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This work presents analytical solutions for bending deformation and stress distributions in functionally graded beams with arbitrarily and continuously variable thicknesses and resting on a two-parameter Pasternak elastic foundation. Based on two-dimensional elasticity theory directly, the general solutions of displacements and stresses which completely satisfy the differential equations governing the equilibrium for arbitrarily varying thickness functionally graded beams are derived for the first time. The undetermined coefficients in the general solution are obtained using Fourier series expansion along the upper and lower surfaces. The accuracy and efficiency of the proposed method are verified through several typical examples. The effects of mechanical and geometry parameters on the stress and displacement distributions of varying thickness functionally graded beams resting on a two-parameter Pasternak elastic foundation are discussed further. [ABSTRACT FROM AUTHOR]

l Approach obtaining analytical solution for vibration of functionally graded beam with arbitrarily varying cross-sections and elastic foundations. l Declare effect of mechanical and geometric parameters on vibration of variable-thickness FGB. l Provides potential alternative for analysing vibration of functionally graded components with variable cross-sections in case of ultra-high precision requirement in modern mechanical system. An efficient approach to obtain accurate solutions for free vibration of functionally graded beams (FGB) with variable cross-sections and resting on Pasternak elastic foundations is presented. The general expressions of displacements and stresses which thoroughly result from the dynamic differential equations and boundary conditions for FGB with arbitrarily variable cross-sections are obtained by using the separate variable method and Laplace transform on the basis of 2-D elastic theory. Meanwhile, the frequency equations for free vibration of FGB with variable cross-sections are derived by applying Fourier series expansion along the lower and upper boundary conditions of the FGB. Validity of the developed approach as well as its effectiveness and accuracy are verified through analysing several typical FGB examples, and this provides a potential alternative approach for analysing vibration of functionally graded components with variable cross-sections in case of ultra-high precision requirement in modern mechanical systems. The effect of geometric and mechanical parameters on vibration frequency and mode shapes of FGB with variable cross-sections resting on Pasternak elastic foundations is further analysed. Image, graphical abstract [ABSTRACT FROM AUTHOR]

A non-local numerical model for large deformation analysis of functionally graded beam (FGB) with variable cross-section is presented by using peridynamic differential operator (PDDO) in this paper. The nonlinear equilibrium equation and boundary condition of FGB with variable cross-section are reformulated from the local differential form into the non-local integral form based on PDDO. By employing variational analysis, Lagrange multiplier method, and Newton-Raphson method, the non-local governing equations can be solved with iterations. The proposed method is applied to the large deformation analyses of a cantilever homogeneous beam and two cantilever FGBs with linearly varying and parabolic lower surfaces, respectively, and the effects of geometric size and material gradient on mechanical behaviour of FGB are investigated further. Comparison with finite element results validates the convergence and accuracy of the proposed non-local model. [ABSTRACT FROM AUTHOR]