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 Is = -1 (where Is = input as ISYM on the MODE command). If = 0, sin β = cos β = 1.0.

Figure 11.9: Axisymmetric Harmonic Solid Elements

Axisymmetric Harmonic Solid Elements

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:

(11–156)

(11–157)

(11–158)

(11–159)

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:

(11–160)

(11–161)

(11–162)

(11–163)

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:

(11–164)

(11–165)

(11–166)

(11–167)

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.

(11–168)

(11–169)

(11–170)

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.

(11–171)

(11–172)

(11–173)

(11–174)