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**Optimal design of a beam subject to bending: a basic application.**
*(English)*
Zbl 1383.49053

Summary: The minimization of both the mass and deflection of a beam in bending is addressed in the paper. To solve the minimization problem, a multi-objective approach is adopted by imposing the Fritz John conditions for Pareto-optimality. Constraints on the maximum stress and elastic stability (buckling) of the structure are taken into account. Additional constraints are set on the beam cross section dimensions. Three different cross sections of the beam are analyzed and compared, namely the hollow square, the I-shaped and the hollow rectangular cross sections. The analytical expressions of the Pareto-optimal sets are derived. As expected, the I-shaped beam exhibits the best compromise in structural performance, which is related on the particular loading considered.

### MSC:

49S05 | Variational principles of physics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74P05 | Compliance or weight optimization in solid mechanics |

49M30 | Other numerical methods in calculus of variations (MSC2010) |

### Keywords:

multi-objective optimization; analytical solution; beam in bending; Pareto-optimality; beam cross section
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\textit{F. M. Ballo} et al., Meccanica 52, No. 15, 3563--3576 (2017; Zbl 1383.49053)

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### References:

[1] | Askar S, Tiwari A (2009) Finding exact solutions for multi-objective optimisation problems using a symbolic algorithm. In: IEEE Congress on evolutionary computation · Zbl 1208.90151 |

[2] | Askar, S; Tiwari, A, Finding innovative design principles for multiobjective optimization problems, IEEE Trans Syst Man Cybern Part C Appl Rev, 41, 554-559, (2011) |

[3] | Benfratello, S; Giambanco, F; Palizzolo, L; Tabbuso, P, Optimal design of steel frames accounting for buckling, Meccanica, 48, 2281-2298, (2013) · Zbl 1293.74343 |

[4] | Björck A (1996) Numerical methods for least squares problems. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0847.65023 |

[5] | Dutta, J; Lalitha, CS, Bounded sets of kkt multipliers in vector optimization, J Glob Optim, 36, 425-437, (2006) · Zbl 1120.90054 |

[6] | EN-10034 (1995) EN 10034: Structural steel I and H sections. Tolerances on shapes and dimensions |

[7] | EN-10219-2 (2006) EN 10219-2: Cold formed welded structural hollow sections of non-alloy and fine grain steels—tolerances, dimensions and sectional properties |

[8] | EN-1993-1-1 (2005) EN 1993-1-1 Eurocode3: Design of steel structures-Part 1-1: general rules and rules for buildings |

[9] | Gobbi, M; Levi, F; Mastinu, G, Multi-objective stochastic optimisation of the suspension system of road vehicles, J Sound Vib, 298, 1055-1072, (2006) |

[10] | Gobbi, M; Levi, F; Mastinu, G; Previati, G, On the analytical derivation of the Pareto-optimal set with an application to structural design, Struct Multidiscip Optim, 51, 645-657, (2014) |

[11] | Gobbi, M; Previati, G; Ballo, FM; Mastinu, G, Bending of beams of arbitrary cross sections—optimal design by analytical formulae, Struct Multidiscip Optim, 55, 827-838, (2017) |

[12] | Kasperska, RJ; Magnucki, K; Ostwald, M, Bicriteria optimization of cold-formed thin-walled beams with monosymmetrical open cross sections under pure bending, Thin Walled Struct, 45, 563-572, (2007) |

[13] | Kim, D; Lee, G; Lee, B; Cho, S, Counterexample and optimality conditions in differentiable multiobjective programming, J Optim Theory Appl, 109, 187-192, (2001) · Zbl 1017.90099 |

[14] | Levi F, Gobbi M (2006) An application of analytical multi-objective optimization to truss structures. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference |

[15] | Levi, F; Gobbi, M; Mastinu, G, An application of multi-objective stochastic optimisation to structural design, Struct Multidiscip Optim, 29, 272-284, (2005) |

[16] | Lütkepohl H (1996) Handbook of matrices. Wiley, New York · Zbl 0856.15001 |

[17] | Mastinu G, Gobbi M, Miano C (2006) Optimal design of complex mechanical systems. Springer, Berlin · Zbl 1099.74002 |

[18] | Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, Dordrecht · Zbl 0949.90082 |

[19] | Ostwald, M; Rodak, M, Multicriteria optimization of cold-formed thin-walled beams with generalized open shape under different loads, Thin Walled Struct, 65, 26-33, (2013) |

[20] | Papalambros P, Wilde D (2000) Principles of optimal design. Modeling and computation. Cambridge Universirty Press, Cambridge · Zbl 0962.90002 |

[21] | Pedersen, P; Pedersen, N, Analytical optimal designs for long and short statically determinate beam structures, Struct Multidiscip Optim, 39, 343-357, (2009) · Zbl 1274.74157 |

[22] | Rondal J, Würker K, Dutta D, Wardenier J, Yeomans N (1992) Structural stability of hollow sections. Verlag TUV Rheinland, Koln |

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