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ARTICLE

A Triangular Plate Element with Drilling Degrees of Freedom, for Large Rotation Analyses of Built-up Plate/Shell Structures, Based on the Reissner Variational Principle and the von Karman Nonlinear Theory in the Co-rotational Reference Frame

Y.C. Cai^1,2, J.K. Paik³, S.N. Atluri²

1 Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai 200092, P.R.China
2 Center for Aerospace Research & Education, University of California, Irvine
3 Lloyd’s Register Educational Trust (LRET) Center of Excellence, Pusan National University, Korea

Computer Modeling in Engineering & Sciences 2010, 61(3), 273-312. https://doi.org/10.3970/cmes.2010.061.273

Abstract

This paper presents an elementary finite element method for geometrically nonlinear large rotation analyses of built-up plate/shell structures comprising of thin members. The tangent stiffness matrix of the element in the updated Lagrangian co-rotational reference frame is developed, based on the von Karman nonlinear theory of plates, and the Reissner variational principle, allowing for unsymmetric stresses and drilling rotations, useful in the analysis of built-up plate and shell structure. The finite rotation of the co-rotational reference frame relative to a globally fixed Cartesian frame, is simply determined from the finite displacement vectors of the nodes of the element in the global reference frame, thus allowing for an elementary transformation of the tangent stiffness matrix from the updated co-rotational reference frame to the globally fixed Cartesian frame. The element employed here is a 3-node plate element with 6 degrees of freedom per node, including 1 drilling degree of freedom and 5 degrees of freedom [3 displacements, and the derivatives of the transverse displacement around two independent axes in the plane of the plate in the co-rotational reference frame]. The (18×18) tangent stiffness matrices of the plate element in the updated Lagrangian co-rotational reference frame are derived, based on the assumptins that: (1) the inplane stress resultants N_αβ (unsymmetric) are constant in each element; (2) the bending moments M_αβ (symmetric) are linear and C⁰ within each element; and (3) the transverse rotations θ_i (including the drilling degrees of θ₃) are linear and C⁰ within each element. When compared to the primal approach wherein C¹ continuous trial functions for transverse displacements over each element are necessary, the trial functions for the transverse bending moments and the rotations are very simple in the current approach, and can be assumed to be linear within each element. The present (18×18) tangent stiffness matrices of the plate, based on the Reissner variational principle and the von Karman type simplified nolinear plate theory in the co-rotational reference frame, lead to analyses, which are much simpler and more physically-based, than many others in the literature for large rotation/deformation analysis of built-up plate/shell structures [such as component plates joined at an angle]. Numerical examples demonstrate the accuracy and robustness of the present method.

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