| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness and Stress Stiffness Matrices; and Thermal and Newton-Raphson Load Vectors | Equation 11–19, Equation 11–20, Equation 11–21, Equation 11–22, Equation 11–23, and Equation 11–24 Around the circumference: Fourier Series | Along the length: 2 |
| Mass Matrix and Pressure Load Vector | Same as stiffness matrix | Along the length: 3 |
| Load Type | Distribution |
|---|---|
| Element Temperature | KEYOPT(1) = 0 Linear thru wall and linear along length |
| Nodal Temperature | Constant across cross-section, linear along length |
| Internal and External Pressures | Constant |
References:
Bathe and Almeida ([366])
Yan, Jospin, and Nguyen ([367])
Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations.
Pipe cross-sectional motions (that is, radial expansion, ovalization, and warping) are modeled with Fourier series. The corresponding unknowns (Fourier magnitudes) are treated as internal degrees of freedom. A higher number of Fourier modes may be required to achieve an adequate level of accuracy in cross-sectional motions. Also, a higher number of integration points around the circumference may be needed for capturing nonlinear material behaviors or ensuring sufficient numerical integration accuracy.
No slippage is assumed between the element layers. Shear deflections are included in the element; however, normals to the center wall surface are assumed to remain straight after deformation, but not necessarily normal to the center surface. Therefore, constant transverse shears through the pipe wall are allowed.