Buckling analysis of non-uniform bars with rotational and translational springs

The governing equation for buckling of non-uniform bars with rotational and translational springs subjected to concentrated axial compressive forces is given by the second order differential equation for bending moment M(x). The closed-form solutions of the governing equations are derived for eleven important cases. The normalized fundamental solutions for the bending moment of one-step non-uniform bars are developed based on the derived exact solutions, then the shear force Q(x), slope y’(x) and displacement y(x) are obtained by differentiating or integrating M(x). By using the fundamental solutions and the recurrence formulas developed in this paper, the eigenvalue equation for buckling of multi-step non-uniform bars with any kinds of two end supports and any number of rotational and translational springs at the ends and intermediate points can be conveniently determined from a second order determinant. The decrease in the determinant order, as compared with previously developed procedures, leads to significant savings in the computational effort. The buckling problem of such non-uniform bars with seven boundary conditions is investigated. Numerical examples show that the proposed procedure for determining the critical buckling force of non-uniform bars with rotational, translational springs is a simple, efficient and exact method.