A functionally graded beam model accounting for thickness stretch

This paper presents an enhanced Reddy-type beam formulation for functionally graded materials, in which the mechanical properties vary continuously through the beam thickness. The proposed model extends the classical third-order shear deformation Reddy beam by introducing additional kinematic fields that allow for the transverse contractions and expansions of the beam cross section. The governing equations are derived in a variational form, from which the energetically conjugate generalised stresses, distributed loads and concentrated forces are consistently defined. The constitutive model for functionally graded materials is formulated by deriving analytical expressions for the entries of the local stiffness tensor, i.e. the higher-order elastic moments, employing an arbitrary polynomial variation of the Young modulus along the beam thickness. The elastic equilibrium equations are consistently derived and the analysis of the functionally graded beam is conducted by formulating a one-dimensional two-node finite element, with linear and cubic Hermite shape functions, for which the displacement and strain operators are explicitly derived and expressed in matrix form. Numerical investigations are carried out on both homogeneous and functionally graded beams. The proposed beam model successfully reproduces the nonlinear stress distributions along the thickness associated with the enriched kinematics and accurately captures localised boundary effects induced by external loads and constraints, yielding a richer and more physically consistent stress state compared to conventional beam models. Comparisons with two-dimensional continuum finite element analyses and benchmark results from the literature confirm the accuracy, robustness and predictive capabilities of the proposed model.