CONTINUUM TOPOLOGY OPTIMIZATION OF BUCKLING-SENSITIVE STRUCTURES

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1 I CONINUUM OPOLOGY OPIMIZION OF BUCLING-SENSIIVE SRUCURES Salam Ramatalla, Graduate Researc ssistant Colby C. Swan, ssociate Professor, member I Civil and Environmental Engineering, Center for Computer-ided esign, e University of Iowa Iowa City, Iowa 52242, US BSRC wo formulations for continuum topology optimization of structures taking buckling considerations o account are developed, implemented, and compared. In te first, te structure undergoing a specified loading is modeled as a yperelastic continuum at finite deformations, and is optimized to maximize te minimum critical buckling load. In te second, te structure under a similar loading is modeled as linear elastic, and te critical buckling load is computed wit linearized buckling analysis. Specific issues addressed include usage of suitable mixing rules'', a node-based design variable formulation, and consistent design sensitivity analysis. e performance of te formulations is demonstrated on te design of different structures. Wen problems are solved wit moderate loads and generous material usage constras, designs using compression and tension members are realized. lternatively, wen fairly large loads togeter wit very stringent material usage constras are imposed, structures utilizing primarily tension members result. Issues tat arise wen designing very ligt structures wit stringent material usage constras are discussed along wit te importance of considering potential geometrical instabilities in te concept design of structural systems. INROUCION N MOIVION Variable topology material layout optimization is a potentially useful tool in te design of structures, mecanical parts, composite materials, and even micro-electro-mecanical systems (MEMS). Its usage and a variety of formulations ave been widely explored over te past fifteen years for a considerable range of applications. Bendsoe and ikuci and numerous subsequent works by a wide range of investigators ave roduced topology optimization metods dealing wit linear elastic material beaviors and geometrically linear structural beavior. o tis po, only a few works ave addressed continuum structural topology optimization of nonlinear systems. mong tese are works using Voigt-Reuss continuum topology formulations and consistent sensitivity analysis tecniques for designing structures 2 and composite materials 3 featuring general materially nonlinear beaviors. More recently, a few works ave also been presented for continuum topology optimization of elastic systems undergoing nonlinear, finite deformations. 4-7 Consideration of geometrical and/or buckling instabilities is an important issue in te conceptual design of sparse spatial structures. For example, in design of long-span bridges, tension structures are typically optimal since tey preclude potential buckling. If continuum topology optimization were applied to obtain concept designs of sparse spatial structures and buckling were not considered, te design results could rely excessively on compression, and tus constitute unacceptable concept designs. In design optimization of discrete truss structures, buckling beaviors can be avoided altogeter by prescription of Euler buckling constras on loads in individual structural members 8,9. In continuum topology design formulations, owever, tis process is muc more callenging since it is very difficult to identify discrete structural members, teir geometrical properties, and teir end support conditions from te vector of design variables. One promising approac to addressing geometrical instabilities in continuum structural topology optimization 0 is to model te structure as a linearly elastic system and to use te minimum critical buckling load computed via eigenvalue analysis in

2 eiter te objective function or as a design constra. On te oter and, in te design of MEMS wit continuum structural topology optimization, it is not uncommon for te system to be designed to undergo finite deformations even before geometric instabilities migt develop,2. Insuccases,itis necessary to base te analysis and design on a more general framework tat addresses finite deformations in te system and te development of potential geometrical instabilities. In recent years, works dealing wit continuum topology optimization of structures to minimize te generalized compliance at finite deformations ave been proposed and demonstrated 4, altoug minimization of nonlinear compliance does not necessarily address potential geometrical instabilities. o tis po in time, only a limited number of works 5-7 ave dealt wit tis important issue of taking finite deformations of te system o account, wile also taking account of te associated instabilities tat arise in te structure in a consistent fasion. In te body of tis paper, a general yperelastic continuum framework is developed for structural analysis and continuum topology optimization. Witin tis framework, te minimum critical buckling load for te system can be addressed. s an alternative, a linear elastic continuum topology formulation is also used along wit minimum critical buckling values computed by linearized buckling analysis. For bot formulations, expressions for design sensitivity of structural responses to material distribution parameters are developed and verified. e two approaces ave been implemented, tested and compared on a variety of relevant design problems. ltoug it is not te central focus of tis work, anoter critical issue to acieving te objectives of tis work is tat of using a continuum topology formulation based on nodal design variables as opposed to element-based design variables. Since node-based design variables feature C 0 continuity, tis new framework is invulnerable to ceckerboarding instabilities, and tus does not require any spatial filtering tecniques to preclude suc instabilities ISRIBUION OF MERILS e design of a structure is ere considered to be te spatial distribution of te structural material in. o describe very general structural material distributions in a volume-fraction approac 2 described earlier 2 and alternatively described as te ``density'' approac by oters, is used. Wile preference is given ere to final material layout distributions, wic are nearly discrete, suc distributions are typically acieved using continuous formulations permitting mixtures to exist trougout te design domain. By permitting mixtures, te structural material and a fictitious void material B are allowed to simultaneously and partially occupy an infinitesimal neigborood about eac Lagrangian po X in. e volume fraction of structural material pase at a fixed Lagrangian po X in te design domain is denoted by φ (X) and represents te fraction of an infinitesimal volume element surrounding po X occupied by material. Natural constras upon te spatial volume fractions for te two-material problem are: 0 φ ; 0 φ B ; φ + φ B () e last pysical constra of () states tat te material volume fractions at X are not independent and so one need only be concerned wit te layout of structural material. Using te same mes and basis functions tat will be used to solve te structural analysis problem described below, te spatial distribution of structural material in is expressed using te following expansion: N φ ( X) å φ N ( X) X i i (2) i were φ i are te nodal volume fractions, and N i (X) are te nodal sape functions, and N denotes te number of nodes in te structural model. e design vector b describing te arrangement of materials in te structural domain tus as te composition b φ, φ, φ,...,. is approac yields a C 0 ( ) 2 3 φ N continuous design variable field immune to ceckerboarding instabilities. NLYSIS FORMULION Hyperelastic nalysis at Finite eformations e strong form of te nonlinear elliptic boundary value problems to be solved is: 3 Find u : ( [0, ]) R, suc tat te ircoff stress field satisfies τ ρ γ ij, j + 0 j 0 X, t [0, ] (3a) subject to te boundary conditions:

3 u (t) g (t) j n τ i ij j j (t) on Γ on g j Γ j t [0, ] t [0, ] (3b) (3c) e ircoff stress tensor τ is related to te Caucy stress tensor σ via te relation τjσ were Jdet(F) and F is te deformation gradient. For a given mes discretization of wose complete set of nodes is denoted {η} te subsequent design formulation is facilitated by roducing a subset of nodes {η }at wic non-vanising external forces are applied, and a subset of nodes {η g } at wic non-vanising prescribed displacements are applied. Since te analysis problem is being solved in te context of topology optimization, it is assumed tat a local microscopic mixture of two generic materials and B resides at eac po X in te structural domain. In (3a) τ represents te macroscopic ircoff stress of te local mixture, wic is dependent upon te constitutive properties of te two material pases and te mixing rule employed B B τ ( X) τ[ τ ( F ); τ ( F ); φ], (4) were F and F B are te respective deformation gradients for materials and B at a given po X, τ and τ B are te stress tensors for te two materials at X, andφ is te volume fraction of material. For simplicity and efficiency, a power law mixing rule wit an iso-deformation condition (2) is used ere. erefore a specific form of (4) is: p p B τ( X) φ τ + ( φ ) τ, (5) were p is a fixed parameter of te mixing rule. In accordance wit te iso-deformation assumption, x F F B F X (6) were xx+u(x). It is assumed tat bot materials and B obey isotropic yperelastic constitutive laws. e particular strain energy function E used ere for bot materials is tat of Ciarlet (988) werein te volumetric U and deviatoric W strain energy functions for te two materials and B are assumed to be decoupled and of te forms: E(F) U(J) + W(θ) U(J) W 2 µ [ 2 (J ) ln(j) ] 2 [tr( θ) 3] 2 (7) (8) (9) 3 In te preceding expression, J is again te determinant of F; is a constant bulk modulus; µ is a constant sear modulus; θ FF is te left Caucy- Green deformation tensor; and θ is its deviatoric part. For tis model, terefore, te ircoff stress in a material τ is tus related to deformation quantities in tat material as follows: W τ JU (J) + 2 dev. (0) θ e virtual work form of te structural equilibrium problem can be restated in egral form as: τ ijδεij d Γ ρ 0 γ j j δu δu j d j dγ + () were te expression on te left represents te ernal virtual work, and tat on te rigt, te external virtual work. e differential of te ernal virtual work can be written as: d ( δw ) δε ij dl v (τ ij ) d + δε τ dε ji im jm d (2) indicating a decomposition o, respectively, a material stiffness term containing te Lie differential of te ircoff stress, and a geometric stiffness term. Usage of a Galerkin formulation, in wic te real and variational kinematic fields are expanded in terms of te same nodal basis functions, and discretization of te time domain o a finite number of discrete time pos, leads to te following force balance equations at eac unrestrained node in te mes as ere at te (n+) t time step: were ( f ( f ) n+ ext ) n+ + ext + + r n ( f ) n ( f ) n 0 ( B ) ρ 0 N n+ : τ n+ d γ n+ d + N Γ n+ dγ (3) (4a) (4b) bove B and N represent, respectively, te spatial infinitesimal nodal strain displacement matrix and te nodal basis function for node. Under finite deformations, (3) represents a set of nonlinear algebraic equations tat must be solved in an

4 iterative fasion for te incremental displacement field for eac time step of te analysis problem. Wen external forces applied to a structure are independent of its response, te derivative of te i t residual force vector component at te t node wit respect to te j t displacement vector component of te B t node is simply: B æ ö ç è il ø B ji c jk B B kld + B ( N ) τ ( N ) δ d (5), j were c ij is te spatial elasticity tensor in condensed form. ssembly of tis nodal stiffness operator for all unrestrained nodes and B gives te structural tangent stiffness matrix. Critical Load and esign Sensitivity nalysis One approac to designing buckling-resistant structures is to apply displacement-controlled loading to te structure and to maximize te critical ernal force tat can be generated in response to tis loading. e magnitude of te applied displacement loading is merely tat wic induces te first instability in te structural model. e first instability po is te first po in te loaddeflection response of te structural model at wic te tangent stiffness operator defined in (5) becomes singular. reliable algoritm for finding suc singular pos tat as been used by te autors in a variety of oter applications (4,5) is presented in Fig.. Essentially, tis algoritm involves gradual and iterative approacing of te structural model s first critical po of instability. Once te critical po is found, te objective is to compute te magnitude of te resistance force generated in te structure, and to design te structure so as to maximize te resistance. For example, if te applied displacement loading g( X) å N ( X). g (6) { n g } is applied to te structure and induces instability, ten te magnitude of te critical resistance force generated in te structure will be: f crit é ê ê n g ë g å { } g B jk τ d,k ù ú ú û il (7) ccordingly, an optimization problem to maximize tis critical force would be as follows: max f crit b r( b, u) 0 ; Vmaterial Vallowable 0 (8) e design derivative of te critical buckling force is computed using adjo sensitivity analysis as follows: a r ù ( u ) (9) df crit é g æ τ ö å d d {η ê çb + b g } ú ë g è b ø b û a were ( u ) is te adjo displacement associated wit te t node at wic prescribed displacements are applied. Specifically, it is te solution of te following adjo problem: ( u ) ( f ) a ucrit (20) were is tangent stiffness operator at te current state of te model. LINERIZE BUCLING MOEL Eigenvalue nalysis e full geometrically nonlinear metod proposed above for maximizing te minimum critical buckling load in continuum structural topology optimization is potentially very computationally expensive. potentially more efficient alternative migt be to use linearized buckling eigenvalue analysis to estimate te critical buckling load for te structure and to ten maximize tis approximation. Neves, Rodrigues, and Guedes 0, ave presented an approac to include te critical load criterion in te continuum topology optimization model. Linearized buckling eigenvalue analysis proceeds as follows: prescribed force loading f ext is applied to te structure wit its magnitude necessarily being less tan tat required to induce geometric instability in te structure. Once te resulting linear, elastostatic displacement solution u R N to te applied loading f ext is obtained ( L uf ext ), were L is te linearized stiffness matrix, ten te following eigenvalue problem is solved ( b) + λg( u, b) (20) Ψ ψ L 0 In te preceding, b R M is te vector of design variables; L is te linear tangent stiffness operator; and G(u,b) is te linearized geometric stiffness 4

5 matrix. It follows from (20) tat te magnitude of te eigenvalue λ is ψ L ψ λ ψ G ψ (2) wic represents te factor by wic f ext must be scaled to create instability in te structure, and ψ is a normalized eigenvector satisfying ψ L ψ. For tis model, it is assumed tat linearized stiffness operator L is real, symmetric, and positive definite, wile G is only assumed to be real and symmetric. o avoid numerical difficulties in te solution of (2) associated wit te indefinite caracteristics of G, it is common (6) to solve te modified eigenvalue problem tat under most circumstances deals wit two positive definite matrices. + G γ ψ (( ) ) (22) L L 0 e eigenvalue of tis problem γ is related to tat of (20) as follows: λ γ. (23) λ esign Sensitivity nalysis Once te linearized eigenvalue problem (22) is posed and solved, te design problem is formulated to maximize te minimum buckling load λ. e objective functional to be minimized for tis problem would simply be te reciprocal of te lowest eigenvalue λ as follows. f ( u, b) (24) min(λ) e optimization problem is tus stated to minimize te reciprocal of te first (or minimum) critical buckling load as follows æ ö æ ö min f min minç ψ G ψ ç max (25) b, u b, u λ, ç è ø b u è ψ 0 ψ L ψ ø For a simple eigenvalue te design gradient of L is written simply as d L f ( a r + u ) (28) d b b b were: f G L ψ ( + ) ψ b b λ b and r b b L f u b ext (29) (30) In te current formulation, a simple eigenvalue as been assumed. In cases were multiple eigenvalues occur, computation of te sensitivities can be somewat more involved since te eigenvectors of te repeated eigenvalues are not unique, rendering te eigenvalues only directionally differentiable 7,8 Frequently, owever, multiple eigenvalues occur in symmetrical structures and are actually attributable to te structural symmetry 9. In suc cases, approaces tat reduce te design space in accordance wit te symmetry 20 can render te repeated eigenvalues fully differentiable in te reduced design space. EMONSRIVE EXMPLES Material Properties and Mixing Rules In all example problems solved below, te initial starting designs always utilized a completely solid structural domain. In addition, te solid structural material in all problems was isotropic wit Young s modulus of 307 GPa and sear modulus of 8 GPa. Furtermore, te powerlaw mixing rule wit p 4 was used in all computations. e nodal design variable formulation of Section 2 was employed witout any spatial filtering of design variables and witout any perimeter control. subject to te normal bound constras on te design variables, te linear structural equilibrium state equation (3), and a constra on material resources. Using adjo design sensitivity analysis, te Lagrangian of te optimization problem (25) is: L f + u a r( u, b) (26) were u a is te adjo displacement vector and te solution of te following linear, adjo problem a G Lu ψ ψ. (27) u e X-Structure Problem 5 In tis problem, an X-saped structure sown in Fig. 2a is considered. e lateral by vertical dimensions of te frame are (x40). displacement loading of d 0 is applied to te center node of tis structure, and for all designs, te algoritm of Fig. is applied togeter wit geometrically nonlinear structural analysis to compute te critical ernal buckling force tat develops in response to te applied displacement loading. e design problem solved is tat posed in (8), were only alf

6 of te original structural volume can be employed in te final, optimized structure. e structure is mesed wit 200 bilinear continuum finite elements, and te optimization problem is solved using sequential linear programming tecniques. e design solution to tis particular version of te problem is sown in Fig. 2b. Since it employs strictly tension as opposed to compression, it is indeed te correct solution tat one would expect. Wile optimization of structures to maximize te minimum critical buckling load based on fully nonlinear analysis was sown to be successful above, te computational expense can be considerable. s a potentially more efficient way to acieve te same objective, linearized buckling analysis can also be used. us, for te same structural model sown in Fig. 2a, a force of magnitude f ext is applied to te center node te structural optimization problem was solved once again as follows: ext min r( b, u) 0 f ; Vmatl - Vall 0 (3) b λ Wit tis linearized buckling criteria, te resulting design (Fig. 2c) is virtually identical to tat acieved wit nonlinear analysis and sown in Fig. 2b. It is wort noting, owever, tat te computational time required wit te linearized buckling analysis was significantly smaller tan tat required wit te geometrically nonlinear analysis. e Circle Problem Continuum topology optimization solutions are frequently used only as starting concepts tat are suggestive of potentially optimal structural forms. at is, te designs produced are ten taken to a second stage were more detailed sape and sizing analysis is performed. Wen used in design of largescale structures continuum topology solutions are frequently quite ``eavy'' in tat te ratio of volume occupied by structural material to te enclosed structural volume can be unrealistically large. Specifically, te global volume fraction constra used in many continuum topology optimization metods is typically in te range of , wereas in many structures, te structural material occupies only approximately 2% of te enclosed structural volume. us, wile buckling of compression members can be a real concern in design of sparse structures, continuum topology metods tat produce unrealistically eavy designs 6 are unable to address suc concerns in te concept design stage. In order to address te development of geometric instabilities associated wit buckling, continuum topology optimum metods must be used wit a finite deformation nonlinear formulation or wit a linearized buckling approac. In addition, sufficiently stringent material usage constras must be imposed wic can in turn require igly refined structural meses. o illustrate tis po, we consider te design of a structure to carry a radial po load applied at te center of te structural domain to te fixed boundary of te circular domain of Fig. 3a. e analysis problem was solved by imposing a finite displacement δ 0.R at te center node of te structure and using a nonlinear formulation to compute te resulting structural response. e design optimization problem was first solved using te minimum critical buckling loads as te objective function. Using te moderately fine mes sown in Fig. 3a, te design problem was solved a number of times. It was first solved wit material usage constras of 25% of te structural volume, and wit applied an applied displacement δ 0.R, resulting in te design sown in Fig. 3b (undeformed) and 3c (deformed). Since te material usage constra is generous, and since te loading is not necessarily large enoug to generate structural instability, a design using bot compression and tension is produced. Wen te design displacement loading is increased to δ 0.8R and wen te material usage constra is reduced to 5% of te total structural volume, te purely tensile designs of Fig. 3d (undeformed) and 3e (deformed) are obtained. Suc designs are clearly preferred since tey will not be vulnerable to buckling instabilities. similar solution (Figs. 3f and 3g) can be obtained by maximizing te minimum critical buckling load obtained using linearized buckling analysis. e Fixed-End Beam Problem In tis problem, a fixed end beam is loaded vertically wit a large displacement applied to te center of te upper edge of te beam as sown in Fig. 4a. e vertical and orizontal dimensions of te beam are (2x40), and te structural material usage constra for tis problem is 0% of te total structural volume. similar problem was roduced and solved by Bul and Sigmund 5 wo sowed tat teir formulation yields completely different designs

7 wen using geometrically linear analysis and wen using geometrically nonlinear analysis. More recently Gea and Luo 7 ave studied a similar problem wit tree concentrated loads applied to te upper edge of te beam, obtaining design results similar to tose presented in 5. In te nonlinear problem formulation, a large displacement of d 9 is applied to te structure, wile a force f is applied in te linearized buckling case. e continuum topology optimized material layout designs obtained are sown in Figs. 4c and 4d. e resulting topologies signify te ability of bot metods to track te geometrical instabilities occurring in te structure, by building two long tension members and two sort compression members. ese results agree wit tose of 5,7. ISCUSSION N CONCLUSIONS In tis paper, te objective as been to develop continuum structural topology optimization formulations tat can be used to detect and avoid buckling instabilities in te conceptual design stage of sparse structural systems. oward tis end, bot a finite deformation yperelastic treatment of te structure, and linear elastic, linearized buckling treatment of te structure ave been developed and implemented. Wit te yperelastic structural treatment, te minimum critical buckling load was considered as objective function. Based on te example problems solved, selection of te minimum critical buckling load as te objective function appears to be effective at consistently acieving stable designs. e minimum critical buckling load as computed from linearized buckling eigenvalue analysis was also considered as an objective function in tis work, and was found to give design results comparable to tose produced from nonlinear stability analysis. Since te formulation is based on linear structural analysis, it is muc less computationally expensive tan te formulations based on nonlinear analysis. It sould be noted tat te nodal volume fraction approac roduced in tis work as been found very effective at eliminating te numerical instabilities tat lead to ceckerboarding designs. CNOWLEGEMENS an NSF/RP Grant in te OPL Program. r. Ricard. Miller is also acknowledged for elpful discussions and recommending te circle problem as a good test problem. REFERENCES Bendsoe, M.P., and ikuci, N., Generating optimal topology in structural design using a omogenization metod, Comput. Met. ppl. Mec. and Engng., Vol. 7, 988, pp Swan,C.C.andosaka,I., Voigt-Reuss topology optimization for structures wit nonlinear material beaviours, Int. J. Numer. Met. Engng. Vol 40, 997, pp Swan, C.C., and rora, J.S., opology optimization of material layout in structured composites of ig stiffness and ig strengt, Structural Optimization, Vol. 3 (), 997, pp Bruns,.., and ortorelli,.., opology optimization of geometrically nonlinear structures and compliant mecanisms, Proc. 7-t Symposium on Multidiciplinary nalysis and Optimization, I/USF/NS/ISSMO, I ,998, pp Bul,. Pedersen, W., and Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidisc. Optim., Vol. 9, 2000, pp Pedersen,C.B.W.,Bul,.,andSigmund,O., opology syntesis of large-displacement compliant mecanisms, Int. J. Num. Met. Engng. Vol. 50, 200, pp Gea, H. C., and Luo, J., opology optimization of structures wit geometrical nonlinearities, Computers & Structures, Vol. 79, 200, pp Rozvany, G.I.N., ifficulties in truss topology optimization wit stress, local buckling and system stability constras, Structural Optimization, Vol., 996, pp is researc was funded in part by a grant from te University of Iowa CIFRE Program, and in part by 7 9 ctziger, W., Local stability of trusses in te context of topology optimization Part II:

8 numerical approac, Structural Optimization, Vol. 7, 999, pp Neves, M.M., Rodrigues, H., and Guedes, J.M., General topology design of structures wita buckling load criterion, Structural Optimization, Vol. 0, 995, pp Sigmund, O., On te design of compliant mecanisms using topology optimization, Mec. of Struct. and Mac., Vol. 25 (4), 997, pp Larsen, U., Sigmund, O., and Bouwstra, S., esign and fabrication of compliant micromecanisms and structures wit negative Poisson s ratio, J. of Microelectromecanical Sys. Vol. 6, 997, pp Cardoso, and rora, J.S., djo sensitivity analysis for nonlinear dynamic termoelastic systems, I J. Vol. 29 (2), 99, pp Swan C.C., and Seo Y.-., Limit state analysis of earten slopes using dual continuum /FEM approces. Int. J. Num. nalat. Met. Geomec. Vol. 23, 999, pp porous medium models, J. Geotec. Geoenv. Eng., Vol. 27 (5), 200, pp Bate,.J., Finite Element Procedures, New Jersey, Prentice Hall Seyranian,.P., Lund, E., and Oloff, N., Multiple eigenvalues in structural optimization problems, Structural Optimization, Vol. 8, 994, pp Osaki, M., and Uetani,., Sensitivity analysis of bifurcation load of finite-dimensional symmetric systems, Int. J. Num. Met. Engng. Vol. 39, 996, pp Pedersen, N.L., Maximization of eigenvalues using topology optimization, Structural Optimization, Vol. 20, 2000, pp osaka, I., and Swan, C.C., symmetry reduction metod for continuum structural topology optimization, Computers & Structures Vol. 70, 999, pp Swan C.C., and Seo Y.-., Stability analysis of embankments on saturated soils using elasto-plastic n0; m0; t 0 0; t P c t n+ t n + m Ye Critical state P Can r n+ 0 be solved, and N t t/4 mm+ nn+ Ye δ c δ a b Fig.. a) critical load and deflection associated wit a structural response; and b) algoritm for finding te first critical po of a structure s response. 8

9 Fig. 2. a) X-frame design problem were te structure of dimensions of by 40 is loaded as sown and is modeled wit a mes of 200 bilinear continuum elements. (b) esign solution obtained by maximizing te minimum critical buckling load; (c) esign solution obtained by maximizing te minimum critical load obtained using linearized buckling analysis. Fig. 3. (a) Mes of circle domain wit rigid boundary restras and central load. (b) Undeformed solution obtained by maximizing minimum critical load (nonlinear formulation) wit material constra at 20% of structural volume and applied displacement loading δ 0.R; (c) eformed configuration of associated design; (d) Undeformed design obtained to maximize minimum critical load (nonlinear formulation) wit material constra at 5% of structural volume and applied displacement loading δ 0.8R; (e) deformed configuration of associated design; (f) esign obtained to maximize minimum critical load obtained wit linearized buckling analysis; and (g) deformation associated wit critical mode. 9

10 Fig. 4. Fixed end beam problem. (a) esign problem wit applied displacement loading for wic structure is to be designed to maximize te minimum critical load computed via nonlinear analysis; (b) asssociated design solution; and (c) deformed sape of design solution. (d) esign problem wit applied force loading for wic structure is to be optimized to maximize minimum critical load computed wit linearized buckling analysis; (e) associated design solution; and (f) deformed sape associated wit buckling eigenmode. 0