Keywords: optimization, nonlinear analysis, beam-column, instability.

In this paper the optimal design of vertical cantilever beam, immersed in an elastic Winkler soil and subjected to several loads: (1) concentrated force at its tip section, (2) self weight and (3) uniform distributed load along its length is analyzed. Typically, [3] and [1], the optimal design was related to find the shape of the column, of a given fixed volume and length, such that its buckling load is maximum. The problem, in this way formulated, may present some numerical and theoretical difficulties, particularly in columns with different boundaries conditions to cantilever, such as clamped in its both ends. In fact, they may appear during the optimization procedure the presence of column cross sections with very small area and the existence of a multiple (double) first mode [4] with the inherent problems of gradient descent direction is not univocally determinate, i.e. according to [2] the problem becomes a nonsmooth one.

An alternative definition to the former column buckling optimization problem is given, that may be some of the commented difficulties can be avoided. The alternative definition of optimal design is concerned with a column of given length that can resist a specified load or load combination, in both ways, as a column compressive load and as beam-column buckling load, with the minimum volume. Contrary to previous formulations the column material strength, namely its admissible stress, is introduced.

In the framework of this optimization problem definition it is possible to study several types of loads, conservative and nonconservative or follower load as well as different beam-column models, Navier-Bernoulli, Timoshenko with shear deformation, etc. With this formulation the existence of a first multiple buckling mode, can be easily detected, because its value is fixed as a constrain instead of being an objective value to be maximized. In this way the correct gradient direction can be identified. Besides, zero areas occurrence is obviously avoided with the introduction of admissible stresses constraints.

This problem is analyzed according to the linear elasticity theory and within different alternative structural models: column, Navier-Bernoulli beam-column, Timoshenko beam-column (i.e. with shear strain) under conservative loads, typically, constant direction loads. The results obtained in each case are compared, in order to evaluate the sensitivity of model on the numerical results. The beam optimal design is identified by the section distribution characteristics (area, second moment, shear area etc.) along the beam span and the corresponding beam total volume. The influence of alternative Finite Element discretization levels, as number of elements, polynomials or beam-column solutions shape functions is discussed. However, other loading situations, some of them very interesting from the theoretical point of view, (including follower loads) are also commented upon, but numerical details and results are left for a future work

1

Bogacz, R., Irreteier, H., Mahrenholtz, O., Optimal design of structures subjected to follower forces. Engineering-Archives, 49:63-71, 1980.

2

Cox, S. J., Oberton, M.L., On the optimal design of columns against buckling. SIAM Journal of Mathematical Analysis, 23:287-325, 1992. doi:10.1137/0523015

3

Hanaoka, M., Washizu, K., Optimum design of Beck's column. Computers and Structures, 11:473-480, 1980. doi:10.1016/0045-7949(80)90054-1

4

Tadjbakssh, I., Keller, J., Strongest columns and the isoparametric inequalities for eigenvalues. Journal of Applied Mechanics, 29:159-164, 1962.

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