Finite difference method for buckling analysis of tapered Timoshenko beam made of functionally graded material

Document Type : Research Article


1 Department of civil engineering, faculty of engineering, University of Kashan, kashan, Iran

2 Department of Civil Engineering, University of Khajeh Nasireddin Tousi

3 Department of Civil Engineering, Faculty of Engineering, Shahr-e-Qods Branch, Islamic Azad University


This paper investigated the linear buckling analysis of tapered Timoshenko beams made of axially functionally graded material (AFGM). Material properties of the non-prismatic Timoshenko beam vary continuously along the length of the member according to power law as well as exponential law. Based on Timoshenko beam assumption and using small displacements theory, the linear equilibrium equations are adopted from the energy principle for functionally graded non-uniform Timoshenko beams related to constant compressive axial load. The resulting system of stability equations are strongly coupled due to the presence of transverse deflection and angle of rotation. The differential equations are then uncoupled and lead to a single homogeneous differential equation where only bending rotation is present. The finite difference method (FDM) is then selected to numerically solve the resulting second-order differential equation with variable coefficients and determine the critical buckling loads. Referring to this approach, all the derivatives of the dependent variables in the governing equilibrium equation are replaced with the corresponding forward, central and backward second order finite differences. The discreet form of the governing equation is then derived in a matrix formulation. The critical buckling loads are finally determined by solving the eigenvalue problem of the obtained matrix. Two comprehensive numerical examples are finally conducted to demonstrate the performance and efficiency of this mathematical procedure. The obtained results are contrasted with accessible numerical and analytical benchmarks.


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