## 11.8. Axisymmetric Harmonic Solids

This section contains shape functions for axisymmetric harmonic solid elements. These elements are available in a number of configurations, including certain combinations of the following features:

- if quadrilateral, with or without extra shape functions (ESF).

• with or without midside nodes.

The shape functions of this section use the quantities sin β and cos β (where = input as MODE on the MODE command). sin β and cos β are interchanged if Is = -1 (where Is = input as ISYM on the MODE command). If = 0, sin β = cos β = 1.0.

### 11.8.1. Axisymmetric Harmonic 3-Node Triangular Solids

These shape functions are for the 3-node axisymmetric triangular solid elements, such as PLANE25 with only 3 nodes input:

 (11–153)
 (11–154)
 (11–155)
 (11–156)

### 11.8.2. Axisymmetric Harmonic 6-Node Triangular Solids

These shape functions are for the 6-node axisymmetric triangular solids elements, such as PLANE83 input as a triangle:

 (11–157)
 (11–158)
 (11–159)
 (11–160)

### 11.8.3. Axisymmetric Harmonic 4-Node Quadrilateral Solids without ESF

These shape functions are for the 4-node axisymmetric harmonic quadrilateral solid elements without extra shape functions, such as PLANE25 with KEYOPT(2) = 1, or PLANE75:

 (11–161)
 (11–162)
 (11–163)
 (11–164)

### 11.8.4. Axisymmetric Harmonic 4-Node Quadrilateral Solids with ESF

These shape functions are for the 4-node axisymmetric harmonic quadrilateral elements with extra shape functions, such as PLANE25 with KEYOPT(2) = 0.

 (11–165)
 (11–166)
 (11–167)

Unless (MODE) = 1, u1 or u2 and w1 or w2 motions are suppressed for elements near the centerline.

### 11.8.5. Axisymmetric Harmonic 8-Node Quadrilateral Solids

These shape functions are for the 8-node axisymmetric harmonic quadrilateral solid elements such as PLANE78 or PLANE83.

 (11–168)
 (11–169)
 (11–170)
 (11–171)