Tensile and compressive buckling of shear deformable damaged beams

In this work the static stability analysis of the Timoshenko beam in presence of an arbitrary number of concentrated cracks is studied. The governing differential equations are formulated by modelling the cracks as a concentrated reduction of the flexural stiffness, accomplished by the use of Dirac’s delta distributions. The adopted model allowed the formulation of exact closed form solutions of the buckling modes and the buckling load equation for an arbitrary number of cracks and under different boundary conditions able to account for the shear deformability. The presented closed from solutions, besides the classical compressive buckling, is also able to capture the tensile buckling phenomenon, i.e. damaged columns subjected to a tensile axial load. The influence of multiple cracks on the stability of shear deformable beams, particularly under the action of tensile loads, has never been assessed in the literature and is here addressed on the basis of an extensive parametric analysis.