## 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:

- 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:

 (11–100)
 (11–101)
 (11–102)
 (11–103)
 (11–104)
 (11–105)
 (11–106)
 (11–107)
 (11–108)
 (11–109)
 (11–110)
 (11–111)

### 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:

 (11–112)
 (11–113)
 (11–114)
 (11–115)
 (11–116)
 (11–117)
 (11–118)
 (11–119)

### 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.

 (11–120)
 (11–121)
 (11–122)
 (11–123)
 (11–124)
 (11–125)
 (11–126)
 (11–127)
 (11–128)
 (11–129)
 (11–130)
 (11–131)

### 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]))

 (11–132)
 (11–133)

Equation 11–132 is adjusted for axisymmetric situations by removing the u1 or u2 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:

 (11–134)
 (11–135)
 (11–136)
 (11–137)
 (11–138)
 (11–139)
 (11–140)

### 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

 (11–141)
 (11–142)
 (11–143)

#### 11.7.6.2. Mapping Functions

 (11–144)
 (11–145)
 (11–146)

### 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

 (11–147)
 (11–148)
 (11–149)
 (11–150)

#### 11.7.7.2. Mapping Functions

 (11–151)
 (11–152)

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