The influence of multiple cracks on tensile and compressive buckling of shear deformable beams

• Timoshenko column in presence of an arbitrary number of concentrated cracks is considered.
• Cracks have been modeled as concentrated reduction of the flexural stiffness.
• Explicit expressions of buckling shapes and the buckling load equations are proposed.
• Tensile buckling phenomenon is captured by the proposed explicit solutions.
• The influence of multiple cracks on the tensile and the compressive buckling is assessed.

In this work the static stability of the uniform Timoshenko column in presence of multiple cracks, subjected to tensile or compressive loads, is analyzed. The governing differential equations are formulated by modeling the cracks as concentrated reductions of the flexural stiffness, accomplished by the use of Dirac’s delta distributions. The adopted model has allowed the derivation of the exact buckling modes and the corresponding buckling load equations of the Timoshenko multi-cracked column, as a function of four integration constant only, which are derived simply by enforcing the end boundary conditions, irrespective of the number of concentrated damage. Since shear deformability has been taken into account, the buckling load equation allows capturing both compressive and tensile buckling. The latter phenomenon has been recently investigated with reference to rubber bearing isolators, modeled as short beams, but it has been shown to occur also in slender beams characterized by high distributed shear deformation, like composite and layered beams. The influence of multiple concentrated 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. All the reported results have been compared with the Euler multi-cracked column in order to highlight its limits of applicability.