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Research Papers

A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

[+] Author and Article Information
Guangyu Shi1

Department of Mechanics, Tianjin University, Tianjin 300072, Chinashi_guangyu@163.com

George Z. Voyiadjis

Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803voyiadjis@eng.lsu.edu

1

Corresponding author.

J. Appl. Mech 78(2), 021019 (Dec 20, 2010) (11 pages) doi:10.1115/1.4002594 History: Received September 04, 2009; Revised September 18, 2010; Posted September 21, 2010; Published December 20, 2010; Online December 20, 2010

This paper presents the derivation of a new beam theory with the sixth-order differential equilibrium equations for the analysis of shear deformable beams. A sixth-order beam theory is desirable since the displacement constraints of some typical shear flexible beams clearly indicate that the boundary conditions corresponding to these constraints can be properly satisfied only by the boundary conditions associated with the sixth-order differential equilibrium equations as opposed to the fourth-order equilibrium equations in Timoshenko beam theory. The present beam theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the authors, a system of sixth-order differential equilibrium equations in terms of two generalized displacements w and ϕx of beam cross sections, and three boundary conditions at each end of shear deformable beams. A technique for the analytical solution of the new beam theory is also presented. To demonstrate the advantages and accuracy of the new sixth-order beam theory for the analysis of shear flexible beams, the proposed beam theory is applied to solve analytically three classical beam bending problems to which the fourth-order beam theory of Timoshenko has created some questions on the boundary conditions. The present solutions of these examples agree well with the elasticity solutions, and in particular they also show that the present sixth-order beam theory is capable of characterizing some boundary layer behavior near the beam ends or loading points.

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Figures

Grahic Jump Location
Figure 1

A cantilevered beam with transverse load at the free end

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Figure 2

A simply supported beam with a concentrated load

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Figure 3

The modifying terms ξ to the central deflections of the elementary beam solution

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Figure 4

The variation of shear force Qx(x) along a three-point loaded beam (L=10h)

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