Buckling analysis of two-directionally porous beam

In this paper, buckling analysis of two-directionally porous beam is conducted. Based on the available results of Young's modulus and mass density via Gaussian random field theory, a new two-directionally porous beam model is developed. With the help of Euler-Bernoulli beam theory and minimum total potential energy principle, the equilibrium equations for nonlinear and linear buckling are derived. The numerical solutions of critical buckling loads for different porosity distribution patterns can be obtained by generalized differential quadrature method. The final numerical results exhibit that more porosities near the middle surface or the two edges of beam can lead to a larger critical buckling load when the same total volume fraction of porosity is in different porosity distribution patterns. The effect of porosity distribution in thickness direction is more dominated on the critical buckling load than that of the axial porosity distribution. Moreover, the critical buckling load becomes more sensitive to aspect ratio of beam and total volume fraction of the porosity when increasing mode number. The critical buckling load of two-directionally porous beam depends not only on bending coefficient (like the one-directionally porous beam), but also on first and second derivatives of the bending coefficient.