Shell elements are 4- to 8-node isoparametric quadrilaterals or 3- to 6-node triangular elements in any 3D orientation. The 4-node elements require a much finer mesh than the 8-node elements to give convergent displacements and stresses in models involving out-of-plane bending. Figure 1 shows some typical shell elements.
The General and Co-rotational shell element is formulated based on works by Ahmad, Iron and Zienkiewicz and later refined by Bathe and Balourchi. It can be applied to model both thick and thin shell problems. Also, the geometry of a doubly curved shell with variable thickness can be accurately described using this shell element.
Figure 1: Sample 3D Shell Element
Figure 2: Typical Shell Elements
The Thin shell element is based on thin plate theory. The bending behavior of the element is based on a discrete Kirchoff approach to plate bending using Batoz's interpolation functions. This formulation satisfies the Kirchoff constraints along the boundary and provides linear variation of curvature through the element. The membrane behavior of the element is based on the Allman triangle which is derived from the Linear Strain Triangular (LST) element. A general curved surface is approximated by this element as a set of facets formed by the planes defined by the three nodes of each element. For these reasons a well-refined mesh is necessary.
The element geometry is described by the nodal point coordinates. Each shell element node has 5 degrees of freedom (DOF) - three translations and two rotations. The translational DOF are in the global Cartesian coordinate system. The rotations are about two orthogonal axes on the shell surface defined at each node. The rotational boundary condition restraints and applied moments also refer to this nodal rotational system. The two rotational axes (V1 and V2) are usually automatically determined by the processor and you do not have to specifically orient them.
The rotational R1 and R2 directions at each node are determined by the cross product of the normal vector Vn and a guiding vector Vg as follows. See Figure 3 (V1 and V2 are the unit vectors along the R1 and R2 rotational direction, respectively):
R1 = V2 x Vn
R2 = Vn x Vg
The default guiding vector Vg is (1,0,0). In other words, the R1 axis is actually the projection of the guiding vector Vg onto the shell surface, and the R2 axis lies on this shell surface but orthogonal to R1 and the normal vector Vn. Figure 3 shows the orientation of R1 and R2 using the default guiding vector (1,0,0).
You can enter two guiding vectors for each shell element part. If the guiding vector has zero projection onto the shell plane, the second guiding vector is used. You can also specify a guiding vector for a specific node.
The processor determines the normal direction of each shell element node by taking the average of different normal vectors from different elements connecting to the node point. The element normals are computed from the element corner nodes data. From this normal vector, two rotational axes are then determined. You can also specify the normal vector direction directly and suppress the processor's calculations. The two rotational vectors can also be oriented in a user-specified direction such that skewed rotational restraints or moments in 3D space can be easily defined.
When multiple shell panels intersect at an area (usually at a line of intersection), it is difficult to specify the rotational DOF at the area. In this case, the nodes at the intersecting area should be designated as 6-DOF nodes.
Figure 3: The Rotational Degrees of Freedom Determined Using Vg
Six DOF nodes will have globally oriented rotational DOF instead of the unique R1 and R2 DOF at each node.
Three-dimensional shell elements are Type 26 elements and are 4- to 8-node isoparametric quadrilaterals or 3- to 6-node triangular elements in any 3D orientation. They can be applied to models for both thick and thin shell problems. Also, the geometry of a doubly curved shell with variable thickness can be accurately described using this shell element. Theoretical aspects of this element formulation can be found in Bathe and Balourchi.
The element geometry is described by the nodal point coordinates. Each shell element node has five degrees of freedom (DOF) - three translations and two rotations. The shell elements' rotational DOF are defined by the shell guiding vectors. The projection of the guiding vector, Vg, onto the shell element will be the first rotational axis (R1). The second rotational axis is orthogonal to both the R1 and the shell normal vector (Vn), using the rule R2 = Vn x Vg. The default guiding vector is (1, 0, 0). If the guiding vector is parallel to the shell normal vector, Vn, a second guiding vector [(0, 1, 0) by default] will be used to define the R1 direction.