Hierarchical one-dimensional finite elements for the thermal stress analysis of three-dimensional functionally graded beams

In this work, the thermoelastic response of functionally graded beams is studied. To this end, a family of advanced one-dimensional finite elements is derived by means of a Unified Formulation that is not dependent on the order of approximation of the displacements upon the beam cross-section. The temperature field is obtained via a Navier-type solution of Fourier’s heat conduction equation and it is considered as an external load within the mechanical analysis. The stiffness matrix of the elements is derived via the Principle of Virtual Displacements. Numerical results in terms of temperature, displacements and stresses distribution are provided for different beam slenderness ratios and type of material gradation. Linear, quadratic and cubic elements are used. Results are validated through comparison with three-dimensional finite elements solutions obtained by the commercial software ANSYS. It is shown that accurate results can be obtained with reduced computational costs.