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Research Papers

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

[+] Author and Article Information
Yinhuan Zheng

School of Mechanical
and Electronic Engineering,
Wuhan University of Technology,
Wuhan 430070, Hubei, China
e-mail: zhengyinhuan@whut.edu.cn

Ahmed A. Shabana

Department of Mechanical
and Industrial Engineering,
University of Illinois at Chicago,
Chicago, IL 60607
e-mail: shabana@uic.edu

Dayu Zhang

National Key Laboratory of
Aerospace Flight Dynamics,
School of Astronautics,
Northwestern Polytechnical University,
Xi'an 710072, Shaanxi, China
e-mail: dyzhang@mail.nwpu.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 3, 2017; final manuscript received June 19, 2017; published online October 31, 2017. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 13(1), 011013 (Oct 31, 2017) (12 pages) Paper No: CND-17-1149; doi: 10.1115/1.4037226 History: Received April 03, 2017; Revised June 19, 2017

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

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Figures

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Fig. 2

Shear angle θ at a middle point (–□– M=1 N⋅m, –○– M=10 N⋅m)

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Fig. 3

Spatial derivative of shear angle θ at a middle point (–□– M=1 N⋅m, –○– M=10 N⋅m)

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Fig. 4

Free falling flexible pendulum

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Fig. 5

Vertical displacement at tip point using 6 ANCF finite elements (–□– method 1, –○– method 2, and –Δ– method 3)

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Fig. 6

Magnitude of the ry vector at tip point using method 2with 6 ANCF finite elements (–□– k=0, –○– k=103, –Δ– k=104)

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Fig. 7

Different curvature definitions at tip point using method 3 with 40 ANCF finite elements (–□– |rss|, –○– ∂θ/∂s)

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Fig. 8

Magnitude of the rss vector at the middle point using 40 ANCF finite elements (–□– method 1, –○– method 2, and –Δ– method 3)

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Fig. 9

Shear angle at middle point using 40 ANCF elements (–□– method 1, –○– method 2, and –Δ– method 3)

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Fig. 10

Energy balance obtained using method 2 with 6 ANCF finite elements (–□– kinetic energy, –○– elastic energy, –Δ– potential energy, and — total energy)

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Fig. 11

Energy balance obtained using method 3 with 6 ANCF finite elements (–□– kinetic energy, –○– elastic energy, –Δ– potential energy, and — total energy)

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Fig. 12

Derivative of θ with respect to the arc length at tip point using 40 bilinear CRBF elements (–□– method 1 and –Δ– method 3)

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Fig. 13

Shear angle at tip point using 24 bilinear CRBF elements (–□– method 1 and –Δ– method 3)

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