Finite Element Analysis of the Limit Load of Straight Pipes with Local Wall-Thinning Defects under Complex Loads

Abstract

Local wall thinning is a common defect on the surface of pipelines, which can cause damage to the pipeline under complex pipeline loads. Based on the study on the limit load of straight pipes with defects, the nonlinear finite element method was used to analyze the limit load of straight pipes with local wall-thinning defects under internal pressure, bending moment, torque, axial force, and their combinations, and the empirical limit-load equations of straight pipes with local wall-thinning defects under single and complex loads were fitted. Based on the allowable load on the equipment nozzles, the influences of torque and axial force on the load-bearing capacity of straight pipes with local wall-thinning defects were quantitatively analyzed. For medium and low-pressure equipment, the load-bearing capacity was reduced by 0.59~1.44% under the influence of torque, and by 0.83~1.80% under the influence of axial force. For high-pressure equipment, the load-bearing capacity was reduced by 10.07~20.90% under the influence of torque, and by 2.01~12.40% under the influence of axial force.

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.

2. Finite Element Model and Limit Load Analysis Method of Straight Pipes with Local Wall-Thinning Defects

2.1. Geometry Parameters of the Calculation Model

Figure 1 shows the model of SP_LWT. The finite element calculation model uses a Φ108 × 8 straight pipe with a length of 600 mm. The shape of the local wall-thinning defect is rectangular, which is ideal, and the local wall-thinning defect is located at the center of the straight pipe’s outer wall. The parameters of straight pipe sizes and defect sizes are shown in

Table 1. The straight pipe size and defect size.

Straight Pipe Size, mm
Ro	Outer radius
Ri	Inner radius
T	Thickness
L	Length
Defect Size, mm
A	Axial half-length
B	Circumferential half-length
C	Defect depth
Dimensionless Defect Size
a	A/(RoT)0.5
b	B/(πRo)
c	C/T

. The internal pressure P is applied on the inner wall of the straight pipe. The axial force F, the torque N, and the bending moment M are applied on the end face of the straight pipe.

2.2. Boundary Condition

ANSYS APDL was used for nonlinear finite element analysis with the small deformation hypothesis. Figure 2 shows the finite element model and mesh of SP_LWT. Considering internal pressure, bending moment, torque, and axial force, the whole model was established. All volumes were meshed into SOLID95 elements via the sweep method, and the number of elements was 94,352, meeting the requirements of grid independence. The material was assumed to be an elastic–perfectly plastic material.

Table 2. The material parameters of the straight pipe.

Material	Elastic Modulus, GPa	Poisson’s Ratio	Yield Strength, MPa
20 g	207	0.3	245

shows the material properties of the straight pipe.

As shown in Figure 2a, the internal pressure was applied on the inner wall of the straight pipe, and the axial equivalent head load was applied on the left end face in the z direction of the straight pipe. The axial equivalent head load can be calculated by Equation (1):

$$P_{\mathrm{c}}=\frac{{\mathit{PR}}_{\mathrm{i}}^2}{R_{\mathrm{o}}^2-R_{\mathrm{i}}^2}$$

(1)

Meanwhile, a node was established at the center of the left end face of the straight pipe in the z direction, which was used for the MASS21 mass element. This node was coupled with all nodes of the straight pipe’s end face to form a rigid region. As shown in Figure 2b,c, the bending moment in the y direction and the torque in the z direction were applied on this node, respectively, and the axial force in the z direction was applied on this node in the same direction as the axial equivalent surface load P_c, as shown in Figure 2d. Circumferential and axial constraints were applied on the right end face of the straight pipe in the z direction.

2.3. Pipe Loads

Pipe loads are usually obtained by pipeline stress analysis. When the data on pipeline stress analysis are lacking, they can be conservatively estimated according to the specified allowable nozzle load. For example, China Huanqiu Contracting & Engineering Co., Ltd. regulations require that equipment nozzles also bear axial force and moment in three directions, in addition to internal pressure. The load direction is shown in Figure 3, and the load size is shown in

Table 3. The allowable load of the equipment nozzle.

Equipment	DN	F/N	TL/N	TC/ N	ML/Nm	MC/Nm	MT/Nm
Medium and low-pressure equipment	80	1500	1500	1500	600	600	600
	100	2100	2100	2100	1100	1100	1100
	150	4600	4600	4600	3400	3400	3400
High-pressure equipment	80	7960	5640	5640	5720	5720	5720
	100	12,200	11,460	11,460	11,640	11,640	11,640
	150	24,920	20,400	20,400	20,800	20,800	20,800

. Generally, the equipment is divided into medium- and low-pressure equipment (i.e., design pressure is less than 10 MPa) and high-pressure equipment (i.e., design pressure is no less than 10 MPa). The maximum allowable working pressure of the calculation model in this paper is 15.90 MPa [24].

2.4. Determination Method of Limit Load

Usually, there are two algorithms to determine the limit load of the structure: limit load analysis, and elastic–plastic stress analysis. Limit load analysis is usually based on the elastic–perfectly plastic constitutive model with small deformation assumption, which often obtains a conservative result. Elastic–plastic stress analysis is usually based on the elastic–plastic constitutive model with nonlinear geometry, which obtains more reasonable results but requires more material information. According to the numerical results, the limit load can be determined by several methods [25], such as the double-tangent criterion, double-elastic-deformation method, double-elastic-slope criterion, 0.2% residual strain criterion, zero-slope criterion, zero-curvature criterion, etc. The limit load analysis is used conservatively in this paper [26], and the load corresponding to the last convergence point of the numerical calculation is taken as the limit load. This algorithm is used to determine the limit load of SP_LWT under single and complex loads.

When determining the limit load of SP_LWT under a single load, the load step time is 1.0, and a sufficiently large load (P, M, N, F) is applied. The load of the last convergence point of numerical calculation corresponds to the limit load (P_LS, M_LS, N_LS, F_LS) of SP_LWT. When determining the limit load of the pipeline under complex loads, such as the combination of internal pressure and bending moment, the following method is often adopted: the internal pressure load is fixed, and the limit bending moment of the pipeline is calculated to analyze the limit load of the pipeline with a local wall-thinning defect [7]. Liu [16] further extended this method to the limit load analysis of straight, pipes with defects under the combination of loads of internal pressure, bending moment and torque. In this paper, the loading method for nonlinear finite element analysis of SP_LWT under complex loads is as follows: In the first calculation step, the internal pressure P, bending moment M, and axial force F are applied in different proportions, and the load step time is 1.0. After the first calculation converges, the second step of nonlinear calculation is performed. Meanwhile, a sufficiently large torque is applied, and the load step time is 2.0. The torque of the last convergence point of numerical calculation corresponds to the limit torque of SP_LWT under loads (P, M, and F), and the limit torque of SP_LWT under complex loads is defined as N_LC.

Using the limit internal pressure P_LS, limit bending moment M_LS, limit torque N_LS, and limit axial force F_LS of SP_LWT under different single loads, the applied internal pressure P, bending moment M, axial force F, and calculated limit torque N_LC can be expressed in dimensionless form. The dimensionless internal pressure, the bending moment, the axial force, and the torque are represented by p_l = P/P_LS, m_l = M/M_LS, f_l = F/F_LS, and n_l = N_LC/N_LS respectively. Then, a dimensionless limit state point (p_l, m_l, n_l, f_l) under complex loads is obtained. Through parametric calculation, the dimensionless limit state response hypersurface under complex loads can be obtained.

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 SP_LWT 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 SP_LWT 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. Orthogonal test levels under single loads.

Level	a	b	c
1	0.2	0.2	0.2
2	1	0.4	0.4
3	3	0.6	0.6
4	5	0.8	0.8

shows the orthogonal test levels, while

Table 5. Calculation scheme of orthogonal tests under single loads.

Test Number	a	b	c
1	0.2	0.2	0.2
2	0.2	0.4	0.4
3	1.0	0.8	0.6
4	0.2	0.6	0.6
5	5.0	0.6	0.4
6	0.2	0.8	0.8
7	5.0	0.2	0.8
8	3.0	0.8	0.4
9	3.0	0.6	0.2
10	5.0	0.8	0.2
11	3.0	0.4	0.8
12	1.0	0.4	0.2
13	1.0	0.6	0.8
14	1.0	0.2	0.4
15	5.0	0.4	0.6
16	3.0	0.2	0.6

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. The calculation results of the limit load of straight pipes with local wall-thinning defects under single loads.

Defect Size			Limit Load
a	b	c	Limit Internal Pressure/MPa	Limit Bending Moment/kNm	Limit Torque/kNm	Limit Axial Force/kN
0.2	0.2	0.2	42.59	18.92	15.66	586.12
0.2	0.4	0.4	40.36	17.04	11.93	538.00
1.0	0.8	0.6	36.16	15.06	10.30	320.43
0.2	0.6	0.6	37.31	15.48	8.61	421.57
5.0	0.6	0.4	22.70	17.08	14.59	456.37
0.2	0.8	0.8	34.58	12.85	6.31	243.84
5.0	0.2	0.8	9.47	14.57	10.28	501.88
3.0	0.8	0.4	27.09	16.38	12.93	416.90
3.0	0.6	0.2	35.60	18.21	15.78	531.68
5.0	0.8	0.2	35.09	17.61	15.55	513.44
3.0	0.4	0.8	14.26	14.59	10.14	402.53
1.0	0.4	0.2	39.71	18.01	16.19	557.07
1.0	0.6	0.8	24.51	14.32	8.83	292.87
1.0	0.2	0.4	34.91	16.88	12.85	561.34
5.0	0.4	0.6	16.02	14.75	11.50	318.98
3.0	0.2	0.6	21.91	15.70	12.37	519.19

.

3.3. Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects

Conveniently, the limit loads of SP_LWT 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 p_LS = P_LS/P_L0, m_LS = M_LS/M_L0, n_LS = N_LS/N_L0, and f_LS = F_LS/F_L0, respectively, where P_L0, N_L0, M_L0, and N_L0 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 P_L0, limit bending moment M_L0, and limit axial force F_L0 for defect-free straight pipes.

$$P_{\mathrm{L}0}=\frac{2}{\sqrt{3}}{\sigma }_{\mathrm{y}}\ln \frac{R_{\mathrm{o}}}{R_{\mathrm{i}}}$$

(2)

$$M_{\mathrm{L}0}{=4r}^2{T\sigma }_{\mathrm{y}}$$

(3)

$$F_{\mathrm{L}0}=\pi \left(R_{\mathrm{o}}^2-R_{\mathrm{i}}^2\right){\sigma }_{\mathrm{y}}$$

(4)

where r denotes the average radius of the straight pipe, and σ_y denotes the yield strength of the material.

Guo [27] gave the calculation equation of the limit torque N_L0 of defect-free straight pipes.

$$N_{\mathrm{L}0}=\frac{2}{\sqrt{3}}{\sigma }_{\mathrm{y}}{r}^{2}T$$

(5)

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 SP_LWT under different single loads are as follows:

(1)

The limit load of SPLWT under internal pressure can be calculated by Equation (6):

$$p_{\mathrm{LS}}=-0.517G_1+0.995$$

(6)

where G₁ denotes the geometric parameters of the local wall-thinning defect, G₁ = a^0.5b^0.1c. 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):

$$m_{\mathrm{LS}}=-0.033G_2^2-0.386G_2+1.039$$

(7)

where G₂ denotes the geometric parameters of the local wall-thinning defect, G₂ = a^0.01b^0.1c. 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):

$$n_{\mathrm{LS}}=-0.416G_3+1.045$$

(8)

where G₃ denotes the geometric parameters of the local wall-thinning defect, G₃ = a^−0.1b^0.7c. 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):

$$f_{\mathrm{LS}}=-1.242G_4^2-0.213G_4+0.985$$

(9)

where G₄ denotes the geometric parameters of the local wall-thinning defect, G₄ = a^0.05b^1.5c^0.05. 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 SP_LWT 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 SP_LWT under complex loads are obtained.

4.1. Orthogonal Calculation Scheme

Under complex loads, the non-dimensional internal pressure p_l, non-dimensional bending moment m_l, and non-dimensional axial force f_l 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. Orthogonal test levels under complex loads.

Level	pl	ml	fl
1	0.1	0.1	0.1
2	0.2	0.2	0.2
3	0.4	0.4	0.4

shows the orthogonal test levels, while

Table 8. Calculation scheme of orthogonal tests under complex loads.

Test Number	pl	ml	fl
1	0.1	0.1	0.1
2	0.1	0.2	0.2
3	0.1	0.4	0.4
4	0.2	0.1	0.2
5	0.2	0.2	0.4
6	0.2	0.4	0.1
7	0.4	0.1	0.4
8	0.4	0.2	0.1
9	0.4	0.4	0.2

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 p_l = 0.4, m_l = 0.4, and f_l = 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 p_l = 0.4, m_l = 0.4, and f_l = 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. The calculation results of the limit load of straight pipes with local wall-thinning defects under complex loads.

Internal Pressure pl	Bending Moment ml	Torque fl	Limit Torque/kNm
			Axial Defect	Circumferential Defect	Large-Area Defect	Small-Area Defect
0.1	0.1	0.1	18.10	13.09	14.67	18.43
0.1	0.2	0.2	17.31	12.28	14.57	17.88
0.1	0.4	0.4	14.93	9.33	13.49	15.01
0.2	0.1	0.2	17.61	12.49	14.49	17.85
0.2	0.2	0.4	15.53	10.67	13.75	15.72
0.2	0.4	0.1	16.33	11.43	13.79	16.83
0.4	0.1	0.4	13.73	10.03	12.75	13.52
0.4	0.2	0.1	16.58	11.79	13.46	16.82
0.4	0.4	0.2	14.95	9.78	12.89	14.64

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 n_l and non-dimensional internal pressure p_l, non-dimensional bending moment m_l, and non-dimensional axial force f_l 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):

$$n_{\mathrm{l}}^{1.6}=0.958-1.016m_{\mathrm{l}}^2-1.284p_{\mathrm{l}}^2-1.028f_{\mathrm{l}}^2+0.054m_{\mathrm{l}}+0.322p_{\mathrm{l}}+0.028f_{\mathrm{l}}$$

(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):

$$n_{\mathrm{l}}^{1.5}=0.989-1.251m_{\mathrm{l}}^2-1.104p_{\mathrm{l}}^2-0.588f_{\mathrm{l}}^2+0.238m_{\mathrm{l}}+0.072p_{\mathrm{l}}-0.407f_{\mathrm{l}}$$

(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):

$$n_{\mathrm{l}}^2=0.856-0.863m_{\mathrm{l}}^2-0.983p_{\mathrm{l}}^2-1.332f_{\mathrm{l}}^2+0.023m_{\mathrm{l}}+0.137p_{\mathrm{l}}+0.424f_{\mathrm{l}}$$

(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):

$$n_{\mathrm{l}}^{1.3}=0.989-1.120m_{\mathrm{l}}^2-1.315p_{\mathrm{l}}^2-1.023f_{\mathrm{l}}^2+0.134m_{\mathrm{l}}+0.308p_{\mathrm{l}}-0.048f_{\mathrm{l}}$$

(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):

$${\left(m_{\mathrm{l}}\right)}^2+ {\left(p_{\mathrm{l}}\right)}^2=1.0$$

(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 n_l and non-dimensional axial force f_l 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 p_l = 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.

5. Discussion

In this section, the effects of internal pressure, bending moment, torque, and axial force on the limit load are discussed, and the effects of torque and axial force on the limit load of SP_LWT are analyzed quantitatively.

5.1. Influence of Torque on the Limit Load Equation

The influence of internal pressure, bending moment, and torque on the limit load of SP_LWT was analyzed. Taking the non-dimensional axial force f_l in Equations (10)–(13) as zero, the limit load equation of SP_LWT under the combination of internal pressure, bending moment, and torque was obtained. Figure 9a–d show the limit state surface of SP_LWT under the combination of internal pressure, bending moment, and torque. It can be seen that the torque has a great influence on the limit load of SP_LWT. When the torque is low, it has little effect on the limit load curves of SP_LWT. With the gradual increase in torque, the influence of torque on the limit load curve becomes larger. When the torque is close to the limit torque of SP_LWT, the influence of internal pressure and bending moment on the limit load can be almost ignored, considering only the influence of torque on the limit load.

The allowable torque of the equipment nozzles is shown in Table 2. The allowable torque is expressed in dimensionless form through the limit torque of SP_LWT under single loads. The non-dimensional allowable torque on medium–low-pressure equipment and high-pressure equipment is represented by n_l-g = N_l-g/N_LS and n_l-h = N_l-h/N_LS respectively. For different diameters of pipe, the average of the non-dimensional allowable torque is n_l-g = 0.10 for medium- and low-pressure equipment, and n_l-h = 0.33 for high-pressure equipment. Figure 10 shows that the effect of the allowable torque on the limit load of SP_LWT for n_l is 0, 0.10, and 0.33. It can be seen that the range surrounded by the limit load curve of SP_LWT is reduced with the increase in torque, which means that the load-bearing capacity of defective straight pipes is reduced, and the influence of torque on straight pipes with different local wall-thinning defects is different.

According to the allowable torque, the relative reduction in the average radius of the limit load curves of SP_LWT was calculated to quantitatively analyze the influence of the torque on the load-bearing capacity of SP_LWT under the combination of internal pressure and bending moment.

Table 10. The relative reduction in the average radius of the limit load curves of straight pipes with local wall-thinning defects under the influence of torque.

Defect Type	Medium- and Low-Pressure Equipment	High-Pressure Equipment
Axial defect	1.28%	9.03%
Circumferential defect	1.44%	9.01%
Large-area defect	0.59%	6.67%
Small-area defect	1.20%	8.41%

shows the relative reduction in the average radius of the limit load curves of SP_LWT under the influence of torque. It can be seen that the load-bearing capacity of SP_LWT in high-pressure equipment is more affected by torque than that in medium- and low-pressure equipment. For medium- and low-pressure equipment, the load-bearing capacity of SP_LWT changed little under the influence of torque, reducing by 0.59~1.44%. The straight pipe with a circumferential defect was the most affected, while the straight pipe with a large-area defect was the least affected. For high-pressure equipment, the load-bearing capacity of SP_LWT was greatly affected by torque, reducing by 6.67~9.03%. The straight pipe with an axial defect was the most affected, while the straight pipe with a large-area defect was the least affected.

5.2. Influence of Axial Force on the Limit Load Equation

The influence of internal pressure, bending moment, and axial force on the limit load of SP_LWT was analyzed. Taking the non-dimensional torque n_l in Equations (10)–(13) as zero, the limit load equations of SP_LWT under the combination of internal pressure, bending moment, and axial force were obtained. Figure 11a–d show the limit state surfaces of SP_LWT under the combination of internal pressure, bending moment, and axial force. It can be seen that there were still some differences in the limit state surfaces of the straight pipes with four local wall-thinning defects, but the trend of overall change was the same. When the internal pressure, bending moment, and axial force were the same, the influence of the three loads on the limit load of SP_LWT was roughly similar. When the axial force was large, the load-bearing capacity of the straight pipes for internal pressure and bending moment was reduced, meaning that the axial force has an impact on the load-bearing capacity of SP_LWT.

The allowable axial force of the equipment nozzles is shown in Table 2. The allowable axial force is expressed in dimensionless form through the limit axial force of SP_LWT under single loads. The non-dimensional allowable axial force on medium–low-pressure equipment and high-pressure equipment is represented by f_l-g = F_g/F_LS and f_l-h = F_h/F_LS, respectively. For different diameters of pipe, the average of the non-dimensional allowable axial force is f_l-g = 0.05 for medium- and low-pressure equipment, and f_l-h = 0.41 for high-pressure equipment. Figure 12 shows that the effect of the allowable axial force on the limit load of SP_LWT for n_l is 0, 0.05, and 0.41. The range surrounded by the limit load curves of SP_LWT is the relative value of the load-bearing capacity. When the range surrounded by the curve decreases, it means that the load-bearing capacity of the defective straight pipe is reduced. The load-bearing capacity of high-pressure equipment is lower than that of medium- and low-pressure equipment.

According to the allowable axial force, the relative reduction in the average radius of the limit load curve of SP_LWT was calculated to quantitatively analyze the influence of the axial force on the load-bearing capacity of SP_LWT under the combination of internal pressure and bending moment.

Table 11. The relative reduction in the average radius of the limit load curves of straight pipes with local wall-thinning defects under the influence of axial force.

Defect Type	Medium and Low-Pressure Equipment	High-Pressure Equipment
Axial defect	0.83%	8.57%
Circumferential defect	1.80%	12.93%
Large-area defect	1.15%	8.07%
Small-area defect	1.01%	7.51%

shows the relative reduction in the average radius of the limit load curves of SP_LWT under the influence of axial force. For medium- and low-pressure equipment, the load-bearing capacity of SP_LWT changed little under the influence of axial force, reducing by 0.83~1.80%. The straight pipe with a circumferential defect was the most affected, while the straight pipe with an axial defect was the least affected. For high-pressure equipment, the load-bearing capacity of SP_LWT was greatly affected by axial force, reducing by 7.51~12.93%. The straight pipe with a circumferential defect was the most affected, while the straight pipe with a small-area defect was the least affected.

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 SP_LWT 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 SP_LWT were analyzed quantitatively. For medium- and low-pressure equipment, the load-bearing capacity of SP_LWT 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 SP_LWT 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.