Seismic vulnerability of multi-span continuous girder bridges 1

with steel fibre reinforced concrete columns 2

Yuye Zhang a,b,, D. Dias-da-Costa b,c 3

a Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing 210094, China 4 b School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia 5

c ISISE, University of Coimbra, Rua Luís Reis Santos, 3030-788 Coimbra, Portugal 6

Abstract: The occurrence of earthquakes during the lifetime of bridges can result in the closure or even 7

the failure of the structure. The vulnerability of multi-span continuous girder (MSCG) bridges strongly 8

depends on the seismic performance of its columns. This paper investigates the seismic vulnerability of 9

MSCG bridges with reinforced concrete (RC), steel fibre reinforced concrete (SFRC) and RC-SFRC 10

columns. Quasi-static tests were conducted on eight different columns to obtain the limit-state 11

capacities and validate numerical models. Numerical analyses were performed on MSCG reference 12

bridges with RC, SFRC or RC-SFRC columns to derive probabilistic models of the demands upon critical 13

components. Fragility curves were established as a function of peak ground acceleration by integrating 14

the capacity distributions with correlated component demand distributions. Results indicate that: i) 15

SFRC columns with 1.0% fibre ratio show a better performance-price ratio for improving the structural 16

capacity, compared with those with 1.5%; ii) MSCG bridges with SFRC and RC-SFRC columns are less 17

vulnerable to earthquakes when compared with those constructed using only RC, and these differences 18

increasing with the earthquake intensity; and iii) the seismic vulnerability of MSCG bridges with SFRC 19

placed only at the plastic hinges is similar to the one found SFRC on the whole column. On a broader 20

perspective, the conclusions drawn could offer a new strategy for the seismic enhancement or retrofit 21

of MSCG bridges in earthquake regions, with optimal use of SFRC. 22

Corresponding author. Tel.: +86 25 84315773; fax: +86 25 84315875.

Email address: zyy@njust.edu.cn (Y. Zhang).

Keywords: Seismic vulnerability; Fragility curves; Girder bridges; Columns; Steel fibre reinforced 23

concrete. 24

1. Introduction 25

Multi-span continuous girder (MSCG) bridges are widely found around the world. Many of these 26

bridges already suffered from seismic effects, such as the Baihua Bridge in the 2008 Wenchuan 27

earthquake [1], Juan Pablo II Bridge in the 2010 Chile Earthquake [2], and the Tohoku-Shinkansen 28

viaducts in the 2011 Great East Japan earthquake [3]. The extent of damage in these examples has raised 29

the public concern towards the seismic safety of MSCG bridges. 30

The columns of MSCG bridges play an important role on the structural performance and potential 31

vulnerability of the structural system. During an earthquake, the columns can dissipate large amounts 32

of energy by means of the plastic hinges [4,5] and hence mitigate the vulnerability of the bridge to 33

damage. Bridge columns can be severely damaged during such an event, particularly in the presence of 34

inadequate flexural strength, ductility and energy dissipation capacity [6,7]. Improving these features 35

can significantly enhance the overall seismic performance. A conventional method for achieving higher 36

ductility and energy dissipation consists in locally increasing the density of stirrups. However, this is a 37

common source of construction difficulties and insufficient concrete pouring quality [8,9]. For this 38

reason, the use of steel fibre reinforced concrete (SFRC) has been pointed out by several researchers 39

[10-12], as a good alternative with advantages over reinforced concrete (RC). 40

The seismic capacity of hollow bridge columns constructed with SFRC was already investigated in a 41

previous study [8] where it was found that SFRC can indeed improve the ductility and energy dissipation 42

of the column. Yet, little research has been done in what concerns the seismic performance of the whole 43

structural system and how SFRC columns may impact on the overall safety of the bridge subjected to 44

earthquakes. This needs to be quantitatively addressed so that enhancement using SFRC in columns can 45

be adopted in future bridge design. 46

In this scope, the development of fragility curves to describe the probability of a bridge exceeding a 47

particular level of damage, as a function of ground motion intensity, can potentially offer an insight to 48

the assessment of the prescribed performance criteria of the bridge [13-15]. Fragility curves are being 49

increasingly used to evaluate the seismic vulnerability of bridge structures due to their probabilistic 50

features and could form a basis for performance-based seismic design of bridges [5,16,17]. In particular, 51

columns, bearings and abutments are relatively vulnerable components in continuous girder bridge 52

systems when subjected to ground motions, and for this reason existing works focused on the definition 53

of the different seismic damage states based on the failure of these components [14,16,18]. Fragility 54

curves are increasingly used to evaluate the seismic vulnerability of bridge structures due to their 55

probabilistic features and could form a basis for performance-based seismic design of bridges [5,16,17]. 56

In particular, columns, bearings and abutments are relatively vulnerable components in continuous 57

girder bridge systems when subjected to ground motions, and for this reason existing works focused on 58

the definition of the different seismic damage states based on the failure of these components [14,16,18]. 59

To the authors’ knowledge, the seismic performance of MSCG bridges with SFRC columns was not yet 60

assessed in the literature. Aiming at closing the gap, this paper addresses the seismic vulnerability of 61

MSCG bridges with different kinds of columns (i.e., RC, SFRC or RC-SFRC columns) using the fragility 62

function method. An innovative solution based on the optimised use of SFRC suitable for construction 63

based on segments in single-column piers is also presented and assessed. A numerical study based on 64

models calibrated with experimental tests from concrete columns is used to highlight the advantages of 65

multi-span continuous girder bridges with SFRC and to develop fragility curves that could be used for 66

future vulnerability tests in these structures and design purposes provided that more experimental data 67

is obtained. 68

The paper is organised as follows. Firstly, Section 2 describes quasi-static tests of eight solid column 69

specimens made of RC, SFRC or RC-SFRC to obtain the characteristic values of the structural capacity for 70

different damage states and calibrate a finite element model [19]. Section 3 presents the numerical 71

models of MSCG bridges used with to analyse the nonlinear time histories and develop probabilistic 72

models of demands on critical components, including columns, bearings and abutments. The fragility 73

curves of the bridge systems are established in Section 3 as a function of peak ground acceleration (PGA) 74

by integrating the capacity distributions with the correlated component demand distributions. The 75

fragility curves obtained for different cases are compared in detail in Section 4 to assess the role of the 76

column enhancements on the structural safety of the MSCG bridges subjected to earthquakes. Finally, 77

the most relevant conclusions are summarised in Section 5. 78

2. Structural behaviour of SFRC columns 79

This section describes the experimental and numerical characterisation of the seismic behaviour of 80

SFRC columns that support the development of the bridge models in the next section. 81

2.1. Concrete strength 82

The strength of concrete with different steel fibre ratios (i.e., 0, 0.5%, 1.0%, and 1.5%) was first 83

assessed using 36 compression tests were conducted on concrete prisms. The specimens, with 84

dimension of 100×100×300 mm3, were divided into 12 groups. For each steel fibre ratio, three groups 85

of specimens comprised of nine prisms were tested. The compression test system and a typical specimen 86

are shown in Fig. 1. Typical compressive stress-strain curves for Groups 2 and 7 are shown in Fig. 2, 87

whereas the concrete strength and corresponding strain are listed in Table 1 for the whole range of tests. 88

As general observation, the strength of concrete increases slightly with the increasing percentage of 89

steel fibres, whereas the impact on the corresponding ultimate strain is relatively more significant. 90

2.2. Quasi-static experimental tests 91

Eight solid column specimens were constructed with different steel stirrup ratios, steel fibre content 92

and height of SFRC region. From these specimens, one RC specimen (S1) was used as reference, whereas 93

the other four SFRC specimens (S2~S5) and three RC-SFRC specimens (S6~S8) are defined in Table 2. 94

In specimens S6~S8, the SFRC was applied only in the region of potential plastic deformation, as this is 95

the most vulnerable area when subjected to ground motions [7,20,21]. 96

The longitudinal reinforcement consisted of four steel bars with 14 mm diameter, whereas the 97

stirrups were 8 mm diameter bars. As shown in Table 2, the stirrups in S5 were spaced 110 mm 98

vertically and its volumetric transverse reinforcing ratio (ρv) was 1.0%. For the other specimens, the 99

stirrups were spaced 70 mm and ρv was 1.5%. The steel fibre contend (ρf) ranged from 0.0% on the RC 100

column to 1.5%. The steel fibres had a diameter of 0.55 mm and length of 35 mm with hooks at both 101

ends, and its tensile strength was 1150 MPa. 102

Fig. 3 illustrates the experimental set-up used in this study, in which the specimens are tested in an 103

inverted position to accommodate frame. A schematic diagram is shown in Fig. 3a with a picture of a 104

representative specimen in the test rig in Fig. 3b. Each specimen was 200×200×800 mm3 and was cast 105

monolithically with a 700×700×300 mm3 foundation block and 700×700×250 mm3 loading block. The 106

geometry of test columns was set at a one-tenth scale to the MSCG bridge columns discussed later on in 107

this paper. The height-to-width aspect ratio for the columns was defined according to Chinese standard 108

JTG/T B02-01-2008, in which this ratio for a regular column shall be defined between 2.5 and 10.0 [22]. 109

The standard also suggests that the aspect ratio for the single-column piers (as used in this study) to be 110

relatively small, so that the main girder is not prone to overturning. Thus, an aspect ratio of 4.0 was 111

selected for the columns in the investigation. To improve the shear capacity of the columns and minimise 112

shear damage, a relatively large volumetric transverse reinforcing ratio (1.5%) was selected for the 113

columns, as for a real bridge in China [23]. The lateral cyclic loading and constant axial loading were 114

applied to the loading block through the steel clevis seen at the bottom of each specimen. The lateral 115

load was applied under displacement control during the test, for gradually increasing lateral 116

displacements of 5, 10, 15, 20, 30, 40, 50, 60 and 75 mm. The axial load was constant and represented 117

10% of the nominal axial capacity of the column, the selected axial load ratio (equals to P/f 'cAg, where P 118

is the axial load, f 'c is the concrete compressive strength and Ag is the gross area of column cross-section) 119

is determined according to previous studies [24-26]. LVDTs were placed to measure the displacements 120

at key locations along the specimens, while the forces were measured by the loading system. 121

2.3. Quasi-static test results 122

The horizontal load-displacement hysteretic curves of the column specimen can be obtained from 123

the data measured by the LVDTs placed at the top. Typical results of load-displacement hysteretic curves 124

for specimens S1 and S7 are shown in Fig.4, where the latter shows much better energy dissipation 125

capacity (seen from the hysteresis loop areas). The envelope curves illustrated in Fig.5 were obtained 126

by connecting the peak responses on the hysteresis curves and can provide important information 127

regarding the seismic capacity of structures. The role of the steel fibre content(Vf) and the length of SFRC 128

region (hf) is shown in Figs.5a and b, respectively. 129

The SFRC columns exhibit higher bearing capacity and an improved behaviour with respect to 130

maintaining its peak capacity (i.e., the increasing ductility) when compared with the reference RC 131

column, as illustrated in Fig. 5a, and this enhancement increases with the steel fibre volume ratio. It 132

needs to be noted, however, that this is not obvious when the fibre ratio increases from 1.0% to 1.5%. 133

Fig.5b shows that the skeleton curves for specimens S7 (hf=200 mm) and S8 (hf=300 mm) are similar to 134

the one obtained for S3 (hf=800 mm). Thus, it can be concluded that adopting SFRC locally, i.e. at the 135

potential plastic hinge region, produces an enhancement similar to the adoption of SFRC for the whole 136

column. In this study, this region extends to circa 200-300 mm at the bottom of the column, which is in 137

the range of about 1-1.5 times the maximum cross-section dimension for ductile bridge columns [25,27]. 138

2.4. FE analysis 139

Numerical models of the columns discussed above were carried out using OpenSees [19] to 140

investigate the seismic response. The model was composed of beam-column and contact (zero-length) 141

elements. Fig. 6 shows a representation, including the cross-sectional mesh and main constitutive 142

models adopted. 143

The numerical model shown in Fig. 6 used ‘concrete07’ and ‘reinforcing steel’ models in OpenSees 144

[19] to, respectively simulate both confined and unconfined concrete, and steel bars. The specific values 145

adopted for these models can be found in a previous study [8]. Since the addition of steel fibers to the 146

concrete structures can also influence the peak bond strength and bond degradation, the ‘bond_sp01’ 147

material model [19, 28] was applied to simulate the differences in bond–slip response between RC and 148

SFRC columns. A zero-length section element using this material model was introduced at the 149

intersection between the pier shaft and foundation, as illustrated in Fig. 6. There are six main mechanical 150

parameters adopted in the 'bond_sp01' material model, i.e. fy, Sy, fu, Su, b and R, as presented in Fig. 6 and 151

detailed in Table 3. In addition, parameter K in the figure denotes the initial stiffness of the stress vs. slip 152

curve, which can be obtained by fy/ Su. 153

The bar slip for both yield (Sy) and ultimate stresses (Su), can be obtained using the equation 154

proposed by Zhao and Sritharan[28] as follows: 155

( ) 34.0)8437/(12)(54.21

++= cyby ffdS (1) 156

yu SS = 35 (2) 157

where db is reinforcing bar diameter, fy and fc are the steel yield and concrete compressive strengths 158

respectively, and α is the parameter defining the local bond-slip relation and herein taken as 0.4. For the 159

SFRC specimens with different steel fibre ratio (ρf), the value of Sy is modified by the equation 160

recommended by Harajli [29], as follows: 161

]}1))(/[(8.1exp{ 2

1 −= SFRCuuSS myy (3) 162

3/2)]/(45.0[78.0 bfffcy ddLcVcfu +=

(4) 163

)()25.0/34.01()( RCudLVSFRCu mfffm −+=

(5) 164

3/2)/(75.0)( bcm dcfRCu =

(6) 165

where uy is the yielding bond stress, um is the maximum bond stress at bond failure, S1 is a parameter 166

that equals 0.15c0 (c0 is the clear distance between the ribs of the reinforcing bar), c is the minimum 167

concrete cover, Vf is the steel fibre volume fraction, and Lf and df are the length and diameter of steel 168

fibres, respectively. Using the above equations, the mechanical properties adopted in 'bond_sp01' 169

material model for the SFRC specimens with different steel fibre ratio can be obtained, as listed in Table 170

3, in which the values of b and R are the recommended values in OpenSees [19]. 171

The hysteretic load-displacement curves of typical specimens (S1 and S7) are compared against the 172

numerical results in Fig. 7. In general terms, it can be highlighted the good agreement between both for 173

the different stages of analysis. The validated RC and SFRC column models were applied in the MSCG 174

bridge model shown in the following sections. 175

3. Seismic fragility analysis of MSCG bridges 176

3.1. Analysis model 177

A MSCG bridge with the geometry shown in Fig. 8 was selected for studying the seismic 178

vulnerability of the structure. The bridge was 109.0 m long and 12.6 m wide, and had three spans 179

(30.5+48.0+30.5 m) supported by two piers of 8.0 m height between the two abutments at the ends. 180

Each pier consisted of one column, with a cross-section of 2.0x2.0 m2. Lead rubber bearings (LRB) 181

connected the superstructure to the column caps, whereas sliding bearings (SLB) connected the 182

superstructure to the abutments. 183

Three different nonlinear finite element models with RC, SFRC or RC-SFRC columns, corresponding 184

to Cases 1, 2 and 3, are respectively listed in Table 4. The columns adopted in Cases 2 and 3 were 185

modelled according to specimens S3 and S7, respectively, in the quasi-static tests described earlier, due 186

to the relatively better performance. The bridge with RC columns (Case 1) was taken as a reference 187

model for the comparative analysis. The bridge models were then analysed concerning the influence of 188

the SFRC and the seismic vulnerability of the structure. 189

The superstructure of the MSCG bridge was modeled with elastic beam-column elements, since it 190

is expected to remain in the elastic stage when subjected to ground motions [30,31]. There were ten 191

elements in the main span, and seven elements in each side span. The identified material properties and 192

models of RC and SFRC columns listed above were adopted in the MSCG bridge model. The bridge 193

columns were modeled by nonlinear beam-column elements, and the section material properties were 194

represented using the three constitutive models illustrated in Fig. 6 (i.e., ‘concrete 07’, ‘reinforcing steel’ 195

and ‘bond_sp01’ material models). The columns and caps were coupled using rigid links as 196

recommended by Seo et al. [32]. Linear translational and rotational springs, located at the base of the 197

columns, were used to represent the behavior of pile foundations, as carried out in previous literature 198

[14,33]. The abutments were incorporated in the model using a rigid element with a length equal to the 199

deck width, which was connected to the superstructure centerline with a rigid joint. As recommended 200

by Pang et al. [34], the rigid element was supported by a gap element at each end, and springs in 201

horizontal and vertical directions using nonlinear zero-length elements. The force-displacement 202

relationship of the zero-length elements, shown in Fig. 8, was simplified into a multi-linear relationship 203

as proposed by Caltrans (2006) [35]. The abutment stiffness constants, maximum carrying capacities 204

and the expansion gap width were obtained from Caltrans (2006) [35] and Pang et al. [34]. 205

The bearings (LRB and SLB), were modelled using zero-length elements with a bilinear constitutive 206

model in both longitudinal and transverse directions. The element type and corresponding mechanical 207

constants can be found in [36] and are shown in Fig. 8. In this figure, μ is the sliding friction coefficient, 208

N is the bearing pressure, Xy is the critical displacement, Fy is the yield strength, and K1 and K2 are the 209

initial stiffness and yield stiffness, respectively. 210

The uncertainties concerning material and geometric parameters related to the structure were 211

considered in the structural safety assessment [13,14]. Random samples of twelve parameters, 212

including material properties and structural parameters, were combined with the bridge models to 213

account for the inherent variability and uncertainties. The parameters were taken as random variables 214

with the mean and standard deviation (variance) presented in Table 5. The concrete strength and 215

corresponding strain for different kinds of concrete, listed in Table 5, were derived from the 216

compression test results presented in Section 2.1. The standard deviation for the concrete strength and 217

corresponding strain were calculated by the following equation: 218

=

−=n

i

ixn 1

2)(1

(7) 219

where β is the standard deviation, n is the number of test values for the concrete strength or 220

corresponding strain, xi is the test value, and is the mean of each parameter. 221

3.2. Fragility analysis 222

Fragility analyses were carried out for the three cases in order to assess the effectiveness of the 223

SFRC in improving the seismic performance of the bridge system. First, a suite of ground motions was 224

selected and the corresponding nonlinear dynamic analysis was performed. This allowed obtaining the 225

probabilistic seismic demand, considering the different ground motion conditions and structural 226

modelling parameters. The Latin-hypercube technique was used to generate random samples of bridges, 227

which were then paired with the ground motion records. The limit-state capacities of the critical bridge 228

components were determined based on the above test results and corresponding analyses. Finally, by 229

integrating the data related to limit-state capacities and seismic demands, the fragility curves were 230

derived for the three bridges. In the following sections, each step into this derivation is further detailed. 231

3.2.1. Ground motion suite 232

A suite of 56 ground motion records from the PEER Ground Motion Database 233

(http://peer.berkeley.edu/smcat) was used to evaluate the seismic response of the bridges in this paper. 234

The effect of the earthquake source and magnitude, and soil was taken into account in terms of ground 235

motion [14,37]. The magnitudes ranged from 6.0 to 7.5, and the epicentral distances from 10 to 100 km. 236

The standard site condition corresponded to dense soil or soft rock with a range of average shear wave 237

velocity as described in previous studies [37,38]. The response spectra with 5% damping ratio are 238

illustrated in Fig. 9 for the selected 56 ground motions, together with the mean response spectrum 239

(shown as black thick line). 240

3.2.2. Seismic Demand Estimates 241

The bridge models for the three cases were paired with the 56 selected ground motions, resulting 242

in a total of 56 earthquake-structure samples for each bridge case. The selected ground motions were 243

applied in the longitudinal direction of the bridge for simplicity. The seismic responses of the different 244

components of the bridge were then analysed. As recommended by Nielson and DesRoches [13], the 245

peak ground acceleration (PGA) was regarded as a good intensity measure for the structure and was 246

herein selected as intensity parameter. The probabilistic seismic demand model (PSDM) was obtained 247

by regression analysis using the response of the critical bridge components [38] (i.e., columns, bearings 248

and abutments). These responses were measured for the three cases by the column displacement 249

ductility ratio (μd), the bearing deformation (δbl for LRB deformation, δbs for SLB deformation) and the 250

abutment deformation (δaa for active abutment deformation, δap for passive abutment deformation) 251

respectively, as listed in Table 6. The relationship between seismic responses and PGA can be 252

represented in a power model [38]: 253

( )=b

  DS a PGA

(8) 254

which can be linearised according to the following expression: 255

= + ln ln lnDS a b PGA

(9) 256

where SD is the median value of the demand in terms of PGA, and a and b are regression coefficients. 257

Then, the column displacement ductility ratio, bearing and abutment deformation versus PGA can 258

be plotted on a log–log scale. The probabilistic seismic demand models (PSDMs) for Case 1 is illustrated 259

in Fig. 9. Table 6 shows the PSDMs of the selected components for the three SMCG bridge cases. 260

Parameter βD|PGA is defined as the logarithmic standard deviation of the demand given the PGA, and can 261

be estimated for each component in the regression analysis as 262

( )

−

−

2

|

ln ln 

2

bi

PGAD

Pd GAa

N (10) 263

where di is the peak component demand and N the number of simulations. 264

3.2.3. Limit-state capacities 265

In probabilistic analyses, the limit-state capacities of bridge components are commonly served as 266

the thresholds of the components entering prescriptive damage states, such as slight, moderate, 267

extensive and complete damage [39,40]. Accordingly, the limit-state capacity (limit state) can be defined 268

as a measure of the capacity of the bridge components to withstand an earthquake without exceeding 269

the prescribed performance level [13]. For different kinds of bridge columns, the damage index (DI) 270

proposed by Karim and Yamazaki [41] was adopted to determine the quantitative limit states for slight, 271

moderate, extensive and complete damage state respectively. The DI is expressed as 272

cu

chcDI

+=

(11) 273

where DI is 0.14, 0.40, 0.60 and 1.00 for slight, moderate, extensive and complete damage respectively 274

[41]; μc is the displacement ductility ratio of the bridge columns; μcu is the ultimate displacement 275

ductility ratio, which is defined as the ratio of the displacement at the ultimate load (Δu) to that at the 276

yield load (Δy), as listed in Table 7; μch is the cumulative energy ductility ratio, which is defined as the 277

ratio of the hysteretic energy to the energy at yield point (derived from the load-displacement hysteretic 278

curves); and β is the cyclic loading factor obtained by the following equation: 279

0( 0.447 0.073 0.24 0.314 ) 0.7 w

t

ln

d

= − + + + (12) 280

where l/d is the shear span ratio, n0 is the normalised axial force, ρt is the reinforcement ratio for 281

longitudinal steel, and ρw is the reinforcement ratio for confining steel, as described in Ghosh et al. [42]. 282

In this study, the displacement ductility ratio (μc) is selected as the indicator for different damage states 283

of the columns, for simplicity. Then, the minimum ductility ratio, μci, for each damage state can be 284

obtained by the equation as follows: 285

chcuci DI −= )(

(13) 286

where μcu = Δu/Δy (listed in Table 7), in which Δu and Δy are obtained from the envelope curves of the RC, 287

SFRC and RC-SFRC columns (S1, S3 and S7 in Fig. 5). 288

The minimum ductility ratio (μci) for RC, SFRC and RC-SFRC columns at different damage states can 289

be obtained from Eq. (13) and listed in Table 8, which are used as the limit-state capacity indexes for 290

different bridge columns. The uncertainty associated with each median (Sc) for the column limit-state 291

(μci) can be characterised by assigning coefficients of variation (COV). The values of 0.25, 0.25, 0.50 and 292

0.50 are usually adopted to slight, moderate, extensive and complete damage states, respectively [16,18], 293

following the assumption that there is more uncertainty in higher damage states. It is also assumed that 294

different kinds of retrofitted bridge columns (at the same damage state) have the same COV value, as 295

conducted in [33,43,44]. The lognormal standard deviation (βC) is calculated using the equation 296

recommended by Ramanathan et al. [33]: 297

)1ln( 2COVC += (14) 298

The median values and the lognormal standard deviations for the column limit states can then be 299

obtained. For the abutments and bearings of the SMCG bridge models, the median values and the 300

lognormal standard deviations of the limit-state capacities are adopted as defined by Zhong et al. [36] 301

and Padgett et al. [45]. Table 8 lists the limit-state capacities for each component and for the three bridge 302

cases in terms of medians (SC) and lognormal standard deviations (βC). In this table, ‘N/A’ indicates that 303

the damage observed would not cause long-term impact on the serviceability of the structure and was 304

not included in the calculation of system vulnerability curves [45,46]. 305

3.2.4. Fragility curve development 306

The fragility curves express the probability of exceeding a limit state for each of the four damage 307

states defined in the previous section. The probability of the bridge component demand exceeding its 308

capacity, Pf, can be given by 309

=   fP P D C PGA (15) 310

where Pf is the fragility, and D and C represent the bridge component demand and capacity, respectively. 311

These were assumed to follow lognormal distributions, in which case the conditional probability of the 312

component demand reaching or exceeding the capacity, for a given PGA, can be computed in a closed 313

form as 314

( )

( )

= +

1/22 2

ln  D C

f

CD PGA

S SP D C PGA (16) 315

where Ф[·] is the standard normal cumulative distribution, SD and βD|PGA are the median and conditional 316

logarithmic standard deviation for the demand, and SC and βC are the median and conditional logarithmic 317

standard deviation for the capacity. 318

Substituting Eq.(9) into Eq.(16) leads to 319

( )

( )

− −

= +

1/22 2

ln ln ln  C

f

CD PGA

PGA S a bP D C PGA

b

(17) 320

where the bridge component fragility curves are presented in terms of their lognormal parameters. 321

The following steps were followed to assess the seismic vulnerability of the bridge system 322

[5,13,38,45]. Firstly, the correlation coefficients of the various component responses were used to 323

generate joint probabilistic seismic demand models (PSDMs) [13,32], in which the interaction among 324

components was considered. Then, the seismic vulnerability of the bridge system was derived from the 325

demand values and limit-state capacities of the components using the Monte Carlo simulation and a 326

system approximation [5,38,45]. The simulation integrates the joint PSDMs over all possible failure 327

modes for the different limit states [37]. The probability of the bridge system reaching or exceeding a 328

specified limit sate (Fs) is given by: 329

=

=

1

  s i

n

i

FPFP

(18) 330

where P[·] represents the probability, and Fi is the probability of each component reaching or exceeding 331

the specified limit state. Finally, the seismic vulnerability of the bridge system (i.e., the probability of the 332

system failure for different damage states) can be obtained for each PGA level [45]. Accordingly, the 333

parameters for the bridge system fragility, such as the median values of the damage probability, were 334

then estimated by regression analyses. The following section presents and discusses these results. 335

4. Results and discussions 336

Using the methodology presented above, the fragility curves of various bridge components were 337

derived for different damage states. The component vulnerability curves for Cases 1, 2 and 3 at 338

moderate damage state are illustrated in Fig. 11. Overall, it can be seen that, by comparing Figs. 9a, b, 339

and c, the bridge components become less fragile when adopting SFRC columns (Case 2) or RC-SFRC 340

columns (Case 3). The results also show that the application of SFRC in columns not only mitigates their 341

own seismic fragility, but also reduces the vulnerability of nearly all other evaluated components. This 342

is mainly because the hysteretic behaviour of the columns and the energy dissipation of the bridge 343

system are increased by the use of SFRC, and this then impacts on the seismic response of other bridge 344

components, such as the abutments and bearings. Comparing the damage probability of the columns for 345

Cases 1, 2 and 3 at 1.0 g PGA, the probability of exceedance at moderate damage is 76.5%, 53.1%, and 346

57.5% respectively. The damage probabilities of the columns for the latter two cases are substantially 347

reduced due to the SFRC. For lead rubber bearings (LRB), the probability of damage exceedance for 348

Cases 1, 2 and 3 is 82.9%, 66.7% and 68.2% respectively. It is shown that the damage probability of LRB 349

for each case changes with similar trend to that of the columns. 350

Fig. 12 shows the seismic fragility curves for the three bridge (system) cases and the four damage 351

states. Overall, the damage probability decreases with the increasing damage state. The bridge with RC 352

columns (Case 1) is more fragile than the one where the columns are made of SFRC and RC-SFRC (Cases 353

2 and 3), whereas the vulnerability is similar between Cases 2 and 3. Taking the slight damage state as 354

example, the probability of damage exceedance at 0.3 g PGA is reduced from 65.1% for Case 1, to 52.3%, 355

57.8% for Cases 2 and 3 respectively; while for 1.0 g PGA, this reduction is even greater, the probability 356

reduces from 98.9%, for Case 1, to 79.3% and 86.2%, for Cases 2 and 3, respectively. At the moderate 357

damage state, the trend is similar as slight damage state, with the probability of exceedance (for 0.3 g 358

PGA) decreasing from 4.9%, for Case 1, to 2.6% and 1.9% for Cases 2 and 3; while the probability (for 359

1.0 g PGA) reduces from 87.6%, for Case 1, to 59.6% and 67.1% for the other two cases. It can be seen 360

that the MSCG bridges with SFRC and RC-SFRC columns are less vulnerable to earthquakes when 361

compared with those with RC columns, and the differences increase with the earthquake intensity (PGA). 362

The median values (for PGA) of the damage probability of exceedance (with 50% exceeding 363

probability) were obtained from the above fragility curves to analyse the vulnerability of the three cases. 364

It should be noted that higher median PGA values for a bridge system indicate less vulnerability to 365

earthquakes. 366

Fig. 13 shows a chart of the median PGA for each damage state and the three bridge cases. In the case 367

of lightly damaged state, the median values of PGA range from 0.23 g (in Case 1) to 0.29 g (in Case 2) 368

and to 0.26 g (in Case 3). For moderate damage, the median increases to 0.61 g in Case 1, and 0.86 g and 369

0.80 g in Cases 2 and 3, respectively. For extensive damage, the median increases even further to 0.71 g 370

in Case 1 and 1.12 g and 0.99 g in Cases 2 and 3, respectively. This trend continues, and for complete 371

damage, the median value is 1.14 g, in Case 1, and 1.84 g and 1.80 g in the other two cases. As the damage 372

state increases, so does the difference between Case 1 and the remaining two cases. The reason for this 373

behaviour stems from the material behaviour. In fact, for light damage, both RC and SFRC columns 374

behave similarly while keeping below a pre-yield displacement level. However, as soon as the column 375

begins to be loaded beyond that level, the steel fibres have an important role in controlling cracking and 376

spalling, gradually enhancing the post-yield performance [8,47]. 377

It should be highlighted that the median values in Cases 2 and 3 are always very close. This indicates 378

that adequate seismic vulnerability can be reached by placing SFRC only at the critical regions, i.e. where 379

the plastic hinges are expected to occur (in this particular example, circa 200 mm at the bottom of the 380

column). This is the area where the columns bear higher demands when subjected to moderate or strong 381

earthquakes [20,48] and is critical for the proper seismic design of bridge columns. The design of RC-382

SFRC can be implemented in the construction of actual bridges based on segments in single-column 383

piers, with SFRC used in the plastic hinge zone at the bottom of the column, and regular concrete used 384

for the remaining part, as shown in Fig. 14. Generally speaking, pier columns are cast segment by 385

segment, in which the upper section is progressively constructed above the lower section, as shown in 386

Fig. 15. We suggest that in RC-SFRC columns, the SFRC zone to be cast as a section at the bottom followed 387

by the RC sections cast above it. This construction process is similar to that of ordinary piers. Thus, the 388

construction effort of RC-SFRC piers would not change significantly compared to RC or SFRC piers. 389

5. Conclusions 390

This paper presented a study on the seismic vulnerability of MSCG bridges with SFRC columns. 391

Quasi-static experimental tests on columns were used for the quantitative assessment of the limit-state 392

capacities for different damage states. The probabilistic models of the demands on several critical 393

components were then obtained based a numerical analysis. The fragility curves for the different bridge 394

systems were established as a function of the peak ground acceleration by integrating capacity 395

distributions with correlated component demand distributions. The following conclusions can be drawn: 396

(1) SFRC columns perform significantly better than RC columns, with the structural capacity 397

(especially for the ductility and energy dissipation capacity) of the columns increasing with the steel 398

fibre content, being the specimens with the highest ratio (1.0%) the ones showing better performance-399

price ratio. 400

(2) The use of SFRC columns in MSCG bridges does not only mitigate their own seismic fragility, but 401

it also positively impacts on other critical components, such as bearings and abutments, by lowering its 402

seismic vulnerability. The MSCG bridges with SFRC or RC-SFRC columns are found to be less vulnerable 403

than bridges with RC columns, for all discussed damage states, ranging from light to complete damage, 404

especially for higher damage states. 405

(3) The system vulnerability of MSCG bridges with SFRC located only at the plastic region of the 406

column shows similar behaviour when compared with the design solution of adopting SFRC for the 407

whole column. This can potentially be used for economically designing the reinforcement of MSCG 408

bridge columns, such that only the minimum quantity of SFRC is used. 409

Acknowledgements 410

This study was supported by the National Natural Science Foundation of China (Grant No. 411

51508276), the Fundamental Research Funds for the Central Universities of China (Grant 412

No.30915011329), the China Postdoctoral Science Foundation (Grant No.2015M570399), the 413

Australian Research Council through its Discovery Early Career Researcher Award (DE150101703) and 414

the Faculty of Engineering & Information Technologies, The University of Sydney, under the Faculty 415

Research Cluster Program. The authors would like to acknowledge their support gratefully. 416

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