applied
sciences
Article
Finite Element Analysis of the Limit Load of Straight Pipes
with Local Wall-Thinning Defects under Complex Loads
Yan Li 1 , Bingjun Gao 1, * , Shuo Liu 1 , Kaiming Lin 2 and Juncai Ding 2
1
2
*
School of Chemical Engineering and Technology, Hebei University of Technology, Tianjin 300130, China;
15122510137@163.com (Y.L.); liushuor@126.com (S.L.)
Guangdong Institute of Special Equipment Inspection and Research Zhongshan Branch,
Zhongshan 300072, China; linkaiming@zsjcy.com (K.L.); dingjuncai@zsjcy.com (J.D.)
Correspondence: bjgao@hebut.edu.cn
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 wallthinning 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.
Keywords: straight pipe; local wall thinning; complex loads; limit load
Citation: Li, Y.; Gao, B.; Liu, S.; Lin,
K.; Ding, J. Finite Element Analysis of
the Limit Load of Straight Pipes with
Local Wall-Thinning Defects under
Complex Loads. Appl. Sci. 2022, 12,
4850. https://doi.org/10.3390/
app12104850
Academic Editor: Marek Krawczuk
Received: 18 April 2022
Accepted: 9 May 2022
Published: 11 May 2022
Publisher’s Note: MDPI stays neutral
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Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
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,
Appl. Sci. 2022, 12, 4850. https://doi.org/10.3390/app12104850
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Appl. Sci. 2022, 12, 4850
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the pipeline will also fail [7]. Some scholars have separately analyzed the failure behavior
of SPLWT under the bending moment, and researched the influence of defect size on the
limit bending moment of pipes [8–10]. Furthermore, many scholars have analyzed the limit
load and failure mode of SPLWT under the combination of internal pressure and bending
moment via numerical and experimental methods [11,12]. The limit load equations of
SPLWT 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 SPLWT [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 SPLWT under internal pressure, bending moment, axial force, and their combinations. The limit load has not been systematically
studied for SPLWT 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 SPLWT 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 SPLWT , along with the limit load analysis strategy for single and complex loads.
Sections 3 and 4 give the limit load analysis of SPLWT under single and complex loads,
respectively. Section 5 analyzes the effects of torque and axial force on the load-bearing
capacity of SPLWT . 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 SPLWT . 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 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 SPLWT . 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
Appl. Sci. 2022, 12, 4850
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Appl. Sci. 2022, 12, x FOR PEER REVIEW
number
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of elements was 94,352, meeting the requirements of grid independence.3 The
material was assumed to be an elastic–perfectly plastic material. Table 2 shows the material
properties of the straight pipe.
Figure
The size
size model
model of
Figure 1.
1. The
of the
the straight
straight pipe.
pipe.
Table
Table 1.
1. The
The straight
straight pipe
pipe size
size and
and defect
defect size.
size.
Straight Pipe
Pipe Size,
Straight
Size,mm
mm
Roo
R
Rii
T
T
L
L
Defect Size, mm
Outer radius
Outer
radius
Inner radius
radius
Inner
Thickness
Thickness
Length
Length
Defect Size, mm
A
Axial half-length
A
Axial
half-length 4 of 19
B
Circumferential half-length
B
Circumferential
half-length
C
Defect depth
C
Defect depth
Defect
applied on this node, respectively, Dimensionless
and the axial force
in Size
the z direction was applied on
Dimensionless
Defect
Size
this node in the same direction
as the axial equivalent surface load PA/(R
c, asoshown
a
T)0.5 in Figure
ab
A/(R
oT)0.5
B/(πR
2d. Circumferential and axial constraints were applied on the right
end
o ) face of the
bc
B/(πR
C/T o)
straight pipe in the z direction.
c
C/T
Appl. Sci. 2022, 12, x FOR PEER REVIEW
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 SPLWT. 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 shows the
material properties of the straight pipe.
Table 2. The material parameters of the straight pipe.
Material
20 g
Elastic Modulus, GPa
207
Poisson’s Ratio
0.3
Yield Strength, MPa
245
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):
Pc =
PR2i
2
2
(1)
Figure 2. The boundary conditions of the straight R
pipe:
o - R(a)
i internal pressure; (b) bending moment;
Figure
2.
The
boundary
conditions
of
the
straight
pipe:
(a)
internal pressure; (b) bending moment;
(c) torque;
(d)
axial
force.
Meanwhile,
a node was established at the center of the left end face of the straight
(c) torque;
(d) axial force.
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
2.3.
Pipe Loads
Figure
moment by
in the
y direction
and the torque
in the
Pipe 2b,c,
loadsthe
arebending
usually obtained
pipeline
stress analysis.
When the
dataz direction
on pipe- were
line stress analysis are lacking, they can be conservatively estimated according to the
Appl. Sci. 2022, 12, 4850
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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
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):
PR2
Pc = 2 i 2
(1)
Ro − Ri
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 Pc , 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. Generally, the equipment is divided into medium- and lowpl. Sci. 2022, 12, x FOR PEER REVIEW
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].
Figure 3. The direction of pipe load.
Figure 3. The direction of pipe load.
2.4. Determination Method of Limit Load
2.4. Determination
Method
of LimittoLoad
Usually, there are
two algorithms
determine the limit load of the structure: limit
load analysis, and elastic–plastic stress analysis. Limit load analysis is usually based on
Usually, there are two algorithms to determine the limit load of the
the elastic–perfectly plastic constitutive model with small deformation assumption, which
str
load analysis, and elastic–plastic stress analysis. Limit load analysis is usua
the elastic–perfectly plastic constitutive model with small deformation
which often obtains a conservative result. Elastic–plastic stress analysis is us
on the elastic–plastic constitutive model with nonlinear geometry, which o
Appl. Sci. 2022, 12, 4850
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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 SPLWT under single and complex loads.
Table 3. The allowable load of the equipment nozzle.
Equipment
DN
F/N
T L /N
T C /N
M L /Nm
M C /Nm
M T /Nm
Medium and
low-pressure
equipment
80
100
150
1500
2100
4600
1500
2100
4600
1500
2100
4600
600
1100
3400
600
1100
3400
600
1100
3400
High-pressure
equipment
80
100
150
7960
12,200
24,920
5640
11,460
20,400
5640
11,460
20,400
5720
11,640
20,800
5720
11,640
20,800
5720
11,640
20,800
When determining the limit load of SPLWT 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 (PLS , MLS , NLS , FLS ) of SPLWT . 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 SPLWT 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 SPLWT under
loads (P, M, and F), and the limit torque of SPLWT under complex loads is defined as NLC .
Using the limit internal pressure PLS , limit bending moment MLS , limit torque NLS ,
and limit axial force FLS of SPLWT under different single loads, the applied internal pressure
P, bending moment M, axial force F, and calculated limit torque NLC can be expressed
in dimensionless form. The dimensionless internal pressure, the bending moment, the
axial force, and the torque are represented by pl = P/PLS , ml = M/MLS , f l = F/FLS , and
nl = NLC /NLS respectively. Then, a dimensionless limit state point (pl , ml , nl , 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 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.
Appl. Sci. 2022, 12, 4850
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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.
Table 4. Orthogonal test levels under single loads.
Level
a
b
c
1
2
3
4
0.2
1
3
5
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
Table 5. Calculation scheme of orthogonal tests under single loads.
Test Number
a
b
c
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.2
0.2
1.0
0.2
5.0
0.2
5.0
3.0
3.0
5.0
3.0
1.0
1.0
1.0
5.0
3.0
0.2
0.4
0.8
0.6
0.6
0.8
0.2
0.8
0.6
0.8
0.4
0.4
0.6
0.2
0.4
0.2
0.2
0.4
0.6
0.6
0.4
0.8
0.8
0.4
0.2
0.2
0.8
0.2
0.8
0.4
0.6
0.6
3.2. Finite Element Analysis Results
The nonlinear finite element calculations of SPLWT 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 SPLWT under different loads are different. The failure position of SPLWT 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 SPLWT 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 SPLWT 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 f LS = FLS /FL0 , respectively, where PL0 , NL0 , ML0 , and
of the defect under the axial force.
Appl.
Appl. Sci.
Sci. 2022, 12, 4850
x FOR PEER REVIEW
7 19
of 19
7 of
NL0failure
are theposition
limit internal
pressure,
limit
force, limit
moment,isand
limit
torque
the
is at the
inner wall
of axial
the pipeline.
Thebending
failure position
at the
middle
of the
a defect-free
straight
pipe,force.
respectively.
of
defect under
the axial
Figure 4. The plastic strain contours of a straight pipe with a local wall-thinning defect under single
loads: (a) internal pressure; (b) bending moment; (c) torque; (d) 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 SPLWT under different
single loads at such points. The load corresponding to the last convergence point of the
numerical
calculation
incontours
the figure
was
taken
as
thea limit
load. The defect
results
of the
analysis
Figure
ofofa astraight
wall-thinning
under
single
Figure 4.
4.The
Theplastic
plasticstrain
straincontours
straightpipe
pipewith
witha local
local
wall-thinning
defect
under
single
of
the(a)
limit
loadpressure;
of SPLWT
under
different
single
loads
listed
loads:
internal
(b)
bending
moment;
(c)
(d)
axial
force.
loads:
(a)
internal
pressure;
(b)
bending
moment;
(c)torque;
torque;
(d)are
axial
force.in Table 6.
150
10
0.00
0.04
0.08
Bending moment/kNm
(a)
0.12
12
5
0
0.00
0.04
0.08
12
0.00
8320
0.04
0.08
(b)
4240
0.04
0.08
(d)
240 0
0
8
12 0
0
160
0.00
320 80
0.12
(c)
(b)
16 4
Axial force/kN
(a)
8
4
Bending moment/kNm
5
20
Axial force/kN
Torque//kNm TorqueInternal
/MPa pressure/MPa
//kNm pressureInternal
The load and plastic strain at the point of16
maximum plastic strain in the time history
20
were obtained. Figure 5 shows the load vs. plastic strain curves of SPLWT under different
15 loads at such points. The load corresponding
12
single
to the last convergence point of the
numerical calculation in the figure was taken as the limit load. The results of the analysis
10limit load of SPLWT under different single8loads are listed in Table 6.
of the
0.0
4
0.4
0.8
1.2
Plastic strain
1.6160
0.00
80
0.02
0.04
Plastic strain
(d)
(c)curves of a straight pipe with a local wall-thinning
Figure 5. The load vs. plastic strain
defect under
single
(c) torque; (d) axial force.
0
0 loads: (a) internal pressure; (b) bending moment;
0.0
0.4
0.8
1.2
Plastic strain
1.6
0.00
0.02
0.04
Plastic strain
Appl. Sci. 2022, 12, 4850
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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
0.2
0.2
1.0
0.2
5.0
0.2
5.0
3.0
3.0
5.0
3.0
1.0
1.0
1.0
5.0
3.0
0.2
0.4
0.8
0.6
0.6
0.8
0.2
0.8
0.6
0.8
0.4
0.4
0.6
0.2
0.4
0.2
0.2
0.4
0.6
0.6
0.4
0.8
0.8
0.4
0.2
0.2
0.8
0.2
0.8
0.4
0.6
0.6
42.59
40.36
36.16
37.31
22.70
34.58
9.47
27.09
35.60
35.09
14.26
39.71
24.51
34.91
16.02
21.91
18.92
17.04
15.06
15.48
17.08
12.85
14.57
16.38
18.21
17.61
14.59
18.01
14.32
16.88
14.75
15.70
Limit
Torque/kNm
Limit Axial
Force/kN
15.66
11.93
10.30
8.61
14.59
6.31
10.28
12.93
15.78
15.55
10.14
16.19
8.83
12.85
11.50
12.37
586.12
538.00
320.43
421.57
456.37
243.84
501.88
416.90
531.68
513.44
402.53
557.07
292.87
561.34
318.98
519.19
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.
2
Ro
PL0 = √ σy ln
Ri
3
ML0 = 4r2 Tσy
FL0 = π R2o − R2i σy
(2)
(3)
(4)
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.
2
NL0 = √ σy r2 T
3
(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 SPLWT under different single loads are as follows:
(1)
The limit load of SPLWT under internal pressure can be calculated by Equation (6):
pLS = −0.517G1 +0.995
(6)
where G1 denotes the geometric parameters of the local wall-thinning defect, G1 = a0.5 b0.1 c.
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):
mLS = −0.033G22 −0.386G2 +1.039
(7)
where G2 denotes the geometric parameters of the local wall-thinning defect, G2 = a0.01 b0.1 c.
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):
Appl. Sci. 2022, 12, 4850
9 of 19
nLS = −0.416G3 +1.045
(8)
where G3 denotes the geometric parameters of the local wall-thinning defect, G3 = a−0.1 b0.7 c.
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 LS = −1.242G42 −0.213G4 +0.985
(9)
where G4 denotes the geometric parameters of the local wall-thinning defect, G4
The least square error R2 of the equation is 0.984.
= a0.05 b1.5 c0.05 .
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 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 shows the orthogonal test levels,
while Table 8 shows the orthogonal test calculation scheme.
Table 7. Orthogonal test levels under complex loads.
Level
pl
ml
fl
1
2
3
0.1
0.2
0.4
0.1
0.2
0.4
0.1
0.2
0.4
Table 8. Calculation scheme of orthogonal tests under complex loads.
Test Number
pl
ml
fl
1
2
3
4
5
6
7
8
9
0.1
0.1
0.1
0.2
0.2
0.2
0.4
0.4
0.4
0.1
0.2
0.4
0.1
0.2
0.4
0.1
0.2
0.4
0.1
0.2
0.4
0.2
0.4
0.1
0.4
0.1
0.2
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), largearea 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 f l = 0.2 is taken as an example to illustrate the calculation 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 reAppl. Sci. 2022,
12, 4850
10 of 19
sults.
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
In the first load step, the internal pressure, bending moment, and axial force corresponding
time is 2.0. Figure 6 shows the plastic strain contours of SPLWT corresponding to the last
to pl = 0.4, ml = 0.4, and f l = 0.2 are proportionally loaded, and the load step time is 1.0. In
convergence pointtheofsecond
numerical
calculation
under
complex
canthe
beload
seen
that
load step,
the value of
applied
torque is loads.
30 kNm,Itand
step
timethe
is 2.0.
variesthe
with
thestrain
shape
of theofdefect.
For the axialtodefect,
the failfailure position ofFigure
SPLWT6 shows
plastic
contours
SPLWT corresponding
the last convergence
of numerical
calculation
complex
loads. Itloads
can because
seen that
failureto
position
ure position is at point
the corner
of the
defect,under
because
complex
thethestress
be
of
SP
varies
with
the
shape
of
the
defect.
For
the
axial
defect,
the
failure
position
is
at the
LWT
concentrated at the corner of the defect. For the circumferential defect, the failure position
corner of the defect, because complex loads cause the stress to be concentrated at the corner
is at the defect position.
For the
large-area
defect,
the failure
position
thedefect
corner
of
of the defect.
For the
circumferential
defect,
the failure
positionisis at
at the
position.
the defect, but the
positions
of the
and theis surrounding
proForother
the large-area
defect,
the defect
failure position
at the corner of position
the defect,also
but the
other
positions of
the plastic
defect and
the surrounding
position also
produce
stress
concentration
duce stress concentration
and
strain.
For the small-area
defect,
the
failure
positionand
For the
small-area
defect, that
the failure
position defects
is at the right
of the
straight
is at the right endplastic
of thestrain.
straight
pipe,
indicating
small-area
haveend
little
effect
pipe, indicating that small-area defects have little effect on the failure of the straight pipe,
on the failure of the straight pipe, and plastic strain also occurs at the defect location,
and plastic strain also occurs at the defect location, because stress concentration occurs at
because stress concentration
occurs at the defect position.
the defect position.
Figure
6. The
plastic strain
of a straight
pipe with
a local
wall-thinning defect
under
complex
Figure 6. The plastic
strain
contours
of contours
a straight
pipe with
a local
wall-thinning
defect
under
loads:
(a)
axial
defect;
(b)
circumferential
defect;
(c)
large-area
defect;
(d)
small-area
defect.
complex loads: (a) axial defect; (b) circumferential defect; (c) large-area defect; (d) small-area defect.
Figure 7 shows the load vs. plastic strain curves at the maximum plastic strain point of
Figure 7 shows
theunder
loadcomplex
vs. plastic
strain
at the
point
SPLWT
loads.
It can curves
be seen that
the maximum
plastic strain plastic
of SPLWTstrain
does not
change
in the firstloads.
load step.
This is
the applied
internalstrain
pressure,
moment,
of SPLWT under complex
It can
bebecause
seen that
the plastic
of bending
SPLWT does
notand
force
are low.
the internal
pressure,internal
bending moment,
andbending
axial forcemochange
change in the firstaxial
load
step.
ThisHowever,
is because
the applied
pressure,
the working conditions of the second load step calculation. The torque corresponding to
ment, and axial force are low. However, the internal pressure, bending moment, and axthe 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 SPLWT under complex loads.
0.2
0.2
0.2
0.4
0.4
0.4
Appl. Sci. 2022, 12, 4850
0.1
0.2
0.4
0.1
0.2
0.4
0.2
0.4
0.1
0.4
0.1
0.2
17.61
15.53
16.33
13.73
16.58
14.95
1.5
Load step time
Load step 2
Load step 1
0.0
Load step 2
0.1
0.2
0.5
(a)
0.3
0.4
0.0
0.5
Load step 1
0.0
0.2
(b)
0.4
0.6
1.5
1.2
Load step 2
Load step 2
1.0
0.8
Load step 1
0.4
0.0
17.85
15.72
16.83
13.52
11 of 19
16.82
14.64
1.0
0.5
0.0
14.49
13.75
13.79
12.75
13.46
12.89
1.5
1.0
Load step time
12.49
10.67
11.43
10.03
11.79
9.78
0.0
0.2
0.4
0.6
0.8
Plastic strain
Load step 1
0.5
(c)
0.0
(d)
0.000 0.006 0.012 0.018 0.024
Plastic strain
Figure 7.
strain
curves
of aof
straight
pipe pipe
with awith
localawall-thinning
defect under
Figure
7. The
Theload
loadvs.
vs.plastic
plastic
strain
curves
a straight
local wall-thinning
defect under
complex
loads:
(a)
internal
pressure;
(b)
bending
moment;
(c)
torque;
(d)
axial
force.
complex loads: (a) internal pressure; (b) bending moment; (c) torque; (d) axial force.
Table 9. The calculation results of the limit load of straight pipes with local wall-thinning defects
4.3.
Empirical Limit-Load Equations of Straight Pipes with Local Wall-Thinning Defects under
under complex loads.
Complex Loads
Torque/kNm
Internal
Bending Torque
The calculation
results are expressed in Limit
dimensionless
form, and the treatment
Pressure
Moment
Large-Area
Axial
Circumferential
Small-Area
fl
method
is
shown
in
Section
2.4.
The
empirical
equations
of
the
relationship
between
pl
ml
Defect
Defect
Defect
Defect
non-dimensional
pl, non-dimensional
non-dimensional
0.1
0.1 torque
0.1nl and 18.10
13.09 internal pressure
14.67
18.43
0.1 moment
0.2 ml, and
0.2 non-dimensional
17.31
12.28 force fl were
14.57
17.88
bending
axial
fitted by SPSS23
software.
0.1
0.4
0.4
14.93
9.33
13.49
15.01
Although
the 0.1
equations0.2were obtained
from12.49
typical dimensionless
defects
0.2
17.61
14.49
17.85results, they
0.2
13.75
15.72
were0.2verified within
the0.4
range of15.53
defect size. 10.67
0.2
0.4
0.1
16.33
11.43
13.79
16.83
0.4 limit load
0.1 of a 0.4
12.75
13.52 can be cal(1). The
straight 13.73
pipe with an10.03
axial local wall-thinning
defect
0.4
0.2
0.1
16.58
11.79
13.46
16.82
culated by0.4Equation
0.4
0.2(10): 14.95
9.78
12.89
14.64
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 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):
2
2
2
n1.6
l = 0.958 − 1.016ml − 1.284pl − 1.028 f l +0.054ml +0.322pl +0.028 f 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,
Appl. Sci. 2022, 12, 4850
12 of 19
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):
2
2
2
n1.5
l = 0.989 − 1.251ml − 1.104pl − 0.588 f l +0.238ml +0.072pl − 0.407 f 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):
n2l = 0.856 − 0.863m2l − 0.983p2l − 1.332 f l2 +0.023ml +0.137pl +0.424 f 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):
2
2
2
n1.3
l = 0.989 − 1.120ml − 1.315pl − 1.023 f l +0.134ml +0.308pl − 0.048 f 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 SPLWT 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):
(14)
(ml )2 + ( pl )2 = 1.0
Figure 8 shows the comparison between the limit load curves of SPLWT 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
f l in Equations (10)–(13) as zero, the limit load equation of SPLWT 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.
1.0
0.8
0.6
ml
Appl. Sci. 2022, 12, 4850
limit load results of Equation (14) in Ref. [28]. But the limit loads of Equatio
(13) in this paper are higher than those of Equation (14) in Ref. [28], within
This is because the values of Equation (14) are the lowest boundaries
13 of 19 of all
fects. Equation (14) cannot accurately evaluate all defects.
0.4
0.2
0.0
0.0
This paper(Eq.10)
This paper(Eq.11)
This paper(Eq.12)
This paper(Eq.13)
Chen2001(Eq.14)
0.2
0.4
pl
0.6
0.8
1.0
Figure 8. Comparison of the limit load equations of straight pipes with local wall-thinning defects
Figure 8. Comparison of the limit load equations of straight pipes with local wall-thi
under the combination of internal pressure and bending moment with Ref. [28].
under the combination of internal pressure and bending moment with Ref. [28].
5. Discussion
In this section, the effects of internal pressure, bending moment, torque, and axial
5. Discussion
force on the limit load are discussed, and the effects of torque and axial force on the limit
this
section, the effects of internal pressure, bending moment,
loadIn
of SP
LWT are analyzed quantitatively.
torqu
force on the limit load are discussed, and the effects of torque and axial force
5.1. Influence of Torque on the Limit Load Equation
load of SPLWT are analyzed quantitatively.
The influence of internal pressure, bending moment, and torque on the limit load of
SPLWT was analyzed. Taking the non-dimensional axial force f l in Equations (10)–(13) as
5.1.
Influence
of Torque
Limit
Load
Equation of internal pressure, bending
zero,
the limit load
equationon
of the
SPLWT
under
the combination
moment, and torque was obtained. Figure 9a–d show the limit state surface of SPLWT under
The influence of internal pressure, bending moment, and torque on the
the combination of internal pressure, bending moment, and torque. It can be seen that the
SPtorque
LWT was
Taking
theload
non-dimensional
in Equations
has aanalyzed.
great influence
on the limit
of SPLWT . When theaxial
torqueforce
is low, fitl has
little
effect
on
the
limit
load
curves
of
SP
.
With
the
gradual
increase
in
torque,
the
influence
zero, the limit load equationLWT
of SPLWT under the combination of internal pre
of torque on the limit load curve becomes larger. When the torque is close to the limit
ing moment, and torque was obtained. Figure 9a–d show the limit state surf
torque of SPLWT , the influence of internal pressure and bending moment on the limit load
under
combination
of internal
moment,
can bethe
almost
ignored, considering
only thepressure,
influence ofbending
torque on the
limit load.and torque. It
Thetorque
allowable
torque
of theinfluence
equipment nozzles
shown
in Table
2. LWT
The. allowable
that the
has
a great
on the islimit
load
of SP
When the torq
torque is expressed in dimensionless form through the limit torque of SPLWT under single
has little effect on the limit load curves of SPLWT. With the gradual increase in
loads. The non-dimensional allowable torque on medium–low-pressure equipment and
influence
of torque
onis the
limit load
becomes
larger. When the torque is
high-pressure
equipment
represented
by nl-gcurve
= Nl-g /N
LS and nl-h = N l-h /N LS respectively.
For different
pipe,
the averageofofinternal
the non-dimensional
torquemoment
is
limit
torque diameters
of SPLWTof
, the
influence
pressure allowable
and bending
nl-g = 0.10 for medium- and low-pressure equipment, and nl-h = 0.33 for high-pressure
load can be almost ignored, considering only the influence of torque on the li
equipment. Figure 10 shows that the effect of the allowable torque on the limit load of
SPLWT for nl is 0, 0.10, and 0.33. It can be seen that the range surrounded by the limit load
curve of SPLWT 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.
Appl. Sci. 2022, 12, x FOR PEER REVIEW
Appl. Sci. 2022, 12, 4850
14 of 19
14 of 19
Figure
limitstate
statesurfaces
surfaces
of straight
pipes
local wall-thinning
under
the comFigure 9.
9. The limit
of straight
pipes
withwith
local wall-thinning
defects defects
under the
combiAppl. Sci. 2022, 12, x FOR PEER REVIEW
15 of 19
bination
internal
pressure,
bending
moment,
and(a)
torque:
(a) axial
defect; (b) circumferential
nation ofof
internal
pressure,
bending
moment,
and torque:
axial defect;
(b) circumferential
defect;
defect;
(c) large-area
defect;
(d) small-area
defect.
(c) large-area
defect; (d)
small-area
defect.
ml
The allowable torque of the equipment nozzles is shown in Table 2. The allowable
1.0
(a) through the limit torque of SP
(b)
torque is expressed in dimensionless form
LWT under single
0.8
loads. The non-dimensional allowable torque on medium–low-pressure equipment and
high-pressure equipment is represented by nl-g = Nl-g/NLS and nl-h = Nl-h/NLS respectively.
0.6
For different diameters of pipe, the average of the non-dimensional allowable torque is
nl-g =0.4
0.10 for mediumand low-pressure equipment,
n1 = 0 and nl-h = 0.33 for high-pressure
n1 = 0
= 0.10
equipment. Figure
10 shows that the effect of then1-gallowable
torque on the limit load of
n1-g = 0.10
= 0.33and 0.33. It can be seen that nthe
SPLWT0.2for nl is 0,nl-h0.10,
range surrounded by the limit load
l-h = 0.33
curve0.0of SPLWT is reduced with the increase in torque, which means that the load-bearing
1.0 of defective straight pipes is reduced, and the influence of torque on straight
capacity
(d)
(c)defects is different.
pipes with different local wall-thinning
0.8
According to the allowable torque, the relative reduction in the average radius of the
limit0.6
load curves of SPLWT was calculated to quantitatively analyze the influence of the
torque on the load-bearing capacity of SPLWT under the combination of internal pressure
0.4
n1 = 0
=0
and bending
moment.
Table 10 shows the relativen1reduction
in the average radius of the
n
=
0.10
n
=
0.10
1-g
1-g
limit load curves of SPLWT under the influence of torque. It can be seen that the
0.2
nl-h = 0.33
nl-h = 0.33
load-bearing capacity
of SPLWT in high-pressure equipment
is more affected by torque
than 0.0
that in medium- and low-pressure equipment. For medium- and low-pressure
0.0
0.2
0.4
0.6
0.8
1.0 0.0
0.2
0.4
0.6
0.8
1.0
equipment, the load-bearing capacity ofp SPLWT changed little under the influence of
l
torque, reducing by 0.59~1.44%. The straight pipe with a circumferential defect was the
Figureaffected,
10. The
The effect
ofofthe
torque
onon
thethe
limit
load
curves
of straight
pipes pipes
with
Figure
10.
effect
torque
limit
load
of straight
with
local affected.
wall-thinning
most
while
straight
pipe
with
a curves
large-area
defect
waslocal
thewall-thinning
least
For
defects
under
the
combination
of
internal
pressure
and
bending
moment:
(a)
axial
defect;
(b)
circumdefects
under
the
combination
of
internal
pressure
and
bending
moment:
(a)
axial
defect;
(b)
high-pressure equipment, the load-bearing capacity of SPLWT was greatly affected cirby
ferential defect;
(c) large-area
defect; defect;
(d) small-area
defect. defect.
cumferential
defect;
(c) 6.67~9.03%.
large-area
small-area
torque,
reducing
by
The (d)
straight
pipe with an axial defect was the most affected,
while thetostraight
pipe with
a large-area
was
affected.
According
the allowable
torque,
the relative defect
reduction
inthe
the least
average
radius of the
Table 10. The relative reduction in the average radius of the limit load curves of straight pipes with
limit load curves of SPLWT was calculated to quantitatively analyze the influence of the
local wall-thinning defects
under the influence of torque.
Defect Type
Axial defect
Medium- and Low-Pressure
Equipment
1.28%
High-Pressure
Equipment
9.03%
Appl. Sci. 2022, 12, 4850
15 of 19
torque on the load-bearing capacity of SPLWT under the combination of internal pressure
and bending moment. Table 10 shows the relative reduction in the average radius of the
limit load curves of SPLWT under the influence of torque. It can be seen that the loadbearing capacity of SPLWT 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 SPLWT 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 SPLWT 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.
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
Circumferential defect
Large-area defect
Small-area defect
1.28%
1.44%
0.59%
1.20%
9.03%
9.01%
6.67%
8.41%
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 SPLWT was analyzed. Taking the non-dimensional torque nl in Equations (10)–(13) as
zero, the limit load equations of SPLWT under the combination of internal pressure, bending
moment, and axial force were obtained. Figure 11a–d show the limit state surfaces of SPLWT
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 SPLWT 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 SPLWT .
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 SPLWT
under single loads. The non-dimensional allowable axial force on medium–low-pressure
equipment and high-pressure equipment is represented by f l-g = Fg /FLS and f l-h = Fh /FLS ,
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 highpressure equipment. Figure 12 shows that the effect of the allowable axial force on the limit
load of SPLWT for nl is 0, 0.05, and 0.41. The range surrounded by the limit load curves of
SPLWT 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 SPLWT was calculated to quantitatively analyze the influence of
the axial force on the load-bearing capacity of SPLWT under the combination of internal
pressure and bending moment. Table 11 shows the relative reduction in the average radius
of the limit load curves of SPLWT under the influence of axial force. For medium- and lowpressure equipment, the load-bearing capacity of SPLWT 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
Appl. Sci. 2022, 12, 4850
16 of 19
Appl. Sci. 2022, 12, x FOR PEER REVIEW
16 of 19
high-pressure equipment, the load-bearing capacity of SPLWT 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.
Figure
11. 11.
The The
limit
state
surfaces
withlocal
localwall-thinning
wall-thinning
defects
under
Figure
limit
state
surfacesofofstraight
straight pipes
pipes with
defects
under
the the
Appl. Sci. 2022, 12, x FOR PEER REVIEW
of 19
combination
of internal
bendingmoment,
moment,
axial
(a) defect;
axial defect;
(b)17circumfercombination
of internalpressure,
pressure, bending
andand
axial
force:force:
(a) axial
(b) circumferential
entialdefect;
defect;
large-area
defect;
(d) small-area
(c)(c)
large-area
defect;
(d) small-area
defect.defect.
ml
The1.0allowable axial force of the equipment nozzles is shown in Table 2. The allowa(a)
ble axial force is expressed in dimensionless
form through the limit(b)
axial force of SPLWT
0.8
under single loads. The non-dimensional allowable axial force on medium–low-pressure
equipment
0.6 and high-pressure equipment is represented by fl-g = Fg/FLS and fl-h = Fh/FLS,
respectively. For different diameters of pipe, then average
of the non-dimensional allown1 = 0
1=0
0.4
able axial force is
l-g0.05
= 0.05 for medium- and low-pressure
equipment, and fl-h = 0.41 for
n1-g = 0.05
n1-gf=
0.2
=
0.41
high-pressure
equipment.
Figure 12 shows that nthe
effect
of
the allowable axial force on
nl-h = 0.41
l-h
the limit0.0load of SPLWT for nl is 0, 0.05, and 0.41. The range surrounded by the limit load
curves of
1.0 SPLWT is the relative value of the load-bearing capacity. When the range sur(d)
(c) that the load-bearing capacity
rounded by the curve decreases, it means
of the defective
0.8
straight pipe is reduced. The load-bearing capacity of high-pressure equipment is lower
0.6of medium- and low-pressure equipment.
than that
According
tonthe
allowable axial force, the relative
reduction in the average radius of
n1 = 0
0.4
1=0
the limit load curve
SPLWT was calculated to quantitatively
analyze the influence of the
n1-g of
= 0.05
n1-g = 0.05
0.2
nl-h load-bearing
= 0.41
axial force
on the
capacity of SPnLWT
under the combination of internal
l-h = 0.41
pressure0.0and bending moment. Table 11 shows the relative reduction in the average ra0.0
0.2
0.4
0.6
0.8
1.0 0.0
0.2
0.4
0.6
0.8
1.0
dius of the limit load curves of SPLWT under
pl the influence of axial force. For medium- and
low-pressure equipment, the load-bearing capacity of SPLWT changed little under the inFigure
12.
Theeffect
effectofofaxial
axialforce
forceon
on
the
limitload
loadcurves
curves
straightpipes
pipes
with
local
wall-thinning
Figure
The
the
limit
straight
local
fluence
of12.
axial
force,
reducing
by
0.83~1.80%.
Theofofstraight
pipewith
with
a wall-thinning
circumferential
defects
under
the
combination
of
internal
pressure
and
bending
moment:
(a)
axial
defect;
cirdefects under the
internal pressure and bending moment: (a) axial defect; (b)(b)
defect
was thedefect;
most(c)affected,
while
the
straight defect.
pipe with an axial defect wascircumthe least
cumferential
large-area
defect;
(d)
small-area
ferential defect; (c) large-area defect; (d) small-area defect.
affected. For high-pressure equipment, the load-bearing capacity of SPLWT was greatly
affected
axial
force,reduction
reducing
byaverage
7.51~12.93%.
straight
pipe of
with
a circumferential
Table by
11. The
relative
in the
radius ofThe
the limit
load curves
straight
pipes with
localwas
wall-thinning
defects
under the
influence
of axial force.
defect
the most
affected,
while
the straight
pipe with a small-area defect was the
least affected.
Medium and Low-Pressure
High-Pressure
Defect Type
Axial defect
Equipment
0.83%
Equipment
8.57%
Appl. Sci. 2022, 12, 4850
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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
Circumferential defect
Large-area defect
Small-area defect
0.83%
1.80%
1.15%
1.01%
8.57%
12.93%
8.07%
7.51%
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.
Author Contributions: Writing—original draft preparation, Y.L.; methodology, B.G. and Y.L.; conceptualization, B.G.; writing—review and editing, Y.L. and S.L.; project administration, B.G.; funding
acquisition, B.G.; data curation, Y.L.; supervision, K.L. and J.D. All authors have read and agreed to
the published version of the manuscript.
Funding: This research was supported by the National Key Research and Development Program of
China, 2018YFC0808600.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available upon request from the authors.
Acknowledgments: The authors wish to express their thanks for the financial support from the
National Key Research and Development Program of China (No.2018YFC0808600).
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature
A
B
C
a
b
c
E
F
f
L
Axial half-length, mm
Circumferential half-length, mm
Defect depth, mm
A/(Ro T)0.5
B/(πRo )
C/T
Young’s modulus, GPa
Axial force, kN
Axial force, dimensionless
Length of straight pipe, mm
Appl. Sci. 2022, 12, 4850
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M
m
N
n
P
p
Pc
Ro
Ri
r
T
σy
µ
Subscripts
Bending moment, kNm
Bending moment, dimensionless
Torque, kNm
Torque, dimensionless
Internal pressure, MPa
Internal pressure, dimensionless
Axial equivalent surface load, MPa
Outer radius of straight pipe, mm
Inner radius of straight pipe, mm
Mean radius of straight pipe, mm
Thickness of straight pipe, mm
Yield strength of the material, MPa
Poisson’s ratio
LS
l-h
Limit load of defective straight pipe under single load
Limit load of defect-free straight pipe
Limit load of defective straight pipe under complex loads
Limit load of defective straight pipe
Allowable load of medium- and low-pressure equipment nozzles
Allowable load of high-pressure equipment nozzles
Abbreviations
SPLWT
Straight pipe with local wall-thinning defect
L0
LC
l
l-g
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