Ratcheting Simulation of a Steel Pipe with Assembly Parts under Internal Pressure and a Cyclic Bending Load

Abstract

The ratcheting behavior of a steel pipe with assembly parts was examined under internal pressure and a cyclic bending load, which has not been seen in previous research. An experimentally validated and three dimensional (3D) elastic-plastic finite element model (FEM)—with a nonlinear isotropic/kinematic hardening model—was used for the pipe’s ratcheting simulation and considered the assembly contact effects outlined in this paper. A comparison of the ratcheting response of pipes with and without assembly parts showed that assembly contact between the sleeve and pipe suppressed the ratcheting response by changing its trend. In this work, the assembly contact effect on the ratcheting response of the pipe with assembly parts is discussed. Both the assembly contact and bending moment were found to control the ratcheting response, and the valley and peak values of the hoop ratcheting strain were the transition points of the two control modes. Finally, while the clearance between the sleeve and the pipe had an effect on the ratcheting response when it was not large, it had no effect when it reached a certain value.

1. Introduction

Steel pipes are widely used in machinery and the marine industry, among other fields [1,2,3,4]. They are subjected to internal pressure and a large cyclic bending load, which may incur a ratcheting effect [5]. The ratcheting effect leads to additional plastic damage [6] and has a great influence on the fatigue performance of the structure [7]. Additionally, ratcheting deformation may not cease until final fracture occurs [8]. Similar to fatigue, creep and other damage mechanisms, the ratcheting effect has been taken into account by many criteria, including RCC-MR [9] and ASME Code Section III [10], for assessing the safety and life prediction of the structure [11].

In recent years, many researchers have studied the ratcheting behavior of pipes by experimental and numerical simulation. Chen et al. [12,13,14] studied the ratcheting behavior of elbow and straight pipes under internal pressure and cyclic loading by experimental and numerical simulation. In their study, several cyclic plasticity models were considered in the ratcheting simulation, including bilinear (BKH) [15], multilinear (MKIN/KINH) [16,17], Chaboche (CH3) [18,19], modified Chaboche (CH4) [20], Ohno-Wang [21,22], modified Ohno-Wang [23,24,25,26], and Abdel Karim-Ohno [27]. They also investigated and analyzed the effect of local wall thinning on the ratcheting behavior. Liu et al. [3] evaluated the thermal effects on the ratcheting behavior of a pressured elbow pipe using the Chen-Jiao-Kim (CJK) kinematic hardening model as a user subroutine of ANSYS. In addition, they adopted more accurate Chen-Jiao-Kim model parameters to explore the ratcheting behavior of a pressurized elbow pipe under two sets of complex loading paths [28]. The fatigue-ratcheting behavior of thin-walled carbon steel elbow and tee joint pipes were evaluated under internal pressure and seismic loading through experimental and numerical analysis by Kiran et al. [29]. Numerical models with a nonlinear isotropic/kinematic hardening model were adopted to evaluate the effects of some key factors—such as the mean stress, loading regime, and material hardening properties—on the ratcheting response of a steel tube with a rectangular defect by Zeinoddini and Peykanu [30]. Finally, Li et al. [31] studied the ratcheting behavior of a 90° single unreinforced mitered pipe subjected to a cyclic in-plane closing moment with a non-zero mean value and constant internal pressure by means of experimental and numerical simulation.

Most of the existing research focuses on the ratcheting behavior of a single pipe. However, the ratcheting response of a pressurized pipe with assembly parts has been not yet been considered and studied. Additionally, in practice, the region near the assembly contact area between the sleeve and the pipe is the most dangerous location, so it is worth examining the ratcheting response of a pipe with assembly parts in that region, as this may lay the foundations for further study on the effect of the ratcheting response on the failure of a pipe. In this work, due to the difficulty of an experimental measurement in the assembly contact area, the ratcheting response of a pipe was studied using an elastic-plastic finite element method with a nonlinear isotropic/kinematic hardening model. Then, the ratcheting responses of pipes with and without assembly parts were compared and analyzed. In addition, the effect of assembly contact between the sleeve and the pipe on the ratcheting response was investigated and is discussed herein. Lastly, the influence of the clearance between the sleeve and pipe on the ratcheting response was also analyzed.

2. Hardening Rule for a Steel Pipe under a Cyclic Load

The hardening rule used in this paper is a nonlinear isotropic/kinematic hardening model which has been implemented in the ABAQUS finite element code [32]. The constitutive model was applied to simulate and predict the ratcheting behavior of the pressurized pipe.

The yield surface is defined by the function:

$$F=f\left(\mathit{\sigma }-\mathit{\alpha }\right)-{\sigma }^0=0.$$

(1)

where $\sigma$ is the stress tensor, ${\mathit{\sigma }}^0$ is the yield stress, and $f\left(\mathit{\sigma }-\mathit{\alpha }\right)$ is the equivalent von Mises stress with respect to the back stress tensor,$\mathit{\alpha }$. The equivalent von Mises stress is defined as:

$$f\left(\mathit{\sigma }-\mathit{\alpha }\right)=\sqrt{\frac{3}{2}\mathit{S}-{\mathit{\alpha }}^{dev}:\left(\mathit{S}-{\mathit{\alpha }}^{dev}\right)}.$$

(2)

where $\mathit{S}$ is the deviatoric stress tensor, defined as $\mathit{S}=\mathit{\sigma }+p\mathit{I}$, where $p$ is the equivalent pressure stress and $\mathit{I}$ is the identity tensor, and ${\mathit{\alpha }}^{dev}$ is the deviatoric part of the back stress tensor.

Kinematic hardening models assume associated plastic flow:

$${\dot{\epsilon }}^{pl}={\dot{\overline{\epsilon }}}^{pl}\frac{\partial F}{\partial \mathit{\sigma }}.$$

(3)

where ${\dot{\epsilon }}^{pl}$ is the rate of plastic flow and ${\dot{\overline{\epsilon }}}^{pl}$ is the equivalent plastic strain rate. The evolution of the equivalent plastic strain is obtained from the following equivalent plastic work expression:

$${\sigma }^0{\dot{\overline{\epsilon }}}^{pl}=\mathit{\sigma }:{\dot{\mathit{\epsilon }}}^{pl}.$$

(4)

in which ${\dot{\overline{\epsilon }}}^{pl}=\sqrt{\frac{2}{3}{\dot{\mathit{\epsilon }}}^{pl}:{\dot{\mathit{\epsilon }}}^{pl}}$ for isotropic Mises plasticity.

In kinematic hardening models, the center of the yield surface moves in stress space due to the kinematic hardening component. In addition, when a nonlinear isotropic/kinematic hardening model is used, the yield surface range may expand or contract due to the isotropic component. These features allow modeling of inelastic deformation in metals that are subjected to cycles of load or temperature, resulting in significant inelastic deformation and, possibly, low-cycle fatigue failure [33].

The evolution law of this model consists of two components: a nonlinear kinematic hardening component, which describes the translation of the yield surface in stress space through the back stress, $\mathit{\alpha }$, and an isotropic hardening component, which describes the change of the equivalent stress defining the size of the yield surface ${\sigma }^0$, as a function of plastic deformation [30].

The kinematic hardening component is defined as an additive combination of a purely kinematic term (linear Ziegler hardening law) and a relaxation term (the recall term), which introduces the nonlinearity. When temperature and field variable dependencies are omitted, the hardening laws for each back stress are [30]:

$${\dot{\mathit{\alpha }}}_k=C_k\frac{1}{{\sigma }^0}\left(\mathit{\sigma }-\mathit{\alpha }\right){\dot{\overline{\epsilon }}}^{pl}-{\gamma }_k{\mathit{\alpha }}_k{\dot{\overline{\epsilon }}}^{pl},$$

(5)

and the overall back stress is computed from the relation:

$$\mathit{\alpha }={\sum }_{k=1}^N{\mathit{\alpha }}_k.$$

(6)

where $N$ is the number of back stresses, and $C_k$ and ${\gamma }_k$ are material parameters that must be calibrated from cyclic test data. $C_k$ are the initial kinematic hardening moduli, and ${\gamma }_k$ determine the rate at which the kinematic hardening moduli decrease with increasing plastic deformation. The kinematic hardening law can be separated into a deviatoric part and a hydrostatic part; only the deviatoric part has an effect on the material behavior. When $C_k$ and ${\gamma }_k$ are zero, the model reduces to an isotropic hardening model. When all ${\gamma }_k$ equal zero, the linear Ziegler hardening law is recovered.

The isotropic hardening behavior of the model defines the evolution of the yield surface size ${\sigma }^0$, as a function of the equivalent plastic strain ${\overline{\epsilon }}^{pl}$. This evolution can be introduced by specifying directly ${\sigma }^0$ as a function of ${\overline{\epsilon }}^{pl}$ by using the simple exponential law [30]:

$${\sigma }^0=\sigma {|}_0+Q_{\infty }\left(1-e^{-b{\overline{\epsilon }}^{pl}}\right).$$

(7)

where $\sigma {|}_0$ is the yield stress at zero plastic strain, and $Q_{\infty }$ and $b$ are material parameters. $Q_{\infty }$ is the maximum change in the size of the yield surface, and $b$ defines the rate at which the size of the yield surface changes as plastic straining develops. When the equivalent stress defining the size of the yield surface remains (${\sigma }^0=\sigma {|}_0$), the model reduces to a nonlinear kinematic hardening model.

In this work, one back stress was considered, meaning that $N$ in Equation (6) was equal to 1. The parameters for the nonlinear isotropic/kinematic hardening model were determined through a monotonic test and cyclic test of the standard specimens according to the method in Ref. [34]. Through the monotonic test, the yield stress at zero plastic strain ($\sigma {|}_0=283\mathrm{MPa}$) and Young’s modulus ($E=195\mathrm{GPa}$) were able to be determined. $Q_{\infty }$, $b$, $C_1$, and ${\gamma }_1$ were evaluated by stabilized cyclic tests. Then, the control variate method was adopted to investigate the effect of each parameter on the numerical stress–strain, with the parameters subsequently adjusted to make the numerical results match well with the experimental results. This allowed us to determine the optimized parameters, as shown in

Table 1. Parameters for the nonlinear isotropic/kinematic hardening model.

$\mathsf{\sigma }{|}_0\left(\mathbf{MPa}\right)$	${\mathbf{Q}}_{\infty }\left(\mathbf{MPa}\right)$	$\mathit{b}$	${\mathbf{C}}_1$ (GPa)	${\mathit{\gamma }}_1$
283	109.8	18	4.4	11

.

3. Validation Experiment for the Finite Element Model (FEM)

In order to verify the validity of the finite element model with the hardening model mentioned above, an experiment was carried out before the ratcheting simulation. The experimental details are shown in Figure 1 and Figure 2. In the experiment, the internal pressure levels were 10 MPa, 21 MPa and 28 MPa, and the displacement load was applied in the form of a sinusoidal function:

$$U_1=U_{a}sinwt.$$

(8)

where $U_1$ is displacement, $U_a$ is the amplitude of displacement and $U_a=5.06\mathrm{mm}$ here.

The actual average outside and inside diameters of the steel pipe adopted in the experiment were 11.95 mm and 10 mm, respectively, and the outside diameter of the pipe near the sleeve end was measured before and after the experiment. After completing the experiment, the results showed that the outside diameter of the steel pipe increased with an increase of internal pressure, as shown in the results presented in

Table 2. Change in the outside diameter of the pipe.

Internal Pressure (MPa)	Outside Diameter (mm)		Change in Outside Diameter
	Before the Experiment	After the Experiment
10	11.95	12	0.418%
21	11.95	12.13	1.473%
28	11.95	12.19	2.033%

. The fracture site was near the sleeve end, as shown in Figure 2. Considering the heating problem effect on the accuracy of the strain gauge, as well as the difficulty in attaching the strain gauge due to the large deformation in the area of concern near the sleeve, two strain acquisition points were chosen to collect axial strain data 30 mm away from the sleeve end. These were located on the top and bottom of the pipe, as illustrated in Figure 1. Strain data were recorded in the experiment and compared with the numerical results.

4. Ratcheting Simulation

In this paper, ratcheting strain is defined as [7]:

$${\epsilon }_r=\frac{{\epsilon }_{max}+{\epsilon }_{min}}{2}.$$

(9)

where ${\epsilon }_{max}$ and ${\epsilon }_{min}$ are the maximum strain and minimum strain, respectively, in a cycle.

4.1. Finite Element Model

In this paper, the nonlinear finite element program ABAQUS was used to establish a finite element model simulating experiment and to further study the ratcheting behavior of the cantilevered pipe with assembly parts under cyclic bending load and internal pressure.

As depicted in Figure 3, the finite element model was simplified without affecting the calculation accuracy. The main geometric dimensions of the model and the loads applied were consistent with the experiment. To reduce the computing time, a half model was constructed according to the symmetry of the geometric model and loading conditions, and symmetric constraints were applied to the X-Z symmetry plane. The left end of the joint was fixed. All interactions between the joint, nut, tapered ends of the sleeve, and pipe were set as tie constraints, and the interaction between the parallel parts of the sleeve and pipe was set as the “surface to surface contact”. In addition, the interaction between the plug and pipe was also set as a tie constraint. Internal pressure was applied to the inner wall of the pipe and a symmetrical cyclic displacement load was applied to the clamp. It should be pointed out that the clearance between the pipe and sleeve was 0.025 mm.

The tapered end of the pipe was fixed when the pipe was without assembly parts and the load condition was the same as the pipe with assembly parts.

In the numerical model, a nonlinear isotropic/kinematic hardening model was adopted to simulate and analyze the mechanical behavior of the cantilevered pipe under cyclic loading. An eight-node element (C3D8R), hourglass control, and reduced integration points were used to simulate the cantilevered pipe model and mesh sensitivity analysis was also carried out to select an appropriate mesh density. There were 38,242 elements in the pipe model. The non-linear geometry change option (NLGEOM in ABAQUS) was adopted.

4.2. Validation of the Numerical Model

In the experiment, a strain gauge was attached to the top of the pipe, 30 mm away from the sleeve end, to collect axial strain data under the cyclic load, as shown in Figure 1. Position 4 in the numerical model presented in Figure 3 corresponds to the axial strain collection point in the experiment. In this position, peak values from numerical and experimental axial strain data were compared under a displacement amplitude of 5.06 mm and different internal pressures. The results are shown in Figure 4 The root mean square error (RMSE) under different internal pressures was found to be in the same order of magnitude, although much smaller, than strain data. This was evidence that the numerical results were in good agreement with the experimental results. Consequently, the finite element model used in this paper was found to be reliable and appropriate to study the mechanical behavior of the cantilevered pipe under cyclic bending load and internal pressure.

5. Numerical Results and Discussion

Since the pipe with assembly parts was most prone to crack initiation and final fracture near the sleeve end (as shown in Figure 2), further consideration and subsequent discussion of the ratcheting behavior of the pipe with assembly parts in that area was needed to facilitate failure analysis and improve maintenance.

In order to examine the ratcheting response of the pipe with assembly parts near the sleeve end, three positions in the longitudinal section of the pipe were selected in this area for comparative analysis, as exhibited in Figure 5. Position 1 was a node on the outer wall of the pipe and inside the sleeve, Position 2 was a node on the outer wall of the pipe and closest to the sleeve end, and Position 3 was a node on the outer wall of the pipe and outside the sleeve end. In Figure 5b, Position 1’, Position 2’, and Position 3’ are on the pipe without assembly parts, and correspond with Position 1, Position 2, and Position 3 on the pipe with assembly parts.

Detailed analysis sections are presented as follows. Firstly, the ratcheting behavior of the pipes with and without assembly parts were compared at the locations set out in Figure 5. The ratcheting behavior of the pipe with assembly parts is then further discussed to explore the assembly contact effect on the ratcheting response. The effect of the clearance between the pipe and the sleeve on the ratcheting behavior at the positions of interest is also examined. It is worth noting that hoop ratcheting strain was the main concern of this paper, as ratcheting strain mainly occurs in the hoop direction [13,35].

5.1. Comparative Analysis of the Ratcheting Response between the Pipe with and without Assembly Parts

In Figure 6, the hoop ratcheting strain of the pipe without assembly parts is monotonous and becomes larger as the bending moment increases from Position 4 to Position 1 during cyclic loading. In contrast, the hoop ratcheting strain curve of the pipe with assembly parts has a valley value near the sleeve end and a peak value on the right of the sleeve end. By comparing the two cases, it is clear that the assembled sleeve changed the trend of the hoop ratcheting strain curve. The hoop ratcheting strain near the sleeve was much greater in the pipe without assembly parts, meaning that the hoop ratcheting strain near the sleeve was suppressed and the existence of the sleeve had a great effect on the ratcheting response of the pipe. That is to say, assembly contact between the sleeve and the pipe suppressed cumulative plastic deformation in the pressurized pipe near the sleeve during cyclic loading.

5.2. Investigation of the Ratcheting Behavior of the Pipe Considering the Assembly Contact Effect

Figure 7 shows the hoop ratcheting strain distribution between Position 1 and Position 4 after 45 cycles. In this figure, it is clear that the hoop ratcheting strain increases with an increase in internal pressure. In addition, it should be pointed out that the valley value of the hoop ratcheting strain is almost on Position 2 under an internal pressure of 10 MPa, while it is on the left of Position 2 under internal pressures of 21 and 28 MPa. This suggests that, as the internal pressure increases, the valley value moves further from Position 2. This is due to axial deformation of the pipe, caused by the increase in internal pressure. As a result, Position 2 moves far away from the sleeve end as the internal pressure increases. However, the valley value of the hoop ratcheting strain is still on the node of the pipe which is closest to the sleeve end. Thus, axial deformation of the pipe with assembly parts had no effect on the above conclusion that assembly contact between pipe and sleeve suppresses cumulative plastic deformation. In addition, the position of the peak value of the hoop ratcheting strain hardly changed during the 45 cycles under different internal pressures.

Figure 8 presents the change in outside diameter of the pipe with assembly parts between Position 1 and Position 4 under different internal pressure levels, for which the trend has high consistency with the hoop ratcheting strain distribution in Figure 7. This indicates that the hoop ratcheting strain’s increase with increasing internal pressure led to the outside diameter of the pipe increasing [33], and also gives a reasonable explanation for the experimental results presented in Table 2. It is important to state that the negative change in the outer diameter of the pipe under the 10 MPa internal pressure in Figure 8 was caused by ovalization of the pipe’s cross-section [36], which reached a maximum at Position 2 due to the large bending deformation. However, an increase of the internal pressure and hoop ratcheting strain resists negative deformation, making the outer diameter increase.

As mentioned above, an increase of the bending moment leads to the hoop ratcheting strain increasing, and assembly contact between the sleeve and pipe can suppress the hoop ratcheting response, thereby changing the monotonic trend of the ratcheting strain. There are peak and valley values of the hoop ratcheting strain and the outside diameter change in Figure 7 and Figure 8. Overall, it can be deduced that the ratcheting response between the valley value and the peak value is controlled and suppressed by assembly contact between the sleeve and pipe. The ratcheting response in other areas was dominated by the bending moment, and was found to increase as the bending moment increased. Thus, the valley value and peak value are the transition points between the two control modes.

Figure 9 exhibits the ratcheting response of the pipe with assembly parts at Position 1, Position 2, and Position 3 under the internal pressures of 10 MPa, 21 MPa and 28 MPa. It can be seen that the axial ratcheting strain is much smaller than the hoop ratcheting strain, indicating that the hoop ratcheting strain plays a dominant role in cumulative plastic deformation. This also conforms with the view presented in Chen and colleagues’ papers [13,35].

As displayed in Figure 9, after 45 cycles and under the internal pressures of 10 MPa, 21 MPa, and 28 MPa, the bending moment at Position 3 is smaller than that at Position 2, although the hoop ratcheting strain at Position 3 is larger than that at Position 2. This is due to the effect of assembly contact between the sleeve and pipe, which plays an important role in inhibiting the hoop ratcheting strain at Position 2.

As change in the outside diameter at Position 1 was found to be less than the initial clearance between the sleeve and the pipe during 45 cycles under an internal pressure of 10 MPa (shown in Figure 8), the ratcheting response at Position 1 does not appear to be greatly affected by the assembly contact between the sleeve and pipe, and is instead mainly controlled by the bending moment (as mentioned above). Further, the bending moment at Position 2 was smaller than that at Position 1. So the ratcheting strain at Position 2 was also smaller than that at Position 1 under the internal pressure 10 MPa during 45 cycles (Figure 9a).

In addition, as a consequence of the increase in internal pressure, Position 2 deviated slowly from the sleeve end in the axial direction; thus, the inhibition effect of the assembly contact on the ratcheting strain at Position 2 was gradually weakened. Moreover, increasing internal pressure led to increasing hoop ratcheting strain, resulting in the increase of outside diameter at Position 1. Hence, the clearance between the sleeve and the pipe decreased, and the inhibition effect of the assembly contact between the sleeve and pipe on the ratcheting response at Position 1 was enhanced by a number of degrees. As a result, the ratcheting strain at Position 2 had a trend beyond the ratcheting strain at Position 1 under the 21 MPa internal pressure over 45 cycles, as shown in Figure 9b. Lastly, the ratcheting strain at Position 2 exceeded the ratcheting strain at Position 1 within 45 cycles under the internal pressure of 28 MPa, as illustrated in Figure 9c.

The ratcheting strain at Position 1 was greater than that at Position 3 due to the larger bending moment at Position 1 under the internal pressure of 10 MPa. However, the inhibition effect of the assembly contact between the sleeve and the pipe on the ratcheting response at Position 1 gradually increased, and thus the hoop ratcheting strain at Position 3 also tended to exceed that at Position 1 under the internal pressure of 10 MPa during 45 cycles. As presented in Figure 9b, the hoop ratcheting strain at Position 3 further exceeded that at Position 1 during 45 cycles when the internal pressure was 21 MPa, and the hoop ratcheting strain at Position 3 exceeded that at Position 1 under 28 MPa internal pressure at a faster rate than under 21 MPa internal pressure.

5.3. The Effect of the Clearance on the Ratcheting Response of the Pipe with Assembly Parts

Figure 10 shows the effect of different clearance between the sleeve and the pipe on the ratcheting response of the pipe with assembly parts at the three selected positions under the internal pressure of 21 MPa during 45 cycles. The average hoop ratcheting strain rates are listed in

Table 3. The average hoop ratcheting strain rate at different positions for different clearance under internal pressure of 21 MPa.

Positions	Clearance between the Sleeve and the Pipe (mm)
	0.000	0.025	0.035	0.07
Position 1	0.00524%/cycle	0.02056%/cycle	0.02544%/cycle	0.03135%/cycle
Position 2	0.00929%/cycle	0.01911%/cycle	0.02240%/cycle	0.02823%/cycle
Position 3	0.03144%/cycle	0.02833%/cycle	0.02711%/cycle	0.02571%/cycle

.

In Figure 10a, it can be seen that the hoop ratcheting strain increases as the clearance increases from 0 mm to 0.07 mm at Position 1, with the corresponding average strain rate also increasing (Table 3). The ratcheting response for different clearances at Position 2 was similar to that at Position 1, as displayed in Figure 10b and Table 3. From Figure 10a,b, it can be deduced that as the clearance increases, the assembly contact effect on the ratcheting response gradually declines, leading to the ratcheting strain and average strain rate increasing as the cycles increase.

However, the ratcheting response at Position 3 was different from the first two positions. When the clearance between the sleeve and the pipe increased, the hoop ratcheting strain and the corresponding average ratcheting strain rate decreased at Position 3, as presented in Figure 10c and Table 3. Under the same displacement load, an increase in the clearance resulted in a weakening of the assembly contact inhibition effect, and the part of the pipe inside the sleeve was able to be subjected to more deformation, which made the deformation at Position 3 decrease. This indicates that the bending moment at Position 3 decreased. Thus, when the clearance increases, the ratcheting response at Position 3 reduces due to a smaller bending moment, as exhibited in Figure 10c.

When the clearance was 0 mm, the average ratcheting strain rate at Position 1, Position 2 and Position 3 increased (as shown in Table 3) because the assembly contact inhibition effect decreased from Position 1 to Position 3. When the clearance was 0.025 mm and 0.035 mm, the average ratcheting strain rate at the selected positions had the same trend, with the average ratcheting strain rate at Position 3 being the largest, at Position 1 being the second largest and at Position 2 being the smallest. Position 1 was located inside the sleeve and the assembly contact effect enhanced gradually at Position 1 during the loading cycles. Position 2 was closest to the assembly contact area and the assembly contact effect was found to be the largest at Position 2. This means that the assembly contact effect on the ratcheting response at Position 1 was smaller than that at Position 2. Thus, the average ratcheting strain rate at Position 1 was larger than that at Position 2. Position 3, located outside the assembly contact area, was the least affected by assembly contact. Hence, the average ratcheting strain rate at Position 3 was the largest. When the clearance was 0.07 mm, the ratcheting responses at the selected positions were mainly dominated by the bending moment, and the average ratcheting strain rate decreased due to the increase of the bending moment from Position 1 to Position 3.

Similar trends for the hoop ratcheting strain from Position 1 to Position 4 under different internal pressures were observed over the clearances used, as shown in Figure 11. The assembly contact appeared to have no effect on the ratcheting response when the clearance was large enough (e.g., 0.07 mm). The ratcheting response trend was the same as the pipe without a sleeve displayed in Figure 6.

6. Conclusions

In this paper, a comparative study was carried out on the ratcheting behavior of a pipe with and without assembly parts under internal pressure and cyclic bending loading through a validated finite element model. The effects of the assembly contact and the clearance between the sleeve and the pipe on the ratcheting response were also considered and discussed. The conclusions can be drawn as follows:

(1)

Assembly contact between the sleeve and the pipe suppressed the ratcheting response of the pipe with assembly parts and changed the monotonic trend of the ratcheting response in the pipe without assembly parts along the pipe axis.

(2)

The hoop ratcheting strain of the pipe was much larger than the axial ratcheting strain under internal pressure and cyclic bending loading, and the hoop ratcheting strain played a leading role in cumulative plastic deformation of the pipe.

(3)

The hoop ratcheting strain increased with an increase in internal pressure, leading to an increase of the diameter of the steel pipe and the average ratcheting strain rate.

(4)

There were peak and valley values of the hoop ratcheting strain in the pipe with assembly parts along the pipe axis. The hoop ratcheting response between the peak value and valley value was controlled and suppressed by assembly contact between the sleeve and pipe, while in other areas it was dominated by the bending moment and increased with an increase of the bending moment. The valley and peak values were the transition points between the two control modes.

(5)

The clearance between the sleeve and the pipe had a different effect on the ratcheting response at positions near the sleeve, such as Position 1 and Position 2, and far away from the sleeve, such as Position 3. Near the sleeve, the ratcheting response increased with an increase of the clearance, whereas it decreased with an increase of the clearance when distant from the sleeve (where the bending moment decreased). When the clearance was large enough, the ratcheting response was no longer affected by the clearance.

In this work, failure of the pipe could not be simulated due to limitations of the constitutive model. Thus, a fatigue life predictive model considering the ratcheting effect will be carried out in future work.