Ductile iron casting has become one of the most widely used materials in municipal water supply networks due to its exceptional combination of strength, ductility, corrosion resistance, and relatively low life-cycle cost. By the end of 2022, the total mileage of water-supply pipelines in China had surpassed 1.1 million kilometres, and a large share of these pipelines are made of ductile iron casting. Among the various connection methods, the socket-and-spigot push-on joint is the dominant joining technique for ductile iron casting pipelines, because it provides reliable sealing through a rubber gasket while allowing a controlled amount of axial and rotational movement. In practice, buried ductile iron casting pipelines are subjected not only to internal water pressure but also to external actions such as temperature variation, non-uniform soil settlement, surface surcharge, excavation-induced ground movement, and traffic loading. Temperature changes cause the inserted pipe to move axially relative to the socket, producing a pull-out displacement at the joint. In the meantime, additional earth pressure caused by surface load or excavation can impose significant bending moments on the same joint. The coexistence of axial displacement and bending deformation defines a typical combined tension-bending loading condition for ductile iron casting socket joints.
Much of the current knowledge on ductile iron casting socket joints has been obtained from experiments and simulations that treat axial loading and bending as independent actions. For example, the axial pull-out behavior of such joints has been studied extensively under quasi-static conditions, and the flexural response has been characterized through four-point bending or cantilever bending tests. These investigations have provided valuable information on the force-displacement envelope, the rotation capacity, the leakage limit, and the ultimate failure mode of the joint. However, the actual service condition of a buried pipeline rarely falls into the category of pure tension or pure bending. The axial displacement induced by thermal contraction or soil movement is almost always combined with a certain level of bending due to uneven soil support. It is therefore necessary to clarify how the bending resistance of a ductile iron casting socket joint degrades when an axial displacement is present. The present study addresses this issue by combining full-scale in-situ mechanical tests and high-fidelity finite element simulations of ductile iron casting socket joints. Two commonly used nominal diameters, DN100 and DN150, are selected for investigation. The test results are used to extract the moment-rotation characteristics, and the numerical model is subsequently employed to reveal the underlying mechanism of the degradation of the bending performance with increasing axial displacement.
The paper is organized as follows. The experimental program is described in Section 2, including the test matrix, the loading apparatus, the measurement procedure, and the definitions of characteristic mechanical parameters. Section 3 presents the measured moment-rotation curves and discusses the influence of axial displacement on the flexural parameters. Section 4 introduces the finite element modeling procedure and validates the model against the experimental results. Section 5 provides a detailed analysis of the degradation mechanism and proposes empirical formulas for predicting the bending capacity and stiffness of ductile iron casting socket joints at different axial displacements. Finally, Section 6 summarizes the main conclusions and emphasizes the practical implications for the integrity assessment of ductile iron casting water pipelines.
Experimental Program
The test specimens were composed of two ductile iron casting pipes connected by a standard socket-and-spigot joint, namely a socket pipe and an inserted pipe, with a T-shaped rubber gasket installed between them. The total insertion depth of the inserted pipe in its fully assembled state was set according to the manufacturer’s recommendations. In the present study, the fully inserted position corresponded to an insertion depth of approximately 9.4 cm for the DN150 pipe and 9.0 cm for the DN100 pipe. A deliberate axial pull-out displacement was produced by withdrawing the inserted pipe from the socket by a prescribed amount before the bending test. The accuracy of the axial displacement setting was approximately 0.5 cm, which accounts for the unavoidable manual operation during the assembly of the specimen. After the desired axial displacement was set, the actual value was measured and recorded before the start of the loading procedure.
The bending test arrangement followed a cantilever-type loading configuration. The socket pipe was firmly clamped by steel fixtures, so that its end remained stationary during the entire loading process. The clamped socket pipe was supported by a pipe saddle, and the clamp was tightened to prevent any translation or rotation of the socket. The inserted pipe was supported by a roller support near its free end, and the free end was connected to a hydraulic jack through a hinged base. When the jack was activated, it pushed the free end of the inserted pipe upward, thereby generating a bending moment at the joint. The distance between the joint and the load-application point was such that the maximum stroke of the jack, about 50 mm, corresponded to a rotation angle of approximately \(21^\circ\) to \(22^\circ\) at the joint. This arrangement enabled the entire moment-rotation response to be captured up to the ultimate state.
The loading procedure employed a displacement-controlled scheme. The rotation angle of the joint was increased in steps of approximately \(1^\circ\). At each increment, the rotation angle was measured by two inclinometers mounted on the socket pipe and on the inserted pipe, respectively. The relative angle between the two pipes was taken as the joint rotation. The hydraulic pressure of the jack was recorded simultaneously and converted into the jack force using the calibrated area of the jack piston. The bending moment at the joint was then calculated as the product of the jack force and the lever arm between the joint and the loading line. The point of application of the jack force, however, changed slightly as the free end moved along a circular arc. To minimize the influence of this geometric nonlinearity, the lever arm was updated at each load step using the instantaneous rotation angle and the initial distance between the joint and the loading point. The moment-rotation curve was then constructed from the sequential measurements.
A total of 17 valid test conditions were performed, including eleven conditions for DN150 ductile iron casting joints and six conditions for DN100 ductile iron casting joints. The test matrix is summarized in Table 1. The axial displacement values cover a range from 0 to 3 cm, representing the range from a completely inserted joint to a joint with a significant pull-out displacement. Repeated tests were carried out for several displacement levels in order to account for the natural scatter of the assembly and the loading procedure.
| Condition | Nominal diameter | 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 moment-rotation curves obtained from the tests were used to define five characteristic mechanical parameters, as illustrated by a typical curve in Figure 1. The initial portion of the curve is a gently rising branch controlled by the deformation of the rubber gasket. The slope of this branch is defined as the initial stiffness \(K_1\). As the rotation increases, the inserted pipe gradually comes into contact with the inner wall of the socket, which produces a significantly stiffer load path. The slope of the steep ascending branch is defined as the secondary stiffness \(K_2\). The transition point between the two ascending branches is marked by the critical rotation \(\theta_1\). After the steep ascending branch, the moment-rotation curve enters a plateau region where the bending moment remains nearly constant or even decreases slightly with increasing rotation. The moment at the onset of the plateau is taken as the ultimate bending moment \(M_u\), and the corresponding rotation is denoted as \(\theta_2\). These parameters provide a complete description of the flexural behavior of a ductile iron casting socket joint.
Experimental Results and Discussion
The measured moment-rotation curves for the DN150 and DN100 ductile iron casting socket joints are presented in Figures 2 and 3, respectively, for selected axial displacement levels. All curves exhibit the same three-stage characteristic shape, but the magnitude of the bending moment and the slope of the steep ascending branch vary considerably with the axial displacement. In the initial stage, the moment grows very slowly with rotation because the resistance is primarily provided by the low-stiffness rubber gasket. Once the metallic contact between the inserted pipe and the socket inner wall is established, the moment increases rapidly until the plastic capacity of the contact region is reached. In the plateau stage, plastic deformation continues to accumulate in the pipe wall near the socket entrance but the resisting moment no longer increases appreciably.
The average values of the characteristic parameters for both pipe sizes are summarized in Table 2. For the DN150 ductile iron casting joints, the ultimate bending moment was about \(15.8\ \mathrm{kN\cdot m}\) when the axial displacement was zero. When the axial displacement was within the range of \(0.5\ \mathrm{cm}\) to \(1\ \mathrm{cm}\), the average ultimate moment decreased to approximately \(14.6\ \mathrm{kN\cdot m}\). When the axial displacement was further increased to the range of \(1.5\ \mathrm{cm}\) to \(3\ \mathrm{cm}\), the average ultimate moment reduced to about \(11.7\ \mathrm{kN\cdot m}\), corresponding to a reduction of approximately \(26\%\) with respect to the fully inserted condition. A similar trend was observed for the DN100 ductile iron casting joints. The ultimate bending moment dropped from \(13.1\ \mathrm{kN\cdot m}\) at zero axial displacement to about \(9.6\ \mathrm{kN\cdot m}\) for an axial displacement between \(1\ \mathrm{cm}\) and \(2\ \mathrm{cm}\), which represents a \(27\%\) reduction. These observations clearly indicate that the axial pull-out displacement has a significant detrimental effect on the load-carrying capacity of ductile iron casting socket joints.
| Pipe type | Axial displacement (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 |
In contrast to the ultimate moment, the initial stiffness \(K_1\) was found to be almost insensitive to the axial displacement. For all tested conditions, the value of \(K_1\) ranged between \(0.1\ \mathrm{kN\cdot m/°}\) and \(0.3\ \mathrm{kN\cdot m/°}\). This insensitivity is reasonable because the initial stage is governed by the deformation of the rubber gasket, which remains in contact with the outer surface of the inserted pipe over nearly the same contact length regardless of the axial pull-out position. The axial displacement does not change the diameter of the inserted pipe, nor does it alter the contact pressure distribution between the gasket and the pipe surface in a significant manner. Therefore, the initial compliance of the joint is virtually unaffected by the pull-out displacement.
The secondary stiffness \(K_2\), on the other hand, exhibited a clear decreasing trend with increasing axial displacement. For the DN150 ductile iron casting joints, \(K_2\) decreased from \(2.1\ \mathrm{kN\cdot m/°}\) at zero displacement to \(1.5\ \mathrm{kN\cdot m/°}\) in the range of \(1.5\ \mathrm{cm}\) to \(3\ \mathrm{cm}\). For the DN100 ductile iron casting joints, \(K_2\) dropped from \(1.8\ \mathrm{kN\cdot m/°}\) to \(0.9\ \mathrm{kN\cdot m/°}\) when the axial displacement increased to \(1\ \mathrm{cm}\)–\(2\ \mathrm{cm}\). This means that the post-contact bending stiffness of a ductile iron casting socket joint can be reduced by more than \(40\%\) when the inserted pipe is pulled out by a relatively small amount. The stiffness degradation is attributed to the reduction of the contact area between the inserted pipe and the socket inner wall, as will be further discussed in the numerical analysis section.
The transition rotations \(\theta_1\) and \(\theta_2\) did not change substantially with the axial displacement. For the DN150 joints, \(\theta_1\) remained in the range of \(5.6^\circ\) to \(6.0^\circ\), while \(\theta_2\) remained between \(12.2^\circ\) and \(14.1^\circ\). For the DN100 joints, \(\theta_1\) varied from \(5.4^\circ\) to \(6.8^\circ\), and \(\theta_2\) varied from \(10.5^\circ\) to \(12.5^\circ\). These two parameters are mainly related to the geometric gap between the outer wall of the inserted pipe and the inner wall of the socket. Because the radial gap is much smaller than the axial pull-out displacement, the rotation at which metallic contact occurs is only weakly dependent on the axial position of the inserted pipe. Consequently, the transition rotations remain approximately constant within the range of axial displacements considered in the present study.
After the bending test, the inserted pipe was removed from the socket for visual inspection. It was observed that the cement-mortar lining at the lower part of the inserted pipe had partially spalled, and the outer surface of the inserted pipe near the contact region exhibited noticeable plastic deformation marks. Such permanent deformation confirmed that the pipe wall at the contact point had been subjected to high local stresses and had yielded before the ultimate moment was reached. The failure pattern is consistent with the plateau stage of the moment-rotation curve, in which the bending moment remains stable while the material at the contact zone continues to deform plastically.
To confirm the reliability of the experimental data, the measured characteristic parameters for the fully inserted DN150 ductile iron casting joints were compared with the values reported in previous studies. The comparison is presented in Table 3. The initial stiffness, secondary stiffness, and transition rotations obtained in the present tests fall well within the ranges reported in the literature. The ultimate bending moment agrees with the reference result within \(6.3\%\), and the rotation at the ultimate point agrees within \(4.0\%\). The good agreement indicates that the present test setup and measurement procedure are able to reproduce the flexural behavior of ductile iron casting socket joints with acceptable accuracy.
| Parameter | Literature range | Current test |
|---|---|---|
| \(K_1\) (kN·m/°) | 0.1–0.21 | 0.2 |
| \(K_2\) (kN·m/°) | 0.5–2.2 | 2.1 |
| \(\theta_1\) (°) | 2.9–7.0 | 6.0 |
| \(\theta_2\) (°) | 13.1 | 12.6 |
| \(M_u\) (kN·m) | 14.8 | 15.8 |
Finite Element Modeling and Validation
To gain deeper insight into the mechanical response of the ductile iron casting socket joints under combined tension-bending actions, a three-dimensional finite element model was developed using the general-purpose software ABAQUS. The model consisted of three solid components: the socket pipe, the inserted pipe, and two T-shaped rubber sealing rings. The geometry of the socket and the inserted pipe followed the dimensions of the tested specimens. Although the actual socket joint has a complicated internal profile, the present model simplified the geometric features that do not participate directly in the load-transfer mechanism, such as small chamfers and fillets, while retaining the critical surfaces that are involved in the contact interaction.
The ductile iron casting material was modeled as an elastic-plastic solid with isotropic hardening. According to the relevant material standard, the yield strength was taken as \(300\ \mathrm{MPa}\) and the ultimate tensile strength as \(420\ \mathrm{MPa}\). The maximum plastic strain used in the material model was set to \(0.1\). The elastic modulus and Poisson’s ratio were assumed to be \(170\ \mathrm{GPa}\) and \(0.28\), respectively, which are typical values for ductile iron casting. The constitutive behavior can be represented by the following piecewise description:
$$ \sigma = \begin{cases} E \varepsilon, & \varepsilon \le \varepsilon_y \\[4pt] \sigma_y + K_p (\varepsilon – \varepsilon_y), & \varepsilon_y < \varepsilon \le \varepsilon_u \end{cases} $$
where \(\sigma_y\) is the yield stress, \(\varepsilon_y\) is the yield strain, \(K_p\) is the plastic modulus, and \(\varepsilon_u\) is the ultimate strain. The plastic modulus was calibrated so that the true stress-true strain curve passes through the ultimate tensile strength at the maximum plastic strain.
The rubber gaskets were modeled using the hyperelastic Mooney-Rivlin strain energy potential, which is widely used for carbon-black-filled rubber materials. The strain energy density is written as
$$ W = C_{10} (\overline{I}_1 – 3) + C_{01} (\overline{I}_2 – 3) + \frac{1}{D_1} (J – 1)^2 $$
where \(\overline{I}_1\) and \(\overline{I}_2\) are the first and second deviatoric strain invariants, \(J\) is the volume ratio, and \(C_{10}\), \(C_{01}\), and \(D_1\) are material parameters. The two T-shaped gaskets were made of rubber compounds with different hardness levels, denoted as THA50 and THA88. For the THA50 compound, the material parameters were \(C_{10}=0.25\), \(C_{01}=0.33\), and \(D_1=0.034\). For the THA88 compound, the parameters were \(C_{10}=0.50\), \(C_{01}=1.06\), and \(D_1=0.012\). These values were selected based on available experimental data for rubber materials of similar hardness.
Interactions between the components were taken into account using surface-to-surface contact with a finite-sliding formulation. The coefficient of friction between two ductile iron casting surfaces was set to \(0.15\), while the coefficient of friction between ductile iron casting and rubber was set to \(0.5\). The normal contact behavior was modeled with the hard-contact algorithm, which prevents penetration of the slave surface into the master surface and transfers compressive normal stress when the surfaces are in contact. The tangential shear stress was transmitted according to the Coulomb friction law.
The boundary conditions were chosen to replicate the experimental setup as closely as possible. The end section of the socket pipe was fully fixed, i.e., all degrees of freedom of the nodes on that section were restrained. The cross-section at the free end of the inserted pipe was kinematically coupled to a reference point located at the centroid of the pipe section. A vertical concentrated force was applied at this reference point to simulate the action of the hydraulic jack. The magnitude of the force was increased gradually until the target joint rotation was reached. To avoid numerical convergence difficulties associated with rigid-body motion, the inserted pipe was initially held by a weak spring restraint in the horizontal direction, and the axial displacement was prescribed by moving the inserted pipe along the pipe axis before the bending load was applied.
The mesh of the model was refined in the regions where severe stress concentration was expected, particularly at the front end of the inserted pipe and at the contact interfaces with the socket wall and the rubber gaskets. A fine mesh was generated in these zones, while coarser elements were used in the parts of the pipe distant from the joint. Eight-node linear brick elements with reduced integration and hourglass control were used for the solid components. A mesh-sensitivity study was carried out to ensure that the computed moment-rotation response was not affected by the mesh density. The selected mesh size yielded convergent results for both the peak bending moment and the post-peak behavior.
Figure 4 shows a typical deformed configuration and the Mises stress distribution obtained from the finite element analysis. At small rotation angles, the inserted pipe remains in contact only with the rubber gasket, and the stress level in the ductile iron casting pipe is low. As the rotation angle increases, the front part of the inserted pipe is pressed against the inner wall of the socket, and a localized stress concentration develops at the contact area. When the local stress reaches the ultimate tensile strength of \(420\ \mathrm{MPa}\), the bending moment starts to level off, signaling the formation of a plastic zone at the contact region. This sequence of events is consistent with the three-stage shape of the measured moment-rotation curves, thereby providing a qualitative validation of the finite element model.
For quantitative validation, the computed moment-rotation curves were compared with the experimental curves for both DN100 and DN150 ductile iron casting socket joints. Good agreement was obtained for all axial displacement levels considered. Table 4 presents a comparison of the characteristic parameters derived from the numerical simulations and the tests. The initial stiffness values from the finite element model and the experiments are both in the range of \(0.1\) to \(0.3\ \mathrm{kN\cdot m/°}\). For the DN150 joints, the transition rotation \(\theta_1\) is about \(6.0^\circ\) in the simulation, which matches the measured range of \(5.6^\circ\)–\(6.0^\circ\). For the DN100 joints, the simulated \(\theta_1\) is approximately \(5.5^\circ\), while the measured values range from \(5.4^\circ\) to \(6.8^\circ\). The plateau rotation \(\theta_2\) from the simulation is slightly lower than the measured values, which can be attributed to the idealized material model and the simplified geometric representation of the socket inner wall. The ultimate bending moment is also reasonably predicted. For the DN150 joints, the simulated ultimate moment ranges from \(17.5\ \mathrm{kN\cdot m}\) at zero displacement to \(8.5\ \mathrm{kN\cdot m}\) at an axial displacement of \(2\ \mathrm{cm}\), whereas the measured values decrease from \(15.8\ \mathrm{kN\cdot m}\) to approximately \(11.7\ \mathrm{kN\cdot m}\). The numerical model tends to overestimate the initial capacity but predicts a more pronounced degradation rate. Nevertheless, the overall trend and the magnitude of the bending moment are satisfactorily captured by the simulation.
| Parameter | DN150 FEA | DN150 test | DN100 FEA | DN100 test |
|---|---|---|---|---|
| \(K_1\) (kN·m/°) | 0.17 | 0.1–0.3 | 0.05 | 0.1–0.3 |
| \(\theta_1\) (°) | 6.0 | 5.6–6.0 | 5.5 | 5.4–6.8 |
| \(\theta_2\) (°) | 9.5–11.8 | 12.1–14.1 | 9.0–12.2 | 10.5–12.5 |
| \(M_u\) (kN·m) | 17.5–8.5 | 15.8–11.7 | 12.2–5.8 | 13.1–9.6 |
Degradation Mechanism and Parametric Analysis
After validating the finite element model, a series of numerical simulations was performed to investigate the evolution of the flexural performance of ductile iron casting socket joints as a function of the axial displacement. The axial displacement was varied from \(0\) to \(2\ \mathrm{cm}\) in increments of \(0.5\ \mathrm{cm}\) for both DN100 and DN150 joints. The computed moment-rotation curves are illustrated in Figure 5. It can be seen that, for both pipe sizes, the initial stage of the curve remains almost unchanged, whereas the steep ascending branch shifts downward as the axial displacement increases. The ultimate bending moment and the secondary stiffness decrease continuously with increasing pull-out displacement.
This degradation can be explained by examining the contact condition between the inserted pipe and the socket inner wall. In the fully inserted state, a long segment of the inserted pipe remains inside the socket, and the bending deformation causes the pipe front to bear against the socket wall over a relatively large contact area. The large contact area promotes a greater contact force and thus a higher resisting bending moment. When the inserted pipe is pulled outward by an axial displacement \(L\), the effective length of the pipe inside the socket is reduced. As a result, the contact point between the pipe front and the socket inner wall moves closer to the socket entrance, and the available contact area is correspondingly diminished. The reduction of the contact area reduces the normal reaction force that can be developed for a given rotation, thereby lowering both the secondary stiffness and the ultimate bending moment. Since the geometric gap between the pipe and the socket is not affected by the axial displacement, the transition rotation \(\theta_1\) remains essentially unchanged, as observed in the experiments.
The parametric numerical results can be used to establish simple empirical relationships between the characteristic mechanical parameters and the axial displacement \(L\). The following linear expressions are proposed for the secondary stiffness \(K_2\) and the ultimate bending moment \(M_u\):
$$ K_2^{\text{DN100}}(L) = 160 – 20L \quad (\mathrm{kN\cdot cm/°}) $$
$$ M_u^{\text{DN100}}(L) = 1220 – 320L \quad (\mathrm{kN\cdot cm}) $$
$$ K_2^{\text{DN150}}(L) = 300 – 50L \quad (\mathrm{kN\cdot cm/°}) $$
$$ M_u^{\text{DN150}}(L) = 1750 – 450L \quad (\mathrm{kN\cdot cm}) $$
where \(L\) is the axial displacement expressed in centimeters. These formulas are valid for the range \(0 \le L \le 2\ \mathrm{cm}\). The corresponding characteristic rotations obtained from the simulations are summarized in Table 5, together with the initial stiffness values. The initial stiffness \(K_1\) can be treated as a constant within the investigated range, being approximately \(5\ \mathrm{kN\cdot cm/°}\) for the DN100 joint and \(17\ \mathrm{kN\cdot cm/°}\) for the DN150 joint. The transition rotation \(\theta_1\) is about \(5.5^\circ\) for DN100 and \(6.0^\circ\) for DN150. The plateau rotation \(\theta_2\) lies in the range of \(10.5^\circ\)–\(12.5^\circ\) for DN100 and \(9.5^\circ\)–\(11.8^\circ\) for DN150.
| Parameter | DN100 | DN150 | Unit |
|---|---|---|---|
| \(K_1\) | 5 | 17 | kN·cm/° |
| \(K_2(L)\) | 160 − 20L | 300 − 50L | kN·cm/° |
| \(\theta_1\) | 5.5 | 6.0 | ° |
| \(\theta_2\) | 10.5–12.5 | 9.5–11.8 | ° |
| \(M_u(L)\) | 1220 − 320L | 1750 − 450L | kN·cm |
It is useful to express the degradation in a normalized form. Let \(\eta_M\) denote the ratio of the ultimate moment at an axial displacement \(L\) to the ultimate moment at zero displacement, and let \(\eta_K\) denote the corresponding ratio for the secondary stiffness. Using the empirical formulas above, these ratios become
$$ \eta_M^{\text{DN100}} = 1 – \frac{320}{1220}L \approx 1 – 0.262L $$
$$ \eta_M^{\text{DN150}} = 1 – \frac{450}{1750}L \approx 1 – 0.257L $$
$$ \eta_K^{\text{DN100}} = 1 – \frac{20}{160}L = 1 – 0.125L $$
$$ \eta_K^{\text{DN150}} = 1 – \frac{50}{300}L \approx 1 – 0.167L $$
The normalized degradation rates are quite similar between the two pipe diameters. A ductile iron casting socket joint that has been pulled out by \(2\ \mathrm{cm}\) is expected to lose roughly half of its ultimate bending capacity and about one quarter to one third of its secondary bending stiffness. This emphasizes the importance of controlling the axial movement of ductile iron casting pipelines in the design and maintenance of water distribution networks.
The finite element results also reveal the evolution of the stress state inside the ductile iron casting socket joint during the combined tension-bending loading process. At a rotation angle below \(\theta_1\), the maximum stress occurs in the rubber gasket, which is compressed between the socket and the inserted pipe. The ductile iron casting components remain essentially elastic, and the contribution of the metallic contact to the bending moment is negligible. As the rotation increases beyond \(\theta_1\), the maximum stress zone jumps to the outer surface of the inserted pipe at the location of contact with the socket inner wall. The stress concentration grows rapidly and reaches the yield strength at a relatively small rotation. At the ultimate state, a large portion of the pipe wall at the contact location has exceeded the yield strength, and some elements have already reached the ultimate tensile strength. The fully plastic region continues to spread with increasing rotation, but the moment capacity does not increase further because the global equilibrium is governed by the plastic limit of the contact section. This mechanism is independent of the axial displacement; however, the axial displacement controls the length of the contact zone and therefore determines the value of the moment at which the plastic limit is reached.
The interaction between tension and bending can be visualized in terms of the load path in the axial force-bending moment plane. For a given level of axial displacement, the bending resistance of the ductile iron casting socket joint decreases as the axial displacement increases. This resembles the interaction curve of a metallic cross-section, but in the case of a socket joint, the interaction is governed by the changing contact condition rather than by the material yielding alone. Therefore, the simple empirical formulas proposed in this paper can be regarded as an interaction envelope for ductile iron casting socket joints in the space of axial displacement and bending moment.
Practical Implications and Discussion
The findings of the present study have direct implications for the safety assessment of ductile iron casting water pipelines. In pipeline condition evaluation, the bending capacity of a joint is usually determined from the moment-rotation curve obtained under a fully inserted condition. The present study demonstrates that if the joint has experienced a certain amount of axial pull-out displacement, the actual bending capacity can be considerably smaller than the capacity evaluated under the intact condition. Therefore, the axial displacement at each joint should be treated as a key inspection indicator. When a large axial displacement is detected at a ductile iron casting socket joint, the remaining bending capacity should be evaluated using the proposed degradation formulas instead of the nominal capacity value.
The degradation formulas can also be used to derive a limit-state criterion for ductile iron casting pipeline systems. For example, if a target safety factor is defined as the ratio of the available bending capacity to the maximum expected bending moment, the allowable axial displacement can be determined by solving the inequality
$$ M_{\text{applied}} \le \frac{1}{\gamma} M_u(L) $$
where \(\gamma\) is the resistance reduction factor and \(M_u(L)\) is the axial-displacement-dependent ultimate moment. Because the proposed formulas are linear in \(L\), the allowable axial displacement can be readily calculated for any given demand moment. This provides a simple yet rational tool for prioritizing inspection and rehabilitation activities in existing ductile iron casting water pipelines.
It should be noted that the empirical formulas were derived from tests and simulations with axial displacements up to \(2\ \mathrm{cm}\) for DN100 and \(3\ \mathrm{cm}\) for DN150. For larger axial displacements, the linear degradation trend may not hold because the inserted pipe may lose sufficient engagement with the socket and the rubber gasket, leading to leakage or even joint separation. The ultimate displacement associated with leakage was not specifically addressed in the present work. Further investigation is needed to incorporate the leakage limit into the combined tension-bending failure envelope of ductile iron casting socket joints.
The present study also highlights the role of the rubber gasket in the flexural behavior of ductile iron casting socket joints. The gasket not only provides water tightness but also contributes to the initial rotational resistance. In the design of ductile iron casting pipelines, the contribution of the gasket is often neglected, and the joint is assumed to rotate freely until metallic contact occurs. The test results, however, show that the initial stiffness is not zero and that the gasket provides a non-negligible resistance at small rotations. For a pipeline subjected to cyclic or dynamic loading, such as traffic-induced vibration, the initial stiffness may affect the stress distribution along the pipeline and the fatigue life of the joint. The numerical model developed in this study can be used as a basis for further dynamic analysis of ductile iron casting pipeline systems with socket joints.
Conclusions
In this paper, the mechanical performance of ductile iron casting socket joints under combined tension-bending actions was investigated through a combination of full-scale in-situ tests and three-dimensional finite element simulations. The key findings and conclusions are as follows.
First, the moment-rotation response of ductile iron casting socket joints can be divided into three stages: an initial gently rising branch, a steeply rising branch, and a post-peak plateau. The three stages are associated with three distinct load-transfer mechanisms: compression of the rubber gasket; metallic contact between the inserted pipe and the socket inner wall; and plastic deformation of the pipe wall at the contact zone.
Second, the axial pull-out displacement has a pronounced effect on the flexural capacity of ductile iron casting socket joints. For DN150 joints, the ultimate bending moment decreased from \(15.8\ \mathrm{kN\cdot m}\) at zero displacement to approximately \(11.7\ \mathrm{kN\cdot m}\) when the axial displacement exceeded \(1.5\ \mathrm{cm}\). For DN100 joints, the ultimate bending moment decreased from \(13.1\ \mathrm{kN\cdot m}\) to approximately \(9.6\ \mathrm{kN\cdot m}\) for axial displacements between \(1\ \mathrm{cm}\) and \(2\ \mathrm{cm}\). The secondary bending stiffness exhibited a similar reduction, while the initial stiffness and the characteristic transition rotations remained nearly unchanged.
Third, the degradation of the bending performance is caused by the reduction of the contact area between the inserted pipe and the socket inner wall as a result of the axial pull-out movement. The finite element simulations reveal that the contact zone is the major source of the bending resistance in the ascending branch of the moment-rotation curve, and a shorter engagement length inevitably leads to a lower resisting moment.
Fourth, simple linear empirical formulas were proposed to quantify the degradation of the secondary stiffness and the ultimate bending moment as functions of the axial displacement. These formulas are well suited for engineering practice and can be used in the safety assessment and condition evaluation of ductile iron casting water pipelines. The normalized degradation rate of the ultimate bending moment is approximately \(0.26\ \mathrm{cm^{-1}}\) for both DN100 and DN150 joints, which implies that an axial pull-out of \(2\ \mathrm{cm}\) can reduce the flexural capacity by more than half.
Finally, the combined experimental and numerical approach adopted in this study provides a reliable framework for characterizing the coupled behavior of ductile iron casting socket joints. The proposed empirical relationships, together with the finite element model, can serve as a basis for developing practical guidelines for the inspection, assessment, and rehabilitation of ductile iron casting pipeline systems subjected to combined soil movement and temperature-induced deformation. Future work should extend the present methodology to other pipe diameters, other gasket materials, and more complex loading paths, including cyclic bending and internal pressure effects.