A99-24922

AIAA 99-1313 Correction response surface design of stiffened composite panel with a crack

Roberto Vitali and Bhavani V. Sankar AeMes Department, University of Florida Gainesville, FL

40th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

AIAAlASME/AHS Adaptive Structures Forum AIAA Forum on Non -Deterministic Approaches

Conference and Exhibit 12-15 April, 1999

St. Louis, Missouri For permission to copy or to republish, contact the American Institute of Aeronautics and Astronautics,

1801 Alexander Bell Drive, Suite 500, Reston, VA, 201914344.

AIAA-99-1313

CORRECTION RESPONSE SURFACE DESIGN OF STIFFENED COMPOSITE PANEL WITH A CRACK

Roberto Vitali’and Bhavani V. Sankart University of Florida

Department of Aerospace Engineering, Mechanics and Engineering Science Gainesville, FL 3261 I-6250

Abstract In designing an aircraft composite wing

panel, two of the most important concerns are crack propagation in the skin of the stiffened panel and instability of the panel itself under compressive loads. This study attempts to solve the weight optimization problem of a stiffened composite panel, subjected to crack propagation and buckling constraints, using the response surface methodology. In particular, a combination of a high fidelity method of analysis and a low fidelity method was utilized to describe the crack propagation constraint. A correction response surface was constructed to relate the high fidelity models to the low fidelity models. This method combines the accuracy of the former with the computational efftciency of the latter. The results indicate that correction response surfaces can be used to improve the computational efficiency of structural optimization problems.

Introduction The next generation of supersonic transport

planes are expected to fly at very high altitudes and are very vulnerable to pressure loss due to large cracks, such as due to turbine blade penetrations. Consequently, accurate prediction of crack propagation is a major issue in the structural design of such aircraft. Exact calculation of crack propagation requires detailed structural models for computing stress intensity factors, which is computationally too expensive to be included in the structural design optimization. The present paper discusses the use of correction response surfaces for alleviating the computational burden.

Because of their computational efficiency response surface techniques have gained

* Graduate Student, Student Member AIAA t Professor, Associate Fellow AIAA, Fellow ASME

popularity in engineering optimization. With this approach, the structural response is calculated at a number of design points, and a simple function, typically a quadratic polynomial, is fitted to the calculated values. However, since complex response patterns are often difftcult to approximate as low order polynomials, and since inexpensive lower fidelity models are often available to the designer, a different way of using response surfaces is gaining popularity. In this approach, the response surface is used to correct the lower fidelity model rather than to approximate directly the expensive higher- fidelity model. For example, Mason et al’, used 2-D finite elements for low fidelity models and 3-D finite elements for high fidelity models to approximate failure stresses in a structure. Similarly, Vitali et al2 used strength-of-materials models to predict stresses and buckling loads in a hat-stiffened panel, and used correction response surface technique to predict the results from the high fidelity model based on STAGS3 FE code. Further, Vitali et a1.4 demonstrated the validity of the use of a correction response surface for calculating the stress intensity factors for cracks in stiffened panels.

In the present paper, in addition to using a correction response surface, the response surface methodology is extended to approximate the low fidelity model also. To demonstrate the use of this approach we consider the weight optimization problem of a stiffened composite panel, subjected to crack propagation and buckling constraints. The stress intensity factors were computed using both high and low fidelity models. The high fidelity model describes the crack in the stiffened panel with great detail and computes the stress intensity factor using classical linear elastic fracture mechanics. The

Copyright 0 1999 by Roberto Vitali Published by the American Institute of Aeronautics and Astronautics, Inc. with Permission

low fidelity model uses coarse finite elements but the crack is not present in the model. The stress intensity factor is calculated using a superposition principle in conjunction with a closed-form solution for stress intensity factors. A quadratic correction response surface is fitted to the values of the ratio of the high fidelity and low fidelity stress intensity factors. Further, a cubic response surface is constructed for the low fidelity stress intensity factors in terms of the design variables. The combination of the two response surfaces is used as one of the constraints (crack propagation constraint) in the design optimization problem. The other design constraint was the buckling load factor of the untracked stiffened panel. Only one cubic response surface was used for the buckling constraint, and this was constructed using the results from about five-hundred STAGS analysis runs.

Problem Descrktion The dimensions of the composite stiffened

panel analyzed are shown in Fig. 1. The geometry of the composite stiffened panel and the size of the crack (2~ = 4.0 inches) were kept constant throughout the study. The stacking sequence of the skin and the stacking sequence of the stiffener were kept the same. The lay-up used for both the skin and the stiffener of the panel was l-&5/90/0],. The thickness of the skin and the blade were the only parameters that were varied. The material used in the panel was AS4/3501-6 graphite/epoxy, and its properties are given in Table 1.

Figure 1: Panel geometry

The composite stiffened panel was subjected to two load cases: a tensile loading of 2000 lb/in and a compressive loading of 800 lb/in. These two load sets are used to represent a 2.5 g maneuver and a -1.0 g. maneuver, respectively.

Table 1: Material properties of AS413501 -6

Material property Young’s modulus Et

Value 20 lo6 osi

Young’s modulus EL 1.4 106* si Shear modulus Glz 0.76 10 IF psi Poisson’s Ratio vr2 0.30 Density p 0.063 lb/in3 Kq” (Fracture toughness) 100.000 psi Jin

The thicknesses of different plies, +35. 90 and 0, present in the skin and in the stiffener were used as design variables. The thicknesses of the +45 and -45 plies in the skin were set to be equal to each other. The same equality was enforced on the +45 and -45 plies in the stiffener. A total of six design variables were used. The six design variables were the ply thicknesses and were labeled as: ~45, p90, ~00. b45, b90, bO0. The last two digits of each design variable indicate the ply angle while the letter ‘p” and the letter “b”, respectively refers to the plate (skin) or the blade (stiffener). All design variable values ranged between 0.005 inches and 0.025 inches. The lower limit value of the design variables was chosen by recognizing that the material system used in this study, the AS/3501-6, is generally available in the form of prepreg tape with a nominal thickness of 0.005 inch. The maximum values represent estimates of the required material based on the papers by Vaidya et a1516. The goal of this study was to obtain the minimum weight design that satisfied both the crack propagation constraint and buckling constraint. The two constraints were modeled using the response surface methodology, as described in the following sections.

Finite Element Models The stiffened panel is simply-supported

along the edges parallel to the x-axis, and free along the edges parallel to the y-axis (see Fig. 1). A multipoint constraint was applied to the skin and to the stiffener, enforcing the same displacement in the y-direction along the simply supported edges. Two different finite element models of the stiffened panel were created using the STAGS program. In both finite element models, symmetry was utilized in order to model only i/4 of the structure, thereby reducing the total number of degrees of freedom.

The first STAGS computer model of the stiffened panel was used to accurately capture the stress gradients that develop in the region

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near the crack tip, when the structure was subjected to the tensile loading. In order to achieve accuracy, a large number of 4-noded elements were employed in the vicinity of the crack tip, as shown in Fig. 2. Referring to Fig. 2, the largest elements shown have a characteristic dimension of G-inch, while the smallest elements, that are hardly visible in Fig. 2, had dimensions in the order of l/256-inch. The refined model consisted of 7,158 elements and 46,680 degrees of freedom.

The second finite element model was coarser since it did not contain a crack. A uniform mesh of 4-noded elements was employed. The size of the elements used in the second finite element model was M-inch. The purpose of the model is to compute the nominal stresses in the location where a crack is supposed to be present. Then, a closed form solution was used to compute the approximate stress intensity factor in the presence of the crack. The low fidelity model used 2080 elements for a total of 13284 degrees of freedom.

Figure 2: Refined finite element model of quarter panel

Stress Inteusitv Factor Calculation The methods of Vaidya and Sun’ were used

for predicting crack propagation in a composite laminate. In this method the average stress intensity factor, K, through the thickness of the plate is computed by modeling the laminate as an equivalent orthotropic plate. Then the stress intensity factor in the zero-degree ply is computed using the formula:

K’=nK (1) where the factor q is defined as the ratio of the nominal stress in the zero-degree ply to the average stress in laminate:

q - %f (2) cs YY

It was assumed that the crack in the skin of the panel would propagate, leading to structural failure, when stress intensity factor in the 0’ plies Kc reached Kq’, fracture toughness of the 0’ ply. The value of the tiacture toughness Ki was taken horn Vaidya and Sun’. Vitali et a14. showed that Ic” could not be computed horn the distribution of stresses in the 0’ plies near the crack tip obtained from the refined STAGS model. Instead, K” must be calculated f?om the stress intensity factor K of an equivalent orthotropic panel, in combination with classical lamination theory.

The theoretical distribution of oyy near the crack tip of the equivalent orthotropic panel is given by:

%=A (3)

In the high fidelity model K was obtained by fitting 6, as a function of the distance from the crack tip, r using Eq(3) in a least square sense. An example of such a lit is presented in Fig. 3.

1 distance from the crack tiD r Cinches)

0 0.05 0.1 0.15 0.2

Figure 3: Calculation of K from a curve fit to finite element results

The value of K obtained by using the aforementioned procedure is called the high fidelity value. Calculation of one value of the high fidelity K required 556 seconds of usage time and 41.87 seconds of CPU time on a 233 MHz DEC Alpha Station 200 4/233.

The stress intensity factor K was also calculated using a low fidelity method, based on borrowing the infinite cracked plate solution for K

K=cs,,fi (4)

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where c&. in (4) are the average stresses present in an untracked equivalent orthotropic panel at the location where the crack is supposed to be present. Calculation of one value of the high fidelity K required 66 seconds of usage time and 7.8 seconds of CPU time on a 233 MHz DEC Alpha Station 200 3/233. The lower fidelity value of K obtained is approximate mainly for two reasons: 1) the finite size of the panel was not taken into consideration in Eq. (4); and 2) the beneficial effect of the stiffener on reducing the stress intensity factor could not be fully captured by this model. However. low fidelity method had the advantage of being computationally inexpensive.

Bucklinp Load Factor Calculation The buckling load factor for the compressive

loading case of 800 lb/in was computed using the STAGS linear bifurcation analysis with the coarser of the two finite element models described in the previous section. The buckling load factors thus obtained were within 1% of values obtained by using refined finite element models.

Stress Intensity Factor Constraint As described in a previous section, the stress

intensity factor K of the equivalent orthotropic structure can be obtained using two methods of analysis, high fidelity and low fidelity models. Both methods were employed in construction of the response surface that described the crack propagation constraint. This was done in the attempt to combine the accuracy of the high fidelity method with the low computational cost of the low fidelity method.

First an orthogonal array using three levels for the six design variables, p45, p90, ~00, b45, b90, and bO0, was used to select a total of 54 design points. The selection of the 54 design points was conducted using the subroutine called “addelkemp3”, written by Owen’ and based on the paper by Addelman et al’, For these 54 design points, the value of the stress intensity factor K of the equivalent orthotropic structure was calculated using both the high fidelity and low fidelity methods. The ratio,

K High Fidelity p= T7

rl Low Fidelity (5)

was then computed at all 54 design points. A quadratic response surface PQuad for the ratio p in the six design variables was fitted through the 54

Parameter Value Mean of Response 22222 Root Mean Square Error 271.8 RSquared Adjusted 0.998

The accuracy of the cubic response surface was assessed by calculating the value of the

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design points. The ratio varied between 1.0 and 1.82 indicating substantial differences between the two models. The largest differences were for design points that had a very thin skin and a very thick stiffener.

Table 3: pQuad response surface fit parameters

Parameter Value Mean of Response 1.317 Root Mean Square Error 0.0191 RSquared Adjusted 0.992

A full quadratic response surface in six design variables normally has 28 terms; however, some of the quadratic terms were eliminated as their exclusion then improved the accuracy of the prediction capabilities of the response surface. In order to choose the terms to be eliminated the mixed mode (both backward and forward elimination) stepwise regession method built in the JMP9 program was used. Using this procedure, a total of 17 terms were retained by the response surface. The fit was quite good as seen from Table 3.

Next, a new orthogonal array using 5 levels in the six design variables was created. In this study, the number of design points selected by the orthogonal array amounted to 250. Using the 250 design points, a cubic response surface KCubic . LF m the six design variables was fitted to the values of K obtained from the low fidelity method. It was observed that a full cubic response surface in six design variables has 84 coefficients. The accuracy in the prediction capabilities of the response surface was improved by dropping some of the quadratic and cubic terms, using the mixed mode stepwise regression elimination method built into the JMP program. There was a total number of 63 terms retained by the cubic response surface. The statistical parameters of the fitting procedure are presented in Table 4.

Table 4: Low Fidelity K cubic response surface fit parameters

stress intensity factor K at some additional design points not used in the construction of the response surface. It was found that the response surface had an error of about 10%. The value of the stress intensity factor in the 0’ plies could then be estimated by using classical lamination theory, in combination with the quadratic response surface of l3 and the cubic response surface of KU as:

KO = q pQuad K;;b” (6)

Buckling Load Factor Response Surface A single response surface was fitted to the

buckling load factor h in the six design variables mentioned earlier. The buckling load factor h was computed at the same 250 orthogonal array design points previously used for the cubic response surface of K$‘” . The cubic response surface of h retained 62 of the possible 84 terms once the mixed mode stepwise regression was performed. Some of the more important statistical parameters of the cubic response surface fit of h called ~c”bic~o Table 5. ’

are presented in

Table 5: hCubicZ5’ response surface fit parameters

Parameter Value Mean of Response 0.576 Root Mean Square Error 0.0147 Rsquared Adjusted 0.999

The accuracy of the ACubic-15’ response surface was checked by performing a linear bifurcation buckling analysis at some extra design points. For some of the test design points the error in the buckling load factor h was about 70%. Thus, the prediction capabilities of the cubic response surface hCubic_15’ were considered insufficient for use in the optimization problem.

Therefore, a new set of design points was added by taking the reflection of the original 250 design points with respect to lhe center of the design space. A cubic response surface hCubic5m was fitted to the value of h obtained by using the available 500 design points. The cubic response surface retained 58 of the original 84 terms when the mixed mode stepwise regression was used to eliminate some of the quadratic and cubic terms. Some of the statistical parameters of the 500 design points cubic fit are presented in Table 6.

Table 6: hCubic-500 cubic response surface fit parameters

Parameter Value Mean of Response 0.583 Root Mean Square Error 0.0352 R-squared Adjusted 0.993

The accuracy of the cubic response hCubic_500 was checked using some design points not included in the fitting procedure. From this test it was found that errors of 18% could be expected. Thus, the cubic response surface of the buckling load factor hCubicJM) , obtained using 500 design points, appeared suitable for use in the optimization problem as a constraint.

Optimization Problem The goal of the optimization problem was to

find the minimum weight design of a stiffened composite panel, as described by six design variables, that would satisfy the crack propagation constraint of K’s &‘and the buckling constraint of h 21.0. The fracture toughness of the 0’ plies was then reduced to Kq” = 90,000 psi&, in order to accommodate the 10% possible error in the estimated value of K$‘ic . Similarly, the buckling load constraint was modified to h 2 1.18, in order to accommodate the possible 18% error.

The optimization problem thus obtained is summarized in Table 7.

Table 7: Optimization problem

Objective Function: Design Variables : Constraints:

MintWeight} P45, P90, PO0 b45, b90, bO0 0.005 < p45 < 0.025 0.005 c p90 < 0.025 0.005 < pO0 c 0.025 0.005 < b45 c 0.025 0.005 < b90 c 0.025 0.005 < bO0 c 0.025 K” < 90,000 psi & h> 1.18

The optimization problem was implemented and solved using Microsoft EXCEL. The EXCEL optimizer did not appear to be very robust since the optimizer got trapped in several apparent local minima. The optimization

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procedure was repeated several times starting from different random points, as well as from some design points selected on the basis of engineering judgment. An optimum solution was found and the corresponding values are presented in Table 8.

Table 8: Optimum solution

Parameter Value P45 0.0239 inches P90 0.0180 inches PO0 0.005 inches b45 0.020 inches b90 0.005 inches bO0 0.025 inches K0 87,356 psi & h 1.18 Weight 4.618 lb

As shown in Table 8, the crack propagation constraint was not active for the minimum weight design. However, the value of K” was very close to the maximum allowable. The value of stress intensity factor K for the optimum design structure was checked using the high fidelity analysis. The K value of the equivalent orthotropic panel obtained from the refined finite element model was determined to be K = 21700 psi&. The value of q for the optimum design was predicted as 7 = 4.114. The stress intensity factor in the 0’ plies K!, could then be calculated as:

K”=q*KHF= 89,274 psi& (7) The value of the buckling load factor h was

also checked by a finite element analysis. The value of h obtained for the minimum weight structure was h = 1.02. Therefore the optimal structure satisfied the original constraint of K” 5 I(q” and h 2 1 .O within the design domain.

Conclusions The problem of weight optimization of a

stiffened composite panel, subjected to crack propagation and buckling constraints, was solved using the response surface methodology. In particular, combination of a high fidelity method of analysis and a low fidelity method was utilized to describe the crack propagation constraint. A correction response surface was constructed to relate the high fidelity models to the low fidelity models, thus combining the accuracy of the Former with the computational efficiency of the latter.

The O-degree ply stress intensity factors were calculated using two models: a high fidelity and a low fidelity model. The high fidelity model employed a large number of finite elements to capture the singular stress near the crack tip. The low fidelity model used coarser finite elements for stress analysis of an untracked stiffened panel, and the stress intensity factors were computed using a closed form expression. A correction response surface was used to estimate the high fidelity stress intensity factor from the low fidelity stress intensity factor response surface. A single response surface was fitted to the buckling load factor h in the six design variables. Using these response surfaces, including the correction response surface, the minimum weight panel was found using the optimizer in the commercial spreadsheet program EXCEL. The result indicates that correction response surfaces can be used efficiently in structural optimization problems.

Acknowledgments This research was supported by a NASA

Langley Research Center Grant, NAG-1-2000.

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and Material Conference Paper AIAA-98- 2047, Proceedings, Long Beach, Florida, 1998 Vol.? pp.2917-2922

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6. R. S. Vaidya, J. C. Hung and C. i? Sun “Effect of Ply Thickness and Crack Tip Damage on Failure of Notched Composite Laminates” AIAA/ASME/ASCWAHS/ASC Structures, Structural Dynamics and Material Conference Paper AIAA-97- 1184, Proceedings, Kissimmee, Florida, 1997 Vol. 1 pp.2224-2237.

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