Optimization of clamped columns under distributed axial load and subject to stress constraints

Abstract

The optimum designs are given for clamped-clamped columns under concentrated and distributed axial loads. The design objective is the maximization of the buckling load subject to volume and maximum stress constraints. The results for a minimum area constraint are also obtained for comparison. In the case of a stress constraint, the minimum thickness of an optimal column is not known a priori, since it depends on the maximum buckling load, which in turn depends on the minimum thickness necessitating an iterative solution. An iterative solution method is developed based on finite elements, and the results are obtained for n = 1, 2, 3 defined as I = α n A n , with I being the moment of inertia, and A the crosssectional area. The iterations start using the unimodal optimality condition and continue with the bimodal optimality condition if the second buckling load becomes less than or equal to the first one. Numerical results show that the optimal columns become larger in the direction of the distributed load due to the increase in the stress in this direction. Even though the optimal columns are symmetrical with respect to their mid-points when the compressive load is concentrated at the end-points, in the case of the columns subject to distributed axial loads the optimal shapes are unsymmetrical.

Optimal shapes of clamped–simply supported columns under distributed axial load and stress constraint

“…It is noted that in the solution of the problem, the bimodality of the buckling is checked by computing the critical load of the second buckling mode and the second buckling mode shape. This method has been used by Cagdas and Adali (2007) to determine the bimodal optimal shapes of clamped-clamped columns. In the present study, the second buckling load of the optimal shape of the clamped-simply supported column remained higher than the first one and the mode shapes remained unimodal in all cases.…”

“…In a column under an axially distributed load, the stress level changes from point to point and increases in the direction of the load. This results in non-symmetrical optimal designs with smaller cross-sections when the total distributed load is low and larger crosssections when the total distributed load builds up, even for columns with symmetrical boundary conditions such as clamped-clamped (Cagdas and Adali 2007) and simply supported (Adali and Cagdas 2009). Another difference is that an area constraint specifies the minimum area a priori and can be included in the solution as an input parameter.…”

“…Previous work on the optimization of columns subject to stress constraints involved columns with symmetrical boundary conditions and in particular clamped-clamped (Cagdas and Adali 2007) and simply supported columns (Adali and Cagdas 2009). In these cases the direction of the distributed load did not affect the optimal shape owing to the symmetry of the boundary conditions, but the optimal shapes were still asymmetrical owing to the presence of distributed axial load, which increases towards the fixed end.…”

Unimodal Optimization of Axially Compressed Shear Deformable Columns

“…Also, this exclusion leads to thinner sections, which is important especially when optimizing statically indeterminate columns as the unimodal optimality condition leads to optimal designs where area vanishes at certain parts of the statically indeterminate columns. Also, stress has been used as a constraint in only a few papers [7][8][9], which becomes important when column length is short or material is weak.…”