1. Introduction
Pressure pipelines are widely used in the petrochemical, nuclear, and other industries. When a pressure pipeline conveying hazardous media leaks or breaks under complex loads, it may lead to major accidents. Under the erosion and corrosion of the medium, the pipeline will have local wall-thinning defects. Local wall thinning is a common defect on the surface of pipelines, which may endanger the integrity of the pipeline. Although ratcheting and fatigue should also be considered in the pipeline integrity assessment, the limit load—which determines the load-bearing capacity of structures—is the basic parameter for structural integrity assessment corresponding to primary stress. Therefore, the effective determination of limit load has attracted the attention of many researchers.
To determine the limit load of pipelines with defects, Goodall proposed the analytical solution of the limit internal pressure of defect-free straight pipes [
1]. Based on the average shear stress yield theory, Zhu [
2] obtained a theoretical solution model of the limit pressure of straight pipes, as the function of pipe diameter, wall thickness, and material parameters. Zhang [
3] analyzed the limit internal pressure of straight pipes with corrosion defects, and compared the limit pressure between straight pipes with semi-elliptical ideal defects and straight pipes with rectangular ideal defects. Mousavi [
4] analyzed the limit pressure of straight pipes with corrosion defects. The results were compared with the results of the ASME B31G standard to analyze the applicability of the standard. Additionally, some scholars have analyzed the influence of defect size on the limit internal pressure of pipelines [
5,
6]. The results show that the internal pressure plays a major role in the influence of the limit load of the pipeline, but the bending moment also affects the limit load of the pipeline. When the internal pressure is low and the bending moment is large, the pipeline will also fail [
7]. Some scholars have separately analyzed the failure behavior of SP
LWT under the bending moment, and researched the influence of defect size on the limit bending moment of pipes [
8,
9,
10]. Furthermore, many scholars have analyzed the limit load and failure mode of SP
LWT under the combination of internal pressure and bending moment via numerical and experimental methods [
11,
12]. The limit load equations of SP
LWT under internal pressure, bending moment, and their combinations are contained in the existing evaluation standards for local wall-thinning pipes, such as ASME B31G [
13], API 579 [
14], and GB/T 19624-2019 [
15].
However, there are not only internal pressure and bending moment in the pipeline, but also torque and axial force. The torque and axial force also have a certain influence on the limit load of SP
LWT [
16,
17]. Chen [
18,
19] analyzed the upper and lower limit loads of straight pipes with part-through slot defects under internal pressure, bending moment, and axial force through finite element analysis and mathematical programming methods. They then discussed the failure mode and load-bearing capacity of straight pipes with four different part-through slot defects, and verified the applicability of the limit load. Zhao [
20] analyzed the influence of different corrosion defect parameters and material parameters on the limit internal pressure of pipelines with defects under compressive axial force and axial tensile force via the nonlinear finite element method. Cui [
21] studied the influence of torque and bending moment on the limit internal pressure of pipelines with local wall-thinning defects. Mondal [
22] specifically determined the influence of the compressive axial force and bending moment on the limit pressure of corroded pipelines. Shuai [
23] developed a three-dimensional nonlinear finite element model verified by blasting experiments, and determined the effects of external load (i.e., bending moment and axial force) and geometric characteristics of corrosion defects (i.e., defect depth, width, length, and location) on the limit internal pressure.
Previous works mainly analyzed the limit load of SP
LWT under internal pressure, bending moment, axial force, and their combinations. The limit load has not been systematically studied for SP
LWT under the combination of internal pressure, bending moment, torque, and axial force together. This paper mainly uses finite element analysis to analyze the limit load of SP
LWT under internal pressure, bending moment, torque, axial force, and their combinations. In this paper,
Section 2 contains the finite element model and limit analysis method of SP
LWT, along with the limit load analysis strategy for single and complex loads.
Section 3 and
Section 4 give the limit load analysis of SP
LWT under single and complex loads, respectively.
Section 5 analyzes the effects of torque and axial force on the load-bearing capacity of SP
LWT.
Section 6 presents our conclusions. It should also be mentioned that the methodology discussed can be used for other pipeline components, such as elbows and tees, which cover openings and other stress concentrations.
3. Limit Load Analysis of Straight Pipes with Local Wall-Thinning Defects under Single Loads
In this section, the nonlinear finite element limit load analysis of SPLWT under internal pressure, bending moment, torque, and axial force is demonstrated, and the plastic strain contour, the load vs. plastic strain curve, and the empirical limit-load equation of SPLWT under single loads are obtained.
3.1. Orthogonal Calculation Scheme
Under a single load, the non-dimensional axial half-length
a, non-dimensional circumferential half-length
b, and non-dimensional depth
c are taken as three influencing factors. The calculation scheme is determined by the orthogonal design method, and four levels are selected for each influencing factor.
Table 4 shows the orthogonal test levels, while
Table 5 shows the orthogonal test calculation scheme.
3.2. Finite Element Analysis Results
The nonlinear finite element calculations of SP
LWT were carried out under single loads such as internal pressure, bending moment, torque, and axial force, and the plastic strain contours and the load vs. plastic strain curves were obtained. Taking the local wall-thinning defect (
a = 0.4,
b = 0.4,
c = 0.6) as an example,
Figure 4 shows the plastic strain contours of the last convergence load substep of numerical calculation. It can be seen that the failure positions of SP
LWT under different loads are different. The failure position of SP
LWT is at the middle of the defect edge under internal pressure. When the torque is applied, the pipeline is twisted, and the failure position is at the corner of the defect. When the bending moment is applied, the defect position of the pipeline fails, and the failure position is at the inner wall of the pipeline. The failure position is at the middle of the defect under the axial force.
The load and plastic strain at the point of maximum plastic strain in the time history were obtained.
Figure 5 shows the load vs. plastic strain curves of SP
LWT under different single loads at such points. The load corresponding to the last convergence point of the numerical calculation in the figure was taken as the limit load. The results of the analysis of the limit load of SP
LWT under different single loads are listed in
Table 6.
3.3. Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects
Conveniently, the limit loads of SPLWT under single loads were normalized. The normalized limit internal pressure, the normalized limit bending moment, the normalized limit torque, and the normalized limit axial force are expressed by pLS = PLS/PL0, mLS = MLS/ML0, nLS = NLS/NL0, and fLS = FLS/FL0, respectively, where PL0, NL0, ML0, and NL0 are the limit internal pressure, limit axial force, limit bending moment, and limit torque of a defect-free straight pipe, respectively.
Liu [
16] proposed the calculation equations of limit internal pressure
PL0, limit bending moment
ML0, and limit axial force
FL0 for defect-free straight pipes.
where
r denotes the average radius of the straight pipe, and
σy denotes the yield strength of the material.
Guo [
27] gave the calculation equation of the limit torque
NL0 of defect-free straight pipes.
The equations between the limit load and the size of the local wall-thinning defect were obtained using SPSS software. The relative error between the equation calculation results and the finite element calculation results was less than 5%. The dimensionless limit load empirical equations of SPLWT under different single loads are as follows:
- (1)
The limit load of SPLWT under internal pressure can be calculated by Equation (6):
where
G1 denotes the geometric parameters of the local wall-thinning defect,
G1 =
a0.5b0.1c. The least square error
R2 of the equation is 0.948.
- (2)
The limit load of SPLWT under bending moment can be calculated by Equation (7):
where
G2 denotes the geometric parameters of the local wall-thinning defect,
G2 =
a0.01b0.1c. The least square error
R2 of the equation is 0.957.
- (3)
The limit load of SPLWT under torque can be calculated by Equation (8):
where
G3 denotes the geometric parameters of the local wall-thinning defect,
G3 =
a−0.1b0.7c. The least square error
R2 of the equation is 0.968.
- (4)
The limit load of SPLWT under axial force can be calculated by Equation (9):
where
G4 denotes the geometric parameters of the local wall-thinning defect,
G4 =
a0.05b1.5c0.05. The least square error
R2 of the equation is 0.984.
4. Limit Load Analysis of Straight Pipes with Local Wall-Thinning Defects under Complex Loads
In this section, the nonlinear finite element limit load analysis of SPLWT under the combination of internal pressure, bending moment, torque, and axial force his demonstrated, and the plastic strain contour, the load vs. plastic strain curve, and the limit load empirical equation of SPLWT under complex loads are obtained.
4.1. Orthogonal Calculation Scheme
Under complex loads, the non-dimensional internal pressure
pl, non-dimensional bending moment
ml, and non-dimensional axial force
fl were taken as three influencing factors. The calculation scheme was determined by the orthogonal design method, and three levels are selected for each influencing factor.
Table 7 shows the orthogonal test levels, while
Table 8 shows the orthogonal test calculation scheme.
To include more types of local wall-thinning defect in the actual working conditions, the local wall-thinning defects were divided into axial local wall-thinning defects, circumferential local wall-thinning defects, large-area local wall-thinning defects, and small-area local wall-thinning defects in this paper.
4.2. Finite Element Analysis Results
To save calculation time, only the typical defects were selected for limit load analysis: axial defects (
a = 3,
b = 0.15,
c = 0.5), circumferential defects (
a = 0.2,
b = 0.4,
c = 0.5), large-area defects (
a = 3,
b = 0.4,
c = 0.5), and small-area defects (
a = 0.2,
b = 0.05,
c = 0.4). The case of
pl = 0.4,
ml = 0.4, and
fl = 0.2 is taken as an example to illustrate the calculation results. In the first load step, the internal pressure, bending moment, and axial force corresponding to
pl = 0.4,
ml = 0.4, and
fl = 0.2 are proportionally loaded, and the load step time is 1.0. In the second load step, the value of applied torque is 30 kNm, and the load step time is 2.0.
Figure 6 shows the plastic strain contours of SP
LWT corresponding to the last convergence point of numerical calculation under complex loads. It can be seen that the failure position of SP
LWT varies with the shape of the defect. For the axial defect, the failure position is at the corner of the defect, because complex loads cause the stress to be concentrated at the corner of the defect. For the circumferential defect, the failure position is at the defect position. For the large-area defect, the failure position is at the corner of the defect, but the other positions of the defect and the surrounding position also produce stress concentration and plastic strain. For the small-area defect, the failure position is at the right end of the straight pipe, indicating that small-area defects have little effect on the failure of the straight pipe, and plastic strain also occurs at the defect location, because stress concentration occurs at the defect position.
Figure 7 shows the load vs. plastic strain curves at the maximum plastic strain point of SP
LWT under complex loads. It can be seen that the plastic strain of SP
LWT does not change in the first load step. This is because the applied internal pressure, bending moment, and axial force are low. However, the internal pressure, bending moment, and axial force change the working conditions of the second load step calculation. The torque corresponding to the last convergence point is taken as the limit torque, and the value of limit torque is the applied torque multiplied by (TIME−1.0).
Table 9 shows the calculation results of the limit load of SP
LWT under complex loads.
4.3. Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects under Complex Loads
The calculation results are expressed in dimensionless form, and the treatment method is shown in
Section 2.4. The empirical equations of the relationship between non-dimensional torque
nl and non-dimensional internal pressure
pl, non-dimensional bending moment
ml, and non-dimensional axial force
fl were fitted by SPSS23 software. Although the equations were obtained from typical dimensionless defects results, they were verified within the range of defect size.
- (1)
The limit load of a straight pipe with an axial local wall-thinning defect can be calculated by Equation (10):
where the applicable range of defect size is
a/
b ≥ 10, 5 ≥
a ≥ 1,
b ≤ 0.3, and the least square error
R2 of the equation is 0.999. Within the applicable range of defect size, the upper boundary point (
a = 5,
b = 0.3,
c = 0.5) and the lower boundary point (
a = 1,
b = 0.05,
c = 0.5) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (10) was 7.56% and 5.08%, respectively.
- (2)
The limit load of a straight pipe with a circumferential local wall-thinning defect can be calculated by Equation (11):
where the applicable range of defect size is
a/
b ≤ 1/2,
a ≤ 0.3, 0.6 ≥
b ≥ 0.3, and the least square error
R2 of the equation is 0.974. Within the applicable range of defect size, the upper boundary point (
a = 0.3,
b = 0.6,
c = 0.5) and the lower boundary point (
a = 0.1,
b = 0.3,
c = 0.5) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (11) was 5.78% and 4.61%, respectively.
- (3)
The limit load of a straight pipe with a large-area local wall-thinning defect can be calculated by Equation (12):
where the applicable range of defect size is 10 >
a/
b > 1/2, 0.6 ≥
b ≥ 0.3, 5 ≥
a ≥ 1, and the least square error
R2 of the equation is 0.995. Within the applicable range of defect size, the upper boundary point (
a = 5,
b = 0.6,
c = 0.5) and the lower boundary point (
a = 1,
b = 0.3,
c = 0.5) were selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (12) was 8.46% and 4.32% respectively.
- (4)
The limit load of a straight pipe with a small-area local wall-thinning defect can be calculated by Equation (13):
where the applicable range of defect size is
b < 0.3,
a < 1, and the least square error
R2 of the equation is 0.999. Within the applicable range of defect size, the upper boundary point (
a = 0.95,
b = 0.25,
c = 0.5) was selected for finite element calculation; the error between the results of the finite element calculation and that of Equation (13) was 3.79%.
4.4. Verification of the Load-Bearing Capacity of Straight Pipes with Local Wall-Thinning Defects under the Combination of Internal Pressure and Bending Moment
Chen [
28] discussed the failure mode of SP
LWT under the combination of internal pressure and bending moment, and obtained the limit load curve under the combination of internal pressure and bending moment, while also summarizing the equation of limit load for straight pipes with all local wall-thinning defect sizes, but the equation was without detail division according to the range of defect size. The limit load can be calculated by Equation (14):
Figure 8 shows the comparison between the limit load curves of SP
LWT under the combination of internal pressure and bending moment in this paper and the limit load equation in Ref. [
28]. Taking the non-dimensional torque
nl and non-dimensional axial force
fl in Equations (10)–(13) as zero, the limit load equation of SP
LWT under the combination of internal pressure and bending moment can be obtained. It can be seen that the limit load curves of Equations (11) and (12) in this paper are in good agreement with the limit load results of Equation (14) in Ref. [
28]. But the limit loads of Equations (10) and (13) in this paper are higher than those of Equation (14) in Ref. [
28], within
pl = 0.3~0.7. This is because the values of Equation (14) are the lowest boundaries of all types of defects. Equation (14) cannot accurately evaluate all defects.
6. Conclusions
Based on previous research on the limit load of straight pipes, this paper analyzed the load-bearing capacities of straight pipes with different types of local wall-thinning defect under internal pressure, bending moment, torque, axial force, and their combinations, via nonlinear finite element analysis. The empirical equations of the limit load of SPLWT were fitted separately for single and complex loads with four defects: axial defects, circumferential defects, large-area defects, and small-area defects. The empirical equations were verified by the results of previous works for some load combinations.
Taking the allowable load on the equipment nozzles as pipe loads, the effects of torque and axial force on the load-bearing capacity of SPLWT were analyzed quantitatively. For medium- and low-pressure equipment, the load-bearing capacity of SPLWT was less affected by torque and axial force. The load-bearing capacity was reduced by 0.59~1.44% under the influence of torque, while it was reduced by 0.83~1.80% under the influence of axial force. For high-pressure equipment, the load-bearing capacity of SPLWT was greatly affected by torque and axial force. The load-bearing capacity was reduced by 6.67~9.03% under the influence of torque, and by 7.51~12.93% under the influence of axial force.