Weak form quadrature element analysis of Bickford Beams

doi:10.1016/j.euromechsol.2010.03.012

European Journal of Mechanics - A/Solids

Volume 29, Issue 5, September–October 2010, Pages 851-858

https://doi.org/10.1016/j.euromechsol.2010.03.012 Get rights and content

Abstract

In this paper the recently proposed weak form quadrature element method is applied to high order beam analysis. Differing from other numerical methods, only one element is needed in the weak form quadrature element analysis as long as the cross-section of the beam is constant or varies continuously. Flexural, vibrational and eigenbuckling analyses of homogeneous beams are performed based on the Bickford beam theory. Numerical examples are presented to illustrate the accuracy and efficiency of the developed method.

Introduction

The Euler–Bernoulli beam theory and the Timoshenko beam theory have been widely used in analysis of structural components. For slender structural members where the transverse shear deformation is often negligible, the Euler–Bernoulli beam theory is able to provide accurate predictions of structural response. The Timoshenko beam theory (Timoshenko, 1921) which assumes constant transverse shear distribution is often used when the need of consideration of transverse shear deformation arises. This is especially true in analysis of stubby beams. To reconcile the simplified shear distribution with the reality, a shear coefficient is needed in the Timoshenko beam theory (Cowper, 1966, Gruttmann and Wagner, 2001). With the development of new materials, beam theories accommodating variable shear deformation through beam height are demanded. Among high order beam theories, the Bickford theory (Bickford, 1982, Levinson, 1985) is variationally consistent and assumes simple parabolic variation of shear deformation through beam height. The Bickford beam theory, as enunciated by Bickford (Bickford, 1982), eliminates the burden of selection of a proper shear coefficient for a beam in the Timoshenko beam theory and secures reliable predictions for high frequency phenomena. It indeed offers a good compromise between accuracy and simplicity (Franciosi and Tomasiello, 2007).

Over the past few decades, numerical methods such as the finite element method have been used to assess behavior of complicated structural components. In finite element analysis, even a homogeneous uniform beam is often discretized into a number of elements to acquire accurate representation of deformation, let alone beams either with variable cross-section or under complex loading. Consequently, dozens of degrees of freedom may be needed to acquire accurate evaluation of a single beam. Recently, the weak form quadrature element method (QEM) proposed by Zhong and his colleagues has been applied to various structural problems (Zhong and Yu, 2007, Zhong and Yu, 2009, Zhong and Gao, 2009, Mo et al., 2009). In the case of a beam with either variable cross-section or constant cross-section, one quadrature element is sufficient. The weak form quadrature element analysis starts with efficient evaluation of the integrals involved in the weak form description of the problem. It is then followed by the approximation of the derivatives at the integration sampling points with differential quadrature analog. As a result, the concept of shape functions that is used in the finite element method is diluted and determination of equivalent nodal forces is rendered unnecessary.

In this paper, the weak form quadrature element method is used to study Bickford beams. The preceding words “weak form” are used to emphasize that the present approach is based on variational principles. It distinguishes with the strong form quadrature element method (Striz et al., 1994, Wang and Gu, 1997) that copes with governing differential equations directly. In the full text of the paper, the term – quadrature element refers exclusively to the weak form approach. Flexural, vibrational and buckling responses of homogeneous beams are studied. Results are compared with other available solutions and those of the commercial finite element code ANSYS (ANSYS, 2003). The convergence and the high computational efficiency of the weak form quadrature element method are demonstrated.

Section snippets

Bickford beam theory

Consider an isotropic elastic beam of length L, width b and height h. The Cartesian coordinate system is shown in Fig. 1. Without losing generality, rectangular cross-section is chosen for the beam. Assume further that the beam is deformed in the Oxz plane only. The three displacement components in accordance with the Bickford beam theory are expressed as $u_{x} (x, z, t) = u (x, t) + z ϕ (x, t) - \frac{4}{3 h^{2}} z^{3} (ϕ (x, t) + \frac{\partial w}{\partial x}), u_{y} = 0, u_{z} = w (x, t)$ where (u_x, u_y, u_z) correspond to coordinates (x, y, z); t is temporal variable; u(x

Example one – flexural analysis of a cantilever beam and a simply supported beam

In this example, flexural analysis of Bickford beams is conducted. Two simple cases whose exact solutions were given by Bickford (Bickford, 1982) are studied to demonstrate the convergence of the quadrature element solution. Case I is a cantilever beam subjected to a concentrated load Q_L at the free tip. The analytical deflection of the beam in this case is given by $w_{c} = \frac{1}{3} \frac{Q_{L} L^{3}}{E I} {(\frac{x}{L})}^{2} (\frac{3}{2} - \frac{1}{2} \frac{x}{L}) + \frac{1}{5} \frac{Q_{L} L^{3}}{E I} (1 + υ) {(\frac{h}{L})}^{2} (\frac{x}{L} - \frac{\sin h η_{1} L - \sin h η_{1} (L - x)}{η_{1} L \cos h η_{1} L}), η_{1} = {(\frac{420}{1 + υ})}^{1 / 2} \frac{1}{h}$ where υ is the Poisson’s ratio of

Conclusion

Bickford beams have been studied using the weak form quadrature element method. The high computational efficiency of weak form quadrature element analysis has been demonstrated. It has been shown that quadrature element techniques work very well for Bickford beams with either constant or varying cross-section. One element is sufficient and the number of nodes is adjustable in accordance with the convergence requirement as long as the cross-section of the beam varies continuously. In addition,

Acknowledgments

The present work was undertaken under the support of National Natural Science Foundation of China (No. 50778104).

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Weak form quadrature element analysis of Bickford Beams

Abstract

Introduction

Section snippets

Bickford beam theory

Example one – flexural analysis of a cantilever beam and a simply supported beam

Conclusion

Acknowledgments

Journal of Mathematical Analysis and Applications

International Journal of Mechanical Sciences

Mechanics Research Communications

Tsinghua Science and Technology

International Journal of Solids and Structures

Applied Mathematical Modelling

Swanson Analysis System

Differential quadrature method in computational mechanics: a review

Applied Mechanics Reviews

A consistent higher order beam theory

Developments in Theoretical and Applied Mechanics