Abstract
Lurie (1994, 1995a, b) proved recently that variabletopology shape optimization of perforated plates in flexure for non-selfadjoint problems leads to rank-2 microstructures which are in general nonorthogonal. An extension of the same optimal microstructures to perforated plates in plane stress will be presented in Part II of this study. Using the above microstructure, the optimal solution is derived in this part for cantilever plates in plane stress, which are subject to two displacement constraints. For low volume fractions the above solutions are shown to converge to the known truss solutions of Birkeret al. (1994). The problem of homogenizing the stiffness of nonorthogonal rank-2 microstructures is also discussed.
This is a preview of subscription content, access via your institution.
References
Bendsøe, M.P. 1995:Optimization of structural topology, shape and material. Berlin, Heidelberg, New York: Springer
Birker, T.; Lewiński, T.; Rozvany, G.I.N. 1994: A well-posed nonselfadjoint layout problem: least-weight plane truss for one load condition and two displacement constraints.Struct. Optim. 8, 195–205
Gibiansky, L.V.; Cherkaev, A.V. 1984: Design of composite plates of extremal stiffness. Peport 914, A.F. Ioffe Inst., Leningrad (in Russian)
Jog, C.; Haber, R.B.; Bendsøe, M. 1994: Topology design with optimized, self-adaptive materials.Int. J. Meth. Engrg. 37, 1323–1350
Kohn, R.V.; Strang, G. 1986: Optimal design and relaxation of variational problems. I, II and III.Comm. Pure Appl. Math. 39, 113–137, 139–182, 353–377
Lurie, K.A. 1994: Direct relaxation of optimal layout problems for plates.JOTA 80, 93–116
Lurie, K.A. 1995a: On direct relaxation of optimal material design problems for plates.WSSIAA 5, 271–296
Lurie, K.A. 1995b: An optimal plate. In: Olhoff, N.; Rozvany, G.I.N. (eds.)Proc. First World Cong. on Structural and Multidisciplinary Optimization, pp. 169–176. Oxford: Pergamon
Lurie, K.A.; Cherkaev, A.V.; Fedorov, A.V. 1982: Regularization of optimal design problems for bars and plates. Parts I–III.JOTA 37, 499–522, 523–543;42, 247–282
Ong, T.G. 1987:Structural optimization via static-kinematic optimality criteria. Ph.D. Thesis, Monash University, Melbourne
Ong, T.G.; Rozvany, G.I.N.; Szeto, W.T. 1988: Least-weight design of perforated elastic plates for given compliance: nonzero Poisson's ratio.Comp. Meth. Appl. Mech. Eng. 66, 301–322
Rozvany, G.I.N. 1989:Structural design via optimality criteria. Dordrecht: Kluwer
Rozvany, G.I.N.; Olhoff, N.; Bendsøe, M.P.; Ong, T.G.; Szeto, W.T. 1985/87: Least-weight design of perforated elastic plates.DCAMM Report No. 306. Techn. Univ. Denmark, Lyngby; also:Int. J. Solids Struct. 23, 521–536, 537–550
Saint Venant, B. de 1955:Mémoires des savants étrangers, Vol. 14
Szeto, W.T. 1989:Microstructure studies in structural optimization. Ph.D. Thesis, Monash University, Melbourne
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Károlyi, G., Rozvany, G.I.N. Exact analytical solutions for non-selfadjoint variable-topology shape optimization problems: perforated cantilever plates in plane stress subject to displacement constraints. Part I. Structural Optimization 13, 119–127 (1997). https://doi.org/10.1007/BF01199230
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01199230