Abstract
The paper deals with the conceptual design of a beam under bending. The common problem of designing a beam in a state of pure bending is discussed in the framework of Pareto-optimality theory. The analytical formulation of the Pareto-optimal set is derived by using a procedure based on the reformulation of the Fritz John Pareto-optimality conditions. The shape of the cross section of the beam is defined by a number of design variables pertaining to the optimization process by means of efficiency factors. Such efficiency factors are able to describe the bending properties of any beam cross section and can be used to derive analytical formulae. Design performance is determined by the combination of cross section shape, material and process. Simple expressions for the Pareto-optimal set of a beam of arbitrary cross section shape under bending are derived. This expression can be used at the very early stage of the design to choose a possible cross section shape and material for the beam among optimal solutions.
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Appendix A: I-shaped cross section
Appendix A: I-shaped cross section
A I-shaped cross section (Group 3 in Table 2) steel beam is considered in the following.
The Pareto-optimal solutions in the objective functions domain are reported in Fig. 7. Each optimal solution in the objective functions domain corresponds biunivocally to a single solution A, ϕ e in the design variables domain, as shown in Fig. 6. Being the I-shaped cross section defined by three parameters, namely h, b, t (see Table 2), each Pareto-optimal solution in the design variables domain can be obtained by more than one combination of the cross section parameters. This is clearly shown in Fig. 5. The optimal design solutions with the same level of grey are defined by different vaues of h, b, t, but they have exactly the same performance in terms of mass m and compliance c (and the same value of A and ϕ e).
Pareto-optimal set in the design variables domain for a I shaped steel cross section. Material data in Table 5. Applied moment 1000 Nm, h max = b max = 0.15 m, t max = 0.01 m
Pareto-optimal set in the objective functions domain for a I shaped steel cross section. Material data in Table 5. Applied moment 1000 Nm, h max = b max = 0.15 m, t max = 0.01 m
Point A I , in Figs. 5, 6, 7 represents the solution with minimum mass; point B I , the switching point between the buckling constraint and the available room constraint; point D, the solution with minimum compliance.
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Gobbi, M., Previati, G., Ballo, F. et al. Bending of beams of arbitrary cross sections - optimal design by analytical formulae. Struct Multidisc Optim 55, 827–838 (2017). https://doi.org/10.1007/s00158-016-1539-6
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DOI: https://doi.org/10.1007/s00158-016-1539-6