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 $\varphi x$ 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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# A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions

Guangyu Shi

,
Guangyu Shi

Department of Mechanics,

shi_guangyu@163.com
Tianjin University

, Tianjin 300072, China
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George Z. Voyiadjis

George Z. Voyiadjis

Department of Civil and Environmental Engineering,

voyiadjis@eng.lsu.edu
Louisiana State University

, Baton Rouge, LA 70803
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Guangyu Shi

George Z. Voyiadjis

Department of Civil and Environmental Engineering,

Louisiana State University

, Baton Rouge, LA 70803voyiadjis@eng.lsu.edu

*J. Appl. Mech*. Mar 2011, 78(2): 021019 (11 pages)

**Published Online:**December 20, 2010

Article history

Received:

September 4, 2009

Revised:

September 18, 2010

Posted:

September 21, 2010

Published:

December 20, 2010

Online:

December 20, 2010

Citation

Shi, G., and Voyiadjis, G. Z. (December 20, 2010). "A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions." ASME. *J. Appl. Mech*. March 2011; 78(2): 021019. https://doi.org/10.1115/1.4002594

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