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Research Papers

Inextensible Beam and Plate Theory: Computational Analysis and Comparison With Experiment

[+] Author and Article Information
Deman Tang

Research Scientist
Department of Mechanical Engineering and
Materials Science,
Duke University,
Box 90300, Hudson Hall,
Durham, NC 27708-0300
e-mail: demant@duke.edu

Minghui Zhao

Bejing University of Technology,
Beijing 100124, China;
Visiting Scholar
Duke University,
Durham, NC 27708
e-mail: zmh850308@163.com

Earl H. Dowell

William Holland Hall Professor
Department of Mechanical Engineering and
Materials Science,
Box 90300, Hudson Hall,
Durham, NC 27708-0300
e-mail: dowell@ee.duke.edu

Strictly speaking, in a linear theory, the axial deflection is neglected or zero. What is shown here is the u computed from the relationship εxx=0=(u/x)+(1/2)(w/x)2.

Comparisons have also been made between inextensible and extensible beam theory, following the Novozhilov formulation with excellent agreement. Due to space limitations, these results are not shown.

1Corresponding author.

Manuscript received June 11, 2013; final manuscript received February 3, 2014; accepted manuscript posted February 11, 2014; published online February 28, 2014. Assoc. Editor: Glaucio H. Paulino.

J. Appl. Mech. 81(6), 061009 (Feb 28, 2014) (10 pages) Paper No: JAM-13-1239; doi: 10.1115/1.4026800 History: Received June 11, 2013; Revised February 03, 2014; Accepted February 07, 2014

A new inextensible theory of beam and plate deformation has been developed. For validation of this inextensible beam and plate theory, computational codes for the static and dynamic nonlinear beam and plate modal equations have been developed. The computations and experiments for static loading and deformation of a beam show that the inextensible theory produces results in excellent agreement with experiment. Also, a comparison of the inextensible theory for a plate with a static experiment is encouraging. Finally, a numerical study of dynamic deflection for an inextensible beam and plate has also been made. The results show a hysteresis dynamic response that depends on whether the excitation frequency is increasing or decreasing for the stiffness nonlinearity only or for the inertia nonlinearity only. The inertia nonlinear force has a significant effect on the dynamic response in the resonant frequency range.

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Figures

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Fig. 1

Picture of the static deflection measurement for a cantilever beam with gravity (a) and the correlations between the computational nondimensional tip flap deflection (b), axial deflection (c), and experiment

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Fig. 2

Picture of the static deflection measurement for a cantilever plate with a gravity load, f0 = 0.16 kg at x¯=y¯=1 (a) and the computational and experimental tip static deflection at x¯=y¯=1 of a plate clamped along one edge versus gravity load acting at x¯=y¯=1 (b)

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Fig. 3

Computational and experimental static deflection versus the nondimensional plate position, x¯ at y¯ = 1 of a plate clamped along one edge for gravity loading f0 = 0.04 and 0.16 kg acting at x¯=y¯=1

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Fig. 4

Computational “time history” of dynamic deflection of a beam for increasing and decreasing excitation frequency. The dynamic loading fs is 0.025 kg acting at x¯=0.7.

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Fig. 5

Dynamic rms response amplitude for increasing and decreasing excitation frequency for the dynamic loading fs 0.025 kg and damping ratio ξ = 0.005

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Fig. 6

Dynamic rms response amplitude for increasing and decreasing excitation frequency when varying the dynamic loading fs from 0.005 kg to 0.05 kg and the damping ratio ξ = 0.005

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Fig. 7

Dynamic rms response amplitude for increasing and decreasing excitation frequency with and without the effect of inertia nonlinear force and the stiffness nonlinear force for the damping ratio ξ = 0.01 and fs = 0.025 kg

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Fig. 8

Computational time history of dynamic deflection of a plate for increasing and decreasing excitation frequency. The dynamic loading fs is 0.01 kg acting at x¯=0.5, y¯ = 0.75 and ξ = 0.01.

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Fig. 9

Dynamic rms response amplitude for increasing and decreasing excitation frequency for the dynamic loading fs is 0.01 kg and ξ = 0.01

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Fig. 10

Dynamic rms response amplitude for increasing and decreasing excitation frequency for the dynamic loading, fs = 0.001 and 0.005 kg and ξ = 0.01

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Fig. 11

Dynamic rms response amplitude for increasing and decreasing excitation frequency including the effects of inertia nonlinear force and stiffness nonlinear force for the damping ratio ξ = 0.01 and fs = 0.01 kg

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