Reliability analysis of carbon fiber rod-reinforced umbilical cable under tension using an improved sampling method

Abstract

The umbilical cable is a vital component of subsea production systems that provide power, chemical agents, control signals et al., and its requirement for reliability is exceedingly high. However, as the umbilical cable is a composite structure comprising multiple functional units, the reliability analysis of such cables involves numerous parameters that can impact calculation efficiency. In this paper, the reliability analysis of a new kind of umbilical cable with carbon fiber rod under tension is analyzed. The global dynamic analytical model is first established to determine the maximum tension load, then the local analytical model of umbilical cable including each unit are constructed by finite element method (FEM). Based on the mechanical analytical model, the reliability of umbilical cable under tension load is studied using response surface method (RSM) and Monte Carlo method. During the calculation process, a new tangent plane sampling method to calculate the response surface function (RSF) is proposed in this paper, which could make sampling points faster come close to the RSF curve, and it is proved that the calculation efficiency increases about 33% comparing with traditional method.

Headnote

ABSTRACT

The umbilical cable is a vital component of subsea production systems that provide power, chemical agents, control signals et al., and its requirement for reliability is exceedingly high. However, as the umbilical cable is a composite structure comprising multiple functional units, the reliability analysis of such cables involves numerous parameters that can impact calculation efficiency. In this paper, the reliability analysis of a new kind of umbilical cable with carbon fiber rod under tension is analyzed. The global dynamic analytical model is first established to determine the maximum tension load, then the local analytical model of umbilical cable including each unit are constructed by finite element method (FEM). Based on the mechanical analytical model, the reliability of umbilical cable under tension load is studied using response surface method (RSM) and Monte Carlo method. During the calculation process, a new tangent plane sampling method to calculate the response surface function (RSF) is proposed in this paper, which could make sampling points faster come close to the RSF curve, and it is proved that the calculation efficiency increases about 33% comparing with traditional method.

Keywords:

Umbilical cable

Carbon fiber rod

Failure analysis

Response surface method

Reliability

(ProQuest: ... denotes formulae omitted.)

1. Introduction

The umbilical cable is a crucial component of underwater control and umbilical cable systems. It is a composite cable consisting of hydraulic pipes, cables, optical fibers, chemical conduit, etc. Its primary function is to establish a connection between the main production platform and underwater system equipment, providing electrical, hydraulic power, control data signals, and chemical injection channels for subsea manifold systems and oil wellheads. It serves as the "nerve center" for oil and gas extraction (Guo, 2016). The umbilical cable is subjected to complicated loads during operation, including gravity, waves, currents, platform motion, and others. As a result, tension failure, fatigue failure, buckling and sail failure may occur in the umbilical cable (Thies et al., 2012). Therefore, it is necessary to study the mechanical properties and reliability of the umbilical cable

Many scholars have conducted research on the mechanical properties of the umbilical cable. Knapp derived the equilibrium equation of geometric nonlinearity, the linear stiffness matrix, the stress equation of armored steel wire inside the umbilical (Knapp, 1975, 1979, 1981). Sævik and Gjøsteen (2012) proposed a threedimensional finite element formulation to predict the performance of complex umbilical sections under tension, torque, internal pressure and external pressure. Lu (2013) developed predictive models for the tensile, bending, and fatigue behavior of umbilical cables using the Abaqus pseudo-static analytical method. Zhu (2017) developed a three-dimensional solid finite element method (FEM) of an umbilical cable, simulated various combined loading conditions, and analyzed the effects of different loads on the cable's mechanical properties. Tang et al. (2014) and Zhao (2017) analyzed the tensile behavior of umbilical cables using finite element analysis and calculated the probability of cable failure under in-situ conditions. Liu et al. (2018) explored the influence of the internal filler and tensile rate of the umbilical cable on its tensile properties. The results show that the filling unit and the tensile rate have a certain influence on the tensile stiffness of the umbilical cable, but the influence on the mechanical properties of the umbilical cable section is small. Zhang et al. (2019) used FEM software to analyze the sensitivity of parameters affecting the static strength of umbilical cables, while considering the effect of friction. Wang et al. (2021) developed a three-dimensional nonlinear finite element model for deep-sea non-metallic armored umbilical cables based on precise geometric beam theory and the principle of virtual work. Yang et al. (2023) investigated the mechanical properties, particularly the top tension, of deep-sea umbilical cables during deployment and retrieval. Their research was based on the integration of three-dimensional potential flow theory and the concentrated mass method, accounting for the combined influences of surface-supported vessel motion and ocean currents.

Due to the complex structure, laborious manufacturing process, and numerous functional units of an umbilical cable, there is randomness in the geometric and material parameters that can affect the cable's reliability. Nilsen and Hanson (1995) established a correlation between the failure probability, derived from the deterministic safety factor design method, and the uncertainty associated with random variables. As an illustrative case, they examined a 15-inch flexible riser located at a depth of 330 m underwater. By considering the uncertainty distribution of environmental load variables and riser structural parameters, they confirmed the existence of discrepancies between the safety factor and the corresponding failure probability. Akpan et al. (2007) and Xia and Das (2008) proposed a fatigue reliability calculation method that considers the uncertainties of material parameters, damage index, and structural load, and used the first-order second moment method to calculate the fatigue reliability. Song et al. (2011) focused on the reliability-based design optimization (RBDO) of the riser support system for a floating production storage and offloading unit (FPSO) during operation, extreme conditions, and in the presence of damage or pipeline failures. Within the RBDO framework, they employed RSM and applied the moving least squares method to obtain the optimal probabilistic solution. Yan et al. (2018) analyzed the strength reliability of the critical section of an umbilical cable using an improved first-order second moment method and Monte Carlo method. Xia et al. (2021) used the fuzzy fault tree method to calculate the reliability of an umbilical cable and concluded that the sheath/armor has the greatest impact on the cable's reliability. Zhang et al. (2021a) analyzed the probability of torsional instability of flexible pipeline by both the first-order second moment method and Monte Carlo method. Xu (2019) employed the equivalent normalization method and Monte Carlo simulation to calculate the reliability of an umbilical cable. Zhang et al. (2021b) employed a multi-stage Bayesian network to predict the reliability of an umbilical cable.

The steel tension armor of the umbilical cables in the above research consists of two layers with opposite helical angles, as shown in Fig. 1. However, as the water depth increases, the weight of the steel tension armor also increases, which may lead to the tensile failure of the umbilical cable. The carbon fiber rod reinforced umbilical cable, shown in Fig. 2, replaces the steel tension armor with carbon fiber rods. This modification significantly reduces the weight of the cable and improves its tensile and bending resistance (Yu and Su, 2008). In order to effectively utilize the new type of umbilical cable, it is essential to analyze its mechanical properties and reliability.

2. The mechanical analysis of carbon fiber rod reinforced umbilical cable

The model of the carbon fiber rod reinforced umbilical cable shown in Fig. 2 was constructed using Abaqus software. The geometric parameters and quantities of each unit in the umbilical cable are shown in Table 1, and the material properties of each unit are shown in Table 2.

In the FEM model of umbilical cable, the electric cable was simplified as a copper core wrapped with high density polyethylene, and the optical fiber unit was simplified as a steel shell wrapped with high-density polyethylene. The simplified model of umbilical is shown in Fig. 3

The contacts between the units in the model were defined as general contacts, which were considered as hard contacts. One end of the model was fixed while the other end was subjected to loads, and a reference point was established at the end to constrain the end section at the reference point. The model utilized hexahedral elements with C3D8R grid type. The meshes for steel tubes, stranded wires, and carbon fiber rods were refined to accurately capture their load-bearing characteristics. Fig. 4 shows the meshed umbilical cable.

The FEM model described above was used to apply tension, bending moment, and torsion loads to the end of the umbilical cable, and obtain its stiffness properties. The resulting tension stiffness (KT), torsional stiffness (KB), and bending stiffness (KN) were 385 MN, 21.9 kN m2 , and 15.3 kN m2 , respectively.

For the study of theoretical stiffness models for helically wound structures, Knapp (1981) provided a theoretical formula (Eq. (1)) for the tensile strength of helical structures, and he presented the expression (Eq. (2)) for torsional stiffness excluded radial deformation for helically wound structures.

... (1)

... (2)

where, ðAEÞ0 represents the tensile stiffness of the central conductor, A represents the cross-sectional area of the structure, E represents the elastic modulus, ðAEÞi represents the tensile stiffness of the ith helical structure, Ri represents the distance from the helical structure to the center of the cylinder, Rc represents the radius of the core, ai is the winding angle of the helical structure, ðIGÞ0 represents the torsional stiffness of the central conductor, I represents the moment of inertia, G represents shear modulus, and Ri denotes the radius of each structure.

Costello (1997) examined a single helical structure extracted from the overall umbilical cable model. By ignoring interactions between elements, he determined the bending stiffness of a single free helical structure under pure bending. Summing contributions from all elements, Costello derived the overall bending stiffness formula (Eq. (3)) for the entire structure.

... (3)

where yi denotes the Poisson's ratio of the helical structure.

Upon substituting the relevant material and dimensional data for each component of the carbon fiber rod-reinforced umbilical cable in Fig. 1(b), theoretical values for tensile stiffness, bending stiffness, and torsional stiffness were obtained. A comparison between these theoretical values and the finite element model calculation results are presented in Table 3.

There is a relative error between theoretical calculations and finite element results, and this discrepancy is attributed to assumptions in the stiffness theoretical model, including the neglect of interactions among internal components. The comparison of tension stiffness, torsional stiffness and bending stiffness between theoretical and finite element results verified the validity of the mechanical model.

To analyze the mechanical properties of umbilical cable under tension loads, the global analysis was conducted to determine the maximum tension load that the cable could withstand. The global model was constructed using OrcaFlex software. The umbilical cable analyzed in this study is designed for use at a water depth of 2000 m and employs a lazy wave configuration (Fig. 5) to effectively mitigate top tension (Chen et al., 2018). The length of umbilical cable from top spar platform to the buoyant cylinder AB ¼ 2300 m, the buoyant length BC ¼ 150 m, and the length from buoyant cylinder to seabed CD ¼ 150 m. The horizontal distance from the suspension point to the touch point E ¼ 600 m, and the depth H ¼ 2000 m. The friction coefficient between umbilical cable and seabed and the seabed stiffness are shown in Table 4.

The description of waves typically involves two types: regular waves and irregular waves. Regular waves are primarily derived from fluid mechanics theory and are characterized by parameters such as wave direction, wave height, period, and starting time. The most commonly used regular wave theories include linear wave theory (Airy wave), Stokes wave theory, Dean wave theory, and elliptic cosine wave theory. On the other hand, irregular waves are typically described based on statistical random spectra. Various methods for superimposing regular waves are often employed to represent irregular waves, such as the PM spectrum and JONSWAP spectrum (Guo, 2016).

When selecting wave types for model analysis, the analysis type and actual sea conditions are the main factors considered. In order to simplify the calculation process, this paper utilizes a regular wave in the model to analyze and calculate the dynamic response of the umbilical cable. The aforementioned regular wave theories are applicable to different water depths. In this model, the seawater depth is 2000 m, which is suitable for the use of linear wave theory (Airy wave).

For specific wave loads and to ensure a more conservative calculation of reliability, the chosen sea state parameters are based on a 100-year return period. The selected location is a specific operating zone in the South China Sea. In this zone, the maximum impact of the ocean currents is observed to flow from east to west by Zhao et al. (2022). Therefore, the current direction was selected as 270 in OrcaFlex software. The corresponding values are presented in Table 5

The velocity of seawater is typically influenced by both the velocity of the ocean surface and the depth of the water. In cases where specific current data is not available, the current velocity VCðZÞ at a given water depth z can be estimated using the profile calculation method 1, as proposed by Xing et al. (2012). This method allows for the estimation of current velocities based on certain assumptions and calculations.

... (4)

By utilizing the aforementioned formula, the seawater velocity at various depths can be calculated for the current load with a 100year return period. The specific parameters related to the current are listed in Table 6.

The tension loads along the axial length of umbilical cable are shown in Fig. 6 including the maximum, minimum and mean values. The findings indicate a decrease in the tension load of the umbilical cable along its axial length, with an increase at the buoyancy section. The highest tension load is observed at the top of the umbilical cable, with a value of 138 kN.

After applying a tension load of 138 kN to the finite element model (Fig. 3) that has been validated in the above, the stress distribution of the umbilical cable was analyzed and is presented in Fig. 7. The results indicate that the highest stress is observed in the carbon fiber rod, followed by the stranded wire and steel tube. The stresses in the sheath and fillings are negligible, as expected.

In finite element analysis, the size and quantity of the mesh have a significant impact on the accuracy of the computational results. Therefore, a mesh-independence validation, as illustrated in Fig. 8, was conducted for the established model. It can be observed that the maximum stress almost stabilizes when the number of grids approaches around 240,000. The maximum stress obtained is listed in Table 7.

The stress utilization ratio N in this paper is the product of the safety factor n (n ¼ 1.5) and the maximum stress divided by the yield stress, indicates the failure priority. If N ¼ 1, the maximum stress equals to the yield stress, which means the failure will happen. Table 7 shows the stress utilization ratio N for each unit of the umbilical cable. It can be seen that the ratio of stress utilization of the steel tube is much larger than other units, so its failure priority is 1, which means that tensile failure may firstly happen in the steel tube unit.

Therefore, the following reliability analysis of umbilical cable in this paper is based on the steel tube mechanical analysis.

3. Reliability analysis of umbilical cable under tension loads

The tensile failure function of umbilical cable is defined as

... (5)

where, Sy represents the yield strength, p represents the maximum stress. As discussed in the above, the first priority of tension failure is steel tube, the parameters of Sy and p are the parameter values of steel tube, and n represents the safety factor, taking 1.5 in this paper. When Z1 >0, it means the umbilical is safe, and tension failure will not happen. The reliability calculation is to calculate the probability of PfZ1 >0g.

From Tables 7 and it is seen that the maximum value is smaller than the yield stress, which looks like the reliability is 100%. However, the geometry parameters and material parameters of umbilical cable have some randomness during manufacture and process, which can induce the randomness of Sy and p. So, the randomness has to be considered during the reliability analysis. In this paper, the thickness of steel tube, diameter of stranded wire, diameter of carbon fiber rod and tensile strength of steel tube are considered as random variables, assumed to follow a normal distribution. The statistical characteristics of each variable are shown in Table 8.

3.1. Classical RSM

To calculate the probability of PfZ1 >0g, it is necessary to know the relationship between the tensile strength of steel and the thickness of steel tube, diameter of stranded wire, diameter of carbon fiber rod, and tensile strength, which cannot be obtained directly. In practical engineering, the RSM is often used to obtain an approximate function instead of the real function (Liu and An, 2020). In the RSM, a shape of the function with several undetermined parameters is first assumed, and then the relationship between the undetermined parameters and the random variables is obtained through FEM simulation. During the calculation process, sample points of the random variables are given, and iterative calculation is performed until the error is satisfied. The process of using the classical RSM to fit the structural performance function is as follows:

(1) The selection of RSF type

The quadratic polynomial function without cross terms (shown in Eq. (6)) was selected as the type of RSF, which has higher fitting accuracy and fewer undetermined coefficients.

... (6)

where, a0; bi; ci is the undetermined coefficient in the RSF.

(2) Select sample points to fit the RSF

Once the form of the RSF has been determined, the selection and fitting process of sample points can be carried out using the following steps:

...

The coefficient f typically varies throughout the iterative process. In the first generation, its value is set to 1, and then it is increased to 3 in subsequent generations.

Based on the provided sample points and outputs, the least squares method can be employed to solve the 2n þ 1 undetermined coefficients in Eq. (6). The fitted response surface function can be calculated using either the moment method or the Monte Carlo method. This enables the determination of the checking point X*ðkÞ and the reliability index bk , where k represents the k th iteration. The convergence condition is assessed as follows:

... (7)

If the aforementioned convergence condition is satisfied, then bkþ1 represents the final reliability index. If the condition is not met, the next generation iterative center point XðkÞ and other sample points are determined in the vicinity of the checking point X*ðkÞ . Eventually, when the convergence condition is satisfied, the calculation comes to a halt.

3.2. The improved RSM based on tangent plane sampling

However, as the shape of RSF is unknown, it is determined through initial sample points and many interaction times to come close the shape of RSF, but the calculation has poor fitting efficiency, and even non-convergence. The fitting process is shown in Fig. 9.

Therefore, we propose a new method to quickly come close to the shape of RSF based on the tangent plane. Firstly, the checking point XD and tangent plane are determined by sequence interpolation along specific direction. Then, the indirect point Xð0Þ i of the sample points are selected based on the tangent plane, and the sample points are calculated by the indirect points. Compared with the conventional method, the sample points obtained in this way are close to the RSF faster, which can make the fitting process more efficient. The calculation process is depicted in Fig. 10, while the fitting process is presented in Fig. 11.

(1) The selection of RSF type

The quadratic polynomial without cross-terms simplifies the computational model, effectively addressing convergence issues. Additionally, the parameters mentioned earlier are mutually independent. Therefore, the response function model of the improved RSM utilizes a quadratic polynomial response without cross terms. The calculated adjusted coefficient of determination is R2 ¼ 0.8541, which further substantiates the strong fitting performance of the chosen model.

(2)The selection of initial sample points and the calculation of reliability index b1.

With the mean point as the center, the test points were arranged along the positive and negative directions of each coordinate axis, and a total of 2n þ 1 test points were obtained:

... (8)

... (9)

... (10)

where, Xm is the mean vector of random variables; mi and si are the mean and standard deviation of random variable xi; n is the number of random variables. b is the reliability index, it is defined as the following Eq. (11):

... (11)

where, pf is the failure probability of umbilical cable.

The response values of 2n þ 1 sample points and each sample point were used to fit the first-generation RSF gðXÞ by least square method, and the reliability index of the first-generation RSF b1 was obtained by Monte Carlo method.

(3) The solution of checkpoint

Using the first-order second moment method to calculate the mean value and standard deviation of the first-generation RSF, the initial check point was obtained, denoted as XDL. The following iterative method was used to select the checkpoint, ensuring a specific range between the checkpoint and the RSF:

... (12)

where, gðXiÞ is the function value of the RSF at point Xi. When i ¼ 1 , X0 is Xm, X1 is XDL. The convergence condition of this sequence interpolation is

... (13)

where, ε is the accuracy of checking point selection, which takes 0.001 in this paper.

The sample points satisfying the convergence condition was recorded as XD, which is the checking points of the final selected RSF.

(4) The selection of new generation sample points

Firstly, constructed a vector w ¼ XD Xm, and its unit vector was denoted as h, which is the normal vector of the tangent plane of the RSF curve at the checking point. Then the tangent plane could be obtained:

... (14)

The tangent plane of response surface at the checking point intersects with the coordinate axes of each random variable, and the intersection point was recorded as XC i , i ¼ 1; 2; /. The unit vector of the reconstructed vector was denoted as xi, and the indirect point Xð0Þ i was arranged along the direction of the vector xi based on the check point XD. Since the area near the checkpoint has a great influence on the accuracy of the fitted RSF, the sample points were selected with the coefficient 0:5si, that is, the sample points from the checkpoint 0:5si were selected on the tangent plane:

... (15)

where, si is the standard deviation of the ith random variable.

...

... (16)

If the two symbols are opposite, the sample points Xð1Þ i are selected based on Xi in the opposite direction of h according to Eq. (17):

... (17)

In addition, the last 2n sample points were selected with Xð1Þ i as the checking point and Xð2Þ i as the midpoint, which can make the final two points faster close to the response surface when the value of the random variable changes greatly, and can greatly increase the fitting efficiency.

... (18)

Finally, a total of 2n þ 1 sample points of XD, Xð1Þ i and Xð2Þ i were obtained.

(5) The determination of RSF and the calculation of reliability

Using the final sample points, the RSF were determined, and the reliability and reliability index were calculated using Monte Carlo method.

The method proposed in this paper, which selects sample points based on the tangent plane, has been shown to improve the efficiency of sample point selection, reduce the number of required iterations, and enhance the fitting efficiency.

3.3. The reliability of umbilical cable

Using the improved RSM introduced in the above, the mean value of each random variable was used as the first-generation fitting center point to construct the sample point, and 7 sample points were obtained (Table 9). Subsequently, the FEMs of the umbilical cable for these 7 sample points are established individually, and the maximum stresses of the steel tube unit with each sample point were obtained in Table 9.

Based on the above seven sample points and their response values, through the improved RSM shown in Fig. 8, the final RSF Z1 were obtained as shown in Eq. (19).

... (19)

where, p is the maximum stress of steel tube unit, t is the thickness of steel tube, d1 is the diameter of stranded wire, and d2 is the diameter of carbon fibre rod.

Finally, the reliability and reliability index of umbilical cable under tension load were calculated, as shown in Table 10. It can be seen that the reliability of the carbon fibre reinforced umbilical cable under tension load is 0.99331, and the reliability index is 2.478. Based on the lack of reliability design guidelines for marine flexible pipelines, this paper makes reference to GB-50153-2008 (2008) and observes that the reliability index of flexible pipelines under the aforementioned conditions is significantly lower than the value of 4.2 specified therein.

Table 11 shows the comparison of results and calculation efficiency between the improved method and conventional method. The reliability index changing with iteration times are shown in Fig. 12. It can be found that the improved method is convergent after 6 iteration times, while it is 9 times for the conventional methods. Each iteration requires 7 sample points, that is, 7 finite element models need to be established in each iteration calculation. Therefore, 21 FEM models are reduced using the improved method, with the calculation efficiency increasing 33%.

In this study, the reliability of the umbilical cable is evaluated through the combination of mechanical analysis, RSM, and Monte Carlo method. The proposed new method for selecting sample points based on the tangent plane in the RSM is shown to be efficient and effective in reducing the number of FEM models required for constructing the RSF and increasing the calculation efficiency. This method could be applicable to other reliability calculation problems, particularly those with a large number of random variables, which may further increase the calculation efficiency. Future work could investigate the generalizability of this method to different types of engineering problems and explore the potential for further efficiency improvements through optimization.

4. Conclusions

In this paper, the mechanical properties and reliability of carbon fiber rod reinforced umbilical cable are analyzed, and the following conclusions are obtained.

(1) The global and local analytical models of the carbon fiber reinforced umbilical cable have been constructed. The maximum tensile load of the cable has been calculated to be 138 kN, and the stress utilization ratio (N) of different units in the umbilical cable are compared. The steel tube has the largest ratio 0.903, indicating that the steel tube is the first unit to fail.

(2) The RSF of the umbilical cable under tension load is obtained considering the randomness of thickness of steel tube, diameter of stranded wire, diameter of carbon fiber rod and tensile strength of steel tube. During the interaction calculation process, the selecting sample point method is improved, which can make the sample points faster close to the response surface curve.

(3) The reliability of umbilical cable calculated by the improved RSM is 0.99331 and the reliability index is 2.478. The interaction times is 6 with 42 sample points, while it is 9 with 63 sample points for the conventional method. It is proved that the proposed method in this paper can greatly improve the calculation efficiency.

CRediT authorship contribution statement

Yu Zhang: Writing e review & editing, Writing e original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Hong-Yu Zhang: Writing e review & editing, Writing e original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation. Ran Xia: Supervision, Software, Resources, Methodology, Investigation, Formal analysis. Si-Ao Jiang: Software. Fang Wang: Resources, Project administration, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Financial support for this research was provided by the National Natural Science Foundation of China (Grant No. 52222111).

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ARTICLE INFO

Article history:

Received 28 June 2023

Received in revised form

23 January 2024

Accepted 28 February 2024

Available online 1 March 2024

Edited by Jie Hao and Meng-Jiao Zhou

Footnote

* Corresponding author. E-mail address: [email protected] (Y. Zhang)

References

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