Digital twin-based life-cycle seismic performance assessment of a long-span cable-stayed bridge

Long-span cable-stayed bridges often have a design service life of more than a hundred years, during which they may experience multiple earthquake events and accumulate seismic damage if they are located in seismic-prone regions. Earthquake occurrence is discretely and randomly distributed over the life cycle of a long-span cable-stayed bridge and often causes sudden drops in the structural performance instead of yearly fixed seismic performance degradation. This study thus proposes a digital twin-based life-cycle seismic performance assessment method for long-span cable-stayed bridges. The major components of this method include: (1) a seismic hazard analysis-based generation method of earthquake occurrence sequence; (2) a digital twin-based structural response prediction method considering lifetime earthquake occurrence and sequence; and (3) a service life quantification method. The proposed method is applied to a scaled long-span cable-stayed bridge with a series of shake table tests. The results show that the digital twin can closely reproduce the life-cycle seismic response of the bridge under sequential earthquakes. The proposed assessment method provides a more intuitive presentation of the life-cycle seismic damage accumulation process and a more accurate estimation of the service life of a long-span cable-stayed bridge.


Introduction
Long-span cable-stayed bridges serve as critical infrastructures in highway transportation systems. If they are located in earthquake-prone zones, seismic action becomes a dominant hazard that threatens their functionality and safety. Long-span cable-stayed bridges are often designed to have a service life of more than a hundred years. They may thus experience multiple seismic events in their service life, and the earthquake shocks may potentially cumulate damage on the bridges. It is therefore vital to carry out the life-cycle seismic performance assessment of long-span cable-stayed bridges in consideration of the importance of resilience and sustainability of highway transportation systems.
The performance assessment of a long-span cable-stayed bridge at the scale of life cycle requires considering the effects of degradation, while the mechanisms causing a bridge to degrade are typically categorized as aging and point-in-time extreme events. Degradation because of aging takes place gradually over time, while point-in-time extreme events, such as earthquakes and typhoons, occur instantaneously with respect to structural life but are safety-threatening (Wen and Kang 2001a). Although the life-cycle performance assessment of a structure shall consider all the mechanisms causing the structure to degrade during its service life, the complexities and uncertainties involved in such an assessment often make it reasonable to first consider individual or some mechanisms in detail as one of the components in the life-cycle assessment (Seo and Caracoglia 2013;Dong and Frangopol 2016). This is particularly true when considering the importance of seismic resilience (Bruneau et al. 2003;Cimellaro et al. 2010). Earthquakes are typically point-in-time extreme events that could cause a sudden drop in structural performance, compared to progressive aging. The occurrences of earthquakes are instantaneous with respect to the structural life so that it is advantageous to model the earthquake-induced cumulative damage process separately from the slowly time-varying aging effects and others. For example, Seo and Caracoglia (2013) and Iervolino et al. (2013) considered strong winds and earthquakes, respectively, in the structural life-cycle assessment without involving aging effects. Regarding life-cycle seismic performance assessment, many researchers focus on quantifying the seismic resilience of structures. Iervolino et al. (2013Iervolino et al. ( , 2015Iervolino et al. ( , 2016 discussed the stochastic modeling of seismic damage accumulation for life-cycle seismic performance assessment of structures. Sharanbaswa and Banerjee (2018), Vishwanath and Banerjee (2019), ,  investigated the lifecycle seismic resilience of aging bridges and/or road networks. Zheng and Dong (2019) analyzed the life-cycle seismic performance of highway bridges equipped with steelshape memory alloy (SMA) reinforced piers. Frangopol and Soliman (2016) presented a brief overview of the research achievements in the field of life-cycle engineering for civil and marine structural systems and discussed the future directions in this research field. Frangopol et al. (2017) further addressed a generalized framework for assessing bridge life-cycle performance and cost in consideration of multiple hazards occurring to the bridge during its service life, and with emphasis on analysis, prediction, optimization and decision-making under uncertainty.
Probabilistic seismic demand and fragility analysis are two key elements in the seismic performance/ resilience assessment of structures (Karamlou and Bocchini 2015), while the accurate seismic performance assessment relies on the accurate quantification of uncertainties involved in both seismic demand and fragility analysis. The structural health monitoring technology may provide a better way in reducing uncertainties in the structural performance assessment (Okasha and Frangopol 2010;Frangopol et al. 2017;Hu and Xu 2019;Xu and Hu 2021). Lin et al. (2020aLin et al. ( , 2021a established the digital twin of a long-span cable-stayed bridge using the concept of structural health monitoring in terms of parallel computing and advanced optimization algorithms. Since the most privileged advance of a digital twin is to closely and timely reflect the actual service state of a physical entity and reinforce information-based services during the entity's life cycle (Tao et al. 2019), the use of the digital twin in the life-cycle seismic performance assessment of long-span cablestayed bridges is naturally expected and taken as the main objective in this study.
Furthermore, most fragility analyses assume a yearly fixed extent of earthquakeinduced performance degradation. However, in real situations, an earthquake is a typical kind of low-probability and high-consequence point-in-time extreme event. Earthquake occurrences are discretely and randomly distributed over the lifetime of a structure. Seismic action thus often causes a sudden drop in the structural performance instead of the yearly fixed extent of seismic performance degradation. Therefore, some researchers (Iervolino et al. 2013(Iervolino et al. , 2015(Iervolino et al. , 2016 modeled the lifetime earthquakes of a structure and assessed the accumulated seismic damage by considering the earthquake sequences chronologically during the service life of the structure in terms of the Gamma degradation model or the Markovian modeling-based seismic damage accumulation process. Nevertheless, whether or not the consideration of earthquake sequence chronologically is prohibited depends on the accuracy of the computation of life-cycle seismic response of a long-span cable-stayed bridge. It is also unknown whether or not the Gamma degradation model or the Markovian modeling-based seismic damage accumulation process is applicable to long-span cablestayed bridges. This study focuses on the life-cycle seismic performance assessment of a long-span cable-stayed bridge as one of the important components of the life-cycle performance assessment of a bridge. A digital twin-based life-cycle seismic performance assessment framework, which includes the digital twin-based seismic response prediction method, the lifetime earthquake sequence generation method and the service life quantification method, is proposed to evaluate the structural life-cycle seismic performance of long-span cablestayed bridges. A scaled long-span cable-stayed bridge and its shake table tests are taken as an example to demonstrate and verify the proposed method. The test data of the physical scaled bridge is used for producing the digital twin of the bridge through nonlinear model updating. The lifetime earthquake sequences are randomly generated based on the seismic hazard analysis of the bridge and the Poisson process sampling. The digital twin-based finite element (FE) model of the bridge is then used for simulating the seismic damage cumulative process of the bridge subjected to multiple earthquake events within its service life. A service life quantification method is finally developed for evaluating the service life of the bridge.

Shake table test of a long-span cable-stayed bridge
The digital twin-based life-cycle seismic performance assessment method proposed in this study relies on the development of a digital twin for a long-span cable-stayed bridge during its service life. A digital twin refers to a digital replica of a physical entity (Grieves and Vickers 2017), such as a physical model of the bridge. In this study, a scaled longspan cable-stayed bridge and its shake table tests are used as an example to demonstrate how to develop the digital twin of the bridge through nonlinear model updating. It is true that using a real long-span cable-stayed bridge will make the proposed method more persuasive. However, the service conditions of a real long-span cable-stayed bridge are much more complicated and the acquisition of the required measurement data during earthquake events within life time is almost impossible. Hence, the physical model of the bridge rather than the real bridge is used in this study at the moment to validate the digital twin-based seismic response prediction and service life quantification methods. The prototype of the scaled bridge is the Sutong cable-stayed bridge that locates in Jiangsu Province of China (Fig. 1a).

Physical scaled bridge and measurement system
A length scale ratio of 1:35 was used to construct the physical scale model. The dimensions of the prototype and scaled bridge are shown in Fig. 1a and b, respectively. The total length of the scaled bridge is 59.66 m and the height of each tower is 8.58 m. The girder was made of a steel box with a hollow rectangular section. Reinforced micro concrete was used to construct the towers and piers. The bases of the two towers and the four piers were fixed on the four shake tables respectively. An advanced measurement system, including accelerometers, displacement transducers, strain gauges and other sensors, was installed on the bridge model to record its structural responses during the tests (Guan et al. 2019). The arrangements of the accelerometers and displacement transducers at critical locations of the physical bridge are depicted in Fig. 1c. Apart from that, several strain gauges were installed along the height of the bridge towers to monitor the micro-level seismic damage to the bridge components. The measured displacement responses were used in the model updating for the data fusion of the digital twin in the virtual space. The simulated macro and micro responses were compared to the measurement results to validate the ability of the updated digital model for capturing the nonlinear seismic response of the physical bridge with damage (Lin et al. 2021b). The typical sectional details of the major components of the bridge are displayed in Fig. 1d.

Test case arrangement
The scaled bridge model was tested using the shake table facility in the Multi-Functional Shaking Table Laboratory of Tongji University (Guan et al. 2019;Zhou et al. 2019), as shown in Fig. 1b. An artificial site ground motion was selected as the input. It was scaled to different magnitudes and then fed into the shake table sequentially (see Table 1) to investigate the earthquake-induced damage progression of the scaled bridge (Guan et al. 2019;Zhou et al. 2019). A white noise signal was also used to input to the shake table after the artificial site ground motion to ascertain the dynamic characteristics and damage states of the physical bridge. According to Guan et al. (2019), the bridge was found to remain elastic from test cases E2 to E5 (see Table 1). After that, damage gradually accumulated from test case E7 to E15 and led to the collapse of the bridge when the peak ground acceleration (PGA) of the input ground motion equaled 1.3 g in test case E15.
To quantify the accumulated seismic damage of a bridge during its service life and thereby assist its life-cycle performance assessment, it is vital to ensure that the digital twin is capable of accurately simulating the structural responses of the physical bridge under sequential earthquakes. In reality, the service state of a physical bridge varies over time.
Hence, its digital twin should be timely updated against the real-time measurement data during its life cycle. Based on test observations and considering the main objectives of this study, two stages of digital twin construction are implemented. The measurement data from the test cases E9 and E13 are respectively used to update the FE models of the physical bridge so as to obtain the digital twins of the scaled physical bridge at the two stages.
In the first stage, the measurement data from the test case E9 is used to update the 1 st -stage digital twin, denoted as E9-DT. This E9-DT is used to elaborate the proposed digital-twin-based framework for predicting the structural responses of long-span cablestayed bridges under future earthquakes. Based on E9-DT, the ground motions from the test cases E9 to E15 are fed to this model sequentially as that happens in reality during the shake table test. The simulation results are compared to the experimental measurement data recorded at the test cases to validate the accuracy of the digital twin technology in predicting the structural responses and damage of the physical bridge.
In the second stage, the test case E13-based 2nd-stage digital twin, denoted as E13-DT, is used to demonstrate the proposed life-cycle seismic performance assessment method for the cable-stayed bridge. The primary reason for selecting the test case E13 for the 2nd-stage digital twin construction is that this test case recorded the most recent nonlinear state of the physical bridge right before the collapse test case (E15).

FE modeling
The original FE model of the scaled bridge was established by Lin et al. (2020a) based on the design drawings of the scaled bridge and using the software MSC. Marc (2012), as shown in Fig. 2.
In the FE model, the two inverted Y-shape towers and the four side piers were modeled using the multi-layered shell elements and fixed on the foundations. Compared to beamcolumn elements in many other studies, shell elements could provide more intuitive modeling of the rectangular hollow section towers of the physical bridge, especially the beam-column conjunction areas. Furthermore, the use of shell elements for simulating seismic behaviors of pylons and piers of bridges has also been widely validated in many other studies (Son and Lee 2011;Li et al. 2018a, b, c;Lin et al. 2021b;Qiu et al. 2022). The shell elements contained both concrete and steel layers to simulate the composite material behavior of reinforced concrete components (Lu et al. 2013), whose thickness was determined based on the sectional reinforcement details as shown in Fig. 1d. For the concrete model, the widely used Kent-Scott-Park model was adopted (Scott et al. 1982). For the reinforcement steel, the material model proposed by Esmaeily and Xiao (2005) was adopted. The element deactivation method of Lu et al. (2013) was adopted for the collapse simulation of the bridge, which has been proved to be capable of simulating the failure of reinforcement concrete components resulting from concrete crushing, rebar fracture and rebar buckling through the material-level failure criteria (Rodriguez et al. 1999;Kunnath et al. 2009;Lu et al. 2013;Lin et al. 2021b). Furthermore, these shell elements were modeled along the neural axis of the wall of the hollow tower section. The changes in the wall thickness were modeled by changing the thickness of the shell elements based on the design drawings of the bridge (Fig. 1d). At the corners of the tower sections, the massoverlapping was avoided by adjusting the density of the corresponding shell elements.
Moreover, the steel box girder, which deformed elastically during the shaking table tests, was modeled by elastic shell elements. In the scaled bridge, additional masses were employed on the towers, girder and piers for achieving the expected mass distribution according to the similitude requirements (Guan et al. 2019). In the FE model, the additional mass was modeled by adding elastic elements with corresponding mass to the numerical model.
The pre-stressed cables were modeled by truss elements with initially added prestresses. Each cable was modeled by one truss element to avoid the matrix singularity during calculation. The sag effects were considered by the Ernst method (Karoumi 1999). In the experimental tests, these cables were anchored to build-in steel frames of the two main towers, which were modeled by elastic beam elements that shared nodes with the shell elements of the towers. The truss elements were connected to the elastic beam elements to model the cable anchorages. The bridge was fixed on the ground by adding constraints to the nodes at the bottom of the tower and pier foundations. A mesh sensitivity analysis was performed to determine the proper mesh size of various components in the numerical model. After meshing, the FE model had more than 16,000 elements and 11,500 nodes. More FE modeling details can be found in Lin et al. (2020a).
During the nonlinear time history analyses of the bridge, the gravity load was applied to the model by adding gravitational acceleration, and the ground motions were input by applying the corresponding acceleration time history to the FE model and running the nonlinear transient analyses (Lu et al. 2013). The widely used Rayleigh damping was adopted by setting the stiffness and mass matrix coefficients at the material level (Chopra 2012). The damping ratio was nonlinearly updated using the measurement data (Lin et al. 2021a).
Regarding the numerical solver, the single-step Houbolt operator, which is unconditionally stable, second-order accurate and asymptotically annihilating, was adopted as the numerical solver for the time history analysis of the bridge (Chung and Hulbert 1994). A fixed step-time of 1/256 s was used in the transient analyses, which was identical to the sampling time interval of the measurement data. A sensitivity analysis was conducted to verify the rationality of the selected step-time. The simulation results from the time history analyses with step-times of 1/128 s and 1/512 s were compared to make sure the selected step-time could provide reliable results.

Cluster-computing aided nonlinear FE model updating
Data fusion is a vital phase of the digital twinning process, which enables the digital twin to closely reflect and predict the actual responses of the physical entity. In data fusion, the measurement data from the sensory system are used to update the original FE model. The cluster-computing aided nonlinear FE model updating program developed by Lin et al. (2020a) was used to update the original FE model so as to obtain the high-fidelity digital twin of the bridge at the two stages (E9-DT and E13-DT) by means of the measurement data recorded during the test cases E9 and E13, respectively. A 115-core cluster, which was built up by five high-performance servers (dual Intel Xeon E5-2670 v3 @ 2.30 GHz processors, 512 GB memory), was used as the computing platform for running the program. More details of the nonlinear FE model updating program, including the response time history-based objective function, the selection of updating parameters, optimization algorithm, and code framework of the program can be found in Lin et al. (2020b). After the nonlinear model updating, the comparisons of the structural responses of the digital twin with the measured responses of the physical bridge in the test cases E9 and E13 are shown in Figs. 3 and 4, respectively, in which the measurement points refer to Fig. 1c. It can be seen from Figs. 3 and 4 that the simulated response time histories match well with the measured ones based on the error index proposed by Lin et al. (2020b), indicating that the digital twin (E9-DT and E13-DT) at both stages represent the actual bridge well.

Validation of digital twins under sequential earthquakes
As mentioned above, whether or not the digital twin can be used to predict the nonlinear structural response of the bridge under sequential earthquakes should be validated so that the digital twin-based life-cycle performance assessment can be conducted. In the shake table test, the PGAs of the input ground motions increased from 0.7 g to 1.3 g chronologically from the test cases E9 to E15. Correspondingly, the 1 st -stage digital twin (E9-DT) is used for the required validation by simulating the structural responses during these designated test cases. Based on the E9-DT, the same ground motion sequence (see Fig. 5) is input to the model to predict the structural responses of the scaled bridge from the test cases E11 to E15. The predicted structural responses are then compared with the measured Fig. 4 Guan et al. (2019), the PGAs of the white noise inputs were strictly controlled at 0.1 g to make sure that the bridge remained unchanged and the measurement data can be used to identify the dynamic characteristics of the bridge after the previous artificial site ground motion. Hence, in this study only the artificial site ground motion inputs are considered in the ground motion sequence as shown in Fig. 5. A time interval of 160 s is inserted between the two different test cases for the bridge to dissipate the input energy and return to stationary condition after each artificial site ground motion input (Lin et al. 2021b).
The simulated damage progression and collapse mode of the bridge are presented in Fig. 6. The photo of the collapse mode of the scaled bridge model after the test case E15 is also shown in Fig. 6. According to the experimental observation of Guan et al. (2019), significant buckling of rebars and crushing of concrete were found in the beam-column conjunction area above the north tower column when the bridge started failure in the test case E15. The comparison of the structural responses of the north tower at the three typical measurement points between the digital twin and the physical entity is shown in Fig. 7. The three typical measurement points are located near the conjunction area of the north tower (TDY-2, TDY-4 and TDY-5 in Fig. 1c), which is the most vulnerable area of the bridge tower (Lin et al. 2021b).
According to the simulated damage progression using the digital twin, the beam-column conjunction of the north tower is severely damaged at the end of test case E13, which is consistent with the experimental observation of Guan et al. (2019). Then, the damage keeps propagating and new damage is formed at the bottom of the north tower. Finally, the E9-DT collapses when the PGA of the input ground motion is increased up to 1.3 g in the test case E15 (Fig. 6). The comparison between the response time histories, as shown in Fig. 7, indicates that the simulated results closely match the measurement data from the test case E9. For the subsequent test cases, the simulation errors gradually accumulate due to the cumulative damage resulting from the test cases, but the simulated response time histories still match satisfactorily with the measured ones as shown in Fig. 7. Note that as more test cases are involved in the response prediction, the errors between the simulated results and measurement data could inevitably increase due to more uncertainties being introduced to the simulation. Moreover, the E9-DT even successfully predicts the collapse of the physical entity in the test case E15 although the collapse mechanism and time point are slightly different between the simulation and measurement but one thinks about the complexity of the problem and the simulated collapse is based on the E9-DT through the test cases from E11 to E15. Note that in practice, the digital twin is developed and used evolutionarily, that is, the nonlinear model updating is conducted after every earthquake using the measurement data so that the cumulative modeling error can be reduced significantly. It can thus be concluded that the proposed digital twin-based simulation can help predict the structural seismic response and damage progression of a long-span cable-stayed bridge under sequential earthquakes if the digital twin is developed and used evolutionarily.

Lifetime earthquake sequence generation
To assess the life-cycle seismic performance of a bridge, it is vital to properly consider the seismic hazard risk of the bridge site. The Monte Carlo simulation (MCS) method had been widely used to generate site-specific earthquake scenarios for the determination of design earthquakes (Musson 1999;Wang and Lu 2018). Hence, the MCS-based method is used in this study together with the assumption that the occurrence of an earthquake follows a Poisson distribution (Cornell 1968;Kang 2001a, 2001b;SAC 2015). The occurrence of earthquakes during a given time period can then be simulated by the MCS-based method when the hazard curve of the bridge site is provided.

Seismic hazard curve of the bridge site
The most commonly-used power law expression (Sewell et al. 1991) is used in this study to depict the seismic hazard curve of the bridge site.
where IM is the intensity measure of a ground motion, which refers to PGA in this study; v is the yearly rate of exceedance of the ground motion with an intensity greater than the given IM; k and k 0 are the two shape parameters that determine the power-law expression and they can be calculated using Eqs. 2 and 3 below.
where E1 and E2 represent the two levels of design seismic action. The seismic hazard information of the prototype bridge is used to generate the lifetime earthquake sequences in this study. According to the design documents, the return periods of E1 and E2 earthquakes are 1000 and 2500 years, respectively, and the corresponding PGAs are 0.138 g and 0.166 g, respectively, where g is the gravitational acceleration. Based on the above design information, the seismic hazard curve of the bridge site can be worked out and shown in Fig. 8. Then the rate (annual probability) of exceedance (v) corresponding to a given k earthquake intensity can be determined, and the occurrence of the earthquakes within a given time period (t) follows the Poisson distribution given by where P(n) is the probability mass function; and n is the occurrence times of the earthquake in the time period t. Furthermore, for a given earthquake event, the time interval (Δt) to the next earthquake can be randomly generated following Eq. 5.

Lifetime earthquake sequence
As described above, in terms of the seismic hazard curve and the Poisson distribution, MCS is performed to generate the lifetime earthquake occurrence and sequence for the life-cycle performance assessment of the bridge following Eqs. 4 and 5. Since this study concerns the life-cycle performance and possible collapse of the bridge, the earthquakes with a large PGA but a small rate of exceedance should be considered. Based on the shake table tests and the FE simulation of the bridge, the prototype bridge is a typical flexible structure with relatively long vibration periods. It is considered that ground motions with PGAs smaller than 0.30 g can hardly induce damage to the bridge. Hence, a time period of 10 9 years, which is much longer than the actual design life of the prototype, and a threshold of 0.30 g, which is larger than the actual design earthquake intensity, are thus considered in generating the lifetime earthquake sequence of the bridge with a PGA increment of 0.01 g. The generated MCS-based earthquake occurrence and sequence in 10 9 years is shown in Fig. 9. Each circle point represents an earthquake that happens in the 10 9 years. The vertical axis is the PGA of the sampled earthquake while the horizontal axis represents the occurrence time of the earthquake. To examine the accuracy of the generated earthquake occurrences, the occurrences of earthquakes with different intensities are counted to obtain the simulated seismic hazard curve, which is then compared with the design seismic hazard curve of the bridge as shown in Fig. 8. The comparative results indicate that when the PGA is smaller than 1.5 g, the proposed MCS-based earthquake occurrence and sequence generation method can well represent the design seismic hazard of the bridge site. However, (4) P(n) = (vt) n e −vt ∕n! (n = 0, 1, 2, ...) (5) Δt = − ln (rand(0, 1))∕v since only limited earthquakes with a PGA greater than 1.5 g occur in the 10 9 years, the seismic hazard curve calculated based on the generated earthquakes slightly differs from the design one but the trend is the same. Note that for the bridge concerned, the rate of exceedance for the earthquake with a PGA of 1.5 g is less than 2 × 10 -8 , which means such an earthquake is unlikely to happen during the design life of the bridge. The MCS for generating the earthquake sequence of the bridge site in 10 9 years is repeated several times to make sure the results fit well with the hazard curve.
Moreover, this study concerns the life-cycle performance assessment and possible collapse of the bridge and would like to demonstrate the feasibility of the proposed method. Because the Sutong bridge locates in Jiangsu Province of China with a relatively low design seismic intensity, to include enough 0.30 g-above ground motions in the generated earthquake sequences, the earthquake occurrences in a 10 5 year time period are considered and randomly selected from those given in Fig. 9 for the life-cycle performance assessment of the bridge. Note that, if the bridge site has a higher design seismic intensity, the time window should be much shorter than the 10 5 years used herein. As a result, a total of 20 lifetime earthquake sequences are randomly selected from the 10 9 -year earthquake sequence as shown in Fig. 10. In Fig. 10, each subfigure represents a randomly generated lifetime earthquake sequence while in the subfigures, each circle point represents an earthquake event that happens during the lifetime of the bridge. The vertical axis represents the PGA of an earthquake event and the horizontal axis is its arrival time. It can be easily derived that the arrival time of the earthquakes is distributed between the initial time and final time following a Poisson process for the selected earthquake sequences. Each of these lifetime earthquake sequences in Fig. 10 is randomly matched with various ground motion records, as discussed in the next subsection, and is then input to the digital twin of the bridge for life-cycle performance assessment.

Ground motion database
To generate the lifetime ground motion inputs for the life-cycle performance assessment, proper ground motion records that reflect the seismic demands of the bridge should be selected to match up with the lifetime earthquake sequences shown in Fig. 10 (Vamvatsikos and Cornell 2004). In this study, a total of 110 natural ground motion records are selected from the NGA-West2 ground motion database of the Pacific Earthquake Engineering Research Centre (PEER) according to the 5% damped design seismic response spectrum of the prototype bridge in accordance with the specifications of earthquake resistant design of highway engineering (Ministry of Transport 1990) of China.
It is noted that because E13-DT is updated based on the measurement data recorded during the shake table test of the scaled bridge, the selected ground motion records should also be scaled according to the similarity law of the shake table test. Since the length scaling factor is 1:35 while the acceleration scaling factor is 1:1 (Guan et al. 2019), the time of the selected ground motions should be compressed to 1/35 0.5 before they are input to the digital twin of the bridge. After scaling, the 5% damped response spectra of all ground motion records are compared to the scaled design response spectrum of the bridge, as shown in Fig. 11. The p-value method is adopted to test the difference between the mean response spectrum of all the scaled ground motion records and the scaled design response spectrum, and as a result, the p-value is calculated as 0.895, indicating that the mean spectrum fits well with the targeted one.
These selected ground motions are then randomly matched up with the lifetime earthquake sequences given in Fig. 10 to generate the sequential ground motion inputs for the life-cycle performance assessment of the bridge. A typical example of the generated Fig. 10 Lifetime earthquake sequences for life-cycle performance assessment of the bridge sequential ground motion input, using the earthquake sequence No.10 in Fig. 10, is shown in Fig. 12. Note that in earthquake sequence No.10, the arrival time of earthquakes No. 1 to 3, earthquakes No. 4 to 5 and earthquakes No. 6 to 7 are close to each other according to Fig. 10. The corresponding time intervals are much smaller than the total time period of the earthquake sequence No.10 (i.e., 10 5 years). Hence, if the arrival time of these earthquakes is plotted in scale, it will be difficult to tell the difference of the earthquakes from each other in Fig. 12. Instead, a constant distance is maintained between adjacent earthquakes in Fig. 12 and the arrival time and PGA information of the earthquake is labeled near it. It is obvious that the ground motions in different scenarios have different arrival time, intensity and acceleration time histories.
Furthermore, it should be noted that during the life-cycle seismic performance assessment and with reference to the existing studies (Iervolino et al. 2013;Seo and Caracoglia 2013), the earthquake-induced cumulative damage process is analyzed separately from the time-varying aging effects and others. Hence, it is assumed that the damage states of the bridge remain unchanged between two adjacent earthquakes. In the FE analyses of the sequential earthquake scenarios, similar to the analysis in Sect. 3.3, a time interval of 160 s is inserted between different ground motions for the bridge to return to stationary condition after being excited by each of the ground motions in the sequence.

Life-cycle seismic performance assessment
The digital twin E13-DT, which is updated using the measurement data from the test case E13, is used in the life-cycle performance assessment of the bridge. The nonlinear time history analysis of the digital twin E13-DT is conducted by inputting the lifetime ground motion sequence generated in Sect. 4. The life-cycle performance of the bridge is then assessed against the time according to the damage states defined.

Definition of damage states
Prior to the life-cycle performance assessment, the nonlinear structural responses of the digital twin E13-DT subjected to the 1.3 g artificial site ground motion are analyzed to define the thresholds for different damage states of the bridge. In many existing studies, the damage states of a cable-stayed bridge are defined in terms of either structural component strains, support movements, cable strain ratios or others (Wei et al. 2020;Pang et al. 2014;Zhong et al. 2016Zhong et al. , 2017Li et al. 2017Li et al. , 2018aLi et al. , 2018b. In this study, the rebar stress-strain relationships of the reinforced concrete components, cable stress-strain relationships and the connection movements at both piers and towers of the bridge are analyzed to define the damage states of the bridge. Note that the accuracy of the updated digital twin for predicting the micro responses of the bridge components had been validated by Lin et al. (2021b). Regarding the reinforced concrete components (piers and towers), the rebar strains of the elements at critical sections of the towers and piers are monitored during the analysis. For the main towers, the base and conjunction area of the towers are selected as the critical sections as shown in Fig. 13 ( i.e.,Bot1,Bot2,Mid1,Mid2). For the piers, the rebar strains of all elements are monitored. Every element within the critical regions is closely monitored during the simulation.
The time-history analysis of E13-DT under the 1.3 g artificial site ground motion considers both material and geometric nonlinearities, and the analysis reveals the most critical states of different components and their corresponding locations, as shown in Fig. 14. It can be seen that the stress-strain relationship of the rebar is strongly nonlinear during the The stress-strain relationship of the cable, however, remains linear, which is the same as observed by Guan et al. (2019) in the test case E15 of the shake table test. Many existing studies also conclude that the stayed cables in a cable-stayed bridge are seldom damaged under seismic actions (Pang et al. 2014;Zhong et al. 2016;Li et al. 2018aLi et al. , 2018b. Hence, the stayed cables of the bridge concerned in this study are also assumed to remain elastic throughout the course. Therefore, the different damage states of the rebar are defined in terms of its strain and stress based on the constitutive model of the reinforcement as shown in Fig. 15. The four damage states of the rebar, which include slight, moderate, severe and complete damage, are defined and listed in Table 2. The different damage states of the critical connection are defined in terms of its deformation (Li et al. 2018b). The four damage states of the critical  Table 2. These defined damage states are used for the life-cycle performance assessment of the bridge based on the time history analyses of E13-DT under the ground motion sequences.

Time history analysis of the digital twin under ground motion sequence
With the above-defined damage states and the generated life-cycle earthquake sequences, the time history analyses of the digital twin E13-DT are conducted to evaluate the lifecycle seismic performance of the bridge. The ground motions are applied to the bridge in the transverse direction uniformly to keep it consistent with the shake table tests. Note that although bi-axial ground motions are much close to the real situation, this study focuses on developing the digital twin-based framework for assessing the life-cycle seismic performance of cable-stayed bridges. Hence, only transverse earthquakes are used for simplicity although bi-axial earthquakes can also be considered by the proposed framework. The response time histories of the digital twin E13-DT under the input of earthquake sequence No.10 are shown in Fig. 16 and used to illustrate the lifetime seismic performance evaluation of the bridge under sequential earthquakes.
The starting year of the earthquake sequence No.10 is 2238.12 × 10 5 year (see Fig. 12). It can be seen from Fig. 16 that the digital twin E13-DT is slightly damaged after the input of EQ No.1 at the year of 2238.44 × 10 5 . No severe damage occurs to the bridge until the occurrence of EQ No.4 at the year of 2238.80 × 10 5 . Under this earthquake, the beam-column conjunction of the south tower is moderately damaged, and one of the pier-girder connections is severely damaged as its deformation exceeds 20 mm as shown in Fig. 16. By assuming that no repair is made to the bridge, the bridge still survives under the occurrence of EQ No.5 according to the defined damage states. However, under the input of EQ No.6 at the year of 2238.93 × 10 5 , the rebar strain at the beam-column conjunction of the south tower exceeds 0.10, leading to the failure of the south tower. Moreover, the south tower tilts to one direction and the bridge is completely damaged according to the displacement time history recorded at the top of the south tower as shown in Fig. 16. It is clear that by resorting to the digital twin-based time history analysis as demonstrated above, the seismic damage accumulation under sequential earthquakes can be well considered and the proposed method can help simulate the seismic damage accumulation process of a long-span cable-stayed bridge during its service life. Of course, this study focuses on the life-cycle seismic performance assessment of a long-span cable-stayed bridge as one of the important components of the life-cycle performance assessment of a bridge. In reality, the effects of deterioration, retrofit, and maintenance due to other hazards during the life-cycle performance simulation should be considered in future studies in order to achieve a more practical and exhaustive assessment.

Life-cycle seismic performance assessment of the bridge
The lifetime seismic responses of the digital twin E13-DT under other earthquake sequences are also analyzed to obtain the exceedance probabilities of the four damage states of the bridge. The times of the bridge exceeding different damage states are first recorded from the starting year, and the number of the exceedance of each damage state is then divided by the total sample number of 20 to obtain the exceedance probability of each damage state against time, as shown in Fig. 17. The recorded results are finally regressed using the lognormal distribution, and the regressed results are also shown in Fig. 17. The regressed logarithmic means and standard deviations (μ and σ) of the exceedance probability of the four damage states are summarized in Table 3. Note that μ is the natural logarithm of the median time (m) of the bridge exceeding the corresponding damage state. It can be seen from Fig. 17 that with increasing time, the exceedance probability of each damage state increases. At the 10 5 years, the bridge exceeds the slight damage state at a probability of almost 100%. However, the exceedance probability decreases significantly Fig. 17 Damage exceedance probabilities and regression results for the moderate, severe and complete damage states. The median time of the bridge to exceed the slight damage state is about 13,763 years but it increases to 22,349, 68,907 and 442,413 years for the moderate, severe and complete damage states, respectively. It should be noted that because only 20 earthquake sequences are used and the number of the bridge exceeding the complete damage state is only 4 out of 20, the service life estimated by the regression results corresponding to this damage state might not be accurate enough.
Furthermore, the proportions of the four damage states plus no damage state within a given time period are calculated using the four regressed lognormal distributions. Figure 18a shows the damage proportions of the five states of the bridge within the time period of 10 7 years. Figure 18b zooms the distribution of the damage states in the first 10 6 years depicted in Fig. 18a. The damage proportions of the five states at the time of 10 4 , 10 5 , 10 6 and 10 7 years are summarized in Table 4. At the time of 10 4 years, the proportions of the damage states of the bridge are 63.88% (no damage), 10.48% (slight damage), 10.32% (moderate damage), 13.02% (severe damage) and 2.30% (complete damage). However, at the time of 10 7 years, the proportions of the damage states of the bridge are 0.00% (no damage), 0.00% (slight damage), 0.42% (moderate damage), 4.62% (severe damage) and 94.96% (complete damage). Clearly, the proposed digital twin-based life-cycle seismic performance assessment method can help estimate the service life of a bridge in terms of the defined damage states.   1 3

Conclusions
The digital twin-based life-cycle seismic performance assessment method has been proposed in this study for long-span cable-stayed bridges. A scaled long-span cablestayed bridge and its shake table tests are used to validate the life-cycle seismic response prediction results and demonstrate the feasibility of the proposed method. The major conclusions from this study are summarized as follows: (1) The digital twin of the bridge for life-cycle seismic performance assessment can be obtained through the evolutionary nonlinear model updating of the FE model of the bridge using the measurement data recorded during the earthquakes.
(2) The lifetime earthquake occurrence and sequence generated by the proposed MCSbased earthquake generation method match well with the design seismic hazard curve of the bridge site.