Keywords: shells, Ahmad elements, rank deficiency, shear locking, three-dimensional solids, 2D/3D element transformation, engineering practice.

The original implementation of the Ahmad shell element is represented by an element with three-dimensional geometry and with a two-dimensional mid-plane curved surface. Depending on deformation approximation, it has 8 or 9 nodes, each with 5 degrees of freedom, (DOFs). These are 3 displacements and 2 rotations. This formulation is fine for structures with shell elements, however a typical finite element model also comprises solid three-dimensional elements etc. They have 3DOFs per node and their connection to Ahmad element's nodes 5 DOFs is difficult.

The presented paper alleviates this problem by implementing Ahmad elements in form of a full three-dimensional element. It resembles three-dimensional hexahedral elements with 8 or 9 nodes at the top and bottom surfaces, with 3 global displacements per node. The process of derivation of this element lies in transformation of the original 5 DOFs (located at a mid-plane node) to 3DOFs of the corresponding nodes (at the bottom and top surface of the new three-dimensional element shape). An inherent part of this process is constraining the excessive sixth DOF to conform with shell theory. A multipoint Dirichlet condition is used to ensure that strain in direction of the shell thickness remains zero. Another advantage of this formulation is that the three-dimensional visualisation of the new element surpasses its two-dimensional form. The element can be processed by any three-dimensional pre- and post-processor, thereby significantly simplifying implementation and use of the element.

Several kinds of the element are derived. They stem from combination of Lagrangian, hetherosis or serendipity approximation of deformations with full, selective or reduced integration schemes. (The layered approach is employed through element's thickness).

The first part of this paper derives the element, particularly it describes the transformation rules for transition from the two to the three-dimensional formulation. The second part of the paper investigates its properties and computational performance. The developed elements differ in accuracy and CPU costs. Some of them suffer from shear locking, the others from spurious energy modes. The elements are investigated in both linear and nonlinear regimes. A special robust eigen-mode solution scheme is employed to perform element modal analysis. Superfluous zero energy eigen-modes are the best indicators of spurious element modes.

The elements are implemented in the finite element package ATENA [1], which is also used to for the analyses presented. The elements include geometrical nonlinearities and can be combined with any nonlinear material model. The elements allow for embedded reinforcement (smeared in some element layers). Discrete bar and plane reinforcement are also supported.

The results obtained for the new three-dimensional shell elements are compared with those for the original Ahmad shell formulation. It is found that their properties are similar, however, the advantage of easier use makes the new three-dimensional form of the element superior. A similar transformation is usable also for other elements and problems, as elements for transport analysis etc.

1

J. Cervenka, L. Jendele, "Atena User's Manual, Part 1-7", Prague: Cervenka Consl., 2000-2006.

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