Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory

In this work, postbuckling response of bi-directional functionally graded (FG) beams with porosities is investigated. The transverse shear deformation is taken into account based on a novel third-order shear deformation theory in which the kinematic of displacements is derived from an elastic formulation. Porosities owing to the technical issue during the preparation of functionally graded materials (FGMs) with even and uneven distributions are considered. Material properties of bi-directional FG beams vary smoothly along the thickness and axial directions simultaneously based on the power law distribution. Geometric nonlinearity is described by employing the von Karman nonlinear theory. Equations of motion are derived utilizing the principle of minimum potential energy. Nonlinear partial differential equations are solved numerically to obtain the critical buckling loads and the postbuckling equilibrium paths under different boundary conditions using the generalized differential quadrature method (GDQM) and the Newton-Raphson iteration. Numerical results demonstrate that FG and axially FG (AFG) indexes, porosities distribution, boundary condition, Young's modulus ratio, aspect ratio, and plane strain and plane stress states have significant influences on the buckling and postbuckling responses of bi-directional FG beams.