A thermal stress analysis of functionally graded beam structures by hierarchical finite elements

In this study, the thermoelastic behaviour of three-dimensional functionally graded beams is investigated. The temperature field is treated as an external load within the mechanical analysis and it is obtained by exactly solving Fourier's heat conduction equation. The three-dimensional beam is modelled through advanced one-dimensional finite elements derived via hierarchical expansion of the displacements over the cross-section. The approximation order of the displacement field is a free parameter that leads to the formulation of a family of several beam elements. The number of nodes per elements is also a free parameter. Linear, quadratic and cubic variations over the beam axis are considered. The governing algebraic equations are obtained via the Principle of Virtual Displacements. Displacements and stresses are evaluated and results are validated towards three-dimensional FEM results as well as analytical solutions. The temperature load results in a three-dimensional stress state that calls for accurate models. Numerical investigations show that the proposed finite elements yield accurate yet computationally efficient solutions.