The element is assumed to be a thin-walled pipe except as noted. The corrosion allowance is used only in the stress evaluation, not in the matrix formulation.

The element stiffness matrix of PIPE16 is similar to that of a 3D elastic beam, except that

(1–29)

(1–30)

(1–31)

and,

(1–32)

where:

π = 3.141592653

Do = outside diameter (input as OD on R command)

Di = inside diameter = Do - 2tw

tw = wall thickness (input as TKWALL on R command)

f = flexibility factor (input as FLEX on R command)

Further, the axial stiffness of the element is defined as

(1–33)

where:

E = Young's modulus (input as EX on MP command)

L = element length

k = alternate axial pipe stiffness (input as STIFF on RMORE command)

The element mass matrix of PIPE16 is the same as for a 3D elastic beam, except total mass of the element is assumed to be:

(1–34)

where:

me = total mass of element

mw = alternate pipe wall mass (input as MWALL on RMORE command)

ρ = pipe wall density (input as DENS on MP command)

ρfl = internal fluid density (input as DENSFL on R command)

ρin = insulation density (input as DENSIN on RMORE command)

Do+ = Do + 2tin

tin = insulation thickness (input as TKIN on RMORE command)

= alternate representation of the surface area of the outside of the pipe element (input as AREAIN on RMORE command)

Also, the bending moments of inertia (Equation 1–30) are used without the C_f term.

The element gyroscopic damping matrix is:

(1–35)

where:

Ω = rotation frequency about the positive x axis (input as SPIN on RMORE command)

G = shear modulus (input as GXY on MP command)

As = shear area ( = Aw/2.0)

The element pressure load vector is

(1–36)

where:

F1 = FA + FP

F7 = -FA + FP

F4 = F10 = 0.0

P1 = parallel pressure component in element coordinate system (force/unit length)

P2, P3 = transverse pressure components in element coordinate system (force/unit length)

The transverse pressures are assumed to act on the centerline, and not on the inner or outer surfaces. The transverse pressures in the element coordinate system are computed by

(1–37)

where:

, the unrestrained axial strain caused by internal and external pressure effects, is needed to compute the pressure part of the element load vector (see Figure 1.4: Thermal and Pressure Effects).

Figure 1.4: Thermal and Pressure Effects

is computed using thick wall (Lame') effects:

(1–38)

where:

ν = Poisson's ratio (input as PRXY or NUXY on MP command)

Pi = internal pressure (input using face 1 on SFE command)

Po = external pressure (input using face 5 on SFE command)

An element thermal load vector is computed also, based on thick wall effects.

The output stresses, computed at the outside surface and illustrated in Figure 1.5: Elastic Pipe Direct Stress Output and Figure 1.6: Elastic Pipe Shear Stress Output, are calculated from the following definitions:

(1–39)

(1–40)

(1–41)

(1–42)

(1–43)

where:

σdir = direct stress (output as SDIR)

Fx = axial force

do = 2 ro

tc = corrosion allowance (input as TKCORR on RMORE command)

σbend = bending stress (output as SBEND)

Cσ = stress intensification factor, defined in Table 1.1: Stress Intensification Factors

σtor = torsional shear stress (output as ST)

Mx = torsional moment

J = 2Ir

σh = hoop pressure stress at the outside surface of the pipe (output as SH)

te = tw - tc

= lateral force shear stress (output as SSF)

Average values of P_i and P_o are reported as first and fifth items of the output quantities ELEMENT PRESSURES. The outside surface is chosen as the bending stresses usually dominate over pressure induced stresses.

Figure 1.5: Elastic Pipe Direct Stress Output

Figure 1.6: Elastic Pipe Shear Stress Output

Stress intensification factors are given in Table 1.1: Stress Intensification Factors.

Table 1.1: Stress Intensification Factors

KEYOPT(2)	Cσ
	at node I	at node J
0
1		1.0
2	1.0
3

Any entry in Table 1.1: Stress Intensification Factors either input as or computed to be less than 1.0 is set to 1.0. The entries are:

= stress intensification factor of end I of straight pipe (input as SIFI on R command)

= stress intensification factor of end J of straight pipe (input as SIFJ on R command)

σ_th (output as STH), which is in the postprocessing file, represents the stress due to the thermal gradient thru the thickness. If the temperatures are given as nodal temperatures, σ_th = 0.0. But, if the temperatures are input as element temperatures,

(1–44)

where:

Equation 1–44 is derived as a special case of Equation 2–8, Equation 2–9 and Equation 2–11 with y as the hoop coordinate (h) and z as the radial coordinate (r). Specifically, these equations

This results in:

(1–45)

or

(1–46)

and

(1–47)

Finally, the axial and shear stresses are combined with:

(1–48)

(1–49)

where:

The maximum and minimum principal stresses, as well as the stress intensity and the equivalent stress, are based on the stresses at two extreme points on opposite sides of the bending axis, as shown in Figure 1.7: Stress Point Locations. If shear stresses due to lateral forces are greater than the bending stresses, the two points of maximum shearing stresses due to those forces are reported instead. The stresses are calculated from the typical Mohr's circle approach in Figure 1.8: Mohr Circles.

The equivalent stress for Point 1 is based on the three principal stresses which are designated by small circles in Figure 1.8: Mohr Circles. Note that one of the small circles is at the origin. This represents the radial stress on the outside of the pipe, which is equal to zero (unless P_o ≠ 0.0). Similarly, the points marked with an X represent the principal stresses associated with Point 2, and a second equivalent stress is derived from them.

Next, the program selects the largest of the four maximum principal stresses (σ₁, output as S1MX), the smallest of the four minimum principal stresses (σ₃, output as S3MN), the largest of the four stress intensities (σ_I, output as SINTMX), and the largest of the four equivalent stresses (σ_e, output as SEQVMX). Finally, these are also compared (and replaced as necessary) to the values at the right positions around the circumference at each end. These four values are then printed out and put on the postprocessing file.

Figure 1.7: Stress Point Locations

Figure 1.8: Mohr Circles

Three additional items are put on the postdata file for use with certain code checking. These are:

(1–50)

(1–51)

(1–52)

where:

MXI = moment about the x axis at node I, etc.