A thermal stress finite element analysis of beam structures by hierarchical modelling

In this study, the thermoelastic behaviour of three-dimensional isotropic and laminated beams is investigated. 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 element is also a free parameter. Linear, quadratic and cubic variations along the beam axis are considered within the element. 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 governing algebraic equations are obtained via the Principle of Virtual Displacements. Displacements and stresses are investigated and results are validated towards three-dimensional FEM 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 results.