| Matrix or Vector | Geometry | Shape Functions | Integration Points |
|---|---|---|---|
| Stiffness Matrix and Thermal Load Vector | Quad | Equations Equation 11–164, Equation 11–165 , and Equation 11–166 or if modified extra shape functions are included (KEYOPT(2) = 0) and element has 4 unique nodes: Equation 11–168, Equation 11–169 , and Equation 11–170 | 2 x 2 |
| Triangle | Equation 11–156, Equation 11–157 , and Equation 11–158 | 3 | |
| Mass and Stress Stiffness Matrices | Quad | Equation 11–122, Equation 11–123 , and Equation 11–124 | 2 x 2 |
| Triangle | Equation 11–101, Equation 11–102 , and Equation 11–103 | 3 | |
| Pressure Load Vector | Same as stress stiffness matrix, specialized to the surface | 2 | |
| Load Type | Distribution |
|---|---|
| Element Temperature | Bilinear across element, harmonic around circumference |
| Nodal Temperature | Bilinear across element, harmonic around circumference |
| Pressure | Linear along each face, harmonic around circumference |
Reference: Wilson([39]), Zienkiewicz([40]), Taylor([50])
Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations.
The material properties are assumed to be constant around the entire
circumference, regardless of temperature dependent material properties or
loading. For
(input as MODE on MODE command)
> 0, the extreme values for combined stresses are obtained by computing these
stresses at every 10/
degrees and selecting
the extreme values.
In general, temperatures have two effects on a stress analysis:
Temperature dependent material properties.
Thermal expansion
In the case of
= 0, there is no conflict
between these two effects. However, if
>
0, questions arise. As stated in the assumptions, the material properties
may not vary around the circumference, regardless of the temperature. That
is, one side cannot be soft and the other side hard. The input temperature
for
> 0 varies sinusoidally around the
circumference. As no other temperatures are available to the element, the
material properties are evaluated at Tref (input on TREF command).
The input temperature can therefore be used to model thermal bending. An
approximate application of this would be a chimney subjected to solar heating
on one side only. A variant on this basic procedure is provided by the temperature
KEYOPT (KEYOPT(3) for PLANE25). This variant provides
that the input temperatures be used only for material property evaluation
rather than for thermal bending. This second case requires that α
x, α
y,
and α
z (input on MP commands)
all be input as zero. An application of the latter case is a chimney, which
is very hot at the bottom and relatively cool at the top, subjected to a wind
load.