Nonlinear bending of FGP beams and frames based on a third-order shear deformable theory

Abstract The nonlinear bending of beams and frames made of functionally graded porous (FGP) material under mechanical loading is presented based on a third-order shear deformable theory. The beams and frames consist of an FGP core and two faces of isotropic material. The porosities of the core are considered to be symmetric distributions in the direction of thickness. The von Kármán nonlinear assumption is adopted to formulate a nonlinear beam element, and then to construct the nonlinear equilibrium equation. Different from the existing third-order shear deformable elements, the present element is derived by using the rotation of transverse shear, not the rotation of cross-section, as a. Explicit expressions for the vector of internal forces and the matrix of tangent stiffness are given. The nonlinear response of various FGP beams and frames to mechanical loading is predicted using the Newton–Raphson iterative method, and the effects of the porosities on the nonlinear behavior are examined. Numerical results show that the beam element proposed herein is efficient, and it can accurately predict the nonlinear response by using just several elements. The effects of the porosity coefficient on the nonlinear behavior of the FGP beams and frames are investigated in detail.