensed stiffness. The technique Is promoted as more efficient than the conventional method of 

rearrangement although no comparisons of computer time are stated for 

the example problems. 

Lopez [2.9J presented POLO-FINITE as the first major system to support multi-level substructurtng as a natural approach to finite element 

modeling. Major advances incorporated In POLO-FINITE include a very 

simple substructure definition process and fully automated solution 

procedures. The dependencies between substructures fn the hierarchy are 

determined completely by the system from basic Input. Any number of 

related and/or Independent hierarchies may be defined within a single 

database, with up to 20 levels of substructurlng permitted In each 

hierarchy. Substructures and finite elements may be mixed at any level; 

the system processors treat finite elements and substructures Identically. The extensive system logic that automatlcaJ Iy controls the 

solution process also enables intelligent reanalysis to incorporate substructure modifications. As Input data describing the modifications are 

processed, substructure results dependent upon the modifications are In-
17 

vcliidateci. For example, changing a load case definition invalidates exIsting displacements for thai· load case but not the structure stiffness 

matrix. Prior to reanalysis, the system traverses the hierarchy to 

dE~termlnE~ the type and order of computations required. During the 

tr'aversa I, dependent resul ts for other substructures are destroyed and 

. tclgged for recomputation. When a substructure at level "I" Is marked 

fc)r computation, dependent subs·~ructures at level "1-1" are also marked 

for computation. In this manner, the effects of substructure modlficati ons arEI automatl ca I I Y propagated upward through the hierarchy. POLOFINITE Elquatlon solvers are based on the hypermatrlx techniques first 

IntroducEtd In the ASKA system. An efficient "partial decomposition" algc)rlthm Is uti I Ized to condense substructure stiffness matrices. 

The most recently publicized substructure system, MISA [2.7J, was 

dE~velopeci by the Japanese ship building Industry. MISA Incorporates 

sE~veral unique concepts. Wavefl·ont, rather than variable bandwidth, 

equation solvers are used to ccmdense substructure stlffnesses although 

nc) computational advantage Is claimed. The software logic deduces the 

substructure hierarchy from parent-child relations Input by the user. A 

"copy" function facilitates repeated use of previously defined substructures. MISA does not support mixed substructures and fInite elements at 

the same level. It currently analyses linear structures for static and 

thermal loads, and performs steady-state heat conduction analyses. 

Dodds and Lopez [2.1J extended the POLO-fiNITE system to support 

multilevEl1 substructured models for static, nonlinear analysis. The 

ana I yst clef I nes I I near reg Ions that are substr·uctured and condensed to 

fc)rm eHectlvely elastic supports surrounding the nonlinear, highest 
18 

level structure. The reduced size of the nonlinear structure analyzed 

with the Iterative technlqu& frequently yields significant cost savings. 

Currently, the nonlinear regIon must be defined prior to beginning the 

analysIs. This Is a major disadvantage of the approach. For the InitIal loading levels, this proves IneffIcient as a much smaller nonlinear 

·reglon develops than Is actually declared. To Improve the situation, It 

Is necessary to consider substructures that can be made nonlinear and 

brought Into the Iterative solution as the loading levels Increase. 

ThIs requires yet another level of sophIsticatIon In the control logic 

and has not been attempted. 

2.4 User-Software Interface 

The user-software Interface for general purpose systems must 

provide sufficient flexibility to Invoke the options, and yet It should 

not discourage the Infrequent user with unnecessary details. Substructuring further complicates this interface with the Introduction of more 

elaborate topology, geometry, and computational algorithms. The 

developers of most software to support substructurlng have not made the 

interface particularly simple for the user. These systems usually incorporate substructurlng as an extension of the original software 

design. Substructurlng Is viewed as a last resort to obtain a solution, 

rather than as a natural modeling approach. Users are required to 

"program" the substructured solution by directly or Indirectly Invoking 

computational modules and by manipulating disk flies that store substructure data. Consequently, only the most experienced users attempt 

substructured solutions; casual users are told to avoid substructurlngc 
19 

The process of defining a substructured model and conducting the 

analysis consists of five logical steps. These are: 

1. The definition of each independent substructure (elements, 

constraints, and loadings). 

2. The elimination of substructure "slave" (condensed) nodal 

freedoms. "Master" nodal freedoms r-emaln after condensation. 

3. The connection of individually defined substructures In a 

topologlc and geometric sense. This requires matching of substructure boundaries to insure displacement compatibi I Ity and 

may also include coordinate transformations of substructure 

matrices from their local system to a common global reference 

frame. Additional boundary constraints may also be required. 

4. The creat Ion of a load I ng set h i erc~rchy th at para I I e I s the 

substructure hierarchy. Equivalent nodal loads computed for 

the loading cases defined In (1) for the Individual substructures are also reduced through condensation. Loads reduced to 

the master nodes serve to define a hierarchy of loading sets 

on the complete model. For example, the reduced loads may be 

appl ied to selected copies of a substructure at the next 

higher level to create a desired pattern of loading. The 

loads may also be carried up through the hierarchy with scalar 

multipliers and possibly combined with other reduced load 

cases to form new loading cases on a higher level substructure. 

5. Requests for computation and output. The compleXity ot computational requests depends on the level of procedural detal I 

required by the software to effect the solution. Output requests become complex when the user designates the hierarchy 

level for which results are desired, for which loading cases, 

elements, nodes, coordinate reference frames, etc. 

The commeln technique to approach these problems Involves the use of multlple programs or a single program executed many times. The programs 

communicate through data stored on disk files (and/or tapes). The 

analyst Is responsible for coordinating the program executions to 

produce the desired results. A typical analysis with one level of substructures might require that the fol lowing tasks be performed by the 

analyst: 
20 

1. Elements, nodal coordInates, topology, constraInts and loads 

are defIned to a fInIte element program for each IndivIdual 

substructure. The analyst provides special Instructions indicating a condensation is desIred and the lIst of master (or 

slave) DOF. Disk files are attached to the program onto which 

the reduced stiffness matrIces and load vectors are wrItten. 

Usually each substructure must be processed In a separate 

program executIon and with unique data files. 

Z. A substructure processing program Is executed wIth al I substructure disk fIles attached. This program is provided with 

disk file numbers for the substructures and the order in 

which they are to be processed. The connectivity relations 

between substructure master DOF and the glubal numberIng 

system are also specified. Additional complexities arise when 

substructure reference axes are not al I paral lei. Since only 

substructures can be processed, It Is not possible with such a 

scheme to mix substructures and simple finite elements at the 

same level, whIch Is inconvenient In nonlinear analysis. The 

substructure processing program assembles the reduced stlffnesses and loads to form a fInal set of equilIbrIum equations 

for solution, then computes the displacements for the highest 

level. Master DOF displacements for user selected substructures are extracted and written onto another set of disk 

files. 

3G Finally, the finite element program Is again Invoked with 

proper disk files attached that contain displacements for the 

substructure master nodes. Backcondensatlon procedures may 

then be performed to recover slave displacements and element 

strains-stresses. This process must be ~epeated for each copy 

of the substructure sInce the computed displacements, unl Ike 

the stiffness matrix, are unique. . 

Each of the above steps involve considerable manipulation of disk files 

and several program executions by the user. The manpower costs can 

easily approach those for model generation. The process Is almost intractable when more than one level of substructurlng is used (If permltted at al I by the software). Nonlinear problems require looping over 

the second and third operations for each load Increment. To place 

numbers on the amount of user effort Involved with this approach, a 

linear model containing four substructures at the first level was 

analyzed with one of the most successful commercial finite element 
21 

programs. Fol lowing the recommended procedures in the user documentation, ninE! separate computer Jobs were required to obtain a solution. 

Absolutely no automatic solution techniques are available in the 

program. Changes to one of the substructures required a complete 

reclnalysis of the entire structure due to the arrangement of data on 

'tape storage. 

Demands placed on the user are greatly simplified when the software 

Is designed to support substrltC"rured analysis In an Integrated fashion. 

Stclndard 2lna I yses become the s r mp I est defau I t procedure in such a 

system. The following example, analyzed wl·~h the POLO-FINITE system, 

illustrates the degree of simplification POSSII)le with a user-orIented 

approach to substructured analyses. This example Is not Intended to 

demonstrate savings of computational effort; the computational aavanta~les arel demonstrated in more comp I ex examp I es at the end of th Is 

chclpter. 

The structure Is a simple two span, planar truss as shown In 

FI g. 2.1. The genera II y non-symmetrl c loads ,-equ I res that a fu II model 

(both spans) be analyzed. Components of the substructured model are 

shclwn In Fig. 2.2, with names assigned to each component for Identiflcatlcln In the model description. Figure 2.3 Illustrates the substructure 

hierarchy in tree form for this simple example. One span Is defined and 

condensed to the four nodes necessary for connection with the adjacent 

substructure. The brIdge Is defined using two copies of the condensed 

substructl.lre and three rod elements to comp I ete the mode I • 
22 

0 

N 

@l 

~ 

::.::: 

8 

..... 

0') 

c 

l. 

~ 

+-

u 

~ 

l. 

+-

If) 

.D 

~ 

U) 

(J) 

+-

ro 

l. 

+-

If) 

::l 

$ 

(J) 

l. 

~ 

+-

u 

~ 

l. 

+-

U) 

(J) 

0') 

U 

l. 

co 

N 

(J) 

l. 

~ 

OJ 

l..L. 
BRIDGE 

OJ 

( 

Y1SVl"\. ~~ 1 2 4 6 8 

SPAN 

... , "s 

...... 

t 

~ 

3 5 7 

1 246 8 

SPAN 

Figure 2.2 -- Substructured Model for Bridge Example 

Xss 

....... 

N 

\..N 
13 Rod 

Elements 

13 Rod 

Elements 

Figure 2.3 -- Tree Representation of Structural Hierarchy for Bridge Example 

~, 

+:> 
25 

The lowest level (sub)str'ucture, SPAN, consists of 8 nodes (each 

with 2 OOF) and 13 rod elements. Input data descrlblrig structure SPAN 

to FINITE aFe listed In Fig. 2.4. Element types, properties, topology, 

and noded coordinates are first specified. The problem or I eni'ed 

language (POL) Input eases data entry by eliminating column and command 

,order rE~str I ct Ions. No natural boundary conditions occur at the nodes 

ellmlnatE~d by condensation. Constraints are thus omitted on this lowest 

I Elvel substructure. Three I ndependent load I ng cases are def I ned to act 

on SPAN. These represent a unl'form load applied over the bottom chord, 

a un I for'm load on the top chord, and a I atara I load act I ng on the top 

chord. The magnitude of each loading Is unity to simplify the definition of elctual loading magnitudes In the higher level structure. 

Structure SPA~CON Is defined as the statically condensed version 

01~ SPAN. FI gure 2.5 II ste the I nput data descr I bing th I s structure. 

Nc)des 1, 3, 7, and 8 are reta I ned after condensation. SPA~CON In th I 5 

e)(ample corresponds to a "super-element" In the terminology used by some 

rE)searchElrs. Ana I ysts exp II cl t I y I ntroduce condensed substructures Into 

the hler'archy through Intermediate structures such as SPAN_CON. Structure SPAN Is referred to as the "child" of the "parent" structure, 

SPAN_CON, which resides at the next higher level In the structure tree. 

This technique has proven to be a natural means of Incorporating the 

ccmdensecl vers I on of the subst'rlJcture I nto the h r erarchy. It e I 1m I nates 

cc,nfuslorl on the analyst's part and maintains a consistent definition of 

st'ructurE)S I n the database. Some structures are simp I y tagged as "condelnsed" which serves to control execution of the processors. The Incidences specified for element one of SPAN_CON designate the nodes 
26 

C 

C PROBLEM UNITS ARE KIPS, FEET 

C 

STRUCTURE SPAN 

NUMBER OF NODES 8 ELEMENTS 13 

ELEMENTS ALL TYPE ROD E 4.32E06 AX 0.0347 

COORDINATES 

X Y 

0 

2 20 

3 20 

4 40 

5 40 

6 60 

7 60 

8 80 

INCIDENCES 

1 1 3 

2 2 3 

334 

LOADING UNIT_TOP 

NODAL LOADS 

0 

0 

20 

0 

20 

0 

20 

0 

3 7 FORCE Y -10 

5 FORCE Y -20 

LOADING UNIT BOTTOM 

NODAL LOADS 

2 4 6 FORCE Y -20 

LOADING UNIT SWAY 

NODAL LOADS 

3 FORCE X 1.0 

Figure 2.4 -- POLO-FINITE Input Data for Structure SPAN 
27 

STRUCTURE SPAN CON 

NUMBER OF NODES 4 ELEMENTS 

ELEMENT 1 TYPE SPAN CONDENSED 

INCIDENCES 

1 1 3 7 8 $ BECOME NODES 1-4 

LOADING UNIT TOP 

EXTERNAL ELEMENT LOADS 

1 UNIT TOP 

LOADING UNIT BOTTOM 

EXTERNAL ELEMENT LOADS 

1 UNIT BOTTOM 

LOADING UNIT SWAY 

EXTERNAL ELEMENT LOADS 

1 UNIT SWAY 

Figure 2.5 -- POLO-FINITE Input Data for Structure SPAN CON 
28 

retained from substructure SPAN. FINITE currently requires that al I OOF 

at a node be either eliminated or retained during the condensation 

process. Loading cases are carried up through the structural hierarchy 

with a loading type desIgnated EXTERNAL ELEMENT LOADS. An EXTERNAL 

loading specifies the names of loading cases on the child substructure. 

·The loads are condensed and placed on the parent substructure nodes 

under the declared load case. There Is a one-to-one correspondence 

between loading case names for the parent and chi Id substructures In 

this example, but this Is not required. Only loading case names within 

a structure must be unique. Any number of loading cases, with optional 

scalar multipliers, may be selected from the child substructure to construct loadings on the parent substructure. After the stiffness and 

loadings on SPAN are condensed during solution, SPAN-CON has the major 

characterIstics of any other structure; It has a stiffness matrix and 

loading cases stored In a standard format. 

Structure BRIDGE Is modeled from two copies of SPAN_CON with three 

additional bar elements to complete the model. Figure 2.6 lists the Input data descrIbing structure BRIDGE. Copies of SPAN_CON (elements 

and 2) are placed In BRIDGE wIth the same orientation relative to the 

coordinate system In which they are defined. In this model, substructure reference axes X -Y and the highest level structure axes Xa-Y are I I ~ 9 

paral lei. In other cases, a rotational transformatIon of substructure 

stiffness matrices and nodal loads may be requIred for proper geometric 

alignment. Coordinates are required for nodes of the three additional 

bar elements. CoordInate values are used to determine element size and 

orientatIon, thus the orIgin locatIon Is Immaterial. For lInear 
29 

STRUCTURE BRIDGE 

NUMBER OF NODES 8 ELEMENTS 5 

ELEMENTS 

1 2 TYPE SPAN CON ROTATION SUPPRESSED 

3-5 TYPE ROD E 4.32E06 AX 0.0694 

COORDINATES 

X Y 

2 0 0 

5 -20 20 

6 0 20 

7 20 20 

INCIDENCES 

1 1 4 5 2 

2 278 3 

356 

467 

5 2 6 

CONSTRAINTS 

1-3 V = 0.0 

1 U = 0.0 

LOADING FULL TOP 

EXTERNAL ELEMENT LOADS: 

1 2 UNIT TOP 2 

NODAL LOADS 

5 7 FORCE Y -20 

6 FORCE Y -40 

LOADING LEFT BOTTOM 

EXTERNAL ELEMENT LOADS 

1 UNIT BOTTOM 

COMPUTE DISPLACEMENTS FOR STRUCTURE BRIDGE 

[ output requests ] 

STOP 

Figure 2.6 -- POLO-FINITE Input Data for Structure BRIDGE 
30 

analysis, mixing of substructures and elements Is more a convenience for 

model definition than a computational necessity. Constraints Imposed on 

the nodes of structure BRIDGE model the simple support boundary conditions. EXTERNAL loads are again used to apply loadings from SPAN_CON to 

BRIDGE. Load cases on BRIDGE are defined by selecting external loads on 

. elements and 2. Nodal loads, standard element loads (e.g., a 

distributed force on an element), and external element loads may be combined In any manner to define the loading cases on BRIDGE. 

A request for analysis has the simple form COMPUTE DISPLACEMENTS 

e80 as shown In Fig. 2e6. FINITE processors traverse the hIerarchy to 

automat I cal Iy determine the order of processing required and to check 

for topological consistency. The solution then proceeds to completion 

wIthout user Intervention. A solution In this context Implies the computation of displacements for structure BRIDGE. The hierarchy traversal 

performed for each COMPUTE DISPLACEMENTS command Insures that only 

needed quantities are actually generated. 

Output requests provide the capability to designate the unique 

copy of a substructure for whIch results are desired In addition to 

other Information such as loading cases, element and node numbers, and 

coordinate systems for stresses-strains. The "substructure stack" Is a 

part of the structure name In the OUTPUT command. Thus, to obtain 

stresses In al I bar elements for the right span of BRIDGE for al I 

loading cases, the following command Is sufficient: 

. , 
31 

OUTPUT STRESSES FOR STRUCTURE BRIDGE/2/1 

In which the 2/1 designates subelement 2 of structure BRIDGE, which Is a 

structure named SPAN_CON, then subelement 1 of SPAN_CON, which Is a 

structure named SPAN. The list of elements and/or nodes (In this case 

an Impl led "al I" elements) refers to the final structure I Isted In the 

stack. FINITE processors examine the substructure stack and 

automatically determine the type and order of backcondensatlon processes 

required to satisfy the request. 

The OUTPUT command provIdes considerable flexIbl1 ity for requestIng 

substructure results. The above command represents an extreme case In 

whIch the stack points to a unique occurence of a substructure In the 

hierarchy. Alternatively, the command 

OUTPUT STRESSES FOR STRUCTURE BRIDGE 

requests the pri nti ng of resu I tSi for a II elements in structure BR I DGE. 

In this case, FINITE processors automatically traverse the complete 

hierarchy below structure BRIDGE, recovering displacements, strains, and 

stresses at every level. The output processor traverses the same stack 

to print the results from the top (BRIDGE) down. 

The major advantages of the above approach to substructured model 

deiflnltion and solution are now apparent. The complete problem definition and solution is accomplished In a single computer run with no user 

Interaction concerning the placement of substructure Information on'to 

data files. In effect, the analyst has access to a structural model 

database which can be defined and modified at a very hIgh (logical) 

level. Substructure model description Is a completely natural extension 
32 

of standard model description procedures. When implemented In this 

manner, computation requests are Identical wIth and without substructuring. The troaversal processes are performed automatically regardless 

of the model complexity. Output commands are qUite flexible In that 

results for a wide or limited range of substructures may be requested. 

'The POL Input, while not required, considerably simplifies data entry 

and provides ready documentation for the model. It Is mandatory for Interactive processing. 

FINITE currentlY requires that the structural hierarchy be defined 

in an inverted order. For example, structure SPAN must exIst In the 

database at the time structure SPAN_CON Is deflned~ SImilarly, structure BRIDGE cannot be defined unless structure SPAN-CON exists. The 

structural tree must be defined from the bottom up; however, each branch 

need not be completely defined before begInning another branch. ThIs 

restrIction Is sometImes InconvenIent In that descriptions of a tree 

from the top down may be more natural. 

2.5 Computational gng Software Issues 

Multilevel sUbstructurlng creates a number of computational and 

software problems not encountered In conventional finite element 

systems. This section provides a brief survey of the major computatIonal and software Tssues e The algorithmic details of stiffness 

condensatIon, load reduction, and displacement recovery for an Isolated 

substructure are first reviewed. The logical control techniques and 

data structures required to automatically process complex substructured 

models are then examined. These non-numerIcal aspects of the software 
33 

dleterm I ne, to a I arge extent, the event-ua I genera I I ty and useracceptance of the system. The final topic discussed In this section 

concerns data structures and algorithms for the solution of very large 

sets of linear equations. The less widely known hypermatrlx procedure 

Is described. Hypermatrlx computations offer a number of advantages 

-compared to other techniques, Including skyline and wavefront, especlal Iy when the Implications of a paged virtual memory system and 

paral lei-pipeline hardware are considered. Hypermatrlx techniques are 

a I so adv·ocated later in th i 5 report for e I genprob I em so I ut I on of very 

I i~rge systems. 

2.5.1 ~bstructure Reduction 

The three major computational activities associated with processing 

an Isolated substructure include: 

1. Condensation of the substructure stiffness matrices to 

eliminate the slave nodal freedoms; 

2. Condensation of the equivalent nodal loads applied to the 

slave nodes; 

3. Recovery of slave node displacements once master nodal 

dIsplacements are known from solutions of higher level structures. 

The classical equations for these operations were first presented by 

Przelmlenlcki [2.15J and are Included here for completeness. The formal 

procedures are inefficient and seldom fol lowed. Two additional condensation procedures, referred to as coordinate transformation and partial 

decompOSition, have better efficiency and are discussed in detail. 
34 

Equilibrium equations for an Isolated substructure are first partltloned Into two sets corresponding to the slave and master nodes as 

where the superscripts m and s denote master and slave nodes respectlvely. The number of slave OOF Is designated by "p", the number of 

master OOF by "q". Constraints forcing a slave displacement to have a 

prescribed value and multi-point constraints between slave displacements 

are assumed to have been Imposed by modification of the coefficient 

matrix and load vector(s). The solution of Eq. (2.1) in partitioned 

form follows the standard procedures summarized below: 

(2.2 ) 

(2.4) 
35 

These equations reveal the form of the condensed stiffness matrix, 

Eq. (2.3), the reduced load vector, Eq. (2.4), and the procedure to 

recover slave node displacements from known master node values, 

[Ks s, -.1 Eq. (2.2). Their Inefficiency derives from the computation of J 

The computational penalty Increases dramatically when [Ks~ has a narrow 

. bandw I dth but a fu I I y popu I ateel I nverse (the most common case). Operation counts for the computations are given later In this section for 

comparing the reductIon methods. 

In the second method for condensatIon, the slave node displacements 

are related to the master node displacements i'hrough a coordinate transformation matrix, [T], such that 

Each column of [T] contains the displacements of the slave nodes for a 

unit value of one master node displacement component, al I other master 

DOF displacements being held zero. Because these dIsplacements 

represent deformed substructure shapes that are analgous to mode shapes 

In dynamics, they are often referred to as "static constraint modes". 

The matrix [T] Is evaluated by substItutIng Eq. (2.5) Into the fIrst row 

partition of Eq. (2.1) which yields, In the absence of external loading, 

{o }. (2.6) 

After eliminating {urn} from both sides of this equation, the product of 

[Ks:J with each column of [T] equals the corresponding column of _[KsrIj. 
36 

Equation solving techniques are thus applicable to compute the columns 

of [TJ. The matrix [KssJ Is triangulated only once using the front, 

Choleskl or some other decomposition scheme. The procedure may be relatively efficient since [KssJ Is often banded (the natural ordering of 

slave OOF Is not altered In forming these equations). The columns of 

'[TJ are obtained by successive forward and backward substitution over 

the columns of _[Ks~. An expression for the condensed stiffness matrix 

Is obtained by equating the strain energy of the substructure with and 

without the coordinate transformation. Thus, 

Using the symmetry of off-dlagonaJ submatrtces, [KS~ and [~sJ, and the 

symbol Ie form of [TJ, the right side of Eqe (2.7) can be expanded and 

slmpJ Ifled. The form of the condensed stiffness for computation becomes 

Eq. (2.8) Is often written in an unexpanded form that Includes an Identity matrix in the definition of [TJ. The condensed load vector for the 

master DOF Is related to the slave OOF load vector by [TJ In a similar 

manner using contragredfence 
37 

A third technique for substructure condensation, which has been 

widely adopted,· employs a ilpartlal decomposition" of the stiffness 

matrix. Wilson [2.19J provides a detaIled explanatIon of the compu"l"atlonal procedures. Equations for an Isolated substructure are assembled and partitioned Into slave and master nodal OaF as In Eq. (2.1). 

·Gauss or the more efficient Choleskl decomposItion Is appl led to complE~tely eliminate the first "p" rows corresponding to the slave OOF. 

Row-wise storage. and decomposition of the lower triangle Is assumed in 

this discussion. Similarly, the master-slave couplIng terms of [f(TlSJ 

arE:! reduced following standard procedures for off-d,agonal terms. The 

Ch<:>leskl I"eductlon formulas applicable to the slave OOF are 

I .. = I I k" " I I 

I " " IJ 

k." - = IJ 

i-1 2 1'/2 

L: lik) ; 

k=1 

j -1 

L: 

k=1 

I ~ " 

JJ 

:: p 

< p (2.10) 

j < i 

LO~ier limits of k=l on the summations Imply a fully populated coefflclent matrix. Extension 1"0 accomodate a variable bandwidth Is 

straightforward. 

A par-tfal decomposition Is then performed on the remaining [Km~ 

submatrlx of master OOF coefficients to eliminate the coupling effect of 

slewe oaF In the matrix [KmsJ. This Is accomplished by terminating the 

normal summations In Eq. (2~10) at column "p" for the terms I,J > p. 

The master" OaF stiffness terms are modified by thIs process to reflect 

thEI or I g I nal coup I I ng between s I ave- s I ave DaF an d s I ave-master DaF entl roe I y with I n the master-master' DaF terms. The mod I fled submatr I x [K mrYJ 
38 

Is the desired condensed stiffness matrix for the substructure. 

Final elimination of the master DOF occurs during solution ot the 

highest level structure. The basis for partial decomposition Is that 

slave nodes within a substructure do not have a topological connection 

with nodes elsewhere in the hierarchy. Thus, standard decomposition of 

·the complete structure stiffness, generated without substructurlng, 

simply skips the summations that Involve zero coupling terms between 

slave DOF In the equivalently substructured model. 

Condensation of substructure loading vectors Is accomplished 

through a forward reduction using the partIally triangulated stiffness. 

Summations for master DOF terms are again terminated at column "p". 

Condensed loads for the master DOF reside In the last "q" rows of the 

load vectors. The first "p" rows contain partial displacements of slave 

DOF reaay for backsubstrtutton. These are termed the "partial slave 

displacements." 

Recovery of slave DOF displacements is performed by completIng the 

backward substitution over the first "p" rows once master DOF displacements are available for the next higher structure. Before the back substitution Is begun, the partial slave displacements generated and saved 

durIng load condensation are placed In the fIrst "p" rows ot the load 

vector. Although this appears to be a trivial task, it becomes a very 

complex logIc problem when users are permitted to define load cases on 

hIgher level structures as combinations of condensed loading cases. In 

effect, users may Impl tcltly define new load cases on the lower level 

substructure for which partial slave displacements are not computed 

during load condensation. The displacement recovery logic must 

" \ 
39 

traverse the hierarchy again from the top, down to the level being 

processed to generate the loading combinations defined Implicitly. Only 

thl:ln can the correct partial slave displacements be computed by summing 

the separate load case values with the Implicitly defined multipliers. 

Computational efficiency Is always of concern when large matrices 

. an;, man I pu I ated. Experience with a large number of analyses has shown 

that generation of the condensed stiffness matrix requires nearly al I of 

thE~ computational effort associated with processing an Individual substructure .• Load case condensations and slave displacement recovery combilled seldom require 10% of the effort for stiffness condensation. Each 

of the a~gorlthms described above Involve extensive summations to 

generatl:J resultant terms. The number of multiplications performed In 

evaluating these summations provides a measure of their relative efflclency. Operation counts \'Iere developed for the stiffness condensatl(m phasE3 of the three algorithms based on the following assumptions: 

1. Matrix [KssJ Is symmetric, has "p" rows, and an average half 

bandwidth of "r"; 

2. Matrix [KmsJ = [K s~ T and Is fully populated as a consequence 

of DOF reordering necessary to partition slave and master OOF 

for condensation; 

3. Matrix [KmffiJ Is fully populatede 

I n the best poss I b I e case matr I x [K m SJ 'I supper tr I angu I ar; the 

fully populated worst case Is assumed here. Matrix [KsSJ wi II nearly 

always be banded. With these assumptions, It Is a simple task to 

estimate the number of multiplications required for each algorithm. The 

resu I ts aroe tabu I ated be I ow: 
40 

1. Classical Inversion: 2 2 p(O.5r + pr + pq + q ) 

2. Coordinate Transformation: p(O.5r 2 + qr + pq+ q2) 

3. Partial Decomposition: 2 2 O.5p(r + pq + q ) 

To Illustrate the effort required for each algorithm, consider a substructure for which p=500, q=50, and r=100; only 10% of the nodal OOF 

are retained after condensation. The operation counts for each algorlthm are listed below: 

1. Classical Inversion: 4.125 X 10 7 

2. Coordinate Transformation: L875 X 10 7 

3. Partial Decomposition: 0.940 X 10 7 

The results demonstrate clearly the Inefficiency of the classical 

Inversion algorithm employed In early substructure software. The 

coordinate transformation algorithm, which Is explicitly required for 

dynamic reduction, and the partial decomposition algorithm are considerably more effIcient. The effort for coordinate transformation 

rapidly approaches that for classical Inversion when "q" nears the value 

for "p", that Is, when a larger percentage of substructure DOF are 

retained after condensation. The results also demonstrate that the partlal decomposition procedure should always be used for static analysis. 
41 

2.5.2 Logl£Bl Control QUQ Data ~tructures 

The numerical algorithms described In the previous section are appi Icable for an Isolated substructure. Before the computations for a 

substructure may begin, the software logic must determine the correct 

order In which to process the substructures. The proper ordering 

depends on the type of operations to be perfor'med (such as stiffness assembly or displacement recovery) and the topologlc relationships between 

substructures, as represented by the hierarchical tree. This Is not 

tr I v I a I task for mode I s thai' cons I st of forty or fifty substructures 

distributed through five and six levels In a hierarchy. Finite element 

models with such characteristics are quite common for the analysis of 

large, geometrically complex structures. This section first outlines 

the familiar techniques used In standard analysiS programs. An overview 

is then given of the techniques required to support general substructuced analysis as Illustrated In the POLO-FINITE example of Section 2.4. 

Programs that support only standard, sta1"lc analysiS have comparatively simple control and data structure problems. A solution Is accomplished by Initiating a set of processing modules In a completely 

pr'edeterm I ned order that does not vary fran one ana I ys I s to another. 

The typical sequence Is: (1) read the Input data, (2) compute the eleme,nt stlffnesses, (3) assemble the structure stiffness and triangulate, 

(4) generate the loading vectors and perform a load pass to obtain 

displacements, (5) compute element strains and stresses, and (6) output 

results. ExtenSions to accomodate new 10adln~1 cases are quite simple. 

MCldlflcatlon of the structural model necessitates a completely new solu-
42 

tion. 

The data Is storea In a sequential manner that reflects the simple 

access mechanism required In the processing modules. A few sequential 

disk files suffice for most programs. Random files are occasionally 

adopted to facilitate stiffness assembly and triangulation for large 

. structures processed with out-of-core techniques. 

In contrast, the automatic solution of a general substructured 

model requires a dynamic control capabi I Ity. Each model solved with a 

general system may potentially require a unique order of module execution. The order of execution cannot be "pre-programmed" In the software 

as It Is for standard analysis. Rather, the software logic must use a 

description of the substructure hierarchy to determine the flow of execution for each particular analysis (or reanalysis). Dynamic control 

logic easily accomodates modifications to substructures that alter the 

flow of execution during reanalysis. More sophisticated data structures 

and access schemes are needed to support the dynamic nature of the solution procedure. The requirement for equal access to the data for any 

substructure el iminates consIderation of sequential file storage (unless 

literally hundreds of files are available). The topoiogic dependencies 

between substructures suggest the use of a hierarchIcal data organization that paral leis the natural substructure hierarchy. Formal data 

base management techniques are necessary to define, maintain, and access 

the data structures. With this approach It becomes feasible to efficiently maintain data for any substructured model, regardless of size 

and complexity, within a single dIsk file. 
43 

For purposes of discussion, the substructured solution Is considered to have two computational phases. These are: 1) processing the 

model through the computation of displacements for the highest level 

structure:, and 2) recovery of substructure displacements and element 

strains-stresses. The separation Into these two phases fol lows from the 

. da"\"a structures and processing logic naturally suited for each task. 

Output generation Is also Important but does not Impact the computational processes or data organization. The first phase Involves 

stIffness and load vector assembly, and requires that substructures be 

processed upward from the bottom of the hierarchy. Data structures must 

support the repeated use of a substructure stiffness matrix and loads to 

form similar matrices for higher level structures. In the second compu"I'atlonal phase, processing of the hierarchy occurs from the top-down 

and then only along user designated paths. Displacements, strains, and 

str'esses are not normally recovered for al I substructures. Data structUl"es mwst reflect the uniqueness of displacements, strains, and 

stresses ·for each occurence of a substructure I n the hierarchy. 

Stiffness matrix and load vector assembly proceed upward from the 

lowest levels of the hierarchy. Assemoly of a structure at level "I" 

cannot begin until all dependent substructures at level "1-1" have been 

completed. Control of the assembly process through a stack driven 

tnlversal of the hierarchy appears, at first, to be a natural approach. 

However, a topo I og I c sort of "~he hierarchy (performed us I ng a stack) t'o 

de'i"ermlne the processing order before computations begin Is far simpler. 

Th!:) topo I og I c sort prov I des thE~ st I ffness modu lew I th a proper I y ordered 

list of substructures to process. Figure 2.7 presents a block flow 
44 

Set STKlEV • 1 

Push Structure "X" 

on to Stack 

Topologic Sort 

>----1 Completed. 

Save List. 

Perform Consistancy Checks 

on Current Structure 

Mark Structure 

as Precessed, 

Decrement STKLEV 

Reset Control 

Variables to Structure 

at STKLEV 

Emit Structure Id 

I nto Sorted List. 

Mark [K1 at STKLEV-l 

Invalid 

'----..-j Save Control Variables I n Stack., 

Increment STKLEV. 

I nitialize Next Stack Level to 

Begin Processing Substructure 

Figure 2.7 -- Topologic Sort to Order Substructures "for Assembly 
45 

diagram for the topologlc sort. A substructure appears only once In the 

sorted list regardless of Its number of occurrences In the hierarchy. 

During reanalysis, the topologlc sort simply omits al I unmodified substructures from the list. Logical control within the assembly module 

becomes quite simple given the sorted list of substructures to process. 

·Flgure 2.8 Illustrates the overal I assembly logic, with the topologlc 

sort shown as the first operation. The first structure In the list Is 

e><:tracted and Its stiffness assembled, followE~d by the second structure 

I n the I 1st, etc. On I y struc1·ures appear I n the sorted I 1st. The assembly module generates matrices for finite elements which occur In the 

structure being assembled. 

When a "condensed" substructure 15 extracted from the sorted list, 

the assembler will have already computed the "uncondensed" stiffness 

matrix since the single child of the condensed substructure occurs one 

level lower In the hierarchy. The assembler suspends execution and Invokes the equation solving module, passing It the Identifier for' the 

substructure to be condensed as Indicated In Fig. 2.8. After the stiffness condensation 15 completed, the assembler resumes execution, 

retrieves the master node matrices, and saves them In data structures 

for the condensed structure. No traditional assembly operations are 

performed. Thereafter, the condensed substructure has a stiffness 

matrix with the same format as any other structure (or finite element). 

The assembler continues execution by selecting the next structure In the 

sortea list. Because the processing modules are dynamically Invoked, 

condensation at any number of levels creates no logical control difficulties with this procedure. 
Database of ( 

Structure j > Detin itions 

[Kl For List 

Structure 'X' Exhausted Completed 

46 

Enter with Assembly Request 

for Specified Structure 'X" 

Topologic Sort of H ierachy 

Starting at Level 'X' 

Get Id of Next Structure 

to Assemble from List 

y 

'A' 

'B' 

'el 

> ID' 

. 

'X' 

Sorted List of 

Substructures 

for Assembly 

Compute [kG] For Each 

Finite Element -in this 

Structure 

r-------~-----------~ I i 

I I 

! Save Assembler I 

'>---+-"-1 Status I 

N 

Assemble [KJ for 

Current Structure 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

Run iSolverl 

Module 

Restore Assembler 

Status 

I Build (Kl for Condensed 

I 

I 

I 

I 

I 

.------t-I -; Structure from Solver I 

I Results I I I 

I I 

I I 

--~~~~~~::~---------~ 

A 

Figure 2.8 -- Multilevel Substructure Assembly Procedure 
47 

Figure 2.9 Illustrates the essential features of a data structure 

for storage of the hierarchy description and stiffness matrices. The 

ELEMENTS table contains one column for every finite element and structure declared by the user. Rows of this table describe attributes for 

thE~ element or structure, for examp I e, the number of nodes "NUM.-NODES". 

'A STRUCTURE table exists only for columns that contain a structure. The 

STRUCTURE table stores pointers to lower level tables that contain data 

applicable only to a structure, for example, subelement Incidences, 

subelement orientations, nodal coordinates, and components. The COMPONENTS table has one entry for each subelement In the structure. The 

da'~a va I we for each sube I ement I s the co I umn number, denoted ECOL on' the 

figure, In the ELEMENTS table that defines the subelement. The subelement may be a structure or a finite element. Given the ECOL for a 

structure, the hierarchy from that level downward may be easily 

traversed. 

The stiffness matrix for a finite element and for a structure have 

Id1cmtlcal storage formats as shown in Fig. 2.9. A stiffness matrix cons Ists of noda I "submatr Ices" wh I ch are stored I n a row-w I se format. Ttle 

column of the first non-zero subl!latrlx In a nodal row Is Indicated by 

th'E~ value of FIRST_COL. For finite elements, this value Is always ono. 

FOir" structures, the value of FIRST_COL reflects the half bandwidth for 

th·e nodal row. The lower triangle of the element stiffness Is stored. 

All subma'trlces from col umn FIRST_COL through the diagonal are generated 

even though there may be Intermediate nul I submatrlces. 
ElEMENTS 

I) 

Column 

Number 

ECOl 

Nu:~~li%~~~~1 t--+ 1-1-1-1-1 +-1 -+-1 -i---i8 ... STRUCTURE ~ 

S1I ffNES S 11--111-11-1-1--1-1--+1 +1 -1-1 -II 

• • • 

1 2 3· NUM NODES 

I • I i I 

FIRST_COLI I I I I I 

NUM_5UB_MAT I I I I I I 

SU BMATR ICES 1-1 -L---I'yl.....JI.----'---'" 

1 

• • • 

NUM_SUB_MAl t--

.....l- Column Number ECOL 

It--

NUMJlEMENTS Lfor i th 5 ubelement 

Fi gure 2.9 -- Data Structures for Hierarchy Representation of St iff ness Ma1-r ices 

1 ' 
49 

Assembly of the equivalent nodal loads for a multilevel substructure model Is logically more complex than stiffness assembly. For example, when displacements for a load case defined on the highest level 

structure are requested, the equivalent loads processor must traverse 

the complete hierarchy while determining which substructures have load 

·cases that contribute to the one specified on the highest level structure. The procedure Is even more Interesting when analysts define load 

cases I n terms of other load cases from onEI substructure I eve I to the 

next. A more flexible data structure than a simple vector of structure 

Identifiers Is necessary to accomodate the load case laentlflers. A 

multiple vector, Inverted list proves adequa1"e. Condensation of the 

load vectors Is performed In the same manner as stiffness condensatlon·--

as an Interrupt In the normal procedure of transferring loads from one 

substructure level to another. 

Phase two computations recover substructure slave displacements and 

element strains-stresses. The analyst generally specifies a path 

thl~ougn the hierarchy that I dentl f I es the occurence of a substructure 

for which results are desired. The particular substructure may be at 

the lowest level of the hierarchy, In which case relatively few resul'ts 

are generated. Alternatively, It may be an Intermediate level substructure, which can be used to Imply dIsplacement recovery for that level 

stl"ucture and al I lower level substructures of the same branch. 

Logical control using a stack technique Is most convenient to drive 

substruc-l"ure displacement recovery. Given the analyst suppl led path and 

thi~ COMPONENTS tab I e shown In FIg. 2.9, the traversa I procedure to reach 

th·e deslr1ed substructure(s) Is quite sImple. The back-condensatIon com-
50 

putatlons to retrieve slave displacements from the master values are 

logically treated as an Interrupt In processing down the hierarchy. In 

a manner paral lei Ing the assembler condensation procedures, the 

displacement recovery module suspends execution, then Initiates the 

equation solving module to compute siave DOF displacements for the cur-

'rent substructure. On resuming execution, the displacement recovery 

module extracts the slave DOF displacements In the solver data structures and reformats them to conform with standard displacement data 

structures. The next lower level In the substructure list may then be 

processed. Element strain-stress computations may be performed immediately after slave DOF displacements are' recovered or a separate 

module may be Invoked fol lowing the completion of al I displacement 

recovery. A stack drIven procedure also facilitates the computation ot 

substructure straIns-stresses. 

The most interesting aspect of phase two involves the development 

of data structures for storage of substructure displacements, straIns 

and stresses. Only one stiffness matrix exIsts In the database for a 

substructure regardless of Its number of occurrences, but the displacements, strains, and stresses are unique for each occurrenceo The data 

structure shown in FIg. 2.9 for phase one processing requires extension 

to support substructure displacement recovery. One solutIon for this 

problem Is Illustrated In Fig. 2.100 The uniqueness of substructure 

results Is recognized only at the highest level structure. The LOADS 

table points to lower level tables that contain the definition and computed results for each load case. Displacements, strains, and stresses 

for the complete hierarchy for a load case are stored under the cor-
ELEMENTS 

LOAD 

SPOIN 

D POIN 

0 

S 

T 

T 

LOAD_CA SEJD 

DISPLACEM 

ST 

STR 

ENTS 

RAINS 

ESSES 

51 

ELEMENTS -

-

LOADS 

-

STRAINS M::-··"If' ,1' _ I~I III 

l 

r-

~ 

100'- '-

Variable Length 

Result Vectors 

••• 

r--

~ 

• • • 

I--

j-

• • • 

t--

--

1 

2 

I Vector I Position I 

:/I S ubelements 

Figure 2.10 -- Data Structures for Storage of Substructure Results 
52 

responding column of the LOADS table (defined for the highest level 

structure). Results are packed into sets of variable length vectors 

that reside under the DISPLACEMENTS, STRAINS, and STRESSES pointer 

tables. The al location and retrieval of result vectors is accomplished 

by a two-level pointer scheme. For example, vector SPOINT shown in the 

·figure contains one entry for each subelement in the structure. For 

subelement~ that are finite elements, the "i" th entry in SPOINT 

defines the vector and position within the set of result vectors at 

which values for the element begin. For subelements that are substructures, the SPOINT entry defines a relative shift In the vector numbers 

from the current level structure's vector to the first relative vector 

for the specific occurrence of a substructure. SPOINT entries for the 

substructure refer to vector numbers that are relative to the first 

result vector for the substructure. Results for multiple occurrences of 

the same substructure thus appear In different stress-strain vectors. 

The absolute vector number for processing at any time Is obtained by accumulatIng relative shifts for each substructure present In the traversal stack. 

The most complIcated aspect of this scheme Involves construction of 

the SPOINT vectors for each substructure. Note that the SPOINT vectors 

are Independent of any load cases. The mapping of substructure results 

onto vectors Is Identical for each load case. An identical scheme Is 

used to store substructure nodal displacements, reactions, Initial 

strains-stresses, and residual loads In nonlinear analysis. 
53 

2 .. 5.3 Linear Equation Solving 

Efficient procedures for linear equation solving are essential In 

fInite element analysis. Reduction of the symmetric stiffness matrix to 

triangular form often requires 50 percent of the computational effort 

for linear analysis and a slightly smaller percentage for nonlinear 

analyses. Many efficient algorithms and the detal Is of their computer 

Implementation have been published. Meyer [2.12J has presented an extEmslve r-evlew of the subject. 

The growing avallabl I Ity of computer hardware desIgned specifically 

for "number crunchIng" applicatIons has spurred renewed Interest In 

equatIon solvIng procedures. The details of data storage and access 

mE~chanlsms comprise the key factors In the ability to utilize advanced 

h(~rdware. Advanced mach I ne ar"ch I tectures I nc I ude para I I e I processors 

(the basis for most supercomputers) and virtual memory superminis. Some 

of the most recent superminI computers also have a pipeline design that 

gr-eatly speeds up numerical operatIons compared to more conventional 

scalar processing. The pipet ine concept uses a single processor but 

with separate prefetchlng of data to minimize processor walt times. It 

thus represents a Intermediate desIgn between a scalar processor and the 

parallel processing machine. The scalar deSign superminis can also be 

enhanced with the addition of an attached array processor. 

The basic requIrements for any equation solving procedure can be 

summar I ZE~d as: 
54 

1. The number of equations should be I imlted only by the amount 

of disk storage available; 

2. Use a reasonable amount of memory so as not to severely Impact 

scheduling of the computer runs; 

3. Minimize the storage of zero coefficients and operations on 

them; 

4. Minimize data transfers to and from disk (I/O); 

5. Exploit capabilities of modern computer architecture. 

Substructure condensation by partial decomposition Is essentially an 

equation solving procedure and therefore the same requirements listed 

above apply. However, stiffness condensation may adversely effect one 

decomposition procedure more than another. With band-based solvers, for 

example, reordering of the equations to facilitate condensatIon may 

produce a border-banded matrix of coefficients. The computation time 

required for partIal decomposition Is greatly Increased compared to the 

time required for complete decomposition of the equations without 

reoraerlng. 

This section briefly surveys three standard equation solving 

procedures, namely (1) Choleski with variable band (skyline) storage, 

(2) the front method of solution, and (3) Choleskl with hypermatrlx 

storage. Each of these Is a direct solution algorithm that Involves a 

three step process: (1) reduction of the coefficient matrix to 

triangular form, (2) forward reduction of the load vector(s), and 

(3) recovery of displacements by a backward pass. The primary Interests 

In the current discussIon include the Impact of reordering the equations 

prior to condensation and the prospects for adopting each procedure to 

new computer hardware. 
55 

yarlabl~ Bandwidth Procedure 

This tecnnlque uses a Choleskl or Crout triangulation algorithm 

with a compact storage scheme to accomodate wide variations In bandwidth. Jennings and Tuff [2.8J originally developed the procedure. 

Mondkar and Powell [2.13J later published thE~ Implementation details and 

results of timing studies. This procedure appears to have been widely 

adopted for "In-house" finite element systems because the data handling 

details are straightforward. 

Cont I guous co I umns of the upper tr I ang I EI are grouped together In 

blocks (Fig. 2.11). AI I coefficients In a block are transferred to and 

from disk In a single, logical operation. The available memory determines t'he number of columns assigned to each block. The first non-zero 

r'ow I n each co I umn I s recorded dur I ng assemb I y. Th I s I nformat Ion Is 

simply appended after the last column of each block. AI I terms from the 

first non- zero row to the d t agona I I n a co I umn are assumed to f I I I'~ In 

during triangulation. 

Reduction of each term in column "I" requires that column "1" and 

elt I east one other co I umn, "J"" where J < I, res I de t n memory. Th us, at 

least two blocks of columns must fit sImultaneously Into the available 

memory. During forward load reduction, one coefficient block and one 

load block are required In memory (multiple load cases are handled very 

efficiently). The same memory requirements exist for the backward pass 

t'o recover dis placements. 
56 

Blocks 

fZl Reduced 

(§J Being Reduced 

o Not Yet Reduced 

{al Typical Variable Band Blocked Coefficient Matrix 

-i I- Retained (Master) DOF 

(]) ®CV® 

Note Large Bandwidth 

(bl Coefficient Matrix Reordered for Condensation 

Figure 2.1 I -- Storage Scheme for Blocked, Variable Bandwidth Procedure 
57 

Over 80 percent of the effort In triangulating the matrIx Involves 

computing Inner products of two non-contiguous columns. This makes the 

algorithm very attractive for all types of advanced hardware. The' 

number of terms In the Inner pr'oducts Is usually large relative to the 

break even point on most haruware (some overhead Is Involved In paral lei 

'processing to Initially align data which requires that inner products 

have a minimum length to recover the overhead tlme--general Iy 5 to' 10 

terms) • 

The variable band procedure has a number of drawbacks. It can be 

very I/O Inefficient during structure stiffness assembly if only one or 

two blocks fit Into memory. Elements are usually processed In seque~· 

tlal order to assemble the structure stiffness. As a consequence, It Is 

possible for a structure to have a very narrow bandwidth and yet require 

considerable swapping of blocks during assembly. ' The classic example Is 

a narrow rectangular grid with nodes numbered In the short direction to 

minimize the bandwidth and elements numbered In the long direction 

(whl~h does not affect the bandwidth). To assemble each sequence at 

elements In the long direction, each coefficient block must be brought 

Into memory. Another drawback occurs when column heights vary slg-· 

nlflcantly. Large numbers of Inactive terms are transferred Into memory 

during triangulation as a consequence of the blocking (Fig. Z.lla). 

Las't I y, the storage scheme Is biased towards access I ng coeff I cl ents 

column-wise, which Is natural for' triangulation and forward load reduC"-

tloo. As noted above, each block Is transferred to memory only once 

during the forward pass. During the backward pass, however, row-wise 

access Is the most natural. A block of columns may be transferred Into 
:38 

memory many times depending on bandwidth variations. 

When executed on virtual memory hardware, the logical memory space 

(that declared In FORTRAN DIMENSION statements) for blocks may be very 

large, for example, sufficient to store seven or eight blocks. The 

program Issues fewer logical I/O requests to transfer blocks than a con-

·ventlonal approach in which space for only two blocks Is al located. The 

virtual memory operating system performs additional I/O required forcing 

the program to execute In a given amount of real memory (working set 

size). This process is normally handled through fragmentation of data 

arrays Into pages which are transferred to and from memory as the array 

elements are referenced In the programe Considerable experimentation Is 

necessary to balance the virtual paging and program block I/O, especlally on virtual machines that permit users to expl icltly declare the 

working set size. 

Condensation causes difficulty only when the bandwidth Increase due 

to reordering becomes excessive. Consider a structure which Is numbered 

to minimize the average bandwidth for conventional analysis. In 

general, the reordering to accomodate condensation produces the borderbanded matrix shown In Fig. 2.11b. This procedure requires that at 

least one complete column fit into a block. For large 3-D structures, 

thIs requirement may lead to large increases In block sizes to accomodate the very long columns corresponding to master degrees of 

freedom. During the partial decomposition process, no computational 

penalty occurs until the equations with the large band in the last 

blocks are reached. Most of the triangulation time may be involved In 

eliminating these terms. However, advanced hardware can be exploited to 

t-- "\ 
59 

the ful lest In processing the very long columns • 

.Er:rulliL p.r:9ce d u r e 

The highly touted frontal procedure was first introduced by Irons 

[2.6J and has since been extended to include condensation. The frontal 

solver Is E~ssentlally Gauss elimination with eXTensive bookkeeping to 

minimize operations on zeroes. The structure stiffness matrix Is never 

explicitly assembled. Rather, element stlffnesses are brought sequenti a II y I n"rO memory, their terms added to the system matr I x and then 

trlangulatE~d, all In one logical <but very complex) process. The 

ordc:lrlng 01f equations In the system matrix Is determined by the element, 

rather than the node, numbering scheme. As elements are 

newly appearing OOF are simply appended as new equations. 

occupied by active equation coefflcents is referred to as the 

processed, 

The memory 

"front. IV 

The front storage space varies dynamically during solution as new equa-· 

tlolns are added and old ones are completely el iminated <eliminated coefficients are transferred to disk). The front Is very similar to the 

"active triangle" concept In band-based solvers. 

The frontal procedure eliminates much of the CPU and I/O costs associated with structure stiffness assembly. The triangulation aspect Is 

no more eH i c i ent than the Cho I esk I process. There are two major d i sad-· 

van'tages c::>f the fronta I procedure for non-substructured ana lyses. 

First, the active front size becomes quite large for 3-D solid analyses. 

The bookkeeping logIc Is very complex when the complete front fits Into 

memory; it appears Intractable when coupled with a "spill" algorithm to 

acc()modate a I arge front that Is parti a I I Y memory res I dent. STorage 
60 

space of 100-200K words for an active front Is common. This may cause 

scheduling problems on some machines and Impose absolute structure size 

I Imitations on others. The second difficulty with the frontal method Is 

Its adaptation to advanced hardware. Unl Ike the Choleski procedure, 

Inner products Involving hundreds of terms are not common In the frontal 

. solver. Operations are performed almost randomly over the active front. 

Paral lei and pipeline hardware offer lIttle advantage. 

The frontal method appears Ideally suited for virtual memory computers with scalar processors. The problem of handling a very large active front Is relegated to the virtual memory operatIng system of the 

computer. An extremely large array space Is dImensioned for the front. 

The operating system pages the array segments as needed to maintain a 

predefined working set. The random accessing Into the front causes no 

penalty on a scalar (sequential) processor. 

The absence of an explicitly assembled structure stiffness Is a 

drawback of the frontal solver for substructured analyses. Substructure 

stiffness matrices (original and reduced forms) must always be available 

for use In defining other structural hierarchies. Stiffness condensation also affects the efficiency of the frontal solver as It does the 

variable band solver. Consider the element grid shown In Fig. 2.12a. 

All nodes on the perimeter are retained after condensation. Conventional node numbering for a band solver produces a border-banded matrix. 

Condensation In the frontal solver Is accomplished by retaining al I 

master nodes In the front until slave nodes are completely eliminated. 

For the mesh shown, the active front size Increases each time a top and 

bottom row element Is processed. The front sIze reaches a maximum when 
toD 

f(4) 

~(3) 

f(2) 

f(O 

61 

All Perimeter Nodes 

/ Retai ned (Master DOF) 

,~?) 

t<!~ 

(8~ 

(a) Worst Numbering for Condensation with Front Procedure 

-- Best Numbering for No Condensation 

I@s I~?) 1@9) lea§) 

1(9) IUID I® 

I~) f(2) 

i~ 1(1) GID f@6) 

l~s @g) @~ C?~ 

(b) Number Scheme to Postpone Maximum Wavefront Until Last 

Elements Processed 

Figure 2.12 -- Effects of Element Numbering on Condensation by Frontal Technique 
62 

element 88 Is processed. Alternatively, elements can be numbered as 

shown In Fig. 2.12b for front solution. The front size remains constant 

while al I slave OOF are completely eliminated. It then grows rapidly to 

the same maximum size for the previous numbering scheme as elements 

coupling the master and slave OOF are processed. The only advantage of 

'the second case is that a maximum front size exists for less execution 

time. This could drastically reduce the paging rate on virtual memory 

machines. The same effect occurs for the variable band solver In that 

the active triangle reaches a maximum size when the master OOF are encountereo. 

Hypermatrlx Procedure 

The basic concept of hypermatrlx storage is Illustrated in 

Figo 2.13a. A block of contiguous columns defined in the variable band 

procedure Is further partItioned row-wise to form rectangular "hypermatrlces"o This storage format overcomes the major problems with 

variable band storage--excesslve column heights and the transfer of 

unused terms during triangulatIon. 

Diagonal hypermatrlces are always square; off-diagonal matrices are 

frequently rectangular. Hypermatrlces with al I zero terms are never 

created. It Is thus a simple matter to omit large numbers of zero 

operations wIth such a storage format. The sizes of hypermatrfces can 

easily be adjusted to fit a particular amount of memory available for 

processing. The use of larger blocks (I.e., greater than 50 x 50) Increases the number of zero terms picked up at the periphery of the band. 

The use of smaller blocks Improves the recognition of zero terms but In- r·· \ 
63 

Non-Zero ~ 

Hypermatrix ~ QJ 

N 

~~~.44441-..,.--+--4---I-. - ;;; 

®~ Zero Hypermatrix D 

(Not Stored) 

~~~,~~~~--- ~ CD .~ """~~~~~-- > @-; "'''''''~o44f,..<;4404 - == 

Hypermatirx Sizes Vary Diagonal I ® t 

10 x 10--50 x 50 Hypermatrices - ~ 

Always Square ® ~ 

t 

(a) Symmetric Partitioning of a Sparse Matrix into Hypermatrices 

r--Hypercolumn POinters--j 

112345 nl 

First Non-Zero --..----------... Row in Column 1 I 1 2 3 •• • H. R.--Hyperrow 

Pointer to Row Pointers • •• 

H. R. (3) ........ 1.......,..., ....... 

H. R. (4) 

H. R. (5) 

Last Pointer for Each Col umn

l 

is to Diagonal Hypermatrix 

(b) Data Structure for Hypermatrix Storage 

FiSlure 2.13 -- Hypermatrix Storage of Sparse Matrices 
64 

creases the data handling overhead. 

A formal data structure for hypermatrlx storage is shown In 

Flg.2.13b. Individual hypermatrlces are stored as separate entitles 

(for example, a logical disk file record). Pointer vectors that locate 

all hypermatrlces within a column emlnate from the header table shown. 

'Polnters to zero hypermatrlces are not maintained In the vectors. Instead, the header table contains an offset locating the first non-zero 

hypermatrlx In each column. 

Hypermatrlx partitioning has been studied extensively by computer 

scientists [2.10J In connection with array operations on virtual memory 

(paging) hardware. It has been demonstrated that hypermatrlx partitioning requires the least working storage and Incurs the least page 

faults for the operations of multiplication and triangulation. The 

usual scalar formulations for matrix multIplication and triangulation 

(Gauss, Cnoleskl, Crout) have very sImilar counterparts when cast In 

hypermatrlx form. Choleski decomposItion In hypermatrtx form requires 

the same number of floating point operations as In scalar form. 80th 

multiplication and triangulation require that only three hypermatrices 

reside simultaneously in memory. Therefore, large problems may be 

solved in a very small memory space, for example, 7500 words If 50 x 50 

blocks are used. Moreover, If memory for only three blocks is available 

It becomes a trivial problem to predict exactly the number of block 

transfers to and from disk. When space Is available In memory for more 

than three blocks, It has been suggested [2.10J that a "least recently 

used" replacement algorithm efficiently utilizes the additional memory. 
65 

Fuchs and Schrem [2.4J devised a Choleskl triangulation scheme In 

hypermatrlx form and Implemented the procedure In the ASKA program 

[2.16J. The data structure In ASKA employs a two level matrix of 

pclinters to the hypermatrlces Instead of the scheme shown In Fig. 2.13b. 

The POLO-FINITE system uses the hypermatrlx procedure with the storage 

,format ~;hown In Fig.2.13b. Both ASKA and POLO-FINITE use demand 

pclg I ng, v I rtua I memory management systems for wh I ch hypermatr I ces are 

Ideally suited. Hypermatrlx sizes are usually defined such that one 

hypermatr-Ix fits onto a page. A page corresponds to a single random 

disk file record. The advantage of al locating one hypermatrlx per page 

Is that totally unbiased access to any block Is obtained~ Triangulation, forward pass, and backward pass operations thus have equal effIciency with respect to data access. As previously noted, the variable 

band schE3me suffers a heavy access pena I ty dur I ng the backward pass. 

As with the variable band procedure, the major effort In hypermc:ttrlx triangulation Involves Inner products of non-contiguous columns o 

Figure 2.14 Illustrates this. However, In hypermatrlx form, the inner 

product Is actually a sequence of matrIx multiplications as IndIcated on 

the figure. In effect, inner products over multiple scalar columns 

proceed simultaneously. Timing studies for large sets of equations have 

shown th'3t the matrix multiplications usually represent 90% of the 

trlangul.3tlon effort. The data handling overhead seldom exceeds 5% of 

the triangulation time. For example, after two 50 x 50 blocks are 

br'ought Into memory, approximately 250,000 floating point multiplications and additions (plus subscripting) are performed to complete the 

matrix multiplication. 
T 

t 

t = i 

c>--m m = i 

Uli 

AI ready Reduced U2i 

-

:-

Kik 

Symmetric Null 

Symmetric Positive Definite 

Hypermatrlx K 

Upper Triangular 

Hypermatrix U 

i-I T 

Typical Operation: [Uikl ;:; [Kid - E [Uti] [Utk] 

t==l 

Figure 2014 -- Schematic of Hypermatrix Choleski Triangulation I 

m = k 

~ 

~2t 

. Ua 

! 

0\ 

0\ 
67 

Hypermatrix triangulation appears wei I suited for adaptation to 

paral lei and pipeline hardware. The order of matrices In the multiply 

operation shown In Fig. 2.14 is particularly important. Because the 

first term Is transposed, the matrix multiply Is simply Inner products 

of columns In the two matrices, rather than the traditional row mul-

'tlplled Into a column. Storage of the lower triangle In hypermatrlx 

for"m eliminates this advantage of column-wise Inner products. The 

equivalent matrix product requires Inner products of two rows, rather 

than two columns. The choice of lower or upper triangle storage is immaterial for computations performed on scalar hardware. However, upper 

triangular storage Is preferred for the most general case. 

The large bandwidth Increase that occurs with equation reordering 

prior to condensation presents no difficulties In the hypermatrlx 

scheme. Large bandwidth fluctuations simply Increase or decrease the 

number of non-zero hypermatrlces In a column. The problem of fitting an 

entIre column Into memory that occurs with the variable band method does 

not occur with hypermatrlx storage. 

206 Exampw..Qf Substructured Analyses 

Two E~xamp I e ana I yses are descr I bed In th I s sect I on to II I ustrate 

typical computational savings achieved with substructured models. In 

the first example, a linear analysiS Is performed for a portion of a Jet 

en!~lne exhaust duct. Condensation Is applied at two levels In the substructure hierarchy -- at the lowest level to reduce the bandwidth and 

at the next higher level to utilize repeated components. The exhaust 

duct repr-esents an exce I I ent test case for substructured dynam 1 c 
68 

analysis with modal synthesis (discussed in Chapter 3) considering only 

linear response. Dynamic analyses using a ful I, unsubstructured model 

and some I imlted experimental data are available In the literature for 

comparison. 

The second example problem consists of a sl ightly curved thick 

'shel I constructed of impact resistant acryl ic. A uniform pressure is 

appl led over a very smal I area at the apex to simulate the Impact of a 

nondeformable object. The magnitude of the pressure is increased In 

each load Increment but the loading area remains constant. Nonl inear 

response due to yielding of the material In the impact region is considered. The problem typifies a large class of structures for which 

substructured models reduce the computation time. The region of nonlinear response Is easily estimated prior to the analysis. Standard and 

substructured models for this example are analyzed to provide data for 

comparisons of computational effort. This problem also provides an excellent test case for substructured, nonlinear dynamic analysts. A 

transient analysis Is required to predict dynamic response fol lowing the 

Impact of a high velocity proJectile. Even under such loading, the nonlinear zone remains smal I relatIve to the overal I structural dimensions. 

Condensation of the linear region should greatly reduce the computational effort for a transient analysis. For comparison, the computational effort within each time step of a transient analysis corresponds to that for a load Increment In static analysis. The results 

presented here for the static solution provide a basis to eSTimate computational savings for transient analysts. 
69 

Each structure has been analyzed with the POLO-FINITE system. Comparisons between substructured and standard model solutions are made on 

the basis of CPU time and I/O. The POLO supervisor performs database 

and memory management functions for FINITE sUbsystems. POLO has extensive Instrumentation that provides detal led summaries of CPU usage among 

'the FINITE subsystems, for example: Input, assembly, and triangulation. 

Within each subsystem. the CPU time expended on data management activities and on actual finite element computations Is also available. 

CPU times are presented In non-dimensional form to eliminate dependenclE~s on the processor execution speed. I/O activity Is measured by the 

number of "page faults" executed by the POLO memory management system. 

A page fault represents the transfer of one 2500 word record from memory 

to disk fol lowed by the transfer of another 2500 word vector to memory. 

For a given structural model, the number of page faults performed by 

POLO Is Independent of the computer hardware; It depends only on the 

dimensioned length of a data vector within POLO. Page faults are thus a 

very simple measure of I/O activity. Printer output and I/O transfel~s 

to sequential card Image files are Ignored. 

2.6.1 .L.J..n?.:.a.r:. Examp I e 

Figure 2.15 shows a portion of a jet engine exhaust duct modeled 

for static stress analysis. The duct consists of a thin circular shroud 

connected to the central core with radial fins. Loadings of Interest 

Include ·torslon, uniform external pressure, and nonuniform temperature 

distributions. For this analysis, the lnner edge of each fin Is camp I «:lte I y f I xed. 
DIMENSIONS 

Sh roud Fins 

R = 170 m m L = 64 m m 

t=1.5mm t=3.0mm 

w = 81 m m w .. 27 m m 

Full, 3600 Model 

70 

Edge Completely 

Fixed (Typical) 

V 

320 Isoparametric Shell Elements 

88 I soparametric Solid Elements 

9504 Degrees of Freedom 

Figure 2.15 -- Jet Engine Exhaust Duct for Example Linear Analysis 
71 

Difficulties In constructing a proper finite element model arise at 

each junction of a fin with the shroud. Accurate determination of the 

bending stresses In the junction vicinity requires a relatively fine 

mesh and the maintenance of element displacement compatibility. Curved 

shl!~11 elements adequately model major portions of the fins and shroud. 

'The 8 node Isoparametrlc shel I element Is used here. Each Junction Is 

modeled as a 3-D solid with shel I-to-sol Id, and sol Id-to-sol Id transition elements employed to maintain displacement compatibility. This 

model provides realistic predictions of stress distributions at each 

Junction without an undue Increase In the number of nodal OaF. Figure 

2.16 Illustrates the four types of elements employed In the analysis. 

A 360 degree model is required for general analysis. For the grid 

shown In Fig. 2.15, the ful I model has 408 elements and approximately 

9500 OaF. The generation of each element stiffness matrix requires 

numerical Integration. And while the lar'ge number of OaF does not 

present major problems for general purpose systems, the grid topology 

does introduce some severe computational penalties. The traditional 

node numbering scheme fol lows the narrow (axlell) direction then the circumferential direction. Each fin causes a lelrge re-entrant area In the 

equation coefficients, but the most severe penalty arises because the 

first and last nodes are coupled. As a result the stiffness matrix is 

border-banded; the last 78 rows have a bandwidth of nearly 9500. This 

drastically Increases the triangulation effort. Substructurlng wl'th 

condensation proves very computationally efficient for this structure as 

demonstrated below. 
a) 8 Node Shell (QSHELL) 

d 16 'Node Solid (TS161 SOP) 

b) Shell-to-Solid Transition 

(SHEll-TRANS) 

d) 18 Node Solid Transition (TS181 SOP) 

Figure 2.16 -- EI ement Types for Exhaust Duct fl.1odel 

--.J 

N 
73 

Division of the ful I model Into Identical 90 degree substructures 

1$ obvious. The selected partitions place the fin at the center of the 

substruci"ure as shown In Fig. 2.17. The location of each type of finite 

element Is also Indicated on this figure. Condensation of this substructurE~ reduces the number of nodes to 26; 13 nodes across each end 

'for connection with adjacent copies of the same substructure. Within 

the 90 dE~gree substructure, the fin I s a I so mode I ed as a substructure 

and condensed to the 13 nodes that connect to the shroud. Condensation 

of the fIn eliminates the re-entrant area In the coefficients described 

above for the fu II mode I. FI gure 2.18 shows the same 90 degree section 

model wIthout substructurlng. 

Considering the analysis of an Isolated 90 degree section, the fin 

condensation reduces CPU time by 14% and I/O transfers by 21% compared 

tl:> an analysis without fin condensation. (Fin stIffness generation clnd 

cl:>ndensatlon time are Included In the comparison.) However, condensation 

to eliminate al I but the 26 boundary nodes Increases the CPU by 53% and 

the I/O transfers by 65%. These Increases reflect directly the penalty 

Incurred by reordering the equations prior to condensation. Reordering 

for this substructure has the same effect on the bandwidth as closing 

the ring on the fut I model. The first and last nodes are effectively 

coupled by the reordering, but final 156 rows have a bandwidth of only 

2400 compared to 9500 for the final 78 rows of the 360 degree model. 

The large bandwidth difference produces the major savIngs for the fut t 

model generated with substructures. (Recal I that the triangulation effort Is proportional to bandWidth squared.) 
74 

13 Nodes on th is Edge Reta i ned 

TS161 SOP ~~ ••• i!? SHELL -TRANS 

13 Nodes on this 

Face Retai ned 

TSl81 SOP -M@ 

SHELL-TRANS 

QSHELL 

Fixed Edge 

Substructure FIN 

13 Nodes on this 

Edge Retai ned 

Substructure Quarter 

Figure 2.17 -- Substructured 90° Model for Exhaust Duct 
75 

~de407 Node 419 

Standard 90° Model 

80 I soparametric Shell Elements 

22 I soparametric Solid Elements 

2402 Oeg rees of Freedom 

Node 1 

Node 13 

Figure 2.18 -- 90° Section of Exhaust Duct Without Substructuring 
76 

Using four properly oriented copies of the reduced 90 degree substructure, the 360 degree model was generated and solved. The final 

structure has only 312 nodal OOF with effectively a ful I bandwidth. 

(The condensed substructure stiffness Is fully populated.) Figure 2.19 

Illustrates the substructure hierarchy for the 360 degree model. Five 

'levels of substructures are present in the model. 

Table 2.1 compares the CPU and I/O for the various solutions. The 

CPU and I/O required for complete solution of the substructured 360 

degree model are assigned values of 1.0. Relative times are given for 

other models o Over 90% of the full model solution effort Is expended In 

the substructure reduction process. This is not too surprising given 

the very large OOF reduction and the very smal I number of OOF at the 

hlgnest level structure. It was not possible to analyze the unsubstructured 360 degree model with POLO-FINITE given the current I Imitation on 

the number of nodes In a structure (the system limits the number of 

nodes In a single substructure to 833). However, accurate estimates of 

the CPU and I/O tIme are possible using the exact tImings obtained for 

solution of the Isolated 90 degree structure without fin condensation. 

The estimated CPU and I/O consider the factor of 4 in element stiffness 

generation effort, the factor of 4 In number of DOF, and and the 

quadratic increase with bandwidth for the final 78 equations. The relative savings with the substructured model are In the 13-14 range for CPU 

and 18-20 for I/O transfers. Most of the saving results from the bandwidth reduction noted above, but a minimum factor of 4 Is real {zed from 

the reduction In nodal OOF and element stiffness generation~ 
Simple Finite 

. Elements 

• 

90° Section 

(Not Condensed) 

Condensed 

Fin 

Fin 

(Not Condensed) 

Si mple Fi nite Elements 

Figure 2.19 -- structural Model Hierarchy for Exhaust Duct 

: 1 

. I 

4 Copies of 

Condensed 90° 

Section 

Same Subtree 

Repeated U nde r 

Each Condensed 

90° Section 

-..J 

-..J 
78 

Model CPU I/O 

( I ) 90° Standard 0.69 0.63 

( 2) 90° Substructured 0.60 0.54 

(3 ) 90° Condensation 0.92 0.89 

(4) 360° Substructured I .0 1.00 

( 5) 360° Standard 13.6 * 18.92 * 

* Estimated using CPU and I/O measures for the 90° 

standard model 

Table 2.1 -- CPU Times for the Exhaust Duct Analysis 
79 

Figure 2.20 provides a detai led breakdown of CPU among the various 

POLO-FINITE subsystems for solution of the 360 degree substructur'ed 

model. Not surprisingly, the major effort is expended in the assembler 

and equation solver modules. It is interesting to note that a relatively smal I percentage of the total job time (14%) Is expended in data 

'management activities. The percentage of data management time In the 

assembler Is larger due to the very large number of smal I matrices that 

must be manipulated. In contrast, the equation solver accesses data in 

larger blocks (50x50) and performs a significant amount of numerical 

ccmputatlon on them. Consequently, percentage of data management time 

In the solver Is very smal I. 

2.,6.2 Non I I near Examp I e 

Figure 2.21 shows one quarter of a thick, shal low shel I structure 

ccmstructed of Impact resistant acrylic. A monotonically Increasing 

pressure Is appl led over a smal I region at the apex to simulate the Impact of a proJectile. Yielding of material In and around the Impact 

zone I s the non I I near behav lor" of Interest. Large geometry changes are 

Ignored In the present analysis. 

Figure 2.22 shows the detal Is of the element grids. The square 

panel rE~presents the element grid projection onto the global X-Y plane. 

The middle surface of the shel I lies on the surface of a sphere having a 

radius o~f 60 In. (152.4 cm). The rise along diagonal A-D Is 3.15 In. (8 

em). Ed!~es C-D and B-D lie on symmetry planes; edges A-C and A-B are 

completely fixed (no translation). A simple von Mlses yield criterIon 

with associated flow rule Is adopted for the nonlinear material 
80 

100 100 

I Fi nite Element 

90 Computations 

I;::;::::;:(!H Data Management - 80 ~ 

Q.) 

E 70 i= 62.9 

::l 

0.. 60 u 

"0 

Q.) 

50 N 

«:I 

E 

'- 40 36.3 0 

z 

30 

20 14.0 

10 0.6 9.3 0.015 

0.8 

0 

0.016 

..... c >. V1C CO'I <'a :::::So -..Q "00 OC: ..... c...- ~.- e_ 0_ Q 

et; E 0 ..... ..... > J-- .....JE C'a - -- Q.) 

V1 :::::sO .." Q.) O"V'> c:: .." 

E < c:: L.I.J 

Q.) 

l- e..:> 

Process 

Figure 2.20 -- CPU Time Distribution for Exhaust Duct Analysis, 

360 0 Substructured Model 
I _ 15.511 (Typical) 

Cia mped Edge ~ 

t Cia mped Edge --.J 

linear Region 1. 0" 

~ (Load Area) .~ ..::; 

Radius of Cu rvatu re = 60" ~~ 

~ 

~.f 

~metryLine 

Material - I mpact Resistant Acrylic 

E = 200 ksi 

v = 0.45 

Oy = 5.4 ksi 

Figure 2.2! -- Sha! low She!! Subjected to Project!! e Impact 

CD 
Boundary Adjacent to 

Condensed S ubst ructu re 

I Xg 

CiJ 

(Typ. ) 

loaded Element 

A I I I I I I·····:;ill' A Yg- L!t~nZy J . Top View - Nonlinear Zone 

Xg 

A IC f i 

fIg 

1/"'---')""'--'---1 l'---'Ir----1,-----j 

y. • 9 

Section A - A 

Yg • ~ iilf",I, o 

Top View .. linear Substructure 

Figure 2.22 -- Substructures for Nonl inear Impact Example 

Q 

(Typ. ) 

e 

(Typ. ) 

. Nonlinear Zone 

CD 

N 
83 

response. The uniaxial stress-strain curve Is Idealized as elastlcperfectly plastic for simplicity. 

Three types of 3-0 solid Isoparametrlc elements are used to model 

tho shel I. Each node of these elements has 3 translational degrees of 

freedom (u, v, w) • I n the I mpaC't zone, 20 node parabo I Ice I ements are em-

'ployed as shown In Fig. 2.22. Two elements are used In the thickness 

dlr-ectlon very near the loaded region. Away from the Impact point, one 

16 node thick shel I element Is adequate through the thickness. A ring 

of 18 node transition elements connect the 20 node and 16 node elements 

without loss of displacement compatibility. A 2x2x2 Gauss Integration 

order Is used In al I elements. 

Figure 2.22 defines the substructured model components. The outer 

I I near reg I on has 483 nodes (1449 OOF), 59 th I ck she II elements, and 8 

transition elements. Condensation reduces the number of nodes to the 43 

(1:29 OOF) that Interface with the non\ Inear region. The natural node 

numbering scheme places retained nodes last In [KJ, which eliminates 

reordering. The highest level structure consists of 56 nonlinear 

par-auol Ic elements In addition to the condensed linear substructure. 

The final nonlinear model has 376 nodes {1128 DOF}. The standard model 

for this structure has exactly the same element grid but without substructuring. This model has 818 nodes (2454 OOF) and 123 elements on 

which the nonlinear solution Is performed. The substructured model thus 

reduces the number of DOF by 54%. 
84 

A total of four load Increments were appl led to each model with the 

modified Newton-Raphson procedure to distribute residual forces. Iterations at constant external load were conducted until the Euclidean norm 

of the residual force vector fel I below 1% of the applied load vector 

norm. Tangent stiffness updates were performed before Iterations 2,5, 

'and 8 In each step. Table 2.2 provides the number of yielded Integration points, the number of stiffness updates, and the number of Iterations for each load increment. The fourth load increment propagated the 

plastic zone into the third element band from the apex. Sufficient CPU 

timing data was col Jected during the first four load steps for the 

desired comparisons. 

Figure 2.23 compares CPU time for the standard and substructured 

models. The normalized CPU time used through each load step Is expressea as a percentage of the total time required for the standard 

model analyzed through load step four. Solution tImes for load step one 

are nearly equal. The standard model requires slightly more time due to 

the larger number of elements for which strains and stresses are computed. The results for step one clearly demonstrate the negligible 

overhead for controlling the substructured solution. The major CPU 

savings with the substructured model begin to occur In load steps three 

and four. At the end of step four, the substructured model used 72% of 

the CPU required for the standard model. Both solutions appear to have 

reached a steady-state condition (constant slopes) at load step three. 

Linear extrapolation of these curves to one hundred load steps shows 

that solutIon of the substructured model would require 64% of the standard model solution time. The Improvement from 72% to 64% arises from 
Load Increment Number of Number of Number of 

Yi elded Po i nts Iterations [KTJ Updates 

2 

-

2 6 ~ 2 

3 II 4 

4 8 3 

Table 2.2 -- Computation Summary for Nonl inear Example (Identical Results 

for Substructured and Standard Models) 

OJ 

VI 
86 

lOO~--------------------------~------

90 

80 

-

~ 

-; 70 

E 

~ 

~ 60 

c.., 

u 

~ 50 

.J::::! 

-rt:J t:40 

o 

z 

30 

20 

• Standa rd Model (2454 dot) 

..... - S ubstructu red Model (1128 dot) 

1 2 3 4 

Load I ncrement Number 

Figure 2.23 -- CPU Time Comparison for the Example Nonl inear Analysis 

5 
87 

the decreasing significance of equal times In step one and equation 

solver savings In the substructured model. 

The bar graph of CPU distribution among processing modules shown In 

Fig. 2.24 Indicates the source of CPU savings. Input translation and 

strain-stress computation (which includes material updating) consume an 

. Insignificant percentage of the CPU time. The equation solver Is most 

dcmlnant fol lowed by the stiffness assembler (which Includes element 

computations) and the residual loads generator. Savings In the equation 

solver fc,r the substructured model are approximately 25% at the end of 

step four. This figure would increase to 40% with additional stiffness 

updates during subsequent load steps. The substructured nonlinear region with 1128 OOF requires 40% less CPU to triangulate than the ful I 

2454 OOF non I I near mode I. 

Little savings accrue In the assembler for the substructured model; 

however, this Is to be expected. After step one, there 15 very little 

difference In the operations performed by the stiffness assembler for 

eclch mode I • The same n umber of non \I near element 51' Iff nesses are 

gElnerated regard I ess of whether or not the mode I Is substructured. The 

smal I sewings result from the reduced number of element stlffnesses assE~mb I ed Un the substructured mode I. Moreover, th is I s an I/O rather 

than CPU Intensive activity. 

Figure 2.25 compares I/O activity for the two models. The I/O actIvity through each load sTep Is again normalized by The TOTal I/O 

througn ~oad step four of the standard model. The curves fol low the 

Si:lme trend as the CPU time In Fig. 2.23. The subsTructured model 

savings are 33% at the end of step four, which Is slightly better than 
100 ..-

90 I80 I-

-

~ 70 I-

-

<1) 

E 

~ 60 I-

::::l 

c.. 

u 50 I-

"'0 

<1) 

N .-- 40 ~ IE 

l.... 

0 

z 30 I20 I10 .... 

0 

88 

1...------11 Standard Model (I ncrease 

Over Substructured Modej) 

[=f!{=!J Substructured Model 

22.8 

17. 7 

·111111111111111 

62.4 

46.9 

10.9 

4.5 

Process 

...... c 

:::10 0..0_ 

c:-n:s --

100 

72.4 

Figure 2.24 -- CPU Time Distribution for the Example Nonl inear Analysis 
89 

laJ ~--------------------------.------~ 

90 

80 

~ 70 

.-~ 

~ 60 

u 

« 

.e 50 

"C 

Q) 

.!::::! 40 

co 

E. 

~ 30 z 

20 

10 • - Standard Model (2454 dot) 

... - Substructured Model (1128 dof) 

1 2 3 4 

Load I ncrement Number 

Figure 2.25 -- I/O Comparison for the Example Nonlinear Analysis 
-

~ -CJ.) 

E 

.-I-

=> 

c... 

u 

"'0 

OJ 

N 

ro 

E 

'-

0 

z 

90 

100 - [r:::::::::::::l Time in Data Management 

100.0 

90-

'''"__--J1 Fi nite Element Computations 80 -

70 -

64. 7 

60 r50 f40 r24.5 30 - 21. 2 

20 - 11.0 

10 - . . . ... .. . 5. 1 I 6. ~ I 33. 5. 6 r ,I. 3

1

. 00 

~.: ':; i.:: ~:: '.~:. ::; .. ~.~:: ;::.~.::~ ':~ ;::.~.j:.:~::; .. ;:; r J' 3.i : .. ~.i :':: .. ..:1.~ : .... ::.:I::.::.!::.:!.;.: ..; ..::!: ... :i ..... :I.~ :: .. ..... :!.::::: ..::~.:.:.: .... ::.!.~. OL-~~~~)~(~)~a~~l~~:::::~:~~~:::::L~_c:'~~---a-J~~ - V') ro-o ::I ('CJ 

-0 0 

';:;:; ......I 

CJ.) 

0.:: 

.. Process 

-c::: 

::10 

Q...-

c:::-ro --

Figure 2.26 -- CPU Time Distribution for the Substructured Nonl inear Example 
the CPU savings. 

sol.vlng file. 

Most of the I/O savings occurred In the equation 

Figure 2.26 provides a breakdown of the CPU time distribution among 

POL.o-FINITE subsystems for the substructured solution. A similar comparlslon for the linear example is given In FIg. 2.20. These two 

'figures Illustrate the Increased percentage of job CPU tIme expended In 

data mana!~ement activities for the nonlinear analysis. While It is not 

strictly valid to compare a linear analysis of one structure with the 

nonlinear analysis of another structure, the same trend of Increased 

da"l"a management time has been observed I n other ana I yses performed with 

POLO-FINITE. For most linear analyses, data management activities 

represent 10-25% of the total CPU time. In contrast, data management 

activities consistently require a larger percentage of the CPU time for 

nonlinear analyses. The Increase Is attributed to the much larger data 

base sizes (generally a factor of 10), the larger number of data structuv'es that must be accessed to obtai n non I I near data, and the I arge 111-

crlease In solution logic that requires movement and duplication of data 

without computation. 
91 

2.7 References 

2.1 Dodds, R.H., and Lopez, L.A., "General ized Software for Nonlinear Analysis," International Journal i,Qr Advances ill 

Engineering Software, Vol. 2, No.4, 1981, pp. 161-168. 

2.2 Dodds, R.H., and Lopez, L.A., "Substructurlng In Linear and 

Nonlinear Analysis," International Journal for Numerical 

Methods ill Engineering, Vol. 15, 1980, pp. 583-597. 

2.3 Egeland, 0., and Araldsen, P., "SESAM-69: A General Purpose 

Finite ELement Program." Computers.an..d. Structures, Vol. 4, 

No.1, 1974, pp. 41-68. 

2.4 Fuchs, G. von, and Schrem, E., "Hypermatrlx Solution of Large 

Sets of Symmetr I c, Pos I t I ve-Def in I te LI near Eq uat Ions." .C0mputer Methods In Appl led MechanIcs ~ EngIneerIng, No.1, 

2.5 

1972, pp. 197-216. . 

Fur u I ke, T. , "Comp uter I zed 

Analysis," Computers.aru1 

695-712. 

Multiple Level Substructured 

Structures. Vol.2. 1972. pp. 

2.6 Irons, B.M., "A Frontal Solution Program for FInite Element 

Analysis," International Journal fQc Numerical Methods In 

Engineering, Vol. 2, No.1, 1970, pp. 5-32. 

2.7 Iwakl, T., Maeda, A., and Ishii. T., "MISA: A General Purpose 

FEM Program," Computecs .9.D..d. Structures, Vol. 10,1979, pp. 

311-322. 

2.8 Jennings, A., and Tuff, A.D., "A Direct Method for the SolutIon of Large Sparse SymmetrIc Simultaneous Equations," In 

Large Sparse ~ ~ LInear EQuations, Ed. J.K. ReId 

(Academic Press, N.Y.), 1971. 

2.9 Lopez, L.A., "FINITE: An Approach to Structural 

Systems," International Journal .f.Q.c Numerical 

Engineering, Vol. 11, No.5, 1977, pp. 851-866. 

Mechanics 

Methods In 

2.10 McKellar, A.C., and Coffman. E.G., "OrganIzing Matrices and 

Matrix Operations for Paged Memory Systems," CommunicatIons 

Qf the ACM, Vol. 12, No.3, March, 1969, pp. 153-165. 
92 

2.11 MacNeal, R. and McCormick, C.W., "The NASTRAN Computer 

Program for Struci"ural Analysis," Computers ill Structures, 

VOl. 1, No.3, 1971, pp. 389-412. 

2.12 Meyer, C., "Solution of linear Equatlons--State of the Art," 

Journal Qf ~ ~uctural Dlylslon, ~, Vol. 99,No. S17, 

1973, pp. 1507-1526~ 

2.13 Mondkar, D.P., and Powell, G.H., "large Capacity Equation 

Solver For Structural AnalysiS," Computers .a.n,g, Structures, 

Vol. 4, 1974, pp. 699-728. 

2.14 Peterson, H. and Popov, E.P., "Substructurlng and Equation 

System Solutions In Finite Element AnalYSiS," Canputers .aruI. 

Structures, Vol. 7, 1977, pp. 197-206. 

2.15 Przemlenleckl, J.S., "Matrix Structural AnalysiS of Substruc:-

tures," AlM JournaL, 1963, pp. 138··147. 

2~16 Schrem, E., "From Program Systems to Programming Systems for 

FInite Element AnalYSis," PrgceedIngs, U.S.-German Symposium 

on Formuiatlonal and Computational Methods In FinIte Element 

Analysts, 1976, MIT Press. 

2.17 Taig, I.C., "Automated Stress Analysis Using Substructures," 

ProceedIngs, 1st WrIght-Patterson Conference, AFFDl TR, 1966, 

pp. 66-80. 

2.18 Williams, F.W., "Comparison of Sparse Matrix and Substructur'e 

Methods," Internatl..Q.1l5ll Journal .t.o.r::. NumerIcal Methods loll 

EngIneering, Vol. 5 p 1973, pp. 383-394. 

2.19 Wilson, E.l., "The static Condensation Algorithm," J..n±ar:-

natlgnal Joyrnal LQC Numerical Methods In EngineerIng, Vol. 8, No.1, 197411 pp. 198-203. 
CHAPTER 3 

DYNAMIC REDUCTION OF STIFFNESS AND MASS MATRICES 

3.1 General 

As demonstrated In Chapter 2, multi Jevel substructurlng provides an 

economical approach to static analysts of very large linear and nonI loear structural models. Size reduction of the substructures, through 

nodal condensation, yields an exact and economical solution to the 

statically loaded problem. Since dynamic analysis of a fInite element 

model requires significantly more computational effort than a statIc 

analysis of the same model, an analogous reduction scheme would be useful In dynamics. 

As an Illustration of the usefulness of substructured models In 

dynamics, consIder the following example. Assume that a partfcular 

structural model contains 5000 OOF and has a half-bandwidth of 500. 

Computation of the 50 lowest natural frequencIes and corresponding mode 

shapes by a method suitable to the problem characteristics reqUires 

roughly 706(10 9) operations. Now suppose that the model can be divided 

Into fIve Identical substructures, each containIng 1200 DOF. Reduction 

of a substructure to 100 Independent DOF whIle retainIng the 10 lowest 

natural frequencies and mode shapes requires roughly 7.5(108) operatlonso Since al I five of the substructures are Identical, the reductIon 

must be performed only once. Assembly of the substructures into final 

form results in the reduced model containing only 300 Independent DOFo 

Since the equations for the substructured model are fully populated, a 

different procedure may be appropriate for computation of natural fre= 

93 
94 

quelncl es. Computation of the 50 lowest frequenci es and mode shapes 

would require only 2.3(107) operations. Thus, a savings In required 

operations of a ful I order of magnitude can be realized by using substructuring on this hypothetical model. These operation counts are, of 

course, highly dependent upon the algorithms used and the model being 

-analysed, but the computational savings Is wei I Illustrated. Additional 

savings can be gained In the solution of the equations of motion and In 

thEl recovElry of substructure displacements, strains, and stresses. 

The floal of dynamic reduc1·lon methods Is to generate stiffness and 

mass matrices that accurately represent the stiffness and Inertia 

chclracterlstlcs of the substructure with the minimum number of 001-. As 

previously stated, reduction in static analysis Is exact and can be 

mathematically viewed as an equation solving technique. In dynamic 

ancllysls, however, exact dynamic reduction of an individual substructure 

Is dependElnt upon the unknown frequencies of the total structural 

system. Since these system frequencies are actually objectives of the 

ancllysls, the analyst must use reduction methods which are either iterative or frequency independent (and therefore approximate). 

Two c I asses of methods for" dynam I c reduct i on have evo I ved for use 

with the FEM. The first class, known as Guyan reduction [3.6J, Is an 

extension of static condensation. It Is currently the technique most 

widely used to reduce the number of oOF prior to frequency or transient 

an<~ I ys I s of standard (nonsubstructured) fin I te element mode Is. The 

method Involves elimination of DOF that are assumed to have a negligible 

effect on mode shapes and thus vibration response of the structure. 

Dynamic results, especially strains and stresses, are generally quite 
95 

sensitive to the choice of OOF to be el imlnated. Although the method 

has proven useful for smaller models, its extension to multilevel substructuring Is expected to be only marginally successful. 

The second class of dynamic reduction techniques, termed modal synthesis, contains methods that rely on a Rayleigh-Ritz transformation of 

·each substructure's geometric coordinates to a smal fer set of 

generalized coordinates. This transformation is usually based on the 

natural frequencies and mode shapes of the Isolated substructures. The 

reduced stiffness and mass matrices for the substructure are defined In 

terms of the general lzed coordlnates~ Assembly of the stiffness and 

mass matrices for the next higher level structure Is based on displacement compatibll ity. The reduction process can then be successively 

repeated for each additional level of substructurlng. 

In the category of modal synthesis have been devl·sed In 

Many techniques 

an attempt to 

select the best combination of substructure modes and displacement compatibility conditions. 

This chapter provides detal led descriptions of the initial formulations of Guyan reduction and modal synthesis. Derivations of the 

governing equations of the methods are fol lowed by a brief review of 

their respective extensions and modifications. Although many more of 

these alternate techniques have been proposed In the open literature 

than are presented here, those discussed effectively encompass the 

breadth of the topic. The chapter concludes with an evaluatfon and comparison of the dynamic reduction techniques and recommendations for implementation In a general purpose software system. 
96 

3.2 Guyan Reduction 

3.2.1 8as.IJ;. Formulation 

Consider an Isolated substructure consisting of simple elements, 

such as the FIN substructure used in the exhaust duct example problem of 

-Chapter 2 (see Figure 3.1). Lei' the Internal boundary of the substructure Identify Its Interface with other substructures while an external 

boundary corresponds to the physical boundary of the entire structure. 

The undamped, free vibration equation of the substructure, partitioned 

to seperate master (m) and slave (s) OOF, Is 

(3.1> 

Master DOF are those that will remain after condensation and are usually 

chosen to I Ie on the Internal boundary of the substructure. They are 

used for connectivity to adjacent substructures. The slave OOF are 

those to be eliminated and usually lie in the Interior of the substructure or o,n its externa I boundary. The natura I frequency w. I s that of 

I 

the complete structural system, not Just the Isolated substructure. The 

presence of nonzero o,ff-diagonal blocks [MsmJ and [MmsJ In Eq. (3.1> 1mpi les the use of a consistent mass formulation. When a lumped mass 

model Is used, the mass matrix Is diagonal. 

The lower half of Eq. (3.1) can be expanded to 

(3.2 ) 

Scdvlng for {us} In terms of {urn} yields a coordinate transformation 
I nterior of 

Substructure-~ 

(Contains Slave OOF) 

97 

--

........... ---.... ...... -....... --..... ........... -..... ...... ..... , -.... ... 

-~-~- I nternaJ Boundary (Contai ns Master DOfl- __ _ 

--- External Boundary (Contai ns Slave DOF)~~,"-~, 

Figure 301 =- Internal and External Boundaries for Substructure FIN 

from Chapter 2 Exhaust Duct Example 
98 

which Is clependent on the unknown system vibra·tlon frequency wi' If It 

Is assumed that the Inertia forces on the slave DOF are smal I compared 

to the static forces, the former can be neglected. Thus, the frequency 

dependence Is eliminated and Eq. (3.2) simplifies to 

(3.3) 

Defining the coordinate transformation [TG

] from {um} to {us} as 

(3.4) 

{us} can be eliminated from Eq. (3.3) resulting In 

[K SS] [T J = _[Ks~. (3.5) 

As dlsicussed In Chapter 2, [Tc}] Is evaluated by standard equation 

solving techniques. It Is impor-tant to recall -that the columns of the 

transformation matrix are composed of the previously defined static constrcllnt modes. The complete substructure displacement vector can be reexpressed, using Eq. (3.4), as 
99 

The substructure strain and kinetic energies are given by 

(3.7a) 

K.E. (3.7b) 

where {u} Is the first time derivative of {u}. These expressions are 

rewritten In terms of the master DOF by substItuting Eq. (3.6) into 

Eqo (307). The resultIng Guyan reduced stiffness [KGJ and mass [MG] are 

given by 

[KG] = [~mJ + [~sJ [TGJ and (3.8) 

[~J = D.fnm] + [TGJT [Ms~ [T ~ + [T GJT [MsrTj + [Mm~[T &1. (3.9) 

For the simpler case of a lumped mass model, Eq. (3.9) reduces to 

In Guyan reduction, the Inertia effects of the slave DOF are not losto 

Instead, the contribution to the mass of these eliminated DOF is 

distributed to the master OOF. It Is assumed that the dynamic response 

of the slave DOF of the substructure is adequately approxImated from 

that of the master DOF by linear combinations of the statIc constraInt 

modes. Note that regardless of whIch mass matrix formulatIon Is used, 

consistent or lumped, the reduced mass matrix, [MG], wll I be fully 

populated. This situation must be considered when choosing algorithms 
100 

for solving the free vibration and transient response problems for the 

assembled structure. 

When loads are appl led to the slave OOF, they too must be condensed. If the substructure Is subjected to an arbitrary virtual 

displacement, {au}, the work done by the substructure forces {p} is 

aW = {oul {Pl. (3.11> 

Th'9 statically equivalent condensed forces, {F}, applied at the master 

OOF must do the same work during a virtual displacement consistent with 

{au}, so 

(3.12) 

Recalling Eq. (3.6), the condensed force vector becomes 

Each substructure has Its stiffness, mass, and loads similarly partitioned and reduced. Assembly of both the reduced substructure mass 

and stiffness Into the next higher level fol lows the procedures outlined 

in Chapter 2. Geometric compatibility between substructures is 

au"romatl ca II y assured by the use of the master OOF as genera I fzed 

coordinates. 

The extension of Guyan reduction to multilevel substructurlng Is 

straightforward. Referring to i"he terminology of Chapter 2, assume that 

all substructures at level "I" have been assembled either from simple 

elements or level "1+1" subsi"ructures. The level "1-1" substructures 
101 

are built up by selecting slave and master DOF for each substructure at 

level "I", condensing these substructures using Eq. (3.8) and Eq. (3.9), 

and assembl ing as described above. When the highest level structure Is 

reached, the fInal condensed stiffness and mass matrices can be used to 

form the equatIons of motion for the entire structure. 

After a free vibratIon problem has been solved for the highest 

level structure, it may be necessary to recover the portion of the 

system mode shapes contained withIn lower level condensed substructures. 

This Is achieved by sImply applying Eq. (3$4) recursively to each substructure to recover the components of the vibration modes at the slave 

DOF from the master DOro Recovery of the dIsplacement patterns after a 

transient analysis is not as elementary. A procedure analogous to the 

computation of partIal slave dIsplacements discussed in Chapter 2 must 

be adopted. During the condensation process, loads applied at the slave 

DOF are transformed Into work-equivalent forces at the master DOF. 

Thus, applicatIon of Eq. (3.4) to displacement patterns at the master 

DOF does not yield the total response of the slave DOF. A separate 

transient analysis of the substructure with Its master DOF fixed is requlrede The displacements of the slave DOF from this analYSis must then 

be superposed with those from Eqo (3.4) to obtain the total response of 

the substructure. Of course, thIs procedure is unnecessary when the In~ 

ternal response of condensed substructures Is of no Interest In the 

analysis of the structureo 
102 

3 .2.2 &ct.Qmat I c Se I ect Ion Qf ~mc .QQf. 

To Insure complete geometr'Ic compatIbilIty, all OOF at the Internal 

boundarIes of a substructure must be Included In the set of master DOF. 

However, i·he set need not be lImIted to the OOF on the Internal boundary 

of the substructure. Other substructure OOF are possIble candIdates for 

rei·ent Ion. 

Henshel I and Ong [3.7J have suggested a sImple method for 

automatln~1 the process of selec"rlng master DOF. The method Is based on 

an assumptIon fundamental to the development of Guyan reduction; that 

thEI mass terms correspondIng tC) the slave OOF have a negligIble effect 

on the mode shapes. ThIs can be rephrased by sayIng that k 1m 55 55 Is 

lar·ge relative to k 1m for any paIr of slave and master OaF. Thus 

mm mm 

thEI obv lOlls cand I dates for master OOF, I n add f ·1-1 on to those on the substructure I nterna I boundary, ar'e those wIth the sma II est ratios k . ./m .. ; II I I 

1=1,2, ••• , n; n=number of substructure I nterna I OOF. 

The method presented In Ref. [3.7J was oriented towards standard 

finite element models. Its extension to multilevel substructurlng requires only the addItion of the Internal boundary OOF tn the set of 

master DOFo 

The clbove dec I s I on cr I ter I on f s eq u I va I ent to preserv I ng the lowest 

vibratIon modes of the substructure. ,The ra"I-lo k . .lm .. Is Interpreted I I I I 

as the square of the vIbratIon frequency assocIated with the "I" th DOF 

while all other substruC"l"ure OOF are held fixed. Retention of the OOF 

with the Sima I I est va I ues of k . .lm.. has shown to produce better accuracy I I I I 

In the lower modes than random selectIon of master DOF. 
103 

3.2.3 Improved Displacement Recoyery 

The frequency dependent transformation from {urn} to {us}, when cast 

In a modified form, Is useful for improving recovery of substructure 

mode shapes and displacements. The technique was Initially presented by 

Kidder [3.15J and was later reintroduced with the addition of some 

numerical results by Miller [3.20J. The technique Is appl led after the 

structure equations have been reduced by standard Guyan reduction~ The 

frequency dependent transformation matrix resulting directly from 

Eq. (3.2) Is 

Expansion of the inverse term gives 

••• (3.15) 

Ignoring as small the terms containing w to powers greater than two, the 

transformation becomes 

(3.16) 

As with [T~, [TwJ can be evaluated by equation solving rather than by 

computing [KssJ-1 • 

.. The transformat f on [T wJ rep I aces [T GJ I n recover r ng the substructure mode shapes and displacements during a transIent analysis of the 

uncoupled linear equations of motron. Upon solution of the modal equa~ 

tions at the hIghest level, mode shape recovery Is achieved via 
i04 

(3.1"7) 

This transformation Is performed for each modal frequency considered so 

as to yield Individual mode shapes within the substructure. Displacement recovery follows the procedure outlined In Sec. 3.2.1 with the 

total displacement vector equal to the sum of Its modal components. 

The Improved displacement recovery technique has no effect on the 

computed system frequencies and highest level structure mode shapes. 

Thl9 argument for Its use Is that frequencies determined from substru(:-

tUlr-e mass and stiffness matrices computed with [TG] are generally 

realistic. However, Improvements are needed In the substructure mode 

shapes fn:xn which strains and stresses are derived. 

Miller's numerical results demonstrated greatly Improved mode shape 

vectors over standard Guyan reduction for frame structures. However, 

the effectiveness of the method Is limited to the lower frequencies of 

vibration. This Is because the size of the truncated terms of the 

series In Eq. (3.15), and thus the error In [Tw]' grows with increasing 

val ues of w. 

In olrder to use Improved d I sp I acement recovery, the eq uat Ions of 

motion of the highest level structure must be solved In their uncoupled 

form so that modal vibration frequencies can be computed. When a transll~~nt analysis of coupled equations Is performed, as In nonlinear 

an(~lysls, the displacements are computed dIrectly and thus, thIs method 

Is not applicable. 
105 

3.2.4 Evaluation .Qf. Guyan Reduction TechniQues 

Guyan reduction techniques have yet to be applIed to multilevel 

substructured models. Therefore, their performance In such an appl Ication remains unknown. Nevertheless, Guyan reduction has some Important 

advantages that could make It an attractive approach to dynamic reduction In partIcular cases. FIrst, its development is taken directly from 

the static condensation approach. This al lows the reduction scheme to 

be readily added to exIsting software which Is currently capable of 

handl fng multIlevel substructured models for statIc analysIs. Secondly, 

reasonably good numerical results have been achieved with Guyan reductIon In computing system frequencies for smal I models. This feature 

makes the technique attractive for prellmtnary vibration analysis. 

Lastly, Guyan reductIon Is the least expensive of al I dynamic reduction 

techniques. 

Some drawbacks of Guyan reduction are evident and must be considered. The success of the method Is highly dependent upon the choice 

of master OaF. This Is further compl leated by the need to Include internal boundary DOF In the set of master OOF for substructured models. 

The result Is likely to be a decrease in the degree of reductIon capable 

for large, multilevel substructured models" Another problem Is the 

quality of the mode shapes. Accurate prediction of strains and stresses 

requires that dIsplacement vectors and mode shapes be well formede The 

ability to achieve this goal with Guyan reduction Is stl! I In doubt. 

However, there are circumstances In whIch It Is not necessary to recover 

straIns and stresses withIn a condensed substructure. In such cases, 

I.e. when the response of only the highest level structure Is of major 
106 

concern, Guyan reduction may perform quite wei I. 

Guyan reduction techniques provide an important first step In the 

development of more sophisticated dynamic reduction methods. As wll I be 

sec:ln I at1er, the procedures der I ved above actua II y represent a 

de!generat1ed case of modal synthesis. Guyan reduction does not appear 

'adequate for general application to dynamic analysis of multilevel substructured models. However, with automatic selection of master OOF and 

improved displacement recovery, it does hold potential as an economical 

approach to dynamic reduction of certain models. 
107 

3.3 MQQsl Synthesis 

3.3.1 Introduction 

Modal synthesis was developed expressly for use with substructured 

models. Although this dIscussion Is limIted to reduction of finIte ele-

.ment models, the techniques have also been appl led to dIstributed 

systems. All modal synthesis transformations are based on Rayleigh-Ritz 

arguments. The procedure Involves the derivation of a transformation 

matrix composed of a truncated set of mode shape vectors that adequately 

describe the dynamIc characteristics of the substructure. This set is 

fewer In number than the number of Independent OOF contained In the substructure. With the transformation matrix, Individual substructure mass 

and stiffness matrices are converted from geometriC coordinates Into a 

reduced set of genera I lzed coordt nates. The general lzed mass and stl ffness matrices for each substructure are then synthesized while maintaIning geometric compatlbil tty along Internal boundaries to form 

similar matrices for the next higher level structure. In a multIlevel 

substructured environment the transformation and assembly processes are 

performed recursively at each level. 

There are two basic operatIons that must be performed with any ap= 

plIcatIon of modal synthesis. First, an approach must be chosen for 

selecting the set of substructure mode shapes from which the· reduced 

substructure matrices are computed. Second, a procedure Is needed to 

enforce geometric compatlbtl Ity along substructure Internal boundaries. 

The numerous modal synthesis technIques proposed In the literature vary 

tn how these concepts are Implemented. 
108 

The ~nitlal formulation of modal synthesis, accredited to Hurty 

[3 .. 11, 3. 12J, has been extens I ve I y mod I fled and enhanced. The fo II ow I rig 

section pr'esents Hurty's formulation, commonly referred to as the flxedIn1"erface method, while the later sections describe suggested improvements or alternatives to the method. 

3.2).2 ~u1:-1 nterface Method 

The origin of modal synthesis techniques lies In Hurty's flxedInterface method. However, details of the procedure presented here 

par'allel the development by Craig and Bampton [3.3J. Hurty's development requfred a distinction between "statIcally determinate" and "redundant" constra I nts. Statl ca I I y determ I nate and redundant constra I nts ar'e 

ar1'Ificiai constraints applied at all master OOF. They are Imposed Independently of any actual physical constraints that may already exist on 

tho substructure at slave OOF. The set of statically determinate constr'alnts serve to restrain any rigid-body motion that may be possible. 

Redundant constraints are those applied to the remaining master OOF. 

FI~~ure 3.2 Illustrates the application of these constraints to a simple 

two-d Imens I ona I p I ate. Three statl cal I y determ I nate constra r nts are 

neE~ded to restral n trans I atl on and rotation. The fIve remal n I ng master 

oor have r'edundant constraints applied. Craig and Bampton treat the two 

sets of constraints simply as boundary constraints. 

Consllder again the undamped, free vIbration equation for an 

Isolated substructure composed of sImple elements and partitioned i'o 

separate master and slave OOF 
109 

~.-~ .. -.~-... -. ~"'-----o-----<J----~J.---......,....-~ 

• 

o 

Master Node (2 OaF/NODE) 

Slave Node (2 DOF/NODE) 

Statically Determinate Constrai nt 

Redundant Constraint 

Figure 3.2 -- Statically Determinate and Redundant Constraints on a 

Simple Two Dimensional Substructure 
110 

(3.18) 

As In Guyan reduction, a static transformatIon from the master to the 

slave OOF can be wrItten 

(3.19) 

As wIth sImple Guyan reductIon, the set of master OOF In modal synthesis 

_._ .. _.-.methods Is not lImIted to those OOF on Internal boundaries. OOF··ln-·the.--

In·~erlor of the substructure may be retained as well and may possibly 

Improve the solution. Inspection of Eq. (3.2-3.4) reveals that [epC] Is 

Identical to [TJ, the static constraint modes. For consistency In the 

fol lowing development, [epc] Is used to represent the static constraint 

modes. 

If the set of master OOF Is restrained from displacement, 

Eq. (3.18) reduces to 

(3.20) 

Thl9 solution of this eigenvalue problem yields the matrix of fixed-fixed 

nOI"mal modes of vibration, [<j>n], having the same order as [KSS] and 

The computed vIbration frequencies, 

I S'D I ated substructure. 

w. , I 

are those of the 

To reduce the substructure mass and stiffness matrices, a transfor-
I II 

mation to general ized coordinates, {q}, is defined as 

{u} = (3.21 ) 

The fixed-interface transformation, [T~, is derived from the static 

constraint and the normal modes as 

(3.22 ) 

[ -n nJ in whIch ¢ ] Is a rectangular matrix of retained modes from [¢ • In 

general, the modes corresponding to the lowest natural frequencies, wi' 

are retaIned in [¢nJ. The slave displacements, {us}, are now dependent 

on both the static constraint modes and the retaIned normal modes of the 

Isolated substructure. 

Two observatIons regarding Eq. (3.21) are noteworthy. First, the 

generalized coordinate sUbvector, {qm}, corresponds precisely to the 

master set of geometric coordinates, {um}. This proves useful In 

guaranteeing geometric compatibility with adjacent substructures. 

Secondly, as the number of mode shapes in [¢n] is reduced, the transformatlon shrinks to just the static constraint modes and thus, the fixedInterface method degenerates to Guyan reduction. 

The reduced stiffness and mass matrices In generalIzed coordinates 

are obtained by maintaining equivalence of kinetic and potential ener-
112 

gies between the two coordinate systems. The resulting forms are 

[K - - T [KG: oj F] - [T ] [K][T ] = ---~---- - F F I -2 ° I w I n 

, and (3.23) 

<3.24) 

FClr a I umped mass formu I atl on of the substructure 

<3.25 ) 

[KG] and [MG

] are the Guyan reduced stiffness and mass matrices. The 

-2 Identity submatrlx In [M F

] and the submatrTx [wn

] In [KF

] result from 

the orthonormallty of [~n]. [w2

] Is a diagonal matrix of natural fran 

quencles corresponding to the modes retained In [~n]. 

Reduction of substructure loads, {P}, follows the same virtual work 

argument used In Guyan reduction. The resulting general ized force vector, {F}, Is 

<3.26 ) 
113 

Before proceeding, a simp' Ifled notation Is Introduced. Let 

and 

mm I -ml 

= ~---!-~-- , M

nm

: 1 

with the relationships to Eq. (3.23) and Eq. (3.24) established by In-

'specti on. 

Although assembly of the reduced substructure stiffness and mass 

matrices Is routine, an Illustration of their form 'Is useful. For an 

assembly of "r" substructures with the master OOF entered last 

-nn 0 0 0 -nm K, ... 0 0 M, 

0 -nn 0 -nm 

K2 0 0 M2 

* * (3.28a) 

[K] = [M] = 

0 -nn 0 0 0 iiiinm (3.28b) K 

r r 

o· 0 *mm -mn 

rvr:::

n Mmn *mm ... K M, M 2 r 

Since the master DOF do not participate In the normal modes, no coupling 

*mm *mm between substructures exists outside the submatrlces [K ] and [M ], 

which are the assembled Guyan stiffness and mass. From a data storage 

and computational vIewpoint, these forms for [K] * * and [M] are Ideally 

suited to hypermatrlx methodso 

The synthesis process for one level of substructurlng Is now complete. If [K] * * and [M] are the stiffness and mass matrices for the 

highest level structure, the differential equation of motion can be 

written and solution for displacements can proceed. If the highest 

level structure has not yet been reached, [K] * * and [M] are reduced just 

like any other substructure. 
114 

In summary, the fixed-Interface method employs static constraint 

modes and a truncated set of fixed-fixed normal modes to achieve a 

reduction In the number of Independent substructure OOF. Geometric 

coordl natE~s at Internal boundaries are retained In the set of 

generallzE~d coordinates to assure displacement compatibility between 

'substructures during assembly. This mixture of geometric and 

generallzE~d coordinates In the substructure equations has resulted In no 

reports of numerical difficulties and none are expected. 

3 ~25.3 .E.cm~ I nterface Method .\ti:tb. I nterfac~ Load I ng 

In the free-Interface method of modal synthesis [3.5, 3.10J, the 

transformation to a reduced set of generalized coordinates rei les on a 

truncated set of free-free normal modes. These mode shapes are computed 

fOI" the Isolated substructure with geometric constraints appl led only as 

thc:lY OCCUI- I n the actua I structure. I nterna I boundary (master) OOF ar"e 

no"l" arbl"trarlly Identified and fixed. This approach allows rigid-body 

modes to appear In the set of substructure normal modes unless suff Ie lent phys I ca I constra I nts actua I I Y ex 1st. The transformation 1'0 

general rZE~d coordInates also neglects the statIc constraint modes common 

to both Guyan reduction and the fixed-Interface method. The transformatlon is simply 

-n {u} :: [cp J{q} (3.29) 

whcere [(fin] contal ns the truncated set of free-free normal modes cor"-

respondln!~ to the lowest substructure natural frequencies. 
115 

The development by Goldman [3.5J includes a distinction between 

rigid-body modes and free-free elastic modes which Is not required In 

Hou's method [3.10J. These two authors use similar approachs to substructure assembly. Geometric compatibility between substructures Is 

enforced by writing equations of constraint In modal coordinates for internal boundary DOF. These constraint equations are used to combine the 

generalized displacement vectors for al I Isolated substructures Into a 

s1ngle general ized displacement vector for the final structure. The 

corresponding reduced stiffness and mass matrices are thus generated. 

A significant difficulty arises with the extension of the freeInterface method to multilevel substructuring. In achieving one level 

of substructuring, at t geometric coordinates are transformed to 

generalized coordInates. Further substructuring is complicated by the 

absence of geometric coordinates which are useful In assuring continuity 

of displacements at substructure Internal boundaries. Beyond the lowest 

level, It wll I be necessary to modify the process of developIng constraint equatrons to I Ink together the substructurese Rather than 

defining the constraints In terms of the general ized coordinates at the 

level being assembled, they must be written In terms of the geometriC 

coordinates at the lowest level. Such a task has yet to be Investigated 

and the requirements for Its Implementation remain unknown. 

The mode shapes of a substructure with free (or fixed) boundaries 

are not totally representative of the substructure's response in the assembled structure. This follows because the stiffness and Inertia effects of the adjoining substructures have not been included In computing 

the dynamic modes of the Isolated substructure. Interface loading [3.2, 
i 16 

3.9J Is a technique which Incorporates some of these effects In an attempt to make the Isolated substructure modes more I ike the modes for 

thc3 entire structural system when the free-Interface method is used. 

Thc3 approach I s to mod I fy the st I ffness and mass of each substructul'e 

prior to extracting Its free-free normal modes. 

Let the OOF of the substructure under consideration reside In set A 

and let a I I rema I n I ng OOF I n the structure res I de inset B. I n genera I, 

set B may contain more than one substructure. Upon assembly, set A wll I 

I n'j-ersect set B over the subset AS. Cons I der subset AB as the master 

OOF for bc:>th sets A and B. Figure 3.3 III ustrates th I s approach as applied to the shallow shell example of Chapter 2. The substructure undelr 

consideration Is the linear zone and It forms set A. The remainder of 

thle structure, the non I I near zone, fa I I sin set B. Subset AB I s the I nter-face between the two sets. 

By 'the same transformation as used In Guyan reduction, the 

dl!;placements of OOF In set B can be written In terms of those In subset 

AB. Recognizing that subset AI3 also defines the master OOF of set A, 

thle d I sp I acements of OOF I n set B are effect I ve I y expressed I n terms of 

th,e master OOF of set A. From these rei atl onsh ips, the I nterface loaded 

stiffness and mass for set A are derived. The effect of the Interface 

loading Is to add the Guyan reduced stiffness and mass of set B to the 

In'~ernal boundary terms (master OOF) of A's stiffness and mass. 

The free-free normal modes of the substructure In set A are compU'~ed using the modified stiffness and mass matrices. Each substructure 

that wil I be reduced Is identified as set A with the remaining substructur-es I umped I nto set B. A new interface load I ng effect I s computed and 
117 

SubsetAB--

(I nterface of A and B) 

'--"---5 ef -ff ..... -.-------------

(Nonlinear Zone) 

Figure 3.3 -- OOF Sets for Interface Loading of the Linear Substructure 

from the Chapter 2 Shel I Example 
i 18 

the free-free analysis performed. Actual geometric constraints are applied as; they occur In either set. After solution for the normal modes 

of' each substructure Is comp I ete, the substructures are a5semb I ed us I ng 

the truncated mode sets and the original, unmodified substructure 5tlffness and mass matrices. The Interface loaded stiffness and mass are not 

'used In the assembly process. Their use Is limited to computing freefree normal modes for the substructures. 

3.3.4 W~ ~ Analysis 

Branch mode analysis [3.4, 3.9J Is a hybrid method of modal synthesis which Incorporates features of both the flxed-Inferface-anu-i"he 

free-lntE~rface methods. The following procedure Is one of several that 

fal I under the class of branch mode analysis [3.13J. 

In branch mode analysis, one substructure Is selected as the main 

bCldy and all adJaclent substructures are designated as branches. The 

-analysis starts with a determination of the fl~ee-free componen--t-modes of 

the main body. Interface loading may be employed prior to computing the 

fr'ee-free modes. The topology ()f the main body is then expanded to Include all substructures on Its boundaries. This Is done byw-Fif-Ing-the 

. displacements for the maIn body and Its adjacent branches In terms of 

the free-free modes of the main body and the fixed-fixed normal modes 

fr'om each branch. Using the transformation to modal coordinates In the 

structure kinetic and potentIal energy equations, the reduced branch 

stiffness and mass matrices are computed. 
i 19 

When substructures are remote from the Initially chosen main body, 

a number of solution schemes may be used. One approach is to Identify 

one main body and Its branches and perform the foregoing reduction. 

This reduced substructure group Is then treated as a branch to a newly 

selected main body. Two more elgenproblems are solved and the process 

Is repeated until each substructure has been Joined. 

3.3.5 Dynamic Stiffness Matrix 

An Iterative method for obtaining an exact solution to the frequency dependent dynamic reduction problem has been developed by Leung 

[3.17J. The method Is closely related to the fixed-interface method but 

Includes the frequency dependence In the transformation to the reduced 

set of coordinates. The first step of the method Is to partition the 

stiffness and mass of each substructure and compute the Guyan reduced 

matrices [KG] and [MG]. With the Internal boundary DOF IdentifIed as 

master DOF, the fixed-fixed normal modes, {¢n}, and associated frequencies, w~, are computed for each substructure. The exact substructure 

I 

dynamic stIffness matrix, [DJ, can then be derived as 

where n

2 = an unknown system frequency 

[G] = [Mms][¢nJ - [Kms][¢nJ[w~J-1 I 

[w-2o] _-

I 

a diagonal matrix of substructure frequencIes 

n2 

[A] = a diagonal matrIx of elements -2 2 

w - n 

(3.30) 

With no effect on the order of [D), all normal modes In [¢nJ may be used 

or the set may be truncated to Include only those associated with the 
120 

lowest fn9quencles. 

Using a theorem stating that the dynamic mass matrix, [M(~)J, 

equals the -2 partial derivative of [0] with respect to w , the substructUlne reduced mass Is gl ven by 

[M(~)] = [MJ + [G][Q][G]T 

-4 

w. 

I whc:lre [QJ = a diagonal matrix of elements 1 - -0::----:::--:::-

(~~ _ ~2)2 I 

(3.31> 

The matrices [KJ and [M(~)] for each substructure at the same 

level are assembled by the method used In Guyan reduction to obtain synthc~s I zed st I ffness and mass matr I ces for the next higher I eve I • The 

process is then repeated until the highest level structure Is reached. 

The solution process for this formulation requires Iteration at 

each lev(~1 of substructures. The unknown system frequency, ~, must be 

InItially estimated and then I'reratlvely Improved until convergence fs 

attained. One Iteration Involves buildIng [M(n>] and [0] for each substructure until the highest level structure Is reached. Then the elgenproblem at the highest level Is solved, 

[D]{\jJ} = {OL (3.32) 

Fr<n this solution, the next value of ~ Is chosen for use In the folI o~d ng I irerat I on. Although the fixed-fixed normal modes need be compu"t'ed only once for each substr"ucture, each Iteration requires that the 

mass matrtx for the Intermediate level substructures be recondensed. 
i 21 

It Is stated that a partial vibration case results when ~ = w. for 

I 

a substructure and thus drives the dynamic mass to Infinity. However, 

there are no clear guidelines given for handling this potential Instab I I Ity In the method. 

The method Is shown to yield good accuracy In computing system frequencles but results for mode shapes are not given. Also, no comparison 

of computational efficiency Is made with the more standard methods of 

modal synthesis. 

3.3.6 Attachment Modes ~ Interface ~ ~ 

Bamford, Wada, Garba, and Chisholm [3.1J Introduced the concept of 

attachment modes as an additional set of static modes used In modal synthesis. An attachment mode defInes the response of the substructure to 

a unit force appl led at an Internal boundary DOF while al I other internal boundary DOF remain free. The motivation for using attachment modes 

Is that their use can be expected to reduce the number of normal modes 

necessary to accurately descrIbe the displacement behavior of the substructuree 

In the transformation from generalized to geometric coordinates for 

the substructure, attachment modes are combined with static constraint 

and normal modes. The normal modes may be computed with Internal boundary DOF fixed, free, or a combination of the twoo Bamford's transformation matrix Includes a set of rigid-body modes, however It Is noted 

that no dIstinction between rfgld-body modes and static constraint modes 

Is required. A drawback In the use of attachment modes Is that they are 

not always linearly Independent of the static constraint and normal 
122 

modes. 

In an effort to establ Ish a more rational approach to substructure 

synthesis, Hintz [3.8J grouped combinations of the four mode classes: 

rigid-body, static constraint, normal, and attachment, Into five different Interface mode sets. The four mode classes are illustrated In 

'FiSlure 3.4 as they apply to the FIN substructure of Chapter 2. Each of 

the! five sets is claimed to be complete In that It precisely represents 

thel original finite element model for static and dynamic response. Implications of truncating a selected Interface mode set are considered 

anc! thus £Iuldellnes are developed for retaining accuracy with a reduced 

size model. Since each of the five mode sets represents the substructur'e differently, truncation of each set has varying Impacts on the substructure model. The guidelines are directed toward establishing a mode 

set' that will al low maximum tr-uncatlon of the normal modes wh i Ie 

ret'alnlng the detail of the original finite E~lement model with respect 

to statically imposed Interface forces and displacements. 

In numerical results preseni-ed by Hintz, the fixed-Interface method 

(defined by one of the five interface mode sets) gave good results over 

a broader range of frequencies than did those methods using attachment 

modes. However, use of attachment modes did produce more accurate 

vibration frequencies In the low frequency range. 
Figure 3.4 

1 

I 

I 

I 

I 

I 

I 

I 

I 

I .... ....... -....... I ....... .J 

123 

Unit Displacement 

Applied at 1 Master DOF 

(All others Fixed) 

a) Rig i d Body ~~ode (Rotat i on) b) Static Constraint Mode 

c) Normal 

-- Mode 

Master DOF Clamped 

for Normal Modes 

Unit Force Applied at 1 

(-': .... _ I nternal Boundary DOF 

I ........ ~.... (All others Free) 

Mode 

;::-' 

~ .... .... '1~ 

" 

Classes " for 

I .... _-""' .... I --("'I 

11".... p .. 1 :1 .... - I' .... - I ........ J.,; 

I 

I 

I 

I 

I 

I 

I 

I 

! 

I 

I 

d) Attachment Mode 

Deformed Configu ration 

Undeformed Configuration 

Actual Physical Constraint 

Interface Mode Sets I I lustrated for 

Substructure FIN 
i 24 

3.:5.7 l!n.Q.roved nlsplacement Re..c9~ In Modal Synthesis 

The frequency dependent transformation previously discussed with 

regard to Guyan reduction, Eq. (3.14), has been appl led In a limited 

fashion to modal synthesis by Kuhar and Stahle [3.16J. The method conslsts of reducing the order of the eIgenvalue problem for the highest 

level structure after modal synthesis has been performed. If r2 Is a 

frequency about which the elgenproblem for the highest level Is to be 

reduced, the synthesIzed matrIces can be reduced with the transformatIon 

(3.33) 

The reduced matrices form an eigenvalue problem of smaller sIze than 

that defined by the synthesized matrices. 

As with Guyan reduction, the elgenproblem Is evaluated more accurately at the frequencies near the reduction value, r2. (In Guyan 

reduction, the reduction frequency Is r2 = 0.) Thus, the choice of master 

OOF for the highest level structure should be directed towards 

preserving the frequency content of the model around r2. Although selection of t'he master OOF Is Initially a matter of Judgement, It can be Improved In the second and later Iterations If necessary. If the frequency range of Interest is broad, It may be advantageous to re-solve the 

problem for different values of r2. 

Upon solution of the reduced elgenproblem at the highest level, the 

set of mode shapes, [¢~, will be In terms of the master OaF. If a computed frequency w. Is equa I to the reduction frequency rG, then the corI 
i25 

responding mode shape Is obtained exactly by 

(3.34) 

,Likewise, the full set of mode shapes Is obtaIned from 

[~J = [-;j [~"'J. (3.35) 

but the vectors in [~J are only approximate. Based on the coeffIcients 

In the mode shape vectors corresponding to frequencIes within the range 

of Interest, new master DOE can be Identified to Improve the accuracy of 

the reduction. 

3.3.8 Restdual ~ ~ Resfdual EJexlbl1 tty 

A hybrId method of substructure reduction wIth the capability of 

modeling a more general condItion of Internal boundary constraInt was 

developed by MacNeal [3.18J and Implemented to a I lmlted extent In 

NASTRAN. The normal modes of the substructure are computed with Internal boundary DOE free, fIxed, or a combination of the two. The decision 

regarding boundary DOE fIxity is based on the nature of the adjacent 

substructures In the model. If an lnterface DOE fn one substructure Is 

held fixed, the correspondIng DOF in the adjacent substructure must 

remain free In order to avofd an overconstralnt problem at the interface. This procedure requires that Information on the topology of the 

substructures be provided by the analyst prior to modal analysis of any 
126 

Individual substructure. MacNeal's formulai"lon also requires that the 

mass at fixed boundary OOF be distributed to free OOF In the substructure since fixed OOF do not participate In the vibration response. 

The representation of the substructure can be Improved by Including 

static approximations to the effects of the tr"uncated higher modes. The 

'Improvement Is In the form of a residual mass matrix for the synthesis 

Involving fixed-fixed normal modes and a residual flexlbl I Ity matrix 

when free-free normal modes are used. For the case of hybrid modes 

(some Internal boundary DOF fixed, others free), both matrices are formulated. Since these matrices are obtained by a static derivation 

(ul = 0), they are val Jd only at" vibration frequencies that are low compared to the lowest mode of the substructure. 

Rubin [3.21J proposed an Improvement to MacNeal's residual flexIbility for substructures wlt"h free-free normal modes only. The approach Is shown to have better convergence than the methods of MacNeal 

and Hurty but I Ike MacNeal's, It's application Is strictly limited to the 

low frequency range. Rubin's method Is further restrIcted to the use of 

free-Interface normal modes and rigid-body modes. Thus, MacNeal's 

residual mass Is not Included In the developmento 

3.3.9 ~~Qrder Polynomial Icans.formatJons 

One of the primary differences among the reduction techniques 

discussed thus far lies In the definition of the transformatIon to 

generalized coordinates. Each method that has been reviewed uses seme 

combination of static constraint and substructure normal modes. 

Melrovltch and Hale [3.19J have recognized that It is not necessary to 
i27 

consider only substructure modes for use In the transformation. They 

have shown that it is sufficient for substructures to be represented by 

admissible functions that are from any complete set. One such set considered useful for continuous systems Is a set of low-order polynomials. 

As appl led to finite element models, the polynomials are defined over 

·the domain of the substructure and satisfy the geometric boundary conditions. The shape vectors are then built by sampling the polynomials at 

the spatial coordinates of each substructure OaF. 

The transformation to generalized coordinates is 

where CTpJ Is the set of shape vectors derived from the low-order 

polynomials. As with the free-Interface method, the general Ized 

coordinates, {q}, contaIn no geometric equivalents for substructure assembly. Compatibility between substructures Is enforced by a weighted 

residual method using spatIal DIrac delta functIons for continuous 

models or equivalently, unit vectors for discrete models. For a fInite 

element model, the effect is achieved by writing equations of constraint 

matching the dIsplacement at the point shared by two substructureso 

Using the constraint equations, a constraint matrix is produced which Is 

used to I Ink together the otherwise Isolated substructures. From this 

operation Is obtained the synthesized stIffness and mass matrices In 

generalIzed coordinates. 
128 

A difficulty lies In development of the constraint matrix. In the 

elE~mentary example problems presented, the matrix was built by Inspectlcm. This Is not an acceptable procedure for general application. As 

observed by the authors, an alternative formulation can be defined which 

re I I eves i"he prob I em of constrLld"I ng the constra I nt matr I x. The approach I c' 

~. simply to use the same transformation matrix as used In the 

fi>(ed-lntE~rface method but with 'fixed-fixed normal modes replaced by 

10~l-order polynomial shapes. Thus 

[T ] = t~ __ L~_j (3.37) p c I P , ¢ I ¢ 

I 

whEire [¢p] I s the set of low-order pol ynoml a Is. For this approach, 

polynomials which have zero values at the Internal boundary nodes must 

be chosen to simulate the fixed-fixed boundary conditions. Geometric 

compatibility is enforced by the presence of the static constraint modes 

[¢cJ and r'etalned Internal boundary nodes In geometric coordinates. 

TherE~ are two trade-offs that must be made I n us I ng the transformation of Eq. (3.37) Instead of that In Eqo (3936). The static constraint 

modes mus1- be evaluated causing some additIonal computational expense. 

Also, thEl size of the set of admissible polynomial functions Is reduced 

by the requirement that the functions be zero-valued at substructure Internal boundaries. 

In contrast to Individual finite elements, the geometry of substructures Is not of a nature that can be easi Iy classified. Thus, 

se~ectlon of the appropriate polynomials Is a matter of experience and 

Judgement.. Because no automated selection method has yet been devised, 
I~ 

this approach to modal synthesis Is currently not capable of performing 

dynamic reduction in a general finite element system. However, the 

procedure remains one of the more actively pursued techniques and does 

hold potentfal. 

,3.3.10 Eyaluatlon Qt Modal SynthesIs Techniques 

In the study of modal synthesis techniques, several advantageous 

characteristics of an Ideal method can be IdentIfied. A discussion of 

these characteristics as evaluation criteria wll I be useful In comparing 

the various methods presentedo 

10 Efficiency Qf ~ ReductIon Method 

The efficiency of a dynamic reduction method is Influenced by a number of factors. First, the method must 

result In an accurate reduction of the substructure stiffness 

and mass. An efficient method will yield synthesized stiffness and mass matrIces that accurately maJntafn the dynamic 

characteristics of the substructure with the mlnfmum number of DOF~ Second, the degree of analyst particfpatlon should be 

I imfted to simply the deflnltfon of the model and speclffcatlon of the solution type. A method should be automatIc once 

the solution process begfns, hence al Imlnatlng the need for 

the analyst to Interpret intermediate results and restart the 

process. This Is not to Imply that the analyst should surrender control of the solution process. Instead~ he should be 

relIeved of the burdensome task of supervising the computational process. Third, the syntheSis method should be efficient in Its use of the computer. Given the problem size, 

algorIthms should be chosen that minimize the requIred computer resources, includIng CPU tIme and I/O~ The number of 

arIthmetic operations performed should be predictable rather 

than dependent upon an arbitrary test for convergence of an 

Iterattve processo 

20 ApplIcabilIty fQ General Problems 

A wide variety of dynamfcs problems exists for which 

modal synthesis ts needed to achieve an economIcal and a~ 

curate solution. A synthesis method used tn a general purpose 

FEM system should be capable of representIng substructures 

over a broad range of geometries and with various types of 

boundary constraint. Also helpful would be the abIlity to Incorporate experimental data, such as mode shapes and natural 
130 

frequencies, Into the substructure model. 

By necess I ty I fin I te el ement ana I yses of non II near struc·~ 

tures are performed incrementally. As the effects of nonI I near mater I a I s and geometry occur, the mode I must be ref or'· 

rnulated to accurately track the true response of the stru~· 

"ture. Therefore, dynamic reduction methods must lend the~· 

selves to this incremental solution process. 

3. J ndependence .Qf. I nd I V l...d.!.ull Substructures 

Often times the analysis and design responsibilities of 

the various components of a structure are distributed among 

different organIzational groups. ThIs separatIon of responsIbil Itles has many advantages and should not be encumbered by 

the synthes I s method. Therefore, the method used shou I d con·' 

$Ider each substructure as an Isolated entIty in evaluatIng 

'Its dynamic response prior to system synthesIs. Topology of 

the substructures should not be considered untIl the equations 

at the next higher level are ready for assembly. 

4. j:ase.Qf. Reana lysis 

The most re I I ab I E~ test for convergence of a dynam I c r~educt I on method Is to re-so I ve the prob I em with a more high I Y 

refined model (more Independent OOF). The additIon of more; 

[)OF to the model can be a relatively simple task, achieved at 

lIttle expense, or It could be as dIfficult and expensive as a 

complete reanalysiS of each substructure. The Ideal approach 

al lows simply the addition of previously neglected terms to 

Improve the accuracy of the reductIon. These terms generally 

take the form of truncated substructure normal modes. 

5. l\ccuracy sruI. Stab II It¥. 

Accuracy of results Is Important in two respects. Wei I 

defined modal response data Is needed to accurately synthesize 

the higher level structures for frequency and transIent 

analysls. Also, In returning to the lower levels for recovery 

of strains and stresses, the quality of the displacement veCa • 

1rors Is critical. Accurate stresses require that displacement ~~radlents be well formed. Intimately tied to accuracy of 

r'esults Is the numerIcal preCision with which computations 

must be performed. Operations such as triangulation In equation solving can have a significant Impact on final accuracy 

cmd the need for such operations should be considered In 

selecting the reduction method. 

The potential for' problems with the stability of operations In the reductIon methods can often be Identified by 

close examInation of the development of the methods. Typical 

problem areas are linear dependence of the vectors comprlsln£1 

a transformation matrix and the dlvlde-by-zero singularIty. 
131 

With the criteria establ ished, each synthesis method can be 

evaluated. The goal of this evaluation Is to Isolate one or n~o methods 

of modal synthesis that wll I be most useful in the general purpose FEM 

system. 

The fixed-Interface method successfully satisfies four of the a,bove 

'five criteria. The method is simple to apply and results In a significant size reduction of properly substructured models. It does not 

require consideration of substructure topology prior to assembly thus 

preserving substructure Independence. Repetitive tests for convergence 

are straightforward, requiring simply the addition of more normal modes 

In selected substructures. The entire problem need not be re-solved to 

Incorporate the additional modes. Thus the cost of convergence tests Is 

low and predictable. Orthogonality of the transformation matrices ensures stability of the method and accuracy has proven favorable for many 

problems. One drawback of the method Is Its limited capabl I Ity to use 

experimental data o Since all master OOF are fIxed during the computation of normal modes, the structural components must be tested In a I Ike 

fashion. 

Interface loading has proven necessary to the use of the freeInterface method. The approximate inertial effects of adjacent substructures serve to Improve the displacement gradients at the Internal 

boundaries, thus yielding more accurate strains and stresses [3.14J. 

Although interface loading is helpful In computing accurate stresses, 

Its use eliminates the ability to maintain substructure Independence. 

As previously mentioned, extension of the free-interface method to multilevel substructuring might be a difficult task to accompllsh0 
132 

Branch mode analysis was originally developed for chain-type structures, e.g. piping networks. As such, Its application to analysis of 

more general structures Is cumbersome. The method requires solution of 

tw·lce as many elgenproblems as the fixed-Interface method and It Is Intimately dependent upon substructure topology. Convergence testing 

'uslng a more refined model becomes Impractical as complete reanalysis Is 

necessary. 

Leung's Iterative method appears limited to eigenvalue extraction 

over a narrow band of frequencies or analysis of transient response to 

harmonic forcing functions. This Is because i"he system equations must 

be synthesized for each modal frequency considered. For multilevel substructured models, the need for repetitive condensations Is expected to 

make the method prohibitively expensive when the frequency range of Interest Is broad. As mentioned In the development of the method, some 

special precautions would also be necessary to maintain stability when 

the system frequency Is very close or equal to a substructure modal frequency. 

Although attachment modes are useful In reducing the number of 

re'ta I ned substructure norma I modes, the I r potentl a I fa tit ngs are not 

we I I I I I ustrated I n the I I teratl.lre. I tis noted that attachment modes 

may cause II I-conditioning In the transformation to generalized 

coordinates [3.1J. What must be recognized Is the difficulty with which 

linearly Independent attachmeni" modes are chosen. The selection of attachment OOF for a substructure Is by no means an Intuitive process. It 

Is expected that even the exper'lenced analyst will require a trlal-anderror approach to their selection. 
133 

Efficient methods for extracting a I imlted number of eigenpalrs In 

large systems effectively make the highest level structure reduction 

proposed by Kuhar and Stahle unnecessary. Their process yields a 

reduced size equation of motion via an Iterative procedure. The same 

results can easily be achieved by solving for selected modes of the unreduced synthesized equations and then formulating a transformation with 

these modes and appropriate static modes. This transformation can then 

be used to reduce the size of the synthesized equations. 

MacNeal's residual mass and flexTbi I Ity method provides an 

automatic approach to modal synthesis which Includes general boundary 

constraint capabrl ttiese However, the method does have some limitations. It Is correctly argued that fixed coordinates do not participate 

In the vibration behavior of the substructure. What Is not considered 

Is that In a multilevel substructured model, the coordinates held fixed 

at level "1+1" may be considered free at level "I". If the mass components of the constrained DOF are redistributed as suggested, the Inertia 

characteristics of those OOF wit I not be accurately preserved. Also, 

the need to define substructure topology and properties prior to assigning Internal boundary constraints hampers the effort to retain independence In computing Individual substructure response. Another drawback is the potential III-conditioning of the residual flexibility 

matrix when many modes are retained in the transformation. The advantages of using residual mass and flexlbll tty matrices appear marginal In 

that they Improve only the low frequency response and that residual mass 

Is Impl icltly contained In Hurty's method [3.18J. 
134 

Polynomial transformation i-echnlques hold tremendous potential as 

dynclm I c rElduct I on methods. A great sav i ngs In computatl ona I effort Is 

possible by evaluating low-order polynomials rather than computing sub-· 

structure vibration modes. Unfortunately though, the approach to 

selElctlng the polynomials has not yet been standardized. As such, the 

techniques are not Immediately useful for general application In FEM 

software. 

3.4 Selection Qi Methods fQc Dyn~lc Reduction 

In VfElW of the above discussion, selection of dynamic reduction 

schElmes becomes simple. The two broad types of structural models, those 

with and without substructurlngp exemplify the need for two dlstlnci' 

methods for- dynamic reduction. The two methods consIdered most suItable 

are Guyan r-eductlon and the f I xed- I nterface method of moda I synthes Is. 

Perhaps tho greatest advantage of both of these methods I s the I r concep·· 

tual sImplicity. This point cannot be overemphasized. It Is ImperativE! 

that the analyst have a complete understandIng of the analysis methods 

he IJses. Without this understanding the chances of achieving a meanIngful dynamilc analysis are remote. The obvious dIsadvantage of this 

philosophy Is the sacrifice In sophistication that the experienced 

analyst makes If he Is to use the general system. One mIght expect, 

howEwer, that spec I a I purpose FEM systems cou I d be used when the need 

for more refined techniques is not satisfIed elsewhere. 
135 

3.5 References 

3.1 Bamford, R., Wada, B.K., Garba, J.A., and Chisholm, J., 

"Dynamic Analysis of Large Structural Systems," SynthesIs .Q.f. 

Vibrating Systems (ASME Booklet Nov. 1971), pp. 57-71 

3.2 Benfield, W.A. and Hruda, R.F., "Vibration Analysis of Structures by Component Mode Sunstltutlon," AlM Journal, Vol. 9 

pp. 1255-1261 (1971) 

3.3 Craig, R.R. Jr. and Bampton, M.C.C., "Coup I Ing of Substructures for Dynamic Analysis," MM Journal, Vol. 6 pp. 1313-1319 (1968) 

3.4 Gladwell, 

Systems," 

( 1964) 

G.M.L., 

Journal 

"Branch Mode Analysis of 

Qi Sound ~ Vlbratloo, Vol. 

Vibrating 

pp. 41-59 

305 Goldman, ReLo, "Vibration Analysis by Dynamic PartItioning," A!8a Journal, Vol. 7 pp. 1152-1154 (1969) 

3.6 Guyan, R.Jcp "Reduction of Stiffness and Mass MatrIces," AlM 

Journal, Vol. 3 pp. 380 (1965) 

3.7 Henshel I, R.D. and Ong, J.H., "Automatic Masters for EIgenvalue Economlzatlon," EarthQuak~ EngIneering W Structural 

Dynamics, Vol. 3 pp. 375-383 (1975) 

3.8 Hintz, R.M., "Analytical Methods In Component Modal Synthesis," AlM Journal, Vol. 13 pp. 1007-1016 (1975) 

3.9 Holze, G.H. and Boresi, A.P., "Free VIbration Analysis Using 

Substructurlng," Journal Qi.::t.b.S! Structural Division <'8's.CE), 

Vol. 101 pp. 2627-2639 (1975) 

3.10 Hou, SoN., "Review of Modal Synthesis TechnIques and a New 

Approach," Shock .Md. Vibration BulletIn, 40(4) pp. 25-39 

(1969) 

3$11 Hurty" W.C." "Vibrations of Structural Systems by Component 

Mode Synthesis," Joyrnal .Qf m EngineerIng Mechanics 

Division (ASCE), Vol. 86 pp. 51-69 (1960) 

3.12 Hurty, W.C., "Dynamic Analysis of Structural Systems Using 

Component Modes," AlM Journal, Vol. 3 pp. 678-685 (1965) 

3.13 Hurty, W.C., "Introduction to Modal Synthesis TechnIques," 

SynthesIs Qt Vibrating Systems (ASME Booklet Nov 1971), pp. 1-13 
i36 

3.14 Hurty, W.C., Collins, ,J.D., and Hart, G.C., "Dynamic Analysis 

of Large structures by Modal Synthesis Techniques," Computers 

gnQ Structures, Vol. 1 pp. 535-563 (1971) 

3.15 Kidder, R.L., "Reduction of Structural Frequency Equations," hl8A Journal, Vol. 11 p. 892 (1973) 

3.16 Kuhar, E.J. and Stahle, C.V., "Dynamic Transformation Method 

for Modal SyntheSiS," AlM Journal, Vol. 12 pp. 672-678 

(1974) 

3.17 Leung, Y-T, "An Accurate Method of Dynamic Substructurlng 

with Simplified Computation," J..o..ternatlonal Journal .f.Q.c. 

Numerical Methods In. Engineering, Vol. 14 pp. 1241-1256 

(1979) 

3.18 MacNeal, R.H., "A Hybrid Method of Component Mode SyntheSiS," 

Computers ~ Structu~, Vol. 1 pp. 581-601 (1971) 

3.19 Melrovltch, L. and Hale, A.L., "On the Substructure SyntheSiS 

Method,"Al8AJournal, Vol. 19 pp. 940-947 (1981) 

3.20 Miller, C.A., "Dynamic Reduction of Structural Models," .J..Q.w:.-

nsl Qf the Structural. Dlylslon (~), Vol. 106 pp. 2097-2108 

3.21 Rubin, S., "Improved Component-Mode Representation for Structural Dynamic AnalYSiS, Al8A Journal, Vol. 13 pp. 995-1006 

( 1975) 
CHAPTER 4 

COMPUTATIONAL ALGORITHMS FOR DYNAMIC ANALYSIS 

4.1 General 

The solutIon of structural dynamics problems by the FEM reqUIres 

computational capabil itles which are not necessary In static analysis. 

The two most important of these, in terms of computatIonal efficiency, 

are elgenproblem solutIon and transient response analysis. These two 

operations form the core of the dynamtc analysIs process. Their proper 

Implementation and use Is essential to the success of the dynamic 

analysis. Other less Important features Include mass matrix and damping 

matrix formulatIon and the use of experimental data. Little computatIonal effort is expended on these operatIons but their use adds 

generality and completeness to the solution strategy. 

Elgenproblem solution Is the single most expensive operation In the 

modal synthesis process. While the approach to modal synthesis will 

control the qual ity of the solution, eigenvalue analysis can be expected 

to control the cost. An understanding of the various methods for eigenvalue analysts Is, therefore, necessary if the analyst is to achieve an 

economical and accurate solution. 

Solution of the differential equations of motion yields the com~ 

plete structural response to the transient loadlnge This solution can 

be obtained by eIther of two approaches: mode superposition for linear 

systems and time-history Integration for both I inear and nonlinear 

systems. 

137 
138 

The remainder of this chapter Is divided Into three sections. The 

first presents a review of elgenproblem solution techniques. The 

methods are evaluated and a selection Is made of those methods deemed 

necessary for incorporation In the general FEM system that employs modal 

synthesis for substructure reduction. The second section discusses the 

methods used to solve the differential equations of motion. Emphasis Is 

placed on the effects that nonl inear structural response and multi level 

substructured modeling have on the solution processes. The final section briefly discusses mass matrix formulation, damping, and the use ot 

experimental data. 
i 39 

4.2 Eigenproblem Solution 

4.2.1 Effects Qf. Multlleyel Substructllrlng 

The Introduction of multilevel substructurlng Into the FEM has 

added significantly to the requirements for a versatile elgenproblem 

solutIon package In the general FEM system. The Importance of accuracy 

In computIng substructure modes and the frequency wlth which the elgenproblem must be solved become maJor conslderatlons In the selection of a 

solution process. Accurate synthesis of the highest level structure 

stiffness and mass is dependent upon the qual fty of the retained modes 

from each substructure. In a model with multilevel substructurlng, the 

stiffness and mass matrices at the highest level are synthesIzed using 

vibration modes from the lower levels~ As each higher level substructure is assembled fram the lower levels, repeated vibration analYSis Of 

synthesized substructures Is performed~ This process will tend to compound any errors that may exist In the vtbratlon modes. Eventual 

degradation of the highest level stiffness and mass can be avoided by 

maintaining good accuracy In the computed mode shapes at each level. In 

the modal synthesis process, a free vibration analysis is performed on 

each substructure. In nonl inear problems, the vibration analYSis ot 

nonlInear substructures Is repeated wIth each update of system properties. For large substructures these requirements for frequent vibration 

analyses could become prohibitively expensive unless very effIcient 

solution meth6ds are used. 
140 

In the evaluation of elgenproblem solution methods, one must consider moroe than the computational efficiency with which a solution Is 

achieved. Also of prime concern Is the stabl I Ity of the method for a 

par'tlcular' problem, given the form of the matrices Involved. If the 

solution method Is inappropria1"e for the given problem, convergence too 

'the) wronfJ solution or divergence can occur. This stability is of particular Interest for finite element models using multi level substructuring. As the synthesis progresses upward through the various levels 

of substructuring, the nature of the eigenproblem changes as outl ined 

below. Thus the analyst must have the capability to select the solution 

me1"hod thclt most closely matches the characteristics of the new problem. 

The changes In the nature of the eigenproblem include variations In 

the! charclcterlstics of the substructure stiffness and mass matrices and 

In the number of elgenpairs needed. For lowest level substructures, the 

mass matrix may be diagonal or banded, positive definite or semidefinite, well formed or III-conditioned. Although the stiffness matrix 

Is normally banded, it may be singular, such as when rigid-body modes 

are) Included. Further complications arise when static condensation Is 

uSEld to eliminate massless degrees of freedom. 

bandform for both the stiffness and the mass. 

This may cause a loss of 

At the lowest substructure levEd, usually only the lowest frequencies and mode shapes are 

necessary to progress with the synthesis process. However, depending 

upon the elgenproblem solution method used, this requirement may Involve 

finding al I elgenpalrs of a reduced system. 
141 

When a structure's stiffness and mass are assembled from lower 

level substructures, the resulting matrices can take on a special form. 

If Guyan reduction Is used at the lowest level, the structure matrices 

are generally fut I or widely banded. If modal synthesis Is used, the 

stiffness matrix wll I be In block diagonal form and the mass matrIx wll I 

be block diagonal and border banded; see Eq. (3.25) and (3.26). It Is 

most desirable to have elgenproblem solution methods that can deal effectively with these special forms when the total number of DOF Is 

large~ 

No single eigenvalue solution method has been developed which Is 

most efficient for all of the expected forms taken by the elgenproblemo 

Within a particular class of problems (e.g. al I elgenpalrs are required 

and the system matrices are narrowly banded) superior solution methods 

can be tdentlfledo An adequate selection of these methods must be 

available In a general FEM system, thus permitting the analyst to tailor 

the solution to the particular characteristics of his problem. 

4.2.2 SolutIon Methods 

For discussion purposes, eigenproblem solution methods are 

seperated Into three categorlesQ These are simultaneous Iteration, 

transformation, and polynomial/vector Iteration methods. Solution techniques within each of these categories rely on one or more of the fundamental propertIes of elgenpalrso Some of these properties are expressed In equation form as 

(4.1) 

This form suggests an Iterative solution by assuming a mode 

shape and successively improving the approximation until convergence to the true mode shape Is attained. 
i 42 

2. [¢JT[KJ[¢J = [AJ 

[¢JT[MJ[¢J = [IJ. 

and (4.2 ) 

(4.3) 

This form suggests that, since the vectors In [¢J are unique 

within a scalar multiple of themselves, successive transformations of [KJ and [MJ to bring them to diagonal form wll I yield 

the matrices [¢J and [AJ. 

3. det(K - AM) = O. (4.4) 

4. 

This form shows that the roots of the characteristic polynomial define the eigenvalues of the problem [KJ{¢} = A[MJ{¢}. 

~(m) < ~(m+1) < ~(m) < ~(m+1) < < ~(m+1) < ~(m) (4.5) 1 - 1 - 2 - 2 - ... - n-m-1 - n-m 

(m) (m+l ) where~. and~. define the eigenvalues of the "m" th and 

j I ' the "m+l" th constraint problem for mode "I". This eigenvalue 

separation property states that the characteristic polynomials 

p (;.\ (m» i m=1,2, ••• , (n-·1) form a Sturm sequence. 

Details regarding these properties may be found In the references [4.2, 

4.25, & 4 .27J. 

4.2.2.1 Simultaneous IteratIon 

Simultaneous Iteration methods Involve successive Improvement of a 

se't of orthogona I vectors chosen to approximate the mode shapes. Each 

of the methods uses one of the basic forms of Inverse Iteration: 

[x. 1J = [AJ[x.J, or 

J+ J 

(4.6) 

(4.7> 

In these forms [KJ Is the (nxn) structure stiffness, [M] Is the (nxn) 

structure mass matrix, [x.J Is the (nxq) mai'rlx of assumed mode shapes J 

for iteration "j" and In symbolic form: 

[A] = [L]-1[M][L]-T, where 

[KJ = [L][L]T. 

(4.8) 

(4.9) 
143 

The differences In the methods I Ie In the processes used to orthogonallze [xj

+1

J and to accelerate convergence. 

Most of the simultaneous Iteration methods reviewed use the standard form of the eigenvalue problem, Eq. (4.6). JennIngs [4.12J 

presented the first such practical method in which the coupling among 

'the vectors of [x.J is evaluated and reduced by an Interaction analysTs. 

J 

The process Involves examInIng the off-dIagonal terms of the product 

(4.10) 

then decoupl fng the vectors in [x. 1J and reorthogonal fzing prio~ to the 

J+ 

next Iteration. 

Rutlshauser [4.23J introduced a procedure In which the matrix of 

modal vectors Is fmproved by solving the reduced (qxq) elgenproblem 

using the matrix 

(4.11> 

No explicit orthogonal Izatlon Is then required. The remaining varlatlons of these methods, primarily attributable to JennIngs and his colleagues [4.5, 4.7, & 4.13J, represent modifications to enhance convergence or to improve the orthogonal fzatton process. 

Dong [4.8J and Bathe [4.2J presented independent developments of 

the method most commonly known as subspace Iteration. This method uses 

the form of Eq. (4.7) and relies on the solution of a reduced elgenproblem similar to that of Rutlshauser. This method has a significant 

advantage In that It does not require conversion of the elgenproblem to 

the standard form of Eq. (4.6). This feature affords the potential to 
144 

solve a wider range of problems ([KJ and/or [MJ II I-conditioned) than 

was previously feasible. Several methods have been proposed to accelerate the convergence of the subspace Iteration process [4.1, & 

4.28J. These include shifting, overrelaxatlon, and the use of Chebyshev 

polynomials. An improved method for selecting starting vectors is also 

'available. Although these techniques were Introduced for application to 

subspace iteration, they are applicable to other simultaneous Iteration 

methods. 

4.2.2.2 Iransformatlon Methods, 

Transformation methods are based on the orthogonal tty property of 

eigenvectors. The approach I s to genera-re a sequence of orthogona I 

transformation matrices, [P.], of order (nxn), that drive the symmetric I 

matrix, [A], also of order (nxn), to diagonal form. For a case requiring "kit Iterations to achieve convergence, the operation takes the 

form 

(4.12) 

The diagonal matrIx [A] contains the complete set of eIgenvalues. The 

corresponding eigenvectors are computed by 

(4.13) 

The eigenpalrs are not ordered as In the case of simultaneous Iteration. 

Further, since the complete eigensystem Is solved, the resulting 

dIagonal matrix and the product In Eq. (4.13) are unique. This uniqueness exIsts only when the transformation matrices are of order (nxn). 
i45 

The generalized Jacobi method Is recognized to be the most efflclent, truely Iterative, transformation method for solving the general 

elgenproblem 

[K]{¢} = A[M]{¢}. (4.14) 

'Each transformation matrix, [P.], performs a rotation which zeros corI 

responding off-diagonal terms In [K] and [M]. One sweep Is completed 

when each off-diagonal term In [K] and [M] has been individually zeroed. 

Since zeroing one off-diagonal term tends to make a previously zeroed 

term nonzero, multiple sweeps are requIred to achieve convergence. 

If the eTgenprobJem can be transformed to standard form: 

(4.15 ) 

where [A] Is defined by Eqo (4.8), transformation of [A] to tridiagonal 

form can be performed without Iteration. Givens Introduced a method of 

plane rotations [4.25J Tn which, for step "j", zeros are Introduced In 

row "j" and column "J" without destroying the zeros from previous steps. 

The efgenpalrs of the resulting tridiagonal matrix can then be found 

with relative ease. Householder developed an Improved approach to 

trIdiagonal fzation using reflection matrices to perform the transformation. Like that of Givens, Householder's method requires (n-2) steps to 

complete the reduction but each step of Householder's method Involves 

roughly half as many multiplications. 
146 

A third method for tridiagonal Izatlon is available and can be appi led to the general eigenvalue problem of Eq. (4.14). The generalized 

Lanczos mE~thod [4.24J Is applicable to problems when [KJ and [MJ are 

syrrunetrlcll equally banded, and at least one Is positive definite. In 

the method, a single orthogonal transformation matrix Is built one 

. co I um n a -~ a t I me. With the evaluation of each new column, {r-}, reol~­ I 

thogonal Izatlon to al I previous columns, {r1} to {r;_1}' Is required to 

ensure stability. 

Solution for the elgenpalrs of a tridiagonal matrix Is effectively 

at-I-alned by using QR Iteration to find the eigenvalues and Inverse veer 

tOI" Iteration to find the eigenvectors. A dlscusslon of lnverse vector 

I tc~rat I on fo I lows l n the next sect I on. 

QR I-~eratlon [4.2J ls simply a factorlzatlon of the trldlagonal 

matrlx, [SJ, lnto the product of an orthogonal matrix, [QJ, and an upper 

triangular matrix, [RJ. 

[SJ :: [QJ[RJ (4.16) 

PrE:!- and post-multlplylng Eq. (4.16) by [Q] and [QJ respectively gives 

[Q] T[SJ[QJ = [RJ[Q] (4.17> 

where the product on the rlgh~'hand slde ls the dlagonal matrix of 

el !~enval UE~S. Jacobl rotatlon matrices are often used to reduce [SJ to 

uppertrlangular form of [R], with [Q] being the product of those rotatIon matl~lces. A shifting strategy, discussed below, can be used with 

QR lteratlon to accelerate convergence. 
i47 

4.2.2.3 Polynomial/Vector Iteration 

Polynomial Iteration techniques Involve an iterative solution of 

the characteristic polynomial 

p(A) = det([K] - A[M]) = O. (4.18) 

In practice the polynomial Is never explicitly evaluated. Instead an 

estimate of the exact eigenvalue Is made and then the determinant Is 

evaluated using Gauss or Choleski factorization. From this evaluation, 

the estimate of the eIgenvalue Is fmproved and the process is repeated. 

Two popular procedures aid In this evaluation: accelerated secant 

Iteration and bisection. Accelerated secant fteratlon fs a superl Inear 

fnterpolatlon method from which an Improved estimate of the eigenvalue, 

Ak+1, Is determined using two previous estimates, Ak- 1 and Ako The Im= 

proved estimate fs 

(4.19) 

where a is an acceleration constant. The polynomial Is evaluated at 

Ak+1 and if required, another lteratfon Is performed. Bisection Is an 

Iterative process In which the Interval between the previously determined upper and lower bounds of Ak Is halved and the sign of the polynomial at that midpoint evaluated. On the basis of the evaluation, the 

new Interval containing Ak ls Identlffed and the process repeated until 

convergence is achieved. 
148 

Vector Iteration Is a special version of simultaneous iteration In 

whl'ch on~y one elgenpalr Is computed at a time. The Iteration sequence 

ta~(es the form: 

(4.20 ) 

If converHence of the eigenvector Is gained In Iteration "k", the elgenvalue can be computed from Rayleigh's quotient: 

A = (4.21> 

ThEl above form of vector iteration, termed Inverse Iteration, will Inltlclily converge to the most dominant elgenpalr (smallest eigenvalue). 

To force convergence to a subdomlnant elgenpalr, the Influence of 

thEl previously computed eigenvectors must be eliminated. If an estimate 

of the el~}envalue Is known, shIfting can be used to make the unknown 

e I ~lenpa I r dom I nant. Letting ~ be the value of the shift, a modified 

elgenproblem Is defined 

[K - ~MJ{¢} = V[MJ{¢} (4.22) 

whElre v = A -~. I nverse Iteration app II ed to th I 5 prob I em produces the 

el fjenpa I r closest to the sh i ft~. When a good estimate to the desl red 

eigenvalue is unavailable, shifting is not effective. The Iteration 

vector must be chosen to be orthogonal to the previously computed elgenvectors. When eigenvalues are multiple or clustered, It Is necessary t·o 

reorthogonal Ize the Iteration vector to avoid convergence to mode shapes 

which havEl already been computed. 
149 

Effective elgenproblem solution techniques using a combination of 

polynomial Iteration and vector Iteration have been developed for 

limited applications. In Gupta's method [4.10J the Individual Intervals 

containing each eigenvalue are defined using sturm sequence properties 

and bisection. A shift Is made to the middle of each Interval and then 

'the elgenpalr Is determined by Inverse Iteration and Rayleigh's 

quotient. The upper bound on the Interval for Ai Is used as the lower 

bound for Ai+1• Bathe and Wilson presented a method cal led determinant 

search [4.2J In which accelerated secant Iteration Is used to accurately 

determine the eigenvalue. Inverse Iteration with shifts Is then used to 

find the eigenvector. If the eigenvalue Is accurately evaluated, a 

shift to that value wll I result In the eigenvector being found In no 

more than two vector Iterations. An advantage of these combined techniques Is that when eigenvalues are not multiple or clustered, each 

etgenpalr is independently evaluated with no need for orthogonal Izatlon. 

Robinson and Harris [4.22J Introduced a Newton-Raphson Iterative 

approach to vector Iteration. For iteration "I", the process Involves 

choosing a vector increment, {6x j }, and a frequency increment, 

which wit I el imlnate the residual vector 

{r .} = [K] {x.} - A .[M] {x. }, I I I I 

6'A., I 

(4.23) 

and thus force convergence to a true elgenpalr. As an additional side 

condition, the vector Increment, {6x.}~ must be orthogonal to the a~ I 

proximate eigenvector, {x.}, with respect to [MJ, I.ee 

I 

{X.}T[M]{t:,x.} = o. (4.24) I I 
150 

For [KJ and [MJ of order (nxn), a set of (n+1) simultaneous equations Is 

thEm solved for {6X.} and 6A .• The process is repeated until some conI I 

vergence criterion is satisfied. Upon convergence, estimates of a dlffer-ent elgenpalr are taken and the process Is restarted. Initial 

estimates to the elgenpalrs are taken from another source, such as sub-

, s pclce I ter"at Ion. 

The above method requires a complete triangulation of an (n+1) 

order set of linear equations for each iteration. Lee and Robinson 

[4.17J presented an alternative method which requires only one 

triangulation for each eigenpalr. Although the convergence rate Is less 

than for the Rob I nson-Harri s method, the overa I I computatl ona I eft I cl ency Is Improved. When eigenvalues are multiple or clustered, al I elgenpairs In the group are found simultaneously by a method using Lagrange 

multlpller"s and the statlonar"y property of the Rayleigh quotient. 11-

lustratlvE~ examples show the Lee-Robinson method to require between 36% 

and 60% as many operations as needed by subspace Iteration. Comparison 

of operation counts Indicates that the method Is also more efficient 

them deter"m I nant search. 

4.2.3 ~ uatl on .Qf. E I genprob I em So I ut I on Mrlhruls. 

As with the evaluation of modal synthesis, It Is useful to 

establ Ish criteria for the evaluation of elgenproblem solution methods. 

Although the categories listed below are similar to those previously 

dlscussed p they differ In the application to elgenproblems. 
151 

1. Computational Effort 

The computational effort required by a solution algorithm 

Is most effectively measured by the number of arithmetic 

operations performed. For elgenproblem solutions, this number 

Is dependent upon the order and bandwidth of the matrices and 

the number of elgenpalrs computed. For large problems, Incore solutIons are not always possible. Therefore, efficiency 

Is further measured by the capabl I lty to solve the elgenproblem with the use of out-of-core data storage. Controlling 

parameters are the number of data blocks requIred to simultaneously reside In core and the frequency with which these 

blocks must be replaced. 

2. Applicability 12 General Problems 

Generality of eigenproblem solution methods Implies the 

abilIty to solve problems of the form given by Eq. (4.14). 

Often the transformation to standard form, Eq. (4015), Is 

ineffective because of an II I-conditioned mass matrix. Such a 

situation Is common with a lumped mass formulation tn which 

rotational DOF are assigned zero Inertia. This limitation Is 

avoided when an elgenproblem solution method Is available that 

can solve the general problem. Generality further applies to 

dealing with the various forms taken by the stiffness and mass 

matrices. As shown in Chapter 3, the dynamic reduction 

process can lead to coefficient matrices that are diagonal, 

banded, full, block-diagonal, or bl'ock-diagonal and border 

banded. 

3. ~ Qf R~analysls 

The need for reanalysis of an elgenproblem arises In two 

Instances. First, the analysis may need to be contInued with 

a tighter convergence limit to attain greater precision In the 

frequencies and mode shapes. Secondly, as the physIcal 

characteristics of the model change during nonl fnear response, 

the vibration characteristIcs also change~ When a previous 

eigensolutfon is available, significant cost savings are 

real lzed If the method takes advantage of this Information In 

reanalysis. 

4. Accuracy ~ StabilIty 

As with modal synthesis, stability of the operations and 

accuracy of results of the elgenproblem solution methods are 

vitale Stability problems tend to be caused by mode shape 

vectors that are not orthogonal while accuracy Is controlled 

by the convergence crlterlae 
152 

The superior elgenproblem solution methods from each of the three 

categories are Identified below. The use of these methods and their Interdependencies wll I be discussed In the next section. 

Subspace Iteration emerges as the superior process In the category 

of simultaneous Interatlon. The primary advantage over the Jennings and 

'Rutshauser approaches Is Its applicability to the general problem form. 

The potentially II I-conditioned transformation to standard form Is 

avoided, thus making the method generally more stable. Other advantclges, wh I ch are common to a I I methods In th i s category, I nc I u de the 

ability to use previous modal Information In reanalysis and the ease 

with which acceleration schemes are Incorporated. 

The most efficient solution process of the transformation methods 

ccltegory Is that employing thE! Householder transformation to tridiagonal 

form followed by QR Iteration (the HQRI method), This method Is effectrve only when the elgenproblem can be written In standard form and when 

a~ I elgenpalrs must be calculated. Although ·the method does not take 

advantagE) of the bandform of the equations, th I sis not necessar II y a 

drawback since the conversion to standard form often results In a fully 

PC)Pu I ated coeff I c rent matr I x. Such f 5 the case w hen a cons I stent or a 

synthesized mass matrix Is used. 

When the above conversion cannot be made, generalized Jacobi Iteration makes an attractive cllternatlve to the HQRI process. Although 

J(~cobl Iteration Is not as computationally efficient as other transformc~tlon methods, It has two distinct advantages. First, as the offdiagonal terms become smaller, the process becomes more efficient, I.e. 

convergence Is more rapid on systems that are almost diagonal. 
153 

Secondly, Jacobi Iteration Is the only transformation method which effectively utilizes a previous solution In reanalysis. Rather than 

starting the process with a standard rotation matrix to zero an offdiagonal element, the previous mode shape set Is used to rotate the 

coefficient matrices close to diagonal forme 

In general, as the order of the elgenproblem and/or its bandwidth 

Increases, polynomial/vector Iteration methods become less economical 

because of their reliance on computing the determinant In Eq. (4.4). An 

exception is the method presented by Lee and Robinson [4.17J. In this 

method only one complete triangulatIon Is performed for each etgenpalr 

that Is evaluated. The required starting solution, In the form of a 

prior modal analysis, can be used to advantage In reanalysis. 

As mentioned previously, operatIon counts provide the best evidence 

regarding the efficiency of a solution a/gorlthmo For comparison purposes, Table 401 lists the operation counts for the methods discussed 

above. It Is assumed that no computational penalty Is paid by Implementing the methods with hypermatrlx data structures. 
Table 4.1 

OPERATION COUNTS FOR EIGENPROBLEM SOLUTION 

Method 

Subspace Iteration~1 ~ 

(2) HQRI .... 

General ized Jacobi Iteration~2!3~ 

Lee-Robinson Vector Iteration .. 

Notes 

(1) Assumes 10 iterations requfred 

(2) Assumes fully populated [K] and [M] 

(3) Total for one sweep only 

Number of operations required [4.2, 4.17J 

2 

nm + nm(4 + 2p) + 4np + 20nq(2m + q + 1.5) 

322 O.67n + fO.5n + pn + 9pn 

3 2 3n + 6n 

2 O.5np(m + 5m + 2) + nT(7m + 6) 

Notation 

n order of [K] and [MJ 

m half-bandwidth of [K] and [M] 

p number of eigenpairs computed 

q min(2p, p + 8) 

T number of iterations required 

Vl 

.j:o> 
155 

4.2.4 ChoIce Qf Eigenproblem SQlution Metbods 

It has been demonstrated that a wIde varIety of elgenproblems can 

arise In dynamic analysis by the FEM. As mentioned earlier, no single 

method can be effectively appl led to the solution of each of these 

problems. Therefore, a selection of methods must be available that permits the analyst to choose the one most suited to his problem. The 

three controlling variables In making the choice are the number of DOF 

In the model, the bandwidth of the matrices, and the number of elgenpairs to be computed. 

In general solution methods from the three catagortes are most efficient when ?pplled to a particular class of problem. When all elgenpairs are required of large, fully populated matrices, a transformation 

method should be used. When only a few frequencies and mode shapes are 

needed and the equations are narrowly banded, polynomial/vector Iteration methods are most efflclent. Simuntaneous Iteration exhibits Its 

superiority In the range of problems between the two mentioned above. 

To fulfil I the selection requirements, It Is proposed that the 

superIor methods Identified In the preceding section be Incorporated Into the general purpose FEM system. These methods are: subspace IteratIon, HQRI, generalIzed Jacobi Iteration, and the Lee-RobInson vector 

iteration. Conveniently, several of these methods can be used in concerto That Is, a starting solution for Lee-RobInson vector Iteration 

may be obtained from subspace Iteration. Also, subspace Iteration effectively utilizes generalized Jacobi IteratIon In the solution of the 

reduced elgenproblem. When the transformation to standard form, required by the HQRI approach, Is not effectrve, generalized Jacobi Itera-
156 

tion becomes the next best alternative. 

Although operation counts provide useful Information In choosing a 

solution method, other factors must be included. The avallabll ity of a 

starting or Initial solution may significantly increase the efficiency 

of some methods. Also, the number of iterations required by some 

'methods is highly dependent on the convergence tolerance that Is 

established. These factors force selection of the appropriate eigenproblem solution method to be made more on the basis of the analyst's 

experience and judgement rather than on published operation counts. 
157 

4.3 Solution Qi the EQuations Qi Motion 

4.3.1 Introduction 

Consider the form of the differential equatIon of motion for a 

damped finite element system: 

[M(t)]{u} + [C(t)]{u} + [K(t)]{u} = {P(t)}. (4.25) 

For many problems the coefficient matrices of Eq. (4.25) are constant 

with respect to time, resulting tn a set of coupled I inear differential 

equations. When material and/or geometric noniinearities do arise, they 

are most often I imtted to the stiffness matrix, [K(t)]. The mass of the 

structure seldom varies with time and the damping characteristics 

generally are not sufficiently wei I understood to warrant any time 

dependent change In the modal damping ratios. 

When multilevel substructured models are used and the coeffiCient 

matrices are derived by modal synthesis, the mass and damping matrices 

can also become time dependent. These dependencIes enter the synthesized mass through the coordinate transformations performed on each 

condensed nonlinear substructure. Likewise, the relationship between 

damping and the structure mass (see Section 4.4) causes a corresponding 

variation tn the damping matrix. Therefore, In Its most general form 

Eq. (4.25) has all three of its coefficient matrices varying with time. 

The tranSient response of a substructured model Is described by the 

displacements {u}, velocities {u}, and accelerations {u} for the DOF tn 

the highest level structure. Two general strategies are available for 

obtaining this solutIon. They are mode superposItion and time-history 
158 

In·~egratlon. Both strategies are fundamental to the dynamic analysis of 

structures and as such are famll iar to most analysts. 

The selection of a particular solution strategy is highly problem 

dependent. The decision variables include the degree of nonlinearity 

(If any), the number of modes participating in the response, the length 

of the time Interval over which the response must be evaluated, and the 

nature of the transient load. A discussion of these variables and of 

the effects of multilevel substructured model ing on the solution 

procedures is presented In the fol lowing sections. 

40:5.2 MQsJ.~ SuperpositIon 

Mode superposition is an economical approach to solving the equatl,ons of motion when the structural response Is linear and Is limited to 

the lower modes of the frequency spectrum. For linear systems, the 

basic approach Is to uncouple the equations by a transformation to modal 

coordinates. The resulting set of equations descrIbe the dynamic 

response of the structure In each mode when excited by the corresponding 

modal component of the appl led load. These modal equations are solved 

Individually by a direct Integration method and then the total response 

Is computed as the sum of the modal responses. 

A computatIonal advantage with modal superposItion over timehistory integration methods (see Section 4.3.3) is realized when the 

domInant portIon of the structural response Is contained In the lower 

modes. This characteristIc Is evident when the loading function has 

very little high frequency content. Examples of such loading types 

might Include earthquake excitation and mechanical vibrations. The 
159 

number of modes that participate In the response and thus must be Included In the modal analysis Is easily estimated by examining the 

loading function after It has been transformed to modal coordinates. 

Loadings that typically have more high frequency content Include blast 

or shock loading and Impact. These later examples are more economically 

,analysed by time-history Integration of Eq. (4.25). 

The extension of mode superposition to nonlinear systems has 

received only limited attention In the literature. Morris [4.18J has 

appl led the concept to cable systems and framed structures. His results 

are not particularly encouraging~ The method performed poorly when applied to structures experiencing more than Just mild nonllnearities. 

Nickel I [4.21J presented a more complete development of nonlinear mode 

superposition. Although mode superposition performed very wei I In his 

example problems, It showed no particular numerical advantage over 

direct Integration schemes. Nickel I does note that some physical insIght Into the behavior of the structure Is gained by observing the 

changes In the modal spectrum as deformation proceeds~ 

The procedure for solving Eq. (4.25) by mode superposition remains 

unchanged when multi level substructured models are IntroducedQ The 

transformations performed on the coefficient matrices are Independent of 

the methods used to build those matrices. Although both geometric and 

modal DOF from the condensed substructures may exist In the structure 

stiffness, damplng 6 and mass matrices, no complication Is expected In 

making further transformations to uncouple the equations of motion. In 

fact, the transformations performed in mode superposition are qUite 

similar to those used In modal synthesis. However, no studies have been 
160 

located that document the effects of substructurlng on the mode superposition solution strategy. 

4.3.3 ILrrte.-History IntegratIon. 

When the dynamic loading on a structure excites many modes ot 

,vIbration, time-history Integration, also known as temporal or direct 

Integration, Is often a more economical solution procedure than mode 

sllperposltlon. The Integration operators work directly on the coupled 

equations of motion without regard for the modal content of the 

re1sponse. 

Oper'ators for time-history Integration vary In the relationships 

between the known and the unknown solution variables. These variables 

ar'e the dIsplacements, velocities, and accelerations of the structure 

DOF at various points in time throughout the response period. The assumed relationship between the known and unknown solution variables 

defines whether the operator Is explicit or Implicit. In explicit 

mElthods [4.2J, the displacements at the end o·f the Interval, {ut+MJ, 

are based on the equilibrium conditions at the start of the Interval: 

Eq. (4.25) evaluated at time t. After a starting solution has been 

established, It Is possible to progress through the solution without 

solving a set of simultaneous equations. This feature of explicit 

meithods presents an advantage in computatl ona I eft I cl ency over the Imp I I cit mEithods discussed be I ow. However, sInce exp I J c J t methods are extrapolatory In terms of the satisfying equilibrium, they are only conditional Iy stable [4.15J at best. A stable Integration operator Is one 

fClr which the error Introduced In each time step Is bounded for the 
161 

chosen step size and thus the total error does not grow without bound as 

the solution progresses. A conditionally stable operator Is one which 

remains stable for only a limited time step size. The central dlfference method [4.9J, when used with a diagonal mass matrix, Is considered to be an accurate and economical expllcft solution method 

. [4.16J. 

Implicit operators rely on the solution of Eq. (4.25) at time 

t + 6t to obtain the solution {u t

+

6tJ• Two significant consequences 

result. First, a set of linear equations must be solved at each Increment In time considered In the solution. Seeondlyp It Is possible to 

develop Impl lelt operators that are unconditionally stable (stable for 

any size of time step 6t). The two most popular Implicit operators are 

the Newmark-B method [4.19J and the WI Ison-8 method [4.26J. 

The Implicit operators which exhibit unconditional stability when 

used on linear problems may not always maIntain this feature when 

general nonlinear response occurs. Proofs of unconditional stability of 

the Newmark-B operator, when appl fed to nonl inear problems g have been 

presented [4.4 & 4.7J but contradictory evidence has also been 

documented [4020J. The question of unconditional stab I I tty is not 

necessarllya drawback to the use of lmpliclt (or even explicit) 

operatorse For nonlinear analYSis relatively smal I time steps (which 

lead to small displacement changes) must be taken to accurately detect 

and Incorporate nonlinear behavior. As a result, frequent equi I Ibrium 

Iterations with updates of the coefficient matrices wll I be required to 

maintain accuracy, regardless of which Integration operator Is employed. 

This process can succeed In arresting any Instability that may develop 
162 

from using a particular Integration operator [4.20J. 

Explicit Integration of the equations of motion has proven to be 

effective for nonlinear problems that do not Involve substructured 

models. Since the frequent equilibrium Iterations prevent Instability 

In the solution, an implicit operator that may be unconditionally stable 

. wll I not show any advantage over explicit operators. Further, It Is not 

always necessary to use the structure stiffness matrix with explicit 

oporators to compute the Internal nodal forces. Computation of the 

elastic force vector can be performed directly by Integration of the 

following expression for each element In the structure [4.3J: 

In which {F } = structure elastic force vector, e 

[LJ = connectivity matrix for the element, 

[8J = strain-displacement relation, 

{cr} = element stress vector, 

v = volume of the element, and 

n = number of elements In the structure. 

(4.26 ) 

In substructured models the element stress vector Is not available 

for condensed lower level substructures. As a result, computation of 

th,~~ e I astl c force vector must I ncorporate at I east a portion of the 

structure stiffness matrix. The resulting elastIc forces are 

= {F }H + [KJ {u} e c c (4.27) 

where {Fe} = structure elastic force vector, 

{Fe}H = Internal forces for uncondensed elements at the 
163 

highest level, computed by Eq. (4.26), 

[K] = stiffness matrix for the adjoining condensed 

c 

substructures, and 

{u} = condensed DOF displacement vector. 

c 

Computation of elastIc forces by Eq. (4.27) requires that the stiffness 

matrix for the condensed substructures be brought Into memory during 

each tIme step. It Is yet unknown whether this requirement wll I reduce 

the computatIonal effectIveness of explIcit Integration of the nonlinear 

equations of motion. 

Since Implicit methods require equation solving wIth the ful I 

structure stiffness matrix, no complications in the algorithms due to 

the use of substructured models are antIcipated. 

4.4 AddItional ComputatIonal Cgnslderatlons 

Several minor computational details regarding the dynamic analysis 

of structures are complicated by the use of substructured fInite element 

models. Details consIdered In this study Include mass matrix formulation, damping, and the use of experimental data~ 

Two approaches are used to assemble the mass matrix of a substructure containing only simple elements [4.6J. The most elementary approach is the lumped mass formulation. With this approach, the mass of 

each element Is lumped at the nodal points. Upon assembly, a diagonal 

substructure mass matrix Is achieved. This form has many computational 

advantages as prevIously noted. The alternate approach Is the conslstent mass formulation. In thIs approach, accelerations wIthin the element are interpolated by the same shape functions used to describe the 
164 

element displacements. Thus, the assembled substructure mass matrix has 

the samE3 banded nature as the substructure stl ffness. Regard I ess of 

Initial formulation, when multi level substructurlng Is Introduced, the 

resulting structure mass matrix Is of banded form. It Is anticipated 

that as the number of levels of substructurlng Increases, the differences between the lumped and consistent formulations wll I diminish. 

The reduction process used, through the various transformations performed, is expected to mask the original mass matrix formulation and 

thus make the two formulations Indistinguishable. 

Structural damping Is a phenomenon which Is not easily modelled by 

analysts. Usually, damping can only be evaluated byexperlmeni"al 

tE:lstlng thus making Its specification for a general finite element model 

dl ff I cu lot. However, there are procedures des I gned to I ncl ude the MfC:lcts of damping In a computationally efficient manner [4.6J. Normally, 

mQdal d;3mping ratios are known (or selected) and used directly In computing structural response In modal coordinates. If it is desirable to 

h;ave an exp I I cl t dampl ng matr Ix In geometr i c coordl nates, one can be 

formulatc3d which Is proportional to both the stiffness and mass 

miatr Ices. In th I s form, termed Ray I e I gh damp I ng, the system v t bratl on 

mode shapes are orthogona I to the damp I ng as we II as the mass and stl ffnc:lSS matrices. In substructured models, application of damping Is 

dC:llayed until the equations of motion for the highest level structure 

a~'e formu I ated. The methods of Guyan reducti on and moda I synthesis do 

not provide for damping at the substructure level. 
165 

Pre-existing experimental data regarding the free vibration characteristics of a substructure could be useful for reducing the computational expenses of a finite element analysis. It Is a simple matter 

to adapt input translators to accept this data and thus el imlnate unnecessary vibratIon analyses. An Important consideration In using experimental data Is the consistency In boundary restraint between the 

test fixture and the finite element model. 
166 

4.5 fuU..er:ences 

4. 1 

4.2 

4.3 

4.4 

4.5 

Bathe, K.J., and Ramaswamy, S., "An Accelerated Subspace 

Iteration Method," Computer Methods...Lo. Appl jed Mechanics .a.ru1 

Engineering, Vol. 23 pp. 313-331 (1980) 

Bathe, K.J., and Wilson, E.l., ~ Methods In flnJ~ 

Element Analysis, Prentice Hal I, 1976 

Belytschko, T., Chlapetta, R.l., and Bartel, H.D •• "Efficient 

large Scale Nonlinear Transient Analysis by Finite Elements," 

International Journal.fQc Numerical Methods In. Englneerln.g, Vol. 10 pp. 579-596 (1976) 

Belytschko, T., anel Schoeberle, D.F., "On the Unconditional 

Stabl I Ity of an Implicit Algorithm for Nonlinear Structural 

Dynamics," Journal QL Appl led Mechanics Vol. 97 pp. 865-869 

(1975) , 

Clint, M. and Jennings A "The Fvaluatlon of Eigenvalues and Eigenvectors of ~eal'Symmetrrc Matr ces by Simultaneous 

Iteration," lli CQmputer Journal, Vol. 13 pp. 76-80 (1970) 

4.6 Clough, R.W. and Penzien, J., Dynamics Qi structures, McGraw 

H II I, 1975 

4.7 Corr, R.B. and Jennings, A., "A Simultaneous Iteration Algor I thm for Symmetr-I c E I genva I ue Prob I ems." I nternat I onal 

Journal fQc Numerical Methods In Engineering Vol. 10 pp. 

647-663 (1976) 

4.8 Dong, S.B., Wolf, J.A., and Peterson, F.E., "On a DlrectIterative Elgensolutlon Technique." International Journal ioc 

Numerical Methods 1.0. Engineering, Vol. 4 pp. 155-161 (1972) 

4.9 Goudreau, G.l. and Taylor, R.l., "Evaluation of Numerical IntegratIon Methods In Elastodynamlcs," CQmputer Methods .Lo. 8jLpiled MechanIcs M9. fnglneerlng, Vol. 2 pp. 69-97 (1972) 

4.10 Gupta. K.K., "Elgenproblem Solution by a Combined Sturm Sequence and Inverse Iteration Technique." International 1Qwc-

~ for Numerical Methods in EngIneering, Vol. 7 pp. 17-42 

( 1973) 

4.11 Hughes, T.J.R., "A Note on the Stabl Iity of 

gorlthm In Nonlinear structural Dynamics," Journal .f.Qr: NumeriCaL Methods III .Englneerlng, 

383-386 (1977) 

Newmark's AIInternational 

Vol. 11 pp. 

4.12 Jennings, A., "A Direct Iteration Method of Obtaining latent 

Roots and Vectors of a Symmetric Matrix." Proceedings Qi 1M 

Cambridge Phi 10sophlcalSocIety, Vol. 63 pp. 755-765 (1967) 
167 

4.13 Jennings, A. and Orr, D.R.L., "Application of the Simultaneous Iteration Method to Undamped Vibration Problems," 10.-

ternatlonal Journal fQc Numerical Methods In Engineering, 

Vol. 3 pp. 13-24 (1971) 

4.14 Johnsen, T.L., "On the Computation of Natural Modes of an Unsupported Vibrating Structure by Simultaneous Iteration," 

Computer Methods In Appl led Mechanics and Engineering, Vol. 2 

pp. 305-322 (1973) 

4.15 Krieg, R.D., "UnconditIonal Stabll lty In Numerical Time Integration Methods, Journal of APDI led Mechanics, Vol. 40 pp. 

417-421 (1973) 

4.16 Krieg, R.D. and Key, S.W., '~ranslent Shel I Response by 

Numerical Time Integration," International Journal .iru:. 

Numer I ca I Methods In Eng I neer I ng, Yo I. 7 pp. 273-286 (1973) 

4.17 Lee, I-W, and Robinson, A.R., "Solution Techniques for Large 

Eigenvalue Problems in Structural Dynamlcs," TechnIcal Report 

H2. lllLll ~~-~, Dept. of CIvil Engineering, Unlv. of 

II I., Urbana, (1979) 

4.18 Morris, N.F., "The Use of Modal Superposition In Nonlinear 

Dynamics," Computers.9.!lJ.t Structures, yol. 7 pp. 65-72 (1977) 

4.19 Newmark, N.M., "A Method of Computation for Structural 

Dynamics," Joyrnal .Qf the EngIneerfng MechanIcs DivIsion 

(ASCE), Yol. 85 pp. 67-94 (1959) 

4.20 Nickel I, R.E., "Direct Integration Methods in structural 

Dynamics," Journal .Qf fu EngIneering MechanIcs DlylsIon (ASCE), Yol. 99 pp. 303-316 (1973) 

4.21 Nickell, R.E., "Nonl inear Dynamics by Mode Superposition," 

Computer Methods in ApplIed Mechanics ~ Engineering, Yol. 7 

pp. 107-129 (1976) 

4.22 Robinson, A.R. and HarriS, J.F., "Improving Approximate 

Eigenvalues and Eigenvectors," Joyrnal .Q.f. the EngineerIng 

Mechanics DIvIsion (ASCE), Yol. 97 pp. 457-475 (1971) 

4.23 Rutlshauser,. H., "ComputatIonal Aspects of F. L. Bauer's 

SImultaneous Iteration Method," Numerische Mathematik, yol. 

13 pp. 4-13 (1969) 

4.24 Weaver, W. Jr. and Yoshida, D.M. '~he Eigenvalue Problem for 

Banded Matrices," Computers lli structures, Vol. 1 pp. 

651-664 <1971> 

4.25 Wilkinson, J.H., The AlgebraIc Eigenyalue Problem, Oxford 

University Press, 1965 
4.26 

168 

WI I son, E. L., Farhoomand, I., and Bathe, K. J., 

DynamIc AnalysIs of Complex Structures," 

Engineering and ~~ctural Dynamics, VOl. 

(1973 ) 

"Non I I near Earthquak~ 

pp. 241-252 

4.27 Wylie, C.R., Adyanced EngIneering .M53thematlcs, 4th EditIon, 

fl-lcGraw Hili, 1975 

4.28 Yamamoto, Y. and Ohtsubo, H., "Subspace IteratIon Accelerated 

by Using Chebyshev Polynomials for Eigenvalue Problems with 

Symmetric Matrices," InternationaL Journal .fQ.!:. Numerical 

Methods In EnglneerlrLq, Vol. 10 pp. 935-944 (1976) 
CHAPTER 5 

MATRIX FORM OF NONLINEAR CONTINUUM MECHANICS 

5.1 Introduction 

The dynamic solution procedures described in the preceeding chapters are independent of the continuum theory used to derive the governi ng non I i near equat ions of mot I on. Th is chapter focuses on the essential aspects of nonl inear continuum theories. This information is inc I uded in the report for three reasons: (1) the formu I at ions are not 

readily available in matrix notation famil iar to finite element workers, 

(2) the exact transformations in matrix form wi I I be required during the 

implementation phase, and (3) a relative assessment of efficiency of the 

formulations is desired. The finite element formulations are developed 

and contrasted as to their suitability for substructured dynamic analysis in the next chapter. 

Initial work on a matrix formulation for nonl inear continuum theory 

was conducted by Nayak [5.1 J. Zienkiewicz (5.2J has been the primary 

advocate of this approach which has been adopted by most researchers at 

Swansea University. This chapter summarizes their work and presents an 

extension of the approach to include an Updated Lagrangian formulation. 

Nonl inear behavior includes that due to constitutive relationships 

(material nonlinearity) and strain-displacement relationships (geometric 

nonl inearity). To achieve maximum generality in the discussion, structures are assumed to exhibit both types of nonl inearity simultaneousI y. The discuss i on of mater i a I non I i near i ty Is directed towards meta I 

plasticity at both infinitesimal and finite strain magnitudes. Whl Ie It 

169 
170 

is recognized that strain-rate dependent behavior may be of importance 

in dynarnic ana!ysis, these effects are not considered in this discussion. Few procedures in the I iterature include strain-rate dependencies, croeep and thermal dependencies of material properties. Most 

often, rate effects are incorporated into the basic concepts of incremental plasticity and thus do not affect the comparison of non! inear 

formulations. 

The dynamic analysis of structures in which nonl inearity is limited 

to the rnaterial stress-strain behavior presents no campi ications in formulating the equations of motion. This follows as all quantities are 

referred to the initial configuration of the structure, with displacements and rotations assumed to be infinitesirnal. The challenge is devE~loping real istic constitutive models for the material behavior under 

cycl ic loading. Geometric nonl inear effects are important when the 

structure undergoes large rotations and/or finite strain magnitudes. 

This occurs, for example, in the following situations: crashworthiness 

of aircraft structures and components, meta! forming processes, necking 

of structural components subjected to tensi Ie overloads, geometry 

changes under service loading that affect performance (e.g., turbine 

blades) and the I oca I I zed deformat i on of mater i a lin the vic i n i ty of 

stress concentrations -- especially notches and cracks. 

The governing equations of nonl inear solid mechanics are summarized 

in the "fo I low i ng sect Ions in terms of Cartes i an reference axes. The 

notat I on adopted here I s a mod i f i cat i on of that first used by Ba-rhe 

{5.3J. Each quantity retains the same symbol throughout the deformation 

process. Su perscr i pts and subscr i pts are used to i nd i cate the con f i gur"at i on of the structure in wh i ch the var i ab I e occurs and to wh i ch con-
171 

figuration it is referenced, respectively. A sing I e I eft subscr i pt 

impl ies an increment of the variable over time ilt with the subscript 

indicating the reference configuration. A summary of the notation is 

presented in Section 5.9. 

All equations are derived using matrix, rather than tensor, notation. The theory of finite deformation is elegantly expressed in tensor 

notation. However, all tensor operations require transformation to 

equivalent matrix form for computer Implementation. This, combined with 

the exclusive use of matrix notation in finite element research, encourages the adoption of matrices at the start. The reader already faml! lar 

with tensor notation wi I I readily recognize the appearance of symmetric 

strain and stress "matrices." 

The following sections contain discussions of coordinate systems, 

strain-displacement relations, stress definitions, and virtual dlsplacements for fin i te deformat Ions. Part i cu I ar emphas lsi s p I aced on the 

finite deformation strain and stress rates that are required for virtual 

work arguments. Two nonl inear formulations arise in the discussion. 

These are termed Total Lagrangian (T. L.) and Updated Lagrangian (iJ. 

L.). In the T. L. formulation al I quantities are referenced to the configuration at time t=O. The difficulty arises with constitutive 

relations for the material which are naturally expressed in terms of the 

deformed configuration, e.g., true stress and true strain. The correct 

transformations of strain, stress, strain rates, stress rates, and 

incremental constitutive relationships between the initial and 

instantaneous configurations are derived in this chapter. The U. L. 

formulation refers all quantities at time t+M to the configuration at 

time t. Numerous attempts have been made to derive an U. L. formulation 
172 

for finite element analysis. Unfortunately, very few have fol lowed the 

rigorous equations of continuum mechanics; most formulations are derived 

by assum i ng certa in non I i near terms are sma I I dur i ng the mot i on from t 

to t+t.t. This has led to considerable misunderstanding in the I ii-er-

(~ture. 

chapter .. 

The exact U. L. formulation is rigorously derived in this 

Only then are several commonly used simpl ifying assumptions 

descr i bE3d. 

An extensive discussion of constitutive models for finite deformation elasto-plasticity is included in this chapter. The var ious 

strain and stress rate definitions and their transformations are 

presentE3d in matrix notation. This material wi II enable the real istic 

mode I i nq of large deformat i on effects that occur near impact zones. It 

Is anticipated that the use of substructuring prior to a transient analysis wi i I show a significant computational advantage in such problems. 

ThE~ chapter concludes with a qual itative assessment of the computation efficiency of the two major formulations. 

~5.2 Coordinate Systems and .Transformations 

The configuration of the body is considered at three times; namely, 

0, t, and t+Ll t. For stat i c ana I ys is, the parameter t may be assoc i ated 

with a loading state rather than time. Motion of the body is described 

with respect to fixed Cartes i an axes hav i ng un it vectors {i }. The 

position vector of any material point at the three times is denoted 

o{x}, t{x}, and t+Llt{x}. Coordinates o{x} remain fixed for a particular 

material point and thus are convected in the present usage. No 

subscripts are necessary for the position vectors and their 

corresponding differentials. 
173 

o Let {dx} represent the components of an in iti a I 

segment,Ods, which deforms into line segment tds at time t. 

nents t{dx} then are given by 

In which t[JJ Is the deformation jacobian with terms defined by o 

t t 

O

J.

1j

• = (x •. ) 

o I,J 

line 

Compo-

(5.1) 

(5.2) 

The subscript outside the expl icitly Indicates the reference 

configuration for differentiation. The deformation jacobian fully 

characterizes motion In the dIfferential neighborhood of a point that is 

displaced o t from P to P. Geometric considerations 

differential volume and area changes may be expressed as 

[5. 11 show that 

(5.3) 

(5.4) 

in which °dV is a differential volume and o{dA} is the vector of 

components for a differential area °dA, both at t=O. Vertical bars are 

used to indicate a determinant. 

The position vector t{x} is written in terms of a displacement 

t vector {u} with components referred to base vectors {i} as 

tot {x} = {x} + {u} 

The deformation Jacobian may then be written in the form 

(5.6) 
174 

Where: [j J is termed the d i sp I acement Jacob i an and [I J is an i dent i ty 

matrix. Terms of :rj) are defined by 

t. 

oj i 1 j :: 

t ( u. . ) o l,j (5.7) 

Equations (5. 1 ) (5.7) provide the basis of strain-stress 

transformations required in the T. L. formulation. 

The finite magnitude changes in volume and area of differential 

elements that occur during the motion from t to t+t:.t may be measur'ed 

with respect to the configuration at time 0 or t. First, the configuration az t+t:.t is expressed in terms of the configuration at t and i-he 

displacement increment (deformation plus rotation). No restrictions are 

imposed on the magnitude of the displacement increment during t:.t. In 

analogy with (5.1) an expression of the form 

(5.8) 

is sought in which t+t:.~[J) is the increment of the deformation Jacobian 

over t:.t but measured with respect to the configuration at time t. The 

terms of the Jacobian increment are given by 

tHtJ 

t ij 

L{~t t {t.u} represent the i ncr-ementa I d i sp I acement vector: 

L~·t.u, tt.v, t;J during motion from t to t+t:.t. Then 

t+~t (J .. ) 

t IJ 

t t 

:: t ( x. .) + t ( ~u. .) I ,J I , J 

More simply, (5.10) may be written 

t+t.~[Jl :: [I I + t+t.~[Vl 

(5.9) 

(5.10) 

(5.11) 
175 

in which t+6~[vJ represents the mean displacement gradient with respect 

to the current (t) configuration (or equivalently the mean velocity 

gradient if divided by 6t). Increments of differential volumes and 

areas from t to t+6t may be simi larly expressed in analogy \'lith (5.3) 

and (5e4) as 

(5.12) 

(5.13) 

Equations (5.8) - (5.13) provide the basis for transformations required 

in the U. L. formulation as first presented by Yaghmai and Popov 

[5.4). The U. L. formulation is generally attributed to Bathe [5.3, 

5.5J due to the numerous papers he has publ ished on the formulation. 

Variables for the configuration at t+6t may also be related to that 

at t=O through a sequence of transformations -- first from t+6t to t, 

then from t to t=O. Consider first that at t+6t, 

Subtraction of (5.1) from the above expression yields 

000 

[6JJ o{dx} = [ t+6t[JJ _ t[JJ ] o{dx} 

(5.14) 

(5.15) 

in which [6J] Is the incremental deformation Jacobian but referred to 

o 

the configuration at t=O. In (5.8), t+6~[Jl is the incremental deformation Jacobian referred to the configuration at time t. USing the 

t incremental displacement vector {6u} over the time increment 6t, the 

terms of [6Jl are 

o 

(6J .. ) o IJ 

t t t 

= o( x·

1 

+ 6u.) .. - (x .. ) I I ,J 0 , ,J (5.16) 
176 

or, more simply 

(~J .. ) = 

o IJ 

t (t:. u) 0 0 

o I,J (5.17) 

Using the chain rule, the derivatives in (5.17) with respect to o{x} are 

transformed to those with respect to t{x}. The operations readi Iy show 

that the result may be written as 

Substitution of (5.18) into (5.15) shows that 

t+L~t{dx} = [ [II + t+t:.~[VI ] t[JI o{dx} 

and since t{dx} = t[JI o{dx}, the above expression simpl ifies to 

o 

(5.18) 

(5.19) 

(5.20) 

which is the same result obtained in (5.8). This procedure illustrates 

the multlpl icative decomposition of deformation that parallels simple 

coordinate transformation (which must be the case when the displacement 

gradients represent a rigid rotation). 

Consider now the imposition of virtual, rather than finite, displacement increments on the configuration at time t. Corresponding 

vi rtua I changes in d i f ferent i a 1 areas and vo I umes are etes ired. Given 

the d i ffE~rent i a I vo I umes and areas at time t, the express ions prev i ous I Y 

dE~rived for finite changes during t:.t are reo-examined for the limiting 

case as t:.t + 0, i.e., the virtual displacement occurs during a vanishing increment of the time (or loading) parameter. Introducing the first 

variational operator,~, the variation of I ine segments is given by 

(5.21) 
177 

in wh i ch terms of the var i at i on of the deformat i on Jacob i an appear i ng 

above are defined by 

t t t oJ .. = (ox .. ) = (ou .. ) o I J 0 I ,J 0 I,J (5.22) 

since the variation of the initial coordinates, o{x}, is zero under the 

imposed virtual displacement. t If the virtual displacements, Iou}, are 

interpreted as occurring over a time dt, the derivatives in (5.22) may 

be considered virtual velocity gradients. Using the chain rule, the 

virtual displacement gradients are referred to the configuration at time 

t with the resulting form 

(S.23) 

in which the variational form of the displacement gradient matrix 

defined in (S~ 17) and (5.18) is used. Substitution of (5.23) into 

(5.21) provides the variation of the deformation Jacobian with respect 

to the current configuration in analogy with (5.8) 

but t[J] o{dx} is simply o 

which impl ies that 

~[OJ 1 

t {dx}, thus the above slmpl ifles to 

(5.24) 

(5.25) 

(S.26) 

Using Euler's theorem for homogeneous functions, it is shown that the 

variation of the deformation Jacobian determinant is given by 

(5.27) 
178 

in wh i ch Tr ( ) denotes the trace of the matr i x. It is customary to 

express the virtual displacement gradient matrix, ~[8Vl, as the sum of a 

t symmetric deformation matrixp t[oel, and an anti-symmetric spin matrix, 

t t[ow], as 

~[Ov] t 

= t[oe] + ~[OW] (5.28) 

in which 

t 1 t [oe] = - ~[ [ov] + [ov] T ] 2 (5.29) 

cmd 

t 1 t[owl = - ~[ [ov] - [ov] T ] 2 (5.30) 

Eloth thel deformat i on and sp i n components are I i near in the vi rtua I d i st placement derivatives. Biot [S.6l has demonstrated that t[ow] 

r'epresents the rotat i on of rig i d-convected axes attached to po I nt t p 

during the sma I I (infinitesimal) virtual displacement. Similarly, 

t 

t[ oe] is shown to constitute the pure deformation caused by the virtual 

cllsplacElments. A null deformation matrix Is therefore a necessary and 

sufficient condition for a rigid virtual displacement. The 6 x 1 vector 

t representat I on of the symmetr I c deformat I on matr I x I s denoted t {oe} 

(shear terms are doubled to form the vector representation). 

Using the above procedure, the fol lowing relationships are derived 

for the U." L. formulation. 

(5.31) 

(5.32) 
179 

These relationships are analogous to (5.21> and (5.23). The vi rtua I 

displacement gradients at time t+~t may again be expressed as the sum of 

. t+~t a symmetric deformation matrix, t+~t[oel, and an anti-symmetric spin 

matrix, ~:~~[OWJI as in (5.28-5.30). 

5.3 Strain-Displacement Relations 

Stra i n measures va I I d for arb I trar i I y large deformat Ions are presented in this section. Corresponding expressions for the variations of 

these measures are der i ved for subsequent use in vi rtua I work equations. The strain measures and variations are written in terms of the 

configuration at times 0 and t corresponding to the T. L. and Uo lo 

approaches. 

The measure of finite deformation Is taken as the difference in the 

squared lengths before and after deformation of a line segment (ds) in 

the infinitesimal neighborhood of a pointe 

(5.33) 

The deformation may be written in terms of the initial line segment como ponents, {dx}, using the deformation Jacobian to yield 

(5.34) 

t The symmetric, Green strain matrix, [eJ 6 appearing above is defined by o 

(5.35) 

Alternatively, the strain may be referred to the final I ine segment 

t components, {dx}, using the inverse deformation Jacobian in the form 

td 2 0d 2 s - s (5.36) 
180 

The symmetric Almansi strain matrix, ~[E:l, appearing above is defined by 

( 5.37) 

in wh i ch 

(5.38) 

Finally, it is verified by direct substitution that the Green and 

Almansi strains are related by 

(5.39) 

The Almansi strain matrix is often referred to as the "true" finite 

stra in measure since the deformed con f i gura"r ion prov i des the reference 

state. When t [J 1 represents a rig i d rotat i on, the trans format i on in 

o 

(5.39) is recognized as a simple change of reference axes. 

The symmetric strain matrix may be written in 6xl vector form 

t t 

as o{e} and t{e} for the Green and Almansi definitions respectively. 

Shear stra i ns from the matr" i x def i nit i on are doub I ed in the vector 

representation. The Almansi to Green strain transformation in (5.39) is 

written in vector form as 

in which the terms of the transformation matrix t[Tl are given In "the 

o 

appendix to this chapter (Section 5.9). 

Both the Green and Almansi strain components correctly describe the 

finite deformation due to a nonhomogeneous displacement field within the 

infinitesimal neighborhood of a point op that displaces to tp during the 

mot ion. The Green stra in components are emp loyed in both the T. Lo and 
181 

U. L. formulations. The incremental form of the Almansi strain is used 

to develop the material constitutive models. 

It is usefu I to separate the AI mans i stra in into I i near and nonI inear components by first writing 

~[Jl = [I I - ~[jl 

in which 

t

t(j .. ) = 

IJ 

t 

t ( u. .) I ,J 

Substitution of (5.41) into (5.37) yields 

(5.41) 

(5.42) 

(5043) 

The I i near d i sp I acement grad i ent terms are now obv i ous and are denoted 

by 

The matrix ~[el is of 'ten 'termed 'the linear "'true" s'trainc The 

simi larity of the above expression with (5e29) is noted. If a virtual 

Almansi strain is derived from a virtual displacement field, the linear 

term of ~[oel is simply ;[oel~ the virtual deformation matrixc 

In the U. L. formulation it is necessary to define the strain at 

time t+~t with respect to the configuration at time t. By the same 

procedure used to develop (5033), the deformation increment Is given as 

(5.45) 

Substitution of (5.8) for t+At{dx} in terms of the Incremental 

deformation Jacobian yields 
182 

(5.46) 

in which 

t+LI tt [ €) = 1 (t+6 t [ J 1 T t+6t [ J 1 - [I 1 ) 2 t t (5.47) 

rE~presents the symmetric Green strain matrix during the motion from t 

to t+6 t but referred to the con f i gurat i on a'~ time t. The 6xl vector 

form of th i s stra i n is denoted t+6t{ } t g • Th i s express i on Is exact; no 

restrictions are placed on the magnitude of the displacement increment 

t{.~u} • 

The strain defined in (5.47) cannot be added to that defined in 

(5.35) to determine the Green strain at t+6t referred to time O. 

However, using (5.8) it is easi Iy shown that 

(5.48) 

or' in vector form 

(5.49) 

This transformation impl ies that, relative to the initial configuration, 

the strain increment employed in U. L. 15 in actuality an Almansl strain 

increment referred to the configuration at t. 

When the body is subjected to a virtual displacement field, a 

corresponding virtual strain field is produced. Application of the 

first variational operator, 0, on the Green strain matrix of (5.35) 

yields 

(5.50) 
183 

Substitution of the variation of the deformation Jacobian from (5.23) 

permits the above equation to be written in the form 

(5.51) 

which may be simpl ified using (5.28) to provide 

(5.52) 

The Green strain rate matrix measured with respect to the initial 

configuration, is symmetric and is given by the transformed, symmetric 

deformation matrix ~[oel (whlch is the linear portion of the virtual 

Almansl strain). Since ~[oeJ vanishes under a rigid virtual displacement, the virtual Green strain also vanishes. Using vector forms of the 

virtual quantities p the above transformation Is written as 

Inverting the above relationship yields an expression for the 

deformation vector 

(5054) 

Using a differential' operator matrix which acts on the virtual 

displacement vector, the virtual Green strain vector is easily evaluated 

as 

t{oe:} = 

o (5.55) 

The terms of th i s operator matr I x are def I ned in the append i x to th i 5 

chapter. 
184 

For the U. L. formulation, the virtual si-rain at time t+f.t referred 

to the configuration at time t is required. Appl ication of the variational operator on (5.47) followed by a SUbstitution for the variation 

of the Incremental deformation Jacablan from (5.32) yields 

(5.56) 

Use of the vector form of the deformat i on and stra In matr I ces perm I ·~s 

the above transformation to be written as 

t+f.t{O€} = t+f.t IT) t+f.t{o} (5.57) t t t+f.t e 

In which t+t.;IT] is defined In the appendix. An expression analagous to 

(5.55) for U. L. Is described In Chapter 6. 

5.4 Stress Measures and Rates 

The most common stress measures emp I oyed in fin i te deformat I on 

th'90ry and the i r correspond I ng rates are descr I bed In th I s sect I on. In 

thl:l defor'med conf I guratl on, the "true" or Cauchy stress components 

provIde the natural measure. As In classIcal elasticIty theory, these 

components are def I ned by cons I der i ng the equ I I i br I um of a different I a I 

tetrahedron extracted from the body at any time t and acted upon by the 

d I Herent'i a I force vector t {dF}. The tetrahedron has three surfaces 

par"allel to the fixed reference axes. The resulting stress matrix, 

t denoted tcrJ, is symmetric In the absence of body couples due to the 

orthogonai Ity of the reference planes. 

A I ter-nat I ve I y, stresses act i ng on the undeformed areas 0 {dA} may 

be defined as those resisting the same differential force vector t{dF}. 
185 

Using (5.4), the stresses can be transformed from deformed to undeformed 

areas such that of the 

in which t rol is the 1st Piola-Kirchoff or Lagrange stress. This matrix 

o 

is in general nonsymmetrlc since the reference areas on which the 

stresses act are not orthogonal. A symmetric stress matrix is obtained 

t if the actua I force vector, {dF}, is cons i dered to be def i ned by a 

transformed pseudo force vector o{dF}. Force o{dF} is assigned a transformation to t{dF} in the t 

same manner that {dx} transforms to 

(5.59) 

Substitution of the transformed force vector into (5~58) yields 

(5.60) 

in which t[SJ represents the 2nd Piola-Kirchoff (2nd P-K) or "pseudo!! o 

stress referred to areas at t=O. Th e above tran s format i on shows that 

this stress matrix Is symmetric; however, it cannot be physically lnterpreted as stress in the usua I manner since it res! sts a trans formed 

force vector and does not act on deformed areas 0 Oef ! n i ng 6xl stress 

t vectors, {a} and and noting the similarity of the above 

equations with (5.39), the transformation in (5.60) may be written in 

vector form as 

(5.62) 
186 

in which matrix t[T) was introduced in (5.39). o 

Now consider the configuration at i'ime t+~t in which Cauchy 

st'resses t+~ t{ ""'} act. Th ~ b f d t d . v ese s tresses may e re erre 0 correspon I ng 

areas at t=o or alternatively at time t as fol lows 

t+~t {S} 

o 

t+~t {S} 

t 

t+~t [T looT t+~t {a} 

o 

t+~~[Tl-·T t+~t {a} 

in which t+~~[T) was introduced in (5.57). 

(5.63) 

(5.64) 

Stress rates and increments are employed in the development of path 

de1pendent, 1 ncrementa.l consti tut i ve mode Is. The 2nd P-K rate is 

obtained by differentiating (!5.60) with respect to time (or the pseudo 

load parameter). The formal stress increment is obtained by multiplying 

the rate by the time increment. The differentiation process parallels 

that used in (5.50) for the variational operator. Denoting a rate (as 

distinguished from a variation) by a dot supersclpt, and using (5.26-

5.30) with a dot rep I ac I ng the var I at I ona I operator, the 2nd P-K stress 

rate is given, after some manipulation, by 

(5.65) 

in which 

(5.66) 

t ·T·t • t - [aJ tel - tel (al + Tr([e)) [a) 

In (5.66), [~l and [~l are at time t. The t • term [a lis the actua I 

Cauchy stress rate. The comb I ned terms t • in (5.66), denoted t[aT), are 

referred to as the Truesde I I str'ess rate I n the literature. Because the 
187 

2nd P-K stress and, therefore, its rate are symmetric, the Truesdell 

rate is symmetric by virtue of the symmetric transformation in (5.65). 

In addition, the Truesdell stress rate transforms to an equivalent 2nd 

P-K rate in the same way that the Cauchy stress transforms to 2nd P-K 

stress, see (5.60). Under a t • rig i d rotat i on, [S J must van Ish since no 

o 

change of the basis unit vectors occurs. The 2nd P-K stresses are thus 

Invariant under a rigid rotation~ This requirement necessari Iy impl ies 

t • that t[aT ] must vanish under a rigid rotation, i.e., 

t • t • T • t [01 = [al - [aJ [wJ - [wI [aJ (5.67) 

Jaumann [S.7J adopted the above terms within the Truesdell stress rate 

as a suitable spin-Invariant stress rate for use in constitutive laws at 

finite deformation, l.e., 

• I n the case of a pure deformat i on J ncrement, [w 1 = 0 and the Jaumann 

rate Is the actual Cauchy rate. The Jaumann stress rate is a I so 

symmetrico In vector form, the above expressions may be written 

(5.70) 

Matrix t[Ol Is defined in the appendix to this chapter. The Jaumann t 

stress rate finds use in fin I te stra in pi ast I city as the quant I ty that 

is I inearly related to the rate of the deformation matrix. As discussed 

by Hi II [5.81, this is equivalent to a true stress vs. logarlthlmlc 

strain relationship in simple tension. Use of the Jaumann stress rate 
188 

as the resu I tant of the consi" i -tut i ve mode I y i e I ds a zero Treusde I I and 

2nd P-K rate under rigid rotai"ion. 

In the course of finite element computations, the equi I ibrium 

configuration at time t+6t is sought using the equil ibrium configuration 

at time t as the initial condition. Increments of the computed stress 

componeni·s are accumu I ated as the I terat I ve so I ut I on converges toward 

ttle correct equl I ibrium configuration at t+6t. In the T. L. approach, 

this process may be written as 

= t{S} + E {6S} 

o 0 (5.71> 

As prevous I y noted, forma I stress increments, {6S}, are obta I ned from 

the corn3sponding stress rate multlpl ied by the time increment, 6t. In 

actual computations, the stress Increments are determined from strain 

I ncremeni"s wh I ch are I nduced by a d I sp I acement Increment {6u}. The 

displacement Increment can be thought of as a velocity multipl ied by the 

time incr"ement. 

The tota I Cauchy stress at time t+6t may be computed us i ng i'he 

transformation In (5.62). The Incremental decomposition of stress above 

Implies that material constitutive models produce increments of 2nd P-K 

stress. However, 2nd P-K stresses are generally unsuitable for materIal 

constitutive relationships which are naturally cast in terms of true 

stress-logarithmic strain. Use of a T. L. formulation necessitates some 

additional strain-stress transformations to obtain the {6S} needed in 

o 

(5.71). These transformations are described In Section 5.6. 

Equation (5.71) also serves as the starting point in developing an 

appropr I ate I ncrementa I stress decompos i t ion for use with the U. L. 

approach .. Both sides of (5.71) are multipl ied by :IJI-1 :[TJT to yield 
189 

(5.72) 

in wh i ch t{S} are 2nd P-K stresses at time t+Llt referred to the configuration at time t. The Truesdell stress increments (t{LlS} ) appear 

naturally in this transformation. The Inverse of (5.64) is applied to 

obtain the Cauchy stresses at time t+Llt from (5.72)9 The above relationshlp is appeal ing from a physical viewpoint if the motion from t 

to t+Llt is a simple rigid rotation. In such a case, the accumulated 

Truesde II increments in (5.72) are zero. The Cauchy stress 

t at t+At, obtained using the inverse of (5.64), becomes {a} transformed 

through the rigid rotation, l.eGe IT! is formed from an orthogonal 

deformation Jacobian. 

5.5 Principle of Virtual Displacements 

The principle of virtual displacements Is applied to derive the 

equilibrium conditions for a body undergoing arbitrari Iy large displacements. The bas I 5 of the argument I s that, for an imposed v I rtua I 

displacement fIeld, the virtual work remains invariant with respect to 

the configuration in which the variables are measured [SG91. 

o t+At Consider a body at time t+At which occupies a volume V and Is 

t+Llt bounded by the surface area Ao Let the body be in equilibrium under 

a set of body forces {p} and surface tractions {q}. For dynam I c 

analysis, the inertia effects are included In the body forces according 

to D'Alembert's prlnclpleo If a virtual displacement field {au} 

consistent with the kinematic boundary conditions is appl led to the 

body, the virtual work done is given by (ignoring thermodynamic effects) 
190 

Jt+~tp {ou}T t+~t{p} t+~tdV + J {ou}T ~:~~{q} t+~tdA 

t+~tv t+AtA (5.73) 

and must be zero. The mass dens I ty I s denoted by p. The body forces 

ar'e in terms of unit mass; the surface tractions are in terms of the 

force per unit area at time t+~t. The appl ied surface tractions {q} are 

written in terms of the equilibrating Cauchy stresses. The classical 

procedurE~ [5.9] I s then emp loyed to convert these I ntegra I s to i-he 

virtual work form using Gauss' theorem. The result is 

t+~tow + t+htoW. 

ext Int = 0 (5.74) 

In which the external virtual work is given by 

t+6toW J t+~tp {ou}T t+~to{p} t+~tdV ext = t+~tv (5.75) 

The internal virtual work is given by 

t+~toW. J t+~t t+~t[J1el) t+~tdV Int = - t+~t Tr( [a) II 

V 

(5.76) 

The i ntE~rna I v I rtua I work measured from the I nstantaneous con f I gurat i on 

f nvo I ves the Cauchy stress and the var i at Ion of the deformat ion matr i x 

(which Is the I inear part of the Almansi strain variation). The above 

expression may be simplified using vector forms of the strain and stress 

to yield 

(5.77) 

To apply this equation, the terms must be expressed as functions of 

variables in a known equilibrium configuration. The obvious choices are 
191 

the configuration at time 0 (T. L.) and time t CU. L.). To develop 

virtual work expressions for the Total Lagrangian formulation, al I 

variables are referred to the undeformed configuration. Use is made of 

the fol lowing transformations previously derived 

t+~t{a} = t+~tIJI-l t+~t[T1T t+6t{S} 

000 (5.78) 

t+~t {oe} = t+~t [T]-1 t+~t {oe} (5.79) 0 0 

t+~tdV = t+~tIJI °dV (5.80) 0 

Upon direct substitution of these transformations into (5076)p the 

Internal virtual work becomes 

t+~toW. = - J 

Int 0v (5.81) 

The Internal virtual work is In terms of the 2nd· Plola-Kirchoff stresses 

and the var I at Ion of the Green stra in. In [5.9] the above Internal 

virtual work expression Is derived by direct transformation of the 

Integrals in (5.76) to time 0 before Gauss's theorem is appl led. 

To obtain the U. l. formulation, the terms in (5.76) are written as 

functions of the known equilibrium configuration at time to The pre= 

viously derived transformations are 

(5.82) 

(5.83) 

Upon substitution of these transformations into (5.76) 9 the internal 

virtual work expression becomes 
192 

(5.85) 

wh i ch is ana I ogous to (5.81) except that i ntegrat i on is performed over 

the configuration at time t rather than time O. 

Transformation of the external virtual work integrals (5.75) to 

either T. L. or U. L. forms introduces another level of complexity. However, the vo I ume integra I for the body forces presents no d i ff i cu I ties 

and is readi Iy transformed to 

t+l'lt oWBF = 

ext 

for" T. L. and to 

(5.86) 

t+~tow BF = 

ext 

J tp {ou}T t+~t{p} tdV (5.87) 

tv 

for U. L. These transformations are particularly simple since the direct Ions of the body force components are not a.! tered under the d i ffer"-

ential volume conversions. Body forces are conservat i ve by def in ition. Inertia effects are simple to incorporate in (5.86) by the substitution of {-u} for {pl. Simi larly, for U. L., inertia effects are 

incorporated by transforming the integral in (5.87) to time 0 by noting 

that tptdV = 0podV (conservation of mass) and letting {p} = {-u}. The 

transformed integral is identical to (5.86). This is of major practical 

consequence as the mass matrix for the U. L. formulation is identical to 

that for T. L. and rema i ns constant for a I I times throughout the 

response. 

The transformation of external virtual work for the appl ied surface 

tractions in (5.75) is more complex due to the different orientation 

of t+~tdA when referred to times 0 and t. This dependence of external 

work on the displacements leads to non-conservative loading. An 
193 

interesting case of appl ied surface tractions is a normal pressure 

loading as considered by Oden (5.101, Nayak [5.11, and Mackay (5.111. 

Thus, 

(5.88) 

t+Llt . where {n} conta, ns the components of a un it outward norma I to the 

d f d t+t.tdA d t+t.t e orme area an qn is the intensity of the appl ied normal 

pressure (suction is positive). Transforming (5.88) to time 0 using 

(5.4) yields 

o where {n} is the unit outward normal at the same material point on the 

undeformed element surface. The surface traction integral of (5.75) 

becomes 

t+LltowST = J {ou}T t+~t[Jl-T o{q*} t+6t 1J1 °dA 

ext 0A 0 0 

(5.90) 

In which o{q*} has been substituted for t+Lltqn o{n}. The analogous 

expression for Updated Lagrangian is 

t+6t6WST 

ext (5.91 ) 

WOrth t{q*} _- t+6tqn t{n}. Wh en th e d" Itt I t ISP acemen s represen arge ro ations and/or finite geometry changes, the use of (5.90) or (5.91) is 

essential to correctly generate residual loads in finite element 

computations. When only concentrated (non-follower) forces and body 

forces are appl jed, the surface integral above vanishes. Similarly, If 

the pressure load i ng is app lied on a surface that undergoes 

infinitesimal displacements, equation (5.91) degenerates to (5.90) In 
194 

mechanics type problems with loading far removed from the intensely 

deformed plastic zone near the crack tip. 

5.6 Constitutive tvbdels for Finite Deformation Elasto-Plasticity 

Constitutive models provide a connection between incremental strain 

and stress changes. There exist no generally appl icable constitutive 

relations for materials subjected to finite strain magnitudes. However, 

two special cases have received considerable attention by finite element 

researchers and have resulted in reasonable engineering approximations for 

material behavior. 

Oden [5.10) has rev I ewed formu I at ions for non r i near e last i c (hyperelclstlc) materials similar to rubber. For such mater i a Is, it may be 

possible to develop an elastic energy function in terms of the total Green 

strain components. The 2nd P-K stress Is obtaln~d by partIal differentiation of the energy function with respect to strain. Thus, 

o t W = W ( {x}, {€} I material constants) (5.92) o 

t aw { S} =--.:;..:.:..-

oat {€} 

o 

(5.93) 

ThEl relatIonship between increments of straIn and stress Is obtained by 

again differentIatIng (5.93) which yields 

d t {s} 

o 

In wh I ch 

(5.94) 

(5.95) 
195 

The matr i x t [D J conta ins the tangent e last i c modu Ii. When the mater i a lis 

o 

isotropic, the function W may be simpl ified through the use of strain Invariants. The commonly used Mooney-Rlvl in model for rubber assumes incompressible behavior with an energy function defined by the first and second 

strain invariants and two material constants. 

The hyper-e last i c mater i a I mode I s are of I itt I e use for most englneerlng materials which experience permanent deformation when subjected 

to stress states outside an elastic domain. The remainder of this section 

focuses on an approximate theory for finite deformation, elastopI ast I city. Th 1 s theory has been adopted by a number of fin I te element 

researchers, especially for applications involving the elastic-plastic 

fracture of metals. The fundamental assumptions of this theory are: 

1 • the mater I ali sin it i a I I y I sotrop I c ; 

2. the material work hardens isotropically; 

3. there exists a linear relation between stress and strain 

increments; 

4. additive decomposition of strain increments into elastic and 

plasti~ parts is val Idj 

5e recoverable elastic strains are infinitesimal; 

6. the stress increment is independent of the rate of rig i d body 

rotatlone 

The bas 1 s for th i 5 mater i a I mode ling theory is prov i ded in HII liS 

work [5.81. More recent I y p Nemat-Nasser [ 5. 121 rev i ewed the status of 

developments of finite deformation plasticity and provided additional 

arguments in support of assumption (4). His argument is that 

Infinitesimal I ncrementa I deformat Ions with ina crysta I line so lid 

occurr I ng from a fin i te I y deformed con f i gurat I on must sat i sfy add I t I v i ty 

as in the infinitesimal deformation case. He further argued that the 

t e symmetric part of the velocity gradients (the deformation vector t{e}) may 

be decomposed into elastic and plastic components. 
196 

Given the above restrictions, the Prandtl-Reuss equations are adopted 

as usua I, but I n terms of true stress and tr~ue stra in. Stra in rate effects, creep, and thermal dependent material constants may be incorporated 

In this theory fol lowing the same techniques used in smal I strain 

plasticity, although I ittle of this work has been reported in the 

, I i'terature. 

Following Hill [5.81, Nemat-Nasser [5.121 and Hibbltt, et al. [5.131, 

the Jaumann stress increment and the increment of the deformat i on vector 

prov I de appropr i ate stress and stra i n increments wh i ch van Ish under rig i d 

body motion. The incremental relation becomes 

(5.96) 

in which ~[DTI is formed exactly as In the case of infinitesimal strain 

plastiCity [5.141, but using true, Cauchy stresses. The above relat'ionship is strictly valid for differential changes of the deformation vector. 

The von l"1ises yield criterion and associated flow rule are adopted for 

metals. For stress states Inside the yield surface, the tangential constitutive matrix, ~[OT1, is composed of simple elastic constants. For an 

elastic-plastic state, the terms of ~IDl are derived from the elastic constants, the current stress state, the material strain hardening characteristlcs, and the history of Cauchy stresses. For uniaxial tension, it is 

simple to show that (5.96) represents a true stress vs. logarithmic strain 

red at 1 onsh 1 p. Numer 1 ca I ref I nements, such as the sub 1 ncrement method, 

used in sma I I strain plasticity are equally appl icable in (5.96) to assure 

satisfaction of the flow rule. 

The incremental relation in (5.96) is adopted in both U. L. and 1". L. 

formulations through appropriate transformation of the terms to the re-
197 

quired reference configuraiton. Considering first the T. L. formulation, 

the deformat ion increment is obta i ned from the Green stra in increment 

us i n g (5. 54) • 

(5.97) 

The assumption here is that the Green strain increment, t{~€}, is actually o 

of differential, not finite, magnitude (see equations 5.50 - 5.55). The 

resu I ti ng Jaumann stress increments obta i ned from (5.96) are tran s formed 

to Truesdell increments with (5.70). 

(5898) 

The Truesdell increment is transformed to the required 2nd P-K increment 

with (5.69). 

(5.99) 

To compute a tangential stiffness, matrix, it is necessary to have a 

direct relationship between Increments of 2nd P-K stress and Green strain 

in the form 

(S.100) 

Combining (5.96-5.99), the tangent material matrix is given by 

(5.101 ) 

However, t because t [Q 1 T s not symmetr ie, the resu I t I ng con st i tut i ve re I at ion is non-symmetr i c wh i ch I eads to an undes i rab Ie non-symmetr i c st j ffness matrix. Osias [5.151 arrived at this same relationship (S.10n and 

retained the non-symmetric terms. Upon examination of the [QJ matrix, the 
198 

non-symmetr I c terms are found to ar I se from the trace of the deformat I on 

matrix which represnts the incremental volume change, see Eq. (5.66). If 

these terms are neglected, t then t[Q) becomes symmetric. Recall i ng 

assumption (5), the rationale to neglect these terms Is that: (1) under 

elastic conditions, the volume change multlpl led by the stresses is a very 

small tet·m compared to the elastic modul i, and (2) under perfect plastlcity conditions the incompr-essibi I ity constraint forces a zero volume 

change. McMeeking and Rice [5.16) omitted the trace term as did Nagtegaal 

and de Jong [5.17), although neither group noted this assumption. Further 

comp I i cat Ions ar I se when the mater I a I has a very low harden I ng mod u I us (i n 

terms of true stress-logarithmic strain). The intermediate result, 

t t· t[OT] - t[Ql, may develop negative terms on the diagonal for large tensile 

strains. One possible remedy is to simply omit ~[Q) in (5.101) when 

forming a suitable constitutive relation for use in computing a tangent 

stiffness. The writers have demonstrated that this technique is 

acceptab I e prov i ded the exact stress increment I s computed us i ng (5.96-

5.99). 

Consider now constitutive relations for use with the U. L. formulation. The deformation increment at time tH.t for use in (5.96) Is computed from the Green strain Increment referred to the configuration at 

time t using (5.57). 

t+At{ } t+Att[TJ-1 t+Att{Ae:;} t+At Ae = Ll (5. 102) 

Again, It is Impl ied that t+A~{Ae:;} Is actually of differential magnit·ude. 

The Jaumann stress Increment is obtained from (5.96) and transformed to a 

Truesdel I Increment by 

(5.103) 
199 

The Truesde I I increment is transformed to an increment of 2nd P-K stress 

referred to the configuration at time t (which is just the Truesdell 

stress transformed to time t) 

(5. 104) 

In U. L., the tangent stiffness matrix is formed for the configuration at 

time t and generally held constant during the increment L\t. Thus, the 

material tangent matrix is given by 

t [0* 1 = t T t[D 1 _ t[QI t T t (5.105) 

In wh I ch both matr I ces on the right side are generated in terms of actua I 

Cauchy stresses at time t. The ~[D*l above relates increments of Green 

strain and Truesdell stress at time t. If the stiffness is updated 

between t and t+L\t (to accelerate convergence) the new constitutive matrix 

is evaluated as in (5.105) using the most current values of Cauchy 

stress. The same difficulty that arises due to the non-symmetry of [QI 

for To L. also occurs here for U. L. 

Approximating assumptions for (5.96-5.105) have been introduced by 

various Investigators as summarized below. Alturl, etc al. [5.18! 

employed an U. L. approach with the approximation 

(5.106) 

wh I ch neg I ects changes in the structura I confl gurat 1 on over the I nterva I L\ t. In forming the constitutive matrix, [Ql was simply ignored. 

Hibbitt, Marcal, and Rice [5.131 fully developed the T. L. equations 

(5.96-5.101) but implemented a simpl ified form val id only for smal I 

strains by neglecting [QI in.ill.. equations. The determ i nant of the 
200 

dE!forrnat I on Jacob I an was a I so assumed to rema I n un I ty wh I ch effect I ve I y 

rElduces [T] to a rotational transformation. Based on these assumptions, 

they claim that the implemented formulation is val id for the case of large 

rotations but infinitesimal strains. McMeeking and Rice [5.161 developed 

an U. L .. approach in which the configuration is continuously updated. 

Under such a procedure, the transformation In (5.102) Is unnecessary, 

i.e., IT] Is an identity matrix. In addition, the transformation of 

Truesdel I stress in (5.104) becomes unnecessary. Bathe, et a I. ! 5.51 

pr'esented both T. L. and U. L. formu I at ions. For T. L., the mater i a I 

constitutive relation in (5.100) was formulated using classical plasticity 

theory but in terms of total 2nd P-K stresses. The proposed U. L. 

constitutive model (5.106) is that adopted by Alturi, et al. [5.181. 

With exception of the Jaumann to Truesdel I transformation in (5.98), 

differences in the various U. L. approaches vanish for sufficiently small 

motion over the interval6t. However, the choice of a "sufficiently" 

sma I I 6t cannot be assured for comp I ex structures undergo i ng large d i splacements; thus, the more exact transformation In (5.104) Is preferred. 

With th I s approach, the T. L. formu I at I on I s recovered as the I 1m i t I ng 

case when no updates of the deformed configuration are performed. The use 

of (5.100) with :roT ) formed In terms of 2nd P-K stresses, as proposed in 

[5.5), Is clearly unacceptable for general applications. 

There continues to exist some question as to the need for the [Q) 

terms appearing in the various transformations. These terms arise from 

the Jaumann to Truesdell stress Increment transformation. In the elastic 

range, these terms are on the order of stress, wh I ch when mu I tip lied by a 

str-ain Increment are clearly negligible compared with elastic modul i. In 

the plastic range, the terms of [Q) become more comparable in magnitude to 
201 

the p r ast i c modu r i (wh i ch may be on the same order as the stress) un less 

the deformation Increment is truly of differential magnitude. Hibbitt, et 

a I. [ 5.13) acknow I edges the ro I e of [0 I then chose to ignore it under the 

restruct ions of sma I I stra i n- large rotat ion. McMeek i ng and Rice [5. 161 

include [01 in their continuously updated formulation as an initial stress 

matr i x and do not report any n umer i ca I d Iff i cu I ties. They conc I ude that 

whenever the slope of the Cauchy (true) stress vs. logarithmic strain 

curve has a magnitude comparable to the current stress level, the 

predicted tangent stiffness cannot be accurate. Both [5.131 and [5.161 

conclude that the relative importance of [01 terms Is not fully known and 

requires further study. 

5.7 Summary and Comparisons of the U.L. and T.l. Formulations 

Using matrix notation, the equations of non I Inear continuum mechanics 

that provide a rational basis for finite element analysis have been 

presented in th i s chapter e Two formu I at ions are descr i bed in deta i I ; 

namely, the Total Lagrangian CT. L.) and Updated Lagrangian (U. l.) The 

in it I a I con f i gurat I on of the body at time t=O serves as the reference 

state for all varTables In the T. L. formulation. The configuration at 

time t serves as the reference state to describe al I variables at time 

t+At In the U. L. formulation. Fu I I deta! I s of the stress measures, 

rates, transformations and nonlinear constitutive models have been 

described that enable analyses involving both infinitesimal and finite 

strain magnitudes. Finite magnitude strains commonly appear in structures 

that experience transient response after being subjected to local ized 

blast or impact loadings. 
202 

The two nonl inear formulations, T. L. and U. L., are shown to derive 

from a common definition of the rate of work per unit mass which leads to 

equivalent virtual work expressions when the corresponding reference 

states are introduced. Analyses derived from each formulation are identical to within truncation errors introduced in the numerical procedures. However, as noted throughout this chapter, various investigators 

have introduced simpl ifying approximations into the formulations that 

I lmit th!~lr appllcabi I ity to the most general class of problems. The U. 

Le, formulation offers the most temptation to simpl ify the equations since 

one might argue that the difference in con f i gurat Ion between time s1"ep t 

and t+~t is sufficiently sma, I I that all second order nonl inear terms may 

be discarded. Stress rates and transformations In the corresponding 

constitutive models are simplified simi larly. No approximations of any 

type are perm i ss I b lei n the T. L. formu I at I on since the deformed and 

rE!ferenCEI configuration are seldom similar. 

The differences In computational efficiency of the two form'ulations 

lies in 'the stress rate transformations and the strain-displacement relations. The mass representation is identical in both formulations and thus 

does not enter into the d! scuss Ion. Simi larly, computations associated 

with a nonl inear constitutive model are identical as the computed results 

rElqu i red by the mode I are I dent i ca lin both formu I at Ions. As shown In 

Section 5.6, the same number and type of transformations to account for 

stress rates are required In each formulation. The U. L. approach appears 

to gain a computational advantage if the simpl ifying approximation Is 

employed that el iminates the Truesdell to 2nd P-K stress transformai"ion. 

However, this advantage is offset by two factors: (1) the number of time 

s1"eps to assure that the change in con f i gurat Ion over a sing let i me step 
203 

is "small" remains unknown, and (2) the stresses at the beginning of the 

step and the increments that occur during the step must be transformed to 

the configuration at the end of the step. This last process is unnecessary for the T. L. formulation in which all stress increments are transformed to the initial configuration as computed. The stress increments 

are simp I y added to the ex i st i ng stress state. Therefore, in terms of 

stress computation, the T. L. approach actua I I Y has a s light eft i c i ency 

advantage over the exact U. L. formulation. 

The computation of an increment of the deformation vector is accompi lshed in the same symbol Ie form (5053 and 5057> for both formulations. 

The differences lie In the effort required to construct the [TI matrix In 

each case. For U. L., the formulation of derivatives in [Tl requires that 

updated coord i nates. be ava II ab I e, whereas In T. L. I der i vat i ves are a I ways 

formed with respect to the initial configuratio~o These derivatives could 

be computed once and saved for re-use. Some effort I s necessary to 

continuously update the coordinates. More importantly, the strain increments produced by U. La cannot be s Imp I y accumu I ated to form the tota I 

strain, as shown in (5.48). This transformation to a common reference 

configuration requires additional computational effort that Is unnecessary 

in the T. L. formulation. 

I n summary, an exam I nat i on of the computat I ona I eff i c i ency of the 

transformations necessary in each approach, reveals that T. L. appears to 

have a sl ight advantage over U. L. ThIs does not include any factors that 

are introduced when the governing equations are cast into a form suitable 

for finite element analysis. In a practical sense, the advantages of 

either the U. L. or T. L. continuum theory appear computationally 
204 

insignifIcant given other costs of computation, for example, equation 

solving and massive data transfers between memory and disk. 
205 

5.8 References 

5.1 Nayak, G. C., "Plasticity and Large Deformation Problems by the 

Fin i te Element Method," unpub I i shed Ph. D. Thes Is, Un i vers i ty of 

Wales, Swansea, 1971 (C/PH/15/1971). 

5.2 Zienklewlcz, O.C., The Finite Element Method in Engineering 

Science, 3rd edition, McGraw-Hi I I, London, 1978. 

5.3 Bathe, Kc J., Ramm, E., and Wilson, E. L., "Finite Element Formulations for Large Deformation Dynamic Analysis," Int. J. for 

Num. Meth. In Engr., Vol. 9, 1975, pp. 353-386. 

5.4 Yaghmai, S. and Popov, E. P., "Incremental Analysis of Large 

Deflections of Shells of Revolution," Int. J. Sol ids and 

Structures, Vol. 7, 1971, pp. 1375-1393. 

5.5 Bathe, K. J. and Ozdernir, Ho, "Elastic-Plastic Large Deformation Static and Dynamic Analysls~1I Computers and Structures, 

Vol. 6, 1976, ppo 81-92. 

5.6 Boit, M. A., Mechanics of Incremental Deformations, WI ley, 

N.Y., 1965. 

5.7 Prager, We, Introduction to Mechanics of Contlnuna, Gin and 

CO. 9 1961. 

5.8 Hill, Ro, "On Constitutive Inequalities for Simple Materials," 

J. Mech. Phys. Sol ids, Vol. 16, 1968, pp.229-242. 

50 9 Dym , C. L. and Sh ames, I ., .;;;So~I..;..I.;:;..d --,-Me;.::;..;;;;c.;.;.h.;;;;.an~i c:::.;s:::.;:~--,-A~V~a:.:.r....;l.;;;a:...;t.-:.l..;:;o.;.;.n-=a..:..1 

Approach, McGraw-Hi I I, 1973. 

5.10 Oden, J. To, "Numerical Formulation of Non I inear Elasticity 

Problems," J. Struct. Dlv. Am. Soc. Clv. Engr., Vol. 93, 1967, 

pp. 235-255. 

5.11 Mackay, D .• C., "Finite Element Formulations for Large Strain 

Plasticity Problems," unpublished M.S. thesis, University of 

Wales, Swansea (C/M/146/80). 

5012 Nemat-Nasser, S., "On F T n I te Deformat ion E I asto-P last i city," 

Int. J. Sol Ids and Structures, Vol. 18, 1982, pp. 857-872. 

5.13 Hlbbitt, H. D., Marcal, Po V., and Rice, J. R. p "A FInite 

Element Formulation for Problems of Large Strain and Large 

Displacement," Int. J. Solids and Structures, Vole 6, 1972, pp. 

1069-1086. 

5.14 Nayak, G. C. and Zienkiewicz, O. C., "Elasto-Plastlc Stress 

Ana I ys i s, A Genera II i zatlon for Var lous Constl tut ive Re I at Ions 

Including Strain Softening," Int. J. Num. Meths. Engr., Vol. 

5,1972, pp. 113-135. 
206 

5. 15 Osias, J. R. and Swedlow, J. L., "Finite Elasto 

Deformat i on - I: Theory and Numer i ca I Examp I es," 

Sol ids and Structures, Vol. 10, 1974, pp. 321-339. 

Plastic 

Int. J. 

5.16 McMeeking, R. M. and Rice, J. R., "Finite Element Formulations 

for Problems of Large Elastic-Plastic Deforrnarion," Int. J. 

Sol ids and Structures, Vol. II, 1975, pp. 601-616. ----

5.17 Nagtegaal, J. C. and de Jong, T. E., "Some Computational 

Aspects of Elastic-Plastic Large Strain Analysis," Report No. 

M.S.10.910. MARC Analysis Research Corp. 

5.18 Alturi, Nakagakl, and Chea, "Fracture Analysis Under LargeScale Plastic Yielding," Flaw Growth and Fracture, ASTM STP631, 

American Society of Testing Materials, 1977, pp. 42-61. 
207 

5.9 Appendix -- Notation 

The fol lowing convention for vector and matrix subscripts and superscripts is used: 

1. a left superscript indicates the discrete time configuration in 

which the variable occurs, 

2. a left subscript in conjunction with a left superscript indicates the discrete time configuration with respect to which the 

variable is measured, 

3. a left subscript by itself indicates an Increment from time t 

to t+&t referred to The configuration at the specified time, 

4. a dot over a symbol denotes a rate quantity. left subscripts 

and superscripts indicate the time and reference configuration, 

5. a "0" symbol denotes a variation of the quantity. Left subscripts and superscripts indicate the time and reference 

configuration. 

Square brackets denote a matrix 

{ } Curly braces denote a column vector 

IT, { }T Right superscript "T" denotes the transpose 

o{x}p t{x}, t+At{x} 

{J} 

t[ J I 

o 

t {u}, t+At{U} 

o t {n}, {n} 

Cartesian coordinates at time 0, t, and t+&t 

unit vectors 

Deformation Jacobian matrix 

Vectors of d i sp I acements from 0 to t and from 0 

to t+&t 

Vector components of the surface norma I at times 

0, t 

Mass density at times 0, t 

Area at times 0, t 
t { p} o 

i"+~t {d I t+~t [€ I 

208 

Volume at time 0 and t 

Vectors of surface force components at- time t per 

unit area at 0 andd t 

Vector of body force components per un it mass at 

time t referenced to configuration at 0 

Cauchy stress vector and matrix at time t+~t. 

(Note that t+~t [01 = ~::~~[Ol). 

2nd Piola-Kirchoff stress vector and matrix at 

time t referenced to the configuration at 0 

2nd Piola-Klrchoff stress vector and matrix at 

time t+6t, referred to configuration at t 

Truesdel I stress vector and matrix at time t 

referred to the configuration at t 

Rates of Jaumann stress vector and matrix at time 

t referred to the configuration at t 

Almansi strain vector and matrix at time t+~t 

(note th t t+~t[ I = N~t[ ]) a € t+~t € • 

Green strain vector and matrix at time t+~t 

referenced to con f i gUI~at i on at 0 

Incremental displacement gradient matrix at t 

referenced to con f I gUI~at i on t 

Deformation matrix at t referred to configuration 

t t (symmetr ie part of .trv]) 

Spin matrix at t referred to configuration t 

t (anti··symmetric part of t[v]) 

Transformation matrix to convert Almansi strains 

at time t to Green strains 
209 

Displacement Jacobian at time t referred to 

configuration at 0 

Different i a I operator to y i e I d Green stra 1 n 

increments from displacement Increments 

Tr( Denotes the trace of a square matrix (sum of 

diagonal terms) 

Transformation matrix that converts Jaumann stress 

increments at time t to Truesdell stress 

increments at time t 

Virtual work quality 

w Elastic energy density function 

ElastIcity matrix at time t referred to 

configuration at 0 

Elastic-plastic matrix at tIme t referred to 

configuration at t 
210 

,Key Transformation Matrices (2-D Case) 

x 

J 

x = x + U 

Y y + v 

oX 

y = Xx = -- y oX 

<111 : -<1 11 1 <11 2 

I I - - - - - - - - - r - - - - - - --- -<122 : <122 1 <112 - - - - - 1- - - - - - 1- -1 - - - - - - ---

o 1 0 1 '2 (<1 11 + (122 ) 
211 

- a - a 

Xx ax 1 Yx ax - - - - --1- - - ---

- - 1-
212 
CHAPTER 6 

NONLINEAR FINITE ELEMENT EQUATIONS 

6.,1 General 

The finite element concept of discretization is combined with the 

continuum mechanics theory in matrix form in this chapter. With the 

addition of solution procedures for nonl inear transient equations, a 

very general analysis capabil ity results. The form of the finite 

e I amen t equat Ions of mot T on are first expressed without regard to a 

particular choice of reference configuration. The use of such a general 

form makes the detailed discussion of nonl inear transient solution 

procedures equally appl fcable for an arbitrary reference 

configuration. Specific forms of the element' stiffness matrices and 

internal resisting force vectors are derived for the Total and Updated 

Lagrangian approaches. The relative computational effort required for 

each approach and the impl ications of a substructured modeling procedure 

are a I so exam i ned. Recommendat Ions are made for appropr i ate 

fermu I atlons and sol utlon procedures necessary to support the ana I ys i s 

of a broad class of problems., 

A number of investigators have contributed to the current state of 

rigorous fInite element formulations derived from the continuum theory 

of the prev i ous chapter. Three approaches appeared a I most 

simultaneously in the literature of the early 1970'so Hfbbftt, Marcal, 

and Rice [6.1] first presented a comprehensive formulation using the 

cenf iguration at time t=O as the reference state <Total Lagrang ian) 0 

213 

, , 
214 

Matrices required for finite element analysis, including the nonsymmetric sti ffness for displacement dependent loading. were 

described. Using matrix notation throughout, Nayak [6.21 independently 

derIved the same element matrices. Both of these studies addressed only 

static analysis and employed the symmetric 2nd P-K stress and Green 

strain as conjugate measures in the configuration at time t=O. Nemat-· 

Nasser and Shatoff [6.31 used a combination of current (time t) and 

initial reference configurations with the non-·symmetrlc 1st P-K stress 

measure. 

Investigators advocating the use of the current configuration 

(convected coord i nates) for the reference state inc I uded Hartzman and 

Hutchinson [6.41, Belytschko and Hsieh [6.51, and Key [6.61. In each 

of these stud i es, the mot Ion of the body over M was decomposed I nto a 

rigid motion and a pure deformation. The procedures worked wei I in each 

case even though different schemes were dev i sed for the decompos I t ion 

Only Key [6.61 focused on the finite strain casej the other 

were concerned primarily with large rotation effects. 

procedure. 

two studies 

Interest i ng I y, each of these stud I es addressed the prob I em of non I i near 

wave prop~lgatlon using explicit integration procedures. McMeeklng and 

Rice [6e 71 presented a rigorous formulation based on a continuously 

updated rE~ference configuration applicable for finite deformation and 

rotation. 

Yaghmal and Popov [6.81 were the first to describe an Updated 

Lagrangian approach which attempts to combine the Total Lagrangian and 

thel updated coord I nate approaches. Th is was done by adopt i ng the 

configuration at any time t prior to t+M as a reference state. The 

ex I st i ng Green stra i ns and 2nd P-K stresses in the reference 
215 

configuration are simply treated as initial strains and stresses to 

wh I ch are added increments of 2nd P-K stress and Green stra in. Th i s 

approach captures the sp I r I t of a convected coord I nate approach but Is 

more appealing for four reasons: (1) the reference configuration does 

not need to be continuously updated, (2) the decomposition of motion 

into deformation and rotation fol lows Lagrangian mechanics, (3) the 

'Total Lagrangian approach Is recovered exactly if the reference 

configuration is not updated, and (4) no I imits on the deformation 

magnitude over At are impl ied due to I inearization if the exact U. L. 

equat Ions of the prev i ous chapter are ut iii zed. Nagaraj an and Popov 

[6.91 subsequently used the U. l. approach to study v!scoplastic 

response of thin shell structures. Bathe, etc alo [6.10, 60111 

formal ized the U. L. procedure for general transient analysis and 

published a large number of papers using the procedure. 

6.2 Nonl inear Equations of Motion 

The virtual work principles derived in Section 5.4 provide the basis 

to generate approx I mate eq uat ions of mot i on us i ng f! n i te element concepts. Considering the motton of the body over the Interval from t to 

t+&t p and without regard to a particular reference coordinate system, 

the principal of virtual work provides that at t+At (see 5.74-5.76) 

oW = oW t + oW. t = 0 ex In (6. 1 ) 

where the t+A t I eft superscr i pt has been dropped in th is sect i on for 

brevity. All quantities are at time t+At unless otherwise noted. 
216 

The I nterna I and externa I v I rtua I work terms are given symbo I I ca I I Y 

by 

oW. t In 

T * = -f V *{OE} *{cr} dV 

* 

T * oWext = f V *p{ou} {p} dV 

* 

(6.2) 

+ T * f A{ou} *{q} dA 

* 

(6.3) 

The asterik (*) is adopted to indicate an unspecified reference 

configuration for the variables. As shown in Section (5.4), the general 

forms of the virtual work equations for both T. L. and U. L. are 

Identical. The differences derive from the choice of reference system 

(*) and the requirement of conjugate strain and stress definitions. 

This notation permits a completely general discussion of the finite 

element process and solution techniques prior to introducing specific 

matrices for the T. L. and U. L. approaches. 

The fin I te element concept is invoked at th i s po i nt to prov i de a 

spatial discretization of the structure. Individual elements are 

connected at discrete points termed nodes. At time t+~t, the continuous 

displacement field, {u}, within each element is approximated by a set 

of i nter'po I at ion 

e displacements {a }. 

e {u} =: IN] {a } 

functions, [N) , 

Thus, 

which act on the nodal 

(6.4) 

In 2-D problems, {u} is a 2 x 1 vector containing u and v displacement 

components as a function of position within the element. Vector" {a e} 1-hen conta i ns (2 * number of element nodes) terms. Terms of 

[N] are usually simple functions of position within the element and do 

noi- depend on its deformed shape or time. 
217 

To faci I itate the development of elements with initially curved 

edges and faces, the shape functions are expressed in terms of an 

intrinsic, convected coordinate system. A one-to-one mapping from the 

parent configuration (usually square) to the actual shape in the 

structure is provided by a set of Interpolating functions and the 

Cartesian coordinates of the element nodes. As wIll be shown, the use 

of Initial or updated Cartesian coordinates for this mapping plays a 

major role in the differences between T. L. and U. L. When 

interpolation functions for the element mapping and those for the 

displacement interpolation are identica I (6.4), the popular 

Isoparametric element family Is obtained. Fortunately, the details of 

transformations between the IntrinsIc isoparametric coordinate and 

9 I oba I Cartes i an coord i nates do not affect the bas i c fin I te element 

solution procedure. 

At time t+At virtual displacements, {oae}, consistent with the 

kinematic constraints are imposed on the nodes. Corresponding 

variations of the element displacement fleld p {OU}, are given by 

e {ou} = [NJ {oa } (6.5) 

Virtual strains caused by the virtual displacement field are denoted 

*{oe}, in which the definition of strain appropriate for the selected 

. reference configuration is impl ied. However, it is completely general 

to express the virtual strains as 

,'1 
218 

in which *(L1Is a differential operator matrix that may be a nonl inear 

function of the deformed element configuration at time t+/lt. Combining 

(6.5) and (6.6) yields 

(6.7) 

in which the conventional notation *(81 of the virtual strain-nodal displacement relationship is introduced. The general form of the internal 

virtual work for an element is written 

e oW . t In (6.8) 

in which the notation {Fe} is introduced to denote equivalent nodal 

for'ces I ncluced by the deformed element. The (*) I eft subscr i pt I s not 

requ i red on such terms since those forces are directed a long g I oba I 

coord in atEl axes. 

Consider now the virtual work of body forces within an element, 

= T * f V *P {ou} {p} dV * e 

(6.9) 

in which *p is the mass density and {p} are the Cartesian components of 

body forcE~ per unit mass. Two types of body forces are considered: (1) 

self-weight and centrifugal, In which the body force Intensity Is 

written in terms of current density as *{F}; (2) Inertia effects using 

'0 

D'Alembert's principle with {u} defining the element acceleration 

fIE:lld. Using (6.4), the body force virtual work may be written in the 

fOl"m 

e T T .0 * - {oa } f V*p(N1 {u} dV * e 

(6.10) 
219 

To simpl ify this expression, let {BFe} represent the first integral 

(equivalent nodal loads due to body force). Again the (*) left 

subscript can be omitted. The second integral is simpl ified by noting 

that in analogy with (6.4) 

.0 .G

e {u} = t N 1 {a } (6e 11) 

.and that the second integral of (6.10) must be invariant with the 

deformed conf iguration (conservation of mass--see Section 5.4). The 

virtual work of element inertia forces may thus be written in the form 

In wh lch 

[Mel = f y[NlT[NlopodY 

o 

Matr 'lx [Me] 1s usually termed th e consls . ttl en . e emen t mass; 

(6.12) 

(6.13) 

it Is 

computed once at the beginning of the solutIon for the Initial 

configuration and recal led from secondary storage whenever needed. 

In a simi lar manner, the virtual work of external tractions appl led 

over the element surfaces may be written In the form 

(6.14) 

The total virtual work for an element is thus written 

(6.15) 

Y I rtua f work for the comp tete structure Is obta 1 ned by summ 1 ng the 

contributions from each element. Element nodal displacements are 

re I ated to the structura I noda f d i sp f acement vector through a simp f e 

,. \ 
220 

Boolean connectivity matrix, which accompl ishes the symbol ic assembly 

process. The resu I t i ng vi rtua I work express ion is ana I ogous to (6.15) 

except that vectors are of structural, rather than elemental, size. 

cW = 0 = {cal T {-{F} + {SF} + {TF} - [M] {;}} (6.16) 

The choice of nodal virtual displacements is arbitrary; thus the summed 

terms in {} must vanish for dynamic equilibrium at time t+~t. The 

result is a set of nonl inear, simultaneous equations in the nodal 

displacements and accelerations which may be written in the form 

{R} -- {P} - {F} - [M1 {a} = {OJ (6.17) 

Vectors {F} and {P} are Impl icit functions of the nodal displacements 

and generally cannot be written as a matrix multipl ied into the current 

displacemEmt vector. For simpl icity, the external appl ied load effects 

an~ comb i ned into {P}. L1 kew i se, damp i ng effects have been neg I ected ; 

• ho~~ever they could be included by a term of the form [el {a} analagous to 

the Inertia term. 

At this point it is Instructive to review the sources of 

nonlinearity in the dynamic equations of motion (6.17): 

1 ) the change in sur face or i entat I on and magn i tud e under load I rig 

causes {P} to become a function of the displacements; 

2) the Internal resisting force vector, {F}, may be nonl inear due 

to the material const-itutive relationship between total stress 

and strain (*{a} and *{e}); 

3) the Internal resisting force vector may be nonl inear due to the 

dependence of *[81 on the nodal displacements. 

The fol lowing section considers general solution procedures. 
221 

6.3 Transient Analysis with Substructuring 

At any given time, 0, ~t, 2~t, ••••• t, t+At ••••• the nodal displacements, velocities, and accelerations are sought such that the nonl inear 

equ i I I br I um equat Ions represented by (6. 17) are sat Is f i ed. A tempora I 

integration operator is employed for the time discretization. Both 

impl icit and expl icit operators are appl icable. With expl icit 

operators, computations are performed directly on the vector form of the 

dynamic equations in (6.17) with {R} taken as zero. Component terms of 

each vector are computed directly from the Integrals In (6.8, 6.10 and 

6c12). No stiffness matrices or mass matrices are ever assembled. The 

advantage of an explicit procedure Is that it provides the computational 

eff i c i ency needed to render feas i b I e the so I ut i on of wave propagat ion 

problems (which require exceptionally smal I ~t). The computations over 

each time step are re I ative I y s imp I e. The major disadvantage is that 

non I I near effects due to the spat i a I var i ab I es cannot be l'i terated" out 

at constant dynamic load, i.eo, no equilibrium iterations within a time 

step are possible. 

With implicit Integration operators, the displacement increment over 

At Is used to predict the accelerations at t+Atc The displacement 

increment is obta i ned by us i n9 an effect i Ve st! ffness matr i x to form a 

set of simultaneous equations. The advantage of an Impl icit approach 

ties I n the capab i I I ty to comp I ete I y correct for non I i near effects in 

the spatial variables that occur over Lit through equil ibrium 

iterations. In addition, corrections 

increment over At during iterations 

predicted acceleration at NAt. At 

in the estimated displacement 

are utilized to improve the 

t+Atp the dynamic equi I ibrium 

equations can be satisfied to within a specified tolerance with an 
222 

implicit procedure. The major disadvantage of the impl icit scheme is 

the increased computational effort required to assemble and triangulate 

the effective stiffness matrix. Thus, a solution procedure based on an 

impl icit scheme is more appropriate for problems in which wave 

propogation is not as important as gross inertia effects. Much larger 

time steps are permitted thus reducing computational costs. If 

necessary, the impl ieit scheme can be used to compute wave propagation 

eHeets. 

The impl ieit solution scheme is adopted for detailed discussion here 

and will be adopted in the software. The implicit scheme builds upon 

thE! exper I ence ga i ned in the so I ut i on of stat i e non I i near prob I ems. 

Stcltie analysis is recovered as the degenerate case when the mass matrix 

is zero. 

The nonlinear equations of motion (6.17) are solved iteratively to 

determine displacements and accelerations at t+~t beginning with a known 

soiution for time t. Suppose that 'mY such iterations have been 

performed and the current estimate of the total nodal displacement field 

is t+~t{a } and 

m 

the acceleration field is t+~t{; }. 

m 

nodal 

t+~t If {R } computed m using (6.17) vanishes for these displacements and 

acee I erat ions, the so I ut I on has converged. 

t -une s t b ep are egun. I f t+~t{R } does not 

m 

Computations for the next 

van I sh, a genera I i zed forGe 

imba lance exi sts at the nodes and is given by the terms i n t+~t {R } .. m 

These are usually termed the residual loads; they must be el iminated by 

suitable changes of nodal displacements and accelerations during the m+l 

iteration. 

necessary to 

solved 

Let {~R } 

m represent the change of the residual loads 

t+~t{R }_ remove 

m ' then the fol lowing equation is to be 
223 

t+~t{R } + t+~t{~R } = {OJ 

m m (6.18) 

The increment of the residual load vector is simply determined by 

summing the increment of each vector in (6.17). 

(6.19) 

. in which {P} and {F} are impl icit functions of the nodal 

displacements. Corrections in the nodal accelerations, {~a}, are 

m 

coupled indirectly to the nodal displacement corrections through the 

integration operator. The finite size corrections impl ied in (6.19) are 

also nonl inear functions of the spatial variables. 8y assuming that an 

approximate correction can be obtained using the differential of {R}, 

(6.20) 

a one term Taylor series about the configuration {a } yIelds a set of 

m 

I inearlzed correction terms for use in (6c19) 

t+.&t {~P } '" t+.&t{dP } 

dP 

= [ --~] {a } = [K1

] {a } (6~21 ) m m da m m m 

t+.&t{~F } '" t+At {dF } 

dF 

= [ __ m] {a } = [KT]{am} (6 .. 22) m m da m 

Vector {a} denotes an increment of nodal displacement. The symbolic m 

differentiations inside the [ lead to the initial load stlffness~ 

and the conventional tangent stiffness~ (KT1~ These two 

matrices are the Jacobians of the functions {P} and {F}~ which are 

nonlinear in the nodal displacements. Explicit forms of [KTJ for the T. 

L. and Uo L. formulations are provided In following sections. The 'm' 

right subscript outside the [ ] denotes the nodal displacements at which 

the stiffnesses are evaluated. The Influence of {Aa } on {AR } is 

m m 
224 

simply a function of the Integration operator and need not be considered 

In (6.21) or (6.22). As discussed In Section 5.4, the [K.ll matrix 

arises due to non-conservative loading. It Is generally non-symmetric 

and thus Ignored in forming {~R }. 

m 

Corrections In the effective 

load i ng, {P}, due to geometry changes are incorporated by occas I ona I I Y 

recomputing {P} using the current geometry. Symmetric forms of [KTl may 

be derived for both T. L. and U. L. formulations as shown In subsequent 

sections. 

Substitution of (6.22) and (6.20) into (6.18) and then into (6.17) 

yledds the Incremental-iterative equations of motion 

(6.23) 

wh i ch can be so I ved for {a } using linear equation solving techniques. m 

t+&t o. 

enters (6.23) through the {a } term. Each 

. m 

Thel Integroati on operator 

vee t or on th erg I ht h an d s, Od eo. f (6 23) o's computed for the mth 

itelratlon estimate of the nodal displacements at t+~t using (6.8), 

(6.10), (6.14) at the element level and (6.16) at the structure level. 

The improved estimate of nodal displacements for t+~t is 

t+~t{ } 

am+1 

= t+~t{a } + {a } (6.24) 

m m 

Corresponding t+~t •• 

accelerations {am

+ 1

} are computed using the 

integration operator. The right side of (6.23) Is then evaluated for 

the improved displacements and accelerations (the R.H.S. is 

t+&t simp I y {R} ) • If this vanishes, the solution has converged; 

otherwise a new correction {a } is computed and the process repeated. m 

Slmpl iflcatlons in solving (6.23) Involve the use of various forms 

of the Newton-Raphson procedure. It Is not necessary to recompute and 

triangulate [KTl during each iteration. Any previously computed IKT1, 
225 

including the I inear elastic [Kl, may be employed. The convergence rate 

of the iterative procedure wil I be affected accordingly. Overrelaxation techniques used in static analysis may also be used here to 

enhance the convergence characteristics. 

In a multi level substructure model, (6.23) is appl ied only at the 

highest level structure which contains the synthesized I fnear substructures and nonl inear finite elements. The effects of substructuring are 

reflected in each term of (6.23). Modal synthesis techniques discussed 

in previous chapters provide procedures to formulate [M] and [Kl at the 

outset of the transient response computation. The external dynamic 

loading t+~t{P} Is generally independent of the nodal displacements m 

(conservative loading) but not independent of time. Thus, if time 

dependent loads are appl led inside a synthesized substructure, the 

equivalent dynamic loads on the nodes (and modes) remaining after 

synthes I s must be recomputed each time step. A simp I If i cat Ion occurs 

when the spat i a I load i ng pattern app I I ed I ns I de the substructure Is 

synthesized once, then a specified time function is ~ppl ied to the 

equivalent loads to yield the variation with time. The i nterna I 

resisting t+~t forces, { F} , are the sum of contributions from the 

individual nonlinear elements and from the synthesized substructures 

that appear in the highest level structure. Nonlinear element 

contributions are given by (6.8); synthesized substructure contributions 

are obta I ned from the product of the I r stl ffness matr Ix wi th 

the mth estimate of the total nodal o t+~t displacements {a }. 

m 

This 

computation requires the retrieval of each synthesized stiffness matrix 

from secondary storage during every equi I ibrium iteration. The major 

computational savings derive from the greatly reduced size 
226 

Initial Calculations 

I. For each I inear substructure (repeat at each level for nested 

substructures): 

a. Form [K] using standard techniques, 

b. Form [M] as consistent or lumped. 

c. Reduce [K] and [M] using any of the dynamic reduction techniques 

described in Chapter 3. 

2. Form the synthesized stiffness, [K], and mass, [M], for the highest 

level structure. 

3. Set initial displacements 0{a}, and velocities, o • {aL 

4. Compute initial accelerations, o{~}, from equil ibrium equation: 

[M] o{:;3} + [K] o{a} = o{PL 

5. Define constants for the specific integration operator, e.g., 

Wi I son,-8, Newmark-S. Denote these constants a., a., a k, etc. 

I J ' 

6. Compute the contribution of the mass to the effective stiffness for 

the highest level structure: [R] = ai[M]. 

7. Triangulate effective stiffness, [KE

] = [K] + [R], of the highest 

level structure, using Choleski decomposition: [KE

] = [L][L]T, 

Table 6.1-- Procedure for Transient Analysis with Substructuring 

(Damping Neglected) 
227 

For Each Time Step t + t+6t: 

I. If specified by the user, update the tangent stiffness, [Kr], of the 

highest level structure. Only elements that are currently nonlinear 

are updated. Condensed subst'ructure st iff nesses are re-used. 

Triangulate the new [KT] = [L][L]T. 

2. Compute the effective load increment vector. Place it in t+6t{R}. 

This expression is derived by subtracting (6.'7) evaluated at time t 

from the same equations at time t+6t as fol lows: 

[M] t+6t{~} + t+6t{F} _ eM] t{;} _ t{F} = t+6t{p} _ tiP}. 

. []{} t+6t{} t{} t+6t{,.} Now substitute ~ 6a = F - F and a = 

a.{6a} + a.t{a} + akt{a} from the integration operator to yield I J 

( [KT

] + ai[M] ){6a} = t+6t{p} - t{P} + [M]( a

j 

t{~} + a/{~} ) 

The right hand side is simply t+6t{R}. It is not necessary to include 

any remaining residual load, t{R}, in the ~ffective load vector for 

the new step if equilibrium iterations wilt be performed. Otherwise 

t{R} should be ~dded to the t+6t{R} derived above. 

3. Solve for the displacement increment vector [L][L]T{ a} = t+6t{R}. 

4. Proceed to step 5 if no equi librium iterations are specified to correct 

the acceleration or to el iminate residual loads due to nonlinear 

response in the spatial variables. 

Table 6.' -- (continued) 
228 

4. (continued) Otherwise, begin equi I ibrium iterations: 

a. Set m = 

b. Compute improved estimates of the total nodal displacements and 

accelerations given values at time t and the most recent estimates 

of change over the step. 

t+L\t t {a } = {a} + {a }; m m 

t+Llt .. t+Llt{} t • t ,. {a } = a. a-a. {a} - a k {a}. m I m J 

c. For the improved total displacements and accelerations, compute the 

total residual load vector1+L\tR } = t+L\t{~ } _ t+L\t{F } - [M] t+L\t{; }. --- m m m m 

d. Solve for a correction to the displacement change over the step: 

e, Update the estima+ed change in nodal displacements for the time 

f. . t+L\t t+L\t Perform checks on convergence us I ng {R }, {L\a}, {a }. If 

m m 

converged, jump to step 5. If not converged and iteration number 

m is less than the maximum al lowed, increment m and go to g; otherwise, terminate analysis as a nonconvergent system exists. 

g. I·f spec if i ed by the user, update [KTJ and tri angu I ate. Go to b. 

5. Compu'te new acce I erat ions, ve I oc i ties, and d i sp I acements for t+Llt to 

serve as initial conditions for the next step: 

t+L\t ... t t • t .. {a} = a. {a} + a. {a} + a k {a}. I J 

t+L\t{~} t • t .. t+L\t .. = {a} + a l {a} + a {a}. m 

t+L\t{a} t = {a} + {aL 

Stresses, strains, reactions, etc. can be computed for the highest level 

structure. Similar results may be recovered for condensed substructures. 

Table 6.1 -- (continued) 
229 

of [KTl possible with substructuring. The frequent triangulations 

of [KTJ required to enhance convergence each consume much less time than 

does the reduction of a [KTJ representing an equivalent unsubstructured 

model. 

Table 6.1 provides a detailed flow of the transient response computation for a multilevel substructured model based on an Impl icit 

integration operator. 

6.4 Total Lagrangian Stiffness 

Specific forms of the elemental stiffnesses for the Te L. approach 

are described in this section. Once element stlffnesses are generated, 

a substructure stl ffness is assemb I ed us I ng standard, we II documented 

techniques [6.121. 

The approach adopted here derives from the' work of Nayak [6.21, 

McKay (6.13], and Dodds [6.14]. As shown in (608), the element nodal 

force vector due to Internal stress and strain is given by 

in which the above Integral may be evaluated at any time to With the 

configuration at time t known, an approximation for {Fe} at t+6t is 

given by 

in which d{a} represents the change in element nodal displacements over 

~t. 

For the T. L. approach, the (*) configuration in (6.25) is taken to 

be that at time O. The integrand is evaluated at time t but referred to 
230 

the! configuration at time 0 using the results derived in Chapter 5, 

sp€lciflcally (5.81). After substituting into (6.25) the differential, 

internal force vector may be written as 

(6.27) 

Using the product rule, differentiation under the integral sign yields 

(6.28) 

From (5.100), the increment of 2nd P-K stress is given by 

wli"h the tangent modulus matrix provided by (5.101). The differential 

of the Green strain is given by (5.55) as 

in which i-he form for t[BJ is repeated in Table 6.2. 

o . 

The differential of the t[Bl matrix in (6.28) with respect to the 

o 

elElment nodal displacements has been derived in a convenient form by 

Nayak £6.21 and Dodds [6e141. The t[Bl matrix Is spl it Into two 

o 

components, the first be I ng independent of the noda I d I sp I acements wh Ich 

vanishes under different I at ion with respect to the nodal 

d I sp I acememts. Differentiation of the nonl inear terms yields an 

expression of the form 

(6.31) 

following rather lengthy but straightforward differentiation of the 

summations. Complete details of the process are given by Dodds 
231 

16.14]. Matrix t lG ) simply contains derivatives of the element shape o 

functions with respect to the initial, o{x}, coordinate system. 

The t [M) matr i x conta ins a regu 1 ar arrangement of the 2nd P-K stress 

o 

components, t{S}. Detal Is of the (Gl and [Ml matrices for the 2-D case 

o 

are given In Table 6.2. 

The complete element tangent stiffness is obtained by combining the 

effects of (6.28-6.31) as 

(6.32) 

in wh I ch fin I te I ncrements of the noda I d j sp I acements, {~a}, are used 

during computation to replace the differential values9 

The initial stress stiffness, [K 1, is given by (6.31). The 

(1 

nonlinear stiffness, [K NL ], is given by 

(6.33) 

and contains the usual I inear stiffness, in addition to displacement 

dependent contributions, l$e., if the nodal displacements are 

zero, t[B] slmpl ifles to the standard linear strain-displacement relao 

tiona 

6.5 Updated Lagrangian Stiffness 

The procedures adopted in the previous section also enable the 

derivation of stiffness matrices for the U. L. formulation. The 

resulting stiffness matrices for U. L. have a form very simi lar to that 

for T. L. which simpl ifles the associated programming when both 

approaches are Implemented. 

c 1 
232 

The Green strain increments in 2-D are: 

t t 

e: = u 

o x o x 

t t 

oe:y 

= v 

o y 

t t 

oYxy = u 

o y 

+ O.5[ tu2 + tv2 l' o x 0 x - , 

t 2 

o ]; 5[ t 2 + v + . u 0 y o Y 

+ tv + tu t 

u + tv 

{aa}T = Lou; avl; variation of total 

displacements 

0 t at 

x = x; u =_u 

o x aOx 

0 

Y = Y 

tv 

o x o x 0 y o x o y 

A virtual or differential change of the strain te: is given by o x 

ate: = otu + tu otu + tv otv with other terms derived simi larly. o x 0 x 0 x 0 x 0 x 0 x 

t a' 

u -

o y ay 

I 

I t a 

I v 

lOX ax 

--1------- --

I 

I 

t a (I + v)-

o y ay 

- - - - -- - - - -I-- - - _. - - - --

t a 

u -

o y ax 

+ 

I 

+ 

Table 6.2 -- Summary of Total l.agrangian Matrices 
233 

Substituting shape functions into the above, the corresponding matrices for 

the Jth node are: 

t[SJ] = 

0 

:[GJ

] = 

( 1 + tu 

dNJ t 3NJ )- vo x dX o x 3x 

I ---------------- - --

dNJ 

I 

t I 

+ tv 

3N

u - J o Y ( 1 dy ) - I o Y 3y 

--4--------

I 

t 3N 1 J 

u 1 ( I 

o y ax I 

+ I + 

+ 

t ) 

dNJ t ( I u v 

o x dy o x 

[ tG I tG2 tGn 

0 0 0 

o o 

dX I ay I I 

I ____ 1_ - - _.J. - - -..j. - - -

1 aN I aN 

00

' 

J I J 

dX I ay 

I 

I I I 

Sxx : Sxy I 0 : 0 

T--~--I---

Sxy I Syy: 0 I 0 

---:--t-- r--

o lOIS IS I xx xy 

-r--~--J..-­ I 

o 0 I sis 

xy I yy 

] 

+ tv 

dNJ ) - o y ax 

dNJ 

dy 

t S .. = S.o 

IJ 0 IJ 

Table 6.2 -- (continued) 
234 

The bas i c equat i on of the U. L. formu I at i on def i nes the i nterna I 

virtual work of an element at time t+~t relative to the configuration at 

time t as 

t+~tow~ = -f vt+~tt{o£}t+~tt{S}tdV tnt t (6.34) 

?eflnlng a suitable differential operator matrix, t+~~[Bl, to yIeld virtual strains from virtual nodal displacements, (6.34) may be written in 

the form of (6.8) as 

t+~t OW:1 = -{oae} T f t+~t [B J T t+M {S} t dV = _{oae} T t+~t {Fe} (6.35) 

tnt tV t t 

in which t+~t{Fe} in (6.35) has the Identical meaning as the same 

quantity In (6.26). 

The elE~ment resisting force vector at t+~t is expressed in terms of 

a known value for the configuration at time t plus a differential change 

over ~t as 

(6.36) 

Tn wh I ch 

The difference between (6.36-37) and (6.26) is simply the reference 

configuration of the tangent stiffness. t e The incremental force,d {F }, 

and nodal displacement Increment, d{a}, have the identical physical 

interpretations In both T. L. and U. L. formulations. 

The specific form obta ined by evaluating the 

differential of (6.35) with respect to the displacements at the value of 

nodal displacements corresponding to time t. Symbolically, 
235 

(6.38) 

From Chapter 5, the fol lowing relationships are used to expand (6.38) 

t t t{S} = {a} (the Cauchy stress) (6.39) 

(6.40) 

The tangent modulus matrix, referred to the current (time t) configuration, ~[DTl, was derived in Chapter 5. 

measured from the current configuration, is 

The differential strain, 

in which ~[Bl is simply t+6:'~[Bl for a zero increment of displacement 

(see Table 6.3). Notice that the resulting form is the conventional 

linear (8) matrix containing derivatives'taken with respect to total 

displacements at time t. Using (6940) and (6.41), the first integrand 

term of (6938) becomes the "nonl inear" stiffness matrix ~[KNL] and is 

given by 

(6.42) 

in which the "L" subscript is used to denote the simple; linear form 

taken by [810 

To obtain the second integrand term of (6~38)p an expression 

for d~[Bl Is requirede This Is found by differentiating t+b,~[Blp with 

respect to the nodal displacements, then evaluating the terms for a zero 

displacement increment. A comparison of t+b,~[Bl In Table 6.3 

with t[BI in Table 6.2 shows a very similar form. The same operations o 

t . t 

that yield dolBI, readily yield dt[Bl as 
236 

The Green strain increments In 2-D are: 

t+L\~{od = t+L\~[BJ{oa}, 

T L °tU OtVl {oa} = ; ; varl at Ion of Incremental displacements that 

produce configuration at t+L\t. 

+ O.5[ 2 2 "] 0 E = u- u- + v- . x = x + u 

t :< t x t x t x . , 

2 oJ + v- . t yo' - 0 

y = y + v 

y = u- + v- + u- u- + v- v- t xy t y txt x t Y txt Y 

dtU 

; u- = t x dX 

A virtual or differential change of the incremental strain tEx is given by 

t+L\t[BJ = . t 

I 

(I + u-) L I v- L 

t x ax I t x ax ________ 1 _______ _ 

a 

u- - t y - dy 

I 

I 

I 

I 

a (I + v-)- t y - ay 

- - - - - - -- - 1- - - - - -

a CI + v-) a 

u- - t Y - t y ax ax 

+ + 

( I a a 

+ u-)- v- - t x - t x -

ay ay 

. Setting u = v = 0 t t 

o 

ax I -- - ~ - --

o 

I ay 

---l---

a a 

ay ax 

Tabte 6.3 -- Summary of Updated Lagrangian Matrices 
237 

Substituting shape functions into the forgoing, the corresponding matrices 

for the Jth node are: 

I aN J 1 aN J 

(I + u-) - v--

t x ax 1 t x ax - - - - - - - _1- _____ --

aN J 

u- - t y - 'dy 

aNJ 

u-- t y - ax 

+ aNJ (I + u-) 

t x ay 

1 

, aNJ 1 (I + t V-) 

1 Y dY - ,- - - - - --

aNJ . (I + v-) 

t Y ax 

1 + 

.,' aN J 

tV;;: -= 

Cly 

I I 

'dNJ I 'dNJ , 0' 0 

-lax I ClY' I __ ..L __ .-J ___ ,_ - - -

I I I 

o 0 aNJ I ClNJ I , 

~ ax ay 

Table 6.3 -- (continued) 

aNJ : 

ax I 

o 

____ 1 ___ _ 

I 

, aN J 

o 1 

J _ ~Y_ 1 

aNJ 1 aNJ 

liy 1 ax 

1 
238 

(6.43) 

in which the details of ~[GJ and ~[MJ are shown in Table 6.3. 

Matrix ~[MJ contains the Cauchy stresses at time t arranged in the Ident i ca I form that the 2nd P-K 

form of ;[GJ is identical to 

stresses appear in t rM1 • Simi larly, the 

o 

that of t[GJ except that derivatives are 

o 

:,"akl3n with respect to the displacements at time t rather than time O. 

The Integral of the product shown in (6.43) Is again termed an initial 

stress stiffness. 

The comp I ete element tangent st i ffness is obta i ned by comb i n i ng 

(6.42) and (6.43) to yield 

(6.44) 

6.6 Compar~ison of Formulations 

Details of a transient solution procedure based upon an Implicl1" 

IntE~gration scheme have been described in this chapter. An impl ici1" 

schE~me is r~ecommended over an explicit scheme for three reasons. First, 

the proced ure5 for stat I c I I near and non I i near an a I y5 i s are recovered 

from the Impl icit scheme by simply omitting the mass matrix; the . ' 

degEmerate case of an expllc!-t- scheme does not yield a formulation 

suitable for static analysis. The capabil ity to perform both static and 

transient analyses with the same software is particularly attractive to 

engineers since a static analysis invariably precedes a dynamic 

ana I ys Is. The second reason to se I ect an imp I I cit scheme is the more 

genora I c I ass of structures that may be ana I yzed. An Imp I i c It scheme 

may be used to compute the deta i I s of I oca I i zed wave propagat I on under 

very high velocity Impact as wei I as the vibration characteristics of a 
239 

massive structure subjected to time dependent loads. The exp I i cit 

scheme may be computationally more efficient for wave propagation 

studies but not necessari Iy more efficient than an impl icit approach for 

the latter class of problems. Thirdly, the equil ibrium imbalance due to 

nonlinear response in the spatial variables can be "iterated out" to 

wlth!n a specified tolerance using the implicit scheme. Dynamic 

equi I ibrium can thus be assured at the end of each time increment. 

Complete details of the element tangent stiffness matrices have been 

presented in the 2-D case for both the T. L. and U. L. formu I at ions. 

For both approaches, stl ffness matr ices have the i denti ca I symbo Ii c 

form, wh i ch is conven 1 ent I y expressed as the sum of an r nit i a I stress 

stiffness, [Ka ], and a nonl inear stiffness, [KNL1. G J ven the common 

rate of work per unit mass expression from which each formulation is 

derived, It Is expected that Identical results for each solution would 

be obtained (provided the full stress rate transformations described in 

Chapter 5 are util fZed). Some computational evidence [6.10, 6.111 does 

demonstrate the agreement of overall load-deflection curves for static 

and dynamic solutions which include plasticity effects. Unfortunately, 

no comparisons are provided of the internal stress distributions, which 

must be Identical If equivalence of the formulations Is to be 

demonstrated. Global agreement of load-deflection does not imply 

identical internal stress distributions. 

The computational efficiency of each approach is addressed at two 

levels; namely, the effort required to compute the element matrices, and 

the number of iterations required to attain the equi I Ibrium 

configuration at time t+~t, assuming ~t is the same in both 

formulations. 

, 1 
240 

I n each formulation the construction of [K ] requires the same 

(J 

number of operations If the expense of coordinate updating In U. L. Is 

neglected. In contrast, construction of the nonlinear stiffness, 

[KNL ], in U. L. requires the same number of 'operations as required for 

a convent I ona I linear st I ffness. The T. L. non I i near st I ffness requ I res 

,more operations due to the absence of zeroes In the IB] matrix (refer to 

Table 6.2). The increase in the operation count depends on the element 

type but eas II y exceeds by a factor of three the number of operat ions 

requ I red to form [KNL 1 In U. L. Eva I uat I on of the I nterna I res I st I ng 

force vector during Iterations requires the Identical number of 

oper'atlons In each case; the U. L. matrix t+A~[B] has the Identical form 

t of [BI in T. L. (no zero terms). The reduced operation count of U. L. 

o 

compared to T. L. dur i ng e lament st I ffness generat I on I s an important 

cons; i derat i on on I y when ill. matr Ices assoc I ated w.i th the ana I ys I s res I de 

In memory during execution. Once the swapping of element stlffnesses to 

and from disk begins, it would appear that the I/O overhead overwhelms 

any advantage of one formu I at I on over the other. Furthermore, the 

timl'ng results shown In Chapter 2 for several large I inear and nonlinear 

analyses rE~veal that element stlHness generation times do not represent 

a major portion of the total solution time. Thus, a reduction of 

stlHness generation time obtained by the selection of U.L over T. L. 

does not yield an equal percentage reduction In total analysis time. 

Perhaps the most Important eff 1 c I ency compar I son between the two 

formulations Is the number of equilibrium Iterations required for 

conl/ergencE~ dur I ng each time stE~p. Us I ng a common program that has both 

U. L. and T. L. capabilities, a structure could be analyzed with each 

formulation for Identical time steps and with an identical convergence 
241 

criterion based on total force equi I ibrium. The resu I ts of such a 

comparison would demonstrate any inherent computational advantage of one 

formulation over the other, a least for a particular problem. 

Unfortunately, no such comparisons have been found in the open 

I fterature. A systematic study of several problem classes using this 

.approach Is necessary to form the basis for any general efficiency 

statements. 

The distorting of elements has led to numerical problems In some 

U. L. finite element analyses. For example, an element that Is square 

at t=O may become a badly distorted quadrilateral at time t+6t. Element 

response characteristics are known to degenerate rapidly as aspect 

ratios Increase; this wi I I influence the response in an U. L. approach 

as shape function derivatives are dependent on the deformed shape at 

time t. In T. Lo I a II such effects are 

multiplier terms of shape function derivatives 

incorporated 

in the t [91 

o 

in the prematrix. The 

shape function derivatives in T. L. are always computed relative to the 

configuration at time t=O and thus remain well-behaved if the aspect 

ratio at t=O is acceptable. The interesting case that demonstrates the 

problem with U. L. Is an a-nodel' 2-D Isoparametric element which Is 

square at t=O but which at time t has the mid-side node displaced toward 

a corresponding corner node. Any movement of the mid-side node toward 

the corner produces a singular point In the corresponding shape function 

when the derivative is evaluated with respect to the current element 

shape. No such singularities are introduced in shape function 

derivatives In T. L. if the mid-side node is properly positioned at t=O. 

A final comparison of the two formulations considers their appllcablltty In a substructured model fng and solution procedure0 At al I times 
242 

t, the st i ffness matr i x of a I i near substructure and its synthes i zed 

form rema in unchanged from that at time t=O. The con f i gurat i on of the 

structure at t=O provides the basis to which all strains and stresses 

are referred. The I i near, stra i n-d i sp I acement re I at Ions and con st i tutive matrix that Is independent of nodal displacements at time tare 

ut iii zed. Cons i der the common nodes a long the boundary of a non I I near 

reg Ion that I s shared by an adj acent I I near substructure. I n an U. L. 

formulation, the coordinates of these common nodes are updated at time t 

to ref I ect the I ncrementa I d I sp I acements over the prev i ous increment 

At. The tangent st I ffness of the non I I near reg Ion is regenerated to 

reflect the new geometry. Thus, there exists a discontinuity of nodal 

positions along the boundary. If the substructure stiffness is 

recomputed and synthesized for the updated geometry along the boundary 

(and the consequent Internal repositioning of nodes) al I the advantages 

of a substr~uctured model are lost. The alternative Is to assume that 

nodes alon!] the interface are sufficiently remote from any effects 

caus I ng non I i near behav lor that the response in non I I near elements on 

the boundar-y is actua II y II near, i.e., d I sp I acements and stra ins- are 

infinitesimal and the matrices involved at the element level revert to 

the linear analysis form. If such an assumption is actually verified In 

the analysis, the Integrity of the results is assured. 

In a substructured T. L. solution, discontinuities of nodal 

pos I t Ions a long a boundary do not occur since noda I coord i nates are 

never updated for the displacement increments. However, the effects of 

deformation and rotation in the nonl inear elements adjacent to the 

boundary al~e Impl icitly incorporated in the tangent stiffness through 

the displacement dependent terms in t[Bl. Thus, the same difficulties 

o 
243 

descr I bed above for U. L. appear to ar I se for T. L. However, the 

maintenance of external geometric continuity is more appeal ing than 

allowing the discontinuities to develop. It could possibly be argued 

that the T. L. formulation would introduce less disturbance in the 

strain field across the boundary than U. L. , but there exists no 

computational evidence to collaborate this speculation. 

In view of the considerations described above, a Total Lagrangian 

approach Is recommended to form the basis of a general software system 

having broad appl icabi I ity. If computational evidence in support of 

U. Lc as a more eff i c I ent approach shou I d be forthcom I ng, on I y minor 

changes in software to support a ToLe formu I at I on are requ ired. In 

fact, nearly all U. L. dependent computations can be Isolated within 

element dependent routines. The reverse situation of implementing a 

T. L. approach in a system designed only for U. L. is not nearly as 

simple" 
244 

6.7 References 

6.1 Hibbitt, H., Marcal, P., and Rice, J., "A Finite Element· 

Formu I at i on for Prob I ems of Large Stra in and Large 

Displacement," Int. J. Sol ids Structures, Vol. 6, 1970, pp. 

1069-1086. 

6.2 Nayak, G., "Plasticity and Large Deformation Problems by the 

Finite Element Method," unpublished Ph.D. Thesis, University 

of Wales, Swansea, 1971. (C/PH/15/1971). 

6.3 Nemat-Nasser, S. and Shatoff, H., "A Consistent Numerical 

Method for the 50·1 ut i on of Non I i near E last i city Prob I ems at 

Finite Strains," SIAM J. of Appl. Math, Vol. 20, 1971, pp. 

462-481. 

6.4 Hartzman, M. and Hutchinson, J., "Nonl inear Dynamics of 

Sol ids by the Finite Element Method," Computers of 

Structures, Vol. 2, 1972, pp. 47-77. 

6.5 Belytschko, T. and Hsieh, B., "Non-Linear Transient Finite 

Element Analysis with Convected Coordinates," Int. J. Num. 

Meth. Engr., Vol. 7, 1973, pp. 255--271. 

6.6 Key, S.W., "A Finite Element Procedure for the 

Deformation Dynamic Response of AxisYmmetric Sol ids," 

Meth. APD' Mech. EnE.!"., Vol. 4, 1974, pp. 195-218. 

Large 

Compo 

6.7 McMeeking, R. and Rice, J., "FlnltE~ Element Formulations for 

Prob I ems of Large E I ast I c-P I ast I c Deformat i on," I nt. J. 

Sol ids Structures, Vol. 11, 1975, pp. 601-616. 

6.8 Yaghmai, S. and Popov, E., "Incremental Analysis, of Large 

Deflections of Shells of Revolution," Int. J. Sol ids Structures, Vol. 7, 1971, pp. 1375-1393. 

6.9 Nagarajan, S. and Popov, E., "Plastic and Viscoplastic 

Analysis of Axisymmetric Shells," _Int. J. Sol ids Structures_, 

Vol. 11, 1975, pp. 1·-19. 

6.10 Bathe, K, Ramm, E., and Wilson E., "Finite Element Formulations for Large Deformation Dynamic Analysis," Int. J. Num. 

Meth. Engr., Vol. 9, 1975, pp. 353'-386. 

6.11 Bathe, K. and Ozdemir, Ho, "Elastic-Plastic Large 

Deformation Static and Dynamic Analysis," Computers & 

Structures, Vol. 6, 1976, pp. 81-92. 

6.12 Cook, R.D., ConceE!s and ApDllcations of Finite Element 

Analysis, John Wi ley & Sons, N.Y., N.Y., 1981, 2nd Ed. 
245 

6.13 McKay, D., "Finite Element Formulations for Large Strain 

Plasticity Problems," unpublished M.S. Thesis, University of 

Wales, Swansea, 1980, (C/M/146/80). 

6.14 Dodds, R.H., "Numerical and Software Requirements for 

General Nonlinear Finite Element AnalYSiS," Civil 

Engineering Studies, Structural Research Series No. 454, 

University of II I inols at Urbana-Champaign, 1978. 
246 
CHAPTER 7 

USER INTERFACE -- INPUT DESIGN 

7.1 General 

The most popular approach to user communication with structural 

analysis software Is the problem oriented language (POL). Virtually al I 

successful software systems use the POL approach, either by Initial 

design or by the use of pre-processors to translate POL input Into fixed 

format, card Images. The POL approach provides the user with greater 

flexIbility by placing him In control of the process rather than forcing 

him to conform to rigid formats and sequences. The self-documenting nature of the input reduces the need for reference to manuals and provides 

a concise description of the structural model for other analysts. The 

POL Is essential for Interactive processing where error recovery Is 

often necessary. 

Dynamic analysis with multilevel substructuring wll I be Implemented 

as an extension of the present POLO-FINITE structural mechanics system~ 

The philosophy establ 'shed during the development of POLO-FINITE was to 

maintain as much Independence as possible among the various components 

of a complete structural model. 

material model specifIcation, 

These components Include nonlinear 

geometric definition of structures p 

parameters control ling the nonlinear algorithms, and requests for computatIon and output. The primary reasons for choosIng this approach are 

to provide maximum flexlbl I Ity in using condensed substructures as elements In higher level structures and to mInimIze the effect of changes 

247 
248 

In the structural model throughout the analysis-design sequence. 

Wherever possible, thIs phIlosophy Is maIntaIned In the extension 

to dynamics. However, one area exIsts in whIch dynamIc solutIon 

parameters must be tIed directly to the geometrIc defInItIon of a substructure. This Is the frequency analysis of a substructure that Is to 

. be condensed by modal synthesIs. Since economical frequency analysis 

depends upon the type of strudure, the number of elgenpalrs requIred, 

and the solution method, It Is not appropriate to select Just one solution algorithm for al I substructures In a complex model. Various substructures wfl I have dffferfng characteristics and may require an unequal number of retafned modes for condensation. It fs also possfble 

tha'\" one substructure cou I d be condensed two or more times In differ I ng 

ways, with varyIng geometric and generalized OOF, for use In separate, 

higher level structures. Thus, it is necessary to tie the selection of 

the elgenproblem solutfon method to the structure deflnftlon. 

The capabllfties selected for general purpose dynamic analysfs, 

along with the various optfons and parameters that control the solution, 

must be deffned accurately and unambiguously by the POL. SectIon 7.2 

presents an explanation of the capabf I Itles to be fncorporated fnto 

POLO-FINITE. SectIon 7.3 lIsts the command str'ucture for dynamics. Examples Illustratfng the use of these commands eIre given In section 7.4. 
249 

7.2 DescrIptIon Qf ~ fQL 

Structure ~ Element ~ 

The mass of a structure can be divIded Into two parts~ primary and 

secondary. PrImary mass Is the mass of the load-carryIng components 

(elements) of the structure. Its definItIon Is easIly added to the 

specifIcatIon of an element through two new element properties. The 

first defInes the type of mass formulation: LUMPED or CONSISTENT. The 

second is the DENSITY of the material of which the element Is composed. 

The element mass matrIx can then be formed using existing shape functIons. Assembly of primary mass for a structure wII I fol Iowa procedure 

Identical to that used In stiffness assembly. The current FINITE system 

accepts up to thirty DOF at each node of an element. These are the 

displacement OOF: U, V, and W, plus theIr first and second derivatIves: 

UX, VX, WX§ UY. etc. Depending upon the particular element formulation p 

it is possible for mass to be assIgned to any or al I of these DOF. 

Secondary mass is the mass of non-load-carrying components. such as 

concentrated and distrIbuted lIve-loads, that are supported by the 

structure. Secondary mass is defIned In a manner simIlar to the application of gravity loads. The secondary mass Is resolved Into eqUivalent nodal masses via the appropriate shape functIons. The result wll I 

always be a lumped mass matrIx whIch Is added to the primary mass of the 

structure. 
250 

There are three types of secondary mass: noda I, element, and pat·· 

tern. Noda I mass I s mass -I-hat I s concentrated at a structure node. 

Element mass Is concentrated or distributed on the surface of and ele-· 

mente As with primary mass, secondary mass may be assigned to any of 

the thirty nodal DOF. The pattern mass Is provided as a convenience to 

. the user. It enables the definition of secondary mass In terms of a 

previously defined loading condition, usually gravity loading. The user 

need specIfy only the name of the loading condition to be used as the 

pat'~ern and a value for the accE~leratlon of gravity to support the ap·· 

proprlate conversion from force to mass. 

Substl~ucture loads can be def t ned at the lower I eve I s and then ap"· 

p II od se I E~Ct I ve I y at the higher I eve I s of the structure hierarchy 

through thE~ "EXTERNAL ELEMENT LOADS" facility. It wou I d be advantageous 

to have a similar capability with respect to substructure mass. The 

analyst may wish to use several copies of a particular substructure, 

each wIth a different mass d I str I but I on (descr i bed by secon dary mass In·· 

put). Thl$ analogy to substructure loading Implies the need for an "EX·· 

TERNAL ELEMENT MASS" facility. This Is not possible In dynamic analysl$ 

since the change In the mass of a substructure changes the natural fre-· 

quencles and mode shapes. Physically distinct substructures exist when 

the mass dIstribution varies. Each of these distinct substructures mus"1-

be uniquely defined at the lower levels of the hierarchy. The mass Is 

then autCllm3tlcai Iy carried through the hierarchy via the condensation or 

synthesis process. 
251 

The commands for computation (assembly) and output of the mass 

matrix for a structure or stand-alone element fel low directly from those 

for the stiffness matrix. 

Structure Damping 

Since the dynamic reduction process recommended here does not include the substructure damping matrix, damping is defined only for the 

highest level structure. Two methods are available for defining structural damping: modal and Rayleigh. Definition of modal damping requires Input of the modal damping ratio for each vibration mode under 

consideration. Modal damping Is applicable only to transient analysis 

by mode superposition. Rayleigh damping involves the definition of two 

damping ratios at two selected frequencies; the frequencies need not be 

eigenvalues of the structure. Rayleigh damping Is applicable to transient analysis by either mode superposition or time-history Integration. 

Use of Rayleigh damping requires that a frequency analysis be performed 

In order to compute the modal damping ratios for mode superposition or 

to expl iCitly form the damping matrIx for time-history Integration. 

Depending upon the method used to define dampfngg either the 

damping matrix or modal ratios can be output for the structure. 

Frequency 8nalysls 

As previously mentioned, the parameters control ling the frequency 

analysis (computation of natural frequencies and mode shapes) must be 

defined explicItly for each structure for which the analysis Is to be 

performed. No default analysis method Is adopted. The syntax for 
252 

specification of the solution method Is simi lar to that for a nonlinear 

material. That Is. the TYPE of solution procedure Is Identified fol100wed by a listing of PROPERTIES unique to that type. When appropriate 

th'9 range of frequencies and the maximum number of modes to cons I der are 

spl9cl f led at th Is time rather than In the computation request. Sol utlon 

,method properti es can 

STRUCTURE. •• seq ue nce • 

be changed via analysis restart and the ACCESS 

If a substructure Is to be condensed by Guyan 

reduction, no frequency analysis specification Is required. 

In the request for computation the analyst may select a nonlinear 

dynamIc loading and time step at which the frequency analysis Is to be 

performed. This al lows the user to interrupt a transient analysis after 

some nonlinear behaviour has occured and compute natural frequencies and 

mode shapes of the structure. Standard output Includes natural frequencIes and modes shapes. 

Prior to a transIent analysis by mode superposItion, the user may 

examine the modal content of a particular dynamic loading condition. A 

special output request facilitates selectIon of the modes that participate In the dynamic response. After a frequency analysis the user 

can request output of MODAL LOADS for the loading condition. The frequency content of the load r ng can then be e)<am i ned and the appropr I ate 

mo,des selected for superposition. 
253 

DynamIc ReductIon 

The procedure to request dynamic reduction paral leis that for 

static condensation. The reduction method Is defIned at the intermediate substructure level; I.e., the substructure with only one element 

. of type: CONDENSED. Guyan reduction is the default method. Automatic 

selection of master nodes, in additIon to the Interface nodes, Is Included by specifying the number of additional nodes to be retained. The 

fixed-interface method Is invoked by specifying which substructure normal modes to retain. The modes specified must be within the range COlliputed In the frequency analysis of the lower-level substructure. The 

retained modes need not be consecutively numbered. As an alternative to 

using normal modes, user-suppl ied modes can be included In the synthesis 

process. These modes could be derived from an experimental analysis or 

some other source, such as low-order polynomials. Input data describing 

these modes must be Included with the defInition of the structure to be 

condensed. 

Dynamic reduction can be expl icltly Invoked with a COMPUTE STIFFNESS.e. or COMPUTE MASS ••• command for the Intermediate level substructure. Reduction Is performed automattcal Iy when required to satisfy a 

request at a higher level. 
254 

In.l.iJ...9l Co n d I t Ion s 

Initial conditions can be defined for a structure prior to transient analysis. They define a starting solution, In terms of displacements and velocities, for the unconstrained physical DOF at time t = o. 

For all other times the displacements and velocities from the previous 

time step are used automatically. 

The user can specify Initial conditions In one of two ways. First, 

he can define numerical values for each DOF with non-zero displacement 

or velocity. The default Inlt'lal conditions are zero displacement and 

ve1loctty for all unconstrained OOF. The second method uses the static 

equilibrium configuration from a previous linear or nonlinear analysis. 

This method allows the structure to be released from some deflected Initial shape with zero Initial velocity. A dynamtc loading can then be 

optionally applied. 

~~ Loading 

The dynamic loading funct-I<:>n, P(x,y,z,t), Is defined such that It 

helS a spatially-varying component, F(x,y,z), and a time-varying componemt, G(t-): 

P(x,y,z) = F(x,y,z) * G(t). (7.1) 

Simply si'ated, the pattern of the load Is fixed and Its magnitude 

changes with time. 
255 

The load pattern, F(x,y,z), can be described as either actual 

forces applied to the structure or as support accelerations. The former 

can be best defined as a static linear loading condition, whl Ie the 

latter requires an additional loading type: NODAL ACCELERATIONS. No 

special provisions are necessary for Input of out-of-phase support aco celeratlons. They can be recognized and handled automatically. 

The time-varying component, G(t), Is combined with other loading 

data to form a dynamic loading condItion. The G(t) vs. t relation may 

be harmonic, Impulsive, or general. The dynamic loading condition must 

also include the loading pattern. F(xoY.z), which Is to be used. More 

than one static I tnear loading condition can be combined to form the 

complete pattern of the dynamic load. Other necessary Input Includes 

the values of time t at which dIsplacements are to be computed (thus 

defIning the step size) and values of time t at which computed results 

are to be retaIned In the data base. This last item Is Important 

because a transIent analysiS of any significant duration could result In 

more data than could be effectively stored. Also, It is likely that 

stresses and strains would be required at only a few of the many time 

points for which displacements are computed. 

Transient AnalysIs 

Transient analysis yIelds the response of the structure, In terms 

of dIsplacements and possibly velocitles R when It Is subjected to tImevarying loading or support accelerations. Two approaches are available 

for performing transient analysis: mode superpositIon and time-history 

Integration. Mode superposition requires that a frequency analysis be 
256 

performed so the equations of motion can be uncoupled. This Implies 

that an appropriate frequency analysis method must be selected prior to 

requesting the transIent analysis. The resulting set of Independent 

equi~ltlons Is easily solved using one of the Lagrange interpolation formu lias. T I me- h I story I ntegrat I on I s performed by a ny one of a n umber of 

, explicit or Implicit operators. Specification of the transient analysis 

method Is similar to that for frequency analysis; the TYPE of method Is 

defined fol lowed by the PROPERTIES list. 

The request for computation Includes the loading condltlon l time 

steps, and optionally Initial conditions and a mode list. The mode list 

Is used with mode superposition i'o specify which modes participate In 

the response. Results available for output Include displacements, 

velocities, strains, and stresses. 

ShQ~ ~icYm Analysis 

The ana I ys I s of shock spectr'um response I s current I y restr I cted to 

lInear structures. The shock spectrum Is Input by defining the functional relationship between a spatial coordinate and a time coordinate. 

The spatial coordinate can be chosen as displacement, velocity, or acceleration, while the time coordinate can be either period or frequency. 

The user Inputs discrete points from the spectrum and the remainder of 

the curve Is constructed by linear Interpolation In four-way logarithmic 

coolrdinates. The direction of application of the shock Is defined usIng 

direction cosines for the translational OOF (UJP V, and W for 3-D structures). The nodes at which the shock Is applied are also defined. 
257 

Prior to computing the spectral response. a frequency analysIs of 

the structure must be performed. Spectral response quantities are computed only after the corresponding output request has been made. 

Results available for output include spectral displacements. spectral 

velocities, spectral strains, and spectral stresses. These quanttties 

. can be output on a mode by mode basis or In some combined form. Methods 

used to combine the modal quantities include SRSS (square root of the 

sum of the squares) and PEAK_SRSS (peak response mode plus SRSS of the 

remaining modes). PEAK_SRSS Is also known as the Naval sum. As a 

measure of the portion of the total mass responding to the shock In each 

mode, the modal PARTICIPATION_FACTORS can also be output. 

Uti I lty Commands 

The dynamic solution process can be monitored by Invoking the 

TRACE ••• command. Messages listing the currently executing module and 

elapsed CPU time are output at various checkpoints. 

To eliminate unnecessary data from the data basesf the DESTROY ••• 

command is expanded to Include results from frequency, transient, and 

spectral analyses. 
258 

7.3 EQL ~ructure 

7.3.1 S¥mQ~ ConventIons Qi th~ Syntax 

The f,ollowlng Is a description of the conventions used In this section to Illustrate the FINITE command syntax. 

A descrIptor Is used to Identify the posli"lon and class of a data 

Item Ina partl cu I ar FIN I TE command lIne. The descr i ptor I s de I 1m Ited 

by "the characters n< >." The command 

NUMBER OF NODES <Integer> 

impl las that the word NODES Is to be fol lowed by an Integer. 

proprlate example Is: 

An ap--

POL. 

NUMBER OF NODES 100 

The fol lowing are definItions of the descriptors used within the 

<Integer> -- a series of digits optionally preceded by a plus or 

minus sign. Examples are 121, +300. -8. 

<real> -- a representation of a floating point number In"eIther 

decimal or exponential form. Reel I numbers must contain a 

decimal points and may be optionally signed. Examples are 

1.0, -3.5, 5.24E-OB. 

<number> -- either an Integer or a real number can be Input. The 

data Item Is converted to a real number. 
259 

<Integer list> -- a sequence of Integers. The sequence may be 

listed explicitly or defined over a range of Integers with a 

constant Increment. The default Increment Is 1. Examples of 

Integer lists are: 1,2,4,5,8,11; 1-10; 2 TO 20 BY 2. 

<real list> -- a sequence of real numbers. Real lists have the 

same form as Integer lists except that there Is no default 

Increment. Examples are: 1.0, 1.5, 2.0, 3.0; 0.0-2.5 BY 

0.25. 

<number list> -- either an Integer I ist or a real list Is input. 

The data Is converted to real. 

<label> -- a series of letters and digits beginning with a letter. 

Examples are: PLANEFRAME~ DEADLOAD10. 

<string> -- any text enclosed wIthin single or double quotes. An 

example Is: "THIS IS A STRING". 

In some Instances a description of the physical meaning of the data Item 

Is added to the class in the syntax of a descriptor. This Is helpful In 

clarifying the use of the data Item. For example a command of the form 

STRUCTURE <structure name:label> 

Implies that the data Item fol lowing the word STRUCTURE Is a label 

defining the name of the structure. 
260 

It is not always necessary to completely spel lout every word on a 

command line In order to have the command correctly translated. Many 

words can be abbreviated and these are Identified In the command syntax 

by underlinIng. The underlined portions of words identify the minimum 

Input necessary for proper command translation. Descriptors are not underlined but are replaced by an item of thE~ specified class when appi Icable. If the command syntax has the form: 

NUMBER OF NODES <Integer> 

thel following Is acceptable as Input: 

NUM OF NODE 10 

When on I y one word from a gr"oup of words may be se I ected as Input, 

thEI choT ces are II sted one above the other' and enclosed I n braces, 

!I{ }". The command syntax 

COMPUTE ~ ffi£.ENESS t 

t DISPLACEMENTS ~ 

Implies that any of the fol lowing commands are acceptable: 

COMPUTE STIFFNES 

COMPUTE 0 I SPlACEMENTS 

COMPUTE DISPL 
261 

When an entire word or phrase in the command is optional, It is enclosed within parentheses. The command with the syntax 

NUMBER (OF) NODES <Integer> 

can be written as 

NUM NODES 100 

When more than one word from a group of words may be selected, the 

group Is enclosed In brackets .. u[ J"o The command 

OUTPUT DISPLACEMENTS 

STRAINS 

STRESSES 

ImplIes that the user may request 

OUTPUT DISPL STRAINS 

Brackets and braces are combined to sl low more flexibfllty In 

designing comrnandso The command syntax 

Implies that the user may enter data of the form: 

x 000 Y 0.0 2 5 e O 

2 X 1.0 2500 
262 

Continuation of an Input line onto a second physical line Is accomplished by placing a comma at the end of the Itne to be continued. 

Comments may be placed In the data by placing a C In column 1 and a 

b liank I n col umn 2 of the commeni' line. 

One method for line termination Is to place a dollar sign "$" on 

. thl9 line. Space on the line following the "$" Is Ignored by the transla'tor and and may be used for comments. 

7.:3.2 .Q.Qmm.Qn.d Syntax 

Ex;ample of the command to specify primary mass: 

ELEMENT 1 TYPE CSTRIANGLE CONSISTENT E 1 NU .3 DENSITY .00074 

Example of the commands to specify secondary mass: 

MASS 

NODAL 

2 U V W 20.0 THETAX THETAY 5.0 

ELEMENT MASS FOR TYPE PLANE FRAME 

3 LINEAR U V W FRACTIONAL LA 0.25 LB 0.75 WA 3.0 WB 8.0 

1 CONCENTRATED U V W L 3.6 M 5.0 

2 CONCENTRATED THETAZ L 3.6 M 3.0 

USE LOADING DEAD~OAD G 386.4 

Assembly command: 

COMPUTE MASS (FOR) { STRUC1URE } < I abe I > 

ELEMENT 

Output command: 

OUTPUT MASS (FOR) {STRUCTURE} < I abe I> 

ELEMENT 
263 

SpecIfIcatIoo Qf DampIng 

Modal damping: 

DAMP I NG MODAL {RATIOS } [<mode Ii st> <number>] PERCENTS 

Rayleigh damping: 

DAMPING RAYLEIGH 

output command: 

[{

FREQUENCIES} <number> <number>] PERIOD 

{

RATIOS} <number> <number> 

PERCENTS 

OUTPUT DAMP I NG { MATR I X } « FOR) STRUCTURE < I abe I» (,) RATIOS 

PERCENTS 

«FOR) MODES <Integer list» 

Units of seconds for time and rad/sec for frequencies wit I be requtredo 

Spec'trcatfon .Qf EriQuency Ana Ivsi s 

DefInItion of the frequency analysis method: 

FREQUENCY ANAL YS I S <TYPE) .t!QB.l 

JACOBI 

SUBSPACE 

NEWTON 

PROPERTIES <defIne properties unique to each type> 

Computation request: 

COMPUTE (NONL INEAR) [(NATURAL) FREOUENCIES] ((FOR) STRUCTURE .. (9) 

(MODE) SHAPES 

<label» «FOR) LOADING <label> (TIME) STEPS <Integer list» 
264 

standard output request: 

OUTPUT < NONL I NEAR) [< NATURAL) FREQUENC I ES] ( < FOR) STRUCTURE < I abe I> (,) (MODE) ,SHAPES 

[

<FOR) LOADING <label> (TIME) STEPS <Integer list>] 

(FOR) MODES <Integer list> 

Mod.:11 loads output request: 

OUTPUT MODAL LOADS « FOR) STRUCTURE < I abe I » ( , ) 

[

(FOR) LOADING <label> J 

(FOR) MODES <Integer list> 

~~:I f I catJ..Qn .Q.f. .lWu:-supp I led .Mo~ Sbapes 

Command sequence: 

ALTERNATE (MODES) <label> «TITL~) <string» 

<specification of DOF order: U V W UX ••• > 

[ 

MODE <mode number: Integer> J 

[<node number:lnteger> [<DOF value:number>]] 

SpecifIcation Qf DynamIc Reduct19n 

Element declaration for intermediate level structure: 

ELEMENT 1 TYPE <structure name> CONDENSED (,) 

t 

RETA INEO {MODES <I nteger II st>} ) NODES <integer> ~ 

USE ALTERNATE (MOQES) <label> j 
265 

Specification Qf Initial CondItions 

Command sequence: 

INITIAL CONDITIONS <label> «TITLE) <string» 

DISPLACEMENTS 

--[<node I 1st> <OOF I 1st> = <number>] 

VELOCITIES 

- [<node I r st> <DOF I t st> = <number>] 

USE (NONLI NEAR) 0 I SPLACEMENTS « FOR) STRUCTURE < I a be I > ) (,) 

(FOR) LOADING <label> (STEP <Integer» 

SpecIfication ~ Dynamic LoadIng Condlt[Qn 

Input of support accelerations as F(x,y,z): 

LOADING <label> «TITLE) <strIng» 

(NODAL> ACCELERAT IONS 

[<node lIst> <OOF list> <number>] 

Definition of loading condition: 

LOADING <label> «TITLE) <string» 

[

DYNAMIC ] NONLINEAR 

DefinitIon of Get): 

For a harmonic variatIon of G(t): 

HARMONIC PERIOD <number> (PHASE (ANGLE) <number» (p) 

(COMBINE) [<label> (FACTOR) <number> (0)] 

For a general variation of G(t): 

GENERAL (COMBINE) [<label >[{~gRS} <number list>] ] 
266 

For an ImpLllslve variation of G(t): 

IMPULSIVE (SHAPE) (HALF_SINE ) DURATION 

) RECTANGULAR ~ } POS TR I ANGULAR ( ~ NEG TR I ANGULAR ) 

(COMBINE) [<label> (FACTOR) <number> ] 

Step sIze definition: 

<number> ( , ) 

[ (TIM£) STEPS <Integer list> «TITLE) <string» <number list> (,) 

(SECONDS) ] 

Definition of results saved In the data base: 

(SAVE (T I ME) STEPS < f nteger" I f st» 

NotEI that the I ast step computed I s a I ways saved, even I f not I n the 

I ntElger I I Sit or I f the conunand I s not g t ven. 

Spec;lflcgtlon Qf. Transient anm~ 

Definition of the transient analysis method: 

TRANSIENT ANALYSIS (TYPE) MODE_SUPERPOSITION 

NEWMARK 

a~NTRAL_DIFFERENCE 

• 

PROPERTIES <define properties unique to each type> 

Computation request: 

COMPUTE [NONLI NEAR] D I SPLAa:MENTS « FOR) ~rRUCTURE <J abe I» 

DYNAMIC 

LOADING <label> (TIMf,) STEPS <Integer list> 

INITIAL CONDITIONS <label> 

INCLUDE MODES <Integer list> 

(, ) 
Output request: 

OUTPUT [DYNAM I C ] NONLINEAR 

267 

[

DI.SPLACEMENTSJ «Integer VELOCITIES 

STRAINS 

STRESSES 

«FOR) STRUCTURE <label» (,) 

list» ( , ) 

(FOR) LOADING <label> (TIME) STEPS <Integer list> 

Specification Qf Shock Spectrum Analysis 

DefinItion of the spectrum: 

(SHOCK) SPECTRUM <label> «TITLE) <string» 

1 

DISPLACEMENTS I VELOCiTIES <number list> 

ACCELERATIONS 

{ PERIODS } FREQUENCIES 

<number I r st> 

DIRECTIONS (,) 

<node list> [1 ~ rdlrectlon COSlne.number>] 

Output request: 

OUTPUT DYNAMIC DISPLACEMENTS «Integer list» (,> 

VELOCITIES 

STRESSES 

STRAINS 

PARTICIPATION_FACTORS 

«FOR) STRUCTURE <label» (,) 

(FOR) (SHOCK) SPECTURM <label> 

(FOR) MODES [< t nteger I f st> ] SRSS 

PEAICSRSS 
268 

~)~ Qf Utility Comman.ds. 

Trace command: 

TRACE~ [ NONL I NEAR] ( SOL UT I ON) DYNAMIC 

Destroy command: 

DESnWY [ NONLI NEAR] RESULTS (FOR) STRUCTURE < I abe I > ( # ) 

DYNAMIC 

{(FOR) LOADING <label> (TIME) STEPS 

{(FOR) (SHOCK) SPECTRUM <label> 

~(FOR) MODES <Integer list> 

< I nteger II st>} 
269 

7.4 Sample Input 

The fol lowing sections contain example Input data illustrating the 

use of the foregoing commands for dynamic analysis of some sImple plane 

structures. Each example problem Is I rberal Iy commented to explain the 

analysiS process. The substructured nonlinear analysIs of Section 7.4.5 

deserves additional discussion and Is described In detail In that sectlon. 

7.4.1 Standard LInear Struyture - Vibration AnalysIs 

*RUN FINITE 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

EXAMPLE INPUT NO.1 FOR DYNAMIC ANALYSIS 

======================================== 

(REFERENCE FIGURE 7.1> 

THIS EXAMPLE ILLUSTRATES THE INPUT NECESSARY TO DESCRIBE 

A LINEAR, THREE ELEMENT. PLANE FRAME AND TO PERFORM 

A FREQUENCY ANALYSIS OF THE STRUCTURE. IN ANTICIPATION 

OF A TRANSIENT ANALYSIS BY MODE SUPERPOSITION. THE 

LOADS ARE OUTPUT IN MODAL COORDINATES. THE PROBLEM IS 

RESTARTED AND THE TRANSIENT ANALYSIS IS INVOKED WITH ONLY 

SELECTED MODES INCLUDED. INITIAL CONDITIONS ARE ALSO 

DEFINED AND INCORPORATED INTO THE TRANSIENT ANALYSIS. 

STRUCTURE FRAME 

NUMBER OF ELEMENTS 3 NODES 4 

ELEMENTS 1,3 TYPE PLANEFRAME LUMPED E 30000 G 12000 AX 20. B 

AY 5.877 IZ 724 DENSITY 0.00074 

ELEMENT 2 TYPE PLANEFRAME CONSISTENT E 30000 G 12000 AX 14.4, 

AY 3.4 IZ 273. DENSITY 0.00074 

COORDINATES 

1 0.0 0.0 

2 0.0 96.0 

:; 9600 96.0 

4 96.0 0.0 

INCI DENCES 

1 1 2 
C 

C 

C 

C 

C 

C 

c 

C 

C 

C 

C 

C 

c 

C 

C 

C 

C 

C 

C 

C 

223 

334 

CONSTRAINTS 

1,4 ALL = 0.0 

LOAD I NG MOTOR 

270 

ELEMENT LOADS FOR TYPE PLANEFRAME 

2 CONCENTRATED FORCE Y P 300.0 L 48.0 

MASS 

NODJ\L MASS 

2.3 U V W 1.0 

ELEMENT MASS FOR TYPE PLANEFRAME 

2 UNIFORM U V W W 0.05 $ THE SECOND W INDICATES AN INTENSITY 

USE LOADING MOTOR G 386.4 

STATIC LOADING PATTERN: FeX,Y,Z) 

LOADING PATTERN 

ELEMENT LOADS FOR TYPE PLANEFRAME 

1 LINEAR FORCE Y LA 0.0 LB 1.0 WA 0.0 WB 1.0 

DYNAMIC LOADING CONDITION: GeT) 

LOADING SHAKE 

DYNAMIC 

HARM)NIC PERIOD 0.04 PHASE 0.0 COMBINE PATTERN 6.0 

TIME STEPS 1-100 0.0 TO 10.0 BY 0.1 

SAVE STEPS 10-100 BY 10 

SPECIFY FREQUENCY ANALYSIS PARAMETERS 

FREQUENCY ANALYSIS TYPE SUBSPACE 

PROPERTIES 

CONVERGENCE TOLERANCE 1. OE-08, 

MAX I MUM ITERAT IONS 13, 

SHIFT EVERY 2 MODES, 

MAXIMUM MODES 4, 

RANGE MIN 0.0 MAX 50~0 

REQUEST COMPUTATION OF FREQUENCIES AND MODES 

COMPUTE NATURAL FREQUENCIES MODE SHAPES STRUCTURE FRAME 

OUTPUT FREQUENCIES SHAPES STRUCTURE FRAME MODES 1-4 

OUTPUT MODAL LOADS STRUCTURE FRAME LOADING SHAKE/MODES 1-4 

STOP 

*RUN FINITE FILES = 20,,22,23 

C 
271 

C 

C RESTART OF THE LINEAR PLANEFRAME TO PERFORM THE 

C TRANSIENT ANALYSIS AFTER EXAMINING THE MODAL LOADS. 

C 

C 

ACCESS STRUCTURE FRAME 

C 

C SPECIFY TRANSIENT ANALYSIS PARAMETERS 

C (USE DEFAULT PROPERTIES) 

C 

·c 

C 

C 

TRANSIENT ANALYSIS TYPE MODE_SUPERPOSITION 

C DEFINE INITIAL CONDITIONS FOR DISPLACEMENTS. 

C (VELOCITIES NOT REQUIRED SINCE WE HAVE NO DAMPING.) 

C 

C 

C 

INITIAL CONDITIONS DEAD_SHAPE 

DISPLACEMENTS 

2,3 U ~ 0$1 

2 THETAZ = -0.085 

3 THETAZ = 0.085 

COMPUTE DYNAMIC DISPLACEMENTS STRUCTURE FRAME LOADING SHAKE, 

TIME STEPS 1-25 INITIAL CONDITIONS DEAD_SHAPE INCLUDE, 

MODES 1,3, 4 

OUTPUT DYNAMIC DISPLACEMENTS 2,3 STRUCTURE FRAME LOADING SHAKE, 

STEPS 10, 20, 25 $ RECALL THAT STEP 25 IS SAVED EVEN THOUGH 

C I DIDN'T REQUEST IT. 

C 

C 

OUTPUT WIDE BY ELEMENT DYNAMIC STRESSES STRAINS ALL STRUCTURE FRAME, 

LOADING SHAKE STEPS 10. 20. 25 

STOP 
272 

2 (1) ~-------------- ._---------------- 3 

CD 

1 

3 - Node Number 

cg)- Element Number 

4 

v 

,t 

NODAL DOF:Ce-u 

8z 

Figure 7.1 -- Three Element, Plane Frame (Example Input #I-~) 
273 

7.4.2 Standard LInear Structure - Shock Spectrum AnalysIs 

*RUN FINITE 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

EXAMPLE INPUT NO.2 FOR DYNAMIC ANALYSIS ======================================== 

(REFERENCE FIGURE 7.1) 

THIS EXAMPLE ILLUSTRATES THE INPUT NECESSARY TO DESCRIBE 

A LINEAR, THREE ELEMENT. PLANE FRAME AND TO PERFORM 

A SHOCK SPECTRUM ANALYSIS OF THE STRUCTURE. THE SHOCK 

SPECTRUM CAN CONTAIN BOTH HORIZONTAL AND VERTICAL 

COMPONENTS. WE CAN GET THE PARTICIPATION FACTORS 

(ALSO KNOWN AS EFFECTIVE MODAL MASS) PRIOR TO REQUESTING 

THAT SPECTRAL DISPLACEMENTS, STRESSES, AND STRAINS BE 

COMPUTED. TWO METHODS ARE AVAILABLE FOR SUMMING THE 

SPECTRAL VALUES FOR EACH MODE: SRSS AND PEAK_SRSS. 

SRSS IS THE SQUARE-ROOT-OF-THE-SUM-OF-THE-SQUARES METHOD. 

PEAK_SRSS TAKES THE MODE WITH THE LARGEST PARTICIPATION 

FACTOR AND ADDS TO THAT THE SRSS OF THE REMAINING MODES 

THAT ARE INCLUDED IN THE ANALYSIS. 

STRUCTURE FRAME 

NUMBER OF ELEMENTS 3 NODES 4 

ELEMENTS 1,3 TYPE PLANEFRAME LUMPED E 30000 G 12000 AX 20. , AY 5.877 IZ 724 DENSITY 0.00074 

ELEMENT 2 TYPE PLANEFRAME CONSISTENT E 30000 G 12000 AX 14.4 ~ 

AY 3.4 IZ 273. DENSITY 0.00074 

COORDINATES 

1 O~O 0.0 

2 0.0 96.0 

3 96.0 96.0 

4 96.0 0.0 

INCIDENCES 

1 1 2 

2 2 3 

3 3 4 

CONSTRAINTS 

1,4 U V = 0.0 

LOADING MOTOR 

ELEMENT LOADS FOR TYPE PLANEFRAME 

2 CONCENTRATED FORCE Y P 300.0 L 48.0 

MASS 
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

c 

c 

c 

c 

C 

274 

NODAL MASS 2.~) U V W 1.0 

ELEMENT MASS FOR TYPE PLANEFRAME 

2 UNIFORM U V W W 0.05 

USE LOADING MOTOR G 386.4 

SHOCK SPECTRUM FAIL_SAFE "5 PERCENT DAMPING" 

VELOCITIES 2 30 30 12 

PERIODS .05 .60 4.5 10. 

DIRECTIONS 1,4 U 0.866 V 0.5 

SPECIFY FREQUENCY ANALYSIS PARAMETERS 

FREQUENCY ANALYSIS TYPE SUBSPACE 

PROPERTIES 

O)NVERGENCE TOLERANCE 1.0E-08, 

M/\XIMUM ITERATIONS 13. 

SH I FT EVERY 2 MODES, 

M/\X I MUM MODES 4, 

RJ\NGE MIN 0.0 MAX 50.0 

REQUEST alMPUTATION OF FREQUENCIES AND MODES 

(~MPUTE NATURAL FREQUENCIES MODE SHAPES STRUCTURE FRAME 

OUTPUT FREQUENCI ES SHAPES STRUCTURE FRAME MODES 1-4 

OUTPUT DYNAMIC PARTICIPATION FACTORS STRUCTURE FRAME SHOCK, 

SPECTRUM FAIL_SAFE MODES ALL 

STOP 

*RUN FINITE FILES = 20,,22,23 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

RESTART OF THE LINEAR PLANEFRAME TO PERFORM THE 

SHOCK SPECTRUM ANALYSIS AND alMPUTE SPECTRAL STRESS 

AND STRAINS. AT THIS POINT, WE HAVE HAD AN OPPORTUNITY 

TO EXAMINE THE PARTICIPATION FACTORS AND SEE THAT ONLY 

THREE OF THE MODES HAVE ANY SIGNIFICANT alNTRIBUTION TO 

THE SPECTRAL RESPONSE. 

ACCESS STRUCTURE FRAME 

C 

C 

C 

OUTPUT DYNAMIC DISPLACEMENTS STRESSES STRAINS STRUCTURE FRAME, 

SPECTRUM FAIL SAFE MODES 1-3 SRSS PEAK SRSS 

STOP 
275 

7.4.3 Standard NonlInear Structure 

*RUN FINITE 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

c 

C 

C 

C 

C 

C 

C 

EXAMPLE INPUT NO.3 FOR DYNAMIC ANALYSIS ======================================== 

(REFERENCE FIGURE 7.1) 

THIS EXAMPLE ILLUSTRATES THE INPUT NECESSARY TO DESCRIBE 

A NONLINEAR, THREE ELEMENT, PLANE FRAME AND TO PERFORM 

A TRANSIENT ANALYSIS OF THE STRUCTURE. AT SOME SELECTED 

TIME STEP, THE TRANSIENT ANALYSIS IS SUSPENDED AND A 

FREQUENCY ANALYSIS IS PERFORMED WITH THE CURRENT STRUCTURE 

STIFFNESS AND MASS. THE GENERALIZED NEWMARK OPERATOR IS 

USED FOR THE TIME-HTORY INTEGRATION. THIS OPERATOR 

CAN BE USED AS EITHER AN EXPLICIT OR IMPLICIT INTEGRATOR 

AND HAS THE ABILITY TO CONTROL SPURIOUS DAMPING. 

MATERIAL STEEL TYPE VON-MISES 

PROPERTIES SIGNAL_YIELD 

USE STRESS-STRAIN FUNCTION SEGMENTAL 

PROPERTIES E 30000. NU 0.3 STRAIN-HARDENING TENSYIELD 30, 

OOMPYIELD 30. TENSIO~SLOPE 3000. COMP_SLOPE 3000. 

STRUCTURE FRAME 

NUMBER OF ELEMENTS 3 NODES 4 

ELEMENTS 1,3 TYPE PLANEFRAME LUMPED E 30000 G 12000 AX 20. f 

AY 5.877 IZ 724 DENSITY 0.00074 

ELEMENT 2 TYPE PLANEFRAME CONSISTENT MATERIAL STEEL E 30000 I 

G 12000 AX 14.4 AY 3.4 IZ 273. DENSITY 0.00074 

COORDINATES 

1 O~O 0.0 

2 0.0 96.0 

3 96.0 ·96.0 

4 96.0 0.0 

INCIDENCES 

112 

223 

334 

CONSTRAINTS 

1.4 ALL = 0.0 

LOADING MOTOR 

ELEMENT LOADS FOR TYPE PLANEFRAME 

2 CONCENTRATED FORCE Y P 300.0 L 48.0 
C 

C 

C 

C 

C 

C 

C 

C 

276 

MASS 

NODAL MASS 

2,3 U V W 1.0 

ELEMENT MASS FOR TYPE PLANE FRAME 

2 UNIFORM U V W W 0.05 

USE LOADING MOTOR G 386.4 

LOADING PATTERN 

ELEMENT LOADS FOR TYPE PLANEFRAME 

1 LINEAR FORCE Y LA 0.0 lB 1.0 WA 0.0 WB 1.0 

LOADING SHAKE 

NONLINEAR DYNAMIC 

HARrJONIC PERIOD 0.04 PHASE 0.0 COMBINE PATTERN 6.0 

TIME STEPS 1-100 0.0 TO 10.0 BY 0.1 

SAVE STEPS 10-100 BY 10 

SPECIFICATION OF NONLINEAR SOLUTION PARAMETERS 

TRACE NONLINEAR SOLUTION 

CONVERGENCE TEST NORM RESIDUAL LOADS TOLER 1.5 INCLUDE TOTAL, 

REACTIONS 

UPDATE STIFFNESS EVERY STEP 

TERMINATE IF NONCONVERGENT 

SPECIFICATION OF TRANSIENT ANALYSIS SOLUTION PARAMETERS 

TRANSIENT ANALYSIS TYPE NEWMARK 

PROPERTIES ALPHA 0.0 BETA 0.5 GAMMA 0.25 

C 

C REQUESTS FOR COMPUTATION AND OUTPUT. 

C 

C 

c 

C 

C 

C 

CnMPUTE NONLINEAR DYNAMIC DISPLACEMENTS STRUCTURE FRAME LOADING, 

SHAKE TIME STEPS 1-30 

OUTP~T DYNAMIC NONLINEAR STRESSES 2 STRUCTURE FRAME LOADING SHAKE, 

STEPS 10.20.30 

STOP 

*RUN FINITE FILES = 20,,22,23 

C 

C 

C 

C 

C 

C 

C 

RESTART OF NONLINEAR PLANE FRAME TO PERFORM THE 

FREQUENCY ANALYSIS AND GET THE NATURAL FREQUENCIES 

AND MODE SHAPES. 

ACCESS STRUCTURE FRAME 

C 

C 

C SPECIFICATION OF FREQUENCY ANALYSIS 
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

FREQUENCY ANALYSIS TYPE JAros I 

PROPERTIES 

MAX_SWEEPS 15, 

CONVERGENCE TOLERANCE 1.OE-OB 

277 

REQUEST roMPUTATION AND OUTPUT OF FREQUENCIES AND MODES 

roMPUTE NONLINEAR NATURAL FREQUENCIES MODE SHAPES STRUCTURE FRAME~ 

LOADING SHAKE TIME STEP 20 

OUTPUT NONLINEAR FREQUENCIES SHAPES STRUCTURE FRAME MODES ALL, 

LOADING SHAKE STEP 20 

STOP 

AT THIS POINT, WE CAN "ACCESS STRUCTURE ••• " AGAIN 

AND CONTINUE WITH THE TRANSIENT ANALYSIS 
278 

7.4.4 Substructured Linear Anal¥~ 

*RUN FINITE 

C 

C 

C 

C 

C 

C 

C 

·C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

EXAMPLE INPUT NO.4 FOR DYNAMIC ANALYSIS 

======================================== 

(REFERENCE FIGURE 7.2) 

THIS EXAMPLE ILLUSTRATES THE INPUT NECESSARY TO DESCRIBE 

A LINEAR, MULTILEVEL SUBSTRUCTURED MODEL AND TO PERFORM 

A FREQUENCY ANALYSIS OF THE HIGHEST LEVEL STRUCTURE. 

FEATURES OF DYNAMICS ILLUSTRATED ARE GUYAN REDUCTION AND 

MODAL SYNTHESIS FOR CONDENSING THE SUBSTRUCTURES AND THE 

METHOD FOR CARRYING FORWARD SUBSTRUCTURE MASS (IE. NO 

SPECIAL CONSIDERATION IS GIVEN TO BRINGING UP MASS). 

THE ENTIRE STRUCTURAL SYSTEM IS BUILT OUT OF ONE STANDALONE ELEMENT. A PLANE FRAME ELEMENT. 

ELEMENT BAR TYPE PLANEFRAME CONSISTENT E 30000. G 12000. AX 20.0, AY 5.877 IZ 724 DENSITY 0.00074 

COORD I NATES 

1 0.0 0.0 

2 96.0 0.0 

C DEFINE LOWEST LEVEL STRUCTURE, A TRUSS WITH THREE BAYS. 

C 

C 

C 

C 

C 

C 

STRUCTURE THREE-BAY 

NUMBER OF ELEMENTS 7 NODES 5 

ELEMENTS ALL TYPE BAR 

1,6,7 ROTATION SUPPRESSED 

2, 4 ROTATION Z 60. 

3. 5 ROTATION Z -60. 

INCIDENCES 

1 1 2 

2 3 1 

3 1 4 

4 4 2 

525 

634 

7 4 5 

MASS 

ADD MASS TO ONE OF THE LOWER OiORDS. 

ELEMENT MASS FOR TYPE PLANEFRAME 

7 UNIFORM U V W W 0.0003 
279 

C 

C CONDENSE OUT NODE 4 USING GUYAN REDUCTION 

C 

C 

C 

C 

'C 

C 

C 

c 

C 

STRUCTURE THREE_CON 

NUMBER OF ELEMENTS 1 NODES 4 

ELEMENT 1 TYPE THREE_BAY CONDENSED 

INCIDENCES 

13152 

STICK TWO OF THE FRAMES TOGETHER AND CLOSE THE GAP 

AT THE TOP WITH A BAR ELEMENT. 

STRUCTURE SPAN 

NUMBER OF ELEr-IENTS 3 NODES 7 

ELEMENTS 

1,3 TYPE THREE_CON ROTATION SUPPRESSED 

2 TYPE BAR ROTATION SUPPRESSED 

INCIDENCES 

1 1 2 3 4 

2 4 5 

33567 

C ADD A LITTLE MORE MASS TO SOME SELECTED NODES 

C 

C 

C 

C 

C 

C 

C 

MASS 

NODAL MASS 

4 p 5 U V W 0.5 

DEFINE FREQUENCY ANALYSIS PARAMETERS FOR THIS STRUCTURE 

SINCE IT WILL BE CONDENSED USING MODAL SYNTHESIS 

FREQUENCY ANALYSIS TYPE SUBSPACE 

PROPERTIES 

CONVERGENCE TOLER 1.0E-08, 

MAX ITERATIONS 10. 

MAX MODES 5 

C CONDENSE SPAN VIA MODAL SYNTHESIS 

C 

C 

C 

C 

C 

C 

C 

STRUCTURE SPAN_CON 

NUMBER OF ELEMENTS 1 NODES 2 

ELEMENT 1 TYPE SPAN CONDENSED RETAINED MODES 5 

INCIDENCES 

1 1 6 

BUILD THE TWO SPAN BRIDGE 

NOTE THAT THE THETAZ DOF AT THE r-ll DOLE SUPPORT KEEPS 

THIS STRUCTURE FROM BECOMING n~o SIMPLE SPANS. 
C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

C 

280 

ALSO NOTE THAT ALTHOUGH THIS STRUCTURE ONLY HAS 3 NODES 

WITH 3 GEOMETRIC DOF EACH, IT HAS 10 GENERALIZED DOF FROM 

THE RETAINED NORMAL MODES FOR A TOTAL OF 19 OOF. 

STRUCTURE TWO_SPAN 

NUMBER OF ELEMENTS 2 NODES 3 

ELEMENTS ALL TYPE SPAN_CON ROTATION SUPPRESSED 

INCIDENCES 

1 1 2 

2 2 3 

CONSTRAINTS 

1 U V = 0.0 

2 3 V = 0.0 

FREQUENCY ANALYSIS TYPE SUBSPACE 

PROPERTIES 

CONVERGENCE TOLER 1.0E-OB, 

MAX ITERATIONS 10. 

MAX MODES 10 

COMPUTE FREQUENCIES STRUCTURE 1lI0_SPAN 

OUTPUT FREQUENCIES SHAPES STRUCTURE 1lI0_SPAN MODES ALL 

STOP 
281 

1 2 

CD 1 .... ----..2 

3 ® 4 CD 5 

Element Bar Structure Three'" Bay 

2 4 5 7 

~----~~~-.----~ 

1 

4_=-_Node Number 

@ ... Element Number 

3 

Structu re Span 

6 

structure Two-Span tV 

NODAL DOF: (:e~z="'''''u 

Figure 7.2 -- Two Span Bridge (Example Input ~4) 
282 

7.4.5 ~;tructured Nonlinear 1\.n.alysls 

The Input listing In this section describes a simple, two-span 

bridge. Nonlinearity Is introduced Into the example by addition of a 

nonlinear bar element over the center support. The example uses six 

levels of substructures with the nonlinear element added at the highest 

level. To facilitate the description of the Input sequence, line 

numbers are placed before each FINITE command line. The numbers are not 

part of the commands. They serve only as reference numbers. Comment 

lines are not numbered. The structure is Illustrated In Figure 7.3. 

Lines 2-6 define a stand-al()ne element, which Is used to construct 

the maJor'lty of the final structure. Lines 7-23 describe the lowest 

I eve I substructure ca I led THREE_BAY. No frequE~ncy ana I ys I s parameters 

are defined since this structure wll I be condensed using Guyan reduction. The condensed version of THREE_BAY Is named structure PIECE. No 

additional Input Is required t() Invoke the condensation process; Guyan 

reduction Is the default procedure adopted by the system. 

Lines 29-48 describe structure HALF which contains two copies of 

substructure PIECE and one copy of stand-alone element BAR. To Il1ustrate Its use In substructures. secondary mass Is applied In Ifnes 

38-40. A frequency analysis method Is de-fined, lines 44-48, so the 

structure can be condensed by modal synthesIs. Structure HALF_CON. 

lines 49-56, I s the condensed vers I on of structure HALF. I tis necessar'y to cCllrry forward the loads from HALF but the mass Is automat I ca I I Y 

Included In the reduction process. 
283 

A non I I near mater i a I mode I Is def i ned in lines 57-61. Th is mode I 

Is required for the nonlinear element used In structure BRIDGE. This 

highest level structure is composed of two condensed substructures, 

HALF_CON. and one simple element, TYPE PLANEFRAME (see lines 62-67). 

Damping and dynamic loading are defIned In lines 78-88 wIth the tran-

. slent analysis specification and requests for computation and output In 

I I nes 89-96. 

*RUN FIN ITE 

C 

C 

C EXAMPLE INPUT NO.5 FOR DYNAMIC ANALYSIS 

C ======================================== 

C 

C (REFERENCE FIGURE 7.3) 

C 

C THIS EXAMPLE ILLUSTRATES THE INPUT NECESSARY TO DESCRIBE 

C A NONLINEAR, MULTILEVEL SUBSTRUCTURED MODEL AND TO PERFORM 

C A TRANSIENT ANALYSIS OF THE STRUCTURE WHEN IT IS SUBJECTED 

C TO A GENERAL DYNAMIC LOADING. THE NONLINEARITY IS 

C RESTRICTED TO A MATERIALLY NONLINEAR ELEMENT AT THE HIGHEST 

C LEVEL. THE STRUCTURE IS THE TRUSS FROM EXAMPLE NO.4 WITH 

C THE GAP OVER THE CENTER SUPPORT CLOSED BY THE NONLINEAR 

C ELEMENT. 

C 

C 

2 ELEMENT BAR TYPE PLANEFRAME CONSISTENT E 30000. G 12000. AX 20.0, 

3 AY 5.877 IZ 724 DENSITY 0.00074 

C 

4 COORDINATES 

5 1 OeO 0.0 

6 2 96.0 0.0 

C 

C DEFINE LOWEST LEVEL STRUCTURE, A TRUSS WITH THREE BAYS. 

C 

7 STRUCTURE THREE_BAY 

8 NUMBER OF ELEMENTS 7 NODES 5 

9 EL Ef.ilENTS AL L TYPE BAR 

10 1,6.7 ROTATION SUPPRESSED 

11 2, 4 ROTATION Z 60. 

12 3, 5 ROTATION Z -60. 

C 

13 INCIDENCES 

14 1 1 2 

15 2 3 1 

16 3 1 4 
284 

17 4 4 2 

18 5 2 5 

19 6 3 4 

20 7 4 5 

C 

C ADD MASS TO ONE OF THE LOWER CHORDS. 

C 

21 MASS 

22 ELEMENT MASS FOR TYPE PLANEFRAME 

23 7 UNIFORM U V W W 0.0003 

C 

C 

C CONDENSE OUT NODE 4 USING GUYAN REDUCTION 

C 

24 STRUCTURE PIECE 

25 NUMBER OF ELEMENTS 1 NODES 4 

26 ELEMENT 1 TYPE THREE_BAY CONDENSED 

C 

27 I NCI DENCES 

28 1 3 1 5 2 

C 

C 

C STICK TWO OF THE FRAMES TOGETHER AND CLOSE THE GAP 

C AT THE TOP WITH A BAR ELEMENT. 

C 

29 STHUCTURE HALF 

30 NUMBER OF ELEMENTS 3 NODES 7 

31 ELEMENTS 

32 1, 3 TYPE PIECE ROTATION SUPPRESSED 

33 2 TYPE BAR ROTATION SUPPRESSED 

C 

34 INCIDENCES 

35 1 1 2 3 4 

36 2 4 5 

37 3 3 5 6 7 

C 

C ADD A LITTLE MORE MASS TO SOME SELECTED NODES 

C 

38 MASS 

39 NODAL MASS 

40 4, 5 U V W 0.5 

C 

C ADD THE PATTERN OF LOAD TO BE USED IN THE DYNAMIC 

C LOADING 

C 

41 LOADING CENTER-SPAN 

42 NODAL LOADS 

43 3 FORCE Y P -1.0 

C 

C 

C DEFINE FREQUENCY ANALYSIS PARAMETERS FOR THIS STRUCTURE 

C S (NCE IT WILL. BE CONDENSED US I NG r.10DAL SYNTHES IS 

C 

44 FREQUENCY ANALYSIS TYPE SUBSPACE 
L85 

45 PRO PERT I ES 

46 CONVERGENCE TOLER 1.0E-08, 

47 MAX ITERATIONS 10. 

48 MAX MODES 5 

C 

C 

C CONDENSE HALF VIA MODAL SYNTHESIS 

C 

49 STRUCTURE HALF_CON 

50 NUMBER OF ELEMENTS 1 NODES 4 

51 ELEMENT 1 TYPE HALF CONDENSED RETAINED MODES 3 

C 

52 INCIDENCES 

53 1 1 2 6 7 

C 

C CARRY FORWARD THE LOADS FROM HALF 

C 

54 LOADING CENTER-CON 

55 EXTERNAL ELEMENT LOADS 

56 1 CENTEFLSPAN 1.0 

C 

C 

C 

C DEFINE THE NONLINEAR MATERIAL 

C 

57 MATERIAL STEEL TYPE VON_MISES 

58 PROPERTIES SIGNAL YIELD 

59 USE STRESS-STRAIN FUNCTION SEGMENTAL 

60 PROPERTIES E 1 NU 0 STRAIN HARDENING, 

61 TENSYIELD 1 TENSION_SLOPE .1 

C 

C 

C 

C 

C 

C 

C 

62 

63 

64 

65 

66 

67 

C 

68 

69 

70 

C 

71 

72 

73 

74 

C 

C 

BUILD THE TWO SPAN BRIDGE 

CLOSE THE GAP OVER THE CENTER SUPPORT WITH A 

NONLINEAR BAR. 

STRUCTURE BRIDGE 

NUMBER OF ELEMENTS 3 NODES 7 

ELEMENTS 

1,3 TYPE HALF_CON ROTATION SUPPRESSED 

2 TYPE PLANE FRAME MATERIAL STEEL CONSISTENT AX 20.0 I 

AY 5.877 IZ 724 DENSITY 0.00074 

COORDINATES 

4 0.0 0.0 

5 96.0 0.0 

INCIDENCES 

1 1 2 3 4 

2 4 5 

33567 
286 

75 CONSTRAINTS 

76 1 U V = 0.0 

77 3 6 V = 0.0 

C 

C APPLY DAMPING TO THE STRUCTURE BY USING RAYLEIGH DAMPING 

C 

78 DAMPING RAYLEIGH FREQUENCIES 2.0 12.0 PERCENTS 1.0 3.4 

C 

C 

C APPLY THE LOAD PATTERN TO EACH SPAN 

C 

. 79 LOAD I NG PATTERN 

80 EXTERNAL ELEMENT LOADS 

81 1,3 CENTER-CON 1.0 

C 

C DEFINE THE DYNAMIC, NONLINEAR LOAD. 

C 

82 LOADING SHAKE 

83 DYNAMIC NONLINEAR 

C 

84 GENERAL COMBINE PATTERN, 

85 FACTORS 0.0 100. 40. -50. 100. o. 50. -90. 0.0, 

86 TIMES 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 

C 

87 TIME STEPS 1-100 0.0 TO 1.0 BY .01 

88 SAVE STEPS 5-100 BY 5 

C 

C DEFINE THE THANSIENT ANALYSIS 

C 

89 lRANSIENT ANALYSIS TYPE NEWMARK 

90 PROPERTIES 

91 ALPHA 0.0, 

92 BETA 0.5, 

93 GAMMA 0.25 

C 

94 COMPUTE NONLI NEAR DYNAM leD I SPLACEMENTS STRUCTURE BR I DGE, 

95 LOADING SHAKE TIME STEPS 1 ... 25 

c 

96 OUiTPUT NONL I NEAR D I SPlAO::MENTS LOAD I NG SHAKE T I ME STEPS 5-25 BY 5 

C 

C 

97 STOP 
287 

1 CD 2 

1 • 

CD 

• 2 

3 ® 4 CD 5 

Element Bar Structure Three - Bay 

2 4 5 7 

~-----.--~--.-----~ 

1 

5 -Node Number 

Q)- Element Number 

3 

Structure Half 

Structure Bridge 

6 

v 

t 

NODAL DOF: ( ·--u 

8z 

Figure 7.3 -- Two Span Nonl inear Bridge (Example Input 115) 
288 
Q-IAPTER 8 

SUMtvlARY AND TOP I CS FOR FURTHER STUDY 

8.1 Summary 

Comprehensive dynamic analysis of complex structural systems by the 

finite element method can be an expensive, If not Impossible undertaking. Existing software systems capable of achieving some economy 

suffer a I imlted scope. The need exists for a general purpose FEM 

system which is capable of dynamiC analYSis of arbitrary structures. 

This capability Includes structures experIencing geometric and material 

nonllnearlties. In order to achieve an economical solution, multi level 

substructurlng Is seen as a reqUisite modeling approach. It is the purpose of this work to bring together the Individual, Isolated topics of 

multilevel substructured modeling, dynamic analYSis by the F~Mg and nonI inear continuum mechanics into the design of a comprehensive, general 

purpose, fInite element package. The resulting software wll I be used to 

perform numerical experiments to explore the behavior of the proposed 

modal synthesis technique In a multi level substructured environment. 

The factors studied wi I I include the economiCS, accuracy, and analyst 

Interaction required to perform modal synthesiS. 

Implementation of multi level substructurlng for static analysis of 

linear and nonl inear structures has been discussed In detal I. The success of the effort Is dependent upon the schemes used for data storage 

and retrieval, equation solving, and user definition of the model. It 

was shown that static results are equivalent for both substructured and 

standard models. Economy In the solution vIa substructured modeling was 

289 
290 

demonstrated in two numerical examples. 

Dynamic reduction of the stiffness and mass matrices has been iden~ 

tlfled as the pivotal process In accurately representing complex struc~ 

tures as simplified models for dynamiC analysis. A number of the 

various methods currently available for dynamic reduction have been 

'Identified by a review of the open literature. Guyan reduction and the 

fixed~interface method have been chosen for Incorporation Into the 

general purpose FEM software system. 

Elgenproblem solution and transient response analysis are the most 

computationally expensive operations In the dynamic analysis of structural systems. Their proper Implementation and use Is essential to the 

success of the dynamic analysis. A brief review of these processes and 

an examination of their use In a multi level substructured environment 

was given. The most effective elgenproblem solution methods have been 

Identified while transient response analYSiS was discussed In more 

general terms. 

Using matrix notation, the nonl inear equations of continuum 

mechanics were derived. Two for-mulatlons, Total Lagrangian (T. L.) and 

Updated Lagrangian (U. L.) were described In datal I. Both formulatIons 

were shown to derive from a common definitIon of the rate of work per 

unit mass and thus should provide identical analysis results. Differences In the computational efficiency of the two formulations were 

shc1wn to arise In the stress ratE~ transformatIons and in the complexity 

of the nonlinear strain-displacement relations. It was concluded that 

the T. L. formulation has a slight advantage in that no question arises 

regarding the significance of certain nonlinear terms, I.e., al I non-
29i 

I inear terms must be included in the formulation. 

The details of a transient solution procedure for a substructured 

nonl inear model based upon an impl icit integration operator were 

presented. An Impl icit scheme was recommended to support dynamic 

analysis since a static solution procedure can be obtained as the 

'degenerate case of dynamic analysis. Details of the elemental stiffness 

matrices were derived for both the T. L. and U. L. formulations. 

Specific matrices were listed for the general 2-D case. Qual itative 

comparisons of computational efficiency were made and a T. L. approach 

was recommended for a general software system. The current absence of 

computational evidence in the literature regarding the performance of a 

finite element solution based on each approach does not enable the 

superior approach to be identified. However, an U. L. approach can be 

easily embedded within a T. L. software system. A T. L. approach cannot be as easily Incorporated into a U. L. based system. Thus, the 

choice of T. L. provides some flexibIlity for future modifIcations. 

The POLO-FINITE input language has been extended to Include the 

computational features recommended In this report for general purpose 

dynamic analysis. Wherever possible, consistency has been maintained in 

the philosophy and method of defining the substructured model. The complete command structure was detal led and examples of Its use were 

presented. 
292 

8.2 Topics .f.Qr:. Further Study 

With the definition of the basic requirements for general dynamic 

analysis now available, efforts can be directed to software design, implementation, and verification. 

The next task to be performed is the design of a prototype software 

system with the specific goal of demonstrating the applicability of multilevel substructurlng in nonl inear dynamic analysis. During the 

literature review, no evIdence was found of this having been attempted 

at any level of sophistIcatIon. AdditIonal software desIgn topIcs Include design of the data structures and processing modules necessary for 

performing the analysis and specification of the formats for convenIent 

and selectIve output of results. 

Later actIvItIes Include ImplementatIon and testIng of the system 

In the POLO-FINITE structural mechanIcs software system. The performance of the system wil I be evaluated over a broad range of structural 

types Including general substructure geometry and I I near/non I Inear 

response. 
End of Document