Semi-analytical elasticity solutions for bi-directional functionally graded beams

Abstract

Elasticity solutions are presented for bending and thermal deformations of functionally graded beams with various end conditions, using the state space-based differential quadrature method. The beams are assumed to be macroscopically isotropic, with Young's modulus varying exponentially along the thickness and longitudinal directions, while Poisson's ratio remaining constant. The state space method is adopted to obtain analytically the thickness variation of the elastic field and, when coupled with differential quadrature, the longitudinal discretization can be analyzed in an approximate manner. This approach is then validated by comparing the numerical results with the exact solutions for a special functionally graded beam and with finite element solutions. The influences of material gradient indices on the response of bi-directional functionally graded beams are finally investigated.

Thermo-Elasticity in Shell Structures Made of Functionally Graded Materials

“…In Fig. 12 the stress distributions at x = L /2 are compared to ANSYS and to the solutions published in [29,30]. The axial thermal stress distribution σ xx is in good agreement compared to the reference solutions (see Fig.…”

“…In this example, originated by Lü et al [29], a bi-directional FGM beam (L = 0.1, h = w = 0.01) depicted in Fig. 10 is analyzed.…”

“…A clamped-clamped situation is analyzed. This example is solved analytically by Lü et al [29] based on a two-dimensional thermo-elastic model. Lezgy-Nazargah [30] analyzed the same problem using a fully coupled isogeometric two-dimensional finite element approach.…”

Bending Analysis of Two Directional Functionally Graded Beams Using A Four-Unknown Shear and Normal Deformation Theory

“…2D steady-state free and forced vibrations of two-directional FGBs by using the Element Free Galerkin Method are analyzed in [26]. The state-space based differential quadrature method is employed to obtain the semi-analytical elasticity solutions for bending and thermal deformations of FGBs with various end conditions [27]. A symplectic elasticity solution for static and free vibration analyses of two-directional FGBs with the material properties varying exponentially in both axial and thickness direction is presented in [28].…”

Static behaviour of two-directional functionally graded sandwich beams using various beam theories

“…For instance, the practical engineering applications of the 2D-FGM were presented in [32], by using the Element Free Galerkin Method, 2D steady-state free and forced vibrations of two-directional FG beams are analysed in [33]. The elasticity solutions are proposed for bending and thermal deformations of FGBs with various end conditions by using the state-space based differential quadrature method in [34]. A symplectic elasticity solution for static and free vibration analyses of two-directional FG beams with the material properties varying exponentially in both axial and thickness direction is presented in [35].…”