Buckling of multi-step cracked columns with shear deformation

The governing differential equation for buckling of a one-step cracked column with shear deformation is established and its solution is found first. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented for solving the entire and partial buckling of a multi-step column with various end conditions, with or without cracks and shear deformation, subjected to concentrated axial loads. The main advantage of the proposed method is that the eigenvalue equation for buckling of a multi-step column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a system of two linear equations. Due to the decrease in the determinant order as compared with previously developed procedures, the computational time required by the present method for solving the title problem can be reduced significantly. A numerical example is given for explaining the proposed procedure and investigating the effects of shear deformation and the number, depth and location of cracks on critical buckling force of a multi-step column. It is shown through the numerical example that the proposed procedure is a simple, exact and efficient method.