Computational layout design optimization of frame structures

Size: px

Start display at page:

Download "Computational layout design optimization of frame structures"

Phyllis Gaines
6 years ago
Views:

Weldeyesus c, Helen Farclough b * Department of Archtecture & Cvl Engneerng, Unversty of Bath, Bath, UK. j.ye@bath.ac.

1 Interfaces: archtecture.engneerng.scence 5-8th September, 017, Hamburg, Germany Annette Bögle, Manfred Grohmann (eds.) Computatonal layout desgn optmzaton of frame structures Jun Ye*, Paul Shepherd a, Lnwe He b, Matthew Glbert b, Buck Davson b, Andy Tyas b, Jacek Gondzo c, Alemseged G. Weldeyesus c, Helen Farclough b * Department of Archtecture & Cvl Engneerng, Unversty of Bath, Bath, UK. j.ye@bath.ac.uk a Department of Archtecture and Cvl Engneerng, Unversty of Bath b Department of Cvl & Structural Engneerng, Unversty of Sheffeld c School of Mathematcs, Unversty of Ednburgh Abstract Engneers often expend consderable effort dentfyng the most effcent cross-secton szes for the ndvdual structural members formng a structure, but may neglect to check whether members are optmally postoned, or are even needed at all. Ths can lead to far more materal beng used to form a buldng structure than s needed. To address ths, layout optmzaton can potentally be used early n the desgn process to dentfy effcent arrangements of structural members. Ths paper ntroduces an nteractve desgn approach that combnes parametrc modellng and layout optmzaton, usng an adaptve member addng technque to allow large scale problems to be solved on a standard desktop PC. Incorporaton of the approach n Rhno-Grasshopper allows ntegraton of geometrc modelng and structural layout optmzaton wthn a sngle nteractve modelng envronment. Ths paper brefly outlnes the underlyng theory and mplementaton detals, and then descrbes applcaton of the approach to benchmark problems and a case study problem, a three-centred space frame arch roof. In ths case t s shown that a 30% reducton n materal usage can potentally be acheved through the use of a layout optmzaton-based approach, but that measures to mprove the practcalty of the solutons for use n practce are requred. Ths s beng addressed as part of a new collaboratve research project nvolvng the Unverstes of Bath, Sheffeld and Ednburgh. Keywords: Layout optmzaton, Grasshopper, parametrc desgn, buldng structures 1. Introducton Space frames are wdely used for long-span structures due to ther nherent lght weght, hgh stffness and constructonal effcency, ncludng off-ste manufacture potental. Desgners generally need to decde very early on n the desgn process the layout of the consttuent structural members, long before detaled szng of members has been undertaken. Once a gven layout s chosen t becomes dffcult and expensve to make changes subsequently, snce many other aspects of the desgn depend on t (such as column postons, façade desgn, roof claddng, etc. It s therefore mportant that practcal tools are made avalable to help the desgner quckly and effectvely dentfy effcent layouts of structural members. Although the theory behnd ground-structure based numercal layout optmzaton has been understood for more than half a century, soluton technques have tended to be computatonally expensve, lmtng ts applcaton n practce. Thus, conceptual desgns for structural forms such as long-span roofs or hgh-rse buldngs have tended to be developed based on a desgner s ntuton. However, nowadays there are demands for ever more complex buldng geometres, and t s normally not obvous what the most effcent structural layout mght be. Thus relyng purely on ntuton s unlkely to lead to the most effcent desgns beng dentfed. There s therefore a need for practcal tools Copyrght 017 by Ye, Shepherd, He, Glbert, Davson, Gondzo, Weldeyesus & Farclough Publshed by the Internatonal Assocaton for Shell and Spatal Structures (IASS) wth permsson.

2 Interfaces: archtecture.engneerng.scence whch can quckly and effcently dentfy an effcent structural layout for a gven set of practcal desgn constrants. Ths paper ntroduces a new plugn for Grasshopper whch can dentfy optmal structural layouts usng the adaptve layout optmzaton algorthm developed by Glbert and Tyas [1]. The approach s capable of takng nto account practcal desgn consderatons such as multple load cases and jont costs. After demonstratng ts applcaton to benchmark problems, the tool s appled to a long-span roof structure, hghlghtng the potental savngs that can be made n terms of materal usage.. Numercal formulaton In order to optmze large-scale truss structures, an effcent layout optmzaton formulaton s requred. Both elastc and plastc formulatons have been explored by the research communty. The elastc formulaton, whch seeks to mnmze complance of a structure, nvolves the use of a stffness matrx. However, durng the optmzaton process the stffness matrx may become sngular f members n the structure have cross-sectonal areas whch approach zero []. In contrast the plastc layout optmzaton formulaton nstead nvolves the use of a force equlbrum matrx and does not suffer from ths problem. The plastc layout optmzaton formulaton has been found effectve to acheve an optmum soluton and has been extended to deal wth problems nvolvng multple load cases, self-weght, and member bucklng [3]. Ths latter formulaton wll therefore be used here..1. The plastc layout optmzaton formulaton Consder a 3-dmentonal pn-jonted truss structure comprsng n nodes, m axally loaded members and M load cases. In ths case the plastc layout optmzaton formulaton desgned to fnd the mnmum volume of materal that satsfes force equlbrum and lmtng stress crtera can be wrtten as follows: subject to: T mn V l a (1) Bq f, 1,...,M () q q a 1,..., M ; 1,..., m, 0 q q a 0 T T where V s the total volume of structural materal, l,,... and a,,... l1 l l m a1 a a m (3) are the vectors of lengths and cross-sectonal areas of the bars, respectvely. B s the nodal equlbrum matrx T of sze 3n m q q, q, q, q,..., q, q, where q, q are respectvely the tensle and, 1 1 m m compressve forces n member. It should be noted that for any actve member n the optmum structure, the nternal force can be ether tensle or compressve (.e. only one value of q, q wll be f s the nodal force vector., non-zero). T = f x 1, f y 1, f z 1,..., f x, y, z,..., x, y, z f f fn fn fn respectvely the lmtng tensle and compressve stresses, whlst s used to denote the ndex of M load cases. are

3 Interfaces: archtecture.engneerng.scence Ths s a lnear programmng problem that can be solved effcently usng modern nteror pont solvers. For a gven nodal dscretzaton a globally optmal soluton s guaranteed. The adaptve member addng technque [1] can be used to sgnfcantly mprove computatonal effcency, wth no effect on the soluton obtaned... Transmssble loads In the standard optmzaton formulaton presented n Secton.1, the nodal loads n the vector f T of equaton () requre that there must be a node present at ther pont of applcaton. Ths means that the locaton of the pont of applcaton of the load(s) has a strong nfluence on the resultng optmal layout. To mtgate ths, so-called transmssble loads can be ntroduced. In ths case a load s appled to a group of nodes along a vertcal lne of acton nstead of to one, and the load s dstrbuted optmally wthn ths group n order to reduce the volume of materal requred. More detals on the transmssble load formulaton and ts applcaton can be found n papers by Glbert et al. [4] and Darwch et al. [5]. 3. Implementaton The formulaton descrbed n secton above has been made avalable by the Unversty of Sheffeld n a plugn for the Rhnoceros-based geometrc modelng tool Grasshopper [6], provdng an nteractve parametrc modellng envronment for users. An example of the components n use s shown n Fgure 1, where they can be grouped nto the followng types: (a) Geometry Defnton. The geometry of the desgn doman s defned usng standard Grasshopper components so as to provde a parametrc workflow. Users are free to defne a desgn space n terms of lnes, polygons, NURBS surfaces and complex BReps. The geometry s then meshed to faces and vertces as nput for the desgn doman. A number of bespoke components have been provded to ad ths process for layout optmzaton. (b) Desgn. A number of components are then used to assgn materal propertes such as yeld stress, Young s modulus, and materal densty. The support and load condtons need to be defned, wth multple load cases specfed as necessary. (c) Layout Optmzaton. Ths component performs the layout optmzaton and provdes dagnostc nformaton on the tme and number of teratons needed to solve. The volume of materal requred n the structure s also reported. (d) Vsualzaton. A component s provded to vsually present the resultng members, jont postons, nternal forces and cross-sectonal areas. 4. Example problems To demonstrate the effectveness of the Grasshopper optmzaton tool presented above, a range of example problems wll now be consdered, startng wth standard benchmark problems and movng on to the desgn optmzaton of a three-centered arch space frame roof structure. The optmzaton problems were solved usng lnear programmng and the adaptve member addng procedure was used n the optmzaton process. Wth respect to the optmzaton solvers employed, all problems were solved usng the MOSEK nteror pont solver [7]. The problems were run on a 64-bt Wndows machne wth 3.4GHz Intel CPU and 16GB of nstalled memory. 3

Interfaces: archtecture.engneerng.scence Fgure 1: Grasshopper layout optmzaton plugn: nputs and components 4.1. Benchmark problems The frst benchmark problem presented here s the cantlever truss prevously consdered by Hemp [8].

A more accurate value for the volume was calculated more recently by Tomasz Lewnsk; V 4.3168 Pl / (personal communcaton).

4 Interfaces: archtecture.engneerng.scence Fgure 1: Grasshopper layout optmzaton plugn: nputs and components 4.1. Benchmark problems The frst benchmark problem presented here s the cantlever truss prevously consdered by Hemp [8]. Hemp solved the problem analytcally and also provded an approxmate value for the volume of V 4.34 Pl /, where P s the magntude of the pont load, l s the span and s the lmtng stress, and where. A more accurate value for the volume was calculated more recently by Tomasz Lewnsk; V Pl / (personal communcaton). Usng a 4141 node dscretzaton ( potental members), as shown n Fgure (a), and takng P and as unty, a numercal soluton of V 4.343l was acheved, whch s wthn 0.5% of the exact analytcal value. The resultng layout s shown n Fgure (b). Note that the same soluton s also obtaned when usng a transmssble load, wth P appled to the group of nodes along the rght hand sde of the desgn doman. Now consder the case when the lmtng tensle stress s ncreased to 3 tmes that of the lmtng compressve stress. Wth the pont load agan appled to the node at the md-heght of the doman, a volume of.8643 was obtaned. It can be seen from Fgure (c) that the compressve members have larger cross-sectonal areas compared to the tensle members. However, when a transmssble load was used n ths case the load was found to move downwards to effectvely reduce the lengths of the compressve members, as shown n Fgures (d). The volume was reduced by 1.6% to.819. It can s evdent from Fgure that the resultng trusses contan numerous jonts and are therefore mpractcal for use n practce, at least when conventonal fabrcaton technques are employed. To address ths two potental ratonalzaton technques have recently been consdered by He and Glbert [9]; here the use of a smple jont cost penalty s used. Ths entals addng a fxed length to each member to account for the expense of fabrcatng jonts, effectvely penalzng short members. Fgure 3 shows the effect of ncludng jont lengths n the cantlever problem. Ths shows that for an ncrease n volume of a few percent, the complexty s sgnfcantly reduced. 4

stress, V 4.343l ; (c) as (b) but wth unequal lmtng stress 3, =.

(a) (b) (c) Fgure 3: Cantlever problem wth jont lengths of: (a) 0.05 ( V 4.477 ); (b) 0.

Barrel-Vault Roof The next problem nvolves the redesgn of a long span three-centered

Such structures are commonly used as coal storage structures n Chna.

of a stacker / reclamer, as shown n Fgure 4.

quadratc pyramd cells wth a sze of 4.35m 4m.

5 Interfaces: archtecture.engneerng.scence (a) (b) (c) (d) Fgure : Benchmark D cantlever problem: (a) desgn doman; (b) equal stress, V 4.343l ; (c) as (b) but wth unequal lmtng stress 3, =.8643 ; (d) as (c) but wth transmssble load, =.819. (a) (b) (c) Fgure 3: Cantlever problem wth jont lengths of: (a) 0.05 ( V ); (b) 0.1 ( V 4.58 ); (c) 0.0 ( V ) 4. Barrel-Vault Roof The next problem nvolves the redesgn of a long span three-centered arch space frame roof structure. Such structures are commonly used as coal storage structures n Chna. A key desgn requrement s to ensure enough space for storage of the coal and the operaton of a stacker / reclamer, as shown n Fgure 4. The orgnal three-centered roof desgn used a grd structural form comprsng repeatng quadratc pyramd cells wth a sze of 4.35m 4m. The outer radus and the central angle of the mddle arc are R 98.m and =51 1, respectvely. The two sde arcs have an outer radus r 7.8 m and a central angle of = 55. The total heght and span of the roof are 3.6m and 11.3m respectvely, whle the depth of the grd s 4m. The standard values of loads for the roof desgn are: statc load D 0. kn / m, lve load L 0.5 kn / m, snow load S 0.3 kn / m and the basc wnd pressure s k k k 5

6 Interfaces: archtecture.engneerng.scence Wk 0.44 kn / m. The terran category s A accordng to the orgnal desgn and the Chnese Standard for actons, whle the pressure coeffcents were determned from wnd tunnel tests and are shown n Fgure 4. The self-weght of the structure used n the optmsaton s taken as Gk 0.4 kn / m. Accordng to the orgnal desgn, the controllng load case combnatons are: Q1 1.( Gk Dk ) 1.4Lk (5) Q 1.( Gk Dk ) 1.4Lk 0.84W k (6) Assumng wnd load from both the left and rght drectons, there are three controllng load cases. The steel usage n the orgnal desgn was 47 kg / m ; for a 100m long buldng the total volume of steel requred was 67.37m 3. Fgure 4: Secton through the orgnal desgn of a three-centered space-frame arch roof for coal storage and the wnd pressure coeffcents used n the desgn 4..1 D roof optmzaton It was ntally assumed that the roof structure comprsed a seres of D plane trusses desgned for the 113.m 3.6m D doman wth a longtudnal length of 1m. The desgn doman was then dscretzed usng a very coarse dscretzaton of 73 nodes, dstrbuted randomly, leadng to 3718 potental members. A grasshopper component was developed to convert surface pressures to pont loads for use n the optmzaton. The computed optmum materal volume V 1 and desgns are shown n Table 1 and Fgure 5. The computatonal tme for each optmzaton was less than 10s, whch shows the effcency of plane truss optmzaton. Usng jont lengths of 0.1m and 0.m led to a reducton n the number of 6

7 Interfaces: archtecture.engneerng.scence bars (by 53.4% and 56.1% respectvely) and number of jonts (by 34.3% and 38.6% respectvely), wth only modest ncreases n volume (0.3% and 0.55% respectvely). Table 1: Comparson between optmum solutons and CPU cost for D layout optmzaton of the roof Jont length (m) V 1 (m 3 ) Numbers of bars Numbers of jonts CPU tme* (s) * Includes the tme to setup the problem and complete the optmzaton process (a) (b) (c) Fgure 5: D layout optmzaton soluton of the roof usng dfferent jont lengths: (a) desgn doman; (b) jont length = 0m; (c) jont length = 0.1m; (d) jont cost = 0.m (the layouts are not symmetrcal partly due to the use of a randomly generated nodal grd) (d) 4.. 3D roof optmzaton A 3D desgn optmzaton was also performed. In ths case boundary surfaces were meshed usng 4m 4m quadrlateral facets, resultng n 096 vertces. Nodes were agan randomly dstrbuted across the desgn doman. Table shows the optmum volume wth and wthout transmssble load (V 1 ). In addton, the volume requred to also wthstand Eurocode 3 [10] bucklng requrements has also been calculated (V ). The member sectons used for the desgn are taken from the lst of standard crcular hollow sectons presented n [11]. Table shows that the mnmum volume of materal s about 6.1m 3, approx. 100 tmes the plane truss optmzaton case as expected (snce a longtudnal of 100m rather than 1m was used). However, the ntal optmzaton neglected bucklng and to address ths the volume of materal requred needed to be ncreased by approx. 9 tmes n a post-processng step; more effectve ways of treatng bucklng are currently under development. However, the volumes are stll less than that of the orgnal desgn presented n Fgure 5 (dependng upon the number of nodes and jont length used, the savng ranges from 5.9%-0.6% n the case of a fxed poston pressure load, and 6.9%-33.9% when transmssble loads are used). Other random dstrbutons of nodes were tred, but the results were found not to vary markedly, ndcatng that the specfc dstrbuton does not have a major nfluence on the outcome obtaned. 7

8 Interfaces: archtecture.engneerng.scence Table : Optmum solutons for 3D layout optmzaton of the roof wth and wthout transmssble loads. (Volume V 1 and V are respectvely the raw volume and the volume after bucklng has been taken account of.) Number of randomly dstrbuted nodes Jont length (m) Volume V 1 (m 3 ) Wthout transmssble loads Numbers of bars Numbers of jonts CPU tme* (mn) Volume V (m 3 ) Volume V 1 (m 3 ) Wth transmssble loads Numbers of bars Numbers of jonts CPU tme* (mn) Volume V (m 3 ) * Includes the tme to setup the problem and complete the optmzaton process Form-fndng of the coal storage roof The three-centered space frame truss was ntally desgned usng a form comprsng three arcs, as shown n Fgure 5(a). However, ths leads to a severely constraned desgn doman. Therefore the optmzaton tool was also used wth a less constraned doman, shown n Fgure 6 (where the extent of the doman s ndcated by red lnes). Ths doman was generated by subtractng a 18.6m hgh trapezod, representng the clear space requred for storage and operatonal space, from a rectangular desgn doman. Supports were defned along the base of the doman. The vertcal load combnaton shown n equaton (5) was dstrbuted horzontally as pont loads every 4m, shared amongst vertcal columns of nodes usng the transmssble load formulaton. Two cases are consdered here: one wth a 3.6m hgh rectangular desgn doman (as per the orgnal desgn) and another wth a 50m hgh doman, to leave the flexblty to dscover a more effcent form. A nomnal jont length was ntroduced to smplfy the resultng desgns, whch are shown n Fgure 7. Fgure 6: Revsed desgn doman for roof, usng transmssble loads 8

Interfaces: archtecture.engneerng.scence (a) (b) Fgure 7: Form-fndng solutons for roof structure: (a) doman heght H=3.6m (b) doman heght H=50m (wth jont length = 0.

level. However, the layout changes markedly when the doman heght s ncreased to 50m, whch leads to a parabolc form beng dentfed.

9 Interfaces: archtecture.engneerng.scence (a) (b) Fgure 7: Form-fndng solutons for roof structure: (a) doman heght H=3.6m (b) doman heght H=50m (wth jont length = 0.1m n both cases) Wth the heght of desgn doman as per the orgnal desgn, the optmum layout resembles a plane truss n the md-span regon, wth nclned members at the edges takng the forces down to ground level. However, the layout changes markedly when the doman heght s ncreased to 50m, whch leads to a parabolc form beng dentfed. In both cases t s evdent that ratonalzaton of the solutons would be requred n order for these to be realsed n practce. (Also, the use of addtonal load cases would be requred to ensure robustness.) 5. Conclusons Ths paper has brefly outlned how numercal layout optmzaton can be used at the conceptual desgn stage to dentfy structurally effcent layouts. A desgn approach that combnes parametrc modellng and layout optmzaton, usng an adaptve member addng technque, has been developed to allow large scale problems to be solved on a standard desktop PC. Incorporaton of the approach n Rhno-Grasshopper allows ntegraton of geometrc modelng and structural layout optmzaton wthn a sngle nteractve modelng envronment. It has been found that ncludng jont lengths n the optmzaton can sgnfcantly smplfy the resultng layout, wth only a small ncrease n materal consumpton. Also, for the practcal case-study problem consdered, t was found that materal savngs of up to approx. one thrd could be acheved; savngs were greatest when transmssble loads were used. However, consderable further work s requred n order to enable the full potental of the layout optmzaton technque to be realsed. For example, there s a need to ncorporate bucklng n the formulaton n a computatonally effcent manner, and to take account of practcal consderatons such as specfed mnmum ntersecton angles between members at jonts. These ssues are currently beng addressed as part of a new collaboratve research project nvolvng the Unverstes of Bath, Sheffeld and Ednburgh. 9

10 Interfaces: archtecture.engneerng.scence Acknowledgements Ths work descrbed s supported by Engneerng and Physcal Scences Research Councl (EPSRC) grants EP/N0369/1, EP/N03471/1, and EP/N01965/1. References [1] Glbert M, Tyas A. Layout optmzaton of large-scale pn-jonted frames. Engneerng Computatons. 003;0: [] Rozvany GIN. On desgn-dependent constrants and sngular topologes. Structural and Multdscplnary Optmzaton. 001;1: [3] Darwch W. Novel computatonal mplementatons for ultmate lmt state analyss and desgn. PhD thess; The Unversty of Sheffeld; 010. [4] Glbert M, Darwch W, Tyas A, Shepherd P. Applcaton of large-scale layout optmzaton technques n structural engneerng practce. 6th World Congresses of Structural and Multdscplnary Optmzaton [5] Darwch W, Glbert M, Tyas A. Optmum structure to carry a unform load between pnned supports. Structural and Multdscplnary Optmzaton. 010;4:33-4. [6] McNeel R. Grasshopper generatve modelng for Rhno. Computer software (011b), [7] MOSEK A. Denmark (016). The MOSEK Optmzaton Tools Manual Verson 7.1. [8] Hemp W. Optmum Structures. Clarendon Press, Oxford [9] He L, Glbert M. Ratonalzaton of trusses generated va layout optmzaton. Structural and Multdscplnary Optmzaton. 015;5: [10] CEN. Eurocode 3: Desgn of Steel Structures. Part 1-1: General Rules and Rules for Buldngs. Brussels: European Comttee for Standardzaton; 005. [11] Cold Formed Crcular Hollow Sectons to BS EN 1019-: 006. URL: 10

Design of Structure Optimization with APDL

Design of Structure Optimization with APDL Desgn of Structure Optmzaton wth APDL Yanyun School of Cvl Engneerng and Archtecture, East Chna Jaotong Unversty Nanchang 330013 Chna Abstract In ths paper, the desgn process of structure optmzaton wth