Plate and shell elements

1. Plate and shell elements All the following elements enable to create FE mesh of a thin-walled body, with the thickness being one of the important input characteristics like material. Stress distribution corresponds to plate and shell theory, with zero normal component and in-plane components with linear distribution along the thickness. This means that both surfaces of such a thin-walled body must be checked to find a critical point of the structure. Another important feature of the plate and shell elements are the rotational degrees of freedom in nodes, added to the displacements. For this reason, coupling of solid and thin-walled elements in a single FE mesh must be done with care. Generally, there are two basic possibilities. The first one is face-to-face attachment (Fig. 6-1 a). Fig.6-1 Coupling of solid and thin-walled elements in case of a) different, b) the same deformation parameters in node

2. In case of the same type of deformation parameters in nodes, the elements can be attached in one line only (Fig.6-1 b). Plate element Plate element can be seen as a generalisation of a beam in two dimensions. The simplest 4-node plate element has 12 parameters, three in each node - one displacement and two rotations: Fig.6-2 Quadrilateral plate element The unknown deflection is approximated by an ordinary way where δT = │w1, φx1, φy1, w2, φx2, ……, φx4, φy4 │, N = │ N1 N2 N3 …… N12 │.

3. The stiffness matrix , where B is a matrix obtained as a second derivative of shape function matrix N. Dm is a bending stiffness . Quadrilateral shell element Plate element according to the previous paragraph can model only bending of plates. Combination of this element with membrane elements like PLANE42 results in elements with combined bending-membrane behaviour, which can be used to model shells of general shapes. The combination according to Fig.6-3 shows the final shell element having six degrees of freedom in node, three displacements and three rotations: Fig.6-3 Shell element as a combination of plate and membrane element

4. In ANSYS, 4-node shell elements can be found under the names SHELL63 (linear behaviour), SHELL43 (nonlinear behaviour) or SHELL181 (strong nonlinearity). All of them can be degenerated into the triangular shape. In case of curved shell surfaces, 8-node SHELL93 element is best to use the complicated shape. The Example 6.1 illustrates application of shell elements for analysis of pipe intersection.