1. Reference problem#
1.1. Geometry#
This test is taken from a study, carried out by Department MMC, which focused on the modeling of industrial piping. It resulted in a fatigue calculation according to the RCC -M B3600.
The line has 10 bends. It is oriented from node N1 to node N2.
Characteristics of the sections:
Straight parts:
\(R\mathrm{=}406.4\mathit{mm}\), \(\mathit{EP}\mathrm{=}32.\mathit{mm}\)
Tubing 1, \(R\mathrm{=}410.\mathit{mm}\), \(\mathit{EP}\mathrm{=}38.\mathit{mm}\);
Tubing2, \(R\mathrm{=}444.4\mathit{mm}\), \(\mathit{EP}\mathrm{=}70.\mathit{mm}\);
Elbows:
Mesh group \(\mathit{POUCT}\): \(R\mathrm{=}406.4\mathit{mm}\); \(\mathit{EP}\mathrm{=}34.\mathit{mm}\);
Flexibility coefficient for all elbows, \(\mathit{cflex}\mathrm{=}6.032\);
Elbow bending radii: \(1220\mathit{mm}\)
In addition, for the seismic calculation, 6 discrete elements (DIS_T) are added at 3 points of the line (1 vertical and one horizontal per anchor point).
For the transitory thermal calculation (which makes it possible to estimate the temperature gradients across the wall), a calculation is not performed on a \(\mathrm{3D}\) bend, but only on two representative sections:
the cross section of the straight tubes (thickness \(32\mathit{mm}\)),
of the elbow section (thickness \(34\mathit{mm}\)).
These calculations are axisymmetric. Since the solution is independent of the axis, the portion of pipe is modelled by a rectangle, finely meshed along the radius, and comprising a single element along the axis.
Two transients are modelled.
1.2. Material properties#
The line is made of standard steel. The force calculations are carried out at various temperature values. We therefore consider the properties of materials as a function of temperature:
Temperature ( \(°C\) ) |
Young’s Modulus ( \(\mathit{GPa}\) ) |
Average Expansion Coefficient (from \(20°C\) ) |
0.0 |
205 |
1.092e-05 |
20.0 |
204 |
1.092e-05 |
50.0 |
203 |
1.114e-05 |
100.0 |
200 |
1.15e-05 |
150.0 |
197 |
1.187e-05 |
200.0 |
193 |
1.224e-05 |
250.0 |
189 |
1.257e-05 |
300.0 |
185 |
1.289e-05 |
350.0 |
180 |
1.324e-05 |
Poisson’s ratio: \(0.3\)
The WOHLER curve is defined by:
Salt ( \(\mathit{MPa}\) ) |
Number of cycles |
0.01 |
1.E15 |
86 |
1000000 |
93 |
500000 |
114 |
200000 |
138 |
100000 |
160 |
50000 |
215 |
20000 |
260 |
10000 |
330 |
5000 |
440 |
2000 |
570 |
1000 |
725 |
500 |
1070 |
200 |
1410 |
100 |
1900 |
50 |
2830 |
20 |
4000 |
10 |
The interpolation is logarithmic, and the extension to the left is linear. Because of the low stress amplitude values for this line, the first point is artificially added:
0.01 |
1.E15 |
The characteristics used for fatigue analysis according to RCC -M are:
\(m\mathrm{=}3\)
\(n\mathrm{=}0.2\)
\(\mathit{Sm}\mathrm{=}133.6\mathit{MPa}\)
The densities include insulation.
The thermal characteristics are provided at the mean temperature of the calculated transient:
Transient 2: mean temperature = \(273.5°C\),
Transient 6: mean temperature = \(281°C\),
Temperature ( \(°C\) ) |
273.5 |
281 |
|
Thermal conductivity (\(W\mathrm{/}\mathit{m.}°C\)) |
46.595 |
46.37 |
|
Heat capacity (\(J\mathrm{/}{m}^{3}\mathrm{.}°C\)) |
4.25 106 |
4.25 106 |
4.27 106 |
The discrete elements used for seismic calculation have the following stiffness:
\(\mathit{K1}\mathrm{=}0.5{10}^{8}N\mathrm{/}m\)
\(\mathit{K2}\mathrm{=}1.0{10}^{8}N\mathrm{/}m\)
1.3. Boundary conditions and loads#
The various elementary mechanical loads considered constitute the stabilized states corresponding to the design situations of the line:
Thermal expansion loads:
A calculation by loading is carried out, which combines the forces of thermal expansion thwarted in the line at the prescribed temperature, with those caused by displacement of the component (the reference temperature is equal to \(20°C\) in all cases):
Load number |
Temperature (\(°C\)) |
\({U}_{x}\) (\(\mathit{mm}\)) |
\({U}_{y}\) (\(\mathit{mm}\)) |
\({U}_{z}\) (\(\mathit{mm}\)) |
|
1 |
10 |
0 |
0 |
0 |
|
2 |
287 |
0.046466 |
—0.0304945 |
0.076 |
|
3 |
274.5 |
0.046466 |
—0.0304945 |
0.072 |
|
4 |
272.5 |
0.046466 |
—0.0304945 |
0.072 |
|
5 |
286 |
0.046466 |
—0.0304945 |
0.076 |
|
6 |
275 |
0.046466 |
—0.0304945 |
0.072 |
|
7 |
290 |
0.046466 |
—0.0304945 |
0.077 |
|
8 |
284 |
0.046466 |
—0.0304945 |
0.077 |
|
10 |
256 |
0.0360129 |
—0.0245167 |
0.067 |
|
12 |
257 |
0.0360129 |
—0.0245167 |
0.067 |
|
14 (hydraulic test) |
20 |
20 |
0 |
0 |
0 |
Boundary conditions: for all previous loads the N2 node is embedded.
Note:
The movements of the next component x and y used do not correspond to those provided in the list of situations in note [2], because they are expressed in another coordinate system (coordinate system linked to the line, such as the local \(x\) axis makes an angle of 25° with the global \(X\) axis). Here we use the displacements expressed as a global coordinate system (the one used to define the geometry of the line).
Hydraulic test: it is defined (apart from the test pressure defined in the list of situations) by the load 14: blocked N1 and N2 ends and forces due to the difference in weight between the solid line and the empty line. In addition, supports - weights are added for this load: they are modeled by a condition \(\mathit{DZ}\mathrm{=}0\), applied in 7 nodes distributed over the line.
Earthquake: the floor spectra corresponding to SNA (earthquake considered for fatigue analysis) are:
The associated anchor movements are:
Node N2: \(\mathit{Dx}\mathrm{=}4\mathit{mm}\), \(\mathit{Dy}\mathrm{=}7\mathit{mm}\), \(\mathit{Dz}\mathrm{=}5\mathit{mm}\)
Node N1: \(\mathit{Dx}\mathrm{=}11.96\mathit{mm}\), \(\mathit{Dy}\mathrm{=}4.35\mathit{mm}\), \(\mathit{Dz}\mathrm{=}1\mathit{mm}\)
Definition of situations:
Situation |
Number of occurres |
Press |
||
( \(\text{bar}\) ) » |
Load Number |
Thermal transient |
||
1 |
190 |
1 71.5 |
1 » |
|
2 » |
||||
2 |
1300000 |
58.9 57.6 |
3 » |
|
4 » |
2 |
|||
3 |
4000 |
70 59 |
5 6 |
6 |
4 |
100000 |
73.4 68.1 |
7 8 |
2 |
5 |
16080 |
71.5 44 |
9 10 |
6 |
6 |
790 |
74.5 44 |
11 12 |
6 |
7 |
10 390 sub-cycles |
Earthquake |
||
11 |
13 |
112 1 |
14 |
Thermal transients: two transients are calculated. They correspond to an exchange condition in the internal skin of the axisymmetric calculation defined by an exchange coefficient \(H\mathrm{=}30000W\mathrm{/}{m}^{2.}°C\) and two fluid temperature histories:
transitory 2:
Time ( \(s\) ) |
Fluid temperature ( \(°C\) ) |
0.0 |
274.5 |
10.0 |
274.5 |
310.0 |
272.5 |
610.0 |
274.5 |
910.0 |
272.5 |
transitory 6:
Time ( \(s\) ) |
Fluid temperature ( \(°C\) ) |
0.0 |
272.0 |
11.0 |
272.0 |
20.0 |
290.0 |
40.0 |
290.0 |
The fatigue analysis is performed on the first node located immediately after the exit of the component (\(\mathit{N80}\) belonging to the \(\mathit{M1}\) mesh).