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This paper presents the elastic buckling and static bending analysis of shear deformable functionally graded (FG) porous beams based on the Timoshenko beam theory. The elasticity moduli and mass density of porous composites are assumed to be graded in the thickness direction according to two different distribution patterns. The open-cell metal foam provides a typical mechanical feature for this study to determine the relationship between coefficients of density and porosity. The partial differential equation system governing the buckling and bending behavior of porous beams is derived based on the Hamilton's principle. The Ritz method is employed to obtain the critical buckling loads and transverse bending deflections, where the trial functions take the form of simple algebraic polynomials. Four different boundary conditions are considered in the paper. A parametric study is carried out to investigate the effects of porosity coefficient and slenderness ratio on the buckling and bending characteristics of porous beams. The influence of varying porosity distributions on the structural performance is highlighted to shed important insights into the porosity design to achieve improved buckling resistance and bending behavior.

a b s t r a c tThe nonlinear free vibration behavior of shear deformable sandwich porous beam is investigated in this paper within the context of Timoshenko beam theory. The proposed beam is composed of two face layers and a functionally graded porous core which contains internal pores following different porosity distributions. Two non-uniform functionally graded distributions are considered in this paper based on the equivalent beam mass, associated with a uniform distribution for purpose of comparison. The elastic moduli and mass density are assumed to vary along the thickness direction in terms of the coefficients of porosity and mass density, whose relationship is determined by employing the typical mechanical characteristic of an open-cell metal foam. The Ritz method and von Kármán type nonlinear strain-displacement relationships are applied to derive the equation system, which governs the nonlinear vibration behavior of sandwich porous beams under hinged or clamped end supports. A direct iterative algorithm is then used to solve the governing equation system to predict the linear and nonlinear frequencies which are presented by a detailed numerical study to discuss the effects of porosity coefficient, slenderness ratio, thickness ratio and to compare the varying porosity distributions and boundary conditions, providing a feasible way to improve the vibration behavior of sandwich porous beams.

This paper investigates the free and forced vibration characteristics of functionally graded (FG) porous beams with non-uniform porosity distribution whose elastic moduli and mass density are nonlinearly graded along the thickness direction. Both symmetric and asymmetric porosity distributions are considered. The relationship between coefficients of porosity and mass density is determined based on the typical mechanical property of an open-cell metal foam. Within the framework of Timoshenko beam theory to include the effect of transverse shear strain and by employing Lagrange equation method together with Ritz trial functions, the equation of motion is derived then solved by using Newmark-β method in the time domain. Natural frequencies and transient dynamic deflections are obtained for porous beams under different loading conditions, including a harmonic point load, an impulsive point load and a moving load with constant velocity. A detailed numerical study is conducted to examine the effects of varying porosity distribution, porosity coefficient, slenderness ratio and boundary condition, shedding important insights into the design of functionally graded porous beams to achieve improved dynamic behaviour.

The nonlinear free vibration and postbuckling behaviors of multilayer functionally graded (FG) porous nanocomposite beams that are made of metal foams reinforced by graphene platelets (GPLs) are investigated in this paper. The internal pores and GPL nanofillers are uniformly dispersed within each layer but both porosity coefficients and GPL weight fraction change from layer to layer, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the beam thickness. The mechanical property of closed-cell cellular solids is employed to obtain the relationship between coefficients of porosity and mass density. The effective material properties of the nanocomposite are determined based on the Halpin-Tsai micromechanics model for Young's modulus and the rule of mixture for mass density and Poisson's ratio. Timoshenko beam theory and von Kármán type nonlinearity are used to establish the differential governing equations that are solved by Ritz method and a direct iterative algorithm to obtain the nonlinear vibration frequencies and postbuckling equilibrium paths of the beams with different end supports. Special attention is given to the effects of varying porosity coefficients and GPL's weight fraction, dispersion pattern, geometry and size on the nonlinear behavior of the porous nanocomposite beam. It is found that the addition of a small amount of GPLs can remarkably reinforce the stiffness of the beam, and the its nonlinear vibration and postbuckling performance is significantly influenced by the distribution patterns of both internal pores and GPL nanofillers.