A generalized numerical method is proposed to derive the static and dynamic stiffness matrices and to handle the nodal load vector for static analysis of non-uniform Timoshenko beam–columns under several effects. This method presents a unified approach based on effective utilization of the Mohr method and focuses on the following arbitrarily variable characteristics: geometrical properties, bending and shear deformations, transverse and rotatory inertia of mass, distributed and (or) concentrated axial and (or) transverse loads, and Winkler foundation modulus and shear foundation modulus. A successive iterative algorithm is developed to comprise all these characteristics systematically. The algorithm enables a non-uniform Timoshenko beam–column to be regarded as a substructure. This provides an important advantage to incorporate all the variable characteristics based on the substructure. The buckling load and fundamental natural frequency of a substructure subjected to the cited effects are also assessed. Numerical examples confirm the efficiency of the numerical method.Key words: non-uniform, Timoshenko, substructure, elastic foundation, geometrical nonlinearity, stiffness, stability, free vibration.
A numerical method for free vibration analysis of beams
