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Research Papers

Post-Buckling Analysis of Curved Beams

[+] Author and Article Information
Zhichao Fan, Qiang Ma, Yuan Liu

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China

Jian Wu

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China
e-mail: wujian@tsinghua.edu.cn

Yewang Su

State Key Laboratory of Nonlinear Mechanics,
Institute of Mechanics,
Chinese Academy of Sciences,
Beijing 100190, China

Keh-Chih Hwang

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China
e-mail: huangkz@tsinghua.edu.cn

1Corresponding authors.

Manuscript received November 16, 2016; final manuscript received December 15, 2016; published online January 24, 2017. Assoc. Editor: Daining Fang.

J. Appl. Mech 84(3), 031007 (Jan 24, 2017) (15 pages) Paper No: JAM-16-1561; doi: 10.1115/1.4035534 History: Received November 16, 2016; Revised December 15, 2016

Stretchability of the stretchable and flexible electronics involves the post-buckling behaviors of internal connectors that are designed into various shapes of curved beams. The linear displacement–curvature relation is often used in the existing post-buckling analyses. Koiter pointed out that the post-buckling analysis needs to account for curvature up to the fourth power of displacements. A systematic method is established for the accurate post-buckling analysis of curved beams in this paper. It is shown that the nonlinear terms in curvature should be retained, which is consistent with Koiter's post-buckling theory. The stretchability and strain of the curved beams under different loads can be accurately obtained with this method.

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References

Figures

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Fig. 1

Schematic illustration of initial and deformed curved beam

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Fig. 2

Schematic illustration of elastic ring under uniform compression

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Fig. 3

The ratio of load to critical load, p¯2/p¯2(0), versus the normalized displacement, U2max/(2R), during post-buckling, which is consistent with the results of Carrier's model

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Fig. 4

Schematic illustration of boundary conditions of circular beam under bending moment load

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Fig. 5

The ratio, γε, with and without the nonlinear terms in curvature: (a) the ratio, γε, versus the Poisson's ratio ν for the normalized angle α¯=1/4, 1/2, and 2/3. (b) The ratio, γε, versus the normalized angle α¯ for gold (i.e., ν=0.42).

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Fig. 6

The normalized maximum principal strain εmaxR/w with and without the nonlinear terms in curvature versus the shortening ratio γd for ν=0.42 and α¯=1/4, 1/2, and 2/3

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Fig. 7

The distributions of the twist angle of the circular beam for the different shortening ratios, γd  = 0.1, 0.2, and 0.3

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Fig. 8

Schematic illustration of boundary conditions of circular beam under uniform pressure

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Fig. 9

The ratio of load to critical load, p¯2/p¯2(0), with and without the nonlinear terms in curvature versus the normalized out-of-plane displacement, U1max/LS, for ν=0.42 and α¯=1/4, 1/2, and 2/3

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