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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 86
Edited by: B.H.V. Topping
Paper 110

Linear and Geometrically Nonlinear Analysis of Shell Elements with Rotational Degrees using the Quadrilateral Area Coordinates Method

L. Kang and Q.L. Zhang

Department of Building Engineering, Tongji University, Shanghai, China

Full Bibliographic Reference for this paper
L. Kang, Q.L. Zhang, "Linear and Geometrically Nonlinear Analysis of Shell Elements with Rotational Degrees using the Quadrilateral Area Coordinates Method", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 110, 2007. doi:10.4203/ccp.86.110
Keywords: shell element, rotational degrees of freedom, quadrilateral area coordinates, generalized conforming conditions, trying function, geometrically nonlinear.

A new shell element with rotational degrees of freedom is proposed in this paper. Instead of the traditional isoparametric coordinates, the quadrilateral area coordinates are used to establish the linear and geometrically nonlinear formulations of the novel element. The transformation between the area coordinates and the Cartesian coordinates is always linear. Thus, the order of the displacement field expressed by the area coordinates will not vary with the mesh distortion. Based on the quadrilateral area coordinate method presented in references [1,2,3], several membrane elements and thin plate-bending element were successfully developed.

During the deductive procedure, the rotational degrees of freedom are added in through utheta,vtheta, the displacement fields generated by angular displacements. The characteristic matrix of the nodal displacements is defined as A and C is defined as the characteristic matrix of the generalized displacements. The transfer matrix, G, shows the relationship between the generalized displacements and the nodal displacements. In order to obtain the shell element with good performance, the function adopted is of quadruplicate order. The displacement field on the boundary and 60 generalized conforming conditions are then introduced to determine all unknown parameters. The different matrix C can be obtained by adopting different compatibility conditions. There are many kinds of association in the generalized displacement compatibility conditions, so long as the compatibility conditions are separated, namely the matrix C is reversible.

The features of the new shell element are as follows: It is a four-node shell element with six DOF at each node. Thus it can be used in place of any four-node shell element. It has better bending performance and is insensitive to mesh distortion. The new element describes bending to a quadratic order and adds in the rotational degrees of freedom, using incompatible shape functions. This higher order description is important to shell structures. The linear and nonlinear results of the element is kind of reliability. In order to work out the elements easily, a universal method with characteristic matrix of nodal displacements, characteristic matrix of generalized displacements and transfer matrix is proposed during the finite element construction procedure.

The efficiency and suitability of the element are tested and demonstrated by applying them to the linear and geometrically nonlinear analysis of some numerical examples. The results obtained by the present element are compared with those available in the literature and show that the new element possesses the advantages of high accuracy and reliability.

Chen X.M., Cen S., Long Y.Q., Yao Z.H., "Membrane elements insensitive to distortion using the quadrilateral area coordinate method", Computers and Structures, 82(1), 35-54, 2004. doi:10.1016/j.compstruc.2003.08.004
Soh A.K., Long Y.Q., Cen S., "Development of eight-node quadrilateral membrane elements using the area coordinates method", Computational Mechanics, 25(4), 376-384, 2000. doi:10.1007/s004660050484
Soh A.K., Long Z.F., Cen S., "Development of a new quadrilateral thin plate element using area coordinates", Computer Methods in Applied Mechanics and Engineering, 190(8-10), 979-987, 2000. doi:10.1016/S0045-7825(99)00457-0

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