System-based probabilistic optimization of fluid viscous dampers equipped in cable-stayed bridges

Abstract

This study presents a methodology to evaluate the optimal parameters of fluid viscous damper for cable-stayed bridges using the system-level fragility assessment approach. Instead of investigating the impact of different isolation devices on the component’s vulnerability separately, this study focuses on evaluating the optimal parameters of fluid viscous damper to achieve the best overall performance of cable-stayed bridge as a system. Numerical model of a cable-stayed bridge with the most common configuration in China is established using OpenSEES that can account for their nonlinear response and uncertainty treatment. A joint probabilistic seismic demand model and Monte Carlo simulation are employed to obtain the system fragility of cable-stayed bridges by accounting for the contribution of multicomponents to the global damage state. The system-level fragility curves and component fragility curves are compared before and after the application of fluid viscous damper with different parameters. The results indicate that a given parameter of the fluid viscous damper may have a negative impact on some components, yet lead to a better performance of the bridge as a system. Thus, in order to obtain comprehensive knowledge of bridge performance and derive the accurate optimal parameters of fluid viscous damper, it is necessary to consider the fragility based on bridge system.

Keywords

cable-stayed bridges fluid viscous damper optimal design probabilistic seismic demand model system-level fragility

Introduction

In recent decades, cable-stayed bridges have been widely constructed around the world because of their aesthetics, efficient use of construction materials, and fast construction period. Many studies focused on the dynamic characteristics of cable-stayed bridges subjected to earthquake (Bruneau, 1992; Ren and Obata, 1999; Wilson and Gravelle, 1991). These studies showed that the seismic demand of towers would experience a significant increase in terms of bending moment and shear forces when restraining the bridge deck completely at tower locations. When there are no additional restraints in the longitudinal direction, larger deck displacement would occur during earthquake, resulting in pounding and unseating (Aiken and Kelly, 1992), which is not desirable for performance-based earthquake engineering. Therefore, there is an agreement among many researchers that mitigation devices should be equipped to allow some sort of deck movement, which would lead to the balance of the force and displacement demand of the cable-stayed bridges (Sharabash and Andrawes, 2009). Many researchers have conducted work on the isolation devices to reduce the overall seismic response of cable-stayed bridges (Ali and Abdel-Ghaffar, 1994; Sharabash and Andrawes, 2009; Zhang et al., 2009; Zhu and Qiu, 2014); however, these studies are determined methods, which could not account for the earthquake uncertainties, because of the unpredictable nature of seismic events (Zhang and Huo, 2009).

In order to consider the uncertainties of earthquake and structure information, it is a widely used approach to develop probabilistic model in the form of fragility curves (Basoz et al., 1999; Choi et al., 2004; Hwang et al., 2001; Nielson, 2005; Padgett and DesRoches, 2008; Pan et al., 2007; Ramanathan, 2012; Shinozuka et al., 2000a, 2000b). Fragility function is quite useful for comparing and selecting mitigation strategies. Casciati et al. (2008) evaluated the seismic vulnerability of components (bending moment, shearing force, deck displacement, cable strain, etc.) of a cable-stayed bridge with passive hysteretic devices in terms of damage exceedance probability. Barnawi and Dyke (2014) developed component (deck displacement, overturning moment, deck shear, etc.) fragility curves of a benchmark cable-stayed bridge equipped with response modification systems. These studies evaluated the effectiveness of the mitigation devices using component fragility separately, rather than the system fragility of the overall bridge performance. For example, some kind of isolation devices might decrease the fragility of bending moment, but would increase the fragility of deck displacement. Hence, a comprehensive evaluation of seismic isolation on bridges should be based on system-level fragility instead of on the component level. It is still worth noting that the above-mentioned studies did not emphasize on selecting optimum mitigation devices to achieve the best performance of cable-stayed bridges.

This article will use the system fragility function method to investigate the efficiency of isolation device and derive the optimal parameters to achieve the best overall performance of cable-stayed bridges. A long-span cable-stayed bridge with the most common configuration in China is chosen and a numerical model is built using OpenSEES (McKenna and Fenves, 2010) that can account for their nonlinear response and the uncertainties in the ground motion and structural properties. This article will mainly focus on the viscous damper, which is a widely used mitigation device equipped in cable-stayed bridges. Fragility curves of bridges at system level are derived from the multi-dimensional damage model considering the correlations between the components of cable-stayed bridge. The fragility at system level of cable-stayed bridge equipped with different viscous dampers is compared. Finally the optimal parameters are derived.

Methodology for optimal design of viscous damper

This study describes the methodology of the optimization and sensitivity analysis of fluid viscous damper (FVD) based on system-level fragility functions. The flow chart is illustrated in Figure 1 and described as follows.

Figure 1.

Flow chart for methodology of optimization and sensitivity analysis of FVD based on system-level fragility functions.

Step 1: establish probabilistic seismic demand model and capacity model

The fragility can be simply defined as the conditional probability that the seismic demand (D) placed upon the structure exceeds its capacity (C) for a given IM level, as shown in the following equation

Fragility = P [D > C | IM]

(1)

The demand model (S_D) is conventionally defined as a linear regression model in a log-transformed space (Cornell et al., 2002), obtained from the simulations

S_{D} = aI M^{b}

(2)

where a and b are regression coefficients. Practically, the fragility function can be expressed assuming that both the demand and capacity follow lognormal distributions

P [D \geq C | IM] = Φ (\frac{\ln (S_{D}) - \ln (S_{C})}{\sqrt{β_{D | IM}^{2} + β_{C}^{2}}})

(3)

where S_D and β_D|IM are the median value and dispersion, respectively, of the demand as a function of an IM; S_C and β_C are the median value and dispersion, respectively, of the structural capacity; and Φ(•) is the cumulative distribution function of the standard normal distribution. The dispersion of the demand model can be expressed as

β_{D | IM} = \sqrt{\frac{\sum {[\ln (d_{i}) - \ln (S_{D})]}^{2}}{(n - 2)}}

(4)

where d_i is the ith realization of the demands obtained from simulations and n is the number of simulations.

Limit state models reflect multiple levels of structural functionality associated with observed damage. As mentioned before, each limit state model follows a distribution with two parameters, S_C and β_C.

Step 2: correlation coefficient matrix calculation and system-level fragility deriving

The general assessment of seismic vulnerability for the bridge system as a whole must be made by combining the effects of various bridge components (Nielson and DesRoches, 2007). The multi-dimensional integral defining the seismic risk and involved in optimization cannot be calculated, or even accurately approximated, analytically. Therefore, joint probabilistic seismic demand model (JPSDM) and Monte Carlo sampling (MCS) are employed to obtain the system-level fragility by the stochastic analysis. JPSDM is developed in a log-transformed state by considering the correlation between transformed demands of various bridge components. MCS is performed in which N random samples are generated from both the JPSDM and limit state model. Each sampling demand is evaluated in the sampling failure domain and tracked by an indicator function (I_F). The indicator function of n components is presented in equation (5)

I_{F} = {\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} if (x_{1}, x_{2}, . . ., x_{n}) \in F_{1, 2, . . ., n} \\ i f (x_{1}, x_{2}, . . ., x_{n}) \notin F_{1, 2, . . ., n} \end{matrix}

(5)

where (x₁, x₂, …, x_n) is the sampling of the n components demand and, F_1,2,…,n is the multi-dimensional failure domain defined by the limit state of various components. At a given IM, the probability of being in the failure domain is estimated as

P [F r_{system} | IM = a] = \frac{\sum_{i = 1}^{N} I_{F_{i}}}{N}

(6)

By performing the MCS from a wide range of IMs, the entire curve of the system fragility can be obtained.

Step 3: optimization and sensitivity analysis based on system-level fragility functions

The system fragility curves are the object function. Conduct the system-level fragility analysis for each cable-stayed bridge equipped with different viscous dampers. Compare the fragility function for each case and derive the minimal system fragility function, and the corresponding viscous damper parameters are the optimal one. The advantage of this approach is the ability to evaluate the impact of retrofit on multiple vulnerable components as well as the bridge’s overall seismic performance.

Case study

Input ground motions

This study selects the suite of ground motions used in Shafieezadeh et al. (2012), which comprises 80 ground motions extracted from the PEER Strong Motion Database and 20 ground motions pertinent to Los Angeles selected from the SAC project database. The 80 PEER ground motions have an even selection of recorded time-histories from four bins that include combinations of low and high moment magnitudes as well as large and small epicentral distances.

Bridge description and numerical model

This study selects Polonggou Bridge, which is the most common type of cable-stayed bridge constructed in China. It has three spans; the main span and side span are 430 and 156 m, respectively. The 17 stay cables are arranged in a fan configuration. The detailed geometry of the cable-stayed bridge is shown in Figure 2.

Figure 2.

Configuration of Polonggou cable-stayed bridge (unit: cm).

Using the OpenSEES analysis platform (McKenna and Fenves, 2010), which is an open-source collaborative software framework for simulating the seismic response of structural and geotechnical systems, a detailed nonlinear three-dimensional model is created for the subject bridge. The superstructure is expected to remain linear and is thus modeled using elastic beam-column elements. A distributed plasticity fiber model is used to represent sections of pylons to account for the axial force-moment interaction and material nonlinearity. Foundations of the pylons are modeled using six spring elements (Zhong et al., 2016). The response of each cable is simulated using a nonlinear tension-only element, which is modeled as a large-displacement truss element using the Ernst method or modified elastic modulus method as a result of the easy usage and the capability to account for the sag effect (Pang et al., 2014). The initial stress of the cable is also considered in the model (Figure 3(d)). The abutment element comprising passive, active, and transverse actions is modeled by the stiffness of the passive resistance of the backfill soil (Figure 3(a)) and the stiffness of the piles (Figure 3(b)) (Choi et al., 2004; Nielson, 2005; Padgett and DesRoches, 2008). The pounding effect (Figure 3(e)) between the superstructure and abutment is modeled using the contact element approach developed by Muthukumar and DesRoches (2006), which includes hysteretic energy loss. The longitudinal response of the pot bearing (Figure 3(c)) is simulated using a nonlinear bilinear element, while the transverse response of the bearing is assumed to be constrained. Further details on the assumptions and analytical models themselves can be found elsewhere (Zhong et al., 2016, 2017).

Figure 3.

Force-deformation relations of various components: (a) abutment passive, (b) abutment active, (c) pot bearing, (d) cables, and (e) pounding.

Engineering demand parameters

Demands placed on critical components are taken into account for the vulnerability assessment of bridges. In this study, peak demands of components are adopted as engineering demand parameters (EDPs), and the critical components are the pylons, bearings, abutments, and the relative displacement of abutment and deck, as listed in Table 1.

Table 1.

Demand parameters of different bridge components considered.

Engineering demand parameters	Abbreviation	Unit
Longitudinal curvature ductility in the Section 1 Pylon 1	μ_φ-P1S1	–
Longitudinal curvature ductility in the Section 2 Pylon 1	μ_φ-P1S2	–
Longitudinal curvature ductility in the Section 1 Pylon 2	μ_φ-P2S1	–
Longitudinal curvature ductility in the Section 2 Pylon 2	μ_φ-P2S2	–
Abutment passive displacement	δ_AP	mm
Abutment active displacement	δ_AA	mm
Pot bearing displacement at Pylon 1	δ_PB	mm
Relative displacement of abutment and deck	δ_AD	mm

Probabilistic seismic demand model

Selecting an optimal IM for seismic fragility assessment is not a trivial matter and has been focused on in numerous studies (Padgett et al., 2008, Shafieezadeh et al., 2012). In some of the pioneering works, various IMs were selected as optimal IM, including Modified Mercalli Intensity Scale (ATC, 1985), peak ground acceleration (PGA) (FEMA 1997), peak ground displacement (PGD), spectral acceleration at a period of 1 s (Ramanathan, 2012), and vector-based IMs (Baker and Cornell, 2005; Luco and Cornell, 2007). However, for long-period structures, especially for long-span cable-stayed bridges, peak ground velocity (PGV) was demonstrated to be the optimal IM (Zhong et al., 2016). Therefore, PGV is chosen as the intensive measure in this study. The linear fit parameters and the dispersion in the log-log space are shown in Table 2.

Table 2.

Demand models and regression parameters.

EDP	b	β	ζ
μ_φ-p1s1	0.896	0.310	0.346
μ_φ-p1s2	0.754	0.230	0.305
μ_φ-p2s1	1.539	0.830	0.539
μ_φ-p2s2	1.348	1.210	0.898
δ_AP	1.296	0.570	0.440
δ_AA	0.728	0.500	0.687
δ_PB	1.012	0.480	0.474
δ_AD	1.011	0.450	0.445

EDP: engineering demand parameter.

Components classification and limit states definition

This study partially employs existing models or definitions of limit states used in previous studies. The limit states for the abutments follow the work of Ramanathan (2012). The design displacement of the pot bearing is ±300 mm. Thus, two limit states are defined for the pot bearing; LS1 is 300 mm and LS2 is 600 mm. Limit states for the abutment seat is intimately linked to the unseating of superstructure. This study uses the moderate (LS2) and extensive (LS3) limit states for the abutment seat determined per the definition of Avşar et al. (2011), which is illustrated in Table 3. The LS3 and LS4 of the abutment and bearing are not defined because these components are regarded as accessory components and thus contribute only to the slight and moderate damage of cable-stayed bridges.

Table 3.

Limit state models for various EDPs.

EDPs	LS1		LS2		LS3		LS4
	S_C	β_C	S_C	β_C	S_C	β_C	S_C	β_C
μ_φ-p1s1	1.00	0.35	1.33	0.35	4.52	0.35	24.8	0.35
μ_φ-p1s2	1.00	0.35	–	–	1.37	0.35	6.27	0.35
μ_φ-p2s1	1.00	0.35	1.33	0.35	4.52	0.35	24.8	0.35
μ_φ-p2s2	1.00	0.35	–	–	1.37	0.35	6.27	0.35
δ_AP	48.8	0.35	170.8	0.35	–	–	–	–
δ_AA	46.5	0.35	155.0	0.35	–	–	–	–
δ_PB	300	0.10	600	0.10		–	–	–
δ_AD	–	–	1980	0.10	2625	0.10	–	–

EDPs: engineering demand parameters.

The column is the most critical component of bridges when performing the fragility analysis of bridges. Several previous studies (Nielson and DesRoches, 2007; Pan et al., 2007; Ramanathan, 2012; Zhang and Huo, 2009) defined different limit states for columns and achieved different conclusions. However, the limit states are related to the geometry of the section, material, and volumetric percentage of the confining steel, and axial compression ratio of the section. Thus, numerical simulation is employed to derive the limit states of the pylon sections. First, a fiber section is created using OpenSEES (McKenna and Fenves, 2010), as shown in Figure 4, and a pushover analysis is then performed to obtain the moment-curvature curve of the sections. The normalized values of the curvature with respect to φ₁ are listed in Table 3.

Figure 4.

Fiber section of sections 1 and 2.

Component and system-level fragility curves

Using the probabilistic seismic demand model (PSDM) and the limit states in the framework of equation (3), fragility curves for each of bridge components are easily calculated and plotted in Figure 5. The pot bearing (δ_PB) appears to be the most vulnerable component at the slight damage state, followed by the abutments (δ_AP and δ_AA). For the moderate damage state, the pot bearing (δ_PB) is still the most vulnerable component, but followed by section 1 of pylon 2 (μ_φ-p2s1). However, section 1 of pylon 2 (μ_φ-p2s1) are the most vulnerable components at the extensive damage state and complete damage state, respectively. For all the damage states, pylon 2 is more vulnerable than pylon 1. This is associated with the fact that pylon 1 is longer and more flexible than pylon 2, and thus pylon 2 experiences larger seismic demands than pylon 1.

Figure 5.

Component and system fragility curves for the cable-stayed bridge: (a) slight damage, (b) moderate damage, (c) extensive damage, and (d) complete damage.

X = (X₁, X₂, …, X₁₂) represents the vector of demands, placed on the EDPs of the cable-stayed bridge, and Y = ln(X) is the vector of demands in the log-transformed space. JPSDM is then formulated in the log-transformed space by assembling the vector of means $μ_{Y} (PGV = pgv)$ and the covariance matrix $σ_{Y}$ . The correlation coefficient matrix of the demand of 8 EDPs is listed in Table 4. MCS is used to compare realizations of the demand and capacity of components to calculate the failure probability of the system using equation (6). Finally, a system fragility curve can be determined as a lognormal cumulative distribution function with two parameters (median and dispersion) by performing a regression analysis, which is shown in Figure 5.

Table 4.

Correlation coefficients between transformed demands.

	μ_φ-P1S1	μ_φ-P1S2	μ_φ-P2S1	μ_φ-P2S2	δ_AP	δ_AA	δ_PB	δ_AD
μ_φ-P1S1	1.000	0.722	0.589	0.471	0.656	0.327	0.665	0.698
μ_φ-P1S2	0.722	1.000	0.347	0.512	0.766	0.136	0.841	0.842
μ_φ-P2S1	0.589	0.347	1.000	0.179	0.370	0.380	0.436	0.468
μ_φ-P2S2	0.471	0.512	0.179	1.000	0.438	0.203	0.539	0.535
δ_AP	0.656	0.766	0.370	0.438	1.000	0.194	0.759	0.764
δ_AA	0.327	0.136	0.380	0.203	0.194	1.000	0.045	0.049
δ_PB	0.665	0.841	0.436	0.539	0.759	0.045	1.000	0.992
δ_AD	0.698	0.842	0.468	0.535	0.764	0.049	0.992	1.000

Optimization of FVD based on system fragility function

Polonggou cable-stayed bridge is located in the border of Sichuan and Tibet of China, which is prone to experience large earthquake. Viscous fluid dampers, deemed as passive energy dissipation devices for seismic protection of structures, are equipped in the cable-stayed bridge which possess lower damper, as shown in Figure 6.

Figure 6.

The Polonggou Bridge during construction and the application of FVD.

Experimental testing has shown that a suitable mathematical model for describing the behavior of viscous fluid dampers is given by the following nonlinear force–velocity relation

f_{D} = C_{D} | \overset{\cdot}{u} |^{α} sgn (\overset{\cdot}{u})

(7)

where f_D is the force developed by the damper; u is displacement across the damper; α is a real positive exponent whose value is determined by the piston head orifice design; C_D denotes the damping coefficient with units of force per velocity raised to the power α; sgn(·) is the signum function; and the over dot indicates differentiation with respect to time, t. The exponent α typically has a value usually ranging from about 0.2 to 0.4 for cable-stayed bridges, 0.3 is selected in this study for α, and investigates the optimal C_D for Polonggou cable-stayed bridge (Figure 7).

Figure 7.

The restore curves of FVD.

Parametric study and optimization of dampers are performed with the exponent α as 0.3 and the damping coefficient of the viscous dampers C_D varying from 1.0 × 10³ to 10 × 10³.

The component fragility curves are compared before and after the application of FVD with different parameters, as illustrated in Figure 8. The results indicate that increasing the additional damper of the cable-stayed bridge could reduce displacement of the bridge, resulting in reducing the probability of damage of displacement related EDPs (the bearing, the abutment, the unseating, etc.) for a given intensity measure. So the optimal C_D is 1.0 × 10⁴ in terms of the bearing abutment and unseating performance. Changing the C_D from 1.0 × 10³ to 1.0 × 10⁴ would increase the seismic force of the bridge, but the bending moment would not increase until C_D is larger than 3.0 × 10³ and 1.0 × 10⁴ for pylon 1 and pylon 2, respectively. The optimal C_D is 3.0 × 10⁴ or 8.0 × 10⁴ if we focus only on the vulnerability based on pylon 1 or pylon 2. The trends are evident and are shown clearly in Figure 9 which depicts the median value of fragility curves of μ_φ-P1S1, μ_φ-P2S1, and δ_PB. Note that higher median PGV in Figure 9 indicates less vulnerability of the given component.

Figure 8.

Moderate fragility curves of components of Polonggou cable-stayed bridge equipped with varied FVD: (a) μ_φ-P1S1, (b) μ_φ-P2S1, and (c) δ_PB.

Figure 9.

The median value of fragility of each component.

The optimal parameters of FVD varied when focused on different components. Hence, the importance of characterizing not only the component vulnerability but also system-level impact of FVD is emphasized. The component fragility curves analysis leads to a need to evaluate the comprehensive insight of the overall system vulnerability of the cable-stayed bridge. The system fragility curves are developed by considering the contribution of multiple vulnerable components. Increasing the C_D from 3.0 × 10³ to 8.0 × 10³ would have a negative impact on pylon 1 (increasing the fragility of μ_φ-P1S1), yet lead to a better performance of the cable-stayed bridge as a system (reducing the system-level fragility). Consideration of the fragility based only on individual retrofitted components, without regard for the system, may lead to over-estimation or under-estimation of the impact on the bridge fragility. The importance of developing system-level retrofitted bridge fragility curves is again recognized. For example, changing the C_D from 8.0 × 10³ to 1.0 × 10⁴ would have reduced vulnerability of unseating, but would increase the probability of damage as a bridge system. As shown in Figure 10, as a system, the Polonggou cable-stayed bridge tends to be less vulnerable when C_D varies from 1.0 × 10³ to 8.0 × 10³ and turns more vulnerable when the C_D becomes larger than 8.0 × 10³. Thus, we can make a conclusion that the optimal parameter of FVD is 8.0 × 10³ to achieve the best performance of Polonggou cable-stayed bridge subjected to earthquake.

Figure 10.

System-level fragility curves of Polonggou cable-stayed bridge equipped with varied FVD: (a) moderate damage and (b) extensive damage.

Conclusion

Seismic retrofit measures tend to focus on individual component response, less research has been carried out on the cable-stayed bridge performance as a system, and the optimal parameters of retrofit devices have not been addressed in the literature. This study presents a methodology to evaluate the optimal parameters of FVD for cable-stayed bridges using the system-level fragility function. JPSDM and Monte Carlo simulation are employed to obtain the system fragility of cable-stayed bridges by accounting for the contribution of multi components to the global damage state. Optimal parameters of FVD are derived by directly evaluating the system-level fragility curves of the cable-stayed bridge equipped with these given FVD.

A case study bridge, Polonggou cable-stayed bridge, retrofit with varied FVD is employed to exemplify the methodology. Details of the numerical modeling include nonlinear analysis, the uncertainty treatment, limit state capacities, contribution of critical components to the bridge performance as a system, and so on. For illustration purposes, component fragility curves are developed, which indicate that retrofit measures alters the relative vulnerability of the different components. The system-level fragility curves as well as component fragility curves are compared before and after the application of FVD with different parameters. The results indicate that a given parameter of the FVD may have a positive impact on some components, yet lead to a worse performance of the bridge as a system. Thus, in order to obtain comprehensive insight of bridge performance and derive the accurate optimal parameters of FVD, it is necessary to consider the fragility based on bridge system. The damper exponent α and coefficient C_D are chosen as 0.3 and 8000, respectively, to achieve the best performance of Polonggou cable-stayed bridge as a system.

This methodology can also be employed to perform the optimization of other mitigation devices, the laminated rubber bearing (LRB) system, friction bearing systems, energy dissipation bearing (EDB), shape memory alloy (SMA), and so on. However, it is time consuming to determine the ideal design parameters when the device has more than two characteristic parameters since the substantial effort for the large number of computations. Future work is to propose an efficient computational framework to determine the best performance for isolation devices with multiparameters.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant 51608161), the China Postdoctoral Science Foundation (2016M602007), and the Fundamental Research Funds for the Central Universities (JZ2016HGBZ0786). The support is gratefully acknowledged.

ORCID iD

Jian Zhong

References

Aiken

Kelly

(1992) Comparative study of four passive energy dissipation systems. Bulletin of the New Zealand Society for Earthquake Engineering 25(3): 175–192.

Ali

Abdel-Ghaffar

(1994) Seismic energy dissipation for cable-stayed bridges using passive devices. Earthquake Engineering & Structural Dynamics 23(8): 77–93.

ATC (1985) Earthquake Damage Evaluation Data for California. Redwood City, CA: Applied Technology Council (ATC).

Avşar

Yakut

Caner

(2011) Analytical fragility curves for ordinary highway bridges in Turkey. Earthquake Spectra 27(4): 971–996.

Baker

Cornell

(2005) A vector-valued ground motion intensity measure consisting of spectral acceleration and epsilon. Earthquake Engineering & Structural Dynamics 34(10): 1193–1217.

Barnawi

Dyke

(2014) Seismic fragility relationships of a cable-stayed bridge equipped with response modification systems. Journal of Bridge Engineering 19(8): A4013003.

Basoz

Kiremidjian

King

et al . (1999) Statistical analysis of bridge damage data from the 1994 Northridge, CA, earthquake. Earthquake Spectra 15(1): 25–54.

Bruneau

(1992) Evaluation of system-reliability methods for cable-stayed bridge design. Journal of Structural Engineering 118(4): 1106–1120.

Casciati

Cimellaro

Domaneschi

(2008) Seismic reliability of a cable-stayed bridge retrofitted with hysteretic devices. Computers & Structures 86(17–18): 1769–1781.

10.

Choi

DesRoches

Nielson

(2004) Seismic fragility of typical bridges in moderate seismic zones. Engineering Structures 26(2): 187–199.

11.

Cornell

Jalayer

Hamburger

et al . (2002) Probabilistic basis for 2000 SAC Federal Emergency Management Agency steel moment frame guidelines. Journal of Structural Engineering 128(4): 526–533.

12.

FEMA (1997) FEMA 274, NEHRP Commentary on the Guidelines for the Seismic Rehabilitation of Buildings. Washington, DC: Federal Emergency Management Agency.

13.

Hwang

Liu

Chiu

(2001) Seismic Fragility Analysis of Highway Bridges. Memphis, TN: The University of Memphis.

14.

Luco

Cornell

(2007) Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions. Earthquake Spectra 23(2): 357–392.

15.

McKenna

Fenves

(2010) Nonlinear finite-element analysis software architecture using object composition. Journal of Computing in Civil Engineering 24(1): 95–107.

16.

Muthukumar

DesRoches

(2006) A hertz contact model with non-linear damping for pounding simulation. Earthquake Engineering & Structural Dynamics 35(7): 811–828.

17.

Nielson

(2005) Analytical fragility curves for highway bridges in moderate seismic zones. PhD Dissertation, Georgia Institute of Technology, Atlanta, GA.

18.

Nielson

DesRoches

(2007) Seismic fragility methodology for highway bridges using a component level approach. Earthquake Engineering & Structural Dynamics 36(6): 823–839.

19.

Padgett

DesRoches

(2008) Methodology for the development of analytical fragility curves for retrofitted bridges. Earthquake Engineering & Structural Dynamics 37(8): 1157–1174.

20.

Padgett

Nielson

DesRoches

(2008) Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios. Earthquake Engineering & Structural Dynamics 37(5): 711–725.

21.

Pan

Agrawal

Ghosn

(2007) Seismic fragility of continuous steel highway bridges in New York State. Journal of Bridge Engineering 12(6): 689–699.

22.

Pang

Shen

et al . (2014) Seismic fragility analysis of cable-stayed bridges considering different sources of uncertainties. Journal of Bridge Engineering 19: 04013015.

23.

Ramanathan

(2012) Next Generation Seismic Fragility Curves for California Bridges Incorporating the Evolution in Seismic Design Philosophy. Atlanta, GA: Georgia Institute of Technology.

24.

Ren

Obata

(1999) Elastic-plastic seismic behavior of long span cable-stayed bridges. Journal of Bridge Engineering 4(3): 194–203.

25.

Shafieezadeh

Ramanathan

Padgett

et al . (2012) Fractional order intensity measures for probabilistic seismic demand modeling applied to highway bridges. Earthquake Engineering & Structural Dynamics 41(3): 391–409.

26.

Sharabash

Andrawes

(2009) Application of shape memory alloy dampers in the seismic control of cable-stayed bridges. Engineering Structures 31(2): 607–616.

27.

Shinozuka

Feng

Kim

et al . (2000a) Nonlinear static procedure for fragility curve development. Journal of Engineering Mechanics 126(12): 1287–1295.

28.

Shinozuka

Feng

Lee

et al . (2000b) Statistical analysis of fragility curves. Journal of Engineering Mechanics 126(12): 1224–1231.

29.

Wilson

Gravelle

(1991) Modeling of a cable-stayed bridge for dynamic analysis. Earthquake Engineering & Structural Dynamics 20(8): 707–721.

30.

Zhang

Huo

(2009) Evaluating effectiveness and optimum design of isolation devices for highway bridges using the fragility function method. Engineering Structures 31(8): 1648–1660.

31.

Zhang

Zhu

(2009) Seismic performance of benchmark base-isolated bridges with superelastic Cu-Al-Be wire damper. Structural Control and Health Monitoring 16(6): 668–685.

32.

Zhong

Jeon

Pang

et al . (2016) Seismic fragility assessment of long-span cable-stayed bridges in China. Advances in Structural Engineering 19(11): 1797–1812.

33.

Zhong

Jeon

Yuan

et al . (2017) Impact of spatial variability parameters on seismic fragilities of a cable-stayed bridge subjected to differential support motions. Journal of Bridge Engineering 22(6): 04017013.

34.

Zhu

Qiu

(2014) Incremental dynamic analysis of highway bridges with novel shape memory alloy isolators. Advances in Structural Engineering 17(3): 429–438.