Keywords: geometric stiffness matrix, higher-order stiffness matrix, rigid body motion, triangular flat plate element.

This paper presents a simple concept to derive the higher-order stiffness matrix of a triangular flat plate element for the analysis of shell or plate structures subjected to heavy loads. This method can be stated as follows; when there is a set of incremental nodal forces acting on the element, let the element undergo a small rigid body motion, the incremental forces should keep their magnitude and follow the rigid body motion just as the initial nodal forces acting on the element behave. In this manner the higher-order stiffness matrix of the element can be derived with the help of the existing geometric stiffness matrix derived by researchers. This physical phenomenon can be interpreted as follows when the incremental forces due to extensions in the element at each incremental step, should also enforce the equilibrium conditions under the rigid body motion. This is an extension of the concept of the so-called rigid body rule. The derived higher-order stiffness matrix of the element has a clear physical meaning, i.e. it is analogous to the properties of the geometric stiffness matrix of the element.

In general, engineers have to consider both the geometric and material nonlinearity in the nonlinear analysis of structures. In order to capture the geometric nonlinearity of structures with large deformations, the geometric stiffness matrix should be taken into account in the tangent stiffness matrix of the element in the analysis of structures. In the literature, based on the various variational principles many nonlinear elements have been derived to conduct the geometric nonlinearity of the structures [1]. Belytschko et al. [2]; Hughes and Liu [3]; Yang and Kuo [4], and others have derived geometrically nonlinear shell or plate elements and used effective numerical methods to solve the nonlinear problems. But the derivation of the geometric stiffness matrix of a flat plate element is somewhat difficult and tedious based on the variational principles because of the integrals involving the coupling effects of the virtual nonlinear strain and initials stresses. For this reason mentioned, Yang, Chang, and Yau [5] proposed a simple concept to reduce the difficulty of deriving a geometric stiffness matrix of a triangular flat plate element. Based on the conventional variational principle the derivation of a higher-order stiffness matrix that associates the virtual nonlinear strain and incremental stresses has been shown to have tedious derivations and lengthy expressions just as the derivations of the geometric stiffness matrix do. For this reason as mentioned above there are two main purposes herein to present in this paper. Because the effect of the higher-order stiffness matrix of a plate element has been less noted in nonlinear analysis, so to investigate the effect in the nonlinear analysis of shell and plate structures is an interesting work. The other purpose is that because the derivation of the higher-order stiffness matrix is tedious and has lengthy expressions by conventional variational principle. So the purpose of an engineering approximation to these integral terms that are associated with virtual nonlinear strain coupling with incremental stress terms is an interesting research topic.

In short based on the concept of rigid body motion, equilibrium conditions of the incremental nodal forces in the element in each incremental step, and the derived geometric stiffness matrix given by researchers, one can derive the higher-order stiffness matrix. One merely changes the initial forces in the geometric stiffness matrix instead of the incremental forces. Then an explicit form of the higher-order stiffness matrix of the triangular flat element can be obtained easily, because both higher-order stiffness matrix and geometric stiffness matrix are similar in expression but different in coefficients. Herein we adopt the geometric stiffness matrix of a triangular flat plate derived by Chang and Huang [6] as a basis for the derivation of the higher-order stiffness matrix. Finally, a numerical example is analysed using the proposed elements in this paper.

1

Washizu, K., Variational Methods in Elasticity and Plasticity, 2nd Edition, Pergamon Press, Oxford, England, 1975.

2

Belytschko, T., Moran, B. and Liu, W. K., Nonlinear Finite Elements for Continua and Structures, John, Wiley, 2000.

3

Hughes, T.J.R., and Liu, W.K., "Non linear Finite Element Analysis of Shells: Part II", Comput. Methods. Appl. Mech. Eng., 27, 331-62, 1981. doi:10.1016/0045-7825(81)90148-1

4

Yang, Y.B., and Kuo, S.R., Theory & Analysis of Nonlinear Framed Structures, Prentice Hall, Englewood Cliffs, N.J.,1994.

5

Yang, Y.B., Chang, J.T., and Yau, J.D., "A Simple Nonlinear Triangular Plate Element and Strategies of Computation for Nonlinear Analysis", Comput. Methods. Appl. Mech. Eng., 178, 307-321, 1999. doi:10.1016/S0045-7825(99)00022-5

6

Chang, J.T. and Hung, I.D. "An Explicit Geometric Stiffness Matrix of a Triangular Flat Plate Element for the Geometric Non-linear Analysis of Shell Structures", In "Proceedings of the Ninth Conference on Civil and Structural Engineering Computing", Topping, B.H.V. (Editor), Civil Comp Press, Stirling, Scotland. Paper 34, 2003. doi:10.4203/ccp.77.34

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