Nonlinear analysis of composite-material shell structures using the finite element method and the search for a satisfactory finite element for shell structural nonlinear analysis has been active for decades. One main approach for formulating shell finite elements is to enhance a linear shell element to perform nonlinear analysis by adding an appropriate geometric stiffness matrix and implementing a full Newton-Raphson procedure to the linear analysis computer program (Argyris et al. [1], Mohan and Kapania [2]). The presented finite element is, in a broad view, a part of this approach.

Mohan and Kapania [2], use an updated Lagrangian formulation of a three node flat triangular shell element for geometrically nonlinear analysis of laminated plates and shells. Unlike the presented paper they obtain the geometrical effects using a nonlinear kinematics assumption.

Argyris et al. [1] presented the TRIC flat triangular shell element for nonlinear postbuckling analysis of arbitrary isotropic and composite shells. Their element is based on the natural mode finite element method. Unlike the presented paper that includes the geometric effects of all the forces due to changes in geometry, their geometric stiffness matrix includes only those natural forces which produce rigid-body moments when the element receives rigid-body rotations.

Here the constant strain triangle (CST) membrane finite element (Zienkiewicz [3]) was chosen for the in-plane behavior of the shell and the discrete Kirchhoff (DKT) theory plate finite element (Batoz et al. [4]) was chosen for the bending behavior of the shell. Both elements are three-noded flat triangular and have been proven to be efficient for linear analysis.

A unique approach for deriving the geometric stiffness matrix is presented. It is based on load perturbation of the linear discrete equilibrium equations and defines the geometric stiffness matrix as the gradient of the element nodal force vector in global coordinates. This approach has been successfully applied to nonlinear analysis of trusses, space frames and membranes by Levy and Spillers [5] as well as for thin isotropic shells Levy and Gal [6].

Whereas the presented approach depends on an a priori chosen linear elastic finite element it is independent of large strain formulations that are essential otherwise and utilizes linear relationships throughout. The formulation is first order complete and truly large rotations small strains one.

Because of obvious difficulties in taking derivatives of the rotation matrix with respect to the nodal coordinates, all evaluations are performed in the local coordinate system. The out-of-plane geometric stiffness matrix is introduced to circumvent the need of rotation matrix derivatives.

The incremental analysis procedure used herein detects buckling by monitoring the tangent stiffness matrix. Snap-through and equilibrium paths are handled with the cylindrical arc length method (Crisfield [7]). Stress retrieval is performed using linear, kinematic and constitutive, relationships because the resulting pure deformations are small due to efficient unique procedures for the removal of rigid body displacements and rotations.

Finally the finite element was coded in FORTRAN and results obtained using the presented formulation are in good agreement with those available in the existing literature.

1

Argyris J., Tenek L.T., Papadrakakis M., and Apostolopoulou C., Postbuckling performance of the TRIC natural mode triangular element for Isotropic and Laminated Composite Shells. Comp. Meth. Appl. Mech. & Engng. 1998; 166:211-231. doi:10.1016/S0045-7825(98)00071-1

2

Mohan P., Kapania R.K. Geometrically Nonlinear Analysis of Composite Plates and Shells Using a Flat Triangular Shell Element. 38th AIAA/ ASME/ ASCE/ AHS/ ASC Structures, Structural Dynamics and Materials Conference. Kissimmee, FL, Paper 97-1233; 1997; 2347-2361.

3

Batoz J.L., Bathe K.J., Ho L.W. A Study of Three Noded Triangular Plate Bending Elements. Int J Num Meth Engng 1980; 15; 1771-1812. doi:10.1002/nme.1620151205

4

Zienkiewicz O.C. The finite element method 3rd edition. McGraw-Hill: New York, 1977.

5

Levy R., Spillers W.R. Analysis of geometrically nonlinear structures. Chapman and Hall: New York, 1994.

6

Levy R., Gal E. Geometrically Nonlinear Analysis of Shells Using Triangular Flat Elements. Computers and Structures 2001; 79: 2349-2355. doi:10.1016/S0045-7949(01)00066-9

7

Crisfield M.A. A Fast Incremental / Iterative Solution Proceedure That Handles "snap-through". Computers and Structures 1981; 13: 55-62. doi:10.1016/0045-7949(81)90108-5

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