## 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:

triangular or quadrilateral.

- 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 I_{s} = -1 (where I_{s} = 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:

### 11.8.2. 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:

### 11.8.3. 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:

### 11.8.4. 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.

Unless
(MODE) = 1, u_{1} or u_{2} and w_{1} or w_{2} 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.