## 13.25. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid

Matrix or VectorGeometryShape Functions Integration Points
Stiffness Matrix and Thermal Load VectorQuadEquations Equation 11–161, Equation 11–162 , and Equation 11–163 or if modified extra shape functions are included (KEYOPT(2) = 0) and element has 4 unique nodes: Equation 11–165, Equation 11–166 , and Equation 11–167 2 x 2
Triangle Equation 11–153, Equation 11–154 , and Equation 11–155 3
Mass and Stress Stiffness Matrices Quad Equation 11–120, Equation 11–121 , and Equation 11–122 2 x 2
Triangle Equation 11–100, Equation 11–101 , and Equation 11–102 3
Pressure Load VectorSame as stress stiffness matrix, specialized to the surface2
Load TypeDistribution
Element Temperature Bilinear across element, harmonic around circumference
Nodal TemperatureBilinear across element, harmonic around circumference
PressureLinear along each face, harmonic around circumference

Reference: Wilson([38]), Zienkiewicz([39]), Taylor([49])

### 13.25.1. Other Applicable Sections

Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations.

### 13.25.2. Assumptions and Restrictions

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.

### 13.25.3. Use of Temperature

In general, temperatures have two effects on a stress analysis:

1. Temperature dependent material properties.

2. 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.

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