Keywords: geometrically nonlinear analysis, flat triangular finite element, shell structures, drilling DOF, tangent stiffness matrix, load perturbation, Newton-Rapson.

This paper examines the performance of Allman's membrane finite element [1] as a supplementary membrane element for geometrically nonlinear analysis of shell structures. The examination is based on an original formulation that is also presented in this paper. The presented shell element is comprised from Allman's membrane element [1] and the discrete Kirchhoff theory (DKT) plate element [2].

The analysis of shells using a method which is based on the superposition of flat elements is sometimes characterized by singularities that are caused because the plate and membrane elements do not provide degrees of freedom for out-of-plane rotation perpendicular to the element's plane. Given a conventional formulation of shells, five degrees of freedom exist for each node at the element level. The local degrees of freedom are three spatial displacements and two rotations at each node. A global coordinates system provides six degrees of freedom at each node, of which three are spatial displacements and three are spatial rotations. If all the elements are located on the same plane, a mechanism would result. With an aim of solving this singularity Allman [1] presented a membrane element with a drilling DOF.

Here Allman's membrane element [1] was upgraded together with the DKT plate element [2] to perform geometrically nonlinear analysis of shell structures. Our formulation is based on load perturbation (see Green et al. [3]) of the linear discrete equilibrium equations of an element in global coordinates that leads to the description of the stiffness matrix as the gradient of the nodal force vector. In comparison to other publications [4,5] the proposed geometrically nonlinear analysis is somewhat different. Calculating the derivatives with respect to global coordinates, is avoided by introducing the out-of-plane effect due to small rigid body rotations (out-of-plane geometric stiffness matrix) as a supplement geometric stiffness to that obtained from the gradient in local coordinates (in-plane geometric stiffness matrix). A derivation of this type of perturbation is first order complete since: the in-plane-geometric stiffness matrix is a first order linearization of the perturbed nodal force vector in local coordinates; the out-of-plane stiffness matrix is a first order correction stemming from the change in the nodal force vector due to small rigid body rotations; Newton's method which is a 1st order Taylor expansion, is used for analysis. This approach has been successfully applied to nonlinear analysis of space frames and membranes by Levy and Spillers [6] and for shells by Levy and Gal [7,8].

The analysis results show (see Figure 132.1) that the suggested formulation does not necessarily converge to the "correct" solution and the results depend on the load steps size/number. Comparing to the performance of a shell element comprised of the constant strain triangle CST (see [9]) membrane element and the DKT elements, which never converge to a "wrong" solution, Allman's element is not the best choice.

1

Allman, D.J., "A compatible triangular element including vertex rotations for plane elasticity analysis", Comp. & Struct., 19, 1-8, 1984. doi:10.1016/0045-7949(84)90197-4

2

Batoz, J.L., Bathe, K.J., Ho, L.W., "A study of three noded triangular plate bending elements", Int. J. Num. Meth. Eng., 15, 1771-1812, 1980. doi:10.1002/nme.1620151205

3

Green, A.E., Knops, R.J., Laws, N. "Large deformation, superimposed small deformations, and stability of elastic robs", Int. J. Solids & Struct., 4, 555-557, 1968. doi:10.1016/0020-7683(68)90065-6

4

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

5

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. & Eng., 145, 11-85, 1997. doi:10.1016/S0045-7825(96)01233-9

6

Levy, R., and Spillers. W.R., "Analysis of Geometrically Nonlinear Structures", 2nd edition, Kluwer Academic Publishers, 2003.

7

Levy, R., and Gal E., "Geometrically Nonlinear three-noded flat triangular shell elements", Comp. & Struct., 79, 2349-2355, 2001. doi:10.1016/S0045-7949(01)00066-9

8

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

9

Zienkiewicz, O.C., "The finite element method", 3 ed, McGraw-Hill, 1977.

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