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DEVELOPMENTS AND APPLICATIONS IN COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
A Modern and Compact Way to Formulate Classical and Advanced Beam Theories
E. Carrera1, G. Giunta2 and M. Petrolo1,3
1Department of Aeronautic and Space Engineering, Politecnico di Torino, Italy
E. Carrera, G. Giunta, M. Petrolo, "A Modern and Compact Way to Formulate Classical and Advanced Beam Theories", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero, (Editors), "Developments and Applications in Computational Structures Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 4, pp 75-112, 2010. doi:10.4203/csets.25.4
Keywords: refined beam theories, finite elements, unified formulation, shell capabilities, variational statement, thin walled sections.
This chapter presents a unified formulation for refined beam theories. This work is embedded in the framework of the Carrera unified formulation (CUF) for higher-order beam  and plate/shell  models. The CUF is hierarchical, that is, the order of the formulation is considered as a free parameter of the analysis. The hierarchical features of CUF permit us to deal with refined beam theories in a unified manner with no need of ad hoc formulations. Moreover, arbitrary cross-section geometries can be analyzed by means of the present beam modeling.
Classical beam models are those by Euler-Bernoulli and Timoshenko. The former does not account for the transverse shear effects on the cross-section deformations. The latter provides a model that foresees a constant shear deformation distribution above the cross-section. These theories work properly when slender and compact structures made of isotropic materials are considered. The structural analysis of thin-walled or short beams made of anisotropic materials requires more sophisticated models. Higher order beam models are able to overcome the limits of classical theories without a severe increase in computational costs.
Different structural problems have been addressed herein. Static and free vibration analyses have been conducted on compact and thin-walled beams by considering various geometries (rectangular, annular, airfoil-shaped, C-shaped, bridge-like), loading conditions (distributed and point loads, bending and torsional), materials (isotropic and FGM). Closed form, Navier type solutions as well as finite element approaches have been exploited. Shell and solid element models have been used for comparison purposes. The effectiveness analysis of the higher order terms is presented as the concluding section of this chapter.
The static analysis of a compact short beam has shown the beneficial effect of higher order terms in cases of bending and torsional loadings. The analysis of conventional and unconventional beam structures has highlighted the capability of the CUF beam in dealing with arbitrary geometries and detecting non-classical effects such as the out-of-plane warping of the cross-section. The effects of a point load on a thin-walled cylinder have been detected as well as the effects of a coupled bending-torsional load on a C-shaped beam. The enhanced features of refined beam models have been pointed out further using the free vibration analysis of an annular cross-section since the shell-like natural modes have been correctly computed. The use of higher-order models is also important to detect the shear stress distribution above the cross-section as shown by the static analysis of the bridge-like structure and the beam made of functionally graded materials. In all the structural problems presented, the computational cost of the higher-order beam models has been significantly smaller than those requested by shell and solid elements.
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