Investigations on Fatigue Life of Tube Connections Based on International Codes of Pressure Vessel
Abstract
:1. Introduction
2. Current Approaches to Fatigue Life Assessment in Codes
2.1. Elastic Stress Analysis and Equivalent StressesMethod 1
2.2. Elastic–Plastic Stress Analysis and Equivalent StrainsMethod 2
2.3. Fatigue Assessment of Welds—Elastic Analysis and Structural Stress—Method 3
2.4. Detailed Assessment of Fatigue LifeMethod 4
3. Differences and Similarities
3.1. Design Principles and Assessment Parameters
3.2. Fatigue Design Curve
3.3. Welding Material
3.4. Correction Coefficient
 (a)
 Plastic correction. In Method 1, the fatigue loss coefficient Ke is introduced to consider the influence of the plastic. Method 2 is about the elastic–plastic analysis. The stable cycle stress–strain curve is applied to analyze the material property, and the plastic correction coefficient is no longer considered because the influence of the material in the plastic stage has been calculated. In Method 3, the Formulas are determined by the Neubers principle and the stress–strain curve model of the material hysteretic property, which is applied to calculate the corresponding local nonlinear structural stress and strain range. On this basis, the calculated structural stress range should be corrected for low cycle fatigue. According to Method 4, for both the weld and the unwelded, if the calculated nominal elastic structural stress range exceeds twice the yield stress of the material, the plastic correction coefficient Ke (referring to the mechanical load) or Kν (referring to the thermal load) should be considered. For the results obtained by the elastic–plastic analysis in Method 4, if the total strain range (including the elastic and plastic strain) caused by all the loads is known, then the plastic correction is unnecessary.
 (b)
 Thickness correction. In Methods 1 and 2, the fatigue curves of the smooth specimens are applied, so the thickness correction has been included in the curves. In Method 3, the corresponding parameter is applied to the formula to analyze the influence of the thickness. Method 4 defines the calculation method of the thickness correction coefficient for the weldments, unweldments and bolts. The formula of the thickness correction coefficient of the nonweldments involves the unknown allowable load cycle, so the calculation needs a hypothesis for the iterative computations, which greatly increases the complexity of the computation [18].
 (c)
 Temperature correction. There are usually differences between the design temperature of the vessel and the test temperature of the fatigue design curve, which will influence the assessment of the actual allowable life. In the ASME VIII2, based on the ratio between the elasticity modulus of the specimen material at the test temperature (the environment temperature) of the design fatigue curve and the elasticity modulus of the specimen material at the design temperature, the allowable cycle index is adjusted. Method 4 determines the calculation method of the coefficient for temperature correction to the weldments, nonweldments and bolts. In the design, we should consider that the strength of the metal at high temperatures is reduced, and creep failure may be caused when the temperature is too high; low temperature can improve the fatigue strength of metal materials to a certain extent, but at the same time, it will also reduce the toughness of the material and increase the brittleness, so toohigh temperature and toolow temperature are harmful to fatigue.
 (d)
 Mean stress correction. In Method 1 and Method 2, the fatigue curve of the smooth specimen is adopted, and the mean stress correction is included in the curves. In Method 3, the relevant parameter is applied to evaluate the influence of the thickness in the formula. Similar to Method 1 and Method 2, the influence of the mean stress is involved in the fatigue design curve of the weld in Method 4; however, for the unweld, the corrections are made separately according to the absolute value of the maximum stress and the range of stress. It uses the full mean stress correction under the purely elastic state and the reducing mean stress correction under the shakedown elastic–plastic state. Under the cyclic plastic state, the plastic correction is considered instead of the mean stress correction. The mean stress correction coefficient is related to the yield strength, the tensile strength, the maximum stress, the stress range, the mean stress, the allowable cycle number, etc., where the initial value for the iteration needs to be assumed.
 (e)
 Surface treatment correction. For the three methods in ASME VIII2, the correction factor of 20 for the number of cycles accounts for factors that actually affect the fatigue life but have not been considered in tests, including surface correction factors. In Method 4, the smooth standard specimen is applied to the assessment of the unweld, but it does not resemble the structure in the real world. Thus, the correction coefficient of the surface roughness is introduced to the assessment. Similar to the calculation of the thickness correction coefficient, because of the unknown allowable cycle number, iterative computation is required in the assessment, which makes the computation more complex [19].
3.5. Histogram Development and Cycle Counting
3.6. Linear Fatigue Damage Cumulation
3.7. Strength Theory
4. Finite Element Analysis of Opening Tubing Connection
4.1. Structure Dimension and Design Parameter
4.2. Finite Element Model
4.2.1. Models and Grids
4.2.2. Loads and Boundary Conditions
 (1)
 Apply the symmetry constraint to the symmetry plane.
 (2)
 In order to avoid the global displacement of the model, the displacement of the Ydirection at the end face of the nozzle is restricted.
 (3)
 Apply the pressure to the internal surface of the model.
 (4)
 Apply the equivalent end force to the shell end: The end force is expressed as the uniform pressure acting on the shell end, which is equal to the axial force generated by the internal pressure at the shell end face, divided by the cylinder crosssectional area.
4.3. Finite Element Calculation Results
4.3.1. Elastic Stress Calculation
4.3.2. Elastic–Plastic Stress Calculation
4.3.3. Stress Intensity Analysis
5. Fatigue Evaluation Result and Analysis
5.1. Evaluation Results of Table 3
5.2. Evaluation Results of Table 4
5.3. Evaluation Results of Table 5
5.4. Result Analysis
6. Conclusions
 (1)
 For the calculation of the elastic stress, Method 1 adopts the effective total equivalent stress amplitude for assessing the fatigue damage. Method 1 is the most widely used traditional method and can be used in both welded structure and unwelded structures. This method has simple operation, safety and reliability.
 (2)
 For the elastic–plastic calculation, Method 2 adopts the effective strain range for assessing the fatigue damage and can be used in both the welded structure and the unwelded structure. This method is with high accuracy, good stability, safety and reliability, but it is difficult to obtain the stable stress–strain cyclic curves of the corresponding materials. Furthermore, the elastic–plastic analysis is very complicated.
 (3)
 For the calculation of the elastic stress, Method 3 adopts the equivalent structure stress for assessing the fatigue damage. This method is applied for the fatigue assessment of the welded. It is suggested to be used for the welded joint without mechanical processing. This method is developed under fracture mechanics, but it is still conservative and unstable, and the procedure is very complicated.
 (4)
 In Method 4, the detailed assessing procedure is performed separately for the welded and unwelded. For the welded, Method 4 applies the hotspot stress obtained from the principal stress by the extrapolation for the assessment. For the unweldment, it applies the notch stress as the assessment parameter. The iterative calculations are required. This method is the most accurate, stable and reliable.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Design Load Combination  Allowable Stress 

(1) P + Ps + D (2) P + Ps + D + L (3) P + Ps + D + L + T (4) P + Ps + D + Ss (5) 0.6 D + (0.6 W or 0.7 E) (6) 0.9 P + Ps + D + (0.6 W or 0.7 E) (7) 0.9 P + Ps + D + 0.75 (L + T) + 0.75 Ss (8) 0.9 P + Ps + D + 0.75 (0.6 W or 0.7 E) + 0.75 L + 0.75 Ss  Determined based on the stress category shown in Figure 1 
Design Load Parameter  Description 

L 

E  Earthquake loads 
W  Wind loads 
W_{pt}  The pressure test wind load case. The design wind speed for this case shall be specified by the owner user. 
S_{s}  Snow loads 
T  The selfrestraining load case (i.e., thermal loads, applied displacements). This load case does not usually affect the collapse load but should be considered in cases where the elastic followup causes stress to be insufficient to redistribute the load without excessive deformation. 
NO.  Cylinder Diameter  Cylinder Thickness  Nozzle Diameter  Nozzle Thickness  Testing Pressure  Testing Cycle Life  

Do/mm  T/mm  do/mm  t/mm  P_{max}/MPa  P_{min}/MPa  n  
1  325  10  44.2  7  1.2931  P_{min} = 0.1 P_{max}  180,200 
2  325  10  44.2  7  1.3272  182,100  
3  325  10  140.2  11  1.1223  150,900  
4  325  10  159.5  15.9  1.3272  104,800  
5  325  10  159.3  22  1.4199  155,000  
6  325  10  159.3  22  1.376  149,700  
7  325  10  252  22  1.1174  90,100  
8  325  10  252  22  1.1711  42,100  
9  325  10  268.3  32  1.1711  79,600  
10  600  20  303.7  34  1.3367  79,400  
11  600  20  303.7  34  1.2839  72,000  
12  600  20  303.7  34  1.2997  72,800  
13  325  20  44.2  7  2.7821  41,200  
14  325  20  44.2  7  2.5503  105,400  
15  325  20  159.3  22  2.2075  171,200  
16  325  20  159.3  22  2.3184  86,400  
17  325  20  244.6  22  2.3083  43,300  
18  325  20  244.6  22  2.0664  119,900  
19  325  20  267.5  32  2.1067  97,700  
20  325  20  267.5  32  1.7338  95,400  
21  325  20  139.1  11  1.9959  91,000  
22  325  20  139.1  11  1.9051  128,000 
NO.  Cylinder Diameter  Cylinder Thickness  Nozzle Diameter  Nozzle Thickness  Testing Pressure  Testing Cycle Life  

Do/mm  T/mm  do/mm  t/mm  P_{max}/MPa  P_{min}/MPa  n  
23  342.90  19.050  50.700  9.525  25.389  0  51,000 
24  25.389  51,300  
25  22.712  87,500  
26  22.712  103,000  
27  46.884  9990  
28  46.884  10,900  
29  39.341  18,800  
30  34.149  46,900  
31  34.149  53,500  
32  34.474  49,100 
NO.  Cylinder Diameter  Cylinder Thickness  Nozzle Diameter  Nozzle Thickness  Testing Pressure  Testing Cycle Life  

Do/mm  T/mm  do/mm  t/mm  P_{max}/MPa  P_{min}/MPa  n  
33  321.4  13  70.9  13  37.296  0  6500 
34  35.317  7600  
35  33.339  9800  
36  29.441  15,600  
37  321.4  13  94.8  13  39.216  8600  
38  35.317  14,800  
39  29.441  20,500  
40  334  20  110.1  20.8  49.010  3460  
41  43.125  6400  
42  39.261  14,300  
43  33.376  20,900 
Model  Structure  Material  Allowable Stress S/MPa  Yield Strength Sy/MPa  Tensile Strength Su/MPa  Elastic Modulus E/×10^{3} MPa  Poisson’s Ratio  

Table 3  1–22  Cylinder  Carbon Steel  180  283  434  201  0.3 
Nozzle  174  262  446  
Table 4  23–26  Cylinder Nozzle  A201A  162  244  388  
27–32  A302B  275  533  660  
Table 5  33–36 40–43  Cylinder  FTW60  285  617  686  
Nozzle  JISSF60  235  353  597  
37–39  Cylinder  FTW60  285  617  686  
Nozzle  302B  269  496  647 
Number of Grid Nodes  Number of Elements  Maximum Stress/MPa  Deviation 

135,117  27,856  518.724  5.41% 
267,338  58,412  492.097  datum 
398,546  88,660  499.350  1.47% 
Material  ε_{offset}  K_{css}/MPa  n_{css}  σ_{yield}/MPa 

Carbon Steel  2.0 × 10^{−5}  757  0.128  189.508 
Carbon SteelWelded  2.0 × 10^{−5}  695  0.110  211.397 
NO.  Carbon Steel  Carbon SteelWelded  

Stress Range/MPa  Plastic Strain Range  Stress Range/MPa  Plastic Strain Range  
1  379.0161  0  422.7933  0 
2  400  0.000020938  450  0.000030515 
3  450  0.00011294  500  0.00014376 
4  500  0.00030834  550  0.00039708 
5  550  0.00069347  600  0.00092403 
6  600  0.0014  650  0.0020 
7  650  0.0027  700  0.0039 
8  700  0.0048  750  0.0073 
9  750  0.0082  800  0.0131 
10  800  0.0137  850  0.0228 
Stress Evaluation Area  P_{m}/MPa  (P_{m} + P_{b}) /MPa  S_{m}/MPa  Stress Evaluation  Results 
1  160.64  193.99  180  ${\mathrm{P}}_{\mathrm{L}}1.0{\mathrm{S}}_{\mathrm{m}}$ ${\mathrm{P}}_{\mathrm{L}}{+\mathrm{P}}_{\mathrm{b}}{3\mathrm{S}}_{\mathrm{m}}$  qualified 
Stress Evaluation Area  P_{L}/MPa  (P_{L} + P_{b}) /MPa  S_{m}/MPa  Stress Evaluation  Results 
2  84.719  144.38  174  ${\mathrm{P}}_{\mathrm{L}}1.5{\mathrm{S}}_{\mathrm{m}}$ ${\mathrm{P}}_{\mathrm{L}}{+\mathrm{P}}_{\mathrm{b}}{3\mathrm{S}}_{\mathrm{m}}$  qualified 
3  249.63  464.58  174  ${\mathrm{P}}_{\mathrm{L}}1.5{\mathrm{S}}_{\mathrm{m}}$ ${\mathrm{P}}_{\mathrm{L}}{+\mathrm{P}}_{\mathrm{b}}{3\mathrm{S}}_{\mathrm{m}}$  qualified 
4  206.19  281.57  180  ${\mathrm{P}}_{\mathrm{L}}1.5{\mathrm{S}}_{\mathrm{m}}$ ${\mathrm{P}}_{\mathrm{L}}{+\mathrm{P}}_{\mathrm{b}}{3\mathrm{S}}_{\mathrm{m}}$  qualified 
Model NO.  Fatigue Evaluation Result  Model NO.  Fatigue Evaluation Result  

Method 1  Method 2  Method 3  Method 4  Method 1  Method 2  Method 3  Method 4  
1  13.047  10.210  24.600  8.270  23  9.032  6.705  9.351  7.492 
2  14.367  11.354  26.973  9.072  24  9.085  6.744  9.406  7.536 
3  8.979  7.764  27.279  5.550  25  9.499  7.600  11.312  8.342 
4  7.642  6.718  20.588  4.699  26  11.182  8.946  13.316  9.820 
5  8.147  7.085  20.444  5.0374  27  12.784  6.945  13.790  4.380 
6  7.081  5.969  17.887  4.367  28  13.948  7.578  15.046  4.779 
7  3.805  3.384  17.775  3.483  29  4.290  5.455  13.972  4.910 
8  2.188  1.776  9.625  1.658  30  6.041  6.974  22.492  6.241 
9  2.038  1.799  12.213  2.662  31  6.891  7.956  25.657  7.119 
10  4.132  5.212  13.391  2.856  32  6.533  7.639  24.284  6.705 
11  3.272  2.281  10.716  2.268  33  16.575  4.820  9.1058  4.499 
12  3.448  2.635  11.255  2.388  34  13.805  4.195  8.662  4.251 
13  3.924  3.237  8.369  2.534  35  12.129  3.940  9.070  4.358 
14  7.544  6.015  16.308  4.957  36  7.257  3.309  9.441  4.157 
15  8.923  7.174  17.788  5.631  37  33.133  12.675  21.466  5.796 
16  5.305  4.504  11.461  3.347  38  30.434  12.295  23.892  6.613 
17  5.075  4.421  10.389  3.122  39  11.086  6.497  16.861  4.306 
18  9.803  8.320  20.345  6.049  40  4.921  1.680  3.867  1.758 
19  5.287  4.523  13.189  3.606  41  3.446  1.602  4.609  1.933 
20  2.817  2.517  6.369  1.785  42  3.150  2.2617  7.578  2.891 
21  6.678  5.581  22.197  4.176  43  1.785  1.611  6.600  1.994 
22  8.050  6.676  26.985  5.055 
Method 1  Method 2  Method 3  Method 4  

ω  3.25  2.57  6.27  1.89 
$\overline{\beta}$  6.43  5.42  16.64  4.21 
CV/%  51.63  48.66  38.55  46.06 
Method 4 and Method 1  Method 4 and Method 2  Method 4 and Method 3  

MAPE/%  51.57  30.32  312.58 
Method 1  Method 2  Method 3  Method 4  

ω  2.92  0.88  5.75  1.63 
$\overline{\beta}$  8.93  7.25  15.86  6.73 
CV/%  32.70  12.10  36.25  24.14 
Method 2 and Method 1  Method 2 and Method 3  Method 2 and Method 4  

MAPE/%  35.01  119.28  16.01 
Method 1  Method 2  Method 3  Method 4  

ω  10.17  3.82  6.38  1.51 
$\overline{\beta}$  12.52  4.99  11.01  3.87 
CV/%  81.20  76.57  57.95  38.95 
Method 4 and Method 1  Method 4 and Method 2  Method 4 and Method 3  

MAPE/%  183.00  32.41  174.19 
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Su, W.; Cao, Q.; Cui, G.; Chen, Z. Investigations on Fatigue Life of Tube Connections Based on International Codes of Pressure Vessel. Materials 2023, 16, 231. https://doi.org/10.3390/ma16010231
Su W, Cao Q, Cui G, Chen Z. Investigations on Fatigue Life of Tube Connections Based on International Codes of Pressure Vessel. Materials. 2023; 16(1):231. https://doi.org/10.3390/ma16010231
Chicago/Turabian StyleSu, Wenxian, Qinqin Cao, Gaoyu Cui, and Zhiwei Chen. 2023. "Investigations on Fatigue Life of Tube Connections Based on International Codes of Pressure Vessel" Materials 16, no. 1: 231. https://doi.org/10.3390/ma16010231