Figure 11.3:  Axisymmetric Harmonic Shell Element

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.

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)

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:

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).

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

hi (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)