Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method.