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.

Figure 11.6: 2-D and Axisymmetric Solid Element

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

Figure 11.7: 4-Node Quadrilateral Infinite Solid Element

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

Figure 11.8: 8-Node Quadrilateral Infinite Solid Element

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