The Computational Generation and Realization of Spatial Truss Structures

Transcription

1 1238 The Computatonal Generaton and Realzaton of Spatal Truss Structures X. Z. Zhao 1, R. H. Wang 2 and K. Shea 3 1 College of Cvl Engneerng, Tongj Unversty, Shangha, Chna, ; emal: x.zhao@tongj.edu.cn 2 School of Cvl Engneerng and Archtecture, Jangsu Unversty of Scence and Technology, Zhenjang, Jangsu, Chna, ; emal: wrhchna@163.com 3 Engneerng Desgn and Computng Laboratory CLA F, ETH Zurch, 8092 Zurch, Swtzerland; emal: kshea@ethz.ch ABSTRACT Computatonal methods have been successfully appled n structural analyss and drawngs of large spatal structures. However, there are stll huge challenges n performng conceptual desgn of structures usng computers as what human bran s able to do. A computatonal method, named performance-based structural synthess has been proposed and valdated to generate and realze planar truss structures. However, t s nsuffcent for more general spatal truss structures whose geometres and load-carryng requrements are more complex compared wth those of planar truss. Therefore, ths computatonal synthess method based on planar shape grammar was extended and presented n ths paper to generate three-dmensonal spatal truss structures through buldng 3D shape grammars and mult-objectve hybrdzed optmzaton algorthm. Meanwhle technques of load-generaton were employed n the structural synthess method to realze ts engneerng applcaton. In the aspect of structural confguraton, the method ntegrates structural topology, geometry and secton sze allocaton; n the aspect of structural desgn and optmzaton, t takes nto account the combnng effects of safety, economy, constructon and archtectural elegance. Thus the structure generated by ths synthess method must be the best desgn n terms of the overall structural performance and cost compared wth that obtaned by tradtonal desgn method. INTRODUCTION The applcaton of computer n conceptual desgn of spatal truss structures remans a huge challenge, even though t has been successfully appled to drawngs and structural analyss n detal desgn. Ths chasm between the two stages s partally due to the extremely complcated soluton wth mult-objectve requrements such as

2 1239 safety, economy and elegance n conceptual desgn, and more mportantly to the lack of suffcent method to numercally descrbe the desgn process. To provde nnovatve alternatves for conceptual desgn of spatal truss structures, a computatonal method has been proposed based on the performance-based synthess method STSA whch has successfully generated planar truss and truss-beam(shea, 1997; Shea and Zhao, 2004). Formal algebra and polyhedron synthess theory of structural morphology have been used to descrbe regular structural form for conventonal frame structures (Nooshn and Dsney, 2000), but the generaton based on those approaches pays more attenton to structural form rather than structural functon. Whle the computatonal desgn synthess (Cagan, et al., 2005) based on shape grammars (Stny and Gps, 1972) ntroduces a new desgn dea to merge the structural form wth structural functon together. Based on analogy between knowledge applcaton of structural desgn and the formng process of human language, a set of shape transformng rules ncludng shape, szng and topology rules were defned as rule syntaxes of desgn language, and used to realze shape composton to descrbe the structural form. And then the computer mplementaton system based on shape grammars nstructs machne execute routne tasks and promote creatve actvtes of the desgner (Chau, et al., 2004), whle more complex factors affectng the structural desgn can be consdered n such synthess system. The performance-based structural synthess research has exhbted that shape annealng (Cagan and Mtchell, 1993) ncorporatng shape grammar and smulated annealng s suffcent to generate and realze nnovatve planar truss (Shea, et al., 1997), dome (Shea and Cagan, 1997), and cantlever spatal truss beam (Shea and Zhao, 2004). In addton, a recently publshed shape annealng study on planar truss structure dentfed such method as an optmzng tool to desgn a full-scale transmsson tower (Shea and Smth, 2006). These nsghts establsh a compellng ratonale to target the computatonal generaton for general spatal truss structures. In the aspect of optmzng algorthm, we recently proved the feasblty of targetng computatonal generaton of spatal truss structures usng mproved shape annealng (ISA) (Wang, et al., 2009) that hybrdzes shape annealng wth fully stressed desgn crtera (FSD) (Patnak and Hopkns, 1998) whch s an effcent mechancal method for szng optmzaton of truss structure wth fxed confguraton. In the aspect of topologcal generaton of spatal truss, 3D shape grammars n addton to planar shape grammars has been explored through addng bracngs, and applyng load-generaton technque for dstrbuted loads ncludng wnd load and the weght of exteror-protected structures. These results ndcate that the computatonal method for spatal truss based on the mproved shape annealng, 3D shape grammars as well as load-generaton technque can successfully produce nnovatve desgns ntegratng structural functon and confguraton.

3 1240 SHAPE ANNEALING ALGORITHM FOR TRUSS GENERATION Introducton of STSA for truss generaton (Shea and Smth, 2006). Shape annealng was frst appled to produce optmally drected shapes usng the language specfed by shape grammar (Cagan and Mtchell, 1993). And then that was ntroduced to desgn planar truss structure usng shape grammars and optmzng algorthm of modfed SA (Reddy and Cagan, 1995). Recently, t was renamed structural topology and shape annealng (STSA) (Shea and Smth, 2006), whch employed more effcent annealng schedule of modfed Lam-Delosm annealng schedule n addton to mproved shape grammars. Optmally drected search of the resultng desgn space s accomplshed through the modfed annealng schedule, where the acceptng rate s defned as the number of accepted moves out of total moves over a fxed statstcal nterval. Once one shape rule has been chosen randomly based on the probablty functon of selectng rule, t s appled to the current desgn at every run. A new desgn wll be created and be evaluated by mult-crteron objectve functon that conssts of a mult-objectve functon and dynamcally weghted constrant volaton penaltes as shown n Eq.1. Snce each move to the current desgn may result n large dsturbance n the performance and evaluaton of the objectve functon, t may not always generate better desgn than the prevous one. Thus whether to accept the move or not s determned n accordance wth the Metropols crtera based on the acceptng probablty. m cost = ( objectve_ weght * objectve_ value ). (1) + = 1 n j = 1 ( constrant _weght j * constant_ volaton Normally, the mult-objectve functon s the weghted summaton of the mass and a penalty of constrants. Yet the cost functon may exst n some dfferent patterns altered wth specal problem. Penalty factor facltates tacklng varant constrants such as stress, bucklng and dsplacement constrants. Note that all constrants are soft constrants whch could be volated as the desgn progresses, but annealng schedule attempts to push the constrant volatons to zero at the end of the teraton process whle mnmzng the objectve cost. Improved STSA algorthm. Compared wth tradtonal desgn method, STSA s capable of generatng nnovatve and effcent alternatve desgn, but emphaszes manly structural confguraton rather than structural performance. Actually the poneers (Reddy and Cagan, 1995) of shape annealng had ponted out such character that the grammar contrbutes to ts neffcency, so that one shape rule applcaton does not always make meanngful change for the desgn. To overcome the dsadvantage descrbed above, FSD s hybrdzed wth STSA to make the heurstc searchng algorthm consder structural mechancal j )

4 1241 behavor, and establshed an ISA algorthm (Wang, et al., 2009). Studes show that FSD can be gnted when relatvely stable topology s obtaned n desgn, where topology rules neffcently modfy the desgn whle szng rules are most effcent than other rules. But the pont s how to confrm that the topology s relatvely stable. The effcency of change to the desgn modfed by each knd of rules can be utlzed to confrm the stablty, n accordance wth the qualty that s the rato of the summaton of cost change varaton to the summaton of the rule-appled numbers as shown n Eq. 2a. Note that the qualty of each type of rules should be evaluated at the end of the outer teraton and before the startng of the next teraton., ( = g / s / t), (2a) Δcost j q j = 1 = j = 1 R Q q g + q + s j q, ( g / s / t) = q t =. (2b) Here, the qualty, q, ndcates the mpact of the th type of grammar rules appled on the tentatve desgns. R j s the number of applcaton of the j th rule wthn g/s/t whch are the abbrevatons for geometry, szng and topology rules respectvely. The fnal qualty, Q, represents the relatve qualty of a certan type of rule compared wth that of other type of rules. Thus the value obtaned va Eq. 2b manfests the real effcency of one type of grammar rules. In our experence, the current canddate s relatvely stable once three types of relatve qualtes comply wth the relaton as Q g < Q s < Q t. 3D SHAPE GRAMMAR EXTENDED BY PLANAR SHAPE GRAMMAR As far as the confguraton of spatal truss beam system s concerned, the basc unt of that s not the trangle but the pyramd. The generaton of spatal truss structure can refer to the transformng approach of planar truss structure, as planar grammar can be appled to descrbe the confguraton of spatal truss beam through sutable geometrcal transformaton and addng bracngs as shown n Fgure 1(Shea and Zhao, 2004). Fgure 1. Synthess of spatal truss beam wth planar truss and bracng. Therefore, 3D shape grammars were proposed n the paper to replace pyramd by extendng planar shape grammar wth bracngs for the descrpton the structural form of spatal truss beam system. Then shape grammars of those can be expressed as planar shape grammars + bracngs, as shown n Fgure 2. In Fgure 2, the brace-n

5 1242 s the nner bracng to generate stable spatal truss beam derved from twn planar trusses va approprate geometrcal transformng, and brace-out s the outer bracng between two stable spatal truss beams. Fgure 2. Shape grammars of spatal truss beam system. LOAD-GENERATION TECHNIQUES FOR DISTRIBUTED LOAD Not only wnd load but also the weght of exteror-protected constructon should be transferred to the load-resstng system. For dome structures, the dstrbuted loads are convenent transform to pont loads as ther forms are entrely composed of trangle shapes; for spatal truss beam system produced by 3D shape grammars, those can also be tackled nto pont loads through approprate geometrcal segmentaton as Fgure 3. Fgure 3. Segmentaton of dstrbuted load. The regon subjected to dstrbuted load can be decomposed frstly nto trangles as ΔABC and ΔADC. Every decomposed trangle can be re-segmented nto three mn trangles as ΔABO, ΔBCO and ΔCAO. Subsequently, the load on every mn trangle s assgned to the nearest component n the shape. Fnally, the assgned loads of all components wll be processed as pont loads to ponts A, B and C respectvely. Note that every pont accomplshes load allocaton durng addng bracngs between two twn planar trusses and between two spatal truss beams. ILLUSTRATION EXAMPLES Spatal dome structure wth multple load condtons A spatal dome (Shea, 1997) s adopted to verfy the load-generaton technque for dstrbuted load and ISA. The desgn space s defned as an ellpsod wth 30 m dameter span and 9.25 m heght. The materal s alumnum wth elastc modulus of 7.1E 6 N/cm 2, allowable tensle and compressve stresses of 2.1E 4 N/cm 2, mass densty of 2.7E -3 kg/cm 3. The desgn objectve s to maxmze the utlty of envelope space whle to mnmze the surface area, under three load condtons as

6 1243 followng: 1. concentrated force of 3.0E 5 N at the center pont of dome and gravty of members; 2. load condton 1 plus wnd load of 3.6E -2 N/cm 2 drectly actng on members wthout exteror-protect structure; 3. load condton 1 plus wnd load of 3.6E -2 N/cm 2 ndrectly actng on members wth exteror-protect structure. After ten runs, the optmal confguraton of three load condtons are shown as Fgure 4 respectvely, lstng the statstc numercal results as Table 1. condton-1 condton-2 condton-3 Fgure 4. Optmal confguraton of dome under three load condtons. Table 1. Statstcal results of multple load condtons. load condton mean weght max weght mn weght standard devaton The average weght of the ten runs s 4644 kg wthout wnd load. Compared wth that wthout wnd load, load condton 2 wth wnd load drectly actng on the members wthout envelope structure ncreased about 15%; whle the average weght ncreased about 19%for load condton 3. The results demonstrate that wnd load has crucal effect on dome structures. Spatal truss beam system. To valdate the computatonal synthess method proposed based on 3D shape grammars and ISA, a spatal truss beam wth a span of 54 m was explored frst. And then t was used to establsh ntal topology for a complex spatal truss beam system, whose man sze derved from the Shangha Stadum n Chna wth approprate smplfcaton through rotatng and scalng transformaton. The two structures use same steel materal wth an elastc modulus of 2.1E 7 N/cm 2, allowable tensle and compressve stresses of 3.1E 4 N/cm 2, and mass densty of 7.85E -3 kg/cm 3. Eght concentrated loads of 9.53E 5 N act on eght fxed ponts of the upper chord. The constrants nclude stress and Euler bucklng. In terms of spatal truss beam, after ten runs, two representatve optmal desgns are selected to llustrate the capacty for conceptual desgn of mproved synthess system, as shown n Fgure 5. The bow-shaped confguratons of the two optmal desgns are n agreement wth the nner moment dstrbuton. The spatal truss beam system consstng of 32 spatal truss beams wth

7 1244 bracng system has symmetrcal confguraton, and ths character facltates the generaton of such structure. The generaton of such complex system undergoes two steps, one s that a par of twn planar trusses construct the frst basc spatal truss beam through geometrcally rotatng and bracng brace-n, the other s that a seres of spatal truss beams consttute the spatal truss beam system through scalng the basc truss beam n accordance wth ther locatons n desgn space and bracng brace-out between two spatal truss beams, as shown n Fgure 6. No.1 No.2 Fgure 5. Generaton of optmal spatal truss beam. ntal desgn fnal desgn Fgure 6. Generaton of spatal truss beam system. Here, only one annealng run was mplemented, but that s enough to acheve the prmary purpose of ths paper, whch s to explore a feasble method to generate computatonally the general spatal truss structures. Although there s stll constrants volaton of Euler bucklng due to the fxed number of moves n annealng schedule (Reddy and Cagan, 1995), the desgn has been obvously mproved compared wth ntal desgn. Our experence fnds that the fnal desgn can be further refned f a larger number of moves are utlzed n the annealng schedule. CONCLUSION Wth the mprovement n optmzng algorthm, shape grammar and load-generaton technque, the mproved computatonal synthess method proposed n the paper accomplshes two complcated desgn tasks, structural form and functon, n conceptual desgn for spatal truss structure. The form of spatal truss structures derved s more regular and meanwhle more ratonal for carryng loads. Thus the structure generated by ths synthess method must be the best desgn n terms of the overall structural performance and cost compared wth that obtaned by tradtonal desgn method.

8 1245 ACKNOWLEDGEMENT The authors wsh to thank the Natonal Natural Scence Foundaton of Chna under grant and Natural Scence Foundaton of Jangsu Provnce of Chna under grant BK for supportng ths work. REFERENCES Cagan, J., Campbell, M. I., Fnger, S., and Tomyama, T. (2005). "A framework for computatonal desgn synthess: Model and applcatons." J. Comput. Inf. Sc. Eng., 5(3), Cagan, J., and Mtchell, W. J. (1993). "Optmally drected shape generaton by shape annealng." Envron. Plannng B, 20, 5-5. Chau, H. H., Chen, X. J., McKay, A., and de Pennngton, A. (2004). "Evaluaton of a 3d shape grammar mplementaton." Proceedngs of Desgn Computng and Cognton, Cambrdge, MA. Nooshn, H., and Dsney, P. (2000). "Formex confguraton processng." Int. J. Space Struct., 15(1), Patnak, S. N., and Hopkns, D. A. (1998). "Optmalty of a fully stressed desgn." Comput Method Appl M, 165(1-4), Reddy, G., and Cagan, J. (1995). "An mproved shape annealng algorthm for truss topology generaton." J Mech Desgn, 117(3), Reddy, G. M., and Cagan, J. (1995). "Optmally drected truss topology generaton usng shape annealng." J Mech Desgn, 117(2), Shea, K. (1997). "Essays of dscrete structure: Purposeful desgn of grammatcal structure by drected stochastc search." Dssertaton, Carnege Mellon Unversty. Shea, K., and Cagan, J. (1997). "Innovatve dome desgn: Applyng geodesc patterns wth shape annealng." Artf. Intell. Eng. Des. Anal. Manuf., 11(5), Shea, K., Cagan, J., and Fenves, S. J. (1997). "A shape annealng approach to optmal truss desgn wth dynamc groupng of members." J Mech Desgn, 119(3), Shea, K., and Smth, I. F. C. (2006). "Improvng full-scale transmsson tower desgn through topology and shape optmzaton." J. Struct. Eng., 132(5), Shea, K., and Zhao, X. Z. (2004). "A novel noon mark cantlever support: From desgn generaton to realzaton." Internatonal Assocaton for Shell and Spatal Structures, Montpeller, France. Stny, G., and Gps, J. (1972). "Shape grammars and the generatve specfcaton of pantng and sculpture." Proceedngs of IFIP Congress, Amsterdam. Wang, R. H., Zhao, X. Z., and Xe, B. Y. (2009). "A novel hybrd algorthm for the topology optmzaton of truss structure." Internatonal Conference on Engneerng Computaton (ICEC), Hong Kong, Chna.