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1 STRUCTURAL OPTIMIZATION OF A HAT STIFFENED PANEL BY RESPONSE SURFACE TECHNIQUES Roberto Vitali, Oung Park, Raphael T. Haftka and Bhavani V. Sankar University of Florida Department of Aerospace Engineering, Mechanics and Engineering Science 231 Aerospace Building, P.O. Box Gainesville, FL Cheryl A. Rose NASA Langley Research Center Hampton, VA Abstract The paper describes a design study for the structural optimization of a typical bay of a blended wing body transport. A hat stiffened laminated composite shell concept is used in the design. The geometry of the design is determined with the PANDA2 program, but due to the presence of varying axial loads, more accurate analysis procedure is needed. This is obtained by combining the STAGS finite element analysis program with response surface approximations for the stresses and the buckling loads. The design procedure results in weight savings of more than 30 percent, albeit at the expense of a more complex design. The response surface approximations allow easy coupling of the structural analysis program to the optimization program in the widely available Microsoft EXCEL spreadsheet program. The response surface procedure also allows the optimization to be carried out with a reasonable number of analyses. In particular, it allows combining a large number of inexpensive beam-analysis stress calculations with a small number of the more accurate STAGS analyses. Introduction Major air carriers have expressed the need for larger airplanes to meet the growing demands for air travel, especially in the pacific rim and on transatlantic routes between major airports in USA and Europe. A blended wing body (BWB) airplane capable of carrying about 800 passengers is one of several configurations currently being considered for satisfying this need. As the name implies, a principal feature of a BWB transport is a wide double deck center body which is blended in the wing. Due to the shape of the airplane the pressured center-body region, which includes the passengers area and the cargo area is non-circular (see Fig. 1). This type of configuration for the center-body region is challenging from the standpoint of structural design, since the upper and lower cabin surfaces carry both internal pressure loading and running loads due to wing bending. Further, to keep structural weight as low as possible, the structure is envisioned to be built of composite materials. The paper describes work which a part of a BWB design study, led by the McDonnell Douglas Corporation in collaboration with NASA and four universities (University of Florida, Stanford University, University of Southern California, and Clark Atlanta University). The study described in the paper involves the optimization of one of the structural concepts considered for the design, a stiffened shell concept for the upper cover panel of a typical passenger bay. The passenger area is divided into several bays, separated by longitudinal ribs, to reduce instability problems in the upper wing skin, which carries compressive wing bending loads. Specifically, results are presented for a composite hat-stiffened skin configuration. The structural optimization problem is formulated using weight as the objective function, with constraints on buckling of the cover panel and maximum stress in the cover panel. The structural analysis is performed by an analysis code which does not have optimization capabilities and is difficult to connect to an optimizer. For such situations, response surface techniques, which create simple approximations of structural response, have been shown to be useful 1. The use of response surface techniques also permits the integration of simple analysis models together with more complex ones in a single approximation. In the present work, both simple and complex stress analyses are integrated in a response surface used for the structural optimization. Copyright 1997 Roberto Vitali. Published by the, Inc with permission

2 Figure 1. Center-body Region of a BWB Transport Table 1. Material Properties for Graphite- Epoxy design specifications value E 11 (msi) 9.25 E 22 (msi) 4.67 G 12 (msi) 2.27 ν σ all (ksi) 50 τ all (ksi) 18 Problem Description The structural configuration considered in the optimization is a hat-stiffened skin, upper cover panel of a typical passenger bay, as shown in the Fig. 2. The panel is assumed to be 150 inches long in the spanwise, or x, direction and 900 inches in the chordwise, or y, direction. The structure was first designed subject to a uniform pressure load of psi applied to the bottom skin surface, and an end load, N x equal to 4319 lb./in. applied to the right end of the panel in the spanwise direction. The left end of the panel is restrained from movement in the x- direction. The right and left ends are clamped and the unloaded sides are simply supported. Figure 2. Hat-stiffened skin upper cover 150" x Structural Optimization PANDA2 Solution The first step in the structural optimization was to use the PANDA2 program 3, to obtain an initial design. PANDA2 is a program specifically designed to find minimum weight design of laminated composite stiffened panels. The philosophy of PANDA2 is to obtain an initial design using simple close form expression and finite difference analyses of a discretized panel module independently to obtain buckling load factors. A panel module and the design variables used in the PANDA 2 optimization are shown in Fig 3. A module includes the cross section of a stiffener plus the panel skin, with width equal to the spacing between stiffeners (b in Fig. 3). A typical stiffened panel consists of several panel modules placed adjacent to each other. 900" The skin and the individual components of the hat stiffener are constructed from graphite-epoxy preforms, which were developed under the NASA ACT program 2. Each preform is a stack of material which is equivalent to nine layers of unidirectional prepeg with 44.9%, 49.2% and 12.9% of 0, 45 and 90 degree fibers, respectively. The nominal stacking sequence of the preforms used in the skin and in all of the components of the hat stiffener is [45/-45/0/90/- 45/45]. Each preform, or stack, has a cured thickness of inches. Nominal material properties for a cured stack and the stress allowables that are se in the designs are provided in Table 1. h Figure 3: Design variables for the PANDA2 optimization w t w t c w 2 b2 b The objective function, the design variables and the side constraints used in the optimization are listed in Table 2. The first constraint ensures that there are enough equally spaced stiffeners in the panel for the single module model to give good approximation to the local skin buckling mode. The second and third constraints guarantee numerical t f t s 2

3 stability. The fourth constraint controls the height of the hat. The upper value reflects a manufacturing limit. The fifth and sixth constraints ensure that the flange is at least 1.3 inches wide, again for manufacturing reasons, and the seventh constraints ensures hats with reasonable proportion. Finally the eight constraint sets the upper and lower thickness bounds of all elements of the stiffener and the panel skin. Stress and buckling constraints were also imposed by PANDA2. Initially, a factor of safety of 1.3 was applied to the stresses, and a factor of 1.15 was applied to the buckling load factor. Table 2. PANDA2 Optimization Problem (dimensions in inches) Objective function : min {weight} Design Variables: b, b 2, w, w 2, h, t s, t f, t w, t c Side Constraints: 6 < b < 24 (1) b b > 0 (2) 4.5 < b 2 < 18 (3) 2 < h < 6.5 (4) b 2 -w 2 >2.6 (5) 2.9 < w 2 <15.4 (6) 2.25 < w < (7) 0.11 < t < 1.1 (8) Buckling and stresses constraints in Panda2 (9) The optimum design obtained by PANDA2 is provided Table 3. For the design shown in Table 3, the active constraints at the mid-length of the panel were fiber compression in the crown, web buckling, and axial strain. At the ends of the panel, the active constraints were local buckling of the panel skin between stringers, local buckling of the panel skin under the hats, and fiber compression in the skin. Table 3. PANDA2 Optimum design Weight (lb/ft2) 4.65 b b w 6.5 w t s 5.8 t f 0.22 t w 0.34 t c 0.20 PANDA2 is capable of handling only continuous design variables. However, the thicknesses of the components are discrete variables, since the thicknesses are limited to integer multiples of the basic laminate stack described earlier. The thickness in Table 3 obtained from the continuous variable optimization were therefore rounded up to the next discrete thicknesses. The design obtained after rounding up the thickness variables and the associated weight of the rounded up design are provided in Table 4. Table 4. Rounded up PANDA2 optimum design Weight (lb/ft2) 4.86 b b h 6.5 w 4.3 w t s 0.22 t f 0.33 t w 0.22 t c 0.33 Initial Response Surface Solution For the loading and boundary conditions of the current problem, PANDA2 leads to a conservative design. The conservative design results from the requirement in PANDA2 that the panel cross section and the properties are constant in the spanwise direction while the loading is variable. Near the ends of the panel the skin is in compression and the crown of the hat is in tension. In the middle of the panel the opposite is true; that is, the skin is in tension and the crown of the hat in compression. PANDA2 finds the location in which each element of the panel module has the highest compressive stresses, and based on these stresses it designs the thickness and size each element. A more efficient design for the panel was obtained by relaxing the constraint imposed by PANDA2 that the cross-section of the panel is constant in the spanwise direction. The cross-section of the panel was varied by dividing the panel into three sections: two identical sections at the ends of the panel and a section in the middle of the panel as shown in Fig. 4. The tendency for the skin to buckle near the ends of the panel, which was an active constraint in the PANDA2 design, was reduced by adding a layer of material equal to the thickness of the flange to the panel skin between the stiffeners and to the panel skin under the hat stiffeners. To prevent buckling of the web crown at the mid-length point of the panel, the thickness of the crown of the hat was increased. The cross-sections of a panel module at the 3

4 ends of the panel and at mid-length are provided in Figs 5a and 5b, respectively. An optimum design for the more complicated panel configuration shown in Figs. 4 and 5 was obtained using response surface techniques. Response surface techniques have been shown to be useful when the design variables are discrete, and when it is difficult to connect the code used to perform the analysis and the code used to perform the optimization 1. The structural analysis was performed using STAGS (Structural Analysis of General Shells) 4. STAGS is a finite element code for non-linear analysis of stiffened shell structures of arbitrary shape and complexity. The STAGS analyses, an additional factor of safety equal to 1.25 was applied to the end load. The end load applied in the STAGS analyses was equal to 5398 lb./in. Figure 4. Division of the panel in three sections d 150" mid section b d Table. 5 Design variables used in STAGS Analyses variable variable meaning t sb Skin Thickness near panel ends t sm Skin Thickness in the middle of the panel t cb Crown thickness the near panel ends t cm Crown thickness near the middle of the panel d distance from panel ends to thickness discontinuity Several methods are available for selecting the design points so that the error in the approximation is minimized. The D-Optimal criterion 1 as implemented in the JMP program 7 was employed for selection the design points. The implementation in JMP finds a D-Optimal set of points from a given set of candidate design points in the design domain. The candidate design points that were used for constructing the response surfaces are provided in Table 6. crown of the hat end section y a a extended flange end section module of width b Table 6. Candidate points for Response Surface Construction Design Variable minimum step maximum d t sb t sm t cb t cm " x skin The response surfaces created were then used as constraints function in an optimization to minimize the panel weight. The optimization was performed using a spread sheet program in Microsoft EXCEL 5 that allows discrete design variables. the design variables used in the optimization are provided in Table 5. the response surfaces were approximated using polynomials. In order to determine the coefficients in the approximating polynomials, the structural response surface was evaluated at a number of design points exceeding the number of coefficients in the polynomials. The size of the design domain was reduced to 740 feasible design points by introducing the following considerations: The lower and upper bounds to the optimal weight were estimated to be 3.0 lb./ft 2 and 4.3 lb./ft 2 respectively. The skin near the ends of the panel was expected to be thicker than the skin in the middle of the panel (t sm <t sb ). The thickness of the crown in the middle of the panel was expected to be greater than the thickness of the crown near the ends of the panel (t cb <t cm ). 4

5 Figure 5. Cross-sectional geometry of variable thickness panel h h t w t w b w (a) End region b w (b) Mid-section t cb t cm Buckling Response Surface The STAGS structural analysis required at each of the design points used in creating the response surface is fairly computationally intensive and time consuming. The STAGS model used in the analysis of the divided panel had approximately 70,000 degrees of freedom. The panel skin and all elements of the stiffeners were modeled with branched shells. One linear stress and linear buckling analysis using this model required approximately 6,500 CPU seconds on a DEC ALPHA 200 4/166 work station One non-linear analysis required approximately 9,000 CPU seconds on the same machine. Because of the long time required for each structural analysis we started with a simple response surface and selected 10 points to build a linear response surfaces for the buckling load factor (design points 2-11 in Table 7). To the 10 points we added one extra structure, design point 1 in Table 7, referred as the nominal design, which we selected based on engineering judgment. However the linear fit obtained was very poor with rms error=0.61 and Ra 2 =0.046 (Ra 2 =1 for a perfect fit and Ra 2 =0 for a very bad fit ) and negative coefficients for some of the thickness variables. In attempt to improve the accuracy of the response surface predictions a new design domain was defined around the nominal structure. The thickness variables were permitted to change by inches from the nominal design and d by +12 inches, thus each variable could have 3 possible levels and a total of t f t f t sb t sm 243 points. Twelve design points were selected out of these 243 using the D-Optimality criterion. These design points are shown in Table 7, as points 12 to 23. The linear response surface obtained for this smaller region had rms=0.54 and Ra 2 =0.56, indicating still a non-satisfactory fit. Moreover the t statistics of the coefficients were very small (t statistic is a parameter that indicates the confidence in the values of the coefficients obtained. The higher the value the better is the confidence in the coefficients). Full Domain Quadratic Response Surface Since the linear response surface was not satisfactory we added 10 design points, shown in Table 7 as points 24 to 33. With 33 points we fitted a full quadratic polynomial over the entire domain. A full quadratic polynomial in 5 design variables has 21 coefficients. We discarded terms in the polynomial with low t statistics as long as Ra 2 kept increasing. The quadratic polynomial retained 11 coefficients and is given as: λ = d t sm d t sb d t sm t sm t sb t sm t cm d t cm t sb For this case rms=0.19 and Ra2=0.90 and the lowest t statistic was 3.15, indicating reasonable confidence in the coefficients. The accuracy of the response surface was also checked by constructing response surfaces with 32 points and then comparing the response surface predictions at the point left out with the STAGS analysis predictions at that point (a procedure known as PRESS) 6. Based upon these results the quadratic response surface predictions for the buckling load factor are expected to have less than 25% error. Stress Response Surface A response surface was also constructed for the compressive axial stresses at the change in thickness. Stress values at other regions of the panel for all the designs analyzed did not exceed the maximum stress allowable. There are only 26 design points available for the maximum stresses as shown in Tables 7. 5

6 Design points where maximum stress data is not available are indicated by dashes in the maximum stress column in the tables. The response surface obtained is: σ = d t sb t sm 18.4 d t sb t sm d Point # Table 7. Structural Designs Used for the Linear Response Surface d t sb t sm t cm t cb weight λ (lb./ft.) σ max (psi)

7 where, Ra 2 =0.96 and rms=2145 psi. The t statistics for all the coefficients except for t sb were satisfactory. We also checked accuracy by the PRESS procedure which indicated that a maximum error of 20% is expected. Optimization Using the Response Surfaces The nominal design had a buckling load factor λ= The response surface for the buckling load factor λ in the box domain had errors of 25%. Therefore the allowable buckling load factor was increased by 25%, from 1.15 to Similarly for the compressive stresses s, we increased the safety factor by 20% from 1.3 to The optimization problem in the box domain was formulated as shown in Table 8. Table 8 Optimization Problem Using Response Surfaces Objective Function: min{weight lb/ft 2 )} Design Variables: d, t sb, t sm, t cm, t cb Constraints: λ > σ < t sb - t sm > 0 t cm - t cb > 0 The optimization was performed using a generalized reduced gradient optimizer available in Microsoft EXCEL. The optimum design obtained, subject to the constraints specified in Table 8, is provided in Table 9. Analysis of the design in Table 9 with STAGS gave a buckling load factor of λ=1.379 and a maximum stress at the discontinuity σ=22,450 psi. As shown by comparing the results in Table 9 with the results in Table 4, using STAGS and response surface approach to perform the optimization, the weight of the upper cover panel was reduced from 4.86 lb./ft 2 to lb./ft 2. Table 9. Optimum Design Obtained using response surfaces Point # 35 d 30 t sb t sm t cm t cb Weight(lb/ft 2 ) λ response surface σ (psi) response surface New Design Loads After the previous optimization was performed the design loads and the required safety factors were updated. based on changes to the overall design of the airplane, as well as to refinement in the selection of load cases. Additionally, the safety factors were reduced, mostly reflecting greater confidence in the design, and allowing local buckling between the limit and ultimate load factors A new load case of pressure (p = psi ) and with no in-plane loads was added, and the combined load case was changed to p = psi combined with in-load of 4319 lb./in. These new load cases had a safety factor equal to 1.0. Additionally the STAGS FEM model was simplified to include only one stiffener section of inches wide, with symmetry conditions imposed on the sides. Also, the calculation of the weights was refined to tremove sb,t sm,,t cm, t cb > 0.11 duplications due to intersecting finite elements. To update the design for the new load case the starting point was the optimum obtained in the previous design cycle. The new pressure-only load case generates high stresses in the crown of the hat because the neutral axis of the section near the ends is very close to the skin and far from the crown of the hat. New Buckling Response From the preceding optimization it was clear that the thickness of the skin in the section near the middle, t sm, was likely to remain at its lower limit therefore it was removed from design variable list. On the other hand, the webs accounted for about the 40% of the total weight and so the thickness of these structural elements was introduced as a new design variable. The thicknesses of the skin at the boundary and of the crown in the middle of the panel were permitted to change by inches from the previous optimum. The thickness of the web was allowed to take only the values of inches or 0.22 inches, since a lower value was sure to violate 7

8 the buckling constraint, and a larger value would lead to very heavy designs. Similarly the thickness of the crown near the ends was also limited to only two values of 0.11 inches or inches. The distance from the panel ends to the thickness discontinuity, that is d, was permitted to change by + 6 inches. Given these restrictions the total of 108 new design points around the previous optimum design point (design number 35 in Table 9) were created. The JMP program was used to pick 25 new design points using the D_Optimality criterion (Table 10) and fit two partially quadratic response surfaces, one for each load case, through them. The response surfaces are quadratic in d, t sb, t cm, t w and linear in t cb and have 16 coefficients. During the fitting procedure terms in the polynomial response surface expressions with low t-statistic were discarded as long Ra 2 kept increasing. The partially quadratic polynomials obtained for the pressure-only load case has the form: λ p = d 4.49 t sb 0.44 t cm t cb t w d t sb d t cm t cm t sb t cm 2 Table 10. Structural Designs Near Design 35 Used for the Partially Quadratic Response Surface Point # d t sb t sm t cm t cb weight (lb./ft. 2 ) λ p λ c σ max (psi)

9 with an Ra 2 = and rms= The partial quadratic response surface expression obtained for the combined load case took the form: λ c = d t sb t cm t w 0.12 t sb d t sb t cm t sb t cm t cm t w with Ra 2 = 0.84 and rms= The accuracy of the response surfaces was also checked using the PRESS procedure. The maximum error for the pressure-only load case was about 15% and for the in-load combined with pressure case the error was about 20%. Stress Response Surface The stress values obtained for the new design points showed that the stress limit was exceeded only in the crown near the ends therefore only the stress response surface for this case was created. The procedure followed to create this response surface was different from the one used for the buckling load factor response surfaces. Using beam analysis we can obtain a good approximation to the stresses, and so the beam approximation was combined with the more accurate STAGS results. Because the beam approximation is inexpensive the stresses in the crown of the hat were calculated for all 108 structures. These stresses were compared with STAGS results for 13 structures (points and point 36 in Table 10), and the differences ranged from 0.1% to 4.86%. These approximate stresses were combined with STAGS analyses for the results obtained for the structures and structure 36 in Table 10. The two sets of results were combined in a single response surface by using a weighted least square fit, with the weight associated with the more accurate STAGS analysis being ten times the weight of the beam-analysis stresses. We started with a quadratic response surface in d, t sb, t sm, t w and linear in t cb and used JMP to discard terms with low t-statistic as long as Ra 2 kept increasing. The response surface thus obtained took the form: σ = d t sb tcm tcb t w t sb d tcm t w with Ra 2 = and rms = psi. The maximum error obtained by the PRESS procedure was 3%. New Optimization Using Response Surfaces The two response surfaces for the buckling load factor λ had errors of 15% and 20%. Accordingly the allowable buckling load factor for the pressure-only load case was increased from 1.0 to 1.15 while the allowable buckling load factor for the combined load case was increased from 1.0 to Following the same procedure, the allowable for the stress response surface was increase from 1.0 to The optimization problem in the box domain was formulated as shown in Table 11. Table 11. New Optimization Problem Using Response Surfaces Objective Function: min{weight lb/ft 2 } Design Variable: d, t sb, t sm, t cm, t cb Constraint: λ p > 1.15 λ c > 1.20 σ < 0 The optimization was performed again with Microsoft EXCEL. The optimum design obtained, subject to the constraints specified in Table 11, is provided in Table 12. Analysis of the design in Table 12 with STAGS gave a buckling load factor λ = 1.61 for the pressure-only case and λ = for the combined load case. The maximum stresses in the crown of the hat are psi. As seen by comparing the results in Table 12 with the results in Table 4, using STAGS and the response surface approach to perform the optimization, the weight of the upper cover panel was reduced from 4.86lb/ft 2 to 3.36 lb/ft 2. Table 12. Optimum Design obtained using Response Surfaces Point # 62 d 24 t sb t cm t cb t w 0.22 Weight(lb/ft 2 ) 3.31 λ p response surface 1.61 λ I+p response surface 1.30 σ (psi) response surface 47,095 Concluding Remarks A design study for the structural optimization of a typical bay of a blended wing body 9

10 transport was presented. A hat stiffened laminated composite shell concept was used in the design. Initial optimization was carried out with the PANDA2 program, but it resulted in a heavy design due to the conservative nature of the analysis and design procedure for the case of buckling under varying axial loads. More accurate analysis and a more flexible design procedure was obtained by combining the STAGS finite element analysis program with response surface approximations for the stresses and the buckling loads. The design procedure reduced the weight by more than 30 percent, albeit at the expense of a more complex design. The response surface approximations allowed easy coupling of a structural analysis program to the optimization program in the widely available Microsoft Excel spreadsheet program. The response surface procedure also allowed the optimization to be carried out with a reasonable number of analyses. In particular, it allowed combining a large number of inexpensive beam-analysis stress calculation with a small number of the more accurate STAGS analyses. Acknowledgement This work was supported in part by NASA grant NAG Helpful discussions with members of the BWB team, George Rowland, Art Hawley and Professor Peter Lissaman are greatfully acknowledged. References 1. Mason, R.T. Haftka and E.R. Johnson Analysis and Design of Composite Channels Frames AIAA Paper CP, Proceedings, AIAA/NASA/USAF/ISSMO 5th Symposium on Multidisciplinary Analysis and Optimization, Panama City, Florida, September 7-9, 1994 Vol.2 pp NASA ACT -preforms 3. Bushnell D. PANDA-Interactive program for minimum weight design of stiffened cylindrical panels and shells Computers and Structures, 16, 1983, pp A. Brogen, C. C. Rankin and H. D. Cabiness Structural Analysis of General Shell version 2.0, Lockheed Research Laboratory, Palo Alto Ca. June Microsoft Corporation, EXCEL Myers R.H., and D. C. Montgomery, Response Surface Methodology, Wiley, New York SAS Institute, JMP version 3.1, Cary NC, February

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