Probabilistic Seismic Evaluation and Experimental Tests of Multi-Direction Damping System on a Super-Long Column-Pylon Cable-Stayed Bridge

ABSTRACT In view of the insufficiency of bidirectional seismic control of long-span cable-stayed bridges as well as the limitations of existing schemes, a multi-direction damping system (MDDS) was proposed and investigated in this manuscript, which can provide seismic resistance bidirectionally. The new damping system owned a unique middle connecting structure, which overcomes the disadvantages that axial instability caused of excessive long rod and the demand of large installing space of traditional dampers. Both the performance stability and constitutive relationship of MDDS were verified by engineering tests, and the equivalent design method was then proposed and proved to be applicable. A detailed finite element numerical model of a super-long (main span over 800 m) column-type pylon cable-stayed bridge which equipped with MDDS in practice was established, the fragility curves of critical components were obtained by non-linear time history analysis, which exhibited the high efficiency of MDDS seismic performance in both transversal and longitudinal directions. Through extracting the median fragility value of 35 sections of the pylon under four damping cases, i.e. setting dampers transversely (TDS), longitudinally (LDS), bidirectionally (BDS), and MDDS, it can be concluded that MDDS performed far better on seismic resistance than LDS and TDS, and made the similar contribution with BDS. On account of half number of damper setting and the reduction in installing space demand, MDDS becomes the optimal choice in practical application for its economic efficiency and superiority. Furthermore, the optimal horizontal installing angle of MDDS was derived through six angle cases which was based on the system-level fragility curves comparison.


Introduction
Large-span cable-stayed bridges are increasingly used in practical engineering applications for their excellent span capacity, stability, and magnificent appearance (Han et al. 2019).With the amount of cable-stayed bridges increasing all around the world, their status in traffic engineering is more vital.Therefore, the seismic research of cable-stayed bridges has become a decisive investigating topic of many scholars and engineers.
For the low structural damping and long natural periods of cable-stayed bridges, their structures are highly flexible and vulnerable under earthquake actions (Ahad et al. 2022).The post-earthquake field observation indicates that one of the outstanding damage sources of long-span cable-stayed bridge is that the relative displacement of the components and adjacent spans is large (Bruneau 1998;Li et al. 2013), especially at the girder-pylon junction (Siringoringo, Fujino, and Namikawa 2013), which makes the deformation capacity of the expansion joint and the reserved clearance cannot meet the seismic needs (Yan and Yuan 2004).Therefore, it becomes particularly critical for the selection of seismic constraint design on cable-stayed bridges.A fully rigid connection has been proven would effectively control the displacement of the girder transversely, but would lead to excessive bending moments and shear forces of the pylons in the longitudinal direction (Li, Guan, and Li 2017).In order to restrict the longitudinal displacement of the girder and weaken responses transferred to the pylon, the elastic displacement limiting device with high damping is often used at the girder-pylon location (Calvi, Sullivan, and Villani 2010).Numerous schemes of longitudinal constraint design (Mirtaheri, Samani, and Zandi 2017;Wei et al. 2021b;Zhong, Jeon, and Ren 2018b) have been proposed and developed.
Relatively, the transversal seismic system of cable-stayed bridge is more complex and diverse.According to many scholars, supplementary damping system (SDS) (Wada, Huang, and Bertero 2004) has become the mainstream design scheme of transversal seismic fortification system for bridge structures.SDS is an auxiliary equipment system set in specific parts of the bridge.The advantages (Camara et al. 2017) of that devices are simple design, convenient repair, low cost, and even guarantee traffic obstruction prevention while maintenance and repairing.Domaneschi et al. updated and extended the benchmark cable-stayed bridge for transverse response under seismic loading (Domaneschi and Martinelli 2013), and it was found when the applied devices are implemented in the transversal direction also, the proposed schemes distinctively mitigate the transversal structural response of the bridge with reasonable efficiency.Aiming at an asymmetric single pylon cable-stayed bridge, He et al. (2018) put forward and designed a cable sliding friction ball bearing support system, and studied its transversal seismic performance under the action of strong earthquake.The result showed its efficiency in reducing the internal force of the structure and the transversal displacement of the members.De Mari et al. proposed a seismic isolation system (Roll-N-Cage isolator) on cable-stayed bridges which incorporates several novel features and mechanisms in a single unit (De Mari et al. 2014).The system proved it can achieve effective reductions in the seismic effects in terms of both internal forces and accelerations.Shen, Camara, and Ye (2015) discussed three design schemes of viscous fluid damper, friction pendulum sliding bearings, and transversal yielding metallic dampers on a large-span cable-stayed bridge, and the conclusion showed that transversal yielding metallic damper does not interact with the longitudinal displacement of the girder while the other two cases were opposite.Guan, Li, and Yan (2010) proposed a damper device scheme with low horizontal maintenance requirements of cable-stayed bridges, and proved its devotion in longitudinal and transversal seismic control.Camara et al. (2017) designed a metal damper consisting of triangular plates connecting cross bridges to the girder plates and bearings, but proved that the damping device was no longer sensitive for bridges whose spans more than 600 m.Niu et al. (2019) developed a new type of variable stiffness oil damper, whose parameters are set to be low stiffness and high damping, which can be installed transversely between the girder-pylon and the girder-pier connecting position.Because of the low stiffness of wind resistance and isolation, this oil damper has better performance in controlling the transversal displacement of the girder.
However, the seismic design methods above are considered separately in one direction without taking the influence of the whole horizontal directions into account.In general, the transversal restraint system needs to be designed on the premise of meeting the longitudinal restraint system.If there is no transversal restraint, the girder displacement may exceed the transversal reserved clearance of the pylon and girder, resulting in the collision between them (Zhang et al. 2022).Therefore, most long-span cable-stayed bridges choose to install resistant bearings on the transversal as a conventional restraint pattern to withstand the relative displacement between the pylon and girder, as shown in Fig. 1a.But the stiffness of the structure is increased due to the consolidation system, and redundant structural responses will be generated under a large earthquake.In view of this problem, scholars (Guan, You, and Li 2019;Xie andSun 2014, 2019;Zhou, Wang, and Ye 2019) proposed various seismic control schemes on super-long cable-stayed bridges.A long-span bridge model was established and used to reevaluate an optimization procedure (Domaneschi, Martinelli, and Perotti 2014) for a passive control strategy, already proven effective with a simplified model of the buffeting wind forces.Domaneschi et al. gave passive and semi-active control strategies, which are implemented in the medium-and long-span cable-stayed bridges models for the bridge protection, can provide resistance to multi-hazard, e.g.wind and earthquakes (Domaneschi and Martinelli 2014).Infanti, Papanikolas, and Theodossopoulos (2003) innovated a seismic scheme on the Rion-Antirion Bridge in Greece with both transversal and longitudinal dampers, which are shown in Fig. 1b.However, both the longitudinal and transversal installation are limited by the reserved gap between the pylon and girder, which leads to great limitations on seismic design of different cable-stayed bridges.Martinelli and Domaneschi (2017) pointed out that there is not only one hazard to be faced when long-span bridges are in service, and the influence caused by multi-hazard should be considered, which puts forward higher requirements for control strategies.
In view of the insufficient consideration of existing seismic design of cable-stayed bridges, a multidirection damping system (MDDS) is proposed and investigated in this manuscript, as shown in Fig. 2a,b, which can simultaneously reduce the transversal and longitudinal seismic responses of bridge structures under the condition of specific structure design and limited number of damper arrangement.As an authorized invention patent, MDDS has been applied to a super-long-span cablestayed bridge with the main span over 800 m in China which is shown in Fig. 2c,d.In this investigation, by conducting experimental performance tests and numerical simulation, the performance of the new damping system and its equivalent design method were verified.And the seismic fragility analysis (Ellingwood 2001(Ellingwood , 2009;;Hwang and Jaw 1990;Pang et al. 2021) was then carried out based on nonlinear time history analysis of the cable-stayed bridge which equipped with the MDDS, the results further confirmed the advantages of the new damping system in the practical engineering application.

Structure Features
At present, the general connecting pattern of viscous dampers on long-span bridges are shown in Fig. 3a, which has the following shortcomings: (1) One end is fixed on the girder, with the other end fixed on the bridge pylon (or pier) which leads to the large demand of installation space for the dampers; (2) As the damper is provided axial load, the long installing length makes the damper have the possibility of axial instability; (3) The damper setting direction is unique but various loads come from different directions to the structure, so that the general installation cannot cope with the seismic actions or other loads comprehensively.Therefore, as shown in Fig. 3b, a new multi-direction damping system (MDDS) is designed in response to the existing deficiency of present seismic design on long-span cable-stayed bridges, and which has been applied in the reality.Firstly, a universal hinge support is set not only to connect the dampers but also to provide the middle support for the system.Its horizontal and vertical designed angle makes the system has self-adapting ability in seismic actions.Secondly, benefit by the middle support, the system reduces the demand of dampers installing space.Meanwhile, the possibility of axial instability due to large slenderness ratio is decreased because the MDDS has relative half stroke comparing with conventional damper.Thirdly, the oblique angle setting of MDDS can reduce bridge responses in both transversal and longitudinal directions simultaneously without increasing the number of dampers to be installed.

Equivalent Design Method
The MDDS is installed at the position that between the bridge pylon and the girder crossbeam.An MDDS contains four diagonally placed viscous dampers, each of which is equipped with a universal hinge support.The damper is at an angle of 25° to the longitude direction, and the layout of MDDS is shown in Fig. 4.
Generally, the force-velocity relationship of the viscous damper is expressed by the following Eq.(1): In the above relationship equation, f D is the damping force, C α is the damping coefficient which ought to be derived by experiments, _ u represents the relative velocity, α is the exponent which is used to simulate the viscous damper non-linear behaviour and sgn() is the sign function (Narges, Mohammad, and Mohammadreza 2021).For the sake of brevity, the Eq. ( 1) can be simplified by substituting C α into C and sgn( _ u) _ u j j into V (Wei et al. 2021a), and the simplified formula is shown in the Eq.(2).As MDDS is provided axial velocity V by wind load or seismic forces, according to force decomposition and geometric relationship conversion that are shown in Fig. 5.The relationships among transversal and longitudinal relative velocities V t and V l , the transversal and longitudinal component forces F t and F l of damping force F, the component damping coefficient C t and C l are respectively exhibited.Therefore, a general equivalent design method of MDDS is put forward.
Through the transformation of damping force F by sin function, the transversal component force of F, i.e.F t can be derived: Where F can be substituted by Eq. ( 2) from which axial velocity V can be further transformed into transversal component V t , as shown in Eq. ( 4).
Hence, by observing the Eq. ( 4), the component damping coefficient C t can also be expressed through the transformed C: In conclusion, F t can be finally expressed by following Eq.( 6) for C t and V t : Similarly, the longitudinal damping component force F l and the the longitudinal component damping coefficient C l can be concluded below in Eqs. ( 7) and (8): Since MDDS has been applied to a super-long-span cable-stayed bridge in practice, its initial design parameters in this study are set as settings of the practical bridge.The damping coefficient C and damping exponent α are given as 3000 kN/mm•s −1 and 0.25, respectively.
Aiming at the verification of the equivalent design method, a static analysis model of viscous damper was established.The model simulates the oblique angle of MDDS, carries on the load through the axial reciprocating load, and extracts the axial local force and node displacement under the local coordinate system to reflect its hysteretic curve.In addition, two models set at 0°(longitude) and 90° (transverse) respectively were established to simulate the longitudinal(y) and transversal(x) components of MDDS.Parameter settings were based on Eqs. ( 6)and ( 7) proposed above.See Fig. 6 for schematic diagrams of the three models.
By comparing the longitudinal and transversal components of the oblique model with the axial data of the other two models, it was found that the component hysteretic curves of the oblique model were consistent with the curves representing their respective simulated conditions.Thus, the verification confirms the correctness of the equivalent design method, which can be applied to the following case analysis.

Experimental Tests
A scale model (1:2) of the universal hinge support was manufactured to testify its working performance.MTS electronic servo system was used to load on the model, which was shown in Fig. 7a.The model was designed according to the similitude theory (Buckingham 1914), for whose parameters settings can be made to form similar constants with those of the structure prototype.The design load of a viscous damper in MDDS was 2200 kN, which can predict the required load of the scale model was 550 kN for the stress similitude constant was 0.25.The test model adopted the same material as the practical structure, and the test site was shown in Fig. 7b.And the measuring points of support A and B were selected as shown in Fig. 7c,d, which also shows the strain rosette of each surface respectively.According to the loading test, the maximum principle stress of support A and support B are σ maxA = 74.8MPa and σ maxB = 163.1 MPa, so as the connecting component, universal hinge support was able to meet the structural strength requirements.
Viscous damper performance tests were followed.A viscous damper was tested under five different velocity loading cases.The design maximum speed of the damper was V max = 289 mm/s.According to the codes of viscous fluid damper for bridges, the loading cases are shown in Table 1.Loaded by sine wave load with three cycles and the test results are shown in Fig. 8.The experimental data revealed that the push and pull axial force of eight single viscous dampers is lower than the 110% theoretical value and higher than the 90%, which proved the rationality of the experiment and the velocity performance of the viscous dampers.Since the working path of the union dampers structure is obviously different from that of the general single damper, a specific test platform ought to be designed to test the performance of the union dampers of MDDS.The schematic diagram of the platform structure is shown in Fig. 9a.
The relationship between force and displacement of the union dampers structure under two slow loading conditions, namely, 5 mm/s and 13 mm/s, was analyzed, and the force-displacement hysteretic curves of each case are shown in Fig. 10b.As can be seen, the union dampers worked smoothly in the loading process, and the overall hysteresis curve of the union dampers was smooth and full, indicating a cooperative working state with reliable performance.
According to the above tests and results, the performance of viscous dampers on MDDS and the connecting component, i.e. the universal hinge support, was verified.Moreover, the union dampers structure also exhibited a high working efficiency, which can be further applied in the following study.

Bridge Description
The selected bridge in this article is a super-long-span cable-stayed bridge which crosses the Yangtze River in China and is equipped with MDDS practically.The bridge has five spans, with the main span being 806 m, and the bridge configuration is shown in Fig. 10a.With 50 stays in a fan configuration, altogether 200 stay cables in 4 fans are arranged.The girder is in the form of split amplitude, with a single width of 18 meters and a distance of 17 meters between the two separated beams which are connected by crossbeams, as shown in Fig. 10b.The pylons are divided into three parts: lower pylon, middle pylon, and upper pylon.The sections that connect the lower-middle pylon and middle-upper pylon (section B-B, section C-C) are seen as the critical sections of the pylon.Meanwhile, the pylon bottom usually takes the first place of the most vulnerable sections under earthquakes.So, the bottom section (section A-A) is also deemed to be a key one, and three sections mentioned are depicted in Fig. 10c.The concrete of pylon is grade C50, and the HRB400 steel bars are equipped.The foundation of the pylon adopts a circular cap, under which 30 bored piles are set, and the design of variable diameter pile is adopted.The group piles spacing is 6.0 m.
The whole bridge contains two types of piers: side piers and transition piers (which are connected with approach bridge), respectively, and all adopts concrete of grade C40 with HRB400 steel bars inside.Side piers are arranged as T-shape single-column piers, with 42.18 meters high, setting a 11 m × 6 m rectangular section.The transition pier is a double-column type pier with 33.9 meters height each single column, and the section is an 8 m × 7 m rectangular size.In view of the lower part of the pier being relatively vulnerable to damage in seismic actions, sections which are located at the bottom, lower-part, and middle part are shown in detail in Fig. 11c.

Finite Element Modeling
In view of the numerous node elements and complex nonlinear behavior of the finite element model established in this paper, in this study, OpenSEES software is selected for the finite element analysis to improve the analyzing efficiency.A detailed finite element numerical model of the study bridge is established and shown in Fig. 11, which considers the nonlinearities of both material and geometry.
Under the earthquake actions, it is generally considered that the probability of damage to the bridge superstructure is low.Thus, this article assumes that the girder is always in the elastic state (Li et al. 2018), and uses the elastic beam-column element to model.All stay cables are made of high-strength steel wires, and the elastic modulus which are modified by Ernst method is used for the consideration of sag effect (Jo et al. 2021;Pang et al. 2014), so large-displacement truss element is adopted to simulate every cable.
As the substructures of the cable-stayed bridge system tend to give non-linear behaviour under seismic actions (Wei et al. 2021b), the non-linear beam-column elements are used to simulate piers, lower pylon, and middle pylon.As the superstructure is often the last part to be damaged during earthquakes, the upper pylon is modeled by the elastic-beam column element under the supposition of its linear state.Sections of piers, lower and middle pylons are discretized into confined concrete, unconfined concrete, and reinforced bar fibers, which are modeling by fiber sections (Zhong et al. 2017).As shown in the pier and pylon finite element diagram of (e), (f), and (g) in Fig. 11, the red line segment and the black nodes on its both sides represent nonlinear elements modeling of the bridge.In order to reflect the nonlinear characteristics of the material and the interaction effect of axial forcemoment, the distributed models are used to represent the sections of lower, middle pylons and piers.Furthermore, this article uses a simplified bi-linear reinforced steel bars model considering the stiffness after yield, and the concrete stress-strain model adopts Kent-Scott-Park (Scott, Park, and Priestley 1982) model of Concrete01Material concrete in OpenSEES platform, as shown by Fig. 11b,c.UniaxialMaterial Concrete01 material was used for all the concrete members' core and cover concrete modeling in this study, steel members modeling adopts UniaxialMaterial Steel01 material.
The sliding bearings are set on the piers' top which can adapt to the structure displacement and deformation caused by the various load or temperature effect (Wu et al. 2020).Through the simulation by using zero length element, and Elastic PP materials are using and given stiffness in three directions.The bearing model is shown in Fig. 11d.In particular, the MDDS is simulated as Two-Node link elements in this finite element model and adopts the unixialMaterial ViscousDamper as its material in OpenSEES.In addition to the damper setting parameters given above, the initial stiffness K is also set to simulate the series material, so that the structural period will not be significantly reduced, which is equivalent to pure damping.As a floating system, in order to simulate the boundary conditions, longitudinal sliding bearings are set on the piers of the study bridge, and MDDS and resisting bearings are arranged at the position of the pylon-girder connection.Through the non-linear element simulation, the soil-pile interactions (Domaneschi and Martinelli 2013) are considered by using six soil springs, which contain three translational ones and three rotational ones, set at each bottom of the piers and pylons, as shown in Fig. 11a.Due to the bridge site which is mainly composed of silty clay, soft soil, fine sand, medium sand, and pebble soil from above, the "m" method (P.R.China Ministry of Transport: Guidelines for seismic design of highway bridges, 2008), a subgrade reaction method which considers soil has linear elastic characteristic, is optimal to obtain the springs' stiffness.However, in view of space limitation and research focus of this study, the specific calculation process of each spring is not described in detail.

Fragility Approach
Through the previous study and calculation on seismic fragility analysis of structures, a set of analytical theories based on probability theory have been summarized.The concept of seismic fragility can be simply summarized as the failure probability of each component and system under earthquake actions.Fragility can be defined as the conditional probability that the seismic demand (D) imposed on the structure exceeds its capacity (C) under a given level of seismic intensity measure (IM).
To derive the PSDM (probabilistic seismic demand model) which is capable of probabilistic estimation of EDPs (engineering demand parameters), Cornell (1996) generalized the average demand of the structure under seismic action by the following Eq.( 10).
Where a and b are statistical regression coefficients.In general, it can be assumed that both demand and capacity follow a logarithmic normal distribution, and the transcendental probability is derived in Eq. ( 11) as followed.
Where Φð:Þ represents the standard cumulative normal-distribution function, S D and β D|IM are the median value of structure response and the logarithmic standard error; S C is the median value, β c is the structural capacity dispersion (Casciati, Cimellaro, and Domaneschi 2008;Nielson 2005;Perotti, Domaneschi, and De Grandis 2013), respectively.β D|IM calculation can be expressed as the following Eq.( 12).The d i is the i th simulation result of demands, and the n represents the number of simulation.And the ultimate purpose of PSDM deduction is to obtain the relationship between peak seismic response and earthquake action intensity.

Input Ground Motions
In considering the uncertainty of ground motions inputting (Lee 2005;Zhong, Zhu, and Han 2023), 100 ground motion records are selected in this study.Eighty of them are chosen from PEER strong motion database with magnitude ranges from 5.8 to 6.9, and the epicentral distance ranges from 10 km to 60 km; The rest of records are selected from SAC engineering database which have 2% and 10% exceedance probabilities in 50 years.The magnitudes of 20 ground motions range from 6.0 to 7.0, and the epicentral distance is less than 15 km, of which the lowest is 2.5 km.The PGV distribution of 100 selected records are shown by the size of points in Fig. 12; the bigger point represents the higher PGV value.The ground motions are input into the model according to the directions, and the input is unidirectional and independent.

Engineering Demand Parameters and Damage States Threshold
Seismic fragility is the convolution of demand model and capability model (Zhong, Mao, and Yuan 2023).Defining EDPs' ability and damage limit states is a indispensable step in generating fragility curves.In general, the substructure of a bridge is much more vulnerable to damage than its superstructure under earthquake action.Therefore, this article selected the transversal and longitudinal curvature ductility of section 3-3 in a side pier, section 6-6 in a transition pier, and section A-A, B-B, C-C of one pylon which are shown in Fig. 10c.The displacements of two bearings on two piers and the displacement of MDDS on a single pylon which were defined by displacement-ductility ratio were also considered; a total of 13 EDPs were chosen and listed in Table 2.
The recorded engineering demand parameters need certain quantitative indicators to evaluate their damage status.Side piers, pylons, and transition pier can be regarded as components mainly composed of concrete.For the definition of damage state of concrete members, pioneers have done a lot of research work, and put forward a variety of evaluation indexes (Azevedo et al. 2010;Kowalsky 2002;Powell 2008;Priestley and Kowalsky 2000;Su, Dhakal, and Wang 2017) In this article, the damage states (Padgett and Desroches 2010) are defined as four levels, i.e. slight (DS1), moderate (DS2), extensive (DS3), and complete (DS4).The pushover analysis of three kinds of concrete members is carried out by OpenSEES.In order to facilitate calculation and comparison, the value of DS-1 of  selected components is normalized, and the ratio is used to reflect the quantitative relationship between various damage states, which are listed in Table 2. Sliding bearing and viscous damper dissipate huge energy mainly by their own displacement in earthquake action, thus generally using the ratio of displacement and ductility to evaluate their damage status.The bearing used in this article is bidirectional sliding spherical steel bearing, and its design displacement is ± 300 mm, so half of the design displacement, i.e. ± 150 mm is regarded as slight damage state, and the design displacement is regarded as moderate damage state.
If the damping effect of the viscous damper on the dynamic response is taken into account, the response of the structure under wind load will not exceed the reserved clearance, so the influence of wind load on the bridge does not need to be considered when designing the damping parameters.According to the relative displacement of pylon-girder does not exceed the reserved clearance of 200 mm and 500 mm under E1 earthquake in transversal direction and E2 earthquake in longitudinal direction respectively, the threshold of slight damage and moderate damage is defined as 200 mm and 500 mm.
As the bearings and dampers are non-critical components, their failure will not lead to serious damage or even collapse of the cable-stayed bridge structure, so this article only defines the slight (DS-1) and moderate (DS-2) damage status of them, which are shown in Table 2.

Fragility Curves of Critical Components
In previous studies, a number of intensity measures (IMs) (Hu et al. 2022) have been investigated and adopted.Wei et al. (2020) and Zhong et al. (2016Zhong et al. ( , 2019) ) put forward that peak ground velocity (PGV) is the IM which was seen as the optimum selection for the multi-pylon cable-stayed bridges.Thus, PGV was chosen to simulate the PSDMs of 13 EDPs through Eqs. ( 9)-( 11) in this section, for which effect can confirm that the above achievements were applicable to this research bridge structure.And the PSDMs (Zhong et al. 2022(Zhong et al. , 2023) ) of 13 selected EDPs on PGV are shown in Fig. 13.
Through the selection of EDPs, the comparison of IM, and the fitting of PSDM above, the fragility curves of each component were obtained, as shown in Fig. 14.In the figure, four lines from top to bottom represent four failure states of DS-1, DS-2, DS-3, and DS-4, respectively.Figure 14a,f shows that the transverse (y direction) failure probability of the side piers of the full bridge model equipped with MDDS is greater than that of the longitude (x direction), while the situation of the transition piers is just the same, as shown in Fig. 14b,g.Figures 14c-e,h-j have shown the fragility in both directions of pylon bottom section (section A-A), the joint-part section between column and crossbeam where MDDS arranged (section B-B) and the section which links middle-column and upper-column (section C-C): the DS-1 and DS-2 curves of section A-A are closer in both transversal and longitudinal directions between 0 and 1.0 m/s.On the other hand, in section B-B, the DS-1 is similar from the former, but the upward trend of DS-2 is relatively slower.The corresponding PGV of section C-C which is at the top of the selected elements, with a 50% probability of reaching DS-1, is more than twice that of the first two sections.A comparison of median values is shown in the histogram.In view of the curve characteristics of all the above ductility sections, under the same PGV, the failure probability of components in the longitudinal direction is always smaller than that in the transversal direction.
The fragility of the sliding bearings at the side pier and the transition pier is shown in Fig. 14k,l.It is found that within 2.0 m/s, the bearings are almost impossible to reach the state of moderate damage, and the same is true of the single-unit damper of MDDS; the failure probability will not exceed 20% when PGV reaches 2.0 m/s, as shown in Fig. 14m.Therefore, the bearings and dampers can be regarded as relatively safe components under earthquake actions.

Evaluation on Seismic Performance of MDDS
In order to testify the MDDS working performance in both longitudinal and transversal directions under seismic actions, three other different cases were set for comparison in this article, i.e.Transversal Damping System (TDS), Longitudinal Damping System (LDS), and Bidirectional Damping System (BDS), as shown in Fig. 15.In addition to the different ways of damper arrangement, the damping coefficient C t of TDS and C l of LDS are set according to Eqs. ( 6) and ( 7) to ensure the control of variables and the accuracy of the results.
By inputting the same ground motions, the responses of each component under different cases are obtained.In this article, the responses of altogether 70 non-linear elements of the pylon were selected and their fragility were analyzed and obtained.For the symmetry structure only the elements on one side are considered, i.e. the ductility section of 35 non-linear elements.The PGV values of the fragility curves corresponding to the 50% probability of slight damage and extensive damage were extracted, and the median value graphs of the pylon were drawn, as shown in Fig. 16.
In the case of which the entire bridge equipped with MDDS, the dark blue lines in Fig. 16 show the median PGV values in longitudinal and transversal directions, respectively.As can been seen, the median PGV lines in Fig. 16a,b have similar distribution and trend from bottom to top.In each direction, the values corresponding to the damage probability from the bottom (Ele 5) to the cross beam (Ele 12) of the pylon increase first and then decrease, and values of elements above the cross beam from Ele 13 to Ele 27 begin to show a sustained growth trend.Then, both longitudinally and transversely, the trend tends to be flat from Ele 27 until Ele 32.Finally, from Ele 32 to the top of the middle pylon, i.e.Ele 39, the values show continuous declination.The graphs also tell that section A-A, section B-B, and section C-C are the relatively most vulnerable ones of the pylon sections, which proves the reliability of selecting the three sections as key research objects in the previous analysis.
Meanwhile, the other three lines in the figure respectively reflect the influence of the rest of damping systems on seismic fragility: Transversely, the LDS whose dampers were only set in the longitudinal direction is undoubtedly the worst damping solution.Similarly, TDS made the least contribution in longitudinal direction.In each of two directions, the dark blue lines representing MDDS and the light blue lines representing BDS have nearly the same distribution and trend.It can be indicated that MDDS achieves the seismic performance and damping effect as BDS with the premise of setting half number of dampers.What's more, it further verified the equivalent design method is applicable in the dynamic model.In contrast, the LDS and TDS scheme with dampers setting only in one direction make decent seismic performance only in their own direction, and are not better than MDDS scheme.Moreover, the lines tell that the effect of the damper is more pronounced in the longitudinal direction than in the transversal direction.

Investigation on Effect of MDDS Installing Angle on Seismic Performance of Bridge System
As the above evaluation only selects the fragility of pylon non-linear elements as the evaluation index, system-level fragility analysis is considered to carry out for it is able to further provide comprehensive results.Due to space limitation, under the four damping cases mentioned above, the system-level fragility curves in slight damage and extensive damage will be given directly (Hu et al. 2021;Zhong et al. 2018a), as shown in Fig. 17.
Though the comparison of the system fragility curves under the four cases set in Fig. 15, as shown in Fig. 17, it is verified that MDDS makes the best effect at the system level.Figure 17a  place in either state, while BDS takes the first one due to its sufficient number of dampers.However, MDDS exhibits the indistinguishable line tendency with BDS.To sum up, MDDS contributes similarly with BDS in reducing the failure probability of the system, which verifies the global working performance and efficiency of itself.
In order to explore the influence of different installation angles on the whole system of MDDS under earthquake actions, the different angle settings need to be discussed and compared.The installation angle of MDDS is the included angle between the single-unit viscous damper and the longitudinal direction of the bridge.In this article, the default initial installation angle is 25°, and five other installing angle cases are set, as shown in Fig. 18.
System-level fragility is still used to evaluate various installation angles setting, as shown in Fig. 19, which shows the impact of MDDS on system fragility at 6 installation angles cases.From DS-1 to DS-4, the light blue line representing the 15°Case is at the far left of the line frame, that is, when the installation angle is 15 degrees, the probability of system damage is the highest.Similarly, the green line representing the 30°Case is always at the far right of all the lines, indicating that this angle of installation will make the bridge system least vulnerable to damage.As the installation angle increases from 15°to 30°, the operating performance of MDDS reaches the best.Then the installation angle continues to increase, the operating performance begins to weaken, namely, the damage probability of the system being protected from failure is gradually reduced.However, the performance of 40°is still better than that of 15°and 20°.This analysis indicates that 30° is the best installation angle for MDDS while 15°is the worst, but only for this bridge case.Other bridge cases requiring the installation of MDDS ought to be analyzed separately.

Conclusions
Conventional damping arrangement is often only considered in longitudinal direction while transversal arrangement scheme is relatively lacking.Meanwhile, the traditional damper installation method has the risk of axial instability and the limitation of unidirectional loading only.This study proposed and investigated a new type of multi-direction damping system, i.e.MDDS which consists of multiple viscous dampers connected by a special connecting structure.Based on the support of engineering performance tests and equivalent design, this article simulated the installation of MDDS on a super-long-span cable-stayed bridge by finite element numerical modeling, and assessed its influence through seismic fragility analysis.Moreover, MDDS was compared with other three dampers arranging cases to verify whether it has superior working performance.The results are as follows: (1) By installing universal hinge support as the central supporting member, MDDS overcomes the defects of axial instability of general dampers structure design.And the oblique combination leaded the system was able to reduce responses in both transversal and longitudinal directions.The engineering experimental test results and constitutive relation curve exhibited MDDS practical performance, and the equivalent design method which was proposed through mechanics relations was convinced.
(2) Through seismic fragility analysis of the full bridge finite element model, the damage probability of selected sections (concrete members) is greater in transverse than that in longitude, and the pylon bottom section (A-A) was most vulnerable while the top section (C-C) of middle pylon was the opposite one.Sliding bearings and viscous dampers are relatively seen as the safer components of all in this study.
(3) From the component-level fragility analysis, by comparing the median PGV of 35 non-linear elements of the pylon under four damping schemes, it was found that MDDS and BDS made the greatest seismic devotion whether in two directions.LDS and TDS contributed favourable performance only in their axial direction, while they took the last place in the orthogonal direction.Meanwhile, from the system-level evaluation, MDDS ranks only second to BDS under four damage states.However, MDDS was considered to be the optimal case due to the same effect of BDS under the premise of without increasing the number of dampers.(4) In the comparative analysis of six installation angles cases, 30 degrees was identified as the best design installation horizontal angle for the MDDS of the bridge example studied in this article by comparing the system-level fragility curves.

Highlights
(1) A new multi-direction damping system (MDDS) is proposed and investigated on a super-long cable-stayed bridge based on the support of engineering tests.
(2) The novel design of MDDS overcomes the defects that traditional damping schemes have, and an equivalent design method is put forward and verified.(3) Seismic fragility analysis on component and system levels is carried out to exhibit the superiority of MDDS on its seismic performance.(4) As an authorized invention patent, the MDDS has been applied in practical engineering which is a super-long cablestayed bridge with main span over 800 m.

Figure 6 .
Figure 6.Verification of equivalent design method.

Figure 8 .
Figure 8. Performance test and results of viscous dampers under different loading velocity.

Figure 9 .
Figure 9.Union dampers structure performance test: (a) test platform, (b) hysteresis curves of 5 mm/s and 13 mm/s loading.

Figure 10 .
Figure 10.Bridge layout illustration: (a) bridge configuration, (b) top view of half bridge and cross section diagram of the girder, (c) pylon and piers configurations and key sections.

Figure 11 .
Figure 11.Bridge Finite Element Model: (a) soil springs simulation; (b)~(d) force-displacement relationships of concrete, steel bars and bearings; (e)~(g) finite element model of side pier, transition pier and pylon; (h)~(i) finite element model and practical application of MDDS on the bridge.

Figure 14 .
Figure 14.Exceeding probabilities of components under different damage states.

Figure 19 .
Figure 19.Effect of installation angle of MDDS on system fragility.

Table 1 .
Loading cases of tests.

Table 2 .
Engineering demand parameters and damage states of various components.
SP and TP represent side pier and transitional pier respectively.