A Dynamic-Bayesian-Networks-Based Resilience Assessment Approach of Structure Systems: Subsea Oil and Gas Pipelines as A Case Study

Under unanticipated natural disasters, any failure of structure components may cause the crash of an entire structure system. Resilience is an important metric for the structure system. Although many resilience metrics and assessment approaches are proposed for engineering system, they are not suitable for complex structure systems, since the failure mechanisms of them are different under the influences of natural disasters. This paper proposes a novel resilience assessment metric for structure system from a macroscopic perspective, named structure resilience, and develops a corresponding assessment approach based on remaining useful life of key components. Dynamic Bayesian networks (DBNs) and Markov are applied to establish the resilience assessment model. In the degradation process, natural degradation and accelerated degradation are modelled by using Bayesian networks, and then coupled by using DBNs. In the recovery process, the model is established by combining Markov and DBNs. Subsea oil and gas pipelines are adopted to demonstrate the application of the proposed structure metric and assessment approach.


Introduction
Resilience of engineering systems in the face of the variety of disruptive events has been attracted an enormous amount of research with the increasing of complexity and size of the systems. Structure system, made up of various mechanical components, is almost the most important part in an engineering system. Under the unanticipated natural disasters, any failure of the structure components may cause the crash of the entire structure system. Building resilience into structure system is the key to protecting against the deleterious influences of system disruption because of natural disasters. It is necessary to apply a comprehensive framework to assess the resilience of structure systems through a quantitative approach from a macroscopic perspective.
For certain specific systems, quantitative resilience assessment approaches have been proposed and reported. In the field of engineering systems, for instance, Henry et al. (2012) proposed generic metrics and its corresponding approaches to quantitatively evaluate the resilience of a road network; Shirali et al. (2013) proposed a novel approach for quantitative assessment of the resilience for the engineering system by using PCA and NT approaches; Hosseini and Barker (2016) quantified the infrastructure resilience with the example of an inland waterway port; Francis and Bekera (2014) contributed a novel metric incorporated adaptive capacity, absorptive capacity, and recoverability to assess the resilience of the engineering system and infrastructure system; Specking et al. (2019) proposed an analysis approach based on multiple objective decision to assess the engineering resilience for systems with multiple performance measures; MacKenzie and Hu (2019) proposed a resilience framework that is based on the value-driven design to en-able the assessment of engineering system resilience; Yodo et al. (2018) presented a resilience modelling and analysis approach through using the fundamental control theory to assess the resilience of complex engineering systems; Cai et al. (2018) proposed an engineering resilience metric based on the availability and its corresponding assessment methodology to assess the engineering resilience; Feng et al. (2019) proposed a resilience metric to describe the additional capacity of an engineering system, evaluate and design the system, and a case study on offshore wind farm is used to demonstrate the application.
Although various resilience metrics and corresponding assessment methodologies are developed, they are only suitable for assessing the resilience of the entire engineering system instead of the structure system. The failure mechanism of structure system is different from that of the entire engineering system. That is because the failure of the structure system is caused by the sudden changes of the fatigue mechanical properties, which is related to the mechanical properties of the structural components. The structural failure is usually caused by multiple factors which influence the structure components at the same time. For an engineering system, it usually consists of various subsystems, such as structure systems, control systems, and electrical systems. The failure mechanism of the engineering systems is complex. The evaluation metrics for engineering systems usually incorporate adaptive capacity, absorptive capacity, and recoverability, which are not related to failure mechanism. The performance modelling for the degradation process and recovery process of the engineering structure system is difficult. To the date, how to evaluate the structure resilience is still a challenging task.
Bayesian networks (BNs) can be described briefly as a directed acyclic graph, and it can graphically represent complex relationships between variables. Meanwhile, in the probabilistic knowledge representation and reasoning fields, it is one of the most useful modelling approaches. BNs are widely used in various fields, such as the assessment of system reliability, safety, risk, and resilience. For example, John et al. (2016) proposed an assessment approach through using BNs to enhance the seaport system resilience; Fu and Khan (2019) proposed a probabilistic framework to evaluate the operational risk quantitatively by using BNs; Cai et al. (2012) evaluated the reliability of the subsea blowout preventer control system by using BNs; Abimbola and Khan (2019) used the dynamic object-oriented BNs approach to assess the resilience of engineering systems. It is hard to model the transformation process between multiple states only using BNs when modelling the system performance changing process. The modelling problem can be solved by combining BNs with Markov model. For instance, Dhulipala and Flint (2020) proposed a semi-Markov process model to capture the interdependencies among events in recovery process when the infrastructure system is attacked by successive hazard events; Rebello et al. (2018) proposed an approach for assessing the system functional reliability by combining dynamic Bayesian networks (DBNs) and hidden Markov model.
The work aims to develop a novel resilience assessment metric and a novel resilience assessment approach of structure systems based on the remaining useful life (RUL) of key components. DBNs and Markov are combined to establish the resilience assessment model. The rest of the paper is organized as follows. Section 2 proposes the resilience assessment approach. Section 3 provides a case study of subsea oil and gas pipelines to demonstrate the application of the proposed metric and approach. Section 4 summarizes the work.

Modelling methodology for structure resilience
A new structure resilience assessment metric and its corresponding assessment methodology are proposed. The RUL of structural components is taken as the new structure resilience assessment metric. The resilience assessment methodology of the structure system is shown in Fig. 1. This methodology consists of three main steps: degradation process modelling, recovery process modelling, and resilience value calculation. In the degradation process modelling, the influences of single factors, such as natural factors and external factors, are modeled by BNs, respectively. The influences of natural factors are caused by nature environment, and the influences of external factors are caused by external disasters. The DBNs model is developed through coupling the single factors with each other to simulate the performance degrad-ation of the system. In the recovery process modelling, the DBNs model is developed by combining BNs with Markov model to simulate the performance recovery of the system. In the resilience calculation, the RUL changing trend is simulated according to the performance changing trends; and then the structure resilience is calculated based on the trend of the RUL.

Degradation process modelling
The factors that affect the performance degradation of structure systems are divided into two parts, namely natural factors and external factors. Natural factors refer to the factors which come from the nature environment, such as corrosion, sand erosion and so on. External factors are aroused by random external disasters, such as typhoon, earthquake, internal wave and so on. Assume that there are two factors which influence the system performance, named A and B, in which A is the natural factor and B is the external factor. The degradation models are shown as follows: (1) where P 1 is the system performance under the influence of natural factor A; P 2 is the system performance under the influence of external factor B; a 1 , a 2 , a 3 , b 1 , b 2 , and b 3 are variables. Firstly, when establishing the degradation model of the structure system, the degradation process of system performance is divided into natural degradation process and accelerated degradation process according to the sources of influencing factors. The performance models of the structure system with the impact of the different single factors are modelled by BNs, respectively. Then the degradation model for structure system with the impact of multi-factors is modelled by coupling the natural degradation model with the accelerated degradation model. The coupled multi-factor model is extended to a DBNs model according to time variation. At last, the performance degradation trend of structure system is obtained through the result of the DBNs model. The coupling equation is shown as follows: where P is the structure system performance with the impacts of multiple factors.

Recovery process modelling
All information which is required to predict a state at time t+∆t is contained at time t, and information about earlier time is not required. That is, the recovery process of the structure system follows Markov law. Markov law is used to describe the system that has multiple states and model the transition relationships among the multiple states. In the recovery process of the structure system, Markov law is used to describe the transition relationships, and the DBNs model is established according Markov model. The transition probabilities among time slices are depended on the failure rates of structure components. As shown in Fig. 2a, the system performance degrades from the working state S 0 to failure state S 1 under the influence of various degradation factors, and then the system performance can recover from S 1 to S 0 by taking the complete maintenance approach. When taking the incomplete maintenance approach, the performance can recover from S 1 to an acceptable state S 2 , as shown in Fig. 2b. According to the state transition diagrams of the system, it can be known that the state transition matrixes of the system are as follows:

Modelling of DBNs
DBNs' modelling is divided into two parts, namely structure modelling and parameter modelling. In the structure modelling process, the structure model is transformed according to the physical model. The nodes in the BNs represent variables in the physical model, and arcs represent the causal relationships among variables. Take the degradation process of the structure system under the influence of multiple factors as an example, In a time slice, the performance variables a 1 , a 2 , a 3 , b 1 , b 2 , b 3 … affect the physical performance nodes P 1 and P 2 , so the variable nodes and performance nodes are linked by using arcs. The physical performance nodes P 1 and P 2 affect the system performance node P, then these nodes are linked by using arcs. Finally, the structure of the static BNs is finished. For DBNs model, it is replication of static BNs from time t to t+(n−1)∆t. The variables, like a 1 , a 2 , a 3 , b 1 , b 2 , b 3 ... affect their state at the next moment, so the corresponding nodes between adjacent time slices are connected by arcs. The structure model of the DBNs model is shown in Fig. 3.
The parameter modelling of DBNs also includes two parts, namely intra time slice parameter modelling and inter time slice parameter modelling. Firstly, the prior probabilities of the root nodes are required to be determined when establishing the intra time slice parameter model. The prior probabilities or distributions are usually obtained from expert experience and historical data. However the trends of prior probabilities are usually the same for structure systems, but the quantitative values are different. According to the actual situation, the prior probabilities can be exact numbers, and the prior distributions can be exponential distribution, normal distribution, etc. Next, the conditional probabilities among child nodes and parent nodes need to be determined. For nodes that have a definite causal relationship, the conditional probabilities among them can be represented by standard values, such as 0 and 1. If there are exact expressions among nodes, the expressions can be transformed into conditional probabilities. When establishing the inter time slice parameter model, the transition probabilities among time slices also need to be determined. The transfer relationship among time slices follows certain rules, such as Markov law, or laws given by physical models.

Resilience calculation
The performance changing trend of the structure system can be integrated according to the DBNs models of the degradation process and the recovery process. The structure performance is determined by the estimated objects. For example, the performance can be reliability, safety, availability of structure systems, or crack depth. The structural performance changing trend is used to calculate the system RUL values. As shown in Fig. 4, the system is attacked by an external disaster at time t 1 , and the RUL degrades over time. At time t 2 , the structure system starts to be repaired and the RUL of the system is gradually increasing. Until time t 3 , the RUL of the structure system returns to a new stable value. The time period from time t 1 to t 2 is the degradation process, the structure system degrades from the working state to the failure state, and the time period from time t 2 to t 3 is the recovery process of the structure system, and it recovers from the failure state to a new stable state. The degradation time and recovery time are determined according to the expert experience and historical data. During the degradation process, a series of activities such as configurations of maintenance equipment and allocations of main-tenance personnel are in need to be completed. The degradation time is longer than the effect time of external shocks, and it may need two days. Recovery time refers to the period from the beginning to the end of the maintenance activity. In this process, the failure components need to be diagnosed, repaired, and installed to original location. The recovery time may need three days. The system resilience is calculated as the area ratio of the shaded part to the rectangular part which is enclosed by the dotted line and abscissa axis. The calculation equation is shown as follows: 3 Case study

Introduction of subsea oil and gas pipelines
To the date, subsea production system gradually becomes a mainstream mode of deep water oil and gas resources development with the development of offshore oil and gas industry extending to deep water. Subsea pipelines are important in the connection among various subsea equipments. What is more, subsea pipelines are also critical for the structure safety of the offshore oil and gas industry. Under the harsh working condition of the subsea, the subsea pipelines are often attacked by various external disasters, such as earthquake and internal waves. At the same time, the subsea pipelines can also be influenced by natural factors from the ocean environment, such as sand erosion and corrosion. The comprehensive influences of natural factors and external factors should be considered in the resilience assessment for the subsea pipelines. The RUL of the subsea pipelines is an important metric to assess the resilience. Crack is the main failure mode of the subsea pipelines, and it determines the RUL primarily. To quantitatively assess the resilience of the subsea pipelines with the impact of external disasters, the crack depths need to be evaluated firstly.
Corrosion and sand erosion are the main natural factors that cause the occurrence and growth of the crack. There are two types of the pipeline corrosion, namely external corrosion and internal corrosion. The subsea oil and gas pipelines  are more susceptible to internal corrosion. For the internal corrosion, it is the electrochemical process which is formed through the presence of contaminants, like carbon dioxide and hydrogen sulfide. The internal corrosion starts in a local part, and it develops slowly. The corrosion rate is influenced by the concentration of carbon dioxide, temperature and pressure. For the external corrosion, the surrounding environment of the pipelines determines its mechanism. The increasing of the concentration of the dissolved oxygen on the metal surface, the increase of current velocity, and the decreasing of seawater temperature can significantly increase the speed of external corrosion (Yang et al., 2017). Sand erosion is another important reason that causes the occurrence of the crack. Sand erosion can arise severe crack in the internal wall for the pipelines, particularly for elbows and tees. The sand that included in the fluid can damage the protective film in the pipeline, so the failure rate of the pipeline is accelerated. The external factors are determined by the types of external disasters. Earthquake is one of the most typical environmental excitations during the service life of the subsea pipeline (Zhang et al., 2019). Many researchers focus on studying the influences of earthquake on the engineering systems. For example, Mazumder et al. (2020) presented a framework to evaluate risk and resilience of water distribution systems with considering both the influence of earthquake and time-variant corrosion of pipelines; Lin and El-Tawil (2020) simulated the resilience of lifeline systems in a test bed subjected to a series of seismic events. Under the influence of the earthquake, the ranges and numbers of cyclic stresses that suffered of the pipeline increase significantly. This phenomenon can lead to the sudden changes of the fatigue mechanical properties, resulting in rapid expansion of cracks and the failure of subsea pipelines.
The proposed resilience assessment metric and methodology are demonstrated through using the subsea oil and gas pipelines (elbows unless otherwise specified). At the same time, the combined influences of natural factors and external factors are studied in the case. Crack depth is selected as the system performance, and the RUL of the subsea oil and gas pipelines is taken as the assessment metric to calculate the resilience value.
3.2 Resilience assessment model of subsea oil and gas pipelines (1) Modelling of earthquake The pipelines are affected by wave load when they are influenced by the earthquake, which can lead to strong vibration of the pipelines. The physical model to demonstrate the influence of vibration is the Paris law (Luque and Straub, 2016;Straub et al., 2009), shown as follows: where dD(n)/dn is the crack growth rate; n is the number of cyclic stresses; C and M are empirically determined material parameters, and the value of C is related to M; D(n) is the crack depth; ΔS e is the equivalent stress range per cycle, which can be empirically represented as: λ where Γ() represents the Gamma function; is the scale parameter of Weibull distribution; K is the shape parameter of Weibull distribution.
Assume that the initial depth of the oil and gas pipelines is D 0 , the crack depth at the n-th stress cycle is calculated as: The physical model of the earthquake is mapped into a BNs model, and the structure model is shown in Fig. 5. The distributions and corresponding values of the variable nodes are obtained through the expert knowledge and historical data, as shown in Table 1. The performance changing trends of the pipelines under the influence of different strengths of external disasters can be modelled by changing the number and range of stress cycles.  (2) Modelling of corrosion The corrosion of the subsea pipelines is mainly caused by CO 2 corrosion. Many companies have developed various corrosion prediction models in the oil and gas industry. The most widely used semi-empirical model is the De Waard95 model (Nešić et al., 2003). This model considers the comprehensive impact of temperature, partial pressure of carbon dioxide, pH value, dynamic process of corrosion, mass transfer process and so on. The corrosion rate is mainly related to the reaction rate and the mass transfer rate, shown as follows: where V corr is the corrosion rate; V r is the reaction rate; and V m is the mass transfer rate. V r and V m are calculated as (De Waard et al., 1991): +0.58 log P CO 2 −0.34(pH act − pH CO 2 ); (13) where t is the temperature of the fluid; P CO2 is the pressure of CO 2 ; pH act is the actual pH value; pH CO2 is the pH value of the CO 2 saturated solvent; U is the liquid flow velocity; d is the pipe diameter; n C is the fraction of CO 2 in the gas phase; p o is the operating pressure.
The physical model of the corrosion is mapped into a BNs model, and the structure model is shown in Fig. 6. According to the expert knowledge and historical data, the value of n C is 0.5 and the value of p o is 52. The corresponding values of other variable nodes are shown in Table 2.  (3) Modelling of sand erosion Sand erosion is caused by the particles contained in the fluid, and the particles have sufficient momentum to impinge the pipe wall. Many parameters can affect the rate of the sand erosion, such as the liquid flow velocity, the fluid density, the size of the sand and the dimensions of the subsea pipeline. So it is hard to forecast the rate of the sand erosion exactly, and the prediction usually relies on semiempirical models. For the subsea oil and gas pipelines, one of the most used physical models is the Tulsa model, as shown below (Vieira et al., 2016), where R e is the rate of the sand erosion; K is the empirical parameter that is related to the material of the pipeline; F s is the particle shape coefficient, and it is equal to or smaller than one for sharp, semi-round and fully-round sand; V p is the particle impingement velocity; n is an empirical con-θ stant; f( ) is the empirical function for incorporating the particle impingement angle, and it is represented as: where is the characteristic value of the particle impingement angle.
The physical model of the sand erosion is mapped into a BNs model, and the structure model is shown in Fig. 7. According to the expert knowledge and historical data, the value of K is 2×10 −9 , and the value of F s is 0.02. The corresponding values of the other variable nodes are shown in Table 3.  (4) Entire DBNs for degradation process of subsea oil and gas pipelines The three BNs models for the degradation process of the subsea oil and gas pipelines under the impacts of natural factors and external factors are integrated, and the integrated model is extended to construct the total DBNs model, as shown in Fig. 8. The performance degradation is affected by various factors, and the effects are quantitatively described by the physical models, like the Paris law, De Waard model, and Tulsa model. In the DBNs model, the transition probabilities between adjacent time slices are de- termined by the physical models. When integrating the entire model, the performance indexes of the three sub-networks need to be unified. In other words, the same time scale should be used to measure the variation of crack depth. Under the influence of the earthquake, the crack depth is calculated as the product of the crack growth rate and the number of cyclic stresses. Under the influence of the corrosion and the sand erosion, the corrosion rate V corr and the sand erosion rate R e need to be transformed into crack depth, and the value is calculated as the product of rate and time. In the integrated BNs model, the structure model is constructed by linking the performance nodes D n , V corr and R e in the three sub-network to the total performance node C. The parameter model is constructed by the relationships among D n , V corr , R e and C. In this case, the performance is represented by the crack depth. Crack is the main failure mode of the structure system, and it is closely related to structural mechanics principles. For the structure system, the crack depth can directly reflect the system performance. The subsea oil and gas pipelines are vulnerable to environmental assisted crack under the harsh working conditions. Environmental factors, such as corrosion and sand erosion, are the main natural factors that cause the occurrence and growth of the crack. RUL is selected to be the middle variable to develop a new resilience metric because it is influenced directly by the failure mechanism of structure systems. The RUL is calculated based on the total crack depth of the elbows, and then it is used to quantitatively assess the resilience of the elbows. Finally, the assessment model of the degradation process representing the performance changing trend is obtained by expanding the integrated model into a DBNs model .
(5) Modelling of recovery process of subsea oil and gas pipelines Maintenance measure is one of the most critical operations in operational activities of the subsea pipelines. According to the operation position, maintenance measure of subsea oil and gas pipelines can be divided into two parts, namely above-water operation method and underwater operation method. For the above-water operation method, it repairs the pipelines after lifting the damaged segment to the support vessel. The technology of this method is more mature and repair time is shorter. For the underwater method, the maintenance activity needs to be completed on the seafloor, and it can be divided into two parts, namely the repair in a dry-type cabin and the repair in water. The maintenance method is decided by the damage degree and the mechanism of the subsea pipelines, pipeline materials and so on.
Markov law is adopted to demonstrate the transition laws among different states when modelling the recovery process of the subsea pipelines. DBNs and Markov are combined to model the recovery process. The structure model is constructed according to the relationships among the sys-tem components; the parameter model is developed according to Markov law. Through considering the actual situation of the maintenance activities for the subsea pipelines, the pipelines are maintained in the 40th hour after the occurrence of the external disaster. A series of activities such as configuration of maintenance equipment and allocation of maintenance personnel are both needed to be completed after the occurrence of the external disaster. When modelling the DBNs model of the maintenance process, the prior probabilities are determined by the final state of the degradation process. For the transition relationships between adjacent time slices, they are represented by Markov law as: μ where p is the transition probability of nodes; is the maintenance rate, and it is determined by the maintenance time of the pipelines.

Structure resilience under different strengths of earthquake
According to the developed DBNs model of the degradation process, the distributions and corresponding probabilities for the crack depth of the elbows are shown in Fig. 9. It can be seen that the peak of probabilities for the crack depths move to the right over time under the combined impacts of natural and external factors. In other words, the crack depths of the elbows are gradually increasing as time goes on. After taking the maintenance activities, the distributions and corresponding probabilities for the crack depth in the recovery process are shown in Fig. 10. It can be seen that the peaks of probabilities for the crack depths move to the left over time, that is, the crack depths of the elbows gradually decrease as time goes on.
In the case study, the performance of the subsea oil and gas pipelines is represented by the crack depth. The RUL value is calculated through quantifying the crack depth of elbows. The value of the crack depth is calculated through the sum of products of the crack depths and the correspond- ing probabilities. The failure threshold is selected as 20.00 mm in the current case.
The structure resilience values under the influence of high, medium and low strength are respectively studied. The changing trends of the crack depths under different earthquake strengths are plotted in Fig. 11. It can be seen that the crack propagation speed becomes faster and the crack depth reaches to the threshold earlier with the increasing of the earthquake strength. In the later period, the crack depth increases almost exponentially, because the earthquake has greater effect on the elbows than others. Since the recovery process of the subsea pipelines follows Markov law, the maintenance time is longer if the crack depth is deeper, and the pipelines need more time to reach a new state. According to quantized crack depth of the elbows, the changing trends of the RUL with different earthquake strengths are calculated, as shown in Fig. 12. The RUL of the pipelines decreases rapidly with the increasing of the earthquake strength. The RUL value degrades to 0 only after 36 hours under the impacts of the high earthquake strength, that is, the pipelines reach a complete failure state. The pipelines will remain in the complete failure state if the maintenance measures are not taken by operators. The RUL of the pipelines will gradually recover to a stable state by taking the maintenance measures. However, the RUL of the pipelines cannot reach the original level because of the ex-istence of the natural factors, such as corrosion and sand erosion.
According to the resilience calculation method in Fig. 4, the resilience of the elbows under the influence of different earthquake strengths are calculated, and the results are shown in Fig. 13. They are 77.17%, 73.61% and 72.02%, respectively. The resilience gradually decreases with the increasing of the earthquake strength. It can be seen that the resilience values are not significantly different. The impact of the earthquake on the subsea pipelines is enormous, and the pipelines can reach the complete failure state in a short time period. The strength changing has little impacts on the pipelines, so that the resilience values are not significantly different.

Structure resilience with different initial crack depths
It can be seen from Fig. 14 that the crack depth grows more rapidly with the increasing of the initial depths. When the initial crack depth of the elbows is 0.1 mm, the crack depth grows very slowly in the early stage of the degradation process, and the crack depth only increases to 5 mm in the 30th hour. In the later stage, the crack depth grows rapidly over time, and the changing trend is almost exponential. When the initial crack depths of the elbows are 0.15 mm and 0.2 mm, the degradation trends are almost similar. The growth rate of the crack is obviously increased in the early period, and the crack depth grows to 5 mm in the 18th hour.  In the later period, the crack depth grows very rapidly, and it increases almost exponentially. The RUL changing trends with different initial crack depths are shown in Fig. 15. The RUL of the elbows decreases more rapidly with the increasing of the initial crack depth. The recovery trends are totally the same because the elbows all reach the complete failure state during the degradation process. According to the changing trends of the RUL, the resilience values for the elbows with different initial crack depths are calculated, and the results are shown in Fig. 16. They are 78.43%, 73.61% and 73.22%, respectively. The structure resilience of the elbows gradually decreases with the increasing of the initial crack depth. When the initial crack depth of the elbows are 0.15 mm and 0.2 mm, the degradation trend and recovery trend are almost similar, so the resilience values have no big difference.

Structure resilience for different pipeline components
Elbows, tees and straight pipes are three significant components of subsea pipelines. In this part, crack depth growths, RUL changing trends and resilience values of tees and straight pipes are studied, and the results are respectively compared with the elbows. It can be seen from Fig. 17 that crack depth of elbows is the largest over time, the crack depth of straight pipes is the smallest, and the value of tees is in between. That is because the effect of the sand erosion on the straight pipes is the smallest, and the tees are generally made of materials which are resistant to corrosion and sand erosion.
The changing trends of the RUL for the different parts of the subsea oil and gas pipelines are shown in Fig. 18. The degradation speed of the RUL for the straight pipes is the slowest, the degradation speed of the elbows is the fastest, and the degradation speed of the tees is in between. These are corresponding to the crack growth trends for the elbows, tees and straight pipes. At the same time, the recovery trends are totally consistent for the elbows, tees and straight pipes because they all reach the complete failure state during the degradation process.    CAI Bao-ping et al. China Ocean Eng., 2020, Vol. 34, No. 5, P. 597-607 According to the changing trends of the RUL, the resilience values of the elbows, tees and straight pipes are calcu-lated, and the results are shown in Fig. 19. They are 73.61%, 74.8% and 75.99%, respectively. The resilience for the elbows is the lowest. The overall pipeline system will fail if the elbows are in failure, so the overhaul cycle for the elbows should be shortened to enhance the resilience of the overall pipeline system.

Parameter sensitivity analysis
Parameter sensitivity analysis is the analysis of how sensitive the results of a belief update is to variations of the value of a parameter of the model. It is measured according to the variance of beliefs and variance reduction. The parameters sensitivity analyses for the different nodes are conducted in the 1st hour, 20th hour and 40th hour, and the results are shown in Fig. 20.
In the early period of the degradation process, i.e. the 1st hour and 20th hour, the corrosion has a great influence on the crack depth of the pipelines. In the later period, i.e. the 40th hour, the earthquake has a great influence on the pipelines. In the early period, the initial depth of the crack is small, and the depth is more sensitive to the mass transfer velocity. The fluid velocity can considerably affect the corrosion and the sand erosion, so the corrosion and sand erosion contribute greatly in the crack growth. When the initial crack depth of the elbows is big, the influence of the earthquake can quickly deteriorate the degradation process, so it dominates the crack depth propagation in the later period. Other variables, such as pipe diameter, operating pressure, and fluid temperature have small sensitivities on the crack depth growth.

Conclusions
This paper proposes a new resilience assessment metric and its corresponding methodology based on the RUL of the key components of the structure systems. The weakest parts, that is, the elbows of subsea oil and gas pipelines are used to demonstrate the application of the metric and methodology. The influences of the strengths of the external disasters, the depths of initial crack, and the types of pipelines on the structure resilience are researched, and the parameter sensit-ivities are analyzed. The results show that the assessment metric and methodology can be used to calculate the resilience of structure systems. Under the integration impact of natural factors and external factors, subsea oil and gas pipelines can reach the complete failure state within 40 hours. The system resilience decreases with the increasing of the strengths of external disasters, and corresponding resilience values are all below 80%. The initial depth also affects the degradation speed greatly. When the depth of the initial crack for the elbows is larger than 0.1 mm, the pipelines are totally failed within 36 hours. Therefore, the initial crack requires to be strictly detected and controlled. The pipelines are sensitive to the mass transfer velocity in the early period, that is, the corrosion has a larger effect on the pipelines, and they are sensitive to the stress range in the later period, that is, the effect of earthquake is larger.