- 

NASA Technical Memorandum 101622 

4 

Global/Local Stress Analysis 

of Composite Panels 

J. B. Ransom and N. F. Knight, Jr. 

June 1989 

(NASA 

AHALP 

Langl 

-TH-101622) GLOBAL/LOCAL STRESS N89 -246 8 1 

ey Research Center) 54 p CSCL 20K 

SIS OF COHPOSITE PANELS [NASAUnclas 

G3/39 0217724 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23665-5225 
GLOBAL/LOCAL STRESS ANALYSIS OF COMPOSITE PANELSt 

Jonathan B. Ransom$ and Norman F. Knight, Jr.$ 

NASA Langley Research Center 

Hampton, Virginia 

Abstract 

A method for pcrforming a global/local stress analysis is described and its capabilities are 

demonstrated. The method employs spline interpolation functions which satisfy the linear plate 

bending equation to determine displacements and rotations from a global model which are used 

as “boundary conditions” for the local model. Then, the local model is analyzed independent 

of the global model of the structure. This approach can be used to determine local, detailed 

stress states for specific structural regions using independent, refined local models which exploit 

information from less-refined global models. The method presented is not restricted to having 

a priori knowledge of the location of the regions requiring local detailed stress analysis. This 

approach also reduces the computational effort necessary to obtain the detailed stress state. 

Criteria for applying the method are developed. The effectiveness of the method is demonstrated 

using a classical stress concentration problem and a graphite-epoxy blade-stiffened panel with a 

discontinuous stiffener. 

Nomenclature 

Vector of unknown spline coefficients of the interpolation function 

Polynomial coefficients of the interpolation function, i = 0,1,. . . ,9 

Cross-scctional area 

Net cross-sectional area, (W - 2ro)h 

Stiffener spacing 

Flexural rigidity 

Young’s modulus of elasticity 

Spline interpolation function 

Natural logarithm coefficients of the interpolation function, i = 1,2,. . . , n 

Panel thickness 

Blade-stiffener height 

Stress colicexitration factor (g*)n~aa 

Number of points on the local model boundary 

Panel length 

Number of points in the interpolation region 

’ (oh),,, 

t Invited Talk at the Third Joint ASCE/ASME Mechanics Conference, University of California 

$ Aerospace Engineer, Structural Mechanics Branch, Structural Mechanics Division. 

at San Diego, July 9 -12, 1989. 

1 
Longitudinal stress resultant 

Average running load per inch 

Maximum longitudinal stress resultant, juz)mazh 

Nominal longitudinal stress resultant, (u=)nomh 

Applied load 

Pressure loading on panel 

Radius of the panel cutout 

Radial coordinate of the i-th node in the interpolation region or 

along the local model boundary 

Radius of the interpolation region 

Radius of the circular local model 

Spline matrix 

Element strain energy per unit area 

Maximum element strain energy per unit area 

Panel width 

x coordinate of the i-th node in the iiiterpolation region or 

along the local model boundary 

y coordinate of the i-th node in the interpolation region or 

along the local model boundary 

Parameter used to facilitate spline matrix computation 

Change in stress 

Longitudinal stress 

Maximum longitudinal stress 

Natural logarithm terms of the spline matrix, i,j = 1,2,. . . ,n 

Biharmonic operator 

Nominal longitudinal stress, A,,, P 

Introduction 

Discontinuities and eccentricities, which are coinmon in practical structures, increase 

the difficulty in predicting accurately detailed local stress distributions, especially when the 

component is built of a composite material, such as a graphite-epoxy material. The use of 

composite materials in the design of aircraft structures introduces added complexity due to 

the nature of the material systems and the complexity of the failure modes. The design and 

certification process for aerospace structures requires an accurate stress analysis capability. 

Detailed stress analyses of complex aircraft structures and their subcomponents are required 

and can severely tax even today’s computing resources. Embedding detailed “local” finite 

element models within a single “global” finite element model of an entire airframe structure is 

usually impractical due to the computational cost associated with the large number of degrees of 

freedom required for such a global detailed model. If the design load envelope of the structural 

component is extended, new regions with high stress gradients may be discovered. In that case, 

2 
the entire analysis with embedded local refinements may have to be repeated and thereby further 

reducing the practicality of this brute force approach for obtaining the detailed stress state. 

The phrase global/local analysis has a myriad of definitions among analysts (e.g., refs. [1,2]). 

The concept of global and local may change with every analysis level, and also from one analyst 

to another. An analyst may consider the entire aircraft structure to be the global model, and a 

fuselage section to be the local model. At another level, the fuselage or wing is the global model, 

and a stiffened panel is the local model. Laminate theory is used by some analysts to represent 

the global model, and micromechanics models are used for the local model. The global/local 

stress analysis methodology, herein, is defined as a procedure to determine local, detailed stress 

states for specific structural regions using information obtained from an independent global stress 

analysis. A separate refined local model is used for the detailed analysis. Global/local analysis 

research areas include such methods as substructuring, submodeling, exact zooming and hybrid 

techniques. 

The substructuring technique is perhaps the most common technique for global/local 

analysis in that it reduces a complex structure to smaller, more manageable components, and 

simplifies the structural modeling (e.g., refs. [3-51). A specific region of the structure may be 

modeled by a substructure or multiple levels of substructures to determine the detailed response. 

Submodeling refers to any method that uses a node by node correspondence for the displacement 

field at the global/local interface boundary (e.g., refs. [6-91). A form of submodeling is used in 

reference [8] to perform a two-dimensional to three-dimensional global/local analysis. Recently, 

the specified boundary stiffness/force (SBSF) method [9] has been proposed as a global/local 

analysis procedure. This approach uses an independent subregion model with stiffnesses and 

forces as boundary terms. These stiffnesses and forces represent the effect of the rest of the 

structure upon the subregion. The stiffness terms are incorporated in the stiffness of the 

subregion model and the forces are applied on the boundary of the local model. 

An efficient zooming technique , as described in reference [lo], employs static condensation 

and exact structural reanalysis methods. Although this efficient zooming technique involves 

the solution of a system of equations of small order, all the previous refinement processes are 

needed to proceed to a new refinement level. An “exact” zooming technique [ll] employs an 

expanded stiffness matrix approach rather than the reanalysis method described in reference 

[lo]. The “exact” zooming technique utilizes results of only the previous level of refinement. 

For both methods, separate locally-refined subregion models are used to determine the detailed 

stress distribution in a known critical region. The subregion boundary is coincident with nodes 

3 
in the global model or the previously refined subregion model which is akin to the submodeling 

technique discussed earlier. 

however, this approach presupposes that the analyst can identify an approxhstion sequence for 

Hybrid techniques (e.g., refs. [12,13]) make use of two or more methods in different domains 

of the structure. In the global variational methods, the domain of the governing equation is 

treated as a whole, and an approximate solution is constructed from a sequence of linearly 

independent functions (i. e., Fourier series) that satisfy the geometric boundary conditions. 

In the finite element method, the domain is subdivided into small regions or elements within 

which approximating functions (usually low-order polynomials) are used to describe the 

continuum behavior. Global/local finite element analysis may refer to an analysis technique that 

simultaneously utilizes conventional finite element modeling around a local discontinuity with 

classical Rayleigh-Ritz approximations for the remainder of the structure. The computational 

effort is reduced as a result of the use of the limited number of finite element clegrees of freedom; 

I 

I 

The aforementioned global/local methods, with the exception of the subn lodeling technique, 

require that the analyst know where the critical region is located before perfolming the global 

analysis. However, a global/local methodology which does not require a priori knowledge of 

the location of the local region(s) requiring special modeling could offer advantages in many 

situations by providing the modeling flexibility required to address detailed local models as their 

need is identified. 

The overall objective of the present study is to develop such a computational strategy for 

obtaining detailed stress states of composite structures. Specific objectives are: 

1. To develop a method for performing global/local stress analysis of composite structures 

2. To develop criteria for defining the global/local interface region and local modeling 

requirements 

3. To demonstrate the computational strategy on representative structural analysis 

problems 

The scope of the present study includes the global/local linear two-dimensional stress analysis 

of finite element structural models. The method developed is not restricted to having a priori 

knowledge of the location of the regions requiring detailed stress analysis. The guidelines for 

developing the computational strategy include the requirement that it be compatible with 

4 
general-purpose finite element computer codes, valid for a wide range of elemcmts, extendible 

to geometrically nonlinear analysis, and cost-effective. In addition, the computational strategy 

should include a procedure for automatically identifying the critical region and defining 

the global/local interface region. Satisfying these guidelines will provide a general-purpose 

global/local computational strategy for use by the aerospace structural analysis community. 

Global/Local Methodology 

I Global/local stress analysis methodology is defined as a procedure to detcrnine local, 

detailed stress states for specific structural regions using information obtained from an 

independent global stress analysis. The local model refers to any structural subregion within 

the defined global model. The global stress analysis is performed independent of the local stress 

analysis. The interpolation region encompassing the critical region is specified. A surface spline 

interpolation function is evaluated at every point in the interpolation region yielding a spline 

matrix, S(z,y), and unknown coefficients, a. The global field is used to compute the unknown 

coefficients. An independent, more refined local model is generated within the previously-defined 

interpolation region. The global displacement field is interpolated producing a local displacement 

field which is applied as a “boundary condition” on the boundary of the local model. Then, a 

complete two-dimensional local finite element analysis is performed. 

The development of a global/local stress analysis capability for structures generally involves 

four key components. The first component is an “adequate” global analysis. In this context, 

“adequate” implies that the global structural behavior is accurately determined and that local 

structural details are at least grossly incorporated. The second component is a strategy for 

identifying, in the global model, regions requiring further study. The third component is a 

procedure for defining the “boundary conditions” along the global/locd interface boundary. 

Finally, the fourth component is an “adequate” local analysis. In this context, “adequate” 

implies that the local detailed stress state is accurately determined and that compatibility 

requirements along the global/local interface are satisfied. The development of a global/local 

stress analysis methodology requires an understanding of each key component and insight into 

their interaction. 

In practice, the global analysis model is “adequate” for the specified design load cases. 

However, these load cases frequently change in order to extend the operating region of the 

structure or to account for previously unknown affects. In these incidents, the global analysis 

may identify new “hot spots” that require further study. The methodology presented herein 

provides an analysis tool for these local analyses. 

5 
I Terminology 

I The terminology of the global/local methodology presented herein is depicted in figure 1 I 

to illustrate the components of the analysis procedure. The global model, figure la, is a finite 

element model of an entire structure or a subcomponent of a structure. A region requiring a 

more detailed interrogation is subsequently identified by the structural analyst. This region 

may be obvious, such as a region around a cutout in a panel, or not so obvious, such as a 

local buckled region of a curved panel loaded in compression. Because the location of these 

regions are usually unknown prior to performing the global analysis, the structural analyst 

must develop a global model with sufficient detail to represent the global behavior of the 

structure. An interpolation region is then identified around the critical region as indicated in 

figure lb. An interpolation procedure is used to determine the displacements and rotations used 

as “boundary conditions” for the local model. The interpolation region is the region within 

which the generalized displacement solution will be used to define the interpolation matrix. The 

global/local interface boundary, indicated in figure IC, coincides with the intersection of the 

boundary of the local model with the global model. The definition of the interface boundary 

may affect the accuracy of the interpolation and thus the local stress state. The local model 

lies within the interpolation region as shown in figure IC and is generally more refined than 

the global model in order to predict more accurately the detailed state of stress in the critical 

region. However, the local model is independent of the global finite element model as indicated 

in figure Id. The coordinates or nodes of the local model need not be coincident with any of the 

coordinates or nodes of the global model. 

The global/local interpolation procedure consists of generating a matrix based on the 

global solution and a local interpolated field. The local interpolated field is that field which is 

interpolated from the global analysis and is valid over the domain of the local model. Local 

stress analysis involves the generation of the local finite element model, use of the interpolated 

field to impose boundary conditions on the local model, and the detailed stress analysis. 

Global Modeling and Analysis 

I 

, The development of a global finite element moclel of an aerospace structure for accuratc 

stress predictions near local discontinuities is often too time consuming to impact the design 

and certification process. Predicting the global structural response of these structures often has 

many objectives including overall structural response, stress analysis, and determining internal 

load distributions. Frequently, structural discontinuities such as cutouts are only accounted for in 

6 
the overall sense. Any local behavior is then obtained by a local analysis, possibly by another 

analyst. The load distribution for the local region is obtained from the global analysis. The 

local model is then used to obtain the structural behavior in the specified region. For example, 

the global response of an aircraft wing is obtained by a coarse finite element analysis. A typical 

subcomponent of the wing is a stiffened panel with a cutout. Since cutouts are known to produce 

high stress gradients, the load distributions from the global analysis of the wing are applied to 

the stiffened panel to obtain the local detailed stress state. One difficulty in modeling cutouts 

is the need for the finite element mesh to transition from a circular pattern near the cutout 

to a rectangular pattern away from the cutout. This transition region is indicated in figure 2 

and will be referred to as a transition square (i.e., a square region around the cutout used to 

transition from rings of elements to a rectangular mesh). This transition modeling requirement 

impacts both the region near the cutout and the region away from the cutout. Near the cutout, 

quadrilateral elements may be skewed, tapered, and perhaps have an undesirable aspect ratio. 

In addition, as the mesh near the cutout is refined by adding radial “spokes” of nodes and 

“rings” of elements, the mesh away from the cutout also becomes refined. For example, adding 

radial spokes of nodes near the cutout also adds nodes and elements in the shaded regions (see 

figure 2) away from the cutout. This approach may dramatically increase the computational 

requirements necessary to obtain the detailed stress state. Alternate mesh generation techniques 

using transition zones of triangular elements or multipoint constraints may be used; however, the 

time spent by the structural analyst will increase substantially. 

The global modeling herein, although coarse, is sufficient to represent the global structural 

behavior. The critical regions have been modeled with only enough detail to represent their 

effect on the global solution. This modeling step is one of the key components of the global/local 

methodology since it provides an “adequate” global analysis. Although the critical regions are 

known for the numerical studies discussed in a subsequent section, this a priori knowledge is not 

required but may be exploited by the analyst. 

Local Modeling and Analysis 

The local finite element modeling and analysis is performed to obtain a dctailed analysis 

of the local structural region(s). The local model accurately represents the geometry of 

the structure necessary to provide the local behavior and stress state. The discretization 

requirements for the analysis are governed by the accuracy of the solution desired. The 

discretization of the local model is influenced by its proximity to a high stress gradient. 

7 
One approach for obtaining the detailed stress state is to model the local region with an 

arbitrarily large number of finite elements. Higher-order elements may be usell to reduce the 

number of elements requircd. Detailed refinement is much more advantageous for use in the 

local model than in the global model. The local refinement affects only the lo(.al model, unlike 

embedding the same refinement in the global model which would propagate to regions not 

requiring such a level of detailed refinement. A second approach is to refine the model based 

on engineering judgement. Mesh grading, in which smaller elements are used near the gradient, 

may be used. An error measure based on the change in stresses from element to element may 

be used to determine the accuracy in the stress state obtained by the initial lcd finite element 

mesh. If the accuracy of the solution is not satisfactory, additional refinement j are required. The 

additional refinements may be based on the coarse global model or the displacement field in the 

local model which suggests a third approach. The third approach is a multi-level global/local 

analysis. At the second local model level, any of the three approaches discussc d may be used to 

obtain the desired local detailed stress state. Detailed refinement is used for local modeling in 

this study. 

Global/Local Interface Boundary Definition 

The definition of the global/local interface boundary is problem dependent. Herein, the 

location of the nodes on the interface boundary need not be coincident with any of the nodes 

in the coarse global model. Reference [7] concluded that the distance that the local model must 

extend away from a discontinuity is highly dependent upon the coarse model used. The more 

accurate the coarse model displacement field is, the closer the local model boundary may be to 

the discontinuity. This conclusion is based on the results of a study of a flat, isotropic panel with 

a central cutout subjected to uniform tension, but it extends to other structures with high stress 

gradients. 

Stresses are generally obtained from a displacement-based finite element analysis by 

differentiation of the displacement field. For problem with stress gradients, the element 

stresses vary from element to element, and in some cases this change, Au, may be substantial. 

The change in stresses, Au, may be used as a measure of the adequacy of the finite element 

I discretization. Large Au values indicate structural regions where more modeling refinement is 

needed. Based on this method, structural regions with small values of Au have obtained uniform 

stress states away from any gradients. Therefore, the global/local interface boundary should 

be defined in regions with small values of Aa (ic., away from a stress gradient). Exploratory 
studies to define an automated procedure for selecting the global/local interface boundary 

have been performed using a measure of the strain energy. The strain energy per unit area is 

selected since it represents a combination of all the stress components instead of a single stress 

component. Regions with high stress gradients will also have changes in this measure of strain 

energy from element to element. 

Global/Lo cal Interpolation Procedure 

The global/local analysis method is used to determine local, detailed stress states using 

independent, refined local models which exploit information from less refined global models. A 

two-dimensional finite element analysis of the global structure is performed to obtain its overall 

behavior. A critical region may be identified from the results of the global analysis. The global 

solution may be used to obtain an applied displacement field along the boundary (;.e., boundary 

conditions) of an independent local model of the critical region. This step is one of the key 

components of the global/local methodology; namely, interpolation of the global solution to 

obtain boundary conditions for the local model. 

Many interpolation methods are used to approximate functions (e.g., ref. [14]). The 

interpolation problem may be stated as follows: given a set of function values fj at n coordinates 

(z;, y;), determine a “best-fit” surface for these data. Mathematically, this problem can be stated 

as 

where S(ti,yi) is a matrix of interpolated functions evaluated at n points, the array a defines 

the unknown coefficients of the interpolation functions, and the array f consists of known values 

of the field being interpolated based on n points in the global model. Common interpolation 

methods include linear interpolation, Lagraiigiaii iiileryolatioii and least-squares techniques for 

polynomial interpolation. Elementary linear interpolation is perhaps the simplest method and 

is an often used interpolation method in trigonometric and logarithmic tables. Another method 

is Lagrangian interpolation which is an extension of linear interpolation. For this method, data 

for n points are specified and a unique polynomial of degree n - 1 passing through the points 

can be determined. However, a more common method involving a least-squares polynomial fit 

minimizes the the sum of the square of the residuals. The drawbacks of least-squares polynomial 

fitting include the requirement for repeated solutions to minimize the sum of the residual, and 
the development of an extremely ill-conditioned matrix of coefficients when tl~e degree of the 

approximating polynomial is large. A major limitation of the approximating 1 olynomials which 

fit a given set of function values is that they may be excessively oscillatory between the given 

points or nodes. 

Mat hematical Formulation of Spline Interpolation 

Spline interpolation is a numerical analysis tool used to obtain the “best” local fit through a 

set of points. Spline functions are piecewise polynomials of degree m that are connected together 

at points called knots so as to have m - 1 continuous derivatives. The mathematical spline 

is analogous to the draftman’s spline used to draw a smooth curve through a itumber of given 

points. The spline may be considered to be a perfectly elastic thin beam resti:ig on simple 

supports at given points. A surface spline is used to interpolate a function of I wo variables 

and removes the restriction of single variable schemes which require a rectangitlar array of 

grid points. The derivation of the surface spline interpolation function used herein is based 

on the principle of minimum potential energy for linear plate bending theory. This approach 

incorporates a classical structural mechanics formulation into the spline interpolation procedure 

in a general sense. Using an interpolation function which also satisfies the linear plate bending 

equation provides inherent physical significance to a numerical analysis technique. The spline 

interpolation is used to interpolate the displacements and rotations from a global analysis 

and thereby provides a functional description of each field over the domain. The fields are 

interp Jlated separately which provides a consistent basis for interpolating glol la1 solutions based 

on a plate theory with shear flexibility effects incorporated. That is, the out-of-plane deflections 

and the bending rotations are interpolated independently rather than calculat, tng the bending 

rotations by differentiating the interpolated out-of--plane deflection field. 

The spline interpolation used herein is derived following the approach described by Harder 

and Desmairis in reference [15]. A spline surface is generated based on the solution to the linear 

isotropic plate bending equation 

DV4w I= q. (2) 

Extensions have been made to the formulation presented in reference [15] to include higherorder polynomial terms (underlined terms in equation (3)). The extended Cai tesisn form which 

satisfies equation (2) is written as 

10 
where 

2 

Tf = (x - Xi) + (y - y;)2 (4) 

and zi,yi are the coordinates of the i-th node in the interpolation region. The higher-order 

polynomial terms were added to help represent a higher-order bending response than was being 

approximated by the natural logarithm term in the earlier formulation. The additional terms 

increase the number of unknown coefficients and constraint equations to n + 10. The n + 10 

unkiiowns (a0 , a1 , a2,. . . , as, F;) are found from equation (3) and by solving the set of equations: 

n n 

i= 1 i=l 

n n 

i= 1 i=l 

n n 

i= 1 

n 

i=l 

n 

i= 1 i= 1 

n n 

i= 1 i= 1 

The constraint equations given in equation (5) are used to prevent equation (2) from becoming 

unbounded when expressed in Cartesian coordinates. The modified matrix equations, still of the 

11 
form Sa = f, are 

0 0 0 ... 0 1 1 ... 1 

0 0 0 ... 0 y1 Y2 Yn 

0 0 0 ... 0 21 22 ... Xn 

0 0 0 ... 0 xq 2; ... x; 

0 0 0 ... 0 z; x; ... x; 

0 0 0 ... 0 xqy1 x;y2 ... XnYn 

0 0 0 ... 0 zly: XIY,~ xnyn 

0 0 0 ... 0 y; y; ... Yn 

1 21 y1 ... y: QII 012 Qln 

1 22 y2 ... y; 021 Q22 R2n 

Yn 3 Qn1 Qn2 Qnn 

0 0 0 ... 0 xly1 ~2~2 .-e xnyn 

0 0 0 ... 0 y: y; ... Yn 

2 

2 

2 

3 

.. .. .. .. 

.. 

1 2n Yn 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

fl 

f2 

fn 

where Rij = T;~ In(T:j + E) for i,j = 1,2,. . . , n and E is a parameter used to insure numerical 

stability for the case when ~ij vanishes. The extended local interpolation function is similar to 

equation (3) except that it is evaluated at points along the global/local interface boundary. That 

is, 

fi = a0 + a1xi + a2yi + a3xi2 + a4~iyi + a5yi2 + agxi3 + a7si2yi+ 

n 

a8ziyi2 + a9yi3 + F~T:~ ln(r:j>; i = 1,2, ..., 1 (7) 

j=1 

Upon solving equation (6) for the coefficients (ao, al, a2,. . . , as, Fj), equation (7) is used to 

compute the interpolated data at the required local model nodes. I 

Computational Strategy 

A schematic which describes the overall solution strategy is shown in figure 3. The 

computational strategy described herein is implemented through the use of the Computational 

Structural Mechanics (CSM) Testbed (see refs. [16] and [17]). The CSM Testbed is used 

I to model and analyze both the global and local finite element models of a structure. Two 

computational modules or processors were developed to perform the global/local interpolation 

procedure. Processor SPLN evaluates the spline coefficient matrix [S(zi, yi)]. Processor INTS 

solves for the interpolation coefficients a = {a; Fi) and performs the local interpolation to obtain 

the “boundary conditions” for the local model. Various other Testbed processors are used in the 

12 
stress analysis procedure. The overall computational strategy for the global/local stress analysis 

methodology is controlled by a high-level procedure written using the command language of 

the Testbed (see ref. [17]). The command language provides a flexible tool for performing 

computational structural mechanics research. 

Numerical Results 

The effectiveness of the computational strategy for the global/local stress analysis outlined 

in the previous sections is demonstrated by obtaining the detailed stress states for an isotropic 

panel with a cutout and a blade-stiffened graphite epoxy panel with a discontinuous stiffener. 

The first problem was selected to verify the global/local analysis capabilities while the second 

problem was selected to demonstrate its use on a representative aircraft subcomponent. The 

objectives of these numerical studies are: 

1. To demonstrate the global/local stress analysis methodology, and 

2. To obtain and interrogate the detailed stress states of representative subcomponents of 

complete aerospace structures. 

All numerical studies were performed on a Convex C220 minisupercomputer. The 

computational effort of each analysis is quantified by the number of degrees of freedom used in 

the finite element model, the computational time required to perform a stress analysis, and the 

amount of auxiliary storage required. The computational time is measured in central processing 

unit (CPU) time. The amount of auxiliary storage required is measured by the size of the data 

library used for the input/output of information to a disk during a Testbed execution. 

Isotropic Panel with Circular Cutout 

An isotropic panel with a cutout is an ideal structure to verify the global/local 

computational strategy, since closed-form elasticity solutions axe available. Elasticity solutions 

for an infinite isotropic panel with a circular cutout (e.g., ref. [IS]), predict a stress concentration 

factor of three at the edge of the cutout. The influence of finite-width effects on the stress 

concentration factors for isotropic panels with cutouts are reported in reference [19]. By 

including finite-width effects, the stress concentration factor is reduced from 1 he value of three 

for an infinite panel. 

The global/local linear stress analysis of the isotropic panel with a circular cutout shown 

in figure 4 has been performed. The overall panel length L is 20 in., the overall width W is 

13 
10 in., the thickness h is .1 in, and the cutout radius TO is 0.25 in. This geometry gives a cutout 

diameter to panel width ratio of 0.05 which corresponds to a stress concentration factor of 2.85 

(see ref. [19]). The loading is uniform axial tension with the loaded ends of the panel clamped 

and the sides free. The material system for the panel is aluminum with a Young’s modulus of 

10,000 ksi and Poisson’s ratio of 0.3. 

Global Analysis 

The finite element model shown in figure 4 of the isotropic panel with a circular cutout is a 

representative finite element model for representing the global behavior of the panel as well as 

a good approximation to the local behavior. The finite element model shown In figure 4, will be 

referred to as the “coarse” global model or Model G1 in Table 1. The finite element model has a 

total of 256 4-node quadrilateral elements, 296 nodes, and 1644 active degrees of freedom for the 

linear stress analysis. This quadrilateral element corresponds to a flat C 

is based on a displacement formulation and includes rotation about the outward normal axis. 

Originally developed for the computer code STAGS (see refs. [20,21]), this element has been 

installed in the CSM Testbed software system and denoted ES5/E410 (see ref. [IS]). 

shell element which 

The longitudinal stress resultant N, distribution shown in figure 5 reveals several features of 

the global structural behavior of this panel. First, away from the cutout, the Y, distribution in 

the panel is uniform. Secondly, the N, load near the center of the panel is much greater than the 

N, load in other portions of the panel due to the redistribution of the N, loall as a result of the 

cutout. Thirdly, the N, load at the edge of the cutout at the points ninety dc,grees away from 

the stress concentration is small relative to the uniform far-field stress state. 

The distribution of the longitudinal stress resultant N, at the panel midlength normalized 

by the nominal stress resultant is shown in figure 6 as a function of the distance from the cutout 

normalized by the cutout radius. The results indicate that high inplane stresses and a high 

gradient exist near the cutout. However, a stress concentration factor of 2.06 is obtained from 

a linear stress analysis using the “coarse” finite element model (see figure 4). This value is 28% 

lower than the theoretical value of 2.85 reported in reference [19]. Therefore, even though the 

overall global response of the panel is qualitatively correct as indicated by the stress resultant 

contour in figures 5, the detailed stress state near the discontinuity is inaccurate. 

Accurate detailed stress distributions require a finite element mesh that is substantially more 

refined near the cutout. Adding only rings of elements (Model G2 in Table 1) does not affect the 

discretization away from the cutout; however, the stress concentration factor is still 22% lower 

14 
than the theoretical value. To obtain a converged solution for the stress concentration factor, a 

sequence of successively refined finite element models were developed by increasing the number 

of radial spokes of nodes and rings of elements in the region around the cutout. A converged 

solution is obtained using a total of 3168 4-node quadrilateral shell elements (ES5/E410) in the 

global model (Model G4 in Table 1). Using an intermediate refined finite element model with 

a total of 832 4-node quadrilateral elements, 888 nodes, and 5156 active degrees of freedom, a 

stress concentration factor of 2.72 is obtained which is within 4.6% of the theoretical solution. 

This finite element model is referred to as the “refined” global model or Model G3 in Table 1. 

Normalized longitudinal stress resultant N, distributions are shown in figure 6 for the “coarse” 

global model (Gl) and the “refined” global model (G3). The stress gradient for this panel 

becomes nearly zero at a distance from the center of the panel of approximately six times the 

cutout radius. 

The distribution of the strain energy for Model G1 is shown in figure 7. The change in the 

strain energy per unit area within the transition square indicates that a high $tress gradient 

exists near the cutout and rapidly decays away from the cutout. These results are consistent 

with the structural analyst’s intuition, and the local analyses described subsequently will further 

interrogate the region near the cutout. 

Local Analyses 

A global/local analysis capability provides an alternative to global mesh refinement and 

a complete solution using a more refined mesh. For this example, the “critical” region is well 

known and easily identified by even a casual examination of the stress resultant distributions 

given in figure 5. The global model, the interpolation region and the local models considered 

are shown in figure 8. The global model corresponds to the “coarse” global model (Gl) and 

the shaded region corresponds to the interpolation region which is used to generate the spline 

coefficient matrix and to extract boundary conditions for the local model. As indicated in figure 

8, two different local models are considered: one square and one circular. Both local models 

completely include the critical region with the stress concentration. The boundary of the square 

local model coincides with the boundary of the transition square in the global model. The 

boundary of the circular model is inscribed in the transition square. That is, the outer radius of 

the circular model is equal to half the length of a side of the transition square. Both local models 

(Models LS1 and LC1 in Table 1) have the same number of 4-node quadrilateral shell elements 

(512), nuxnber of nodes (544) and number of degrees of freedom (3072). Both local models have 

only 62% of the elements used in the refined global analysis. The global/local interpolation for 

15 
both local models is performed from the data obtained from the “coarse” global model analysis. 

The radius of the interpolation region RI is 5.7~0 which includes the 48 data points within the 

transition square of the coarse global model. 

The distribution of the longitudinal stress resultant N, at the panel midlength normalized 

by the nominal stress resultant obtained using the square local model is shown in figure 9a as 

a function of distance from the cutout normalized hy the cutout radius. These results indicate 

that the global/local analysis based on the coarse global solution and the square local model 

accurately predicts the stress concentration factor at the cutout as well as the distribution at the 

global/local interface boundary. A stress concentration factor of 2.76 is obtained which is within 

1.5% of the “refined” global model (G3) solution arid 3.2% of the theoretical solution. A contour 

plot of the longitudinal stress resultant distribution is given in figure 9b and indicates that the 

local solution correlates well overall with the global solution shown in figure 5. 

The distribution of the longitudinal stress resultant N, at the panel midlength normalized 

by the nominal stress resultant obtained using the circular local model is shown in figure 10a as 

a function of distance from the cutout normalized by the cutout radius. These results indicate 

that the global/local analysis based on the coarse global solution and the circiilar local model 

accurately predicts the stress concentration factor at the cutout. A stress concentration factor 

of 2.75 is obtained which is within 1.5% of the “refined” global model (G3) solution and 3.2% 

of the theoretical solution. At the global/local interface boundary, the results from the circular 

local Todel differ slightly from the results obtained from the refined globid model analysis. 

This Jifference is attributed to interaction between the “coarse” global model, the interpolation 

region, and the location of the global/local interface boundary. A contour plot of the longitudinal 

stress resultant distribution is given in figure 10b and indicates the overall correlation with the 

distributions obtained using the global models. 

The interaction between the global model, the interpolation region, and the location of the 

global/local interface boundary is now assessed. This assessment involves refining the global 

model in the transition square, varying the radius of the interpolation region R I and varying 

the radius of the local model RL. Using an interpolation region defined as RI = 14~0 (72 data 

points from the global model), several local models are considered in which R 1, is increased from 

twice the cutout radius to five times the cutout radius. The results in figure 1 la are based on 

using the coarse global model (Gl) for the global solution. These results indicate that the local 

solution deteriorates as the global/local interface boundary is moved closer to the cutout. The 

results in figure llb are based on a slightly more refined global model(G2) for the global solution. 

16 
This global model has two additional rings of elements in the transition square. Comparing the 

results in figures lla and llb reveals the interaction between the global model and the location 

of the interface boundary. To obtain an accurate local solution for the case when the global/local 

interface boundary is located within a region with a high stress gradient requires that sufficient 

data from the global model be available in the area to provide accurate “boundary conditions” 

for the local model. These results indicate that by adding just two rings of elements near the 

cutout (Model G2), the extraction of the local model boundary conditions from the spline 

interpolation is improved such that the global/local interface boundary may be located very near 

the cutout. 

The influence of the radius of the interpolation region on the local solution was determined 

to be minimal provided the global model discretization is adequate. For the cases considered, 

identical local solutions are obtained using an interpolation region larger than the local model or 

an interpolation region which coincides with the local model. 

Computational Requirements 

A summary of the computational requirements for the global and local analyses of the 

isotropic panel with the circular cutout is given in Table 2. The computational cost in central 

processing unit (CPU) seconds of the local analyses is approximately 74% of the CPU time of the 

refined global analysis. The CSM Testbed data libraries for the local analyses are 59% smaller 

than the data library for the refined global analysis. The local models have 60% and 16% of the 

total number of degrees of freedom required for the refined model (G3) and converged global 

model (G4), respectively. 

Usage Guidelines 

Usage guidelines derived from the global/local analysis of the isotropic panel with a circular 

cutout are as follows. An “adequate” global analysis is required to ensure a sufficient number 

of accurate data points to provide accurate “boundary conditions” for the local model. When 

the global/local interface boundary, RL is within the high stress gradient (ie., within a distance 

of two times the cutout radius from the cutout edge), the importance of an “adequate” global 

analysis in the high gradient region is increased. The interpolation region should coincide with 

or be larger than the local model. To satisfy the compatibility requirements at the global/local 

interface boundary, the local model boundary RL sliould be defined sufficiently far from the 

cutout (Le., a distance of approximately six times the radius from the cutout). 

17 
Composite Blade-Stiffened Panel with Discontinuous Stiffener 

Discontinuities and eccentricities are common in aircraft structures. For example, the 

lower surface of the Bell-Boeing V-22 tilt-rotor wing structure has numerous cutouts and 

discontinuous stiffeners (see figure 12). Predicting the structural response of such structures 

in the presence of discontinuities, eccentricities, and damage is particularly difficult when 

the component is built from graphite-epoxy materials or is loaded into the nonlinear range. 

In addition, potential damage of otherwise perfect structures is often an important design 

consideration. Recent interest in applying graphite-epoxy materials to aircraft primary 

structures has led to several studies of postbuckling behavior and failure characteristics of 

graphite-epoxy components (e.g., ref. [22]). One goal of these studies has been the accurate 

prediction of the global response of the composite structural component in the postbuckling 

range. In one study of composite stiffened panels, R blade-stiffened panel was tested (see ref. 

[23]). A composite blade-stiffened panel was proof-tested and used as a “control specimen”. The 

panel was subsequently used in a study on discontiriuities in composite blade-stiffened panels. 

The global structural response of these composite blade-stiffened panels presented in reference 

[24] correlate well with the earlier experiment data. The composite blade-stiffened panel with 

a discontinuous stiffener shown in figure 13 is representative of an typical aircraft structural 

component and will be used to demonstrate and assess the global/local methodology. This 

problem was selected because it has characteristics which often require .a globsl/local analysis. 

These characteristics include a discontinuity, eccentric loading, large displacements, large stress 

gradients, high inplane loading, and a brittle material system. This problem represents a generic 

class of laminated composite structures with discoiitixiuities for which the interlaminar stress 

state becomes important. The local and global finite element modeling and analysis needed to 

predict accurately the detailed stress state of flat blade-stiffened graphite-epoxy panels loaded in 

axial compression is described in this section. 

The overall panel length L is 30 in., the overall width W is 11.5 in., the stiffener spacing b is 

4.5 in., the stiffener height h, is 1.4 in., and the cutout radius TO is 1 in. The three blade-shaped 

stiffeners are identical. The loading is uniform axial compression. The loaded ends of the panel 

are clamped and the sides are free. The material system for the panel is T300/5208 graphiteepoxy unidirectional tapes with a nominal ply thickness of 0.0055 in. Typical lamina properties 

for this graphite-epoxy system are 19,000 ksi for the longitudinal Young’s modulus, 1,890 ksi for 

the transverse Young’s modulus, 930 ksi for the shear modulus, and 0.38 for the major Poisson’s 

18 
ratio. The panel skin is a 25-ply laminate ([f45/02/ F 45/03/ f 45/03/ 

and the blade stiffeners are 24-ply laminates ([f45/020/ 

45/03/ f 45/02/ 7 451) 

451). 

End-shortening results are shown in figure 14 for the ‘‘control specimen” and for the 

configuration with a discontinuous stiffener. These results indicate that the presence of the 

discontinuity markedly changes the structural response of the panel. The structural response 

of the “control specimen” is typical of stiffened panels. Two equilibrium configurations are 

exhibited; namely, the prebuckling configuration and the postbuckling configuration. The 

structural response of the configuration with a discontinuous stiffener is nonlinear from the onset 

of loading due to the eccentric loading condition and the cutout. The blade-stiffened panel with 

a discontinuous stiffener was tested to failure. Local failures occurred prior to overall panel 

failure as evident from the end-shortening results shown in figure 14. 

Global Analysis 

A global linear stress analysis of the composite blade-stiffened panel with a discontinuous 

stiffener was performed for an applied load corresponding to P/EA of 0.0008 (ie., an applied 

compressive load P of 19,280 pounds normalized by the extensional stiffness EA). At this load 

level, the structural response of the panel is essentially linear. Out-of-plane deflections are 

present, however, due to thc eccentric loading condition caused by the discontinuous stiffener. 

Several global finite element models are considered as indicated in Table 3 to obtain a converged 

solution for comparison purposes since a theoretical solution is not available. The value of the 

longitudinal stress resultant at the edge of the cutout changed less than 2% between Models G2 

and G4. Therefore, Model G1 will be referred to as the “coarse” global model (see figure 15), 

Model G2 will be referred to as the “refined” global model, and Model G4 will be referred to as 

the “converged” global model. 

The distribution of the longitudinal stress resultant N, normalized by the average applied 

running load (Nz)avg (ie., applied load divided by the panel width) as a function of the lateral 

distance from the center of the panel normalized by the radius of the cutout is shown in figure 

16 for both the “coarse” (Gl) and the “refined” (G2) global models. These results are similar 

to those obtained for the isotropic panel with a cutout. The maximum longitudinal stress 

resultants (N,),,, normalized by the average applied running load (Nr)avg are given in Table 

3. The results obtained using the coarse global model adequately represent the distribution away 

from the discontinuity but underestimate (by 24%) the stress concentration at the edge of the 

discontinuity. 

19 
An oblique view of the deformed shape with exaggerated deflections is shown in figure 17 

for the coarse global model. A contour plot of the longitudinal inplane stress resultant N, is also 

shown in this figure. The distribution indicates that the model provides good overall structural 

response characteristics. The N, distribution reveals several features of the global structural 

behavior of this panel. First, away from the disconlinuity, the N, distribution in the panel skin 

is nearly uniform and approximately half the value of the N, in the outer two blade stiffeners. 

Second, load is diffused from the center discontinuous stiffener into the panel skin rapidly such 

that the center stiffener has essentially no N, load at the edge of the cutout. Third, the N, load 

in the outer stiffeners increases towards the center of the panel and is concentrated in the blade 

free edges (Le., away from the stiffener attachment line at the panel skin). Fourth, the N, load 

in the panel skin near the center of the panel is much greater than the N, load in other portions 

of the panel skin. 

The distribution of the strain energy for Model G1 is shown in figure 18. The change in 

the strain energy per unit area within the transition square again indicates that a high stress 

gradient exists near the discontinuity and rapidly decays away from the discontinuity. These 

results are consistent with the structural analyst’s intuition, and the local models described 

subsequently will further interrogate the region near the discontinuity. For this load level, the 

skin-stiffener interface region has not yet become heavily loaded. However, this region will also 

be studied further to demonstrate the flexibility of the global/local stress analysis procedure 

presented herein. 

Local Analyses 

A global/local analysis capability provides an alternative approach to glolial mesh refinement 

and a complete solution using a more refined mesh. For this example, one “critical” region 

is easily identified by even a casual examination of the stress resultant distribution given in 

figure 17. A second critical region that may require further study is indicated by the slight 

gradient near the intersection of the outer blade-stiffeners and the panel skin at the panel 

midlength as shown in figure 17. Skin-stiffener separation has been identified as a dominant 

failure mode for stiffened composite panels (see ref. [22]). The global model, the interpolation 

regions and the local models considered are shown in figure 19. The global model corresponds to 

the “coarse” global model (Gl) and the shaded regions correspond to the interpolation regions 

which are used to generate the spline coefficient matrix and to extract boundary conditions for 

the local model. As indicated in figure 19, two different critical regions are considered. One 

region is near the discontinuity and a circular local model is used. The boundary of the circular 

20 
model is inscribed in the transition square. That is, the outer radius of the ci. cular model is 

equal to half the length of a side of the transition square. The other region is near the skinstiffener interface region at the panel midlength for one of the outer stiffeners. The global/local 

interpolation for the local models is performed front the data obtained from the “coarse” global 

model analysis. Two interpolation regions were used in each of the local analyses. The first 

interpolation region, specified in the plane of the panel skin, is used to obtain the boundary 

conditions on the global/local interface boundary of the panel skin. The second interpolation 

region, specified in the plane of the stiffener, is used to obtain the boundary c*,nditions on the 

global/local interface boundary of the stiffener. The boundary conditions for i he panel skin and 

the stiffener’ were interpolated separately. Compatibility of displacements and rotations at the 

skin-stiffener intersection on the global/local interface boundaries was enforced by imposing the 

boundary conditions obtained for the panel skin. 

The local model (Model LC1 in Table 3) of thc: first critical region near t; te discontinuity 

has 576 4-node quadrilateral shell elements, 612 nodes, and 3456 degrees of frc edom. This local 

model has only 56% of the elements used in the refined global analysis. The distribution of the 

longitudinal stress resultant N, at the panel midlength normalized by the average running load 

(Nz)avg is shown in figure 20a as a function of the lateral distance from the cutout normalized by 

the cutout radius. These results indicate that the global/local analysis based on the coarse global 

solution accurately predicts the stress concentration at the cutout as well as the distribution 

at the global/local interface boundary. A contour plot of the longitudinal stress resultant 

distribution is given in figure 20b. The results indicate that the local solution correlates well with 

the global solutions shown in figure 17. 

The second critical region near the intersection of the outer blade-stiffener and the pariel 

skin at the midlength is studied further. Three cliffcrent local finite element n odels of this 

critical region arc considered as indicated in Table 3. The first, Model LRl, 11,~s the same nuniber 

of nodes (25) and number of elements (16) within the critical region as the coarse global model 

(Gl). The second, Model LR2, has the same number of nodes (45) and number of elements (32) 

within the critical region as the refined global model (G2). The third and most refined model, 

Model LR3, has 117 nodes, 96 elements and 462 degrees of freedom. The longitudinal stress 

resultant N, distributions obtained for the local models (LR1 and LR2) correlate well with the 

NE distributions for the coarse and refined global models (Models G1 and G2). However, these 

models are not sufficiently refined in the skin-stiRener interface region to accurately predict the 

gradient at the skin-stiffener intersection. The distribution of the longitudinal stress resultant 

21 
N, normalized by the applied running load as a function of the lateral distance from the center 

of the panel normalized by the radius of the cutout is shown in figure 21. These results indicate 

that the global/local analysis using the local model Model LR3 predicts a higher gradient at 

= -4.5 in the skin-stiffener interface region than the other global and local analyses. A To 

third global model Model G3 is used to investigate the local structural behavior predicted by 

Model LR3. The global analysis performed with Model G3 predicts the same local behavior as 

the analysis performed with Model LR3 as indicated in figure 21. Several factors should be borne 

in mind. First, the local analysis revealed local behavior at the skin-stiffener interface region 

that was not predicted by either of the global models. Second, the global modeling requirement 

for examining the skin-stiffener interface region are substantial. Because the c;lobal models were 

generated to predict the stress distribution around the discontinuity, additiond radial “spokes” 

in the transition square are required to refine the panel skin in the skin-stiffel er interface region 

in the longitudinal direction. Third, the global/local analysis capability provi(1es the analyst with 

the added modeling flexibility to obtain an accurate detailed response at multiple critical regions 

(ie., at the discontinuity and at the skin-stiffener interface region) with mini:nal modeling and 

computational effort. 

Computational Requirements 

A summary of the computational requirements for the global and local analyses of the 

graphite-epoxy blade-stiffened panel with the discontinuous stiffener is given in Table 4. 

The computational cost in CPU seconds of the local analyses around the discontinuity is 

approximately 57% of the CPU time of the refined global analysis. The CSM Testbed data 

libraries for the local analyses are half of the size of the data library for the refined global 

analysis. The local models have 55% and 16% of the total number of degrees of freedom required 

for the refined model (G2) and converged global model (G4), respectively. The CPU time for the 

refined local analysis (LR3) of the skin-stiffener interface region is 24% of the CPU time required 

for the global analysis with Model G3. The size of the data library for the 10c.d analysis is 5% of 

the size of the data library required for the analysis with Model G3. 

UsaEe Guidelines 

Usage guidelines derived from the global/local analysis of the blade-stiffened panel with 

a discontinuous stiffener are as follows. An “adequate” global analysis is required to ensure a 

sufficient number of accurate data points to provide accurate “boundary conditions” for the 

local model. When the global/local interface boundary, RL is within the high stress gradient 

22 
(i.e., within a distance of two times the cutout radius from the cutout edge), the importance 

of an “adequate” global analysis in the high gradie1,t region is increased. The interpolation 

region should coincide with or be larger than the local model. To satisfy the compatibility 

requirements at the global/local interface boundary, the local model boundary RL should be 

defined sufficiently far from the cutout (;.e., a distance of approximately six times the radius 

from the cutout). For the blade-stiffened panel, two interpolation regions should be specified, 

one for the interpolation of the boundary conditions on the boundary of the panel skin and a 

second for the interpolation of the boundary conditions on the outer edges of the stiffeners. 

Conclusions 

A global/local analysis methodology for obtaining the detailed stress state of structural 

components is presented. The methodology presented is not restricted to having a priori 

knowledge of the location of the regions requiring a detailed stress analysis. The effectiveness 

of the global/local analysis capability is demonstrated by obtaining the detailed stress states of 

an isotropic panel with a cutout and a blade-stiffened graphite-epoxy panel with a discontinuous 

stiffener. 

Although the representative global finite element models represent the global behavior of 

the structures, substantially more refined finite element meshes near the cutouts are required to 

obtain accurate detailed stress distributions. Embedding a local refined model in the complete 

structural model increases the computational requirements. The computational effort for the 

independent local analyses is less than the computational effort for the global analyses with the 

embedded local refinement. 

The global/local analysis capability provides the modeling flexibility required to address 

detailed local models as their need arises. This modeling flexibility was demonstrated by 

the local analysis of the skin-stiffener interface regions of the blade-stiffened panel with a 

discontinuous stiffener. This local analysis revealed local behavior that was not predicted by the 

global analysis. 

The definition of the global/local interface boundary affects the accuracy of the local 

detailed stress state. The strain energy per unit arca has been identified as a ineans for 

identifying a critical region and the location of tile associated global/loc:al interface boundary. 

The change in strain energy from element to element indicates regions with high stress gradients 

(ie., critical regions). A global/local interface boundary is defined outside of i~ region with large 

changes in strain energy. 

23 
The global/local analysis capability presented provides a general-purpose analysis tool 

for use by the aerospace structural analysis community by providing an efficient strategy 

for accurately predicting local detailed stress states that occur in structures discretized with 

relatively coarse finite element models. The coarse model represents the global structural 

behavior and approximates the local stress state. Independent, locally-refined finite element 

models are used to accurately predict the detailed stress state in the regions of interest based on 

the solution predicted by the coarse global analysis. 

Acknowledgements 

This work represents a portion of the first author’s Master’s thesis submitted to the faculty 

of the Old Dominion University. 

Refer e nc e s 

1. Knight, Norman F., Jr.; Greene, William H.; and Stroud, W. Jefferson: Nonlinear Response 

of a Blade-Stiffened Graphite-Epoxy Panel with a Discontinuous Stiffener. Proceedings of 

NASA Workshop on Computational Methods in Structural Mechanics and Dynamics, W. J. 

Stroud; J. M. Housner; J. A. Tanner; and R. J. Hayduk (editors), June 19-21, 1985, NASA 

CP-3034 - PART 1, 1989, pp. 51-66. 

2. Noor, Ahmed K.: Global-Local Methodologies and Their Application to Nonlinear Analysis. 

Finite Elements in Analysis and Design, Vol. 2, No. 4, December 1986, pp. 333-346. 

3. Wilkins, D. J.: A Preliminary Damage Tolerance Methodology for Composite Structures. 

Proceedings of NASA Workshop on Failure Analysis and Mechanisms of Failure of Fibrous 

Composite Structures, A. K. Noor, M. J. Shuart, J. H. Starnes, Jr., and J. G. Williams 

!,:ompilers), NASA CP-2278, 1983, pp. 61-93. 

4. €Ian, Tao-Yang; and Abel, J. F.: Computational Strategies for Nonlinear and Fracture 

Mechanics, Adaptive Substructuring Techniqucs in Elasto-Plastic Finite Element Analysis. 

Computers and Structures, Vol. 20, No. 1-3, 1985, pp. 181-192. 

5. Clough, R. W.; and Wilson, E. L.: Dynamic Analysis of Large Structural Systems with 

Local Nonlinearities. Computer Methods in Applied Mechanics and Engineering, Vol. 17/18, 

Part 1, January 1979, pp. 107-129. 

6. Schwartz, David J.: Practical Analysis of Stress Raisers in Solid Structures. Proceedings of 

the 4th International Conference on Vehicle Structural Mechanics, Warrandale, PA, Nov. 

1981, pp. 227-231. 

7. Kelley, F. S.: Mesh Requirements of a Stress Concentration by the Specified Boundary 

Displacement Method. Proceedings of the Second International Computers in Engineering 

Conference, ASME, Aug. 1982, pp. 39-42. 

8. Griffin, 0. Hayden, Jr.; and Vidussoni, Marco A.: Global/Local Finite Element Analysis 

of Composite Materials. Computer-Aided Design in Composite Material Technology, C. 

A. Brebbia, W. P. de Wikle, and W. R. Blain (editors), Proceedings of the International 

24 
Conference for Computer-Aided Design in Coinposite Material Technology, Southampton, 

UK, April 13-15, 1988, pp. 513-524. 

9. Jara-Almonte, C. C.; and Knight, C. E.: The Specified Boundary Stiffness/Force SBSF 

Method for Finite Element Subregion Analysis. International Journal for Numerical 

Methods in Engineering, Vol. 26, 1988, pp. 1!167-1578. 

10. Hirai, Itio; Wang, Bo Ping; and Pilkey, Walter D.: An Efficient Zooming Method for Finite 

Element Analysis. International Journal for Numerical Methods in Engineering, Vol. 20, 

1984, pp. 1671-1683. 

11. Hirai, Itio; Wang, Bo Ping; Pilkey, Walter D.: An Exact Zooming Method. Finite Element 

12. Dong, Stanley B.: Global-Local Finite Element Methods. State-Of-The-Art 

Analysis and Design, Vol. 1, No. 1, April 1985, pp. 61-68. 

Surveys on Finite Element Technology, A. K. Noor and W. D. Pilkey (editors), 

ASME, 1983, pp. 451-474. 

13. Stehlin, P.; and Rankin, C. C.: Analysis of Structural Collapse by the Reduced Basis 

Technique Using a Mixed Local-Global Formulation. AIAA Paper No. 86-0851-CP, May 

1986. 

14. Gladwell, G. M. L.: Practical Approximation Theory, University of Watcrloo Press, 1974. 

15. Harder, Robert L.; and Desmarais, Robert N.: Interpolation Using Surface 

Splines. Journal of Aircraft, Vol. 9, No. 2, February 1972, pp. 189-191. 

16. Stewart, Caroline B., Compiler: The Computational Structural Mechanics Testbed User’s 

Manual. NASA TM-100644,1989. 

17. Gillian, R. E.; and Lotts, C. G.: The CSM Testbed Software System - A Development 

Environment for Structural Analysis Methods on the NAS CRAY-2. NASA TM-100642 , 

1988. 

18. Timoshenko, S. P.; and Goodier, J. N.: Theory of Elasticity. McGraw-Hill Book Company, 

19. Peterson, R. E.: Stress Concentration Design Factors. Wiley-International, New York, 1953, 

New York, 1934, pp. 90-97. 

pp. 77-88. 

20. Alnlroth, B. 0.; and Brogan, F. A.: The STAGS Computer Code. NASA CR-2950,1978. 

21. Almroth, B. 0.; Brogan, F. A.; and Stanley, G. M.: Structural Analysis of General Shells - 

Volume 11: User Instructions for STAGSC-1. NASA CR-165671, 1981. 

22. Starnes, James H., Jr.; Dickson, John N.; and Rouse, Marshall: Postbuckling Behavior 

of Graphite-Epoxy Panels. ACEE Composite Structures Technology: Review of Selected 

NASA Research on Composite Materials and Structures. NASA CP-2321, 1984, pp. 137- 

159. 

23. Williams, Jerry G.; Anderson, Melvin S.; Rhodes, Marvin D.; Starnes, James H., Jr.; and 

Stroud, W. Jefferson: Recent Developments in the Design, Testing, and Impact-Damage 

Tolerance of Stiffened Composite Panels. NASA TM-80077, 1979. 

25 
24. Knight, Norman F., Jr.; McCleary, Susan L.; Macy, Steven C.; and Aminpour, Mohammad 

A.: Large-Scale Structural Analysis: The Strnctural Analyst, The CSM Testbed, and The 

NAS System. NASA TM-100643,1989. 

26 
Table 1 Finite Element Models of Isotropic Panel 

with Circular Cutout. 

Number of 

Rings 

of Nodes 

2 

4 

16 

32 

16 

16 

Model 

Designation 

Number of Tot a1 Tot a1 

of Elements Elements Nodes 

Radial Spokes Number of Number of Kt 

16 256 296 2.06 

16 288 328 2.23 

32 832 888 2.72 

80 3168 3272 2.81 

32 512 544 2.76 

32 512 544 2.75 

G1 

G2 

G3 

Model 

Designat ion 

G1 

G2 

G3 

G4" 

G4" 

Measures of Computational Effort 

degrees of freedom CPU, seconds library, Mbytes 

1644 64.0 5.2 

1836 69.7 6.0 

5156 183.4 22.2 

19340 1167.2 160.0 

Total number of Size of data 

LS1 

LC1 

a G4 is the converged model 

Table 2 Summary of Computational Requirements. 

LS1 I LC1 

3072 

3072 

135.4 

121.6 

13.0 

12.6 

a G4 is the converged niodel 

27 
Table 3 Finite Element Models Blade-Stiffened Panel 

with Discontinuous Stiffener. 

Number of 

Radial Spokes 

of Elements 

16 

32 

32 

Model 

Designation 

Total Tot a1 

Number of Number of (N,),,, 

Elements Nodes (Nz 

376 424 2.22 

1024 1088 2.88 

1408 1488 2.88 

G1 

G2 

G3 

16 

- 

- 

- 

G4" 

32 576 612 2.92 

- 16 25 - 

- 32 45 

96 117 - 

LC1 

Total number of 

degrees of freedom 

2316 

6252 

8460 

21084 

3456 

LR1 

LR2 

LR3 

Size of data 

CPU, seconds library, Mbytes 

99.7 8.7 

255.3 28.9 

329.6 38.5 

1006.0 140.6 

188.7 13.1 

Number of 

Rings 

of Nodes 

32 I 80 I 3472 I 3584 1 2.94 

a G4 is the converged model 

Table 4 Summary of Computational Requirements. 

Model 

Designation 

GI 

G2 

G3 

G4" 

LC1 

LR1 

LR2 

LR3 

54 

126 

462 

58.0 

62.0 

78.7 

0.5 

0.8 

1.8 

a G4 is the converged model 

2s 
V 

(a) Global model (b) Interpolati,on region within global model 

Interpolation region 

(c) Global/local interface boundary (d) Independent local model 

Fig. 1 Terminology of the global/local methodology. 

29 
T pica1 

element 

s i! ewed 

Radial 

spokes 

of nodes 

Typical 

tapered 

element 

- Transition 

square 

- Rings of 

elements 

Global regions 

affected by embedding 

local I y-ref i ned 

mesh 

Fig. 2 Terminology associated with modeling cutouts. 

30 
Global modeling and analysis 

Perform 

global analysis 

Global/local interface definition ---11------1-_------11__1_1_1___ I 

+ I 

I 

I 

I 

I 

I 

I 

I 

1 Identify 

I critical region 

I 

I ' --------------------, 

I Specify 

I Interpolation 

: Local modeling and analysis 

I 

I 

I 

:I I 

model boundary I local model I 

I 

I 

I $ I 

displacements e conditions I 

Define local Generate 

-I 

I 

I 

I 

-I 

I1 I 

I 

I 

I 

I Generate local Impose disp Generate I spline matrix 

',-,,,-----------------------------------J I I 

+ I interpolated as boundary I 

Global/local interpolation procedure I I 

I Perform 

I local analysis 

I 

I. I8 I 

I I 

I I 

I 

I I 

I 

I I 

L--,--------,,,----J 

Fig. 3 Schematic of overall global/local solution strategy. 

31 
Fig. 4 Coarse global finite element model of isotropic 

panel with a circular cutout. 

32 
2.0 - 

1 .o 

X N 

(Nx) nom 

0 

6' 

-1 .o 

Fig. 5 Longitudinal stress resultant N, distribution for coarse global 

finite element model of isotropic panel with a circular cutout. 

33 
4.0 

3.0 

NX 

1 .o 

0 

'I 

I 

I 

-1 

Model G3 I 

I Edge of cutout 

5.0 10.0 15.0 20.0 

Fig. 6 Longitudinal inplane stress resultant N, distributions at 

panel midlength for coarse and refined global finite element models 

34 
, 

ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

1 .o 

0.5 

B X 

Y 

0.0 - 

Fig. 7 Distribution of the strain energy measure for coarse global 

model of the panel with a circular cutout. 

35 
Global model 

Local models Circular Square 

I Fig. 8 Global/local analysis models for isotropic panel with circular cutout. 

36 
4m0r I 

Model G3 

1 .o 

\-Model LS1 

Edge of cutout 

I I I 

0 5.0 10.0 15.0 20.0 

- Y 

‘0 

(a) Distribution at panel midlength. 

Fig. 9 Longitudinal stress resultant N, distributions for square local 

finite element model of isotropic panel with a circular cutout. 

37 
NX 

I” 

2.0 - 

1.0 - 

0- 

-1.0 - 

(b) Contour plot. 

Fig. 9 Concluded. 

38 
4.0 (N x’nom 

3.0 

NX 

1 .o 

0 

H 

I \Model LC1 

Edge of cutout 

I I I 

5.0 10.0 15.0 20.0 

- Y 

‘0 

(a) Distribution at panel midlength. 

Fig. 10 Longitudinal stress resultant N, distributions for circular local 

finite element model of isotropic panel with a circular cutout. 

39 
ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

(b) Contour plot. 

Fig. 10 Concluded. 

40 
'-Or i 

I I I I I I 

0 1 2 3 4 5 

Y 

'0 

- 

(a) Interpolation from global Model G1. 

- Y 

'0 

(b) Interpolation from global Model G2. 

Fig. 11 Longitudinal inplane stress resultant N, distributions at panel 

midlength for varied local model boundaries. 

41 
ORIGINAL PAGE 

BLACK RND WMTE PHOTOGRAPH 

Fig. 12 Bell-Boeing V-22 wing panel. 

42 
ORIGINAL' PAGE 

BLACK AND WHITE PHOTOGRAPH 

13 Composite blade-stiffened panel with a discontinuous stiffener. 

43 
Normalized 

applied 

load, 

P/EA 

.0030 

.0025 

.0020 

.0015 

.0010 

.0005 

.oooo 

f U 

/ 

/ 

Extension of , / 

prebuckling ,’ pathw Panel without 

hole 

P/EA .0008 hole 

Local failures 

occurred 

I I I r 

.0005 .0010 .OW5 .0020 .0025 

Normalized end shortening, u/L 

Fig. 14 End-shortening results for composite blade-stiffened panel. 

44 
‘G X 

Fig. 15 Global finite element model of blade-stiffened panel with 

discontinuous stiffener. 

45 
3.00 

2.50 

2.00 

1 .oo 

SO0 

0 

- 

Blade-stiffener Blade-stiffener 

i I I I I I I I 

-6.00 -4.00 -2.00 0 2.00 4.00 6.00 

- Y 

‘0 

Fig. 16 Longitudinal inplane stress resultant N, distributions at panel 

midlength for coarse and refined global models. 

46 
ORIGINAL PAG€ 

BLACK AidD WHITE PHOTOGRAPH 

2.0 - 

Fig. 17 Deformed geometry shape with N, distributions for coarse global 

model of blade-stiffened panel with discontinuous stiffener. 

47 
ORIGINAL PAGE 

BLACK AND WHITE PKOTOGRAPH 

1 .o a 

0.5 

=kY 

X 0.0 

Fig. 18 Distribution of the strain energy measure for the coarse model 

of the blade-stiffened panel with discontinuous stiffener. 

48 
ORIGINAL PAGE 

BLACK AND WHITE PHOTOGRAPH 

Local model Local model 

at skin-stiffener at discontinuity interface (model LR3) (model LCI) 

Fig. 19 Global/local analysis models for blade-stiffened panel with discontinuous stiffener. 

49 
3.00 

2.50 

2.00 

NX 1.50 

(N X) avg 

1 .oo 

SO0 I I Edge 

h I Blade-stiffener I cutout I Blade-stiffener A I 

- r of 9 

-6.00 -4.00 -2.00 0 2.00 4.00 6.00 

- Y 

‘0 

(a) Distribution at panel midlengt h. 

Fig. 20 Longitudinal stress resultant N, distributions for circular local 

finite element model of blade-stiffened panel with discontinuous stiffener. 

50 
ORIGINAL PAG€ 

BLACK AND WHITE PHOTOGRAPH 

(b) Contour plot. 

Fig. 20 Concluded. 

51 
3.00 

2.50 

2.00 

1 .oo 

,500 

0 

- 

I 

I 

I 

I 

I 

I 

I 

I 

I I o ModelGI 

* Model LR3 

0 ModelG3 

I 

I 

I I Edge 09 i 

cutout , 

I 

I 

Blade-stiffener If 

I I I I I I I 

-6.00 -5.00 -4.00 -3.00 -2.00 -1 .oo 

y 

‘0 

Fig. 21 Longitudinal inplane stress resultant N, distributions at 

panel midlength for skin-stiffener interface region. 

52 
IL___pw 1& 

Naliondl Aeronautics and Report Documentation Page 

1. RepoTNo. 

NASA TM-101622 

2. Government Accession No. 3. Recipient’s Catalog No. 

Global/Local Stress Analysis of Composite Panels 

4. Title and Subtitle 

6. Performing Organisation Code 

5. Report Date 

8. Performing Organisation Report No. 7. Author(s) 

Jonathan B. Ransom and Norman F. Knight, Jr. 

10. Work Unit No. 

9. Performing Organization Name and Address 505-63-01-10 NASA Langley Research Center 

Hampton, VA 23665-5225 11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 12. Sponsoring Agency Name and Addreas 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 14. Sponsoring Agency Code 

19. Security Classif.(of this report) 20. Security Classif.(of this page) 

Unclassified Unclassified 

15. Supplementary Notes 

Presented at the Third Joint ASCE/ASME Mechanics Conference, University of California at San Diego, 

Julv 9-12. 1989. 

21. No. of Pages 22. Price 

53 A03 

~~~~ 

16. Abstract 

A method for performing a global/local stress analysis is described and its capabilities are demonstrated. The method employs spline interpolation functions which satisfy the linear plate bending equation 

to determine displacements and rotations from a global model which are used as “boundary conditions” 

for the local model. Then, the local model is analyzed independent of the global model of the structure. 

This approach can be used to determine local, detailed stress states for specific structural regions using 

independent, refined local models which exploit information from less-refined global models. The method 

presented is not restricted to having a priori knowledge of the location of the regions requiring local 

detailed stress analysis. This approach also reduces the computational effort necessary to obtain the 

detailed stress state. Criteria for applying the method are developed. The effectiveness of the method is 

demonstrated using a classical stress concentration problem and a graphite-epoxy blade-stiffened panel 

with a discontinuous stiffener. 

17. Key Word8 (Suggested by Authors(8)) 

Global/Local Analysis 

Detailed Stress Analysis 

Computational Structural Mechanics 

Surface Splines 

18. Distribution Statement 

Unclassified-Unlimited 

Subject Category 39