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Research Papers

On the Effectiveness of Higher-Order Terms in Refined Beam Theories

[+] Author and Article Information
Erasmo Carrera1

Department of Aeronautics and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italyerasmo.carrera@polito.it

Marco Petrolo

Department of Aeronautics and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italymarco.petrolo@polito.it

1

Corresponding author.

J. Appl. Mech 78(2), 021013 (Nov 10, 2010) (17 pages) doi:10.1115/1.4002207 History: Received June 23, 2009; Revised July 15, 2010; Published November 10, 2010; Online November 10, 2010

This work deals with refined theories for beams with an increasing number of displacement variables. Reference has been made to the asymptotic and axiomatic methods. A Taylor-type expansion up to the fourth-order has been assumed over the section coordinates. The finite element governing equations have been derived in the framework of the Carrera unified formulation (CUF). The effectiveness of each expansion term, that is, of each displacement variable, has been established numerically considering various problems (traction, bending, and torsion), several beam sections (square, annular, and airfoil-type), and different beam slenderness ratios. The accuracy of these theories have been evaluated for displacement and stress components at different points over the section and along the beam axis. Error-type parameters have been introduced to establish the role played by each generalized displacement variable. It has been found that the number of terms that have to be retained for each of the considered beam theories is closely related to the addressed problem; different variables are requested to obtain accurate results for different problems. It has, therefore, been concluded that the full implementation of CUF, retaining all the available terms, would avoid the need of changing the theory when a problem is changed (geometries and/or loading conditions), as what happens in most engineering problems. On the other hand, CUF could be used to construct suitable beam theories in view of the fulfillment of prescribed accuracies.

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Figures

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Figure 1

Coordinate frame and geometry of a rectangular beam

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Figure 2

Coordinate frame and geometry of a rectangular cross section

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Figure 3

Coordinate frame and geometry of an annular cross section

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Figure 4

Coordinate frame and geometry of an airfoil cross section

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