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Research Papers

Modeling of a One-Sided Bonded and Rigid Constraint Using Beam Theory

[+] Author and Article Information
Peter J. Ryan

Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115; Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115

George G. Adams

Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115adams@coe.neu.edu

Nicol E. McGruer

Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115

J. Appl. Mech 75(3), 031008 (Apr 08, 2008) (6 pages) doi:10.1115/1.2839898 History: Received February 09, 2007; Revised August 28, 2007; Published April 08, 2008

In beam theory, constraints can be classified as fixed/pinned depending on whether the rotational stiffness of the support is much greater/less than the rotational stiffness of the freestanding portion. For intermediate values of the rotational stiffness of the support, the boundary conditions must account for the finite rotational stiffness of the constraint. In many applications, particularly in microelectromechanical systems and nanomechanics, the constraints exist only on one side of the beam. In such cases, it may appear at first that the same conditions on the constraint stiffness hold. However, it is the purpose of this paper to demonstrate that even if the beam is perfectly bonded on one side only to a completely rigid constraining surface, the proper model for the boundary conditions for the beam still needs to account for beam deformation in the bonded region. The use of a modified beam theory, which accounts for bending, shear, and extensional deformation in the bonded region, is required in order to model this behavior. Examples are given for cantilever, bridge, and guided structures subjected to either transverse loads or residual stresses. The results show significant differences from the ideal bond case. Comparisons made to a three-dimensional finite element analysis show a good agreement.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) A beam with a classical fixed support. (b) A cantilever beam with a one-sided fixed support. (c) A beam with one-sided fixed supports at each end, i.e., a bridge structure.

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Figure 2

(a) A one-sided support subjected to an applied moment M0. (b) A thin film under a uniform tensile prestress before release. (c) The anchor and the free portions of a cantilever beam after release from the substrate.

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Figure 3

Differential elements for a modified beam theory that includes extensional, shear, and bending deformations

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Figure 4

Correction factors due to shear deformation and rotational compliance for a cantilever beam with an end load

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Figure 5

Correction factors due to shear deformation and rotational compliance for a bridge structure with a load at its midpoint

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Figure 6

Deformation of different regions of a bridge structure due to prestress

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Figure 7

Bridge structure midpoint deflection versus compressive prestrain for different L∕h. The dashed lines represent the arbitrary buckling deflections given by the classical beam theory.

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Figure 8

Bridge structure midpoint deflection versus tensile prestrain for different L∕h

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