2. Thick Pipe Thermal Gradient
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2.1. Problem Description
Consider a long thick-walled steel pipe with a complex duty cycle of thermal loading that results in various temperature gradients through the thickness at different times.  At room temperature (77°F) the inner radius of the pipe is 10 in. and the thickness is 1.0 in.  The pipe material properties are given in Table 2.1.  The fluid temperature ranges from 65°F to 600°F, and the maximum operating pressure is 1000 psi.  Figures 2.1a through 2.1e show five different thermal gradients that can occur at various times during operation of the pipe.  Table 2.2 summarizes the loading requirements. 

Table 2.1.  Material Properties

Material
Designation
(Reference)
Tensile
Modulus (psi)
Poisson
Ratio
CTE
(/°F)
Ref. Temp.
(°F)
Density.
(lb/in3)
Steel
301 Stainless
(MIL-HDB-5H)
28e6
0.27
8.55e-6
77
0.286

graphic
Figure 2.1a.  Thermal Gradient 1 (Linear Down)

graphic
Figure 2.1b.  Thermal Gradient 2 (Linear Up)

graphic
Figure 2.1c.  Thermal Gradient 3 (Convex Up)

graphic
Figure 2.1d.  Thermal Gradient 4 (Convex Down)

graphic
Figure 2.1e.  Thermal Gradient 5 (Complex)


Table 2.2.  Loading Requirements

Pressure
(psi)
Temperature
(°F)
Axial Constraint
0 to 1000
65 to 600
Plane Strain (ez = 0)

2.2. Objectives
1. Use Concyl to compute the maximum tensile and von-mises stress for each of the specified thermal gradients with and without the maximum pressure loading applied.  Also tabulate the radial deflections at the inner and outer surfaces of the pipe for each load case. 
2. Verify the Concyl results using FEA. 

2.3. Key Assumptions
1. Linear, elastic, and isotropic material
2. Linear kinematics
3. No temperature or pressure variation in axial direction
4. Fully constrained in axial direction (plane strain)

2.4. Concyl Analysis
Table 2.3 lists the ten load cases that were analyzed.  Nineteen equally spaced layers were used to model the temperature variation through the thickness.  The temperature variation was simulated by assigning a delta temperature to each layer.  This was done by properly varying the stress-free temperature that is assigned to each layer in accordance with a uniform applied temperature of 0°F. 
The first step in determining the proper stress-free temperature for each layer was to project the temperature gradients plotted in Figures 1a through 1e onto the 19-layer model.  This amounts to computing the temperature at each radial interface via linear interpolation of the thermal gradient plot data.  Table 2.4 shows the results of this linear projection for each load case.  Next, a centroidal temperature value was computed for each cylinder as the average of the inner and outer interface temperatures.  Then, the stress-free temperature of 77°F was subtracted to yield the delta temperatures for each layer as shown in Table 2.5.  Since Concyl can only apply a single uniform temperature to the entire model, the delta temperature values listed in Table 2.5 were assigned to each layer as negative stress-free temperatures, and a temperature of 0°F was applied to model.  This gives the proper overall delta temperature distribution through the pipe for each load case. 

Table 2.3.  Load Cases

Case
Pressure
(psi)
Temperature
(°F)
1
0
Gradient 1, Fig. 2.1a
2
0
Gradient 2, Fig. 2.1b
3
0
Gradient 3, Fig. 2.1c
4
0
Gradient 4, Fig. 2.1d
5
0
Gradient 5, Fig. 2.1e
6
1000
Gradient 1, Fig. 2.1a
7
1000
Gradient 2, Fig. 2.1b
8
1000
Gradient 3, Fig. 2.1c
9
1000
Gradient 4, Fig. 2.1d
10
1000
Gradient 5, Fig. 2.1e


Table 2.4.  Radial Temperature Variation For Each Load Case

Radius
Case1
Case2
Case3
Case4
Case5
10.00000
600.0
65.0
600.00
65.00
600.00
10.05263
571.8
93.2
521.05
149.21
573.68
10.10526
543.7
121.3
439.47
233.68
544.21
10.15789
515.5
149.5
334.21
320.53
486.32
10.21053
487.4
177.6
237.47
404.21
439.92
10.26316
459.2
205.8
174.84
475.26
439.54
10.31579
431.1
233.9
122.95
535.26
439.15
10.36842
402.9
262.1
96.11
569.47
420.99
10.42105
374.7
290.3
76.84
592.11
391.75
10.47368
346.6
318.4
68.95
597.37
310.20
10.52632
318.4
346.6
68.95
597.37
262.12
10.57895
290.3
374.7
76.84
592.11
266.37
10.63158
262.1
402.9
96.11
569.47
250.55
10.68421
233.9
431.1
122.95
535.26
115.51
10.73684
205.8
459.2
174.84
475.26
71.32
10.78947
177.6
487.4
237.47
404.21
66.05
10.84211
149.5
515.5
334.21
320.53
113.42
10.89474
121.3
543.7
439.47
233.68
173.95
10.94737
93.2
571.8
521.05
149.21
246.32
11.00000
65.0
600.0
600.00
65.00
320.00


Table 2.5.  Applied Cylinder Delta Temperature For Each Load Case

Cyl
DT1
DT2
DT3
DT4
DT5
1
508.921
2.079
483.526
30.105
509.842
2
480.763
30.237
403.263
114.447
481.947
3
452.605
58.395
309.842
200.105
438.263
4
424.447
86.553
208.842
285.368
386.119
5
396.289
114.711
129.158
362.737
362.729
6
368.132
142.868
71.895
428.263
362.342
7
339.974
171.026
32.526
475.368
353.068
8
311.816
199.184
9.474
503.789
329.368
9
283.658
227.342
-4.105
517.737
273.975
10
255.500
255.500
-8.053
520.368
209.162
11
227.342
283.658
-4.105
517.737
187.244
12
199.184
311.816
9.474
503.789
181.460
13
171.026
339.974
32.526
475.368
106.033
14
142.868
368.132
71.895
428.263
16.414
15
114.711
396.289
129.158
362.737
-8.316
16
86.553
424.447
208.842
285.368
12.737
17
58.395
452.605
309.842
200.105
66.684
18
30.237
480.763
403.263
114.447
133.132
19
2.079
508.921
483.526
30.105
206.158

Table 2.6 lists the maximum tensile and von-mises stresses in the pipe for each load case.  The maximum stress locations are also indicated, with (i) denoting the inner surface, (m) denoting the interior of the pipe, and (o) denoting the outer surface.  The direction of the tensile stress is also indicated, with R for radial, H for hoop, and Z for axial.  All the maximum tensile stresses are in the hoop direction as shown.  Von-mises stress is a scalar, so it has no direction. 
Table 2.6 shows that the maximum von-mises stress is approximately 128 ksi for load case 3. Load cases 3 and 5 both have a very similar von-mises stress maxima.  In fact, it can be clearly seen that the maximum von-mises stress value does not vary much with load case (only 5% variation).  This indicates that the maximum von-mises stress is fairly independent of the shape of the temperature gradient.  However, the location of  the maximum von-mises stress does vary with the load case.  In fact, the maximum von-mises stress always occurs at the location of the maximum temperature. 
The maximum tensile stress, on the other hand, does have much more dependence on the shape of the thermal gradient, with a variability of 45% over the 10 load cases.  This variability is computed by taking the maximum and minimum values and dividing the absolute value of their difference by their average.  Therefore, if we are most interested in tension, then the shape of the thermal gradient is very important.  However, if we are most interested in distortion (von-mises stress), then the shape of the thermal gradient is not important. 
Table 2.6 shows that the maximum tensile stress of 110 ksi occurs for load case 9.  Note that the location of the maximum tensile stress is always at the location of the minimum temperature, which is the opposite of the von-mises stress.  Also, note that the 1000 psi internal pressure load has minimal effect on the maximum von-mises stress.  However, the internal pressure load does have a significant effect on the maximum tensile stress. 

Table 2.6.  Concyl Stress Results

Case
Loading
Tensile Stress
(ksi)
Von-Mises Stress
(ksi)
1
graphic
81.8 H    (o)
125.8    (i)
2
graphic
84.5 H    (i)
125.0    (o)
3
graphic
84.5 H    (m)
127.7    (i,o)
4
graphic
99.8 H    (i,o)
122.4    (m)
5
graphic
82.2 H    (m)
127.3    (i)
6
graphic + Pi
91.6 H    (o)
122.1    (i)
7
graphic + Pi
95.0 H     (i)
122.4    (o)
8
graphic + Pi
69.4 H    (m)
123.8    (i,o)
9
graphic + Pi
110.5 H  (i,o)
121.0    (m)
10
graphic + Pi
91.9 H    (m)
123.5    (i)
 
Variability
45%
5%

Table 2.7 lists the computed the radial displacements at the inner and outer surface of the pipe for each load case.  All of the radial displacement values in this table are positive which indicates expansion of the pipe for all load cases.  The radial displacement does vary significantly with load case as shown.  The variability is around 70% at both surfaces.  Load case 9 has the maximum predicted internal and external radial displacement.  It is also evident that most of the radial deformation is due to the thermal loading, since the addition of the pressure load only gives a slight increase to the deformation for each load case.   

Table 2.7.  Concyl Displacement Results

Case
Loading
Inner (E-2 in.)
Outer (E-2 in.)
1
graphic
2.73
3.00
2
graphic
2.82
3.10
3
graphic
1.88
2.06
4
graphic
3.63
4.00
5
graphic
2.59
2.85
6
graphic + Pi
3.10
3.35
7
graphic + Pi
3.18
3.45
8
graphic + Pi
2.24
2.41
9
graphic + Pi
3.99
4.34
10
graphic + Pi
2.96
3.20
 
Variability
72%
71%

Plots of the hoop stress, radial stress, von-mises stress, and radial displacement were generated from the Concyl plot database file for several of the load cases.  These plots are shown in Figures 2.2 through 2.5 for load cases 1, 3, 5, and 9. 

graphic
Figure 2.2a.  Hoop Stress For Load Case 1

graphic
Figure 2.2b.  Hoop Stress For Load Case 3

graphic
Figure 2.2c.  Hoop Stress For Load Case 5

graphic
Figure 2.2d.  Hoop Stress For Load Case 9

graphic
Figure 2.3a.  Radial Stress For Load Case 1

graphic
Figure 2.3b.  Radial Stress For Load Case 3

graphic
Figure 2.3c.  Radial Stress For Load Case 5

graphic
Figure 2.3d.  Radial Stress For Load Case 9


graphic
Figure 2.4a.  Von-Mises Stress For Load Case 1

graphic
Figure 2.4b.  Von-Mises Stress For Load Case 3

graphic
Figure 2.4c.  Von-Mises Stress For Load Case 5

graphic
Figure 2.4d.  Von-Mises Stress For Load Case 9

graphic
Figure 2.5a.  Radial Displacement For Load Case 1

graphic
Figure 2.5b.  Radial Displacement For Load Case 3

graphic
Figure 2.5c.  Radial Displacement For Load Case 5

graphic
Figure 2.5d.  Radial Displacement For Load Case 9

2.5. Finite Element Analysis
Confirmation of the Concyl results was done using the TEXLESP-2D finite element program (Ref. 1).  Ten runs were done to analyze the load cases listed in Table 2.3.  The FEA results agree well with the Concyl exact solution. 
2.5.1.  Model Description
A 2D axisymmetric finite element analysis was done to generate results for comparison with the Concyl results.  Nineteen separate layers were modeled and assigned the same material properties and delta temperatures as the Concyl model (listed in Tables 2.1 and 2.5).  Each of the 19 layers has three equally spaced elements which gives a total of 57 elements.  The elements are quadratic and isoparametric with nine nodes per elements.  Since this problem has no axial stress variation, only one element is required in the axial direction.  Figure 2.6 shows the mesh and materials.  Axial constraints were applied to the forward and aft edges to simulate plane strain. 

graphic
Figure 2.6.  Finite Element Mesh and Boundary Conditions

2.5.2. TEXLESP Results
The TEXLESP results show excellent agreement Concyl.  All the stresses, strains, and displacements agree to within 2% for all load cases.  Figure 2.7 through 2.9 show some typical plots of hoop stress, radial stress, and radial displacement for several load cases.  These figures show the TEXLESP results with the Concyl results overlayed for direct comparison.  Note that TEXLESP always slightly under predicts the stress while slightly over predicting the displacement. 

graphic
Figure 2.7a.  TEXLESP Hoop Stress For Load Case 1

graphic
Figure 2.7b.  TEXLESP Hoop Stress For Load Case 3

graphic
Figure 2.7c.  TEXLESP Hoop Stress For Load Case 9

graphic
Figure 2.8a.  TEXLESP Radial Stress For Load Case 1

graphic
Figure 2.8b.  TEXLESP Radial Stress For Load Case 3

graphic
Figure 2.8c.  TEXLESP Radial Stress For Load Case 9

graphic
Figure 2.9a.  TEXLESP Radial Displacement For Load Case 1

graphic
Figure 2.9b.  TEXLESP Radial Displacement For Load Case 3

graphic
Figure 2.9c.  TEXLESP Radial Displacement For Load Case 9


2.6. Conclusions
This sample demonstrates how Concyl can be used to analyze thick-walled pipes with various thermal gradient loads combined with internal pressure loading. The Concyl results have excellent agreement with finite-element results.  This verifies that Concyl is an accurate and efficient tool for analyzing complex thermal gradient loading in thick-walled pipes. 

References
1. "TEXLESP-2D Users Manual", Eric B. Becker, et al, NASA MSFC-RPT-1564, 1988. 


Concyl Load Case 1 Listing

!Concyl Input File
!---------------------------------------------------
Sample 2, Load Case 1
!---------------------------------------------------
! Thermal Gradient 1, 19 cylinders
! Pressure = 0
!---------------------------------------------------
!
19    ! number of cylinders
1     ! number of materials
!
!---------------------------------------------------
! cylinder definitions
!   also defines the thermal gradient
!---------------------------------------------------
!
10.00000000,10.05263158,10.10526316,10.15789474,10.21052632,10.26315789 &
10.31578947,10.36842105,10.42105263,10.47368421,10.52631579,10.57894737 &
10.63157895,10.68421053,10.73684211,10.78947368,10.84210526,10.89473684 &
10.94736842,11.00000000
!
1,-508.921
1,-480.763
1,-452.605
1,-424.447
1,-396.289
1,-368.132
1,-339.974
1,-311.816
1,-283.658
1,-255.500
1,-227.342
1,-199.184
1,-171.026
1,-142.868
1,-114.711
1,-86.553
1,-58.395
1,-30.237
1,-2.079
!
!---------------------------------------------------
! material definitions
!---------------------------------------------------
!
1, Steel, 28.0e6,  0.27, 8.55e-6, 0.286
!
!---------------------------------------------------
! load definitions
!---------------------------------------------------
!
0., 0.     ! pressures
0.         ! temperature
0.         ! axial strain
!
!---optional plot definitions---
!
1
ALL
10

TEXLESP Load Case 9 Listing

$ Concyl Sample 2 Test Case
$ Plane Strain Single Cylinder
$ English Units: in,lb,sec,psi
$ Load Case 9
$------------------------------------------------
$ Thermal Gradient Loading 9, Nonlinear Gradient
$ Pressure = 1000
$------------------------------------------------
axisym
thermal,0
$
$ material definition
$
setup,2,ij,115
  iso,steel, 1,   27e6, 0.27, 8.55e-6, -30.105
  iso,steel, 2,   27e6, 0.27, 8.55e-6, -114.447
  iso,steel, 3,   27e6, 0.27, 8.55e-6, -200.105
  iso,steel, 4,   27e6, 0.27, 8.55e-6, -285.368
  iso,steel, 5,   27e6, 0.27, 8.55e-6, -362.737
  iso,steel, 6,   27e6, 0.27, 8.55e-6, -428.263
  iso,steel, 7,   27e6, 0.27, 8.55e-6, -475.368
  iso,steel, 8,   27e6, 0.27, 8.55e-6, -503.789
  iso,steel, 9,   27e6, 0.27, 8.55e-6, -517.737
  iso,steel, 10,  27e6, 0.27, 8.55e-6, -520.368
  iso,steel, 11,  27e6, 0.27, 8.55e-6, -517.737
  iso,steel, 12,  27e6, 0.27, 8.55e-6, -503.789
  iso,steel, 13,  27e6, 0.27, 8.55e-6, -475.368
  iso,steel, 14,  27e6, 0.27, 8.55e-6, -428.263
  iso,steel, 15,  27e6, 0.27, 8.55e-6, -362.737
  iso,steel, 16,  27e6, 0.27, 8.55e-6, -285.368
  iso,steel, 17,  27e6, 0.27, 8.55e-6, -200.105
  iso,steel, 18,  27e6, 0.27, 8.55e-6, -114.447
  iso,steel, 19,  27e6, 0.27, 8.55e-6, -30.105
end,material
$
$ mesh definition
$
1,1, 115,3
  10.00,11.00,11.00,10.00
  0.00,0.00,0.01,0.01
end,grid
$
$ element definitions
$
iloop,3
  qq,,1, 1,1
  qq,,2, 7,1
  qq,,3, 13,1
  qq,,4, 19,1
  qq,,5, 25,1
  qq,,6, 31,1
  qq,,7, 37,1
  qq,,8, 43,1
  qq,,9, 49,1
  qq,,10, 55,1
  qq,,11, 61,1
  qq,,12, 67,1
  qq,,13, 73,1
  qq,,14, 79,1
  qq,,15, 85,1
  qq,,16, 91,1
  qq,,17, 97,1
  qq,,18, 103,1
  qq,,19, 109,1
iend
$
$ boundary conditions
$
iloop,57
  bc,uy, 1,1, 1
  bc,uy, 1,1, 3
iend

bc,pressure,1,1,4, 1000.

end,elements
$
$ linear solver
$
solve
$
$ post processing
$
stress
   option,linear,17
end,stress
stop