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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.
Available online at www.sciencedirect.com
ScienceDirect
journal homepage: www.elsevier .com/locate/ j t te
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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502494
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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502 495
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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502496
(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
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502 497
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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502498
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$m) 4y 4u
Value 554 4.0E-3 3.9E-2
Fig. 10 e Finding the demand point for Shuangying Bridge
before consolidation.
Table 2 e Values of the converting parameters in Eqs. (37)and (38).
c1 c2 cR cT
1.35 0.95 0.75 0.20
Fig. 9 e The design spectrum for Shuangying Bridge
associated with the E2 earthquake.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502 499
Fig. 11 e Consolidation of Shuangying Bridge piers. (a) Construction scheme. (b) Enhanced pier for axes 1 and 4. (c) Enhanced
pier for axes 2 and 3.
Table 3 e Parameters for the momentecurvaturerelationship of Shuangying Bridge piers afterconsolidation.
Parameter My (kN$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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502500
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.
r e f e r e n c e s
Applied Technology Council (ATC), 1996. Seismic Evaluation andRetrofit of Concrete Buildings. SSC 96-01. AppliedTechnology Council, Redwood City.
Aydino�glu, M.N., 2003. An incremental response spectrumanalysis procedure based on inelastic spectral displacementsfor multimode seismic performance evaluation. Bulletin ofEarthquake Engineering 1 (1), 3e36.
Borzi, B., Pinho, R., Crowley, H., 2008. Simplified pushover-basedvulnerability analysis for large-scale assessment of RCbuildings. Engineering Structures 30 (3), 804e820.
Bradley, B.A., 2013. Practice-oriented estimation of the seismicdemand hazard using ground motions at few intensitylevels. Earthquake Engineering & Structural Dynamics 42(14), 2167e2185.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502 501
Camara, A., Astiz, M.A., 2012. Pushover analysis for the seismicresponse prediction of cable-stayed bridges under multi-directional excitation. Engineering Structures 41 (3),444e455.
Chopra, A.K., Goel, R.K., 2000. Evaluation of NSP to estimateseismic deformation: SDF system. Journal of StructuralEngineering 126 (4), 482e490.
Der Kiureghian, A., 1996. Structural reliability methods forseismic safety assessment: a review. Engineering Structures18 (6), 412e424.
Der Kiureghian, A., Liu, P.L., 1986. Structural reliability underincomplete probability information. Journal of EngineeringMechanics 112 (1), 85e104.
Du, X., Qiang, H., Li, Z., et al., 2008. The seismic damage of bridgesin the 2008 Wenchuan Earthquake and lessons from itsdamage. Journal of Beijing University of Technology 34 (12),1270e1279.
Fajfar, P., 2007. Seismic assessment of structures by a practice-oriented method. In: Ibrahimbegovic, A., Kozar, I. (Eds.),Extreme Man-made and Natural Hazards in Dynamics ofStructures. Springer, Dordrecht, pp. 257e284.
Freeman, S.A., 1978. Prediction of Response of Concrete Buildingsto Severe Earthquake Motion. American Concrete Institute,Detroit.
Freeman, S.A., Nicoletti, J.P., Tyrell, J.V., 1975. Evaluations ofexisting buildings for seismic risk - a case study of PugetSound Naval Shipyard. In: 1st U.S. National Conference onEarthquake Engineering, Bremerton, 1975.
Ghobarah, A., Aly, N.M., El-Attar, M., 1998. Seismic reliabilityassessment of existing reinforced concrete buildings. Journalof Earthquake Engineering 2 (4), 569e592.
Goda, K., Hong, H.P., 2008. Estimation of seismic loss for spatiallydistributed buildings. Earthquake Spectra 24, 889e910.
Kalkan, E., Kunnath, S., 2004. Lateral load distribution innonlinear static procedures for seismic design. In:Constructures Congress, Nashville, 2004.
Krawinkler, H., Rahnama, M., 1992. Effects of soft soils on designspectra. In: The 10th World Conference on EarthquakeEngineering, Rotterdam, 1992.
Krawinkler, H., Seneviratna, G.D.P.K., 1998. Pros and cons of apushover analysis of seismic performance evaluation.Engineering Structures 20 (S4/6), 452e464.
Kolias, B., Fardis, M.N., Pecker, A., 2005. Design of Structures forEarthquake Resistance-Part 2: Bridges. EN 1998-2. Institutionof Civil Engineers, London.
Lee, R., Kiremidjian, A.S., 2007. Uncertainty and correlation forloss assessment of spatially distributed systems. EarthquakeSpectra 23, 753e770.
Li, Q., Ellingwood, B.R., 2007. Performance evaluation and damageassessment of steel frame buildings under mainshockeaftershock earthquake sequences. EarthquakeEngineering & Structural Dynamics 36 (3), 405e427.
Li, Q., Ellingwood, B.R., 2008. Damage inspection and vulnerabilityanalysis of existing buildings with steel moment-resistingframes. Engineering Structures 30 (2), 338e351.
Li, Q., Wang, C., 2015. Updating the assessment of resistance andreliability of existing aging bridges with prior service loads.Journal of Structural Engineering 141 (12), 04015072.
Ministry of Housing and Urban-Rural Development of thePeople’s Republic of China, 2011. Code for Seismic Design ofUrban Bridges. CJJ 166-2011. China Agriculture and BuildingPress, Beijing.
Ministry of Transport of the People’s Republic of China, 1989.Code for Seismic Design of Highway Bridges. JTJ 004-89.China Agriculture and Building Press, Beijing.
Muntasir Billah, A.H.M., Shahria Alam, M., 2015. Seismic fragilityassessment of highway bridges: a state-of-the-art review.Structure and Infrastructure Engineering 11 (6), 804e832.
Noh, Y., Choi, K.K., Du, L., 2009. Reliability-based designoptimization of problems with correlated input variablesusing a Gaussian copula. Structural and MultidisciplinaryOptimization 38 (1), 1e16.
Panandikar, N., Narayan, K.S.B., 2015. Sensitivity of pushovercurve to material and geometric modelling - an analyticalinvestigation. Structures 2, 91e97.
Qin, S., 2008. Static Nonlinear Analysis Methods for EstimatingSeismic Performance for Bridges (PhD thesis). DalianUniversity of Technology, Dalian.
Stewart, M.G., Val, D.V., 1999. Role of load history in reliabilitybased decision analysis of aging bridges. Journal ofStructural Engineering 125 (7), 776e783.
Vidic, T., Fajfar, P., Fischinger, M., 1994. Consistent inelasticdesign spectra: strength and displacement. EarthquakeEngineering and Structural Dynamics 23 (5), 507e521.
Vitoontus, S., 2012. Risk Assessment of Building InventoriesExposed to Large Scale Natural Hazards (PhD thesis). GeorgiaInstitute of Technology, Atlanta.
Wang, C., Bao, Q., Li, Q., 2014. Simplified pushover analysismethod for simply supported concrete bridges. Technologyfor Earthquake Disaster Prevention 9 (3), 439e446.
Wang, C., Zhang, H., Li, Q., 2017. Time-dependent reliabilityassessment of aging series systems subjected to non-stationary loads. Structure and Infrastructure Engineering 13(12), 1513e1522.
Wight, J.K., MacGregor, J.G., 2012. Reinforced Concrete: MechanicsandDesign,sixthed.PearsonEducation, Inc.,UpperSaddleRiver.
Zordan, T., Briseghella, B., Lan, C., 2011. Parametric and pushoveranalyses on integral abutment bridge. Engineering Structures33 (2), 502e515.
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.
J. Traffic Transp. Eng. (Engl. Ed.) 2018; 5 (6): 491e502502