Influence of shear deformations on the buckling of columns using the Generalized Beam Theory and energy principles

The paper presents a procedure for the stability analysis of columns that are sensitive to shear deformations, the so-called “weak-in-shear” columns that can be found, in the engineering practice, in build-up or composite columns, or in elastomeric bearings. Two distinct formulas are commonly used to compute the critical load for shear sensitive columns: the Engesser and the Haringx formulae, the latter enabling significantly higher loads. They differ on the choice of the cross section's shear stresses resultant, and this duality has been object of much passionate discussion during the last decades. This problem is analysed here under the perspective of the Generalized Beam Theory, and a specific mode for shear deformations was developed using two distinct strategies: i) a linear shear formulation, corresponding to the Timoshenko beam theory for which cross sections remain plane after deformation, and ii) a nonlinear shear formulation, for which shear warping is allowed in order to accomplish with the condition of null shear distortions at the section's contour. A total potential energy is defined, assuming a linear elastic behaviour, and the correspondent functional is rendered discrete by means of the Rayleigh-Ritz method. Finally, the traditional stability procedures are applied and the critical loads are computed. The Engesser critical load is derived by applying the stability procedures to the total potential energy associated with the linear shear formulation. A parametric study on the critical behaviour of a shear deformable column and a set of conclusions end the paper.