Bending-bending coupled static analysis of functionally graded and porous pretwisted cantilever beams using initial values method

In this study, the bending-bending coupled static behavior of functionally graded and porous pretwisted cantilever Timoshenko beams is reported. Symmetric power-law distribution is employed as the material profile and it is assumed that porosity can disperse over the cross-section in two forms namely, even and uneven distributions. Two different yet equivalent forms of governing equations are obtained by formulating the problem in the fixed and curvilinear coordinate systems, respectively. The initial values method is adopted as the solution methodology. While the fundamental matrix is computed numerically by using the multiplicative integral of Volterra to obtain the solution in the fixed coordinate formulation, the matrix exponential which allows for obtaining exact expressions is utilized in the curvilinear coordinate formulation. Exact analytical deflection expressions are presented for space-fixed vertical tip force and body-fixed uniform distributed force cases. It is shown that both formulation approaches yield consistent results with each other and those obtained from the finite element analyses. In addition, several parametric studies that investigate the effects of the volume fraction index, porosity coefficient, and porosity distribution on the tip deflections are presented.