TOPOLOGY DESIGN OF STRUCTURE LOADED BY EARTHQUAKE. Vienna University of Technology

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1 Bluchr Mchanical Enginring Procdings May 2014, vol. 1, num. 1 TOPOLOGY DESIG OF STRUCTURE LOADED BY EARTHQUAKE P. Rosko 1 1 Cntr of Mchanics and Structural Dynamics, Institut for Building Construction and Tchnology, Vinna Univrsity of Tchnology (pr@allmch.tuwin.ac.at) Abstract. Th contribution dals with optimal topology dsign of civil structurs in dynamics. Earthquak loading is considrd. Th arthquak xcitation has multi-frquncy contnt. Th dynamic input is dfind on th bas of Eurocod 8. Th push-ovr curv is applid in dynamic analysis. Th practic in civil nginring rquirs cost minimization and fulfillmnt th safty conditions. Th optimal topology dsign of structur loadd by arthquak is focusd. Th topology problm is dfind as a matrial distribution problm. Dnsitis in discrtizd lmnts of th structur ar th variabls of th optimization. Th rlationship btwn mass and th stiffnss is dfind on th bas of th SIMP mthod. Th objctiv of th structural optimization is th minimum wight of th structur. Th dsign spac is constraind by conditions basd on natural frquncis and on th maximum displacmnt dfind on th push-ovr curv. Kywords: Topology optimization of structurs, Dynamics, Earthquak. 1. ITRODUCTIO Th civil nginring and mchanical nginring practic rquirs solution of topology problms of structurs in dynamics. Th contribution proposs optimal topology dsign of structurs in dynamics loadd by xcitation with multi-frquncy contnt. Th xcitation with multi-frquncy contnt appars in civil nginring during arthquaks, by wind loading, loading inducd by human activitis. In mchanical nginring th multifrquncy xcitation is causd by machins. Th papr focuss th arthquak xcitation. 2. DYAMICS Th dynamic analysis of th structur is rpatd in ach optimization stp. Th nonlinar analysis in tim domain is vry xpnsiv and not accptabl in th optimization procdur. Th frquncy domain nabls th fficint way of th solution. Th rcourss providd in Eurocod 8 [3] ar applid. Th considration of risk is in this approach includd.

2 Th analysis in dynamics consists of following stps: 1. Solution of th ignvalu problm 2 K ρ i M Φ i 0, i = 1,...,. (1) with stiffnss matrix K ρ, mass matrix M ρ, natural frquncy i and mods Φ i. 2. Th psudo-acclration spctrum is from Eurocod 8 [3] slctd. Figur 1. Th psudo-acclration spctrum 3. To th structural priod T o th corrsponding spctral acclration S a is found. 4. Th quivalnt static forc vctor fs is calculatd as follows ([4], [5]): max f s M ΦΓ S. (2) a whr L. (3) m T L M m. (4) m i 1 T M 2 mi i i 1 i i. (5) with static forc vctor f s, mass m, mods, modal participation factor, cofficint vctor L, spctral acclration S a.

3 5. Applying quivalnt static forcs, displacmnts u can b calculatd Figur 2. Equivalnt forcs and displacmnts. 6. Th displacmnt constraint is on th push-ovr curv chckd. Figur 3. Push-ovr curv.

4 1. Solution of th ignvalu problm: atural frquncis and mods of th structur 2. Th psudo-acclration spctrum is from Eurocod 8 slctd 3. To th structural priod th corrsponding spctral acclration is found 4. Th quivalnt forc vctor is calculatd from mass, mods, modal participation factor and spctral acclration 5. Applying quivalnt forcs, displacmnts ar (th rspons) calculatd 6. Th displacmnt constraint is on th push-ovr curv chckd Figur 4. Flowchart of th dynamic analysis. 3. TOPOLOGY OPTIMIZATIO 3.1. Topology problm dfinition Th topology problm is dfind as a matrial distribution problm. Dnsitis discrtizd lmnts () of th structur ar th variabls of th optimization. 0 < 1, = 1,...,. (6) min in

5 Th rlationship btwn mass and th stiffnss is dfind on th bas of th SIMP mthod [1], [2]. M ρ V matr 1 Th objctiv is th minimum wight of th structur. p K ρ K. (7) matr 1 min V v, = 1,...,. (8) ρ 1 Th constraints ar dfind according to topology dnsitis (6) and maximum displacmnts u u PL. (9) 3.2. Optimization procdur Th gntic algorithm as optimization procdur controls th topology dnsity variabls in th optimization procss Exampl Figur 5. Optimal topology.

6 Th 2-D xampl illustrats th prsntd thory. 4. COCLUSIO Th topology dsign of civil structur with arthquak loads was prsntd. Th dynamic loading according Eurocod 8 was considrd. Prsntd procdur can b usd for dynamic loading with multifrquncy contnt: in civil nginring by wind loading, loading inducd by human activitis.and in mchanical nginring by xcitation causd by machins. Acknowldgmnts Th support of Vinna Univrsity of Tchnology is acknowldgd. 5. REFERECES [1] Bndsø M.P., Sigmund O., Topology optimization, Thory, Mthods and Applications, Springr Vrlag, Brlin Hidlbrg, [2] Z Zhou X., Chn L., Z. Huang Z, Th SIMP-SRV Mthod for stiffnss topology optimization of Continuum Structurs, Structural and Multidisciplinary Optimization 21: , [3] Eurocod 8: Dsign of structurs for arthquak rsistanc, [4] Chopra A.K., Dynamics of structurs, 3rd Edition, Prntic Hall, [5] Chopra A.K., Gol, R.K, A modal Pushovr analysis procdur to stimat sismic dmands for buildings: thory and prliminary valuation, Pacific Earthquak Enginring Rsarch Cntr rport CMS , 2001.