This section describes shape functions for axisymmetric shell elements under nonaxisymmetric load and for general axisymmetric surfaces.
This section describes shape functions for 2-node axisymmetric shell elements under nonaxisymmetric (harmonic) load. These elements may or may not have extra shape functions (ESF).
The shape functions for axisymmetric harmonic shells use the quantities sin β and cos β, where = input quantity MODE on the MODE command. The sin β and cos β are interchanged if I_{s} = -1, where I_{s} = input quantity ISYM on the MODE command. If = 0, both sin β and cos β are set equal to 1.0.
These shape functions are for 2-node axisymmetric harmonic shell elements without extra shape functions, such as SHELL61 with KEYOPT(3) = 1.
(11–35) |
(11–36) |
(11–37) |
These shape functions are for 2-node axisymmetric harmonic shell elements with extra shape functions, such as SHELL61 with KEYOPT(3) = 0.
(11–38) |
(11–39) |
(11–40) |
This section contains shape functions for 2- or 3- node-per-plane general axisymmetric surface elements such as SURF159. These elements are available in various configurations, including combinations of the following features:
With or without midside nodes
A varying number of node planes (N_{np}) in the circumferential direction (defined via KEYOPT(2)).
The elemental coordinates are cylindrical coordinates and displacements are defined and interpolated in that coordinate system, as shown in Figure 11.4: General Axisymmetric Surface Elements (when N_{np} = 3).
When N_{np} is an odd number, the interpolation function used for displacement is:
(11–41) |
where:
i = r, θ, z |
h_{i} (s, t) = regular Lagrangian polynominal interpolation functions like Equation 11–6 or Equation 11–19. |
= coefficients for the Fourier terms. |
When N_{np} is an even number, the interpolation function is:
(11–42) |
All of the coefficients in Equation 11–41 and Equation 11–42 can be expressed by nodal displacements, using u_{r} = u, u_{j} = v, u_{z} = w without midside nodes, and N_{np} = 3:
(11–43) |
(11–44) |
(11–45) |
Similar to the element without midside nodes, the u, v, and w with midside nodes are expressed as:
(11–46) |
(11–47) |
(11–48) |