Optimization of Composite Sandwich Cover Panels Subjected to Compressive Loadings

An analysis and design method is presented for the design of composite sandwich cover panels that includes transverse shear e ects and damage tolerance considerations. This method is incorporated into a sandwich optimization computer program entitled SANDOP. As a demonstration of its capabilities, SANDOP is used in the present study to design optimized composite sandwich cover panels for transport aircraft wing applications. The results of this design study indicate that optimized composite sandwich cover panels have approximately the same structural e ciency as sti ened composite cover panels designed to satisfy identical constraints. The results also indicate that inplane sti ness requirements have a large e ect on the weight of these composite sandwich cover panels at higher load levels. Increasing the maximumallowable strain and the upper percentage limit of the 0 and 45 plies can yield signi cant weight savings. The results show that the structural e ciency of these optimized composite sandwich cover panels is relatively insensitive to changes in core density. Thus, core density should be chosen by criteria other than minimum weight (e.g., damage tolerance, ease of manufacture, etc.). Introduction Composite materials are being widely considered for application to heavily loaded primary aircraft structures such as wing cover panels. To date, much of the research conducted on aircraft wing cover panels has focused on sti ened plate designs. The analysis of sti ened cover panels is well-understood, and tools exist to perform analysis and design optimization of these panels (refs. 1 and 2). Relatively less emphasis, however, has been placed on cover panels of sandwich construction. The present paper describes an analysis and design method that has been developed for composite sandwich cover panels loaded in compression, including damage tolerance considerations. The analysis and appropriate design variables have been incorporated into a constrained sandwich optimization program entitled SANDOP. This program utilizes weight per unit area as the objective function to be minimized subject to several constraints. SANDOP is written in sizing and optimization language (SOL), a high-level computer language developed speci cally for the application of numerical optimization methods to design procedures. (See refs. 3 and 4.) As a practical demonstration of SANDOP, composite sandwich cover panels for transport aircraft wing applications have been designed subject to constraints appropriate for this kind of structure. These composite sandwich cover panels are compared with composite sti ened cover panels that were designed to satisfy identical constraints using a panel analysis and sizing code (PASCO). (See refs. 1 and 2.) Furthermore, the e ect of changing the constraint values on the structural e ciency of these composite sandwich cover panels is investigated. Symbols Amn modal amplitudes (see eq. (4)) A11; A66 inplane sti nesses of cover panel A11;min; minimum-required inplane sti ness A66;min of facesheets A11;min;bl; baseline values of minimum-required A66;min;bl inplane sti ness of facesheets a cover-panel length (see g. 1) b cover-panel width (see g. 1) DQx; DQy transverse shear sti nesses of cover panel (see eqs. (2) and (3), respectively) D11; D12; bending sti nesses of cover panel D22; D66 Ecz sandwich core modulus in z-direction Ef e ective facesheet modulus in longitudinal direction EL; ET lamina modulus in longitudinal and transverse directions, respectively GLT lamina shear modulus 1 Gxz ; Gyz sandwich core transverse shear modulus in xand y-directions, respectively kA scaling factor for minimum-required inplane sti ness L0; L45; L90 lower percentage limit of 0 , 45 , and 90 plies, respectively m number of longitudinal half-waves for cover-panel buckling mode Nx applied longitudinal stress resultant (see g. 1) N b x longitudinal stress resultant at buckling Nw x longitudinal stress resultant for facesheet symmetric wrinkling Nxy applied shear stress resultant Ny applied transverse stress resultant n number of transverse half-waves for cover-panel buckling mode tc core thickness (see g. 1) tf facesheet thickness (see g. 1) t0; t45; t90 thickness of facesheet 0 , 45 , and 90 plies, respectively U0; U45; U90 upper percentage limit of 0 , 45 , and 90 plies, respectively W weight per unit area of cover panel w out-of-plane displacement of cover panel x; y; z Cartesian coordinate system (see g. 1) "x longitudinal strain of cover panel "x;max maximum-allowable longitudinal strain LT lamina major Poisson's ratio C=E carbon/epoxy material density core core density Analysis, Design, and Optimization Methodology This section describes the analysis and design used in this study of sandwich cover panels with composite material facesheets. Dominant response mechanisms for composite sandwich cover panels are presented and analyzed. The analysis is combined with Figure 1. Panel geometry and loading. All edges simply supported. an optimization procedure to obtain structurally efcient designs. The objective function, design variables, and constraints for the structural optimization problem are explained in this section. The sandwich cover panel considered in the present study is shown in gure 1. This sandwich panel is rectangular, at, and simply supported on all four edges. A single, uniform longitudinal stress resultant Nx is applied at opposite ends of the panel as shown in gure 1. The facesheets are symmetric composite laminates with specially orthotropic material symmetry. The sandwich core also exhibits specially orthotropic material symmetry in its transverse shearing sti nesses. The corresponding transverse shearing sti nesses of the core are denoted by Gxz and Gyz . The principal directions of the core material are assumed to coincide with the x and y coordinate directions. (See g. 1.) Response Mechanisms Three response mechanisms are included in SANDOP for designing composite sandwich cover panels loaded in compression. These mechanisms are global buckling (including transverse shear deformation), symmetric facesheet wrinkling, and material failure. A brief description of each of these mechanisms is presented as follows: Global buckling. The equation governing global buckling of sandwich panels, including transverse shear e ects, is derived in reference 5 and is given by 2 "D11D66 1 DQy + Nx DQxDQy!# @6w @x6 + "D11D66 DQx + D11D22 D2 12 2D12D66 1 DQy + Nx DQxDQy!# @6w @x4@y2 + "D22D66 1 DQy + Nx DQxDQy!+ D11D22 D2 12 2D12D66 DQx !# @6w @x2@y4 + "D22D66 DQx # @6w @y6 "D11 +Nx D11 DQx + D66 DQy!# @4w @x4 "2(D12 + 2D66) +Nx D22 DQy + D66 DQx!# @4w @x2@y2 D22 @4w @y4 +Nx @2w @x2 = 0 (1) where the transverse shear sti nesses for an orthotropic core material DQx and DQy are given in reference 6 by, respectively, DQx = Gxz tc + tf 2 tc (2) and DQy = Gyz tc + tf 2 tc (3) Solutions to the buckling equation for sandwich panels are determined directly by assuming a buckling mode shape that satis es both the di erential equation (eq. (1)) and the boundary conditions (simply supported on all four edges). A buckling mode shape that meets this criterion is expressed as w = Amn sin m x a sin n y b (4) where m = 1; 2; 3; : : : (0 x a) n = 1; 2; 3; : : : (0 y b) Substituting this mode shape into equation (1) yields a homogeneous linear algebraic equation that depends on the wave numbers m and n, and thus constitutes an eigenvalue problem. For nontrivial solutions, the resulting equation can be solved for Nx as a function of m and n. The global buckling stress resultant N b x is obtained by minimizing Nx with respect to m and n. This formulation for global buckling includes shear crimping as a response mechanism for sandwich plates. Shear crimping is given by the degenerate case of global buckling for which the wave parameter m is very large. Facesheet wrinkling. Another stability-related response mechanism for sandwich structures is facesheet wrinkling. For this mechanism, the facesheets buckle locally with a wavelength of the same order as the thickness of the sandwich core. Facesheet wrinkling can be symmetric or antisymmetric in form as shown in gure 2. In the present study, only symmetric facesheet wrinkling is included. Since wrinkling in sandwich panels with honeycomb cores is usually of the symmetric type (ref. 7), the current wrinkling analysis is valid for honeycomb cores. The current wrinkling analysis may not be valid for sandwich panels with foam cores since they may buckle in an antisymmetric wrinkling mode. Figure 2. Symmetric and antisymmetric facesheet wrinkling. 3 The equation used in the present study to determine the onset of symmetric facesheet wrinkling (ref. 7) is given by Nw x = 0:67tfEf Ecz tf Ef tc !1=2 (5) Material failure. For a given panel design, the facesheet material may fail before the onset of either of the stability mechanisms previously discussed. Material failure is determined by specifying a maximum-allowable longitudinal strain criterion. Speci cally, the onset of material failure is assumed to occur when the axial strain "x exceeds a maximum strain value "x;max. This maximum strain value is based on a lower limit compression-after-impact failure strain of the composite facesheet that was experimentally determined. The use of this allowable strain criterion implicitly incorporates a damage tolerance constraint into the design process. Objective Function and Design Variables Structural e ciency is de ned by a minimum cover-panel weight for the given design loads. The objective function used in this study is the weight per unit area W of the cover panel. The design variables used in this study are classied as either facesheet design variables or core design variables. The composite facesheets are considered to be homogeneous through the thickness and to consist of 0 , 45 , and 90 plies only. These two assumptions allow the facesheets to be completely de ned by using only the three design variables t0, t45, and t90, which are the thicknesses of the 0 , 45 , and 90 plies, respectively, in the facesheet laminates. Both facesheets are symmetric, specially orthotropic, and identical. The sandwich core is de ned by the two design variables tc and core, the core thickness and core density, respectively. The three core material properties used in the analysis, Gxz , Gyz , and Ecz , are determined by the core type, core material, and core density. Constraints The constraints used to perform the optimization are based on the response mechanisms for sandwich panels previously described and on current design practices for composite facesheets and sandwich cores. A brief description of the constraints is presented as follows: Response mechanism constraints. The coverpanel designs for the present study are constrained to have buckling and wrinkling stress resultants N b x and Nw x greater than the applied stress resultant Nx. In addition, the longitudinal strain "x due to the applied Nx is constrained to be less than the maximumallowable longitudinal strain "x;max. This maximumallowable strain corresponds to the inherent residual compressive strength for an impact-damaged composite laminate, and it is an empirical value. Facesheet and core constraints. Constraints are placed on the laminate and the inplane sti nesses of the composite facesheets. The laminate is constrained by placing upper and lower limits on the relative thicknesses of each ply group (plies with the same orientation) with respect to the total facesheet thickness. These constraints are written as L0 < t0 t0 + t45+ t90 < U0 (6) L45 < t45 t0 + t45+ t90 < U45 (7) L90 < t90 t0 + t45+ t90 < U90 (8) where L and U denote the lower and upper percentage limits, respectively, for a given ply group. These constraints are used to exclude laminate designs that are dominated by one ply orientation. Practical laminate designs are often required to have bers oriented in several directions to satisfy requirements not speci cally considered herein, e.g., repair requirements (ref. 8). The composite facesheet designs are also required to satisfy minimum inplane sti ness constraints. The facesheet sti nesses A11 and A66 are required to be greater than some speci ed minimum sti nesses A11;min and A66;min, respectively. The minimum sti nesses used in this study are discussed in the \Results and Discussion" section. The sandwich core density is constrained to a range of densities that are practical for aircraft cover panels. Upper and lower limits for core density are speci ed for the present study. SANDOP The design and optimization method described above has been incorporated into a sandwich optimization computer program entitled SANDOP. This program is written in sizing and optimization language (SOL), a high-level computer language developed speci cally for the application of numerical optimization methods to design problems. (See refs. 3 4 and 4.) SANDOP allows the user to optimize composite sandwich cover panels. The input parameters available to the user are the facesheet and core material properties, the panel dimensions, the design stress resultant Nx, and the parameter values for the various constraints. SANDOP can be modi ed to expand the present analysis and constraints. Results and Discussion As a demonstration of the capabilities of SANDOP, the program was used to design optimized composite sandwich cover panels for transport aircraft wing applications. These optimized sandwich panels are compared with sti ened composite cover panels designed to satisfy identical constraints. The e ect of the constraints on the optimal design is also investigated. Baseline Design A baseline set of design parameters and constraints was selected to establish a reference design for subsequent comparison. These design parameters and constraints are typical of those used to design sandwich cover panels for transport wing applications. All the cover panels considered in the study are assumed to be square, with 30-in. side dimensions. Cover panels were optimized for load levels ranging from 3000 to 24 000 lb/in. The unidirectional composite material properties used for the facesheets are those of Hercules IM6 carbon bers and American Cyanamid 1808I epoxy interleaved material given in reference 9 as shown in table 1. The core material used in this study is Hexcel 5052 aluminum-alloy hexagonal honeycomb, whose properties were obtained from reference 10; a few typical values are shown in table 2. Since core material properties are available only for speci c values of the core density, SANDOP interpolates these data to obtain core properties at densities other than those given in reference 10. Table 1. Properties of IM6/1808I Carbon/Epoxy Tapea Longitudinal Young's modulus, EL, Msi . . . 18.5 Transverse Young's modulus, ET , Msi . . . . 1.09 Shear modulus, GLT , Msi . . . . . . . . . . 0.70 Major Poisson's ratio, LT . . . . . . . . . 0.33 Density, C=E, lb/in3 . . . . . . . . . . . . 0.058 aValues were obtained from reference 9, except for the density which is estimated. Table 2. Properties of Hexcel 5052 Aluminum-Alloy Honeycomb Corea core, lb/ft3 Ecz , ksi Gxz , ksi Gyz , ksi 1.0 10.0 12.0 7.0 6.0 235.0 96.0 40.5 9.5 420.0 105.0 53.0 aValues were obtained from reference 10. Table 3. Constraint Values of Baseline Design "x;max = 0:0045 in/in. A11;min = f(Nx) A66;min = g(Nx) See gure 3 L0 = 0:125 U0 = 0:375 L45 = 0:125 U45 = 0:375 L90 = 0:125 U90 = 0:375 1:0 lb=ft3 < core < 9:5 lb=ft3 The constraints used for the baseline design are shown in table 3. The minimum-required inplane sti nesses A11;min and A66;min are functions of the load level as indicated by gure 3. This correlation between the minimum-required inplane sti ness and Nx is based on historical data for transport aircraft wings that were presented in reference 11. The limits on the relative thickness of each ply group, with respect to the total facesheet thickness, is based on the recommendations of reference 8. These recommendations are designed to yield laminates suitable for bolted and riveted joints. A maximum-allowable strain of 0.0045 in/in. was selected to provide acceptable damage tolerance capability consistent with current composite material systems. Figure 3. Minimum-required inplane sti nesses for cover panels (ref. 11). 5 Figure 4. Weight comparison between sandwich and sti enedplate composite cover panels. Comparison With Sti ened Cover Panels The structural e ciency of composite sandwich cover panels optimized with SANDOP is shown in gure 4. In this gure the weight per unit area of the cover panel W is shown as a function of Nx. In addition, the structural e ciency of hatand bladesti ened composite panels optimized with PASCO (refs. 1 and 2) is shown in gure 4 for comparison. Optimum designs for both the sandwich and the sti ened cover panels were determined using the baseline material properties and constraints. The composite sandwich cover panels have approximately the same structural e ciency as the composite sti ened cover panels when designed to identical constraints. This behavior is to be expected since the maximumallowable strain and the inplane sti ness requirements are the active constraints for the optimum designs. These two constraints determine the amount of composite material required by both the sandwich and sti ened cover panels. Since the weight of the composite material constitutes the major component of the cover-panel weight, the structural e ciencies of both the sandwich and the sti ened cover panels are approximately equal. E ect of Varying Constraints on Optimum Design To assess the sensitivity of the structural e ciency of composite sandwich cover panels to changes in the constraints, new sets of optimum composite sandwich cover panels were designed while varying the constraints one at a time. By comparing these new cover-panel designs with the baseline designs, the e ect of varying the constraints is identi ed. Figure 5. E ect of maximum-allowable longitudinal strain on structural e ciency of composite sandwich cover panels. E ect of varying maximum-allowable strain. The e ect of varying the maximum-allowable strain constraint on the structural e ciency is shown in gure 5. This gure shows the structural e ciency W of optimized sandwich cover panels as a function of Nx for three values of the maximumallowable strain "x;max. For the baseline design, "x;max = 0:0045 in/in. This maximum-allowable strain is an active constraint for Nx greater than 15 000 lb/in. Increasing the maximum-allowable strain to 0.006 in/in. yields signi cant improvements in the structural e ciency at load levels above 15 000 lb/in. Increasing the maximum-allowable strain beyond 0.006 in/in. yields little or no further improvements since "x;max is replaced by the minimum inplane sti ness requirements as one of the active constraints. If "x;max is decreased to 0.003 in/in., the maximum-allowable strain becomes the active constraint for load levels of 7500 lb/in. and above. The weight of sandwich cover panels designed with a maximum-allowable strain of 0.003 in/in. increases for load levels above 7500 lb/in. as compared with the baseline design. E ect of varying minimum inplane sti ness requirements. The e ect of varying the minimum inplane sti ness requirements on structural efciency is shown in gure 6. This gure shows the structural e ciency W of optimized sandwich cover panels as a function of Nx for three values of the inplane sti ness requirements. In this gure, kA is a scaling factor for the baseline values of A11;min and A66;min. When kA = 1:0, A11;min and A66;min are 6 Figure 6. E ect of minimum-required inplane sti ness on structural e ciency of composite sandwich cover panels. the baseline values. When kA has a value other than 1.0, the baseline values of A11;min and A66;min are multiplied by kA at all load levels. Since the minimum inplane sti ness constraint is active for the baseline design at load levels below 15 000 lb/in., letting kA = 0:5 reduces the weight of the cover panels at load levels below 15 000 lb/in. Further reductions in the minimum inplane sti ness requirements yield little or no further improvements since "x;max replaces A11;min and A66;min as one of the active constraints. Letting kA = 2:0 increases the weight of the cover panels at all load levels considered. The minimum inplane sti ness requirements become an active constraint at all load levels, thus replacing "x;max as the active constraint at load levels above 15 000 lb/in. This is an important trend since the inplane sti ness requirements are likely to increase for newer technology transport aircraft with higher aspect ratio wings. For such a wing, sti ness may become a more important consideration than a higher "x;max for improved damage tolerance in the selection of appropriate materials for future transport aircraft. E ect of varying upper percentage limit of all ply group thicknesses. The results of this study indicate that the upper limit on the percentage of 0 and 45 plies (U0 and U45, respectively) is an active constraint at all load levels. The fact that this constraint is active indicates that the structural e ciency of these cover panels can be increased by allowing laminates with higher values of U0 and U45. Figure 7. E ect of upper percentage limit of all ply orientations on structural e ciency of composite sandwich cover panels. The structural e ciency W of optimized sandwich cover panels is shown in gure 7 as a function of Nx for two values of the upper percentage limits of all ply groups (U0, U45, and U90). The upper curve in gure 7 is for the baseline value of this constraint (U0 = U45 = U90 = 0:375), whereas the lower curve shows the e ect of setting U0 = U45 = U90 = 1:0. In both cases the lower percentage limits for all ply angles (L0, L45, and L90) are equal to 0.125. Figure 7 shows that the weight of all cover panels is reduced by allowing higher values of U0, U45, and U90. For the loading case investigated (Nxy = Ny = 0; Nx 6= 0), the optimum percentage of 0 layers lies between 48 and 54 percent, whereas the optimum percentage of 45 layers lies between 33 and 40 percent. The optimization procedure always drives the percentage of 90 layers to its minimumallowable value, 12.5 percent in this case. The weight savings achieved by using higher values of U0, U45, and U90 indicate the importance of developing ways to understand and utilize laminates in which a high percentage of the plies are oriented in one direction. E ect of varying core density. The optimum core density at all load levels is quite low, typically about 1.0 lb/ft3. For reasons other than minimum weight, it may be preferable to use cores with a higher density. Thus, the e ect of increasing the core density on the structural e ciency was investigated. The results in gure 8 indicate the structural e ciency W of sandwich cover panels using cores of two di erent densities: core = 1:0 and 9.5 lb/ft3. 7 Figure 8. E ect of core density core on structural e ciencyof composite sandwich cover panels.As can be seen from this gure, the weight of thesesandwich cover panels is not very sensitive to changesin the core density; a ninefold increase in core densityincreases the weight by approximately 11 percent.There are two reasons for this behavior. First, thecore is only a small percentage of the total weightof the sandwich cover panel; large di erences in thecore density have a small e ect on the total weight.Second, as the core density is increased, so are itstransverse shear sti nesses Gxz and Gyz . Thus, thecore thickness required to prevent global bucklingfrom occurring is reduced. As can be seen from thedata in table 4, the core thickness is reduced by up to33 percent when the core density is increased from 1.0to 9.5 lb/ft3. Also note that the facesheet thicknessdoes not vary as the core density is increased. Sincethe facesheet thickness tf is mainly determined bythe maximum strain and inplane sti ness constraints,changing the core density has no e ect on tf .Since weight is relatively insensitive to changesin the core density, the selection of core density isprobably best made based on criteria other thanminimum weight, e.g., damage tolerance and ease ofmanufacture.Concluding RemarksAn analysis and design method has been devel-oped for the design of composite sandwich cover pan-els, including transverse shear e ects and damagetolerance considerations. This method has been in-corporated into a sandwich optimization computerprogram entitled SANDOP.A set of optimized sandwich cover panels was de-signed with SANDOP with input values typical ofthose used for transport aircraft wing applications.Based on the designs generated by SANDOP, sev-eral observations can be made about the use of com-posite sandwich cover panels for transport aircraftwing applications. The composite sandwich coverpanels considered in this study have approximatelythe same structural e ciency as composite sti ened-plate cover panels designed to identical constraintswhen the dominant design load is axial compres-sion. Increasing the maximum-allowable strain from0.0045 to 0.006 in/in. decreases the weight of com-posite sandwich cover panels at the higher load levelsconsidered while having no e ect on weight at thelower load levels. Increasing the maximum-allowablestrain beyond 0.006 in/in. has little or no e ect on theweight of the composite sandwich cover panels con-sidered in this study. Decreasing the inplane sti -ness requirements reduces the weight of compositesandwich cover panels at the lower load levels whilehaving no e ect on weight at the higher load levels.Increasing the inplane sti ness requirements inducesa weight increase at all load levels. Increasing theupper limit on the percentage of 0 and 45 plies ofthe facesheet laminate reduces the weight of compos-ite sandwich cover panels. The weight of the sand-wich cover-panel designs in this study is not very sen-sitive to changes in the core density. The core densityTable 4. Core and Facesheet Thicknesses Values of Nx, lb/in., of|Thickness element3000750