| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness Matrix | No shape functions are explicitly used. Rather a flexibility matrix similar to that developed by Chen is inverted and used. | None |
| Mass Matrix | No shape functions are used. Rather a lumped mass matrix using only translational degrees of freedom is used. | None |
| Thermal and Pressure Load Vector | Equation 11–15, Equation 11–16, and Equation 11–17 | None |
| Load Type | Distribution |
|---|---|
| Element Temperature | Linear thru thickness or across diameter, and along length |
| Nodal Temperature | Constant across cross-section, linear along length |
| Pressure | Internal and External: constant along length and around the circumference Lateral: varies trigonometrically along length (see below) |
PIPE16 - Elastic Straight Pipe covers some of the applicable stress calculations.
The geometry in the plane of the element is given in Figure 1.9: Plane Element.
The stiffness matrix is developed based on an approach similar to that of Chen. The flexibility of one end with respect to the other is:
 | (1–53) |
where:
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and where:
| R = radius of curvature (input as RADCUR on R command) (see Figure 1.9: Plane Element) |
| θ = included angle of element (see Figure 1.9: Plane Element) |
| E = Young's modulus (input as EX on MP command) |
| ν = Poisson's ratio (input as PRXY or NUXY on MP command) |
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| Do = outside diameter (input as OD on R command) |
| Di = Do - 2t = inside diameter |
| t = wall thickness (input as TKWALL on R command) |
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| Pi = internal pressure (input on SFE command) |
| Po = external pressure (input on SFE command) |
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The user should not use the KEYOPT(3) = 1 option if:
 | (1–54) |
where:
| θc = included angle of the complete elbow, not just the included angle for this element (θ) |
Next, the 6 x 6 stiffness matrix is derived from the flexibility matrix by inversion:
 | (1–55) |
The full 12 x 12 stiffness matrix (in element coordinates) is derived by expanding the 6 x 6 matrix derived above and transforming to the global coordinate system.
The element mass matrix is a diagonal (lumped) matrix with each translation term being defined as:
 | (1–56) |
where:
| mt = mass at each node in each translation direction |
| me= (ρAw + ρflAfl + ρinAin)Rθ = total mass of element |
| ρ = pipe wall density (input as DENS on MP command) |
| ρfl = internal fluid density (input as DENSFL on RMORE command) |
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| ρin = insulation density (input as DENSIN on RMORE command) |
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| Do+ = Do + 2 tin |
| tin = insulation thickness (input as TKIN on RMORE command) |
The load vector in element coordinates due to thermal and pressure effects is:
 | (1–57) |
where:
| εx = strain caused by thermal as well as internal and external pressure effects (see Equation 1–38 ) |
| [Ke] = element stiffness matrix in global coordinates |
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is computed based on
the transverse pressures acting in the global Cartesian directions
(input using face 2, 3, and 4 on SFE command) and curved beam formulas
from Roark. Table 18, reference no.
(loading) 3, 4, and 5 and 5c was used for in-plane effects and Table
19, reference no. (end restraint) 4e was used for out-of-plane effects.
As a radial load varying trigonometrically along the length of the
element was not one of the available cases given in Roark, an integration of a point radial load was
done, using Loading 5c.
In the stress pass, the stress evaluation is similar to that for PIPE16 - Elastic Straight Pipe. The wall thickness is diminished by the corrosion allowance, if present. The bending stress components are multiplied by stress intensification factors (Cσ). The "intensified" stresses are used in the principal and combined stress calculations. The factors are:
 | (1–58) |
 | (1–59) |
 | (1–60) |
where:
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| te = t - tc |
| do = Do - 2 tc (where tc = corrosion allowances, input as TKCORR on the R command) |