Keywords: geometrically nonlinear analysis, triangular finite elements, shell structures, load perturbation, Newton-Rapson method, co-rotational approach.

The co-rotational approach is one of the ways to describe the response of a finite element in a continuum. In this approach the local displacements and the local stiffness matrix of the element are described using a flat coordinate system which is rotated with the element edges during deformation. Since the local deformations are measured with respect to this coordinate system, they are usually assumed to be small even for large rotations analysis. Wempner [1] who may be regarded as the first to introduce this approach indicated that the rigid part of the motion is involved with the geometrical effects (geometric stiffness matrix) whereas the deformational part is involved with the constitutive laws of the elements. The main advantages of the co-rotational procedure are its simplicity and generality.

In the literature there are various applications of this approach e.g. Argyris et al. [2], Bathe and Ho [3], Levy and Gal [4], Levy and Spillers [5], Pacoste [6], Peng and Crisfield [7], Yang and Chang [8]. In some papers the local displacements are considered as small while in others they are considered as moderate or large. Furthermore, in some papers the local displacements are considered as pure (i.e. without rigid body components) while in other cases rigid body motion is first removed. Since the co-rotational formulation does not enforce large strain assumptions, various types of assumptions (e.g. small/moderate/large displacements and small/moderate/large rotations) may be found in the literature for the derivation of the geometric stiffness matrix. In some papers the geometric effects stem from rigid body motion considerations only, while in others this is not the case. In some papers the finite rotations are considered as small (as a vector) while in others they are taken as finite (as a pseudo-vector).

This paper presents a unique formulation methodology for flat triangular shell finite elements using the perturbation method (see also Levy and Spillers [5]). Two main features are concerned with the proposed formulation. The first is the derivation of the geometric stiffness matrix in the element local coordinate system and the second is stress retrieval. The derivation of the geometric triangular shell elements separates the in-plane and the out-of-plane geometric effects, thus enabling gradients of the nodal force vector that define the geometric stiffness matrix to be taken in local coordinates. Calculation of stresses using small strain assumption is enabled because rigid body motion is removed from the total deformation with rotations treated as finite using a special procedure which is introduce for that purpose. This approach is complete in the sense that all contributions to the response of the same order of magnitude are included. It depends on an a priori chosen linear elastic finite element it is independent of large strain formulations that are needed otherwise.

This formulation has been successfully applied to perform analysis of thin [4], thick [9] and composite material [10] shell structures with results being in good agreement for a diverse set of benchmark examples it was tested on.

1

Wempner, G., "Finite Elements, Finite rotations and small strains of flexible shells", Int. J. Solids Struct., 5, 117-153, 1969. doi:10.1016/0020-7683(69)90025-0

2

Argyris, J.H., Tenek, L., Olofsson, L., "TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells", Comp. Meth. Appl. Mech. & Engng., 145, 11-85, 1997. doi:10.1016/S0045-7825(96)01233-9

3

Bathe, K.J., Ho, L.W., "A simple and effective element for analysis of general shell structures", Computers & Structures, 13, 673-681, 1980. doi:10.1016/0045-7949(81)90029-8

4

Levy, R., Gal, E., "Triangular shell element for large rotations analysis", AIAA Journal. 41(12), 2505-2508, 2003. doi:10.2514/2.6854

5

Levy, R., Spillers. W.R., "Analysis of Geometrically Nonlinear Structures", 2nd edition, Kluwer Academic Publishers, Dordtrecht, 2002.

6

Pacoste, C., "Co-rotational flat facet triangular elements for shell instability analyses", Comp. Meth. Appl. Mech. & Engng., 156, 75-110, 1998. doi:10.1016/S0045-7825(98)80004-2

7

Peng, X., Crisfield, M.A., "A consistent co-rotational formulation for shell using the constant stress/ constant moment triangle", Int. J. Num. Meth. Engng., 35, 1829-1847, 1992. doi:10.1002/nme.1620350907

8

Yang, Y.B., Chang, J.T., "Derivation of a geometric nonlinear triangular plate element by rigid-body concept", Bulletin of The International Association for Shell and Spatial Structures, 39, n. 127, 77-84, 1998.

9

Levy, R., Gal, E., "The Geometric Stiffness of Thick Shell Triangular Finite Elements for Large Rotations", International Journal for Numerical Methods in Engineering, (in print).

10

Gal, E., Levy, R., "The Geometric Stiffness of Triangular Composite-Materials Shell Elements", Computers & Structures, (in print).

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