Advertisement

Structural optimization

, Volume 7, Issue 3, pp 160–169 | Cite as

A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory

  • G. E. Stavroulakis
  • M. A. Tzaferopoulos
Technical Papers

Abstract

The theory of optimal plastic design of structures via optimality criteria (W. Prager approach) transforms the optimal design problem into a certain nonlinear elastic structural analysis problem with appropriate stress-strain laws, which are derived by the adopted specific cost function for the members of the structure and which generally have complete vertical branches. Moreover, the concept of structural universe (introduced by G.I.N. Rozvany) permits us to tackle complicated optimal layout problems.

On the other hand, a significant effort in the field of nonsmooth mechanics has recently been devoted to the solution of structural analysis problems with “complete” material and boundary laws, e.g. stress-strain laws or reaction-displacement laws with vertical branches.

In this paper, the problem of optimal plastic design and layout of structures following the approach of Prager-Rozvany is revised within the framework of recent progress in the area of nonsmooth structural analysis and it is treated by means of techniques primarily developed for the solution of inequality mechanics problems. The problem of the optimal layout of trusses is used here as a model problem. The introduction of general convex, continuous and piecewise linear specific cost functions for the structural members leads to the formulation of linear variational inequalities or equivalent piecewise linear, convex but nonsmooth optimization problems. An algorithm exploiting the particular structure of the minimization problem is then described for the numerical solution. Thus, practical structural optimization problems of large size can be treated. Finally, numerical examples illustrate the applicability and the advantages of the method.

Keywords

Variational Inequality Piecewise Linear Layout Problem Optimal Layout Optimal Design Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Achtziger, W.; Bendsøe, M.; Ben-Tal, A.; Zowe, J. 1991: Equivalent displacement based formulations for maximum strength truss topology design.Rep. No. 338. Schwerpunktprogramm der DFG, Anwendungsbezogene Optimierung und Steuerung, Universität BayreuthGoogle Scholar
  2. Allaire, G; Kohn, R.V. 1993: Topology optimization and optimal shape design using homogenization. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 207–218. Dordrecht: KluwerGoogle Scholar
  3. Antes, H.; Panagiotopoulos, P.D. 1992:An integral equation approach to the static and dynamic contact problems. Equality and inequality methods. Boston: BirkhäuserGoogle Scholar
  4. Delfour, M.C. (ed.) 1992:Shape optimization and free boundaries. NATO ASI Series C 380. Dordrecht: KluwerGoogle Scholar
  5. Aubin, J.-P.; Ekeland, I. 1984:Applied nonlinear analysis. Chicester: John Wiley and SonsGoogle Scholar
  6. Bendsøe, M.P.; Ben-Tal, A.; Zowe, J. 1993: Optimization methods for truss geometry and topology design.Rep. No. 449. Schwerpunktprogramm der DFG, Anwendungsbezogene Optimierung und Steuerung, Universität BayreuthGoogle Scholar
  7. Ben-Tal, A.; Bendsøe, M.P. 1993: A new method for optimal truss topology design.SIAM J. Optimization 3, 323–358Google Scholar
  8. Clarke, F.H. 1975: Generalized gradients and applications.Trans. Amer. Math. Soc. 205, 247–262Google Scholar
  9. Clarke, F.H. 1983:Optimization and nonsmooth analysis. New York: John Wiley & SonsGoogle Scholar
  10. Dorn, W.S.; Gomory, R.E.; Greenberg, H.J. 1964: Automatic design of optimal structures.J. Mécan. 3, 25–165Google Scholar
  11. Drucker, D.C.; Prager, W. 1952: Soil mechanics and plastic analysis or limit design.Quart. Appl. Math. X, 157–165Google Scholar
  12. Duvaut, G; Lions, J.L. 1972:Les inéquations en méchanique et en physique. Paris: Dunod (English translation 1970:Inequalities in mechanics and physics. Vienna: Springer)Google Scholar
  13. Fletcher, R. 1987:Practical optimization methods (2nd edition) New York: John Wiley & SonsGoogle Scholar
  14. Fremond, M. 1988: Yield theory in physics. In: Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G. (eds.)Topics in nonsmooth mechanics, pp. 187–240. Boston: BirkhäuserGoogle Scholar
  15. Glowinski, R.; Lions, J.L.; Tremolieres, R. 1981:Numerical analysis of variational inequalities. Studies in Mathematics and its Applications 8, Amsterdam, New York: North-Holland, ElsevierGoogle Scholar
  16. Hill, R.H.; Rozvany, G.I.N. 1985: Prager's layout theory: a nonnumeric computer method for generating optimal structural configurations and weight influence surfaces.Comp. Meth. Appl. Engrg. 49, 131–148Google Scholar
  17. Hlavaček, I.; Haslinger, J.; Nečas, J.; Loviček, J. 1988:Solution of variational inequalities in mechanics. Appl. Math. Sci. 66. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  18. Höfler, A.; Leyßner, U.; Wiedemann, J. 1973: Optimization of the layout of trusses combining strategies based on Michell's theorem and the biological principles of evolution.AGARD Conf. Proc. 123, A.1-A.8Google Scholar
  19. Jog, G.; Haber, R.; Bendsøe, M.P. 1993: A displacement-based topology design method with self-adaptive layered materials. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 219–238. Dordrecht: KluwerGoogle Scholar
  20. Koltsakis, E.K. 1991:Theoretical and numerical study of structures with nonmonotone boundary conditions. Applications in glued joints. Ph.D. Thesis, Aristotle University, ThessalonikiGoogle Scholar
  21. Lagache, J.M. 1980: A geometrical procedure to design trusses in a given area.Eng. Opt. 5, 1–12Google Scholar
  22. Maier, G. 1977: Limit design in the absence of a given layout: a finite element zero—one programming problem. In: Gallagher, R.H.; Zienkiewicz, O.C. (eds.):Optimum structural design/theory and applications, pp. 223–239. New York: John Wiley & SonsGoogle Scholar
  23. Mandel, J. 1978:Propriétés mécaniques des matériaux/réologie — plasticité. Paris: EyrollesGoogle Scholar
  24. Michell, A.G.M. 1904: The limits of economy of material in framestructures.Phil. Mag. 8, 589–597Google Scholar
  25. Mistakidis, E. 1992:Theoretical and numerical study of structures with nonmonotone boundary and material laws. Algorithms and applications. Ph.D. Thesis, Aristotle University, ThessalonikiGoogle Scholar
  26. Moreau, J.J.; Panagiotopoulos, P.D. (eds.) 1988:Nonsmooth mechanics and applications. CISM Lecture Notes 302, pp. 82–176. Vienna, New York: SpringerGoogle Scholar
  27. Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G. (eds.) 1988:Topics in nonsmooth mechanics. Boston: BirkhäuserGoogle Scholar
  28. Panagiotopoulos, P.D. 1976: Convex analysis and unilateral static problems.Ing. Arch. 45, 55–68Google Scholar
  29. Panagiotopoulos, P.D. 1983: Nonconvex energy functions. Hemivariational inequalities and substationarity principles.Acta Mech. 42, 160–183Google Scholar
  30. Panagiotopoulos, P.D. 1985:Inequality problems in mechanics and applications/convex and nonconvex energy functionals. Boston: Birkhäuser (Russian translation by MIR Publishers, Moscow, 1989)Google Scholar
  31. Panagiotopoulos, P.D. 1988a: Nonconvex superpotentials and hemivariational inequalities. Quasi—differentiability in mechanics. In: Moreau, J.J.; Panagiotopoulos, P.D. (eds.)Nonsmooth mechanics and applications, pp. 82–176. Wien: SpringerGoogle Scholar
  32. Panagiotopoulos, P.D. 1988b: Hemivariational inequalities and their applications. In: Moreau, J.J.; Panagiotopoulos, P.D.; Strang, G. (eds.)Topics in nonsmooth mechanics, pp. 75–141. Basel: BirkhäuserGoogle Scholar
  33. Panagiotopoulos, P.D. 1993:Hemivariational inequalities and their applications in mechanics and engineering. Berlin, Heidelberg, New York: SpringerGoogle Scholar
  34. Pedersen, P. 1970: On the minimum mass layout of trusses.AGARD Conf. Proc. 36, 11.1–11.17Google Scholar
  35. Pedersen, P. 1973: Optimal joint positions for space structures.AGARD Conf. Proc. 123, 12.1–12.14Google Scholar
  36. Pedersen, P. 1993: Topology optimization of three dimensional trusses. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology optimization of structures, pp. 19–30. Dordrecht: KluwerGoogle Scholar
  37. Prager, W. 1973: Necessary and sufficient conditions for global structural optimality.AGARD Conf. Proc. 123, 1.1–1.12Google Scholar
  38. Prager, W.; Shield, R.T. 1967: A general theory of optimal plastic design.J. Appl. Mech. 34, 184–186Google Scholar
  39. Rockafellar, R.T. 1970:Convex analysis. Princeton: Princeton PressGoogle Scholar
  40. Rozvany, G.I.N. 1976:Optimal design of flexural systems: beams, grillages, slabs, plates and shells. New York: Pergamon PressGoogle Scholar
  41. Rozvany, G.I.N. 1981a: Variational methods and optimality criteria. In: Haug, E.J.; Cea, J. (eds.)Optimization of distributed parameter systems, Volume 2, pp. 82–111. Alphen aan den Rijn: Sijthoff and NoordhoffGoogle Scholar
  42. Rozvany, G.I.N. 1981b: Optimal criteria for grids shells and arches. In: Haug, E.J.; Cea, J. (eds.)Optimization of distributed parameter systems, Volume 2, pp. 112–151. Alphen aan den Rijn: Sijthoff and NoordhoffGoogle Scholar
  43. Rozvany, G.I.N. 1985: Generalization of Heyman's and Foulkes' theorems using dual formulations.Int. J. Mech. Sci. 27, 347–360Google Scholar
  44. Rozvany, G.I.N. 1989:Structural design via optimality criteria/the Prager approach to structural optimization. Dordrecht: KluwerGoogle Scholar
  45. Rozvany, G.I.N. 1992: Optimal layout theory: analytical solutions for elastic structures with several deflection constraints and load conditions.Struct. Optim. 4, 247–249Google Scholar
  46. Rozvany, G.I.N.; Hill, R.H. 1978: Optimal plastic design: superposition principles and bounds on the minimum cost.Comp. Meth. Appl. Mech. Eng. 13, 151–173Google Scholar
  47. Rozvany, G.I.N.; Zhou, M; Gollub, W. 1990: Continuum-type optimality criteria methods for large finite element systems with a displacement constraint. Part II.Struct. Optim. 2, 77–104Google Scholar
  48. Stavroulakis, G.E. 1991:Analysis of structures with interfaces/formulation and study of variational — hemivariational inequality problems. Ph.D. Thesis, Aristotle University, ThessalonikiGoogle Scholar
  49. Strang, G. 1987: A framework for equilibrium equations.Num. Anal. Rep. 87-4, Dept. Math., MIT, CambridgeGoogle Scholar
  50. Strang, G.; Kohn, R.V. 1985: Hencky-Prandtl nets and constrained Michell trusses.Comp. Meth. Appl. Mech. Engrg. 36, 207–222Google Scholar
  51. Strang, G.; Kohn, R.V. 1986: Optimal design in elasticity and plasticity.Int. J. Num. Meth. Engrg. 22, 183–188Google Scholar
  52. Tzaferopoulos, M.A. 1991:Numerical analysis of structures with monotone and nonmonotone, nonsmooth material laws and boundary conditions: algorithms and applications. Ph.D. Thesis, Aristotle University, ThessalonikiGoogle Scholar
  53. Tzaferopoulos, M.A. 1993: On an efficient new numerical method for the frictional contact problem of structures with convex energy density.Comp. & Struct. 48, 87–106Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • G. E. Stavroulakis
    • 1
  • M. A. Tzaferopoulos
    • 2
  1. 1.Lehr- und Forschungsgebiet für MechanikRWTH AachenAachenGermany
  2. 2.Materials Laboratory, Engineering DepartmentCambridge UniversityCambridgeU.K.

Personalised recommendations