## 11.5. Axisymmetric Harmonic Shells and General Axisymmetric Surfaces

This section describes shape functions for axisymmetric shell elements under nonaxisymmetric load and for general axisymmetric surfaces.

### 11.5.1. Axisymmetric Harmonic Shells

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 Is = -1, where Is = input quantity ISYM on the MODE command. If = 0, both sin β and cos β are set equal to 1.0.

#### 11.5.1.1. Axisymmetric Harmonic Shells without ESF

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)

#### 11.5.1.2. Axisymmetric Harmonic Shells with ESF

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)

### 11.5.2. General Axisymmetric Surfaces

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 (Nnp) 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 Nnp = 3).

When Nnp is an odd number, the interpolation function used for displacement is:

 (11–41)

where:

 i = r, θ, z hi (s, t) = regular Lagrangian polynominal interpolation functions like Equation 11–6 or Equation 11–19. = coefficients for the Fourier terms.

When Nnp 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 ur = u, uj = v, uz = w without midside nodes, and Nnp = 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)