Optimal Design of Trusses With Geometric Imperfections: Accounting for Global Instability

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1 Cleveland State Unversty Cvl and Envronmental Engneerng Faculty Publcatons Cvl and Envronmental Engneerng -5- Optmal Desgn of Trusses Wth Geometrc Imperfectons: Accountng for Global Instablty Mehd Jalalpour Cleveland State Unversty, Takeru Igusa Johns Hopkns Unversty James K. Guest Johns Hopkns Unversty, Follow ths and addtonal works at: Part of the Cvl Engneerng Commons, and the Structural Engneerng Commons How does access to ths work beneft you? Let us know! Publsher's Statement NOTICE: ths s the author s verson of a work that was accepted for publcaton n Internatonal Journal of Solds and Structures. Changes resultng from the publshng process, such as peer revew, edtng, correctons, structural formattng, and other qualty control mechansms may not be reflected n ths document. Changes may have been made to ths work snce t was submtted for publcaton. A defntve verson was subsequently publshed n Internatonal Journal of Solds and Structures, 48,, (October 5, ) DOI:.6/j.jsolstr..6. Orgnal Ctaton Jalalpour M., Igusa T., and Guest J.K. Optmal desgn of trusses wth geometrc mperfectons: Accountng for global nstablty, Internatonal Journal of Solds and Structures 48(): 3-39, Ths Artcle s brought to you for free and open access by the Cvl and Envronmental Engneerng at EngagedScholarshp@CSU. It has been accepted for ncluson n Cvl and Envronmental Engneerng Faculty Publcatons by an authorzed admnstrator of EngagedScholarshp@CSU. For more nformaton, please contact lbrary.es@csuoho.edu.

2 Optmal desgn of trusses wth geometrc mperfectons: Accountng for global nstablty Mehd Jalalpour, Takeru Igusa, James K. Guest Department of Cvl Engneerng, Johns Hopkns Unversty, Baltmore, MD 8, Unted States. Introducton Structural optmzaton offers a systematc approach to materal layout n engneerng desgn. Its most general branch s topology optmzaton where both structural component szes and system connectvty are smultaneously optmzed (Bendsøe and Sgmund, 3). Structural optmzaton naturally drves desgn towards sparse and slender structures. Such structures are typcally more susceptble to the deleterous effects of fabrcaton errors, ncludng decreased resstance to bucklng n the presence of geometrc mperfectons. Ths work presents a topology optmzaton algorthm that ncludes the effects of geometrc varablty n the manufacturng process, wth the goal of mprovng desgn robustness. We focus on truss structures, and extend a recently developed perturbaton-based approach (Guest and Igusa, 8; Asadpoure et al., ) by accountng for nonlnear structural behavor. Truss topology optmzaton typcally follows a ground structure approach. The desgn doman s dscretzed wth a nodal mesh that s connected by a dense set of potental structural members. Boundary condtons and appled loads are assumed known, and optmzaton s used to determne the dstrbuton of cross-sectonal areas (Krsch, 989; Bendsøe et al., 994; Achtzger et al., 99; Bendsøe and Sgmund, 3). Members wth areas below a Correspondng author. E-mal address: jkguest@jhu.edu (J.K. Guest). certan threshold are deemed neffcent and are removed from the ground structure, thereby changng connectvty of the system. There exsts a rch lterature on the desgn of trusses usng optmzaton. We are concerned here wth those works that are related to bucklng. Ths area has recently generated sgnfcant nterest among researchers due to technologcal advancements, ncludng mproved materal strengths and manufacturng capabltes, that allow for desgn of more slender structural components. Bucklng can be vewed as a combnaton of local (Euler) and global (system) bucklng. One natural approach for developng desgns that resst local bucklng s to nclude the Euler bucklng crteron n the constrants (Neves et al., 995; Stolpe, 4). However ths formulaton poses several fundamental and numercal challenges (Duysnx and Bendsøe, 998; Krsch, 996; Zhou, 996). Guo et al. (), for example, descrbe how ths formulaton may lead to a dvson of the feasble doman nto dsjont subdomans, wth the optmal solutons at the boundares makng them dffcult locate wth conventonal optmzers. Cheng and Guo (997) proposed the method of epslon relaxaton to overcome a smlar dffculty. Whle these methods account for the effects of local bucklng, solutons can stll be globally unstable, as n the common case of a chan of collnearly connected elements. Whle these collnear elements can be merged nto one longer element through a method known as node cancelaton, Zhou (996) demonstrated that ths ncreases the potental for Euler bucklng, leadng to suboptmal solutons. Achtzger (999) crcumvented ths usng local bucklng constrants that account for the node cancelaton effect va an exact

3 3 modelng method to consder the full length of solated collnear chans. In terms of global bucklng, Guo et al. (5) attempt to crcumvent collnear chans by usng overlappng members n the ground structure whle Rozvany (996) explored the use of system stablty constrants and geometrc mperfectons to facltate creaton of the knd of bracng needed to prevent global bucklng behavor. Ben-Tal and Nemrovsk (997) proposed usng artfcal nodal loads to stablze the system, whch Tyas et al. (6) updated wth a scalng based on the nternal force magntudes of the members connected to the node. Ben-Tal et al. () and Kočvara () formulated an approach for determnstc truss desgn that ncluded lnearzed global bucklng n the complance formulaton. As suggested by the works n the precedng paragraphs, topology optmzaton often exhbts numercal challenges assocated wth the underlyng governng mechancs, such as local and global nstablty. Hence, topology optmzaton research has focused prmarly on determnstc desgn problems, wth the ncluson of uncertanty typcally lmted to the loadng, treated probablstcally or through multple load cases (Bendsøe et al., 994; Daz and Bendsøe, 99; Lógó, 7; Lógó et al., 9; Yonekura and Kanno, ). Uncertantes assocated wth structural stffness, however, can be mportant n desgn. Ths s partcularly true n optmzed structures that tend to be lght, slender and senstve to uncertantes n geometrc and materal propertes. In the presence of such uncertantes, the global stffness matrx becomes a random matrx, complcatng the senstvty analyss. Sandgren and Cameron () crcumvented ths by takng a smulatonbased approach, usng a genetc algorthm as the optmzer and Monte Carlo smulaton to represent uncertantes n geometry and materal propertes. Calafore and Dabbene (8) proposed relaxed approxmatons to the formulaton to optmze under the condton of materal property randomness. The work heren follows a recently proposed method for structural optmzaton consderng small uncertanty n nodal locatons (Guest and Igusa, 8) and/or materal propertes (Asadpoure et al., ). Ths method used second-order perturbatons of the stffness matrx to transform the uncertantes n nodal locatons to a set of mathematcally equvalent random forces. Although smlar n concept to the aforementoned work of Ben-Tal and Nemrovsk (997) and Tyas et al. (6), the method s unque n that the loads were a pure mathematcal transformaton of the uncertanty and that the desgn senstvty analyss accounted for ths transformaton, allowng jonts wth uncertan locaton to beng removed or braced n fnal desgns. The perturbatons, however, were based on lnear elastc structural behavor; thus, the optmzed desgns were nvarant to the magntude and drecton of the load. The current work extends ths method by ntroducng frst-order nonlnear effects assocated wth bucklng so that the fnal resultng materal dstrbutons wll be dfferent under compressve and tensle forces. In ths manner, the newly proposed method s capable of handlng both nodal locaton uncertanty and, to frst order, global bucklng effects. To the best of the authors knowledge, such a method does not exst n the lterature. Ths paper s structured as follows. The expected (mean) complance s derved for a truss structure wth mperfectons n nodal locatons and wth potental global bucklng. It s then shown how ths complance expresson can be used n topology optmzaton. Fnally, several desgn examples are presented, llustratng how the effects of geometrc mperfectons and global nstablty can result n substantal changes n the desgn.. Geometrc mperfectons In ths secton we derve the formulatons for convertng the problem of trusses wth geometrc mperfectons to an equvalent random forces problem. These are then extended to consder a frst-order approxmaton to geometrc nonlnearty. Appled loads are assumed determnstc. The detaled dervaton that follows s general and apples to structures defned n any number of spatal dmensons havng any number of nodal locaton uncertantes... Expected value of the complance Structure geometry s defned by the node locatons of the fnte element dscretzaton. Geometrc uncertantes may then be represented by addng randomness to the spatal coordnates of the nodes. Mathematcally, ths s expressed as X ¼ X o þ DX where X s the value of nodal coordnate for the case where there s no geometrc uncertanty and DX s the random varable that quantfes the uncertanty n ths coordnate. The uncertantes DX are modeled h as zero mean, uncorrelated random varables wth varance E DX ¼ r. Guest and Igusa (8) showed that perturbaton can be used to decompose the nodal uncertantes nto a mathematcally equvalent system of random loads. These equvalent random loads are assocated wth the frst-order terms of the perturbaton of stffness matrx and are expressed as Df ðþ ¼-K ; ddx Here, the bold lower- and upper-case letters represents vectors and matrces, respectvely, and standard ndcal notaton s used, where repeated ndces mples a summaton. Also, K s the determnstc global stffness matrx, the subscrpt, ndcates the dervatve wth respect to coordnate, and d s the vector of dsplacements due to the appled loads f. The superscrpt () used for the equvalent random load vector Df () s needed n the followng dscusson. Heren, we extend the above, prevously developed perturbaton result by partally ncludng the nonlnear effect of geometrc mperfectons. Ths s done by teraton. To ntalze the teraton process, we use the magntude of the nodal coordnate random varables as the orgnal random dsplacements, gven n vector form as Dd ðþ ¼ e DX where e s the unt vector assocated wth coordnate. The frst teraton for the random dsplacements would be the sum of ths ntal dsplacement feld and the addtonal dsplacements due to the random forces Df () : Dd ðþ ¼ Dd ðþ þ K - Df ðþ ¼ Dd ðþ - K - K ; ddx ð4þ n whch we have used the expresson for Df () from Eq. (). Ths can be expressed n terms of the ntal random dsplacement Dd () by usng the Kroneker delta, d j ¼ e T e j : Dd ðþ ¼ Dd ðþ - K - K ; de T e jdx j ¼ðI þ UÞDd ðþ where I s the dentty matrx and U ¼-K - K ; de T s a dmensonless matrx. If ths teraton process s contnued, then we obtan, for teraton k Dd ðkþ ¼ðI þ U þ þu k ÞDd ðþ wth the lmt for an nfnte number of teratons gven by: Dd ¼ ði - UÞ ð-þ Dd ðþ ¼ ADd ðþ where A ¼ ði - UÞ - ðþ ðþ ð3þ ð5þ ð6þ ð7þ ð8þ ð9þ

4 33 provded the nverse exsts. The matrx A can be thought of as an amplfcaton matrx as t essentally propagates the effect of a random varable through the structure. The equvalent force assocated wth ths random dsplacement s smply an extenson of Eq. (): Df ¼ -K ; de T Dd ¼ K UDd ðþ A standard approach n topology optmzaton s to maxmze structural stffness by mnmzng external work done by the appled loads. Ths s commonly referred to as the mnmum complance desgn. Guest and Igusa (8) showed that the expresson for the complance that ncludes the effects of geometrc uncertantes up to second order s c ¼ f T d þ Df T K - Df - Dd T dt K ;j ddd j ðþ where Dd j s the jth component of the vector Dd. However, only the frst teraton was consdered, so that Dd = Dd () and Df = Df (). Heren, we use the terated form for the random dsplacement and equvalent force vectors n Eqs. (8) and (), whch, when substtuted nto Eq. (), yelds c ¼ f T d þ Dd T U T K UDd - Dd T e d T K ;;j de T j Dd ¼ f T d þ Dd ðþt A T U T K UADd ðþ - Dd ðþt A T e d T K ;j de T j ADdðÞ ðþ To obtan the expected value of the complance, t s necessary to know the correlaton structure of the random varables. Geometrc randomness s generally uncorrelated n trusses by nature, and thus we express correlaton n terms of the Kroneker delta as E½DX DX j ]¼r d j wthout the summaton over ndex. It s useful to rewrte ths n terms of the ntal random dsplacement Dd () : E½Dd ðþ Dd ðþt ]¼C ð3þ where C () s the dagonal covarance matrx of the coordnate random vector wth dagonal elements r. To evaluate the expected value of the complance, E[c], t s necessary to put the random vectors together n the expresson for the complance, c. Ths s done by usng the trace operator and by usng the commutatve property of matrces and vectors multpled wthn the trace operator: [ { ( ) }J E½c] ¼f T d þ E tr Dd ðþt A T U T K U - e d T K ;j de T j ADd ðþ {( ) h } ¼ f T d þ tr U T K U - e d T K ;j de T AE Dd ðþ Dd ðþt A T j {( ) } ¼ f T d þ tr U T F - e d T K ;j de T j C ð4þ where F ¼ K U ¼-K ; de T s a matrx of normalzed equvalent forces, and we have defned: C ¼ AC A T.. Topology optmzaton formulaton ð5þ It s now possble to express the topology optmzaton desgn problem n terms of the expected value of the complance: {( ) } mn E½CðqÞ] ¼ f T d þ tr U T F - e d T T K ;j de j C q 8 K d ¼ f > < K U ¼ F such that ð6þ > R e q e m e 6 V : q e P q mn where q e s the cross-sectonal area of member e stored n the vector of cross-sectonal areas q, v e s the correspondng volume for unt magntude of q e (member length for trusses), V s the allowable volume of materal, and lower bound q mn s a small postve number ( -3 ). The frst two equalty constrants are for the equlbrum condtons under the real appled loads f and normalzed equvalent random forces F. The matrces K, K,, and K,j are assembled va standard fnte element assembly from ther respectve element matrces; the expressons for these for truss elements can be found n Guest and Igusa (8). Snce these matrces are functons of q e, t follows that A, F, d and U are also functons of q. To make the senstvty computatons for the gradent-based optmzaton process computatonally effcent, the dervatves of the objectve functon wth respect to q may be found usng drect dfferentaton (or the adjont method). The resultng expressons are straghtforward and effcent to compute. They are, however, nvolved and are thus presented n the appendx..3. Optmzaton algorthm The above expressons are used n the desgn optmzaton algorthm as follows.. Intalze q (e.g., areas are unformly dstrbuted to satsfy the volume constrant).. Calculate the dsplacements d due to load f by solvng K d = f. 3. Calculate F and U usng F ¼-K ; de T and solvng K U = F. 4. Holdng the dsplacements fxed, calculate A usng Eq. (9), the objectve functon usng Eq. (4), and the senstvty of the objectve functon usng equaton (A.3) of the Appendx. 5. Update q usng a gradent-based optmzer. 6. If not converged, go to step ; otherwse, q gves the fnal desgn. The steps above requre soluton of several lnear systems. However, a key advantage of the perturbaton-based methodology s that each of the systems n steps and 3 have the same left-hand sde: the determnstc global stffness matrx K. Therefore, the soluton of d and U smply requre solvng a lnear system of equatons wth multple rght-hand sdes (real load case f and equvalent load cases F). Ths cost can be effectvely mtgated wth proper solver selecton, such as L-U factorzaton. The prmary computatonal dfference between ths new methodology and the orgnal perturbaton-based algorthm s the computaton of A. It should be noted that topology optmzaton allows for the removal of structural elements. Thus, n step 5, elements that acheve a cross-sectonal area below a prescrbed threshold are removed from the doman and the structural connectvty s updated (see Bendsøe et al., 994; Krsch, 989; Bendsøe and Sgmund, 3). 3. Numercal examples In ths secton we llustrate the effects of geometrc uncertantes on the desgn of several truss examples usng the proposed formulaton. In these examples, all truss members have unt Young s modulus and total avalable materal volume of 79 unts. Sequental Quadratc Programmng, as mplemented n the MATLAB Optmzaton Toolbox, s used for the gradent-based optmzer. The examples were also solved usng the Method of Movng Asymptotes (Svanberg, 987) wth no sgnfcant dfference n soluton qualty. 3.. Smple column We begn wth the smple ground structure shown n Fg. (a), also studed n Guest and Igusa (8), wth horzontal and vertcal

5 34 Fg.. Smple column example. (a) Ground structure geometry, boundary condtons and appled load, and (b) optmal soluton under determnstc desgn condtons. nodal spacng of 4 and 3 unts, respectvely. The horzontal load s appled md-heght at the rght boundary, as ether a compressve or tensle load. The soluton for the determnstc desgn condton of perfectly allgned nodes s gven by the four colnear bars shown n Fg. (b). Ths soluton s ndependent of the load drecton and magntude. Ths structure appears effcent, but under any perturbaton n nodal locatons t becomes () knematcally unstable under a compressve appled load or () neffcent under a tensle load (becomng effcent only after the bar becomes straghtened under the load). We now consder the llustratve case of randomness n a sngle node usng the proprosed algorthm. Both spatal coordnates of node A (Fg. (a)) are consdered uncertan, wth randomness quantfed by the standard devaton of r A n each coordnate drecton. Fg. dsplays results for a compressve load of magntude P =.5 unts wth two levels of node locaton uncertanty:r A =.L x and.l x, where L x = 4 s the length of a sngle horzontal truss member. The desgn under the smaller magntude uncertanty, shown n Fg. (a), ncludes bracng of node A, as well as the nodes connected to node A. Ths s expected due to the vertcal nternal forces nduced by the msalgnment of node A. Ths topology s dentcal to the soluton presented n Guest and Igusa (8) wth only mnor dfferences n the dstrbuton of materal. The desgn for the larger uncertanty of r A =.L x, shown n Fg. (b), ncludes dstrbuted bracng throughout the system. Ths s consstent wth the fact that bucklng s a global phenomenon. The dfferences n desgns are algorthmcally drven by the amplfcaton matrx A and correspondng dfferences n the global equvalent forces F. For small uncertantes, these equvalent forces are localzed around node A, and become very smlar to the localzed equvalent forces derved by Guest and Igusa (8) (ths explans smlarty of Fg. (a) to ther solutons). For larger uncertantes, the equvalent forces F become sgnfcant for nodes that are farther from node A. Ths global effect results n more extensve truss bracng as shown n Fg. (b). When the appled load s changed from compresson to tenson, the structural desgns reduce to a straght bar for both levels of randomness, as shown n Fg. 3. Analytcally ths s due to the fact that the global equvalent forces are dependent on the sgn of the appled load. In the prevous work (Guest and Igusa, 8), the magntudes of localzed equvalent forces were ndependent of the sgn of the appled loads due to the assumpton of lnear elastcty, so that the desgn under compresson and tenson were both gven by Fg. (a). As suggested n Eq. (6), optmal topologes are also dependent on the force magntude. We brefly examne the changes n the desgn as the tensle force s reduced from P to P/ and P/3, for the case of % geometrc varablty at node A and P of.5 unts as before. The results, shown n Fg. 4(a) and (b), show that as the force s reduced, the bracng ncreases. To understand ths, t s Fg. 3. Optmzed desgn for the smple column example under tensle load P for both % varablty and % varablty n locaton of node A. P (a) (b) Fg.. Optmzed desgn for the smple column example under compressve load P and (a) % varablty and (b) % varablty n locaton of node A. All crosssectonal areas are normalzed by the maxmum member area n (a). Fg. 4. Optmzed desgn for the smple column example consderng % varablty n locaton of node A and tensle loadng of magntude (a) P/ and (b) P/3. Compared to the soluton under load P (Fg. 3), smaller tensle loads spur ncreased bracng as the perturbed mddle bar can be prevented from reachng a collnear deflected state. All cross-sectonal area magntudes are normalzed by the maxmum area of a member n case (a).

6 35 noted that the desgn of Fg. 3 wth geometrc varablty at node A has no axal stffness untl t reaches a collnear state. As complance s the product of force and total dsplacement, ths ntal free moton leads to ncreased total deflecton and thus complance. If the load s relatvely s small, then t s possble to reduce the complance by addng the bracng shown n Fg. 4(a) and (b), whch reduces the tendency of the mddle truss elements to become collnear. For suffcently large loads, however, the mddle elements wll tend to become collnear even wth bracng. Hence, the optmal desgn s to use all of the materal n the mddle elements, as shown n Fg. 3. Ths smple example llustrates two mportant propertes of the new algorthm. Frst, the effect of nonlnearty s evdent n that solutons are dependent on both the magntude and drecton (tensle or compressve) of the appled loads. Second, geometrc uncertanty s spatally propagated through structurally nonlnear behavor (such as global bucklng). Addtonal examples are examned In the followng subsectons to further explore these nonlnear characterstcs. 3.. Cantlever beam, center load The ground structure for ths example s shown n Fg. 5. The load magntude s.3 unts and desgn doman dmensons are 45 and 3 unts n the horzontal and vertcal drectons, respectvely. In ths example, and n all of the remanng examples of ths secton, the lne thckness s used to ndcate relatve crosssectonal area. The determnstc desgn (no geometrc uncertantes) s shown n Fg. 6. As the fgure suggests, the optmzed topology would contan several segments of collnear elements, yeldng a knematcally unstable structure. These collnear elements have been merged nto a sngle element to elmnate these nstabltes. Ths s a standard approach n topology optmzaton but tends to ncrease the length of compressve members, makng them more prone to local (Euler) bucklng (Zhou, 996). Ths s contrasted wth the non-determnstc desgn shown n Fg. 7, found usng the proposed algorthm under the assumpton of geometrc varablty of r =.5L x at all nodal locatons. Although every node has geometrc uncertanty, the bracng s used predomnantly around the Fg. 7. Optmzed cantlever desgn obtaned usng the new methodology and consderng varablty of 5% n all nodal locatons. regons of the structure under compressve forces, as these sectons are more prone to collapse due to geometrc mperfectons. It s also clear that the large, prmary compressve load paths n Fg. 6 have been redesgned nto redundant subsystems to create alternate load paths. The compressve column and dagonal member share the same bracng, leadng to an economcal desgn. It s noted that the tensle members could be manufactured as long elements, whle the compressve members would be braced, as the topology desgn suggests. We now compare the nonlnear structural performance of the determnstc and non-determnstc desgns n the presence of geometrc uncertantes. As mentoned, the determnstc desgns contan collnear elements, and thus wll fal under near-zero magntude appled loads. To make ths comparson meanngful we must merge collnear elements. Tradtonally ths s done by node cancelaton, a process where nteror nodes of the merged elements are elmnated. Such a procedure, however, would change the structural mesh and consequently the random varable feld on whch the topology optmzaton was based. Therefore, collnear elements of both desgns are smply merged heren by replacng nteror frctonless hnges wth rgd connectons. Monte Carlo smulaton s used to generate a hundred ndependent samples for each desgn usng unformly dstrbuted random node locatons wth zero means and standard devaton r =.5L x. Fg. 8 shows (a) Fg. 5. Ground structure and boundary condtons for the cantlever beam example. (b) Fg. 6. Optmzed cantlever desgn under determnstc condtons. Fg. 8. One realzaton of the Monte Carlo generated samples for the cantlever beam structure. (a) Determnstc desgn and (b) desgn usng new methodology.

7 36 Load. New methodology Determnstc desgn.5..5 % of samples faled New methodology Determnstc desgn Dsplacement (a) Load (b) Fg. 9. Performance dagrams of the Monte Carlo generated samples usng desgns n Fg. 6 (determnstc) and 7 (new methodology). (a) Averaged load vs. dsplacement plot and (b) percentage of samples faled vs. appled load. The desgn found usng the new methodology clearly offers mproved performance when consderng geometrc nonlnearty and geometrc mperfectons. one of these Monte Carlo samples for both desgns and llustrates the presence of geometrc randomness along the length of the merged elements. Ths s consstent wth the ntal ground structure geometry and represents manufacturng errors. Truss member cross-sectons are assumed sold cylnders, and the load-deflecton curve of each Monte Carlo sample s computed usng the nd-order elastc analyss tool n Mastan software (Zeman and McGure, ). The average of load-deflecton curves s shown n Fg. 9(a), where the horzontal axs represents the vertcal dsplacement at the locaton of load and the vertcal axs represents the load magntude. It can be seen that the two desgns have comparable performance at low loads, but the desgn consderng uncertantes clearly outperforms the determnstc desgn when the load exceeds. unts. Another way to assess and compare desgns s to examne the structural relablty. Here, we defne structure falure as tp dsplacement exceedng 8 unts. Fg. 9(b) dsplays the percentage of samples faled as a functon of load magntude. All of the samples for the determnstc desgn fal before the load reaches.3 unts, whle for the desgn consderng uncertantes, only 4 of the Monte Carlo samples fal at ths load, all due to local (Euler) bucklng. Addtonal faled samples do not occur untl over three tmes ths load, wth % falure only after the load has ncreased by nearly a factor of sx. The relatvely hgh second-order nonlnear performance of the non-determnstc desgn s due to the braced compressve load paths and the exstence of multple redundant load paths. A consequence of ths desgn s that the structure s (on average) slghtly more flexble than the determnstc soluton under very small loads and deflectons. Ths s not surprsng, as the structure response s near lnear elastc n ths regme. Another nterestng property of the desgn n Fg. 7 s that structure undergoes stffenng wth the applcaton of the load. Ths behavor can be seen n the load-dsplacement curve plotted n Fg. for the realzaton shown n Fg. 7(b), and s due to the force-nduced algnment of the tenson members as the structure s subjected to small loads. It s noted that ths ncrease n tangent stffness s not as evdent n the averaged curve n Fg. 9(a) as not all the realzatons behave n ths manner. Fnally, t s noted that the rregulartes n the curves of Fg. 9 are due to the relatvely low number of Monte Carlo samples, and that ncreasng the number of samples would lkely lead to smoother curves. Load Dsplacement Fg.. Load-dsplacement curve for the realzaton shown n Fg. 8, showng structure stffenng wth applcaton of load. Fg.. Ground structure, appled load and boundary condtons for the tall cantlever example.

8 37 Outer Regon Inner Regon Inner Regon Fg.. Determnstc soluton to the tall cantlever example. Outer Regon Fg. 5. Ground structure, appled loads and boundary condtons for the L-shaped structure example. Fg. 3. Soluton to the tall cantlever example obtaned usng the new methodology and consderng randomness of 5% varablty n all nodal locatons Cantlever beam, corner load The next example s a cantlever structure subjected to a load of.5 unts appled at a corner, as shown n Fg.. The horzontal and vertcal doman dmensons are 3 and 6 unts, respectvely. The optmzed desgn under determnstc condtons s shown n Fg., where agan nteror hnges have been replaced wth rgd connectons at the nteror nodes of collnear patterns. As n the precedng example, the desgn consderng uncertanty assumes geometrc varablty of r =.5L x at every node n the structure. The optmzed desgn found usng the proposed algorthm s shown n Fg. 3. The compressve sde of the structure features multple load paths that share a farly dense bracng system, ultmately leadng to an economcal desgn. To evaluate the performance of these desgns n the presence of geometrc uncertantes, Monte Carlo smulaton s agan used to generate realzatons of each of the desgns n Fgs. and 3, and the load deflecton curve for each realzaton s computed usng nd-order elastc analyss. Averages of the load-deflecton curves and the structural relabltes of each desgn are shown n Fg. 4(a) and (b) n the same way they were shown n Fg. 9(a) and (b) for the precedng example. The results follow the same trends dentfed n the prevous example: the desgns created by the new methodology offer superor stffness and lower falure rates because of the bracng and multple load paths n the nondetermnstc desgn. Load. New methodology Determnstc desgn % of samples faled 6 4 New methodology Determnstc desgn Dsplacement Load (a) Fg. 4. Performance dagrams of the Monte Carlo generated realzatons usng desgns n Fg. (determnstc) and 3 (new methodology). (a) Averaged load vs. dsplacement plot and (b) percentage of samples faled vs. appled load. (b)

9 38 Fg. 6. Determnstc soluton to the L-shaped structure example. Fg. 7. Soluton to the L-shaped structure example obtaned usng the new methodology and consderng randomness n all nodal locatons wth % varablty L-shaped structure The fnal example s the L-shaped structure shown n Fg. 5. The overall dmensons are by unts wth the upper rght quarter of the doman removed from the ground structure. The appled load has a magntude of. unts, leadng to compresson and tenson zones n the nner and outer regons respectvely ndcated n the fgure. The optmal desgn under determnstc desgn condtons s shown n Fg. 6. Snce the nner vertcal member carres twce the load of the outer vertcal member, the nner member s twce the cross-sectonal area. Even wth collnear element mergng, however, the structure s knematcally unstable as t can freely rotate under any horzontal load. Fg. 7 dsplays the desgn consderng geometrc uncertanty assumng varablty of r =.L x n all node locatons. The tenson zones receve relatvely lght, f any bracng, whle alternate load paths and shared bracng are ncorporated nto the compresson zones. The structure s also knematcally stable, a known byproduct of consderng geometrc uncertantes n truss desgn (Guest and Igusa, 8). 4. Concludng remarks Ths paper extends a recently developed structural optmzaton algorthm for desgn under geometrc uncertantes to nclude a frst-order approxmaton to geometrc nonlneartes. Geometrc mperfectons are modeled by consderng the locaton of truss nodes to be uncertan, whch ultmately leads to uncertanty n structural stffness. Perturbaton s used to transform these uncertantes nto a system of equvalent random loads. These loads are enhanced n ths work to account for the ncreased potental of global bucklng n structures wth mperfectons. Key characterstcs of ths approach not seen n the orgnal algorthm are that () optmal desgns are dependent on the magntude of the load, () optmal desgns are dependent on the drecton of the load (tenson or compresson nducng), and (3) the mpact of an uncertanty source may propagate to regons far from the source. Several examples were consdered where the goal was to mnmze the expected value of complance (maxmze stffness) n the presence of geometrc uncertantes. Optmzed desgns consstently featured () compresson regons wth multple load paths and shared bracng and () tenson regons wth lttle to no bracng, even n the presence of geometrc mperfectons. Ths more closely reflects good desgn practce, as tenson usually cancels the effects of mperfectons whle compresson amplfes these effects. Improved performance of the optmzed solutons was also confrmed usng second-order elastc analyss. Solutons found usng the new methodology sgnfcantly outperformed those found under determnstc desgn condtons n terms of maxmum load carred and stffness n the nonlnear regme. Mnmzng expected value of complance s a frst step towards robust desgn and t s clear that the new methodology results n desgns that are less senstve to manufacturng errors. Ths could sgnfcantly reduce the analyss cost n the frst stages of desgns, and furnshes a relable startng pont for desgn engneers. A key beneft of the proposed algorthm s computatonal effcency. Uncertantes n the global stffness matrx, the left-hand sde of the lnear system, are transformed nto equvalent loads, the rght-hand sde. Ths means we must solve, at every desgn teraton, a lnear system wth a sngle left-hand sde and multple rght-hand sdes (load cases). Ths s n contrast to Monte Carlobased optmzaton approaches, whch requre, at each desgn teraton, soluton of multple global stffness matrces for a sngle load case. The effcency of such an approach s apparent and has been dscussed n related works (Guest and Igusa, 8; Asadpoure et al., ). Although the algorthm s mechancs-drven, the prmary dsadvantage of the proposed algorthm s that t s heurstc. The perturbaton-based methodology offers mathematcal equvalence n the lnear elastc regme, but offers only a lnear approxmaton to nonlnear behavor. Although the algorthm clearly produces solutons wth mproved nonlnear performance n the presence of geometrc mperfectons, the mprovement n performance cannot be predcted wthout a full nonlnear analyss. Acknowledgments Ths work was supported by the Natonal Scence Foundaton under Grant No. CMMI-9863 wth Dr. Chrstna Bloebaum servng as program offcer. Ths support s gratefully acknowledged. Appendx A. Senstvty dervaton Consder the expected value of the mnmum complance functon, whch s repeated here for convenence: {( ) } E½c] ¼f T d þ tr U T K U - e d T T K ;j de j C ða:þ In the followng, prmes denote the dervatve wth respect to desgn varables. Usng the chan rule yelds the followng:

10 39 { ( E½c] ¼ f T d þ tr U T K U þ UT K U ( ) ) } - e d T K ;j d þ dt K ;j d e T C þ K U C j ða:þ where we have defned: K U ¼ U T K U - e d T K ;j de T j Dfferentatng Eq. (5) yelds: C ¼ A C A T þ AC A T and dfferentatng Eq. (9) yelds: A ¼ AU A Combnng the precedng three equatons would lead us to: trfk U C g¼trfk U A C A T g¼trfck U AU g ða:3þ ða:4þ ða:5þ ða:6þ Substtutng (A.6) nto (A.), collectng terms, and usng the trace operator propertes yelds: n o E½c] ¼½f T - C j d T K ;j ]d þ tr CðU T K þ K U AÞU n o þ tr CU T K U - C j d T K ;j d ða:7þ The dervatves of the stffness matrces wth respect to the desgn varables are straghtforward to compute (Guest and Igusa, 8). The dervatves of the dsplacements d and U are found usng drect dfferentaton. Recall that: K d ¼ f K U ¼-K ; de T So we have: ða:8þ ða:9þ d ¼-K- K d ða:þ T U ¼-K - K ; de T - K - K ; det - K - K ; d e ¼ K - ðk K- K ; - K K ; þ K ;K - ÞdeT ða:þ Substtutng the above results nto (A.7) would lead us to: h E½c] ¼ f T - C d j d T K ;j - C j d T K ;j d n ( ( ))o þ tr CðU T K þ K U AÞ K - -K U - K ; det - K ; d e T n o þ tr CU T K U ða:þ Usng the propertes of trace operator and collectng terms gves the fnal form of the senstvtes: n ( )o E½c] ¼½f T - C j d T K ;j ]d - tr C U T K U þ K UAK - K U n (( )( ))o - tr e T C U T þ K U AK - K ; d þ K ; d References - C j d T K ;j d ða:3þ Achtzger, W., Bendsøe, M., Ben-Tal, A., Zowe, J., 99. Equvalent dsplacement based formulatons for maxmum strength truss topology desgn. Impact of Computng n Scence and Engneerng 4, Asadpoure, A., Tootkabon, M., Guest, J.K.,. Robust topology optmzaton of structures wth uncertantes n stffness Applcaton to truss structures. Computers and Structures 89, 3 4. Bendsøe, M.P., Sgmund, O., 3. 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