Keywords: thermal load, beam structure, refined models, closed form solution, unified formulation, principle of virtual displacements.

In this paper the deformations of simply supported, isotropic and orthotropic beams subjected to thermo-mechanical loadings are analysed. Higher-order theories formulated on the basis of the principle of virtual displacements are derived. The majority of the works found in the literature concerning the thermal analysis of beams, consider classical theories or higher-order theories but with two-dimensional displacements fields. In this paper, the three-dimensional kinematic field is written in a compact form according to Carrera's unified formulation [1,2], which allows the formulation of several displacement-based theories. The displacements field depends on the approximation order N that is a free parameter of the formulation. Governing differential equations and boundary conditions can be derived in terms of 'fundamental nuclei'. The complexity related to higher than classical approximation terms is tackled and the theoretical formulation is valid for the generic approximation order.

In the present model, the temperature is considered as an external loading. The required temperature field is not assumed a priori. The temperature variation over one coordinate of the beam cross-section is computed using Fourier's heat conduction equation. In fact, even for a thin orthotropic layer the temperature profile is not linear. The temperature profile is described in the same way as the displacements, splitting the contribution relative to the beam axis from the contribution related to that of the beam cross-section.

A Navier-type, closed form solution is adopted. Square cross-section beams are investigated. Numerical results for displacement and stress distributions are provided for isotropic and ortotropic multilayered beams as well as for different slenderness ratios of the beam. Comparisons with three dimensional finite elements models are given. The paper's major conclusions are:

1. As temperature's effect, relatively high expansion-orders are necessary to identify the mechanical behavior of the beam, for both isotropic and orthotropic ones.
2. The proposed formulation leads to good results and computational time is reduced if compared to those of a finite element model, albeit for multilayer laminates it might be better a layer-wise approach, to avoid high errors at the layers' interface.

1

E. Carrera, "Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking", Archives of Computational Methods in Engineering, 10, 215-296, 2003. doi:10.1007/BF02736224

2

E. Carrera, G. Giunta, M. Petrolo, "Beam Structures: Classical and Advanced Theories", John Wiley and Sons, 2011.

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