First order solutions for the buckling loads of weakened Timoshenko columns

In this work, closed-form expressions for the buckling loads of weakened Timoshenko columns with different boundary conditions and shear force approaches (proportional to the bending rotation or to the total slope) are presented. The crack model used promotes discontinuities in both transversal displacements and rotation due to bending. To solve the buckling problem, the perturbation method is used, considering that the solutions for both the cracked and the uncracked columns are slightly different. This procedure leads to first-order closed-form expressions for the buckling loads of the Timoshenko cracked column, which were compared with those found by directly solving the corresponding eigenvalue problem, establishing validity limits for these solutions.