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Research Papers

A Planar Beam Finite-Element Formulation With Individually Interpolated Shear Deformation

[+] Author and Article Information
Vesa-Ville T. Hurskainen

School of Energy Systems,
Lappeenranta University of Technology,
Skinnarilankatu 34,
Lappeenranta 53851, Finland
e-mail: vesa-ville.hurskainen@lut.fi

Marko K. Matikainen

School of Energy Systems,
Lappeenranta University of Technology,
Skinnarilankatu 34,
Lappeenranta 53851, Finland
e-mail: marko.matikainen@lut.fi

Jia J. Wang

School of Mechatronic Engineering,
Harbin Institute of Technology,
Harbin 150001, China
e-mail: wangjia_hit@163.com

Aki M. Mikkola

School of Energy Systems,
Lappeenranta University of Technology,
Skinnarilankatu 34,
Lappeenranta 53851, Finland
e-mail: aki.mikkola@lut.fi

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 1, 2016; final manuscript received November 23, 2016; published online January 20, 2017. Assoc. Editor: José L. Escalona.

J. Comput. Nonlinear Dynam 12(4), 041007 (Jan 20, 2017) (8 pages) Paper No: CND-16-1313; doi: 10.1115/1.4035413 History: Received July 01, 2016; Revised November 23, 2016

This paper introduces a new planar gradient deficient beam element based on the absolute nodal coordinate formulation. In the proposed formulation, the centerline position is interpolated using cubic polynomials while shear deformation is taken into account via independently interpolated linear terms. The orientation of the cross section, which is defined by the axial slope of the element's centerline position combined with the independent shear terms, is coupled with the displacement field. A structural mechanics based formulation is used to describe the strain energy via generalized strains derived using a local element coordinate frame. The accuracy and the convergence properties of the proposed formulation are verified using numerical tests in both static and dynamics cases. The numerical results show good agreement with reference formulations in terms of accuracy and convergence.

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Figures

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Fig. 1

Two-node ANCF-based beam element

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Fig. 2

Description of shear deformation and cross section orientation for the proposed element

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Fig. 3

Static example: geometry and load description of a cantilever beam of length L under vertical tip load Fy

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Fig. 4

Comparison of static response: convergence of the vertical displacement of the free end of a beam in a large deformation case. Dotted reference lines indicate a convergence rate of O(n4). The relative error is calculated according to Eq. (26).

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Fig. 5

Dynamic example: geometry and load description of the free beam [15,16]

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Fig. 6

Comparison of dynamic response: vertical position of point B versus time

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Fig. 7

Comparison of dynamic response: absolute error of vertical displacement of point B versus time. The reference set is composed of results computed using the element formulation by Gerstmayr et al. [18].

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