Analytical and FE modeling of a bi-directional functionally graded Timoshenko beam under thermomechanical loading

Abstract Functionally graded materials (FGMs) have variable volume fractions in different directions, making structural analysis challenging due to shear deformation. This study investigates the behavior of a two-directional functionally graded material (TDFGM) Timoshenko beam under thermomechanical loads. Varied material properties in both axial and transverse directions are analyzed using a mathematical approach and a finite element model. The mathematical approach involves transforming the system of two ordinary differential equations to a single fourth-order differential equation to determine deflection and rotation. Additionally, a bilinear quadrilateral element is used to validate the mathematical solution. The results show that increasing grading parameters reduced transverse deflection, with a maximum reduction of 38.97% for CF beams and 39.26% for SS beams when the gradation index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> </mml:msub> </mml:math> increased from 0 to 1 under mechanical loading. Under combined mechanical and thermal loading, higher temperatures on the beam’s hot side lead to increased deflection, with notable increases at higher gradation indices. Moreover, increased maximum deflection by 6.29% for CF beams and 4.05% for SS beams when considering temperature-dependent properties. It was observed that when TDFGM Timoshenko beams were heated for thermal loading, the SS beam deflected in the opposite direction of the applied mechanical load. The results agree with previous research and indicate alignment between mathematical and finite element solutions. The impact of boundary conditions on the deflection of Timoshenko beams made of TDFGM is discussed. The introduced methodology enhances TDFGM beam bending analysis, allowing control over behavior through grading parameters and boundary conditions selection.