probability-based practice-oriented seismic behaviour

Original Research Paper

Probability-based practice-oriented seismic behaviourassessment of simply supported RC bridgesconsidering the variation and correlation inpier performance

Long Zhang a, Cao Wang b,c,*

a CCCC Highway Consultants CO., Ltd., Beijing 100088, Chinab School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australiac Department of Civil Engineering, Tsinghua University, Beijing 100084, China

h i g h l i g h t s

� A method is developed for seismic reliability assessment of simply supported RC bridges.

� The pushover curve is obtained considering the nonlinear deformation of bridge piers subject to moderate to high seismicity.

� The applicability of the proposed method is demonstrated through an application to a realistic simply supported bridge.

� The roles of variation and correlation of pier performance in bridge failure probability are investigated.

a r t i c l e i n f o

Article history:

Received 19 March 2018

Received in revised form

6 July 2018

Accepted 8 July 2018

Available online 9 October 2018

Keywords:

Seismic assessment

Simply supported RC bridge

Nonlinear analysis

Simplified method

Variation in pier behaviour

Correlation in pier behaviour

a b s t r a c t

Probability-based seismic performance assessment of in-service bridges has gained much

attention in the scientific community during the past decades. The nonlinear static

pushover analysis is critical for describing the seismic behaviour of bridges subjected to

moderate to high seismicity due to its simplicity. However, in existing analysis methods,

the generation of pushover curve needs tremendous amount of computational costs and

requires skills, which may halter its application in practice, implying the importance of

developing practice-oriented analysis method with improved efficiency. Moreover, the

bridge pier performance has been modelled as deterministic, indicating that the variation

associated with the pier material and mechanical properties remains unaddressed. The

correlation in the performance of different piers also exists due to the common design

provisions and construction conditions but has neither been taken into account. This paper

develops a simplified pushover analysis procedure for the seismic assessment of simply

supported RC bridges. With the proposed method, the pushover curve of the bridge can be

obtained explicitly without complex finite element modelling. A random factor is intro-

duced to reflect the uncertainty in relation to the pushover curve. The correlation in the

pier performance is considered by employing the Gaussian copula function to construct the

joint probability distribution. Illustrative examples are presented to demonstrate the

* Corresponding author. School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia. Tel.: þ61 424003086.E-mail addresses: [email protected] (L. Zhang), [email protected] (C. Wang).

Peer review under responsibility of Periodical Offices of Chang'an University.

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j o u rn a l o f t r a ffi c a nd t r an s p o r t a t i o n e n g i n e e r i n g ( e n g l i s h e d i t i o n ) 2 0 1 8 ; 5 ( 6 ) : 4 9 1e5 0 2

https://doi.org/10.1016/j.jtte.2018.10.0062095-7564/© 2018 Periodical Offices of Chang'an University. Publishing services by Elsevier B.V. on behalf of Owner. This is an openaccess article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

applicability of the method and to investigate the impact of variation and correlation in

bridge pier behaviour on the seismic performance assessment.

© 2018 Periodical Offices of Chang'an University. Publishing services by Elsevier B.V. on

behalf of Owner. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Bridges play a critical role in the traffic network, providing

physical support to a region's transportation capacity. While

the design provisions in current codes and standards are

enforced to guarantee adequate levels of serviceability for

the bridges, the severe damage or loss of function posed by

hazardous events is continuously a great concern to the

asset owners and civil engineers, which may lead to sub-

stantial economic losses and even ripple effect in the sur-

rounding community (Li and Wang, 2015; Wang et al., 2017).

Earthquakes are among the hazardous events responsible

for the damage and failure of bridges. For instance, during

the 2008 Wenchuan Earthquake in Sichuan Province, China,

about 24 highways, 6140 bridges and 156 tunnels were

severely destroyed, resulting in 67 billion RMB of losses to

the traffic and infrastructure system (Du et al., 2008).

Moreover, many bridges that were constructed according

to historical codes and standards with insufficient safety

levels are still in use today due to the socio-economic

constraints. As a result, it is essentially important to

assess and maintain the safety levels of these in-service

bridges subjected to earthquake hazards so that their

service reliability may be guaranteed beyond the baseline

in the context of probability.

Significant efforts have been made in the scientific com-

munity during the past decades regarding the seismic per-

formance assessment of civil infrastructures (Der Kiureghian,

1996; Ghobarah et al., 1998; Li and Ellingwood, 2007, 2008;

Muntasir Billah and Shahria Alam, 2015). The nonlinear static

pushover analysis method, originally developed by Freeman

et al. (1975) and Freeman (1978), has been widely accepted

and used to estimate the seismic response for structures

because it provides a simple yet practical description for the

structural elastoplastic behaviour in relation to moderate to

high seismicity. Borzi et al. (2008) proposed a simplified

pushover-based method for vulnerability analysis and loss

estimate of large-scale RC buildings. Zordan et al. (2011)

conducted parametric and pushover analyses on integral

abutment bridges, where a 2-D finite element (FE) model was

established to examine the role of soil property variation

and the temperature change in the bridge safety. Camara

and Astiz (2012) investigated the seismic response of cable-

stayed bridges subjected to multi-directional excitation

using Modal pushover analysis. Panandikar and Narayan

(2015) studied the sensitivity of pushover analysis to the

geometric and material property by means of comparing the

analytical results with the experimental data available from

a test frame. However, in these analyses, the generation of

pushover curve needs tremendous amount of computational

costs and requires skills, which may halter the application

of pushover analysis in practice. The University of Ljubljana

developed a simplified technique for seismic analyses

named N2 method, which is further implemented in the

European standard Eurocode 8 (Fajfar, 2007). However, the

bridge performance has been modeled as deterministic, with

which the variation associated with the structural material

and mechanical properties remains unaddressed.

Subsequently, the correlation between the behavior of

different components, arising from the common design

provisions and construction conditions, yet has neither been

taken into account in existing works (Goda and Hong, 2008;

Lee and Kiremidjian, 2007; Vitoontus, 2012).

In this paper, a practice-oriented method is developed for

the seismic performance assessment of RC bridges. Taking

into account the variation associated with the bridge pier

performance, a random factor is introduced to reflect the

uncertainty in relation to its pushover curve. The correlation

between the performance of the bridge piers is also consid-

ered. This paper chooses an in-service bridge to demonstrate

the application of the proposed method and to investigate the

role of bridge pier variation and correlation in the estimate of

bridge seismic performance.

2. Pushover analysis for simply supportedRC bridges

2.1. Nonlinear static pushover analysis

The nonlinear static procedure (NSP) offers an insight to the

structural nonlinear (inelastic) seismic behaviour for the engi-

neers who are familiar with the linear seismic response of

structures, especially in the era with an emphasis on the inelas-

tic-deformation-based design for structures subjected to mod-

erate tohighseismicity.Thenonlinearpushoveranalysiswithan

outcome of “pushover curve” is key in the NSP, which is repre-

sentative of the relationship between the base shear and the roof

displacement. The ultimate objective of NSP is to compare the

peak elastoplastic deformation with the critical value so as to

judge the displacement-based structural seismic behaviour

(Aydino�glu, 2003; Chopra and Goel, 2000). A brief summary of

the nonlinear pushover analysis procedure is as follows.

Step 1 Obtain the pushover curve by establishing the rela-

tionship between the roof displacement uN and the base shear

Vb. Required is the vertical distribution of the lateral loads

(Kalkan and Kunnath, 2004; Krawinkler and Seneviratna, 1998).

Step 2 Transform the pushover curve to the capacity dia-

gram. That is, the displacement uN becomes uN=G14N1 while

the base shear Vb is converted to Vb/M1*.

J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502492

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

G1 ¼

PNj¼1

mj4j1

PNj¼1

mj42j1

M*1 ¼

0@XN

j¼1

mj4j1

1A

2

PNj¼1

mj42j1

(1)

where mj is the lumped mass of the jth floor, N is the number

of floors, and 4j1 is the jth floor element associated with the

fundamental mode.

Step 3 Convert the elastic response spectrum to the in-

elastic spectrum. There are generally twomethods accounting

for this conversion. The first one is to find an equivalent linear

(elastic) system to replace the nonlinear (inelastic) system

(ATC, 1996), and the second one is by multiplying an

equivalent reduction factor (Chopra and Goel, 2000;

Krawinkler and Rahnama, 1992; Vidic et al., 1994).

Step 4 Transform the inelastic response spectrum from the

A-Tn form to the A-D form, where A is the pseudo-accelera-

tion, Tn is the natural period, and D is the deformation spec-

trum ordinate.

Step 5 Draw the demand and capacity diagrams in the

same coordinate system, and obtain the displacement de-

mand by finding the displacement point, as shown in Fig. 1.

Step 6 Convert the displacement demand obtained in Step

5 to both global (roof) and local (individual component) dis-

placements and then compare them with the critical

displacement values.

2.2. Momentecurvature relationship of concrete crosssections

The flexural behaviour of a typical cross section of concrete

members such as beams or columns can be represented by the

momentecurvature relationship. Usually, one can find the

relationship between the section curvature and the moment

applied to the structure with the help of the plane section

assumption. Here, we consider the singly reinforced rectangle

section as illustrated in Fig. 2; for the case of doubly reinforced

section, the method to obtain the momentecurvature rela-

tionship is similar. According to Fig. 2(b), since the strain

distribution through the depth of the section is linear, the

section curvature, 4, is obtained by

4 ¼ 3c

xn¼ 3s

h0 � xn(2)

where h0 is the effective flexural depth of the section, 3c is the

maximum compression strain at the extreme compression

fibre of the section, 3s is the tensile strain of the reinforced

steel, and xn is the depth of the compression zone.

According to the section equilibrium condition, we have

�C ¼ Tc þ Ts

M ¼ Cyc þ Tcyt þ Tsðh0 � xnÞ (3)

where C is the resultant compressive force of the compression

zone, Ts is the tensile force of the reinforced steel, Tc is the

resultant force of the tensile zone, yc and yt are the distances

between the resultant compressive/tensile forces and the

depth of the neutral axis.

Eqs. (2) and (3) establish the relationship between the

section curvature and the applied moment. One may obtain

the moment-curvature curve by continuously increasing 4

(i.e., the slope of the strain diagram as in Fig. 2(b)) and

calculating the moment M according to Eq. (3) (Wight and

MacGregor, 2012). For a typical bridge pier, the linearized

relationship between the moment and curvature of the cross

section takes the form of

Mð4Þ ¼�EIeff4 0 � 4 � 4y

My 4y <4 � 4u(4)

where EIeff is the section effective stiffness, 4y and 4u are the

yield curvature and ultimate curvature, respectively.

2.3. Plastic hinge at bridge piers

The bridge behavior is expected to be ductile in sites associ-

ated with moderate to high seismicity due to the consider-

ation of both economic and safety reasons, implying that the

bridge components should dissipate a considerable amount of

the input earthquake energy themselves. The presence of

flexural plastic hinges provides physical support to this bridge

performance goal, which can be found in the bridge piers

accessible for routine inspection and repair. In this paper, only

the longitudinal seismic response is taken into account, with

which the formation of plastic hinge (PH) only occurs at the

bottom of the bridge piers, for the case of transverse seismic

response, the derivation of the proposed method is similar,

with the exception that the PH may occur at both the bottom

and the roof of the bridge piers. The PH can be modeled as a

rotary spring at the middle of the effective length Lp. While Lpshould generally be determined according to experimental

tests, it can be still approximated by some empirical formu-

lations in the absence of appropriate laboratory test results.

For instance, for the case where the PH occurs at the bottom

junction of the pier, Eurocode 8 EN 1998-2 (Kolias et al., 2005)

gives

Lp ¼ 0:10Lþ 0:015fydb (5)

where L denotes the distance from the zero-moment-section

to the plastic hinge, fy is the characteristic yield stress of the

Fig. 1 e Finding the demand point by drawing the demand

and capacity diagram in the same coordinate system.

J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502 493

longitudinal reinforcement in MPa, and db is the diameter of

steel bar.

In Chinese code for seismic design of urban bridges

(Ministry of Housing and Urban-Rural Development of the

People's Republic of China, 2011), a similar relationship is

given by

Lp ¼ maxn0:08Lþ 0:022fydb; 0:044fydb

o(6)

Generally, the horizontal displacement at the bent cap,Dtot,

consists of four components, as shown in Fig. 3 (Qin, 2008).

(1) the displacement posed by the elastoplastic behavior of

the pier, Du;

(2) the displacement posed by the shearing stiffness of the

bearing pad, Db;

(3) the displacement posed by the translational move-

ment, Dt;

(4) the displacement posed by the rotational movement,

Dr.

In practice, we may ignore Dt and Dr (both are due to the

soil-structure interaction) by assuming that the junction be-

tween the bridge pier and the ground is rigid. With this, Dtot is

as follow.

Dtot ¼ Du þ Db (7)

Now we suppose that the yield curvature and the ultimate

plastic curvature at the PH area are respectively 4y and 4p,u,

as shown in Fig. 4. Obviously, 4u ¼ 4y þ 4p,u. The local ductility

of the PH affects the global ductility of the bridge pier

significantly since the bridge pier excluding the PH area is

expected to remain elastic under high seismicity. Initially,

during the elastic range, the roof displacement, Dela, is

obtained by

Dela ¼Z �Z

4ðxÞdx�dx (8)

where 4(x) is the pier's elastic curvature along the height. At

the yielding state (i.e., the beginning of the plastic hinge for-

mation), 4(x) can be approximated as a linear function of

x, that is

4ðxÞ ¼ xL4y (9)

Substituting Eq. (9) into Eq. (8) gives

Dy ¼ 134yL

2 (10)

where Dy is the pier's yield displacement.

Next, within the plastic range, the plastic rotation capacity

at the PH area, qp, is estimated by

qp ¼ Lp�4� 4y

��1� Lp

2L

�zLp

�4� 4y

�(11)

where 0<4� 4y � 4p;u.

Fig. 2 e Analysis of moment and curvature of a singly reinforced section. (a) Basic section. (b) Strain distribution. (c) Stress

distribution and internal forces.

Fig. 3 e Roof displacement of a realistic bridge pier.

Fig. 4 e Curvature distribution along the height of bridge

pier with the PH occurring at the bottom junction. (a) Load

condition. (b) Moment diagram.


With Eq. (11), the roof plastic displacement Dp is obtained

by

Dp ¼ qp�L� 0:5Lp

� ¼ Lp�L� 0:5Lp

��4� 4y

�(12)

2.4. Proposed simplified pushover analysis procedure

For a well-design simply supported concrete bridge, the su-

perstructure such as the bent cap and the deck contributes to

the majority of bridge mass. For instance, consider a bridge

pierwith length of L and cross section area ofAb. Theweight of

the pier, G, equals to LAbrc, where rc is the density of the

concrete structure. The weight of the superstructure, P, is

hAbfc, where fc is the concrete compressive strength, and h is

the axial compression ratio, which is typically less than 0.3.

For C40 concrete, the relationship between the ratio of P to G

and the length of the pier is plotted in Fig. 5, from which it is

seen that P is far more than the gravity of the bridge pier. As

a result, one may simplify the multiple degree of freedom

(MDOF) system to a single degree freedom (SDOF) system

when performing seismic analysis for a simply supported

concrete bridge pier (Wang et al., 2014).

As introduced in Section 2, the pushover curve represents

the relationship between the roof displacement and the base

shear force. Consider the bridge pier as shown in Fig. 6, the

objective is to find the relationship between the roof

displacement Dtot and the shear force F. With the

mechanical equilibrium condition, we have

M ¼ FLþ PDtot ¼ FLþ PðDu þ DbÞ (13)

Note that in Eq. (13), Du differs before and after the

formation of the plastic hinge. As a result, we will discuss

the F-Dtot relationship respectively for both of the two stages.

Stage 1: Elastic range.

Within the elastic range, Du ¼ Dela ¼ 134L

2, and M ¼ EIeff4.

Since Db ¼ FK according to the well-known Hooker's law, where

K is the shearing stiffness of the bearing pad, Eq. (13) is

rewritten as

M ¼ F

�Lþ P

K

�þ PDela (14)

Thus,

F ¼ M� PDela

Lþ PK

(15)

Note that

Dtot ¼ Dela þ Db ¼ Dela þ EIeff4� PDela

KLþ P(16)

With which the relationship between Dela and Dtot is

obtained as

Dtot ¼ Dela

1þ 3EIeff � PL2

L2ðKLþ PÞ

(17)

Substituting Eq. (17) into Eq. (15), we have

F ¼ 3EIeff � PL2�L3 þ PL2

K

�h1þ 3EIeff �PL2

L2ðKLþPÞ

iDtot (18)

Eq. (18) works when the bridge pier section is within the

elastic range.

Since Dela � 134yL

2;

0<Dtot � 134yL

2

1þ 3EIeff � PL2

L2ðKLþ PÞ

(19)

Stage 2: Plastic range.

After the formation of the plastic hinge at the bottom of the

bridge pier,M ¼ My according to Eq. (4). Thus, Eq. (13) becomes

My ¼ FLþ PDtot (20)

with which the relationship between F and Dtot is obtained

as

F ¼ My � PDtot

L(21)

Fig. 5 e Relationship between the ratio of P to G and the

length of the bridge pier.

Fig. 6 e Force diagram for the bridge pier.


Since DyþDp�134yL

2þLp4p;u

�L�0:5Lp

�;

134yL

2

1þ3EIeff �PL2

L2ðKLþPÞ

<Dtot�

KLh134yL

2þLp4p;u

�L�0:5Lp

�þMy

KLþP

(22)

Finally, by noting that

Dtot ¼ Du þ FK¼ Du þMy � PDtot

KL(23)

The relationship between Du and Dtot is

Du ¼ Dtot

�1þ P

KL

��My

KL(24)

Eq. (21) implies that after the pier bottomyields, Fdecreases

with the increase ofDtot. This observation interestingly reflects

the basic idea of theperformance-baseddesign: the ductility of

structures may reduce the structural stiffness and enhance

the viscous damping ratio and as a result mitigate the

earthquake effect. Now we consider the conversion of the

pushover curve to the capacity diagram as introduced in

Section 2. With the assumption of SDOF for the simply

supported bridge, N ¼ 1, with which Eq. (1) becomes G14N1 ¼1; M1

* ¼ Pg, where g is the gravitational acceleration. Thus,

the capacity diagram for a single bridge pier is obtained as

8>>><>>>:

uN

G14N1

¼ Dtot

Vb

M*1

¼ gFP

(25)

Obviously, Eq. (25) is beneficial for determining the demand

point as defined in Fig. 1 due to its simplicity. Moreover, by

noting that the major drawback of nonlinear static pushover

analysis method basically relies on the single-mode

response (Aydino�glu, 2003), it is seen that the NSP works

with Eq. (25) since the simply supported bridge pier can be

reasonably assumed as a SDOF as mentioned above. Further,

we consider the bridge pier inventory as a whole, where the

roof displacement of the bridge deck is identical for each

pier, as shown in Fig. 7. In such a case, the pushover curve

for the whole bridge is obtained according to Eq. (26).

FtotðDÞ ¼Xn

i¼1

FiðDÞ (26)

where n is the number of bridge piers, and Fi is the pushover

curve function associated with the ith pier.

Similar to Eq. (25), the simplified capacity diagram is

obtained by

8>>>>><>>>>>:

uN

G14N1

¼ D

Vb

M*1

¼ gFtotðDÞPni¼1

Pi

(27)

where Pi is the vertical load associated with the ith pier.

3. Variation and correlation associated withthe bridge pier performance

Note that the aforementioned pushover curve has been

modeled as deterministic, withwhich the variation associated

with the bridge pier performance remains unaddressed.

Practically, the uncertainties arise in the non-exact structural

performance modelling, the variability in material properties,

geometry, environmental conditions and deterioration pro-

cess (Stewart and Val, 1999). In order to reflect the uncertainty

associated with the structural property, we introduce a

random factor L which satisfies

~FðDÞ ¼ LFðDÞ (28)

where ~FðDÞ is the random pushover curve function of the

bridge pier.

L is assumed to follow lognormal distribution with a mean

value of 1 and a standard deviation of sL. Note that the

random factor L indeed represents both the uncertainties

associated with the bridge pier and the probabilistic behavior

of the bearing pad (the shearing stiffness); in this paper, we do

not distinguish these two types of uncertainties and refer

them to as the pier performance uncertainty for the purpose

of simplicity.

Further, we consider Eq. (26), which is revised as

~FtotðDÞ ¼Xn

i¼1

~FiðDÞ ¼Xn

i¼1

LiFiðDÞ (29)

where Li is the modification factor associated with the ith

bridge pier.

Note that each Li may be correlated due to the correlation

in the bridge pier performance. We usually use the linear

Fig. 7 e Considering the bridge pier inventory as a whole.


(Pearson) coefficient of correlation to model the correlation

between two random variables, X and Y, which is defined as

rij ¼cijsisj

(30)

where rij is the coefficient of correlation between X and Y, cij is

the covariance of X and Y, si and sj denote the standard de-

viation of X and Y, respectively.

Generally, the joint CDF (cumulative density function) of {Li}

is required to describe the probabilistic behavior and further

generate a sequence of samples for {Li}. Unfortunately, the

joint CDF is often inaccessible due to the limit of available data

and knowledge on it. Alternatively, we can use the Gaussian

copula function to help construct the joint CDF for {Li} provided

the marginal distributions and the correlation matrix, with

which the Nataf transformation can be used to generate sam-

ples for {Li} in simulation-based studies (Noh et al., 2009).

Mathematically, if X is a n-dimensional correlated random

vector with marginal CDF of FXiðxÞ and correlation coefficient

matrix of P ¼ ðrijÞn�n, we firstly convert X into a dependent

standard normal vector Y according to Der Kiureghian and Liu

(1986).

Yi ¼ F�1�FXi

ðXiÞ�

(31)

where F is the CDF of standard normal distribution.

The correlation coefficient matrix of Y, P' ¼ ðr0 ijÞn�n, can be

determined with P by

r0ij ¼ Rijrij (32)

where Rij is determined by Eq. (33) for the case where both Xi

and Xj follow lognormal distribution.

Rij ¼ln�1þ rijdidj

�rij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln�1þ d2i

�ln�1þ d2j

�r (33)

where di and dj are the coefficients of variations (COV, defined

as the ratio of the standard deviation to the mean value) of Xi

and Xj, respectively.

Next, we perform the Cholesky factorization for the defi-

nitely positivematrix P0 to find the lower triangularmatrixA¼{aij} by

P0 ¼ AAT (34)

where AT denotes the transposition of A. With this, Y can be

further converted into an independent standard normal var-

iables U by

Y ¼ AU (35)

The relationship between X and U has been established by

Eqs. (31) and (35), with which

8>><>>:

X1 ¼ F�1X1ðFða11U1ÞÞ

X2 ¼ F�1X2ðFða21U1 þ a22U2ÞÞ

«Xn ¼ F�1

XnðFðan1U1 þ an2U2 þ/þ annUnÞÞ

(36)

Eq. (36) presents the generation method for correlated

random vector X. Applying Eq. (36) in the sampling of {Li},

we can obtain the sampled ~FtotðDÞ in Eq. (29).

4. Illustrative examples

The proposed simplified method is illustrated in this section

through an application to a realistic simply supported RC

bridge, with which the role of variation and correlation in pier

behavior in the seismic performance assessment is investi-

gated. The analytical results may be further used to guide the

design of new bridges and the consolidation plans of existing

bridges with insufficient safety levels.

4.1. Bridge configuration

Shuangying Bridge, located over the Liangshui River in Beijing,

China, has a service life of 28 years since the completion year of

1990. It is a continuous five-span reinforced concrete bridge

with an overall length of 82 m and spans of 17 m, 16 m � 3 and

17 m as shown in Fig. 8(a). This bridge has four bent caps and

each of them contains 6 columns. The bridge piers at axes 1

and 4 have a height of 5 m and the piers at axes 2 and 3 have

a height of 9 m. There are 19 T beams at each span with a

height of 0.8 m, and 38 neoprene bearing pads at each bent

cap with a shearing stiffness of 7.395 � 104 kN/m. All the

bridge piers are concrete filled tubes with a diameter of 0.8 m,

whose cross section is illustrated in Fig. 8(c). Shuangying

Bridge was designed and constructed following the 1989

Chinese code for seismic design of highway bridges (Ministry

of Transport of the People's Republic of China, 1989) and may

have an unsatisfied safety level as required in the currently

enforced code (Ministry of Housing and Urban-Rural

Development of the People's Republic of China, 2011). As a

result, the seismic behavior of the bridge is verified in this

section according to the latter code provisions, where the

displacement-based verification is only required in relation to

the E2 earthquake with a return period of 2000 years. The

design spectrum for E2 earthquake is plotted in Fig. 9.

4.2. Seismic performance assessment beforeconsolidation

First, the key parameters for the moment-curvature curve of

the bridge pier are calculated and listed in Table 1. The

pushover curve for the whole bridge is obtained according to

Eqs. (18), (21) and (26) and is shown in Fig. 10. In order to

find the demand point as introduced in Section 2, we

convert the elastic design spectrum as in Fig. 9 into the

inelastic spectrum by introducing a reduction factor R (Vidic

et al., 1994), which takes the form of

R ¼

8><>:

c1ðm� 1ÞcR TT0

þ 1 T � T0

c1ðm� 1ÞcR þ 1 T>T0

(37)

T0 ¼ c2mcTTg (38)

where Tg is the characteristic period, which equals to 0.65 s for

the illustrative bridge, c1, c2, cR and cT are the hysteretic

behaviour and damping related parameters and are found in

Table 2.

Transforming the inelastic response spectrum from the A-

Tn form to theA-D form and drawing the demand and capacity


diagrams in the same coordinate system (Fig. 10), it is seen

that the two diagrams have no intersection, implying that

the bridge will collapse subjected to E2 earthquake and

needs consolidation measures to improve its seismic safety.

4.3. Seismic performance assessment after consolidation

It was recognized from Fig. 10 that the bending moment

capacity of the bridge piers is not satisfied in relation to the

E2 earthquake seismicity following the currently enforced

code (Ministry of Housing and Urban-Rural Development of

the People's Republic of China, 2011). A preliminary

consolidation scheme is to enhance the bridge piers with

larger diameter, as illustrated in Fig. 11. Here, the seismic

performance assessment is performed employing the

method proposed in this paper for the consolidated bridge

piers aimed at evaluating this consolidation scheme.

The key parameters for themoment-curvature curve of the

consolidated bridge piers are found in Table 3, with which the

pushover curve is obtained according to Eqs. (18), (21) and (26)

and is plotted in Fig. 12. It is seen that the demand

displacement is determined as 0.12 m. Emphasized is that

this figure is not the roof displacement of the bridge pier but

the displacement of the bridge deck. The maximum rotation

Fig. 8 e The Shuangying Bridge. (a) Side view. (b) Bridge section. (c) Pier section.


capacity for the bridge pier at axis 1 or 4 is 1/45 and 1/60

for that associated with axis 2 or 3. With this, the critical

value for the bridge deck displacement, is found as 0.14 m

(min{0.14 m, 0.17 m} ¼ 0.14 m) referring to Eq. (24). Clearly,

the consolidation scheme as illustrated in Fig. 11, satisfies

the displacement requirement subjected to E2 earthquake.

Note that the bridge pier performance has been modelled

as deterministic in Fig. 12. To reflect the variation and

correlation associated with the consolidated bridge piers, we

consider the correlated random factors in relation to each

bridge pier as introduced in Section 2. As there is no enough

evidence on the variation and correlation associated with

the pier performance, we assume that the standard

deviation of each L is identically sL and the coefficient of

correlation between different piers equals to r. The values of

the two parameters can be obtained with practical

investigation on the realistic pier properties, and can be

substituted to the present analysis once they become

available. Fig. 13 plots the sample curves for the pushover

curve associated with the whole bridge inventory for the

case of sL ¼ 0.2 and r ¼ 0.5. Obviously, the variation and

correlation affects the demand point and further has an

impact on the demand displacement. For the most sampled

curves they intersect with the demand diagram; however, if

the sample curve is associated with low seismic capacity,

there may be no such intersection, implying that the bridge

pier ductility is not satisfied subjected to the E2 earthquake.

Fig. 14 plots the probability distribution of the demand

displacement associated with different sL for the case of

r ¼ 0.3 using 100,000 replications. Here, if there were no

intersection between the capacity and demand diagrams, the

demand displacement is set to be infinite. The increase of

standard deviation associated with each L does not affect the

mean value of the demand displacement, which equals to

0.13 m for all the three cases. However, the variability of the

demand point increases as a result of the increase of the

random factor. Keeping in mind that the critical displacement

for the consolidated bridge is 0.14 m, the probabilities that the

demand displacement exceeds this critical value are 0, 0.35%

and 9.4% corresponding to the cases of sL ¼ 0.05, 0.1 and 0.2

respectively, implying that increase of the pier performance

variation leads to greater seismic risk significantly.

Next, in order to investigate the role of pier performance

correlation in the seismic performance assessment, Fig. 15 plots

the probability distribution of the demand displacement

associated with different r for the case of sL ¼ 0.2. The increase

of correlation between each L has no impact on the mean

value of the demand displacement as before since all the three

curves intersect at the same point. However, the increase of

the correlation leads to greater variability of the demand point

and longer upper tail behaviour. The probabilities that the

demand displacement exceeds the critical value 0.14 m are

1.8%, 9.4% and 15.4% respectively corresponding to r ¼ 0.1, 0.3

and 0.5, indicating that the increase of correlation in bridge

pier performance results in greater failure probability for the

bridge subjected to E2 earthquake.

The above observations from Figs. 14 and 15 can be

explained as follows. According to Eq. (26), for a given value of

D, i.e., D ¼ d,

E�~FtotðdÞ

� ¼ Xn

i¼1

EðLiÞFiðdÞ ¼Xn

i¼1

FiðdÞ (39)

Var�~FtotðdÞ

�¼Var

"Xni¼1

LiFiðdÞ#¼Xn

i¼1

F2i ðdÞs2

Liþ2

Xn

i<j

FiðdÞFjðdÞrijsLisLj

¼s2L

24Xn

i¼1

Fi2ðdÞþ2r

Xn

i<j

FiðdÞFjðdÞ35

(40)

where E ($) and Var($) denote the mean value and variance of

the random variable in the bracket respectively.

Table 1 e Parameters for the momentecurvaturerelationship of Shuangying Bridge piers beforeconsolidation.

Parameter My (kN$m) 4y 4u

Value 554 4.0E-3 3.9E-2

Fig. 10 e Finding the demand point for Shuangying Bridge

before consolidation.

Table 2 e Values of the converting parameters in Eqs. (37)and (38).

c1 c2 cR cT

1.35 0.95 0.75 0.20

Fig. 9 e The design spectrum for Shuangying Bridge

associated with the E2 earthquake.


Fig. 11 e Consolidation of Shuangying Bridge piers. (a) Construction scheme. (b) Enhanced pier for axes 1 and 4. (c) Enhanced

pier for axes 2 and 3.

Table 3 e Parameters for the momentecurvaturerelationship of Shuangying Bridge piers afterconsolidation.

Parameter My (kN$m) 4y 4u

Axes 1 and 4 2122 3.1E-3 3.6E-2

Axes 2 and 3 2382 2.3E-3 3.0E-2

Fig. 12 e Finding the demand point for Shuangying Bridge

after consolidation.

Fig. 13 e Sample curves of the capacity diagram with

sL ¼ 0.2 and r ¼ 0.5.

Fig. 14 e Effect of pier variation on the probability

distribution of demand displacement with r ¼ 0.3.

Fig. 15 e Effect of pier correlation on the probability

distribution of demand displacement with sL ¼ 0.2.


Eqs. (39) and (40) demonstrate that the increases of both sL

and r have nothing to dowith the expected pushover curve for

the whole bridge but contribute the variability of the pushover

curve significantly and thus increase the seismic risk. More-

over, it is seen from Eq. (40) that Var½~FtotðdÞ� ismore sensitive to

sL than r, implying the relative importance of reducing the

pier performance variation during the construction process.

It is noticed that the aforementioned exceeding probabili-

ties obtained from Figs. 14 and 15 are indeed conditional on

the assumption that the design spectrum represents the

realization of the random earthquake demand. The uncer-

tainty associated with the earthquake demand can be further

considered by employing the total probability theorem

(Bradley, 2013). Nevertheless, the analytical results herein

qualitatively suggest that developing proper construction

program with the objective of reducing the variation and

correlation associated with the pier performance is of

significant importance in practical engineering. These

probabilities may be further utilized to help develop a

quantitative indicator representing the construction quality

as soon as the practical knowledge on the bridge pier

variation and correlation is accessible.

5. Conclusions

A probability-based method has been proposed in this paper

for the seismic behaviour assessment of simply supported RC

bridges. The proposed method enables the bridge pushover

curve to be obtained explicitly without complex finite element

modelling. Moreover, both the uncertainty associatedwith the

bridge pier performance arising from the variability in mate-

rial and mechanical properties and the correlation in the

performance of different piers due to common design pro-

visions and construction conditions are taken into account in

the proposed method. An illustrative bridge is chosen to

demonstrate the applicability of the proposed method and to

investigate the role of variation and correlation in bridge pier

performance in the seismic behavior assessment. The

following conclusions can be drawn from the illustrative

results.

(1) A simplified method is presented for obtaining the

pushover curve of simply supported RC bridges. This

method can be used for the inelastic-displacement-

based seismic performance assessment in relation to

moderate to high seismicity. The pushover curve for a

single bridge pier is first obtained considering the

shearing stiffness of the bearing pad with a two-stage

explicit formula, and then the pushover curve for the

whole bridge inventory is determined by the linear

combination of the curves associated with each bridge

pier. The proposed method also takes into account the

variation and correlation in the pier properties.

(2) The applicability of the proposed method is demon-

strated through an application to a realistic simply

supported bridge. The analytical results can be used to

help guide and optimize the design of new bridges and

the enhancement of existing old bridges with insuffi-

cient seismic safety levels.

(3) The variation and correlation in bridge pier perfor-

mance contribute to the failure probability of the bridge.

The bridge seismic behavior is more sensitive to the

former one, implying the relative importance of con-

trolling the construction quality by enhancing the con-

struction management. Moreover, the analytical results

reveal that the reduction of pier performance correla-

tion is beneficial for the bridge seismic safety, suggest-

ing the importance of optimizing the construction

programaiming at reducing the correlation between the

performance of different piers.

It is noticed, finally, that themethod proposed in this paper

was derived on the assumption of a SDOF system and thus is

applicable for simply supported RC bridges only; future work

may include an improved method that can account for the

more sophisticated cases where the higher-order mode

shapes and other lateral load patterns are considered.

Conflicts of interest

The authors do not have any conflict of interest with other

entities or researchers.

Acknowledgments

This research has been supported by the National Natural

Science Foundation of China (Grant Nos. 51578315, 51778337),

the National Key Research and Development Program of

China (Grant No. 2016YFC0701404) and the Faculty of Engi-

neering and IT PhD Research Scholarship (SC 1911) from The

University of Sydney. These supports are gratefully acknowl-

edged. The authors would like to thank Dr. W. Tang and Dr. L.

Huo for their constructive comments on this paper. The

thoughtful suggestions from the three anonymous reviewers

are acknowledged, which substantially improved the present

paper. The second author gratefully appreciates the support

from Beijing General Municipal Engineering Design and

Research Institute, Beijing, China.

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Long Zhang received his M.S. degree in July2015 from the Department of Civil Engi-neering, Tsinghua University. Currently, heis an assistant engineer in the CCCC High-way Consultants CO., Ltd., Beijing, China. Hisresearch interests include bridge design andbridge condition assessment.

Cao Wang received his M.E. degree in July2015 from the Department of Civil Engi-neering, Tsinghua University. Currently, heis pursuing a PhD degree in the School ofCivil Engineering, The University of Sydney,Australia. His research interests includestructural reliability analysis and naturalhazards assessment.


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