Wave-induced response and fatigue analysis of free-spanning submarine cables with nonlinear bending stiffness

Introduction: Under long-term wave–current interactions, the free-spanning submarine cables are prone to fatigue damage. Owing to the presence of helically wound layers within the submarine cable, its bending stiffness exhibits pronounced nonlinear characteristics, which in turn exert a substantial influence on its dynamic response and fatigue life. Consequently, this study investigates the influence of nonlinear bending stiffness on the wave-induced response and fatigue characteristics of submarine cables.

Methods: First, a full-scale local finite element model is developed in ABAQUS to derive the nonlinear bending stiffness of the cable, which is validated against theoretical formulations. Subsequently, the slip stiffness, stick stiffness, and nonlinear bending stiffness models are incorporated into OrcaFlex to perform global dynamic analyses under combined wave–current conditions. The effects of the three stiffness models on the cable’s motion response, load transfer characteristics, and fatigue performance are then systematically evaluated.

Results: The results indicate that the motion and load response of short-span cables are highly sensitive to nonlinear bending stiffness. As the free-span length increases, the system behavior gradually transitions from a bending-dominated regime to a tension-dominated one. Moreover, the lead sheath is the most susceptible to fatigue failure. Compared to the nonlinear bending stiffness model, employing slip stiffness underestimates the fatigue life by 39%, whereas the use of stick stiffness leads to an overestimation of fatigue life by 25%.

Discussion: Accounting for nonlinear bending stiffness is crucial for enhancing the long-term operational reliability of free-spanning submarine cables in structural design and fatigue assessment.

1 Introduction

As one of the most critical components in power transmission systems, submarine cables are widely employed in island power supply, intergrid connections, and offshore wind power transmission (Taormina et al., 2018). With the continuous development of inter–island power grids and the growing exploitation of marine resources, the laying length of submarine cables has been steadily increasing (Purvins et al., 2018). After being laid on the seabed, submarine cables may develop free–span sections due to complex seabed topography and the scouring effects under waves and currents (Li et al., 2023). For example, in the Qiongzhou Strait cross–sea power transmission project, the free–span lengths of the oil–filled submarine cables range from several meters to several tens of meters (Hedlund, 2015; Zhu et al., 2023). If the free–spanning sections of a submarine cable are not identified and mitigated in time, prolonged exposure to hydrodynamic loads can induce wave–induced motions and vortex–induced vibrations. These dynamic effects may lead to progressive mechanical degradation and fatigue failure of structural components, such as the lead sheath, ultimately threatening the long–term integrity and operational reliability of the cable system (Worzyk, 2009; Hedlund, 2015). Considering the high construction and maintenance costs associated with submarine cables, any failure during the operational stage can lead to substantial expenditure of human and material resources for repair and reconstruction (Yang et al., 2026). Moreover, such failures may also pose a significant risk of environmental pollution (Yoon and Na, 2013). Therefore, it is of great importance to investigate the wave–induced global response and fatigue behavior of free–spanning submarine cables.

As a primary cause of submarine cable failures, free–span formation has long been a focal point of research. The dynamic response of a free-spanning submarine cable can be broadly classified into high-frequency, small-amplitude vortex-induced vibrations triggered by vortex shedding, and low-frequency, large-amplitude wave-induced motions driven directly by the wave orbital kinematics (Zhu et al., 2023). Extensive studies have been conducted on the vibration mechanisms and dynamic response characteristics of free–spanning cables under current–induced excitation. Tan et al. (2023) conducted a fluid–structure interaction of a free–spanning submarine cable through finite element analysis. Their study revealed that the cable exhibited galloping and lock–in vibrations at different flow velocities. When resonance occurred, the transverse vibration amplitude increased significantly. Zhang et al. (2023) established a three–dimensional finite element model of a ±500 kV submarine cable with a suspended span to investigate the effects of ocean current scour and vortex–induced vibration, providing valuable insights for the evaluation of submarine cable operational conditions. Li et al. (2024) investigated the vortex–induced vibration characteristics of free–spanning submarine cables using a three–dimensional numerical model that combines the finite difference and finite element methods, accurately capturing the transverse vibration response of the cable. Zhu et al. (2023) conducted experimental investigations on the vortex–induced vibrations of freely spanning submarine cables under steady currents, revealing that the sag has a significant impact on the modal characteristics and vibration response of the cable. To date, researches by scholars worldwide has predominantly focused on the vortex–induced vibrations of free–spanning submarine cables, whereas studies on the wave–induced global response of free–spanning submarine cables remain comparatively scarce. Given that nonlinear bending stiffness becomes most pronounced under large geometric deformation and predominantly governs the structural response, the present study concentrates on the wave-induced dynamic behaviour.

Accurate analysis of the dynamic response and fatigue characteristics of submarine cables requires knowledge of their true bending stiffness. Yang et al. (2018) investigated the dynamic response patterns and fatigue behavior of cables under combined wave–current loading using systems with varying linear bending stiffness. Their study demonstrated that the bending stiffness parameters significantly influence the curvature distribution and stress response of the cables. Munro et al. (2024) developed a multiphysics–coupled numerical model of submarine cables to predict their dynamic response, damage evolution, and fatigue life under combined wave–current loading. However, their model did not account for the nonlinear characteristics of cable bending stiffness. Hedlund (2015) showed the limitations of conventional vortex–induced vibration analysis in predicting fatigue life. This conventional approach assumes constant bending stiffness. In contrast, an improved model that accounts for viscoelastic effects provides more accurate predictions. Most previous studies have simplified cable bending characteristics as linear, without fully accounting for their true bending behavior.

The submarine cable primarily consists of helically wound armor layers and multiple cylindrical layers, resulting in a complex internal structure that imparts intricate mechanical characteristics (Fang et al., 2025). Due to the residual stress generated during manufacturing and the radial water pressure sustained in the marine environment, the armor wires exhibit a distinct stick–slip behavior during cable bending (Fang et al., 2023). This causes the cable to exhibit pronounced nonlinear bending characteristics (Ramos Jr and Pesce, 2004; Coser et al., 2016; Wu et al., 2022; Ménard and Cartraud, 2023). The study of nonlinear bending behavior of cables can draw on research conducted on flexible pipes and flexible risers. Ramos Jr and Pesce (2004) investigated the structural behavior of flexible risers under combined bending, torsion, axial tension, and internal/external pressure. They defined four pipeline configurations and focused on the fully slipping bending configuration. By linearizing the nonlinear equations, they derived a system of linear equations with unknowns and formulated expressions for the equivalent bending stiffness, clarifying the contributions of individual layers. Wu et al. (2022) mapped the overall structural curvature into the local coordinate system of the steel wires, obtaining three curvature components in different directions and the relationship between curvature and bending moment. Based on these relationships, they further derived analytical expressions for bending stiffness in both the no–slip and fully slipping segments. Li et al. (2022) incorporated the nonlinear hysteretic relationship between bending moment and curvature into the global response analysis of flexible risers. By accurately evaluating the bending stiffness within the corresponding curvature range, they proposed an efficient fatigue analysis approach for unbonded flexible risers.

In addition, incorporating nonlinear bending stiffness in the mechanical analysis of submarine cables enables a more accurate prediction of deformation and internal stress distribution, which is essential for fatigue assessment, structural design, and reliability evaluation (Zhang et al., 2021). Therefore, accurately determining the bending stiffness of the submarine cable is essential when analyzing its wave–induced global response and fatigue life. In current studies on the global response and fatigue characteristics of free–spanning cables, the bending stiffness of the submarine cable is mostly assumed to be linear, based on either the fully slipping or fully bonded assumption, rather than reflecting its actual nonlinear behavior (Van Dongen, 2025). This may lead to discrepancies between the analyzed wave–induced response and fatigue predictions of the submarine cable and its actual behavior. Therefore, this study incorporates both linear and actual nonlinear bending stiffness into the analysis of the global dynamic response of free–spanning cables. By comparing the response under different stiffness assumptions, the effects on cable fatigue behavior are examined, enabling a more rational fatigue life prediction method under long–term wave–current conditions.

This study establishes a local full–scale finite element model of the cable and a global dynamic response model through numerical simulation to analyze its wave–induced response and fatigue behavior. This study also systematically investigates the sensitivity of the submarine cable’s tension, bending moment, and curvature to variations in bending stiffness. The organization of this paper is as follows: Section 2 establishes a local finite element model of the cable using ABAQUS to obtain its axial and nonlinear bending stiffness. Section 3 inputs the axial and nonlinear bending stiffness obtained in Section 2 into OrcaFlex to build a global dynamic response model of the free–spanning submarine cable. Section 4 performs a comparative analysis using stick stiffness, slip stiffness, and nonlinear bending stiffness to investigate the effects of different bending stiffness models on the cable’s dynamic response and fatigue life predictions. The overall computational workflow of this study is illustrated as shown in Figure 1.

Figure 1

Figure 1. Flowchart of the computational procedure.

2 Local modeling

In this section, a full–scale local finite element model is developed to determine the axial tensile stiffness and nonlinear bending stiffness of the submarine cable, which is validated by comparing with the results obtained by theoretical formulae. The axial tensile stiffness and nonlinear bending stiffness of the submarine cable obtained in this section will be implemented into the analysis of subsequent global dynamic response.

2.1 Structure and material parameters

The research focuses on a self–contained oil–filled submarine cable with a composite structure. The cross–sectional structure primarily consists of annular components fulfilling various functions and helically wound armored copper wires, as shown in Figure 2. In order to reduce the computational costs in the calculation of stiffness by finite element method, the cable structure is simplified by neglecting several thin and low–strength layers, such as the conductor shield, insulation shield, and anti–corrosion layer. Table 1 lists the structural and material parameters of the simplified cable model. The armor layer consists of 52 copper wires with a rectangular cross section of 7.4 mm $\times$ 2.5 mm, each wounds helically around the central axis of cable with a lay angle of 8.86° and a pitch of 2,570 mm, corresponding to the axial advance of one complete turn of the helix.

Figure 2

Figure 2. Structure of submarine cable. (a) Cross–section of cable. (b) A real photo of cable.

Table 1

Table 1. Structural and material parameters of submarine cable.

2.2 Finite element model

Considering both geometric and contact nonlinearities, a full–scale local finite element model of the submarine cable is developed in ABAQUS to capture the cable’s nonlinear bending response. The model length is 3 m, exceeding one helical pitch of the armor wires. All layers of the cable are modeled using C3D8I elements to prevent hourglass modes. The cable geometry and mesh configuration are shown in Figure 3, which contains a total of 113,400 elements and 186,032 nodes.

Figure 3

Figure 3. Finite element model of the submarine cable. (a) Geometric model. (b) Mesh.

The tangential and normal contact interactions between the layers are fully considered in the model. To enhance convergence of the finite element model, a node–to–surface discretization scheme is adopted for simulating interlayer contact. The normal contact behavior is defined using the penalty method, with a linear contact stiffness of 2000 N/mm³ to prevent excessive penetration. In the tangential direction, a friction coefficient of 0.20 and an elastic slip limit of 0.005 are applied to all contact definitions. To ensure uniform end loading, two reference points are created at the centers of the cross–sections at both ends of the model. The degrees of freedom (DOFs) of all cross–section nodes are coupled with the corresponding reference points, so that all cable layers and wire ends are rigidly connected to them. The boundary conditions for the cases of tensile and bending loads are summarized in Table 2, where “1” indicates the corresponding DOF is fixed and “0” denotes it is free.

Table 2

Table 2. The boundary conditions for tension and bending case.

Two different mesh sizes are tested, as shown in Figure 4. The simulated tensile stiffness based on the coarse mesh and the fine mesh size are 428.5 $\text{MN}/\mathrm{m}$ and 430.9 $\text{MN}/\mathrm{m}$, respectively, and the corresponding bending stiffnesses are 162.0 $\text{kN}·{\mathrm{m}}^2$ and 165.8 $\text{kN}·{\mathrm{m}}^2$, respectively. The error between two mesh sizes regarding both tensile stiffness and bending stiffness is less than 2.3%. Therefore, in order to minimize the computation time, the coarse mesh size is used in the study.

Figure 4

Figure 4. Two different mesh sizes for the finite element model. (a) Coarse mesh. (b) Fine mesh.

As shown in Figure 5a, an axial tensile load is applied to the left–end reference point in the Z–direction, and the resulting axial tensile response of the model is plotted in Figure 6a. The mechanical response under tension is essentially linear. To account for the effects of residual stress and radial hydrostatic pressure within the cable, an external pressure of P = 1 MPa is applied to the outer surface of the cable. In the bending case (Figure 5b), a rotation about the X–axis is imposed at the left–end reference point. The resulting nonlinear bending response is given in Figure 6b. The nonlinear bending stiffness arises primarily from the stick–slip behavior of the armor wires. Under the applied external pressure and at small bending curvatures, the armor wires remain bonded and do not slip, so the bending stiffness ($K_{stick}$) is maximal. As the curvature increases, the wires progressively begin to slip. The bending stiffness ($K_{slip}$) reaches its minimum value once all wires have slipped. Crucially, the tensile and nonlinear bending data obtained from these finite element simulations are used as input for the subsequent global cable model to capture the true global response of the suspended submarine cable.

Figure 5

Figure 5. Boundary conditions and loading. (a) Tension. (b) Bending.

Figure 6

Figure 6. Load response under different cases. (a) Tension. (b) Bending.

2.3 Validation

To validate the local finite element model, based on the structural and material parameters of the submarine cable in Table 1, the tensile stiffness and slip stiffness of the cable is calculated in this section through theoretical formulae. The tensile stiffness $K_A$ is computed using the analytical formula derived by Knapp (Knapp, 1979):

$K_A={\displaystyle \sum _{i=1}^n}\left[A_i{E}_i\left(1-\frac{\mathrm{\Theta }}{2}\frac{R_c}{R_i}{\tan }^2{\alpha }_i\right){\cos }^3{\alpha }_i\right]+E_c{A}_c(1)$

where $A_i$, $E_i$, $R_i$ and ${\alpha }_i$ are the cross–sectional area, elastic modulus, helical winding radius, and helix angle of the i–th wire, respectively, $n$ denotes the total number of wires, $\mathrm{\Theta }$ is the equivalent Poisson’s ratio parameter, $R_c$ and $E_c{A}_c$ denote the radius and tensile stiffness of the circular ring components.

The slip stiffness is calculated using the Witz model (Witz and Tan, 1992), which disregards frictional effects and assumes uniform bending deformation among components within the same layer. The calculation formulas are as follows:

$K_{slip}={\displaystyle \sum _{i=1}^n}\frac{1}{2}\left[E_i{I}_n+E_i{I}_b{\cos }^2{\alpha }_i\right]+E{I}_0(2)$

where $E{I}_0$ represents the bending stiffness of the ring component, $E_i{I}_n$ and $E_i{I}_b$ represent the bending stiffness of the i–th wire in the normal and bi–normal directions respectively.

The tensile stiffness and slip stiffness of the submarine cable obtained from the numerical model are compared with the theoretical model results in Table 3. The theoretical model outcomes are calculated using Equations 1, 2. The errors between the theoretical and finite element results for both tensile and slip stiffness are within 3%, illustrating that the stiffnesses from these two methods agree quite well with each other. This validates the accuracy of the finite element model and confirms that it provides reliable stiffness data for computing the global response of the cable.

Table 3

Table 3. Comparison between finite element model and theoretical data.

3 Global modelling

Based on the mechanical parameters obtained from the local finite element model in Section 2, this section establishes a global dynamic model for the free–spanning submarine cable using OrcaFlex. This model is developed to investigate the influence of different types of bending stiffnesses on the cable’s global motion behavior, stress distribution, and fatigue performance under combined wave–current loading. Thus, the global dynamic model first incorporates $K_A$, $K_{slip}$, $K_{stick}$, as well as the nonlinear bending moment–curvature curve obtained from the finite element model in Section 2. On this basis, the hydrodynamic response of submarine cables is simulated under the conditions of slip stiffness, stick stiffness, and nonlinear bending stiffness, which corresponding to the results of $K_{slip}$, $K_{stick}$, as well as the nonlinear bending moment–curvature curve, respectively.

In Figure 7, the x–axis represents the axial direction of the submarine cable, the y–axis points horizontally and perpendicular to the cable in the direction of wave and current propagation, and the z–axis is oriented vertically upward. As shown in Figure 7, the cable with a free–spanning section is laid on the seabed. The detailed geometric parameters and mechanical properties of the submarine cable are presented in Table 4. Considering the complexity of the mechanical behavior of the free–spanning cable resting above the seabed, the model is appropriately simplified, and the following assumptions are adopted.

1. The cross–sectional properties of the submarine cable are assumed to be uniform along its entire length, and the self–weight is considered constant per unit length.

2. Except for the local seabed faults that forms the free–span, the remaining seabed is assumed to be flat and aligned along the same horizontal level.

3. In OrcaFlex simulations, applying hinged boundary conditions at the cable ends neglects the transmission of bending moments at the boundaries. However, this factor has minimal impact on the study of dynamic response and fatigue characteristics. Therefore, the hinged assumption simplifies the model without significantly compromising accuracy.

Figure 7

Figure 7. Numerical model developed in OrcaFlex.

Table 4

Table 4. Basic parameters of the submarine cable.

To accurately evaluate the wave–induced dynamic response of the submarine cable, it is essential to specify the hydrodynamic coefficients with precision, namely, the drag coefficient $C_D$ and the inertia coefficient $C_M$. According to DNV–RP–C205, for rough cylindrical bodies such as submarine cables with sheathing, $C_D$ is taken as 1.0 and $C_A$ as 1.0. The relationship between $C_M$ and $C_A$ can be expressed as Equation 3 (Det Norske, 2017):

$C_M=1+C_A(3)$

where $C_A$ is is the non–dimensional added mass. $C_M$ is calculated to be 2.0 $.$

The JONSWAP spectrum is employed to generate the wave sequence in this study. Two wave–current conditions are considered in this study, and the detailed parameters are listed in Table 5.

Table 5

Table 5. Wave and current conditions for the present simulation.

4 Results and discussion

Based on the global dynamic model established in Section 3, this section examines the effect of nonlinear bending stiffness on the wave–induced response and fatigue behavior of the submarine cable.

4.1 Motion response

Taking the 1–year return period wave condition as an illustrative case, Figure 8 presents the displacement amplitude curves of the free–spanning submarine cable under different stiffness conditions. Figures 8a,b corresponds to the cases of free–span length of 5 m and 10 m, respectively. As shown in the figure, within the xOy plane, the displacement amplitudes associated with the stick stiffness and nonlinear bending stiffness are relatively close to each other and remain consistently lower than those observed under the slip stiffness condition. The reason for this phenomenon is that under 1–year return period wave–current condition, the curvature of the cable is small and the cable is in the stick stage, resulting in a close approximation between the nonlinear bending stiffness and the stick stiffness. In addition, the slip stiffness of the cable is the smallest, making the cable more prone to lateral deformation. Further analysis reveals that the difference in lateral deformation between different stiffnesses is smaller in Figure 8b than that in Figure 8a. This is because the increased free–span length enhances the influence of axial stiffness on the cable geometry, weakening the effect of the nonlinear bending stiffness.

Figure 8

Figure 8. Displacement amplitude curves of the free–spanning cable under different stiffness conditions. (a) $L=5\mathrm{m}$. (b) $L=10\mathrm{m}$. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

The time–history curves of maximum von mises stress for the free–spanning submarine cable under different stiffness conditions are shown in Figure 9, in which wave–current conditions are consistent with the results in Figure 8. Under the nonlinear bending stiffness and stick stiffness conditions, the maximum von mises stress of the submarine cable fluctuates around approximately 6,350 kPa. Under the slip stiffness condition, it fluctuates around 5,100 kPa. The stresses for the stick and nonlinear bending stiffness conditions are similar and both higher than those for the slip stiffness condition. This phenomenon is consistent with the analysis presented in Figure 8. Under the combined wave–current condition with a 1–year return period, the cable predominantly remains in the stick regime, where bending deformation is strongly constrained and the sliding has not yet occurred. Therefore, under this operating condition, the case of the nonlinear bending stiffness is nearly identical to the stick stiffness and they both exceed the slip stiffness. Consequently, the stress distributions of the two models are almost the same, while the slip stiffness model exhibits lower stress.

Figure 9

Figure 9. Time–history curves of maximum von mises stress for the free–spanning submarine cable under different stiffness conditions. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$; $L=10\mathrm{m}$).

4.2 Load analysis

Figure 10 depicts the distribution of the maximum axial tension in the free–spanning submarine cable under different stiffness conditions. As shown in the figure, the maximum tension under the slip stiffness condition is markedly higher than that under both the stick and nonlinear bending stiffness conditions. The difference between the maximum tensions for the stick and nonlinear bending stiffness cases is small, at about 1.3%. This result highlights the dominant role of the stick state in the stress–bearing behavior of the free–spanning cable. For the case of $L=5\mathrm{m}$, the axial tension variation among different types of slip states reaches nearly 113%, whereas for $L=10\mathrm{m}$, the variation decreases to about 15.5%. This indicates that larger spans are less sensitive to stiffness changes induced by the slip state. Figure 15 further confirms this phenomenon.

Figure 10

Figure 10. Distribution of maximum tension along the axial direction of the free–spanning submarine cable under different stiffness conditions. (a) Maximum tension for $L=5\mathrm{m}$. (b) Maximum tension for $L=10\mathrm{m}$. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Figure 11 illustrates the axial distribution of maximum curvature and bending moment under different stiffness conditions. Specifically, Figures 11a,b correspond to the case of $L=5\mathrm{m}$, while Figures 11c,d represent $L=10\mathrm{m}$. As shown in the figure, the slip stiffness condition yields the largest curvature. At the mid–span under a free–span length of 10 m, the maximum curvature reaches 0.008 $\text{rad}/\mathrm{m}$. Under the stick stiffness condition, the bending moment attains the highest value. The nonlinear bending stiffness model shows a similar magnitude and distribution to case of the stick stiffness condition. In contrast, the slip stiffness condition produces the smallest bending moment, since the bending stiffness under the slip condition is much smaller than that for the stick condition. A comparison between Figures 11a,c shows that the axial distribution pattern of curvature is generally consistent for both $L=5\mathrm{m}$ and $L=10\mathrm{m}$. However, the curvature magnitude for $L=10\mathrm{m}$ is greater than that for $L=5\mathrm{m}$. It can also be observed from Figures 11a,c that, for the case of $L=5\mathrm{m}$, the curvature extremum under the slip stiffness condition is approximately 40% greater than that under the nonlinear bending stiffness condition. For $L=10\mathrm{m}$, this difference decreases to about 11%. This finding indicates that, under comparable axial tension, the curvature response becomes less sensitive to variations in the bending stiffness as the free–span length increases. This phenomenon can be explained by the dominance of bending effects at shorter spans. When the free–span length is small, changes in bending stiffness strongly affect the local deflection and curvature distribution, leading to marked differences among different types of stiffness models. As the span length increases to 10 m, the axial tension becomes dominant in the overall dynamic behavior, and the system gradually transits into a tension–dominated regime. Consequently, the influence of bending stiffness on the curvature distribution is substantially weakened, leading to nearly consistent curvature profiles under different stiffness conditions. Figure 16a further confirms this phenomenon.

Figure 11

Figure 11. Axial distribution of the bending behavior under different stiffness conditions. (a) Maximum curvature for $L=5\mathrm{m}$. (b) Maximum bending moment for $L=5\mathrm{m}$. (c) Maximum curvature for $L=10\mathrm{m}$. (d) Maximum bending moment for $L=10\mathrm{m}$. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Figure 12 presents the axial distribution of the maximum curvature and bending moment of a 120 m free–spanning cable under different stiffness conditions. As shown in Figure 12a, the maximum curvature under different stiffness conditions all exhibits similar distribution trends. The curvature at the end of the suspended span is zero, and the maximum curvature increases rapidly as it moves towards the mid–span. Except for the end region, the maximum curvature of the suspended span remains basically at the same level. The change in the bending stiffness has little effect on the maximum curvature distribution, because the increase in the free–span length causes the mechanical behavior of the cable to gradually shift from bending–dominated to tension–dominated. Combined with the analysis of Figure 11, the maximum curvature of the $L=120\mathrm{m}$ is approximately 0.0016 $\text{rad}/\mathrm{m}$, which is significantly smaller than those of $L=5\mathrm{m}$ and $L=10\mathrm{m}$. This is also because the longer span length enhances the contribution of axial tension to the overall structural stiffness, effectively suppressing local bending deformation.

Figure 12

Figure 12. Axial distribution of the bending behavior of a 120 m free–spanning cable under different stiffness conditions. (a) Maximum curvature distribution. (b) Maximum bending moment distribution. ($L=120\mathrm{m}$; $U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Taking the fifty–year return period wave–current condition as an example, Figure 13 illustrates the distribution of maximum tension along the axial direction of the free–spanning submarine cable. Under this condition, the effective tension increases significantly, with the maximum value reaching 348.5 kN. Moreover, compared with the case of $L=5\mathrm{m}$, the maximum effective tension at $L=10\mathrm{m}$ is also markedly higher. This increase occurs because both the self–weight and hydrodynamic loads act over a longer span. As the free–span length extends to 10 m, these loads become more significant. Consequently, the cable exhibits greater mid–span deflection and overall curvature, leading to higher axial tensile strain and, ultimately, an increased effective tension.

Figure 13

Figure 13. Distribution of maximum tension along the axial direction of the free spanning submarine cable under different stiffness conditions. (a) Maximum tension for $L=5\mathrm{m}$. (b) Maximum tension for $L=10\mathrm{m}$. ($U=1.4\mathrm{m}/\mathrm{s}$; $H=12.6\mathrm{m}$; $T=24.1\mathrm{s}$).

Under the fifty–year return period wave–current condition, Figure 14 illustrates the axial distributions of maximum curvature and bending moment for different stiffness models. As shown in the figure, for $L=5\mathrm{m}$, the curvature distribution remains similar to that under the one–year return condition. However, the maximum curvature increases to 0.035 $\text{rad}/\mathrm{m}$. As observed from the figure, the curvature under the nonlinear bending stiffness condition becomes closer to that under slip stiffness in the mid–span region (2–3 m). This occurs because the critical curvature for the transition from the stick state to the slip state is 0.025 $\text{rad}/\mathrm{m}$. Once the cable curvature exceeds this threshold, it shifts toward the slip regime, leading to similar response between the slip and nonlinear bending stiffness conditions. For $L=10\mathrm{m}$, the curvature distributions under the three stiffness conditions are relatively similar. The nonlinear bending stiffness case exhibits a distribution pattern closer to that of the stick stiffness condition. This is because the maximum curvature at this span remains below 0.025 $\text{rad}/\mathrm{m}$, indicating that the cable as a whole is still in the stick regime. For both the 5 m and 10 m free–span lengths, the bending moment distributions show a clear distinction among different stiffness models. The stick stiffness condition yields the largest bending moment, followed by the nonlinear bending stiffness, while the slip stiffness condition produces the smallest.

Figure 14

Figure 14. Axial distribution of the maximum curvature and bending moment under different stiffness conditions. (a) Maximum curvature for $L=5\mathrm{m}$. (b) Maximum bending moment for $L=5\mathrm{m}$. (c) Maximum curvature for $L=10\mathrm{m}$. (d) Maximum bending moment for $L=10\mathrm{m}$. ($U=1.4\mathrm{m}/\mathrm{s}$; $H=12.6\mathrm{m}$; $T=24.1\mathrm{s}$).

Taking the one–year return period wave–current condition as an example, Figure 15 illustrates the variation of maximum tension with free–span length under different stiffness conditions. As shown in the figure, the effective tension of the submarine cable increases with the free–span length. This is because as the span length increases, both the cable’s self-weight and the hydrodynamic load increase, leading to the increased tension. It can also be observed that the maximum effective tension remains nearly unchanged across the different bending stiffness models. This indicates that the bending stiffness has only a negligible influence on the overall tension response of the cable. This negligible influence can be attributed to the fact that the free–spanning submarine cable operates in a tension–dominated regime. Under the same load conditions, axial tension is mainly determined by geometric configuration and axial stiffness, rather than local bending stiffness. Consequently, variations in the bending stiffness mainly affect the local curvature and bending–moment response, while exerting only a minimal impact on the global axial force distribution. The overall tensile response of the cable is mainly governed by axial extension and boundary constraints, with minimal sensitivity to bending stiffness.

Figure 15

Figure 15. Variation of maximum tension with free–span length under different stiffness conditions. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Figure 16 illustrates the variation of maximum curvature and bending moment with free–span length under different stiffness conditions. As shown in the figure, the extrema of curvature and bending moment of the cable first increase and then decrease with the growth of the free-span length. In Figure 16a, it can also be found that the slip stiffness condition yields the largest curvature, reaching 0.00834 $\text{rad}/\mathrm{m}$. The difference in maximum curvature among different stiffness models is pronounced for free–span lengths between 0 m and 20 m. However, the values become nearly identical as the span increases from 20 m to 120 m, which is consistent with the preceding analysis. In Figure 16b, the bending moment under the slip stiffness condition is evidently smaller. The stick and nonlinear bending stiffness conditions produce similar and comparatively larger bending moments. This behavior arises because the cable curvature remains below the critical threshold required for the transition to the slip regime.

Figure 16

Figure 16. Variation of maximum curvature and bending moment with free–span length under different stiffness conditions. (a) Curvature distribution. (b) Bending moment distribution. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

4.3 Fatigue analysis

The fatigue analysis of free–spanning submarine cables is based on metal fatigue theory and employs the stress–life (S–N) approach. The stress–life method is based on the S–N curve, which serves as a fundamental model in the fatigue analysis, characterizing the fatigue behavior of materials or structures under cyclic loading. This curve describes the relationship between the stress amplitude (S) and the number of cycles to fatigue failure (N). Its principle is simple: lower stress amplitudes correspond to a greater number of cycles before failure. Its mathematical form is typically expressed as a logarithmic representation of a power function, exhibiting a log–linear relationship, and can be expressed as Equation 4:

$\mathit{lg}{N}_i=a-mlg\left(\Delta {\sigma }_i\right)(4)$

where $\Delta {\sigma }_i$ denotes the alternating stress amplitude, $N_i$ represents the number of cycles to failure corresponding to the stress amplitude $\Delta {\sigma }_i$, $a$ and $m$ are material parameters typically determined from fatigue tests.

In the submarine cable, the copper conductor and armor wires bear the mechanical loads and are thus critical to the cable’s structural integrity. The lead–alloy sheath is more prone to fatigue damage. Accordingly, this section focuses on three key components of the submarine cable: the copper conductor, the armor wires, and the lead–alloy sheath. All three components are modelled under the assumption of linear elasticity. Both the conductor and armor wires are made of copper, while the sheath is composed of lead. The S–N curve parameters for lead and copper are presented in Table 6.

Table 6

Table 6. S–N curve parameters for copper and lead.

To obtain the spatiotemporal variations of stress in each layer of the submarine cable, Equation 5 is employed:

$S\left(x,y\right)=K_{t}T\left(x,t\right)+K_c\left[C_y\left(x,y\right)\sin \theta -C_z\left(x,t\right)\cos \theta \right](5)$

where, $T\left(x,t\right)$ denotes the axial tension of the submarine cable, $C_y\left(x,t\right)$ and $C_z\left(x,t\right)$ represent the curvatures of the cable’s cross–section in the y and z directions, $\theta$ is the circumferential angle of the cable cross–section. $K_t$ and $K_c$ are the tensile and curvature stress coefficients of each material layer.

The coefficients $K_t$ and $K_c$ are determined by fitting the stress–load curves obtained from the local model established in Section 2. The stress–load curves under different stiffness conditions are presented in Figures 17–19. As shown in Figures 17–19, the stresses of all components vary linearly with the axial tension. Except for the armor wires, the stresses of the other components also exhibit a linear relationship with curvature. The stress–curvature relationship of the armor wires exhibits nonlinear behavior. For computational convenience, it is approximated by two linear segments. Specifically, $K_c$ is taken as 1,149.12 $\text{MPa}/\left(\text{rad}/\mathrm{m}\right)$ for curvatures exceeding 0.025 $\text{rad}/\mathrm{m}$, and 4,505.56 $\text{MPa}/\left(\text{rad}/\mathrm{m}\right)$ for curvatures below 0.025 $\text{rad}/\mathrm{m}$. The resulting $K_t$ and $K_c$ values for each material under the various stiffness conditions, obtained via linear regression, are summarized in Table 7.

Figure 17

Figure 17. Stress–Load curves of cable components under slip stiffness. (a) Stress–Tension curve. (b) Stress–Curvature curve.

Figure 18

Figure 18. Stress–Load curves of cable components under stick stiffness. (a) Stress–Tension curve. (b) Stress–Curvature curve.

Figure 19

Figure 19. Stress–Load curves of cable components under nonlinear bending stiffness. (a) Stress–Tension curve. (b) Stress–Curvature curve.

Table 7

Table 7. The $K_t$ and $K_c$ values of each material under different stiffness conditions.

To assess the fatigue damage of the structure under variable–amplitude loading, the load–time histories are processed using the rainflow counting method. The rainflow counting method is particularly suitable for assessing the cumulative damage of structures under complex loading conditions. The hydrodynamic loads acting on the submarine cable and the resulting stress response exhibit characteristic quasi–periodic behavior, making this method well suited for the statistical analysis of its fatigue life. The core principle of the rainflow counting method lies in decomposing a continuous load–time history into a series of discrete load cycles. The method identifies the peaks and valleys in the stress–time curve. From each peak, it traces downward to the nearest valley and upward to the next peak. Each trace defines a stress cycle, representing a full stress fluctuation.

Based on the cumulative fatigue damage theory, stress amplitudes and the corresponding number of cycles can be converted into damage. According to Miner’s rule, the damage caused by loads of different stress amplitudes is independent and can be linearly accumulated to obtain the total damage. Following this principle, the fatigue damage of the submarine cable can be expressed as Equation 6:

$D={\displaystyle \sum _{i=1}^N}\frac{n_i}{N_i}(6)$

where N is the number of loading conditions, $D$ denotes the cumulative fatigue damage (with $D=1$ indicating fatigue failure and $D=0$ indicating no fatigue damage), and $n_i$ represents the actual number of cycles.

The fatigue life of the structure, $n_{life}$, is given by Equation 7:

$n_{life}=\frac{1}{n_{AFD}}(7)$

where $n_{AFD}$ denotes the accumulated fatigue damage value.

Figure 20 illustrates the fatigue life of the cable components under different stiffness conditions. As shown in the figure, the shortest fatigue life occurs at the mid–span region of the cable. Among the three stiffness models, the slip stiffness condition yields the lowest fatigue life, while the stick stiffness condition results in the longest. Among the structural components, the lead sheath exhibits the shortest fatigue life, whereas the conductor and armor wires show significantly longer fatigue life. Therefore, it can be concluded that the lead sheath at the mid–span region of the free–spanning submarine cable is the most susceptible to fatigue.

Figure 20

Figure 20. Fatigue life of cable components under different stiffness conditions. (a) Conductor. (b) Armor wire. (c) Lead sheath. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Figure 21 presents the minimum fatigue life of the conductor, armor wires, and lead sheath under different stiffness conditions. As shown in the figure, the shortest fatigue life for all three components occurs at the free–span length of 10 m. The conductor and lead sheath exhibit their minimum fatigue life under the slip stiffness condition. In contrast, the armor wires reach the lowest fatigue life under the stick stiffness condition. This is because, under the stick stiffness condition, interlayer sliding is restricted, causing the entire cross–section to deform as an integral body under bending. The absence of frictional energy dissipation accelerates the accumulation of fatigue damage. Under the slip stiffness condition, the minimum fatigue life of the lead sheath is 11.89 years, whereas it increases to 25.90 years under the stick stiffness and 19.41 years under the nonlinear bending stiffness. The fatigue life predicted using the slip stiffness model differs by 39% from that obtained with the nonlinear bending stiffness, while that based on the stick stiffness differs by 25%. These results indicate that the selection of stiffness model leads to considerable discrepancies in the predicted fatigue life. According to DNV-RP-F401, the fatigue life safety factor for all cable segments shall not be less than 10. Therefore, a safety factor of 10 is adopted in this case study. After applying the safety factor, the minimum fatigue life of the copper alloy sheath decreases to 1 year, which poses a significant risk.

Figure 21

Figure 21. Minimum fatigue life of cable components under different stiffness conditions and free–span lengths. (a) Conductor. (b) Armor wire. (c) Lead sheath. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$).

Based on the analysis presented in Figure 22, a free-span length of 10 m is identified as the critical condition associated with the shortest fatigue life. The stress distribution along the cable length for this case is illustrated in Figure 22a. The distribution exhibits a parabolic profile, characterized by peak stresses at the mid-span location and diminishing magnitudes towards the ends. Furthermore, the stress time history at the mid-span (the location of maximum stress) is extracted and plotted in Figure 22b to demonstrate the dynamic fluctuation details.

Figure 22

Figure 22. Stress distribution and mid-span stress time history of the lead alloy sheath. ($U=1.3\mathrm{m}/\mathrm{s}$; $H=1.3\mathrm{m}$; $T=7.0\mathrm{s}$; $L=10\mathrm{m}$). (a) Position along the cable. (b) Simulation time (s).

5 Conclusion

This study developed a local refined model of a submarine cable in ABAQUS to derive a nonlinear bending stiffness that can accurately reflect its true mechanical behavior. The reliability of the model is validated against theoretical formulations. The obtained stiffness is then incorporated into OrcaFlex to systematically investigate the wave–induced dynamic response and fatigue behavior of a free–span submarine cable under three different bending stiffness models: stick, slip, and nonlinear. The main conclusions are as follows:

1. The dynamic response of the submarine cable exhibits a strong sensitivity to the bending stiffness model in the short-span regime. However, this sensitivity diminishes rapidly as the free-span length increases. This phenomenon reveals a fundamental mechanical transition: as the span length increases, the dominant factor governing the cable’s motion shifts from bending stiffness to axial stiffness. Consequently, for long-span cables, the influence of the local cross-sectional stiffness model becomes negligible compared to the global geometric stiffness.

2. The motion and stress response of the submarine cable are strongly influenced by bending stiffness under short–span conditions, while this sensitivity gradually decreases as the free–span length increases. When the span length exceeds 20 m, the curvature distributions under different stiffness conditions become nearly identical, indicating that the system transitions from a bending–dominated to a tension–dominated regime.

3. Under identical wave and current conditions, the slip stiffness model produces the largest displacements and curvatures. When the cable curvature is below the critical value, the results of the stick and nonlinear bending stiffness models are similar. However for curvatures exceeding the critical threshold, the slip and nonlinear bending stiffness models yield comparable outcomes. The axial tension exhibits little variation across the three stiffness models, indicating that bending stiffness has a limited effect on the overall tension response.

4. As the free-span length increases, the maximum curvature first rises and then decreases, reaching its peak at a span of 10 m. This trend indicates that an excessive free-span length enhances the dominance of axial tension in governing the overall dynamic behavior of the submarine cable, thereby constraining local bending response.

5. Different stiffness conditions significantly influence the predicted fatigue life. The armor wire and conductor exhibit superior fatigue resistance, while the lead sheath is the most susceptible to fatigue failure. For the lead sheath, the fatigue life obtained using stick and slip methods differs by 39% and 25% respectively from the results obtained using nonlinear bending stiffness. Therefore, to accurately assess the fatigue life of submarine cables, it is necessary to consider the nonlinear bending stiffness of the submarine cable during fatigue analysis.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

YC: Data curation, Formal Analysis, Writing – original draft. DZ: Investigation, Writing – original draft. SZ: Methodology, Writing – original draft. XL: Project administration, Writing – original draft. ZC: Resources, Writing – original draft. QQ: Validation, Writing – review and editing. ZW: Supervision, Writing – review and editing. ZG: Visualization, Writing – review and editing. HS: Project administration, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. The authors would like to acknowledge the supports from China Southern Power Grid, Science and Technology Project (CGYKJXM20240182).

Conflict of interest

Authors YC, DZ, SZ, XL, and ZC were employed by Haikou Sub-Bureau, Guangzhou Bureau, EHV Transmission Company of China Southern Power Grid Co., Ltd.

The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that this work received funding from the China Southern Power Grid. The funder had the following involvement in the study: study design, data collection and analysis, the decision to publish, and the preparation of the manuscript.

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The author(s) declared that generative AI was not used in the creation of this manuscript.

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