The geometric non-linear shell elements which are currently available in XFINAS is based on the Updated Lagrangian method. The formulation of these elements uses Mindlin-Reissner theory, which the formulation of the linear and geometrical stiffness is exactly defined by incorporating membrane forces, bending moment and transverse shear resultant force. In order to remove rigid body rotations, the co-rotational method is used. The material is assumed to be isotropic, laminated composite and elasto-plastic material. Plasticity is handled by applying the von Mises yield condition and Prandl-Reuss flow rule to discrete points through the shell wall. By using six degrees of freedom per node, the present element can model stiffened plate and shell structures.

The transverse shear stiffness in the laminated composite materials is defined by an equilibrium approach instead of using the shear correction factor. Four macroscopic progressive failure criteria for fiber-reinforced composite under loading were implemented. These criteria may be used to check for first ply failure of composite structures or in determining ultimate loads for complete laminate failure and getting risk assessment.


An assumed strain quasi-conforming shell element

A new Quasi-conforming formulations of 4-node stress resultant shell element ( XSHELL41 ) is developed and implemented in the XFINAS for the solution of stability problems of stiffened plates and shells. The shear-locking behaviour is eliminated by using the Quasi-conforming method. The stiffness matrices for the present elements are explicitly expressed and the stresses are taken accurately at the nodal points. Compared to elements using Gauss integration, where the stresses are most accurate at the integration points, the extrapolation procedure needed for post processing is eliminated in the present shell element.

Mid-Surface Geometry and Local Coordinates of 4 Node Shell

An assumed natural strain shell element

A 4-nodes assumed strain shell element ( XSHELL42 ) and 8 nodes assumed strain shell element ( XSHELL82 ) are developed and implemented in XFINAS for the solution of stability problems of stiffened plates and shells. The element is free of both membrane and shear locking behaviour by using the assumed strain method such that the element performs very well in the thin shells.

Mid-Surface Geometry and Local Coordinates of 8 Node Shell

Example Linear analysis of Pinched cylinder (Fig. 4) with rigid end diaphragms using 4-node quasi-conforming shell element is carried out. This example is the most severe benchmark test for 4-node shell element. In addition large deformation analysis of the pinched elastoplastic cylinder with strain hardening using the same examples is carried out.

Load-deflection curve of pinched cylinder

Deformed shape of elastoplastic cylinder

Nonlinear analysis of clamped cylinder

Example Large deformation of elasto-plastic analysis of Agelidis's imperfect stiffened shell under compression is carried out using 8-node shell element.

Preprocessing of stiffened panel

The 4-nodes and 8- nodes nonlinear resultant shell elements are available for the solution of nonlinear dynamic problems of composite plates and shells with failure criteria. The transverse shear stiffness is defined by an improved equilibrium approach instead of using the shear correction factor.

The progressive static and dynamic failure are incorporated in XFINAS using Four failure criteria (Maximum stress, Tsai-Wu, Tsai-Hill and Modified Puck Criterion Theory). The nonlinear static and dynamic failure analysis is done by computing for the inter-laminar stresses in each stress point in an element. Having obtained the stresses in each layer, checking for failure can be done based on a chosen failure criterion.

Example In order to show the performance of finite rotation and deformation of XSHELL83, the nonlinear analysis of the fiber-reinforced composite hyperbolic shell with (0°/90°/0°) lay-ups under pinched loading are carried out.

Geometry and material of composite hyperbolic shell