As a researcher engaged in lifeline engineering, I have been consistently concerned with the mechanical behavior of buried water supply pipelines. In my recent work, I have focused on the performance degradation of socket joints in ductile iron castings subjected to combined tension and bending actions. This paper presents a comprehensive investigation involving in-situ mechanical tests and finite element simulations, aiming to quantify how axial displacement of the inserted pipe affects the bending capacity and stiffness of the joint system.
Ductile iron castings have been widely adopted in urban water distribution networks. According to official statistics, the total length of water supply pipelines in China has exceeded 1.1 million kilometers. Among various pipe materials, ductile iron castings account for a large proportion due to their excellent mechanical properties, corrosion resistance, and relatively low cost. The socket-and-spigot joint is the most common connection method for ductile iron castings in water supply systems. However, these joints are vulnerable to combined loading conditions: temperature variations induce axial displacement of the spigot pipe, while additional earth pressure from surface surcharge, excavation activities, and traffic loads causes bending of the joint. This combined tension-bending action may cause severe degradation of the joint’s mechanical performance, ultimately threatening the structural integrity and serviceability of the entire pipeline system.
Previous studies have predominantly focused on either pure axial behavior or pure bending behavior of socket joints in ductile iron castings. Several researchers have conducted axial pull-out tests and bending tests on various pipe diameters. However, a systematic understanding of the coupled tension-bending behavior, especially the underlying mechanism of performance degradation, remains insufficient. This gap motivated me to conduct a series of experiments and numerical analyses to explore the synergistic effect of axial displacement and bending moment on the mechanical response of socket joints in ductile iron castings.
Experimental Program
In this study, I designed and conducted in-situ bending tests on socket-and-spigot joints of ductile iron castings with two nominal diameters: DN100 and DN150. The test program was designed to capture the bending behavior of the joints under different axial displacement levels of the inserted pipe. A total of 17 test conditions were established, as summarized in Table 1. For each test, the axial displacement of the spigot pipe relative to the socket was precisely measured after assembly, with an accuracy of 0.5 cm, considering the inherent variability of manual assembly operations.
| Condition No. | Pipe Type / Axial Displacement (cm) |
|---|---|
| 1 | DN150 / 0 |
| 2 | DN150 / 0 |
| 3 | DN150 / 0 |
| 4 | DN150 / 0.5 |
| 5 | DN150 / 1 |
| 6 | DN150 / 1 |
| 7 | DN150 / 1 |
| 8 | DN150 / 1.5 |
| 9 | DN150 / 1.5 |
| 10 | DN150 / 2.5 |
| 11 | DN150 / 3 |
| 12 | DN100 / 0 |
| 13 | DN100 / 0 |
| 14 | DN100 / 1.5 |
| 15 | DN100 / 1.5 |
| 16 | DN100 / 2 |
| 17 | DN100 / 2 |
The test setup is schematically illustrated in Figure 1 (not shown here). The socket pipe was firmly clamped to prevent any movement during loading. The spigot pipe was supported on a roller support connected to a hinge, allowing free rotation. A hydraulic jack was used to apply an upward vertical displacement to the spigot pipe, thereby inducing a bending moment at the joint. For some specimens, the spigot pipe needed to be cut and the cut end was ground into an arc shape before being inserted into the socket for testing. The pipe was filled with water during the test to simulate real service conditions, and the internal water pressure was maintained within normal operating range.
Load was applied in a stepwise manner, with each step corresponding to approximately 1° of rotation. At each load step, I recorded: (1) the angles of the socket and spigot pipes relative to the horizontal plane, and (2) the hydraulic pressure of the oil pump, which was converted to the jack force and subsequently to the bending moment at the joint. The maximum loading height of the hydraulic jack was approximately 50 mm, corresponding to a rotation angle of about 21° to 22° of the spigot pipe.
Experimental Results and Analysis
The rotation angle-bending moment curves obtained from the tests are presented in Figures 2 and 3 (not shown here) for DN150 and DN100 pipes, respectively. These curves consistently exhibit a three-stage pattern, which I denote as: (1) a slow-rising stage characterized by low stiffness, (2) an ascending stage with significantly higher stiffness, and (3) a plateau stage where the bending moment stabilizes. Figure 4 (not shown) schematically illustrates these characteristic stages and defines the key mechanical parameters: $K_1$ is the stiffness of the slow-rising stage, $K_2$ is the stiffness of the ascending stage, $\theta_1$ is the critical rotation angle at the transition from the slow-rising stage to the ascending stage, $\theta_2$ is the critical rotation angle at the transition from the ascending stage to the plateau stage, and $M_u$ is the ultimate bending moment, defined as the bending moment at the junction of the ascending and plateau stages.
Table 2 summarizes the average values of the key mechanical parameters for both pipe diameters, categorized by axial displacement ranges. Several important observations can be made from these results:
| Pipe Type | Axial Disp. (cm) | $M_u$ (kN·m) | $K_1$ (kN·m/°) | $K_2$ (kN·m/°) | $\theta_1$ (°) | $\theta_2$ (°) |
|---|---|---|---|---|---|---|
| DN150 | 0 | 15.8 | 0.2 | 2.1 | 6.0 | 12.6 |
| 0–1 | 14.6 | 0.2 | 2.2 | 5.7 | 12.2 | |
| 1–3 | 11.7 | 0.2 | 1.5 | 5.6 | 14.1 | |
| DN100 | 0 | 13.1 | 0.1 | 1.8 | 6.8 | 13.9 |
| 1–2 | 9.6 | 0.1 | 0.9 | 5.4 | 14.3 |
For DN150 pipes, when the spigot was fully inserted (zero axial displacement), the ultimate bending moment ranged from 13.4 to 19.4 kN·m. When the axial displacement was between 0.5 and 1 cm, the ultimate moment decreased to 13–18.5 kN·m. For axial displacements in the range of 1.5–3 cm, the ultimate bending moment further decreased to 10.4–12.4 kN·m. For DN100 pipes, the ultimate bending moment was approximately 13.1 kN·m when fully inserted, and decreased to 8.5–10.5 kN·m when the axial displacement was 1–2 cm.
The slow-rising stage stiffness $K_1$ remained relatively constant at approximately 0.1–0.3 kN·m/° regardless of the axial displacement, as shown in Table 2. This is because, during the slow-rising stage, the spigot pipe does not come into contact with the socket wall; the bending resistance is solely attributed to the rubber gasket. Since the contact area between the gasket and the spigot pipe is not significantly affected by the axial displacement within the tested range, $K_1$ remained unchanged.
In contrast, the ascending-stage stiffness $K_2$ decreased notably with increasing axial displacement. For DN150 pipes, when fully inserted, $K_2$ varied between 1.7 and 2.3 kN·m/°; for axial displacements of 0.5–1 cm, $K_2$ was 1.6–2.3 kN·m/°; and for axial displacements of 1.5–3 cm, $K_2$ decreased to 1–1.6 kN·m/°. Similarly, for DN100 pipes, $K_2$ decreased from 1.5–2.1 kN·m/° (fully inserted) to 0.7–1.4 kN·m/° (1–2 cm axial displacement).
The critical rotation angles $\theta_1$ and $\theta_2$ remained essentially unchanged with increasing axial displacement. For DN150 pipes, $\theta_1$ was approximately 5.5°–6° and $\theta_2$ was 12.2°–14.1°. For DN100 pipes, $\theta_1$ was approximately 5.4°–6.8° and $\theta_2$ was 10.5°–12.5°.
After the tests, I examined the spigot pipes. The cement mortar lining at the lower portion of the spigot pipe had partially spalled, and clear deformation marks were visible on the outer surface of the spigot pipe at locations where it contacted the socket. These observations confirmed that the spigot pipe engages with the socket inner wall during the ascending stage, generating significant contact reaction forces.
To validate the reasonableness of my test results, I compared the mechanical parameters obtained from the fully inserted condition (zero axial displacement) with values reported in the existing literature. As summarized in Table 3, the comparisons show excellent agreement. The maximum discrepancy was 6.3% for the ultimate bending moment, indicating that my experimental procedure and results are reliable.
| Parameter | Literature Values | Literature Sources | My Test | Discrepancy |
|---|---|---|---|---|
| $K_1$ (kN·m/°) | 0.1–0.21 | [8][9][16] | 0.2 | Within range |
| $K_2$ (kN·m/°) | 0.5–2.2 | [8][9][16] | 2.1 | Within range |
| $\theta_1$ (°) | 2.9–7.0 | [8][9][16] | 6.0 | Within range |
| $\theta_2$ (°) | 13.1 | [16] | 12.6 | 4.0% |
| $M_u$ (kN·m) | 14.8 | [16] | 15.8 | 6.3% |
Finite Element Numerical Simulation
To gain deeper insight into the mechanical response of socket joints in ductile iron castings under combined tension-bending actions, I developed finite element models using the ABAQUS software package. The model comprised three components: the socket (bell), the spigot pipe, and the T-shaped rubber gasket rings. The geometry of the models was based on the standard dimensions specified in the national code GB/T 13295-2019.
The ductile iron castings materials were modeled with von Mises plasticity. The yield strength was set to 300 MPa, the maximum plastic strain was 0.1, and the ultimate tensile strength was 420 MPa. The friction coefficient between ductile iron castings surfaces was taken as 0.15, while the friction coefficient between ductile iron castings and rubber was set to 0.5. The T-shaped rubber gasket is composed of two different hardness materials: THA50±5 and THA88±3. These were simulated using the Mooney-Rivlin hyperelastic constitutive model. For THA50, the material parameters were $D_1 = 0.034$, $C_{10} = 0.25$, and $C_{01} = 0.33$. For THA88, the parameters were $D_1 = 0.012$, $C_{10} = 0.50$, and $C_{01} = 1.06$.
To ensure computational accuracy, I refined the mesh in the front portion of the spigot pipe where significant deformation and contact were expected. For the boundary conditions, the socket end section was fully fixed. The spigot pipe end section, corresponding to the location of the hydraulic jack in the test, was coupled to a reference point. A concentrated vertical upward force was applied at this reference point to simulate the loading condition of the experimental setup.
The finite element model was validated by comparing the predicted rotation angle-bending moment curves with the experimental curves for both DN100 and DN150 pipes. The comparisons, shown in Figures 5 and 6 (not shown here), demonstrate excellent agreement between the numerical simulations and the experimental data. This validation confirms that the finite element model is capable of accurately predicting the tension-bending mechanical behavior of socket joints in ductile iron castings.
Figure 7 (not shown) displays the Mises stress distribution contours of the DN150 socket joint at rotation angles of 4°, 8°, 12°, and 16°. A comparison with the post-test observations of the spigot pipes revealed that the deformation patterns predicted by the finite element model closely match the experimental observations, further confirming the accuracy of the numerical model in simulating stress distribution and deformation behavior.
From the finite element simulation, I can clearly identify the deformation mechanism at each loading stage. At small rotation angles (e.g., 4°), the spigot pipe front only contacts the rubber gasket, and the contact stress is relatively low. This corresponds to the slow-rising stage of the rotation angle-bending moment curve, where the bending stiffness is low and the moment increases slowly. As the rotation angle increases (e.g., 8°), the spigot pipe gradually comes into contact with the inner wall of the socket. This corresponds to the ascending stage, where the bending stiffness substantially increases and the moment rises rapidly. When the stress at the front of the spigot pipe reaches 420 MPa, the bending moment reaches its ultimate value. With further rotation, the inelastic deformation at the contact locations between the spigot pipe and the socket inner wall and outer edge increases, and the bending moment remains nearly constant or shows a slight declining trend, corresponding to the plateau stage.
Mechanism of Performance Degradation
Based on the combination of experimental observations and finite element analysis, I have systematically analyzed how increasing axial displacement affects each of the five key mechanical performance indicators of the socket joints in ductile iron castings.
(1) Slow-rising stage stiffness $K_1$: As previously noted, $K_1$ remains essentially constant with increasing axial displacement. This is because, during the slow-rising stage (typically at rotation angles less than 4°), the spigot pipe does not contact the socket wall. The bending resistance is provided solely by the rubber gasket, and the contact area between the gasket and the spigot pipe is not significantly influenced by the axial displacement within the tested range. Therefore, $K_1$ is independent of the axial displacement.
(2) Slow-rising stage critical rotation angle $\theta_1$: This parameter corresponds to the rotation angle at which the spigot pipe first comes into contact with the socket inner wall. Since the gap between the spigot pipe outer wall and the socket inner wall is much smaller than the axial displacement, the influence of axial displacement on this critical angle is negligible. The value remains approximately 5°–6° for both pipe diameters.
(3) Ascending-stage stiffness $K_2$: As illustrated in the stress contour maps, the ascending-stage stiffness is primarily governed by the contact reaction forces between the spigot pipe and the socket inner wall. The magnitude of these reaction forces depends on the contact area between the spigot pipe and the socket inner wall. When the axial displacement increases, the spigot pipe is pulled outward, reducing the overlapping length between the spigot and the socket. Consequently, the contact area during bending decreases, leading to a reduction in $K_2$. This mechanism is confirmed by both the experimental and numerical results.
(4) Ascending-stage critical rotation angle $\theta_2$: This parameter corresponds to the rotation angle at which the contact portion of the spigot pipe wall with the socket inner wall reaches the yield state. The value typically falls in the range of 9°–12° and does not exhibit a strong dependence on the axial displacement.
(5) Ultimate bending moment $M_u$: The ultimate moment corresponds to the bending moment at the critical rotation angle $\theta_2$, when the contact portion of the spigot pipe with the socket inner wall is fully yielded. Since the yielding condition is determined by the stress developed over the contact area, and the contact area decreases with increasing axial displacement, the ultimate bending moment also decreases with increasing axial displacement. The reduction is approximately linear within the range of axial displacements studied (0–2 cm), consistent with both experimental and numerical findings.
Numerical Simulation at Varying Axial Displacements
To systematically investigate the degradation law of the bending performance of socket joints in ductile iron castings under combined tension-bending actions, I conducted a parametric study using the validated finite element model. Simulations were performed for DN100 and DN150 pipe joints with axial displacements ranging from 0 to 2 cm. Figure 8 (not shown) presents the predicted rotation angle-bending moment curves for both pipe diameters at various axial displacement levels.
Table 4 summarizes the evolution of the mechanical performance indicators with axial displacement, along with the fitted empirical formulas. The key observations are as follows:
| Mechanical Indicator | DN100 Joint | DN150 Joint | Unit |
|---|---|---|---|
| Slow-rising stiffness $K_1$ | 5 | 17 | kN·cm/° |
| Ascending stiffness $K_2$ | $160 – 20L$ | $300 – 50L$ | kN·cm/° |
| Slow-rising critical angle $\theta_1$ | 5.5 | 6.0 | ° |
| Ascending critical angle $\theta_2$ | 10.5–12.5 | 9.5–11.8 | ° |
| Ultimate bending moment $M_u$ | $1220 – 320L$ | $1750 – 450L$ | kN·cm |
Note: $L$ is the axial displacement in cm. The initial values in the formulas (e.g., 160 for DN100 ascending stiffness) are in kN·cm/° and the initial values for ultimate moment are in kN·cm. All formulas are valid for $0 \leq L \leq 2$ cm.
From Table 4, it can be observed that:
(1) With increasing axial displacement, the slow-rising stage stiffness $K_1$, the slow-rising critical rotation angle $\theta_1$, and the ascending critical rotation angle $\theta_2$ remain essentially constant for both DN100 and DN150 socket joints.
(2) The ascending-stage stiffness $K_2$ and the ultimate bending moment $M_u$ decrease with increasing axial displacement. For DN100 joints, $K_2$ decreases by $20L$ kN·cm/° and $M_u$ by $320L$ kN·cm, where $L$ is the axial displacement in cm. For DN150 joints, $K_2$ decreases by $50L$ kN·cm/° and $M_u$ by $450L$ kN·cm.
(3) Comparing the finite element simulation results with the experimental results, the discrepancies are acceptable. For the slow-rising stiffness $K_1$, the finite element results are in the range of 0.1–0.3 kN·m/°, consistent with experiments. For the ascending critical angle $\theta_1$, the finite element predictions are approximately 6° for DN150 (experimental range: 5.6°–6°) and approximately 5.5° for DN100 (experimental range: 5.4°–6.8°). For the plateau critical angle $\theta_2$, the finite element predictions for DN150 are 9.5°–11.8° (experimental range: 12.1°–14.1°), and for DN100 are 9°–12.2° (experimental range: 10.5°–12.5°). For the ultimate bending moment, as axial displacement increased from 0 to 2 cm, the finite element results for DN150 decreased from 17.5 kN·m to 8.5 kN·m, while the experimental results decreased from 15.8 kN·m to 11.7 kN·m. For DN100, the corresponding finite element results decreased from 12.2 kN·m to 5.8 kN·m, while the experimental results decreased from 13.1 kN·m to 9.6 kN·m.
Statistical Analysis and Empirical Modeling
To provide a quantitative basis for engineering practice, I conducted statistical regression analysis on the finite element simulation results. The objective was to establish empirical degradation formulas that can predict the bending mechanical performance of socket joints in ductile iron castings at various axial displacement levels.
For the ascending-stage stiffness $K_2$, the linear regression model can be expressed as:
$$K_2(L) = K_{2,0} – \alpha L$$
where $K_{2,0}$ is the ascending stiffness at zero axial displacement, $\alpha$ is the degradation coefficient, and $L$ is the axial displacement in centimeters. For DN100 joints, $K_{2,0} = 160$ kN·cm/° and $\alpha = 20$ kN·cm/(°·cm). For DN150 joints, $K_{2,0} = 300$ kN·cm/° and $\alpha = 50$ kN·cm/(°·cm).
Similarly, the ultimate bending moment $M_u$ can be expressed as:
$$M_u(L) = M_{u,0} – \beta L$$
where $M_{u,0}$ is the ultimate bending moment at zero axial displacement, and $\beta$ is the degradation coefficient. For DN100 joints, $M_{u,0} = 1220$ kN·cm and $\beta = 320$ kN·cm/cm. For DN150 joints, $M_{u,0} = 1750$ kN·cm and $\beta = 450$ kN·cm/cm.
The coefficient of determination ($R^2$) for all the fitted formulas exceeded 0.95, indicating excellent goodness of fit. These empirical formulas collectively provide a convenient tool for engineers to estimate the degraded bending performance of socket joints in ductile iron castings under combined tension-bending conditions, which is essential for the safety assessment of existing water supply pipeline systems.
Discussion
The results of this study clearly demonstrate that axial displacement significantly degrades the bending performance of socket joints in ductile iron castings. This finding has important implications for the safety assessment and maintenance of water supply pipelines, especially in regions with significant temperature fluctuations or in areas prone to ground settlement and traffic-induced vibrations.
It is noteworthy that while the ascending-stage stiffness and ultimate bending moment decrease with axial displacement, the critical rotation angles $\theta_1$ and $\theta_2$ remain relatively stable. This suggests that monitoring the rotation angle alone may not be sufficient to detect performance degradation; instead, engineers should also consider the joint’s axial displacement state when evaluating its structural integrity.
The degradation mechanism is fundamentally attributed to the reduced contact area between the spigot pipe and socket inner wall during bending when the spigot is pulled outward. This geometric effect is intrinsic to the socket-and-spigot connection configuration of ductile iron castings and cannot be easily mitigated without modifying the joint design. However, the empirical formulas developed in this study provide a rational basis for estimating the remaining bending capacity of joints in service, thereby enabling more informed decisions regarding repair, rehabilitation, or replacement priorities.
I also acknowledge certain limitations in the present study. The axial displacement range explored in this study was limited to 0–2 cm. In extreme scenarios, such as those caused by severe ground deformation, larger axial displacements may occur. Future studies should extend the range of axial displacement to investigate whether the linear degradation trends persist or become nonlinear. Additionally, the effect of internal water pressure on the bending performance degradation warrants further investigation. Although the pipes in my tests were filled with water, the pressure was not systematically varied. Cyclic loading conditions, simulating realistic repeated traffic or thermal loads, also deserve further exploration.
Conclusions
In this study, I conducted a comprehensive investigation of the mechanical performance of socket-and-spigot joints in ductile iron castings under combined tension-bending actions. The key conclusions are summarized as follows:
(1) I designed and conducted in-situ bending tests on DN100 and DN150 socket joints of ductile iron castings. By deliberately introducing different axial displacement levels, I obtained the rotation angle-bending moment curves of the joints for various axial displacement conditions. The test setup involved fixing the socket pipe, applying vertical displacement to the spigot pipe via a hydraulic jack, and measuring both the hydraulic pressure and pipe rotations to determine the bending moment and rotation angle.
(2) The experimental results showed that increasing axial displacement significantly reduces both the ultimate bending moment and the ascending-stage bending stiffness. For DN150 socket joints, when the axial displacement reached 2 cm, the ultimate bending moment decreased from 15.8 kN·m to 10.4–12.4 kN·m, and the ascending-stage stiffness decreased from 2.1 kN·m/° to 1–1.6 kN·m/°. For DN100 socket joints, axial displacement of 1–2 cm reduced the ultimate bending moment from 13.1 kN·m to 8.5–10.5 kN·m, and the ascending-stage stiffness from 1.8 kN·m/° to 0.7–1.4 kN·m/°. Meanwhile, the slow-rising stage stiffness and the critical rotation angles remained relatively unchanged.
(3) The reasonableness of my experimental results was verified by comparison with existing literature for the zero-axial-displacement condition, with maximum discrepancies of approximately 6.3%. The finite element model I established accurately reproduced the experimental rotation angle-bending moment curves for both pipe diameters, and the deformation patterns predicted by the numerical model closely matched the post-test observations.
(4) Combining experimental and numerical results, I identified the underlying mechanism of performance degradation. Axial displacement primarily affects the contact area between the spigot pipe and the socket inner wall during bending. As axial displacement increases, this contact area decreases, leading to a linear reduction in both the ascending-stage stiffness and the ultimate bending moment.
(5) Based on the finite element parametric study, I developed empirical degradation formulas for predicting the bending mechanical performance of DN100 and DN150 socket joints at various axial displacement levels, applicable for axial displacements in the range of 0–2 cm. These formulas provide quantitative guidance for the safety assessment and operational maintenance of water supply pipelines constructed with ductile iron castings.
In conclusion, this study advances the understanding of the coupled tension-bending behavior of socket joints in ductile iron castings, offering both experimental evidence and numerical insights that are directly applicable to engineering practice. The empirical formulas presented herein serve as practical tools for estimating the residual bending capacity of in-service joints, supporting the rational allocation of maintenance resources and the long-term resilience of urban water infrastructure systems.