Engineering Structures

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1 Engneerng Structures 37 (2012) Contents lsts avalable at ScVerse ScenceDrect Engneerng Structures journal homepage: Topology optmzaton for braced frames: Combnng contnuum and beam/column elements Lauren L. Stromberg a, Alessandro Beghn b, Wllam F. Baker b, Glauco H. Paulno a, a Department of Cvl and Envronmental Engneerng, Unversty of Illnos at Urbana-Champagn, Newmark Laboratory, 205 N. Mathews Avenue, Urbana, IL 61801, USA b Skdmore, Owngs & Merrll, LLP, 224 S. Mchgan Avenue, Chcago, IL 60604, USA artcle nfo abstract Artcle hstory: Receved 15 Aprl 2011 Revsed 13 December 2011 Accepted 14 December 2011 Keywords: Topology optmzaton Hgh-rse buldngs Materal layout Braced frames Lateral systems Contnuum and dscrete elements Optmal braced frame geometry Ths paper descrbes an ntegrated topology optmzaton technque wth concurrent use of both contnuum four-node quadrlateral fnte elements and dscrete two-node beam elements to desgn structural braced frames that are part of the lateral system of a hgh-rse buldng. The work explores the analytcal aspects of optmal geometry for braced frames to understand the underlyng behavor and provdes a theoretcal benchmark to compare numercal results. The nfluence of the ntal assumptons for the nteracton between the quadrlaterals and the frame members are dscussed. Numercal examples are gven to llustrate the present technque on hgh-rse buldng structures. Ó 2011 Elsever Ltd. All rghts reserved. 1. Introducton Correspondng author. Tel.: ; fax: E-mal addresses: paulno@llnos.edu, paulno@uuc.edu (G.H. Paulno). Topology optmzaton s common n mechancal and aeronautcal engneerng, and t has been, n recent years, progressvely embraced for structural engneerng applcatons. Examples are the mult-story buldng desgn or long span brdge desgn applcatons presented n Stromberg et al. [1], Allahdadan and Boroomand [2], Neves et al. [3], or Huang and Xe [4]. Despte a varety of applcatons wthn the cvl engneerng feld, the focus of ths work s towards hgh-rse buldngs, where engneers are faced wth the challenge of dentfyng the optmal topology of the lateral bracng system that mnmzes materal usage and correspondng cost. Therefore, the scope of ths work s to ntroduce a methodology usng topology optmzaton for sotropc, homogeneous materal that enables engneers to develop the lateral system from the conceptual optmal bracng angles to the fnal szng of the members. The methodology presented n Stromberg et al. [1] represents an ntal attempt at dentfyng optmal bracng angles. However, t presents some lmtatons as llustrated n the problem of Fg. 1, whch shows a schematc for a hgh-rse buldng subject to wnd loadng. The prevous work (see Fg. 1b) was lmted due to hgh concentratons of materal towards the edges of the doman, consstent wth the flange versus web behavor, descrbed n Secton 4 of Stromberg et al. [1]. Such concentratons mpede the dentfcaton of the workng ponts of the column to the dagonal ntersectons. In addton, the columns are so wde that they possess hgh flexural stffness. In practce, ths s not realstc because the columns are relatvely narrow compared to the wdth of the buldng. Moreover, snce the contnuum topology optmzaton problem has a constrant on the volume fracton and a large amount of materal forms the column members, a relatvely low volume s avalable for the dagonals. As a result, there s an ncomplete dagonalzaton n the frame (.e. mssng dagonals at the base of the frame). Thus, one would have to ntroduce an addtonal constrant to dstrbute materal between the columns and the dagonals to prevent concentratons at the edges. Ths paper ntroduces a combnaton of dscrete (beam/column) members and contnuum quadrlateral members to overcome the aforementoned ssues. In Fg. 1c and d, sx dscrete (truss) members are added to model each column whle mantanng the same total volume of materal as the problem n Fg. 1b. As a result, the concentraton of materal at the edges s elmnated, and a complete dagonalzaton wth clear workng ponts emerges Motvaton for braced structural systems Braced frame and moment frame structural systems are commonly deployed n the lateral desgn of hgh-rse buldngs. Braced /$ - see front matter Ó 2011 Elsever Ltd. All rghts reserved. do: /j.engstruct

2 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Nomenclature A 0 cross-sectonal area of column szed for constant stress A cross-sectonal area of member B half the wdth of a frame c complance of the desgn E Young s modulus computed through SIMP E 0 Young s modulus of sold materal E Young s modulus of member f global load vector F nternal force n member from real system f nternal force n member from vrtual system H overall heght of a frame I moment of nerta k proportonalty constant K global stffness matrx L length of member m number of dagonal members n ndex of module n a frame N total number of modules p penalzaton factor for SIMP pont load at pont P r mn u u v V V s W ext W nt x x y z D D req h k m q r X mnmum radus of projecton global dsplacement vector horzontal dsplacement of node vertcal dsplacement of node total volume maxmum volume constrant external work of a frame nternal work of a frame a pont n the desgn doman x-coordnate of node y-coordnate of node heght of the th bracng ntersecton pont deflecton target or allowable deflecton stran pseudo-rotaton at node Lagrange multpler Posson s rato densty stress n member desgn doman Fg. 1. Comparson of exstng topology optmzaton technques wth technque proposed n ths work consderng the same total volume of materal: (a) problem statement for contnuum approach, (b) topology optmzaton result usng quadrlateral elements, (c) problem statement for combned approach, and (d) topology optmzaton result wth quadrlateral and dscrete column elements. frames have been used n several noteworthy buldngs lke the John Hancock Center (Chcago, IL), Broadgate Tower (London, UK) and Bank of Chna Tower (Hong Kong), as shown n Fg. 2. The desgn of such systems s tradtonally based on dagonal braces arranged accordng to a 45 or 60 angle and varatons n-between these two angles. However, there have been few engneerng studes n the past to dentfy the optmal bracng angle and the parameters affectng such angles [5]. The scope of ths paper conssts of explorng optmal bracng layouts to maxmze structural performance whle mnmzng materal. Varous measures of structural

3 108 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 2. Exstng buldngs featurng remarkable braced frame systems: (a) John Hancock Center n Chcago, IL ( (b) Broadgate Tower n London, UK ( and (c) Bank of Chna Tower n Hong Kong ( performance could nclude tp dsplacement, frequency, complance, crtcal bucklng load, etc. The examples of ths paper focus on mnmzng the complance and relatng ths quantty to buldng behavor and desgn. The utlzaton of the optmzaton technques descrbed n ths paper n the ntal conceptual phase of desgn nforms engneers of the most effcent layout of materal. Desgn decsons on the topology of the lateral system can therefore be streamlned wth savngs n materal costs and mnmzaton of mpact on natural resources On exstng frame optmzaton technques Currently, structural engneerng optmzaton technques can be classfed nto two dstnct categores: dscrete member optmzaton usng beam or truss elements [6 10] and contnuum methods [11 13]. Wthn the class of dscrete member technques for structural systems, Takezawa et al. [14] proposed a method for frame elements where the desgn varables consst of the cross-sectonal propertes, ncludng prncple drecton of the second moment of nerta. Fredrcson [15] used a jont penalty and materal selecton approach wth flexble jonts. Kaveh and Shahrouz [16] employed the deas of graph theory to determne the member connectvty between the supports and load paths for bracng systems. Wang [17] optmzed frame structures usng the maxmum bendng moment as the desgn crtera. On the other hand, n Bendsoe and Sgmund [13] several examples are gven for the contnuum topology optmzaton problem where beams are added by creatng a long row of sold elements across the desgn doman, as n the case of a two-dmensonal brdge where the sold area represents the deck. Smlarly, Allahdadan and Boroomand [2] proposed a technque usng contnuum elements to determne the optmal bracng system for dynamc response n desgnng or retrofttng structures. In that method, the floor levels were modeled as sold rgd elements. Another technque, explored n the work of Neves et al. [3], talors the topology optmzaton desgn framework for stablty problems. In such formulaton, the objectve functon s the crtcal bucklng load, rather than mnmum complance. The paper by Neves et al. [3] consders the desgn of a portal frame and a fvestory frame, smlar to the examples presented n ths work. However, ths prevous approach models the structural frames wth sold quadrlateral (Q9) elements, nstead of dscrete (beam) elements, as presented n what follows. Furthermore, structural frame studes were presented wth a specfc natural frequency as the objectve n the work of Daz and Kkuch [18]. Whle each of the aforementoned technques n the lterature s valuable n tself, better structural engneerng tools may be developed by combnng such deas. Several attempts have been made at proposng an ntegrated structural optmzaton framework. For nstance, Lang s technque [19,20] uses a performance ndex based on stran energy densty for the optmzaton of multstory steel buldng frameworks. An exstng frame s modeled of dscrete steel elements wth an underlyng contnuum mesh of quadrlaterals, whch are removed based on the lowest performance ndces. Mjar et al. [21] uses Reuss and Vogt mxng rules for effectve stffness wth topology optmzaton to desgn bracng systems. Beam elements are used to model an unbraced system and contnuum elements model the bracng layout. In Lagaros et al. [22], optmzaton has been taken a step further nto the structural engneerng ndustry by combnng szng, shape and topology optmzaton to desgn three-dmensonal steel structures wth web openngs n complance wth modern desgn codes. Sze optmzaton was used to determne the cross-sectonal area of the beams and columns, whle shape and topology optmzaton was mplemented for the number and sze of web openngs. Here, we use a topology optmzaton approach wth Sold Isotropc Materal wth Penalzaton (SIMP) where beam elements are ncluded n the fnte element analyss porton to acheve more meanngful bracng layouts. These layouts are derved analytcally to verfy the numercal results as well Paper scope and organzaton Ths paper s organzed as follows: n the next secton, we dscuss the man concepts of an energy-based method to effcently sze structural frames usng the least amount of materal. Followng, energy methods, n conjuncton wth the Prncple of Vrtual Work (PVW), are used to mathematcally derve the optmal geometry of a dscrete braced structure n Secton 3. Then, we outlne the man concepts behnd the combnaton of Q4 and beam elements, ncludng several methods to attach the two types of elements. In Secton 5, the topology optmzaton framework s ntroduced and extended to nclude smultaneous use of several element types n the context of structural framng systems. Some fundamental modelng aspects assocated wth the combnaton of beam and quadrlateral elements on a sngle module of a frame are nvestgated n Secton 6 and compared to the results from Secton 3. Fnally,

4 L.L. Stromberg et al. / Engneerng Structures 37 (2012) numercal results are llustrated n Secton 7 and conclusons are drawn on the applcaton of the proposed methodology n the last secton. 2. A szng technque for frame optmzaton In ths secton, energy methods and the PVW are explored to complment the methodology for topology optmzaton of structural braced frames by ntroducng a szng technque for the fnal beam, column and bracng members Applyng energy methods to sze braced frames Baker [23] derved a method to calculate the optmal cross-sectonal area for a statcally determnate frame to lmt the tp dsplacement of a buldng under wnd load to a target deflecton, D, by combnng the PVW and the Lagrangan multpler method. Ths methodology s based on the assumpton that gven a frame wth axal forces due to a lateral load (e.g. wnd load), F, length of members, L, and cross-sectonal area, A, the target deflecton can be acheved through strategc szng of the cross-sectonal areas. In ths procedure, two load cases are analyzed: the real (wnd) load case to calculate the strans and dsplacements (Fg. 3 (left)), and the vrtual (unt) load case to calculate the stresses and forces (Fg. 3 (rght)). Usng the PVW, the work done by the vrtual (unt) system for the dsplacement and deformaton of the real (wnd load) system can be wrtten as follows: D 1 ¼ X Z L 0 fdx ¼ X Z L 0 f F EA dx ¼ X FfL EA where ¼ F= ðeaþ represents the stran n the real system, and f s the nternal force of a member n the vrtual (unt) system. Thus, the vrtual work yelds the followng expresson: D ¼ X FfL ð2þ EA Combnng the PVW wth the Lagrangan multpler method, one obtans D ¼ X F f L þ k X! A j L j V ð3þ E A j ð1þ where k s the Lagrange multpler, A s the unknown cross-sectonal area of a member and P j A jl j V s a constrant on the volume V of materal. We note that addtonal constrants could be ntroduced usng addtonal Lagrange multplers. By dfferentatng the above expresson wth respect to A and performng several numercal manpulatons, the optmal crosssectonal area of a member for the target deflecton, D req, s determned from " ða Þ req ¼ 1 ð D req E F f Þ X # 0:5 0:5 L j F j f j ð4þ j The above expresson provdes optmal cross-sectonal areas for a statcally determnate braced frame. As shown n Baker [23], expressons smlar to Eq. (4) can be derved for moments, shear and torque. Therefore, ths szng technque can be extended to moment frames provded that the moment of nerta, I, of the member s a lnear functon of the area, A, n the form I ¼ ka, where k s a proportonalty constant Overall desgn process The optmzaton technques descrbed prevously help streamlne the desgn decsons at varous stages of a project from the conceptual characterzaton of a braced frame layout to the fnal szng of the members. Once the overall shape of the buldng s known, the optmal brace layout could be establshed assumng that frame columns are arranged at ts outer permeter at a regular spacng to ensure that the trbutary areas for the columns are smlar. At each floor level, a horzontal beam (spandrel) would span between two adjacent columns. Beams and columns would be modeled usng beam elements whle the space bounded by two columns and two beams would be meshed usng quadrlateral elements. After the fnte element mesh s completed the followng steps can be appled n sequence n the desgn flow process (Fg. 4): sze vertcal lne elements (columns) accordng to gravty load combnatons (accountng for dead, supermposed dead and lve loads); Defne gravty load structural system Sze beams and columns n system for gravty loads Perform topology optmzaton over quadrlateral mesh for lateral loads Use topology optmzaton results to develop conceptual bracng system Sze fnal members u sng energy methods Check convergence Fg. 3. Illustraton of the PVW: real (wnd) load case (left) and vrtual (unt) load case (rght). Fg. 4. Schematc for the overall optmzaton process of a braced frame.

5 110 L.L. Stromberg et al. / Engneerng Structures 37 (2012) run topology optmzaton on the quadrlateral elements for lateral load combnatons (accountng for wnd and sesmc loads); dentfy the optmal bracng layout based on the prevous step and create a frame model consstng of beam elements; optmze the member szes usng the vrtual work methodology The above steps ndcate a potental path from a conceptual desgn to the fnal szng of a braced frame. However, each optmzaton step could be appled ndependently dependng upon the specfc need of the engneer. Notce that the dstrbuton of loads shfts n the frame throughout the optmzaton process due to re-szng of the members. Therefore, the process s teratve and should be repeated untl convergence s acheved. 3. Optmal braced frames analytcal aspects Important analytcal aspects of optmal braced frames are explored n ths secton to establsh a benchmark for comparson of the numercal results later presented n ths paper Fully stressed desgn and optmal frames The energy-based desgn method presented n Baker [23] and descrbed n the prevous secton mples that any frame wth optmal cross-sectonal members subject to a pont load at the top s under a state of constant stress (fully stressed desgn) as demonstrated n the followng dervaton. By takng the dervatve of Eq. (3) wth respect to the areas A and solvng for the Lagrangan multpler, we obtan: k ¼ F f EA 2 Consderng a lnear analyss, for the case of a pont load at the top of the frame f ¼ k F, where k s a proportonalty constant, the followng expresson holds: k ¼ F 2 k E ¼ const ð6þ A In the above expresson the Lagrangan multpler s a constant, therefore the stress n the th member, r ¼ F =A s constant. The latter concluson apples to any member of the frame, thus the stress level s constant throughout the structure. In the context of the statcally determnate braced frame systems consdered n ths paper, the equvalence between a constant state of stress and mnmum complance s generalzed from the sngle pont load descrbed above to multple pont loads P appled to the frame. Assumng that the dsplacements at each pont of load applcaton are u, the complance can be expressed as: W ext ¼ X P u ¼ X j ð5þ F 2 j L j EA j ¼ W nt ð7þ where W ext and W nt are the work done by the external and nternal forces respectvely. By ntroducng the Lagrangan multpler constrant on the areas of the members, W ext ¼ X j F 2 j L j EA j þ k X j A j L j V! In order to mnmze the complance of the system wth varous member szes, the rght-hand sde of ths equaton s dfferentated wth respect to the areas A and solved for the Lagrangan multpler k to obtan the followng: k ¼ F 2 1 E ¼ const ð9þ A ð8þ The above result s smlar to Eq. (6) and confrms that, n the present context, mnmum complance leads to constant stresses. In general, for the complance mnmzaton problem, a state of constant stran energy densty represents the condton of optmalty [13]. Snce the stran energy densty s related to the Von Mses stress [24,25], the effectve stresses n optmal structures are constant. Addtonally, for the case of a sngle pont load, mnmum tp dsplacement concdes wth mnmum complance. The constant stress condton s verfed later for the contnuum approach n Secton Optmal sngle module bracng Usng the deas from the prevous secton, we study the optmal geometry of the braced frame shown n Fg. 5, where the overall heght of the frame s gven as H, the total wdth as 2B, and the heght of the bracng ntersecton pont as z. Notce that the problem n Fg. 5 (left) s smplfed nto the problem n Fg. 5 (rght) by takng advantage of symmetry. Lettng the heght of the bracng pont, z, be the desgn varable, we are lookng for the optmal locaton that mnmzes the deflecton at the top of the structure usng the PVW. The frame shown n Fg. 5 s statcally determnate, so by applyng a unt load at the locaton of unknown deflecton, D, the nternal forces of the members can be solved for as follows: f 1 ¼ H z p B ffffffffffffffffffffffffffffff þ z f 2 ¼ 2 B and qffffffffffffffffffffffffffffffffffffffffffffffffffffffff f 3 ¼ þ ðh zþ 2 B ð10þ ð11þ ð12þ Note that the forces n the frame nduced by a wnd load P appled at the same locaton as the unt load would smply be F ¼ Pf. Now, usng Eq. (2) and assumng each member to have a constant stress, r ¼ F =A, the tp deflecton s D ¼ F X f L ¼ rb EA E X f L B ð13þ The tp deflecton of the frame s mnmal when the followng ¼ X f L ¼ 0 B ¼ H H z þ B2 þ z 2 þ B2 þ ðh zþ 2 ¼ 0 ð14þ ¼ rb H E B þ 2z 2 H z ð Þ ¼ 0 B B Thus, the brace work pont heght for mnmal deflecton s z ¼ 3 4 H ð15þ Ths result s not surprsng f we consder the problem n Fg. 6 (top). In ths problem, a pont load representng the wnd (lateral) force actng on the frame s appled at the top left corner and symmetry s enforced. The topology optmzaton of the contnuum mesh does not lead to a smple 45 bracng angle due to the nteracton of shear and axal forces n a smlar fashon to the one descrbed n Secton 4 of Stromberg et al. [1]. The 45 bracng angle would be the outcome of a pure shear problem as shown n Fg. 6 (bottom). However, the cantlever problem (used to model

6 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 5. Geometry and notaton for the sngle module frame optmzaton problem wth even number of dagonals. Fg. 6. Illustraton of the dfferences between the case of a cantlever structure (top) and the pure shear problem (bottom). a hgh-rse) s never pure shear because the overturnng moment PH does not appear n a pure shear problem. Therefore, the topology optmzaton results n a hgh-wasted cross bracng. The actual locaton of the ntersecton pont of the braces at 75% of the heght H as shown n Eq. (15) s confrmed n Fg. 6 (top rght). Ths result has been further confrmed by runnng a smple Matlab code for dscrete members as shown n Fg. 7. In the Matlab code, the ntersecton of the bracng was constraned to move along the centerlne of the module due to symmetry, the heght rato, z=h, was vared from 0.5 to 1 (z beng the dstance of the brace work pont from the base) and the correspondng tp deflecton was calculated (see Fg. 7b). The optmal z=h rato (.e. the one that mnmzes the deflecton at the top of the frame) s shown to be 0:75H n Fg. 7b. The results here are contngent upon the assumpton of constant stresses n the dscrete members, whch was demonstrated n the prevous secton Optmal multple modules bracng for pont load The analyss conducted for a sngle module braced frame can be extended to a frame wth multple modules along the heght and a sngle load appled at the top by observng the relatonshps between the geometry of the frames and the forces n ts members as descrbed by Fgs. 5 and 8. The forces f n the dagonal members due to a unt pont load at the top are (Fg. 5 (rght) and Fg. 8 (rght)): f ¼ L B whle the forces n the columns are gven by f ¼ ðh z Þ B ð16þ ð17þ where ðh z Þ ndcates the moment arm of the unt force n the module under consderaton (see Fgs. 5 (rght) and 8 (rght)). Accordng to Eq. (16), the forces n the braces are dependent upon the length of the members and, n turn, are a functon of the coordnates of the nodal elevatons z. Combnng Eq. (13) wth Eqs. (16) and (17), the dsplacement at the top of the frame s D ¼ rb X f L E B "! ¼ rb X L 2 þ X # H z j Lj E braces j columns ð18þ Ths expresson s only a functon of the nodal elevatons z. Therefore, the frame of mnmal tp deflecton s obtaned by takng the

7 112 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Optmal Geometry 6 Deflecton for Varous z/h Ratos Δ Rato of z/h Fg. 7. Results for the optmzaton of the bracng angle for the cantlever problem: (a) optmal geometry for the truss and (b) plot of deflectons versus brace heght ntersecton rato. Fg. 8. Geometry and notaton for the sngle module frame optmzaton problem wth odd number of dagonals: (a) free body dagram for a dagonal member and (b) free body dagram for a column member. partal dervatves of the above functon wth respect to the elevatons z. For the frame n Fg. 8, as an example, the dsplacement s "! D ¼ rb X L 2 þ X # H z j Lj E braces j columns " # ¼ rb ðh z 2 Þ 2 ð þ z 2 z 1 Þ 2 þ z2 1 E B þ ð H z 1Þz 2 ð19þ 2 The frame wth mnmal top dsplacement s defned by the 1 ¼ 0 ) 3z 2 þ 4z 1 2 ¼ 0 ) H þ 4z 2 3z 1 ¼ 0 Therefore, ð20þ z 1 ¼ 3 4 z 2; z 2 ¼ 4 7 H ð21þ We observe that n the above equatons the brace work pont z 1 s stll located at 75% of the heght of the module z 2, smlarly to the example descrbed n Fg. 5. In addton, the top brace s parallel to the lower one, whch hnts to the presence of a pattern n the optmal soluton Applcaton to hgh-rse buldng patterns The equatons for the optmal work pont elevatons n a frame can be generalzed to the case of the nth module of such a frame (see Fg. 9 for notaton), where the top dsplacement of the frame can be expressed n terms of the dmensonless quantty, ED=ðrBÞ, as follows: ED rb ¼ XN n¼1 ðz 2n z 2n 1 Þ 2 þ ð þ z 2n 1 z 2n 2 Þ 2 þ ð þ H z 2n 1Þ 2 ðz 2n z 2n 2 Þ ð22þ Here N s the total number of modules and t s assumed that z 2n 2 < z 2n 1 < z 2n. By dfferentatng wth respect to the nodal elevatons z 2n (column work pont) and z 2n 1 (brace work pont):

8 L.L. Stromberg et al. / Engneerng Structures 37 (2012) These equatons can be rewrtten as follow: z 2n ¼ z 2n 1 þ z 2nþ1 z 2nþ1 z 2n z 2n 1 ¼ z 2n 2 þ z 2n þ z 2n z 2n ð24þ From the above expressons, two mportant geometrc features of optmal braced frames are nferred: 1. The braced frame central work pont z 2n 1 s always located at 75% of the module heght. 2. The module heghts are all equal. The last geometrc property s easly verfed n Fg. 10 where, after substtuton, we obtan the relatonshp z 2 ¼ z 4 =2 for the two lowest modules. Smlar relatonshps can be derved for the other modules ED ¼ 0 ) 3z 2n þ 4z 2n 1 z 2n 2 ¼ 0 ED ¼ 0 ) z 2nþ1 þ 4z 2n 3z 2n 1 ¼ 0 2n 2n Fg. 9. Notaton for the nth module of a frame (e.g. hgh-rse buldng). ð23þ To compare the valdty of our results wth those presented prevously n the lterature, we consder the optmum frameworks gven n Hemp [26] based on the mathematcs of optmal layouts frst ntroduced n Mchell [8]. In these prevous works of lterature, the authors am to fnd the mnmum volume requred for a gven structural framework and derve the condtons assocated wth such layouts. Usng these condtons, Hemp derved the optmal geometry for the strp 0 6 y 6 h consstng of cyclods. Ths problem can then be appled to the optmum desgn of shear bracng of a long cantlever under a tp shear, F (gven n 4.17 of Hemp [26]). The results of ths study (see Fg. 11) are compared wth Fg. 10. Notaton and geometrc proportons for a frame wth multple modules and a sngle pont load.

9 114 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Therefore, by mnmzng the tp deflecton D, the volume of the frame s also mnmzed. In summary, the optmal frame for a pont load s characterzed by mnmum tp deflecton, mnmum complance, mnmum volume and constant stress n the members. The problem for the optmal number of modules s formulated n terms of m, the optmal number of dagonals n the frame, as llustrated n Fg. 12. For example, wth the geometry shown n Fg. 5 (or m ¼ 2nFg. 12) the volume s V ¼ PB X f L r B ¼ PB H 2 r 4B þ B 2 ¼ PB r 2 þ 7! 2 H 8 B 2 þ 9H2 16 H 2 16 þ þ B2! ð29þ Smlarly to the example above, the dmensonless frame volume Vr=ðPBÞ s derved for the other geometrc confguratons of Fg. 12 and the result s generalzed for systems wth m dagonal members n Table 1. The dmensonless volume from Table 1 s computed and plotted n Fg. 13 for varous aspect ratos H=B. The plot llustrates when the frame structure should transton, for example, from 1 to 3 dagonal members (see Fg. 13a), or 2 to 4 dagonal members (see Fg. 13b) and so on, by whch volume curve s the lowest. The transtonal aspect ratos are shown n the fgure wth a dashed vertcal black lne. The transton ponts can be derved analytcally by equatng the dmensonless volume of the frame wth m dagonals to the one wth m þ 2 dagonals as follows: Vr PB mdagonals ¼ Vr PB mþ2dagonals ð30þ Fg. 11. Comparson of results wth those of the lterature: (a) Dscrete truss showng the optmum shear bracng smlar to that wth a contnuous array of orthogonal cyclods gven n Hemp [26] (rotated by 90 ) and (b) optmal geometry of a sngle module of the truss. those presented n ths work. Based on the angles of the optmal geometry derved by Hemp, H ¼ B ffffff pffffff p 3 3 þ B 3 ¼ 4 pffffff 3 3 B ð25þ we obtan z ¼ B p ffffff 3 p ffff 4 3 B ¼ 3 ð26þ 4 3 thereby verfyng the approach used by the authors Optmal number of modules for sngle pont load The results presented n the prevous secton dentfy geometrc prncples for optmal frames of mnmum tp deflecton and are ndependent of the aspect rato H=B of the frame (see Eq. (24)). Therefore, one may wonder what s the optmal number of modules for a frame of gven aspect rato. Ths queston s answered by mnmzng the volume of the frame, whch s wrtten as follows: X X V ¼ X A L ¼ X F r L ¼ P r f L ¼ PB r f L B ð27þ where P s the magntude of the unt load appled at the top of the frame, B s the wdth of each (symmetrc) frame, and r s the constant stress n each member. Notng the smlartes between the above equaton and Eq. (13), t follows: V ¼ PE r 2 D ð28þ Usng the formulas derved n Table 1 for frames wth an odd number of dagonals, m þ m þ 2 2 H ¼ m þ 2 þ m þ 4 2 H ) H 2m þ 1 B 2m þ 5 B B rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff ð2m þ 1Þð2m þ 5Þ ¼ ð31þ 3 Smlarly for a frame wth an even number of dagonals, rffffffffffffffffffffffffffffffffffffffffffffffffffffff H B ¼ 2 3 mð2m þ 4Þ ð32þ In concluson, the desgner can frst dentfy the optmal number of modules for a braced frame dependng on the H=B rato, then later dentfy the optmal bracng layout accordng to the geometrc relatonshps descrbed by Eq. (24). Usng ths methodology, a conceptual desgn for a competton entry featurng optmal bracng work pont locatons was proposed by Skdmore, Owngs & Merrll, LLP as shown n Fg Optmal multple modules bracng for multple loads The results for a frame wth a sngle pont load appled at the top are here generalzed to the case of multple pont loads. Wthn ths context, the optmalty crtera followed s complance mnmzaton, whch leads to a fully stressed desgn as descrbed prevously. The complance (or external work of the appled forces W ext ) s wrtten n the followng dmensonless form: EW ext rbf ¼ XN n¼1 " ð ðn n þ 1Þ z 2n z 2n 1 Þ 2 þ ð þ ðn n þ 1Þ z 2n 1 z 2n 2 Þ 2 þ þ XN j¼n z 2j z 2n 1! ðz 2n z 2n 2 Þ # ð33þ

10 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 12. Geometry and notaton for optmal braced frames wth m dagonal members. Table 1 Frame volumes for varous numbers of dagonals. Number of dagonals, m m (odd) m (even) Dmensonless frame volume, Vr=ðPBÞ 2 1 þ H B 2 þ 7 H 2 8 B 3 þ 5 H 2 7 B 4 þ 11 H 2 16 B 5 þ 7 H 2 11 B m þ m þ 2 H 2 2m þ 1 B m þ 1 2 þ 3 H 2 4m B The above equaton s very smlar to Eq. (22) derved for the case of a sngle pont load. The mnmum complance s obtaned by takng partal dervatves of ths equaton, as gven for the case of the bracng work pont locaton z 2n 2n 1 EW ext rbf ¼ 0 ) N n þ 1 ð 3z 2n þ 4z 2n 1 z 2n 2 Þ ¼ 0 ð34þ Ths equaton yelds the same results presented n Eq. (23). Therefore, t s confrmed that even n the case of multple pont loads appled to the frame, the optmal bracng work pont s located at 75% of the heght of the module. Furthermore, the optmal heght of a module can be derved by takng the partal dervatve of Eq. (33) wth respect to the column work pont elevaton z 2n. The frame modules are consdered to be of constant heght n what follows. 4. Combnng Q4 and beam elements In ths secton, the ntegraton of beam and Q4 elements for two-dmensonal problems s dscussed wth emphass on the node-to-node connectons or, more specfcally, on the nteracton among the concdent degrees of freedom. Fg. 13. Plot of dmensonless volume versus heght to wdth rato, H=B: (a) odd number of dagonals and (b) even number of dagonals.

11 116 L.L. Stromberg et al. / Engneerng Structures 37 (2012) end rotaton of the beam has no nfluence on the quadrlateral fnte elements because the rotatonal degree of freedom s decoupled. Addtonally, all the nteror nodes along the length of the beam are free to move ndependently of the quadrlateral node translatons (see Fg. 16). Here, the effect of the beam elements s only global on the mesh, that s, the beams provde translatonal and rotatonal stffness at the column node locatons (Fg. 16) Beam and quadrlateral elements attached contnuously along beam lne Fg. 14. Renderng of a competton entry showng an optmal bracng system (mage courtesy of Skdmore, Owngs & Merrll, LLP) Element combnaton alternatves In the proposed technque, the element types used are the standard two-node beam elements wth sx degrees of freedom (two translatons and one rotaton at each node) and the four-node blnear quadrlateral elements wth eght degrees of freedom (two translatons per node). In order to effectvely connect the fnte elements, the nteracton between the rotatonal and translatonal degrees of freedom must be taken nto account. Ths nteracton can be carred out usng the three methods outlned n Fg. 15: the beam element s attached only at the extreme ends of the quadrlateral mesh so nteror nodes of the quadrlateral mesh along the beam move ndependently of the nteror nodes of the beam element (Fg. 15a), the beam s dscretzed nto many smaller beam elements whch are attached at every node of the quadrlateral mesh along the beam lne, forcng the quadrlateral nodes to translate together wth the beam elements (Fg. 15b), the beam elements share all the degrees of freedom wth the enrched quadrlaterals along the beam lne, meanng each node of both quadrlaterals and beams must translate and rotate concurrently as opposed to the prevous methodologes where the quadrlaterals were lmted to pure translatons (Fg. 15c). Detals of these mplementatons are dscussed next, and a comparson of results based on these technques s gven later, n Secton Beam and quadrlateral elements connected at extreme ends only The frst method for combnng contnuum and dscrete fnte elements conssts of smply connectng the beam ends to the extreme corners of the quadrlateral mesh. As dsplayed n Fg. 15a, the beam elements share two translatonal degrees of freedom at each end (hghlghted n red 1 ) wth the quadrlaterals. Thus, the 1 For nterpretaton of color n Fgs. 2, 6, 14, 15, 21, 24, and 27 29, the reader s referred to the web verson of ths artcle. In the second method for connectng the dscrete and contnuum elements (as shown n Fg. 15b), the horzontal beam s dscretzed nto beam elements wth nodes that concde wth the nodes of the quadrlateral mesh. Consequently, the translatonal degrees of freedom of both beam and quadrlateral elements are shared throughout the beam s length (shown n red). Thus, the quadrlateral elements are constraned to move jontly wth the beam elements when the frame deforms. Ths behavor s llustrated n the sketch shown n Fg. 17. Note that the connecton between beams and columns n a structural steel frame can be desgned for varous degrees of moment transfer (.e. shear connectons, flexble moment connectons, moment connectons (see Fg. 18), whch correspond to varous rotatonal stffness levels for the connecton. The nfluence of the connecton stffness on the topology optmzaton results s studed later through numercal examples Beam and enrched (drllng) quadrlateral elements attached contnuously along beam lne Blnear quadrlateral (Q4) elements behave poorly n n-plane bendng; however, ncluson of addtonal drllng degrees of freedom allows the enrched elements (Q4D4), llustrated by Fg. 19, to perform better than the four-node quadrlateral elements (Q4) whle usng less degrees of freedom than the eght-node quadrlateral (Q8) [27]. Here, the two translatons at the mddle nodes n the Q8 are converted to one rotaton at each corner n the Q4D4 element (see Fg. 19). The equatons for the drllng degrees of freedom can be derved from the basc Q8 formulaton [27] as follows: u m ¼ 1 u þ 1 u j þ h j h y j y ð35þ v m 2 v 2 v j 8 x x j where u and v are the horzontal and vertcal translatons at node, h s the pseudo-rotaton at node, m represents a md-span node, and and j are the corner nodes of the element. Though the addtonal drllng degrees of freedom allow the quadrlateral elements to rotate or bend wth the beam elements along the beam lne, as observed n Fg. 20, no sgnfcant nfluence has been observed on the optmzaton (complance) results. Ths behavor can be explaned by observng that the addtonal rotatons provded by the drllng degrees of freedom capture only a local bendng effect. Therefore, standard Q4 elements are suffcently accurate to represent the structural behavor of the examples shown n ths paper snce the translatonal behavor s domnant. Note also that the Q4D4 mplementaton s computatonally more expensve than that usng ordnary Q4 elements. 5. Topology optmzaton formulaton The ntegraton of beam and quadrlateral elements descrbed n the prevous secton can be ncorporated nto the classcal topology optmzaton formulaton by ntroducng a few modfcatons as descrbed below.

12 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 15. Connecton types for beam and quadrlateral fnte elements: (a) attached at global beam ends, (b) attached at all concurrent mesh nodes, and (c) attached at all concurrent mesh nodes wth enrched (drllng) Q4 elements Problem statement Despte several objectves (tp dsplacement, frequency, bucklng, complance, etc.) n topology optmzaton, n ths work we choose to maxmze the overall stffness of a buldng; thereby, mnmum complance s used as the objectve functon of the optmzaton. The desgn doman of the buldngs consdered n ths paper s the outer skn or shell. The optmal layout n terms of mnmum complance can be stated n terms of the densty, q, and the dsplacements, u, as follows: mn q;u cðq; uþ s:t: KðqÞu ¼ f Z q dv 6 V s X qðxþ 2½0; 1Š 8x 2 X ð36þ

13 118 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Beams rotate ndependently Quadrlateral nodes free to move ndependently Fg. 16. Smple moment frame demonstratng a sample dsplacement feld where beam elements rotate ndependently of the nodal translatons of the quadrlateral elements along the beam lne. Quadrlateralstranslate wth beams fracton constrant whch represents the maxmum volume of materal permtted for the desgn of the structure, and q s the materal densty for each desgn varable where q ¼ 0 sgnfes a vod and q ¼ 1 represents sold materal. The ll-posedness of the topology optmzaton problem, or lack of a soluton n the contnuum settng [28 30], can be overcome through relaxaton. A contnuous varaton of densty n the range ½q mn ; 1Š s appled n relaxaton rather than restrctng each densty to an nteger value of 0 or 1 thereby guaranteeng the exstence of a soluton. A small parameter greater than zero, q mn, s specfed to avod sngulartes of the global stffness matrx, KðqÞ. The topology optmzaton problem s solved by means of the SIMP model [31,32,12,33], however, other materal models may be used, such as the Ratonal Approxmaton of Materal Propertes (RAMP) [34,13]. In the SIMP formulaton, a power-law relaton between the stffness and element densty s ntroduced n the form: EðxÞ ¼qðxÞ p E 0 ð37þ where E 0 descrbes Young s modulus of the sold materal and p s a penalzaton parameter wth p P 1. Ths formulaton prescrbes that the materal propertes contnuously depend on the materal densty at each pont. The penalzaton parameter, p, forces the materal densty towards 0 or 1 (vod or sold respectvely) as opposed to formng regons of ntermedate denstes (gray zones) where q assumes a value somewhere between 0 and 1. The optmzaton procedure presented n ths work uses contnuaton, where the penalzaton parameter, p, s ncreased over the range of 1 to 4, n ncrements of 0.5 untl convergence at each value s acheved Projecton methodology wth contnuum and dscrete elements Fg. 17. Example of a dsplacement feld wth beam and quadrlateral elements attached contnuously along beam lne. In the above equatons c represents the overall complance of the structure whle KðqÞ represents the global stffness matrx whch depends on the materal denstes, u and f are the vectors of nodal dsplacements and forces, respectvely, V s s the volume To avod the common problem of checkerboardng over the quadrlateral mesh, a projecton technque, smlar to that of Guest et al. [35], was mplemented. In addton to elmnatng the checkerboardng patterns, projecton s used as a means to specfy the mnmum member sze (characterstc length) n a structure. The projecton method n ths work was performed only over the quadrlateral mesh snce the dscrete members already have a gven cross-sectonal area. Moreover, the presence of the beam or column elements should have no nfluence over the topology optmzaton of the bracng members snce they are already members as Pnned connecton Sem - rgd connecton Rgd connecton Inflecton pont Inflecton pont Inflecton pont Fg. 18. Analytcal representaton of the beam to column connectons of varous stffness and correspondng moment dagrams. From left to rght: shear connecton, flexble moment connecton, moment connecton.

14 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 19. Addton of drllng degrees of freedom usng the md-sde dsplacements: (a) 8-node quadrlateral element (Q8) and (b) 4-node quadrlateral element wth addtonal rotatons (Q4D4). Quadrlaterals translate and rotate wth beams 6. Dscusson on fnte element modelng assumptons In ths secton, the behavor of a sngle module of a hgh-rse buldng s studed to understand some fundamental modelng aspects that arse from combnng dscrete and contnuum elements. Furthermore, we dscuss how varous modelng assumptons affect the fnal topology of the lateral bracng system for a hgh-rse buldng. For the followng problems, we assume an equal heght and wdth of 10 ft wth W1030 steel columns (as specfed). For the topology optmzaton problem, the volume fracton s 30% wth a projecton radus of 6 n Influence of pont load applcaton n the context of symmetry Fg. 20. Example of a dsplacement feld wth enrched (drllng) quadrlateral elements (Q4D4). The applcaton of the symmetry constrant s studed for the load case n Fg. 22. If only one load s appled at the top left corner of the mesh and symmetry s enforced, the top member s crucal to transfer the load to the column on the far sde. Ths concluson holds when the mesh has quadrlateral elements only (Fg. 23a), as well as when beam elements are ntroduced for the columns (Fg. 22a). Wth the applcaton of two loads (one at each top corner n the same drecton) there s no need for the horzontal member to transfer the load to the far sded column. Therefore, such members dsappear from the topology optmzaton layout (Fg. 22b and 23b). Moreover, n Fg. 23, column elements were absent from the mesh and the resultng K-brace n Fg. 23a shows almost dsappearng columns. Ths s consstent wth the statc equlbrum at the node llustrated n Fg. 24 (left). Smlarly, the result n Fg. 23b s consstent wth the free body dagram n Fg. 24 (rght). Moreover, n the presence of vertcal loads the column members would always be requred to transfer such loads from the buldng structure to ts foundaton. In the numercal examples n ths paper, the symmetry condton was appled usng the schematc llustrated n Fg. 22b snce the lateral load consdered s a wnd load whch has a wndward and a leeward component Effect of beam to quadrlateral element connecton on the optmal topology Fg. 21. Mnmum length scale for projecton technque over quadrlateral elements; beam and column elements that le wthn the radus have no effect on contnuum topology optmzaton. llustrated n Fg. 21. The optmalty crtera [36] s used for the optmzaton process. The effect of the varous beam to quadrlateral element connectons as descrbed n Secton 4 are nvestgated n Fg. 25. Fg. 25a corresponds to the stuaton descrbed n Fg. 16 where only the extreme beam ends (black nodes) are attached to the Q4 mesh. Snce the nodes are unattached along the beam lne and the column lne, and the moment s transferred from the beam to the column, the bracng developed stffens the moment frame. Fg. 25b corresponds to the stuaton descrbed n Fg. 17 where the beam and column dsplacements are ted to those of the

15 120 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Zero force member Force transferred to both members Fg. 24. Effect of asymmetrc versus ant-symmetrc load applcaton on column elements: (left) K-brace develops zero-force columns ; (rght) presence of forces n column members of hgh-wasted brace. Fg. 22. Influence of symmetry constrant on mesh wth 6400 quadrlateral (Q4) elements and 2 beam (W1030) elements: (a) sngle pont load wth symmetry appled to the optmzaton and (b) ant-symmetrc pont loads wth symmetrc result. Fg. 23. Influence of symmetry constrant on mesh wth 6400 quadrlateral (Q4) elements (no columns): (a) sngle pont load wth symmetry appled to the optmzaton and (b) ant-symmetrc pont loads wth symmetrc result. quadrlaterals (at the black nodes). Thus, the optmal bracng engages the frame at ntermedate workng ponts along the lengths of the beams and columns. Fg. 25c represents a stuaton smlar to Fg. 25b, the only dfference beng the moment release (drawn as a hollow crcle) at the extreme ends of the beam. As a consequence, the moment s no longer transfered between the beam and column and a stffenng pattern for the corners develops. In order to evaluate the soluton wth the best structural performance, we consder the fnal complance of the three frames n Fg. 25: (a) , (b) , and (c) Snce the volume of materal s the same for all the frames, from an engneerng standpont, f no other constrants are present, the best performng frame would be the braced frame n Fg. 25a Influence of the column stffness on the bracng layout The effect of varyng the column area whle keepng the contnuum mesh unchanged subject to ant-symmetrc pont loads at the top corners (see Fg. 22b) s demonstrated n Fg. 26. The dmensons of ths module are taken to be 48 m by 41:5 m. An ant-symmetrc pont loadng of P ¼ 2 MN s appled to the top corners. The area, A 0, of the column elements n Fg. 26a were szed to acheve a unform stress n accordance wth the condtons of optmalty for complance descrbed n the prevous sectons. Thus, we select an area of A 0 ¼ 0:0021 sq m for the column elements, a thckness of t ¼ 0:002 m for the Q4 elements and E ¼ 200; 000 MPa (steel). For the topology optmzaton, a volume fracton of 20% s used wth a projecton radus of r mn ¼ 3m. As the area, A, s ncreased from the optmal area, A 0, the ntersecton of the cross-brace (workng pont) moves vertcally downward towards the 45 bracng soluton. Fg. 26 shows the mportance of proper szng of the columns to obtan the theoretcal optmal soluton. Correspondngly, the proportons between the radus of gyraton of the columns and the overall wdth of the doman would also be of nfluence to the bracng pont,.e. the hgher ths rato, the lower the bracng pont. However, n practce the columns would frst be szed for gravty loads and later desgned for lateral loads. Therefore, the column area may be hgher than the optmal area, and furthermore demonstrate a lower work pont than the 75% soluton Verfcaton of the constant stress condton As descrbed earler n Secton 3.1, the constant stress condton s verfed n the contnuum approach for the prevous structure n Fg. 27 (left) whch was derved usng a Q4 element mesh. As shown n Fg. 27 (rght), the Von-Mses stresses are nearly constant wthn each optmzed member. 7. Optmal braced frames numercal results Numercal applcatons of the methodology developed n ths paper are presented n ths secton for the case of a

16 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 25. Effect of dfferent beam to quadrlateral connecton types as presented n Secton 4: (a) attached at global beam ends wth mesh of 6400 Q4 and 3 W1030 beam elements, (b) attached at all concurrent mesh nodes wth mesh of 6400 Q4 and 120 W1030 beam elements, and (c) attached at all concurrent mesh nodes wth moment release for mesh of 6400 Q4 and 240W1030 beam elements. Nodes shown n black ndcate beam to quadrlateral connecton. Fg. 26. Effect of varyng the stffness of the column elements on the optmal bracng layout wth mesh of 6400 Q4 elements and two beam elements: (a) A ¼ A 0, (b) A ¼ 2A 0, (c) A ¼ 5A 0, and (d) A ¼ 10A 0. Intersecton of bracng approaches 50% of heght as column area s ncreased from the optmal soluton. two-dmensonal hgh-rse buldng frame. Such examples portray several features of optmal frames that were descrbed n the prevous sectons. Frst, the problem gven n Hemp [26] s solved usng the combned approach n Fg. 28. In ths problem, the overall dmensons of the structure are gven as H ¼ 276 m by 2B ¼ 41:5 m. The loadng consdered s a lateral load of P ¼ 1000 kn appled at the top center wth a symmetry constrant across the y-axs. Ths structure s assumed to be made of steel, wth E ¼ 200 GPa. Usng the vrtual work methodology to satsfy the drft lmt requrements as descrbed n Secton 2 and assumng constant stress as descrbed n Secton 3.1 appled to the analytcal soluton for 11 dagonals based on the truss geometry of Fg. 12, the total volume of the structure was computed to be 48 m 3, where the columns account for 35 m 3 of ths value and the bracng accounts for 13 m 3. The column szes establshed usng the analytcal soluton were carred over to the numercal soluton. Usng the dscrete/contnuum element combnaton, the topology optmzaton problem s run wth contnuaton on the penalzaton from p ¼ 1 to 4 n steps of 0:5 wth a projecton radus of r mn ¼ 3. As seen n Fg. 28b the thck areas of materal are no longer concentrated at the edges of the doman. Furthermore, the braces are now complete and clearly defned and the fnal geometry produces the same angles as shown n the benchmark example of Fg. 11. Moreover, the resultng dagonal members are equal n sze and the stresses are nearly constant throughout the heght. As was stated prevously, for ntermedate denstes the Von Mses stresses wll be constant snce the stran energy s constant. The regons where the stresses are hgher (or lower) are when the denstes are at the endponts of the [0,1] range (.e. a densty of 1 gves a hgher stress and a densty of 0 gves a lower stress than the constant). Ths example verfes the numercal methodology to dentfy the optmal bracng layout. Next, n reference to the results shown n the ntroducton, we study the addton of dscrete truss elements (columns) from Fg. 1c n more detal here. In ths problem, the overall dmensons of the structure are gven as H ¼ 288 m by 2B ¼ 41:5 m. The loadng consdered s a lateral load of P ¼ 2000 kn appled at each module wth a symmetry constrant across the y-axs. Ths structure s assumed to be made of steel, wth E ¼ 200 GPa. Usng the vrtual work methodology to satsfy the drft lmt requrements as descrbed n Secton 2 and assumng constant stress as descrbed n Secton 3.1 appled to the analytcal soluton for sx modules based on the truss geometry of Fg. 12, the total volume of the structure was computed to be 240 m 3, where the columns account for 35% of ths value and the bracng accounts for 20%. The column

17 122 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 27. Topology optmzaton of a frame usng 6400 quadrlateral elements (left) and correspondng plot of Von Mses stresses (rght). Fg. 28. Topology optmzaton for braced frame gven n Hemp [26]: (a) problem statement, (b) topology optmzaton result, and (c) stress dstrbuton. szes establshed usng the analytcal soluton were carred over to the numercal soluton. Smlarly, the volume fracton for the topology optmzaton problem was then taken to be 20% of sold materal. The results of the topology optmzaton problem wth the combned approach usng contnuaton and a volume fracton of 20% wth a projecton radus of r mn ¼ 3 are shown n Fg. 29b. Smlar to the prevous example, the thck areas of materal are no longer

18 L.L. Stromberg et al. / Engneerng Structures 37 (2012) Fg. 29. Topology optmzaton for braced frame: (a) problem statement, (b) two-dmensonal result, (c) stress dstrbuton, and (d) three-dmensonal renderng of result. concentrated at the edges of the doman and the braces are complete and clearly defned. One nterestng feature of ths result shows the denstes ncrease for the bracng members as the load ncreases throughout the heght ndcatng that the szes of the fnal members should ncrease accordngly. The Von Mses stresses for ths geometry featurng modules of the same heght are plotted n Fg. 29c, whch are not constant throughout the structure due to the ncreasng denstes along the heght. In Fg. 29d a threedmensonal renderng of ths result s gven to show how these fndngs mght be used to desgn a hgh-rse buldng. 8. Concludng remarks The methodology presented n ths work for developng a lateral braced frame system n a hgh-rse buldng enables the structural engneer to quckly and effcently dentfy the optmal dagonal layout. In summary, the man contrbutons of ths work are as follows: Several methodologes to connect dscrete and contnuum elements were explored. A technque was proposed for the desgn of an optmal braced frame system. The constant state of stress n an optmzed frame under certan condtons was verfed. The relevance of ths new methodology n the context of hghrse buldng mechancs was demonstrated. The optmal geometry for a braced frame was analytcally derved and numercally confrmed. As an extenson of the work presented n ths paper, the use of shell elements wth three-dmensonal beam elements for large structural systems s currently under exploraton by the authors. Acknowledgment The frst author gratefully acknowledges the support from the Natonal Scence Foundaton Graduate Research Fellowshp Program (GRFP). References [1] Stromberg LL, Beghn A, Baker WF, Paulno GH. Applcaton of layout and topology optmzaton usng pattern gradaton for the conceptual desgn of buldngs. Struct Multdsc Optm 2011;43(2): [2] Allahdadan S, Boroomand B. Desgn and retrofttng of structures under transent dynamc loads by a topology optmzaton scheme. In: Proc 3rd Int Conf Sesm Retroft, October, p [3] Neves MM, Rodrgues H, Guedes JM. Generalzed topology desgn of structures wth a bucklng load crteron. Struct Multdsc Optm 1995;10(2):71 8. [4] Huang X, Xe YM. Topology optmzaton of nonlnear structures under dsplacement loadng. Eng Struct 2008;30(7): [5] Huang C, Han X, Wang C, J J, L W. Parametrc analyss and smplfed calculatng method for dagonal grd structural system. Janzhu Jegou Xuebao/ J Buld Struct 2010;31(1):70 7. [6] Prager W. Optmal layout of trusses wth fnte number of jonts. J Mech Phys Solds 1978;26(4): [7] Prager W. Nearly optmal desgn of trusses. Comput Struct 1978;8(3 4): [8] Mchell AGM. The lmts of economy of materal n frame-structures. Phl Mag 1904;8(47): [9] Ben-Tal A, Nemrovsk A. Robust optmzaton G methodology and applcatons. Math Program 2002;92(3):453. [10] Mazurek A, Baker WF, Tort C. Geometrcal aspects of optmum truss lke structures. Struct Multdsc Optm 2011;43(2):