This section contains shape functions for 3-D solid elements. These elements are available in a number of configurations, including certain combinations of the following features:
Element shapes may be tetrahedra, pyramids, wedges, or bricks (hexahedra).
- If wedges or bricks, with or without extra shape functions (ESF)
With or without rotational degrees of freedom (RDOF)
With or without midside nodes
The wedge elements with midside nodes (15-node wedges) are either a condensation of the 20-node brick element or are based on wedge shape functions.
These shape functions are either a direct 4-node tetrahedral such as SOLID285 or a condensation of an 8-node brick element such as SOLID5, FLUID30, or SOLID98.
The resulting effective shape functions are:
(11–172) |
(11–173) |
(11–174) |
(11–175) |
(11–176) |
(11–177) |
(11–178) |
(11–179) |
(11–180) |
(11–181) |
(11–182) |
(11–183) |
These shape functions are for 10-node tetrahedron elements such as SOLID98 and SOLID227, or by condensation for SOLID90.
(11–184) |
(11–185) |
(11–186) |
(11–187) |
(11–188) |
(11–189) |
(11–190) |
This element is a condensation of an 8-node brick element.
The resulting effective shape functions are:
(11–191) |
These shape functions are for 13-node pyramid elements which are based on a condensation of a 20-node brick element:
(11–192) |
(11–193) |
(11–194) |
(11–195) |
(11–196) |
(11–197) |
The 6-node wedge elements are a condensation of an 8-node brick such as SOLID5 or FLUID30. These shape functions are for 6-node wedge elements without extra shape functions:
(11–198) |
(11–199) |
(11–200) |
(11–201) |
(11–202) |
(11–203) |
(11–204) |
The 6-node wedge elements are a condensation of an 8-node brick such as SOLID5. (See Figure 11.14: 6-Node Wedge Element.) These shape functions are for 6-node wedge elements with extra shape functions:
(11–205) |
(11–206) |
(11–207) |
These shape functions are for 15-node wedge elements such as SOLID90 that are based on a condensation of a 20-node brick element Equation 11–232. or are computed directly.
Elements in a wedge configuration use shape functions based on triangular coordinates and the r coordinate going from -1.0 to +1.0.
(11–208) |
(11–209) |
(11–210) |
(11–211) |
(11–212) |
(11–213) |
These shape functions are for 8-node brick elements without extra shape functions such as SOLID5 with KEYOPT(3) = 1 or FLUID30:
(11–214) |
(11–215) |
(11–216) |
(11–217) |
(11–218) |
(11–219) |
(11–220) |
(11–221) |
(11–222) |
(11–223) |
(11–224) |
(11–225) |
(11–226) |
(11–227) |
(11–228) |
(Please see Figure 11.16: 8-Node Brick Element) These shape functions are for 8-node brick elements with extra shape functions such as SOLID5 with KEYOPT(3) = 0:
(11–229) |
(11–230) |
(11–231) |
These shape functions are used for 20-node solid elements such as SOLID90:
(11–232) |
(11–233) |
(11–234) |
(11–235) |
(11–236) |
(11–237) |
(11–238) |
These shape functions and mapping functions are for the 3-D 8-node solid brick infinite elements such as INFIN111:
These shape functions and mapping functions are for the 3-D 20-node solid brick infinite elements such as INFIN111:
This section contains shape functions for general axisymmetric solid elements. These elements are available in a number of configurations, including certain combinations of the following features:
A quadrilateral, or a degenerated triangle shape to simulate an irregular area, on the master plane (the plane on which the quadrilaterals or triangles are defined)
With or without midside nodes
A varying number of node planes (N_{np}) in the circumferential direction (defined via KEYOPT(2))
The elemental coordinates are cylindrical coordinates and displacements are defined and interpolated in that coordinate system, as shown in Figure 11.20: General Axisymmetric Solid Elements (when N_{np} = 3).
When N_{np} is an odd number, the interpolation function used for displacement is:
(11–263) |
where:
i = r, θ, z |
h_{i} (s, t) = regular Lagrangian polynominal interpolation functions like Equation 11–120 or Equation 11–134. |
= coefficients for the Fourier terms. |
When N_{np} is an even number, the interpolation function is:
(11–264) |
The temperatures are interpolated by Lagrangian polynominal interpolations in s, t plane, and linearly interpolated with θ in circumferential (θ) direction as:
(11–265) |
where:
= node plane number in circumferential direction |
T_{n} = same as Equation 11–128 and Equation 11–138. |
All of the coefficients in Equation 11–263 and Equation 11–264 can be expressed by node displacements. Using u_{r} = u, u_{j} = v, u_{z} = w, and take N_{np} = 3 as an example.
(11–266) |
(11–267) |
(11–268) |