Closed form buckling and sensitivity analysis of isotropic and orthotropic columns with uncertain material and geometric properties via convex modeling Part I: Theory and formulation

•Closed-form uncertain buckling formulas for isotropic and orthotropic columns.•Material and geometric fluctuations affecting the critical buckling loads.•Degree of sensitivity, most important parameters for the buckling load.•Higher performance by result of improved structural safety; reduced safety factor. Unlike deterministic approaches, implemented in several column buckling studies, it is important to consider the uncertainty effects, since material properties and geometric dimensions may vary due to imperfections and defects in the production stage which may significantly affect the buckling load, and they should be considered as uncertain. Variations in the material properties and dimensions can lead to buckling at lower design loads and this, in turn, may cause human and/or material losses. Taking material uncertainties into account in the design stage of structures, and in particular, in the design of columns, improves the structural safety and this can allow the use of a reduced safety factor. In part I of this study, closed form solutions are developed for the buckling of columns with material and geometric uncertainties for isotropic and orthotropic columns for several cross-sectional shapes using symbolic computation, mainly, Maple software. Solutions correspond to worst-case buckling loads for different uncertainty levels based on convex modelling of uncertainty. Formulations are based on Euler - Bernoulli beam theory for isotropic columns, and first order shear deformation theory for orthotropic columns. In part II, numerical results and discussions are presented to investigate the effects of uncertain parameters such as the aspect ratio, cross-sectional dimensions, and material properties on the buckling load. An important issue, involving the design uncertainties, is the degree of sensitivity of the objective function (buckling load in the present study) to the level of uncertainty. This issue is studied by performing a sensitivity analysis which provides numerical data to identify the problem parameters affecting the buckling loads the most. Symbolic computation solutions are verified by implementing the finite element solution of the problems. It is noted that the type of uncertainties investigated in the present study are common types of data uncertainties encountered in the design of columns.