Variational Analysis of Bi-Directionally Functionally Graded Timoshenko Beams Incorporating Shear Correction Effects

This study develops a Gâteaux differential-based mixed finite element formulation for bi-directionally functionally graded (BDFGM) Timoshenko beams graded along both the length and the thickness. The local effective properties are evaluated by the Mori–Tanaka scheme using two independent gradation indices, nx and nz, so that the section stiffnesses vary continuously along the beam axis. The axial strain is condensed under the zero-axial-force condition, and extension–bending coupling is retained through a spatially varying condensed bending stiffness. A two-node element with four independent fields (transverse displacement, rotation, bending moment, and shear force) is derived, and its variable bending and shear compliances are integrated by Gauss–Legendre quadrature. In the through-thickness limit (nx = 0), using the rule of mixtures of the reference solutions, the present deflections agree with refined-theory benchmarks to within about 0.1%; mesh and quadrature studies confirm convergence for the bi-directional case. For nz = 1 and ks = 5/6, increasing nx from 0 to 5 increases the maximum nondimensional deflection by about 59%, from 0.69465 to 1.10563 for L/h = 4 and from 0.61036 to 0.97469 for L/h = 16. For nx = nz = 1, replacing ks = 1 by 5/6 increases the deflection by about 2.3% for L/h = 4 but by only 0.16% for L/h = 16.