Structural Engineering and Mechanics

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Multi-objective topology and geometry optimization of statically determinate beams

  • Received : 2017.02.26
  • Accepted : 2019.03.04
  • Published : 2019.05.10
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Abstract

The paper concerns topology and geometry optimization of statically determinate beams with arbitrary number of supports. The optimization problem is treated as a bi-criteria one, with the objectives of minimizing the absolute maximum bending moment and the maximum deflection for a uniform gravity load. The problem is formulated and solved using the Pareto optimality concept and the lexicographic ordering of the objectives. The non-dominated sorting genetic algorithm NSGA-II and the local search method are used for the optimization in the Pareto sense, whereas the genetic algorithm and the exhaustive search method for the lexicographic optimization. Trade-offs between objectives are examined and sets of Pareto-optimal solutions are provided for different topologies. Lexicographically optimal beams are found assuming that the maximum moment is a more important criterion. Exact formulas for locations and values of the maximum deflection are given for all lexicographically optimal beams of any topology and any number of supports. Topologies with lexicographically optimal geometries are classified into equivalence classes, and specific features of these classes are discussed. A qualitative principle of the division of topologies equivalent in terms of the maximum moment into topologies better and worse in terms of the maximum deflection is found.

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References

  1. Biro, I. and Cveticanin, L. (2016), "Procedures for determination of elastic curve of simply and multiple supported beams", Struct. Eng. Mech., 60(1), 21-30. https://doi.org/10.12989/sem.2016.60.1.021
  2. Branke, J., Deb, K., Miettinen, K. and Slowinski, R. (2008), "Multiobjective optimization, Interactive and Evolutionary Approaches", Lecture Notes in Computer Science, Springer-Verlag, Berlin Heidelberg, Germany.
  3. Coello Coello, C.A.C., Veldhuizen, D.A. and Lamont, G.B. (2002), Evolutionary Algorithms for Solving Multi-Objective Problems, Springer Science & Business Media, New York, NY, USA.
  4. Dems, K. and Turant, J. (1997), "Sensitivity analysis and optimal design of elastic hinges and supports in beam and frame structures", Mech. Struct. Mach., 25(4), 417-443. https://doi.org/10.1080/08905459708905297
  5. Ehrgott, M. (2005), Multicriteria Optimization, Springer, Berlin Heidelberg, Germany.
  6. Imam, M.H. and Al-Shihri, M. (1996), "Optimum topology of structural supports", Comput. Struct., 61(1), 147-154. https://doi.org/10.1016/0045-7949(96)00087-9
  7. Jang, G.W., Shim, H.S. and Kim, Y.Y. (2009), "Optimization of support locations of beam and plate structures under selfweight by using a sprung structure model", J. Mech. Design, 131(2), 0210051-02100511.
  8. Kaveh, A., Kalateh-Ahani, M. and Fahimi-Farzam, M. (2013), "Constructability optimal design of reinforced concrete retaining walls using a multi-objective genetic algorithm", Struct. Eng. Mech., 47(2), 227-245. https://doi.org/10.12989/sem.2013.47.2.227
  9. Kozikowska, A. (2011), "Topological classes of statically determinate beams with arbitrary number of supports", J. Theor. Appl. Mech., 49(4), 1079-1100.
  10. Kozikowska, A. (2014), "Topological classes of statically determinate beams with arbitrary number of supports under the most unfavourably distributed load", J. Theor. Appl. Mech., 52(1), 257-269.
  11. Kozikowska, A. (2016), "Geometry and topology optimization of statically determinate beams under fixed and most unfavorably distributed load", Lat. Am. J. Solids Stru., 13(4), 775-795. https://doi.org/10.1590/1679-78252306
  12. Loureiro, D., Loja, M.A.R. and Silva, T.A.N. (2014), "Optimal location of supports in beam structures using genetic algorithms", 6th World Congress on Nature and Biologically Inspired Computing, NaBIC, Porto, Portugal, October.
  13. Munk, D.J., Vio G.A. and Steven G.P. (2015), "Topology and shape optimization methods using evolutionary algorithms: A review", Struct. Multidisc. Optim., 52, 613-631. https://doi.org/10.1007/s00158-015-1261-9
  14. Ostwald, M. and Rodak, M. (2013), "Multicriteria optimization of cold-formed thin-walled beams with generalized open shape under different loads", Thin Wall. Struct., 65, 26-33. https://doi.org/10.1016/j.tws.2013.01.002
  15. Parmee, I.C. (2001), Evolutionary and Adaptive Computing in Engineering Design, Springer-Verlag, London, United Kingdom.
  16. Rychter, Z. and Kozikowska, A. (2009), "Genetic algorithm for topology optimization of statically determinate beams", Arch. Civ. Eng., 55(1), 103-123.
  17. Wang, D. (2004), "Optimization of support positions to minimize the maximal deflection of structures", Int. J. Solids Struct., 41, 7445-7458. https://doi.org/10.1016/j.ijsolstr.2004.05.035
  18. Wang, D. (2006), "Optimal design of structural support positions for minimizing maximal bending moment", Finite Elem. Anal. Des., 43, 95-102. https://doi.org/10.1016/j.finel.2006.07.004