## 11.7. 2-D and Axisymmetric Solids

This section contains shape functions for 2-D and axisymmetric
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.

### 11.7.1. 2-D and Axisymmetric 3-Node Triangular Solids (CST)

These shape functions are for 2-D 3-node and axisymmetric triangular
solid elements, such as PLANE13 or PLANE182 with only 3 nodes input:

### 11.7.2. 2-D and Axisymmetric 6-Node Triangular Solids (LST)

These shape functions are for 2-D 6-node and axisymmetric triangular
solids, such as PLANE35:

### 11.7.3. 2-D and Axisymmetric 4-node Quadrilateral Solid without ESF
(Q4)

These shape functions are for the 2-D 4-node and axisymmetric
quadrilateral solid elements without extra shape functions, such as PLANE13 with KEYOPT(2) = 1 or LINK68.

### 11.7.4. 2-D and Axisymmetric 4-node Quadrilateral Solids with ESF (QM6)

These shape functions are for the 2-D 4-node and axisymmetric
solid elements with extra shape functions, such as PLANE13 with KEYOPT(2) = 0. (Taylor et al.([49]))

Equation 11–132 is adjusted for axisymmetric
situations by removing the u_{1} or u_{2} term for elements near the centerline, in order to
avoid holes or “doubled” material at the centerline.

### 11.7.5. 2-D and Axisymmetric 8-Node Quadrilateral Solids (Q8)

These shape functions are for the 2-D 8-node and axisymmetric
quadrilateral elements such as PLANE77 and PLANE183:

### 11.7.6. 2-D and Axisymmetric 4-Node Quadrilateral Infinite Solids

These Lagrangian isoparametric shape functions and “mapping”
functions are for the 2-D and axisymmetric 4-node quadrilateral solid
infinite elements such as INFIN110:

#### 11.7.6.1. Shape Functions

#### 11.7.6.2. Mapping Functions

### 11.7.7. 2-D and Axisymmetric 8-Node Quadrilateral Infinite Solids

These Lagrangian isoparametric shape functions and “mapping”
functions are for the 2-D and axisymmetric 8-node quadrilateral infinite
solid elements such as INFIN110:

#### 11.7.7.1. Shape Functions

#### 11.7.7.2. Mapping Functions

The shape and mapping functions for the nodes N, O and P are
deliberately set to zero.