[Kaviani Et Al., 2012]
Content
Engineering Structures 45 (2012) 137–150
Contents lists available at SciVerse ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Seismic behavior of reinforced concrete bridges with skew-angled
seat-type abutments
Peyman Kaviani a, Farzin Zareian a,⇑, Ertugrul Taciroglu b
a
b
Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA
Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USA
a r t i c l e
i n f o
Article history:
Received 30 July 2011
Revised 10 April 2012
Accepted 5 June 2012
Available online 20 July 2012
Keywords:
Skewed bridges
Performance-based earthquake engineering
Shear key
Seat-type abutment
a b s t r a c t
This study focuses on identifying trends in seismic behavior of reinforced concrete bridges with seat-type
abutments under earthquake loading, especially with respect to abutment skew angle. To that end, a
detailed approach for modeling skew-angled seat-type abutments is proposed; and a comprehensive variety of bridge configurations are considered. Specifically, three short bridges located in California are
selected as seed bridges, from which different models are spawned by varying key bridge structural
parameters such as column-bent height, symmetry of span arrangement, and abutment skew angle.
Through extensive nonlinear time-history analyses conducted using three suites of ground motions, it
is demonstrated that demand parameters for skew-abutment bridges, such as deck rotation and column
drift ratio, are higher than those for straight bridges. By investigating the sensitivity of various response
parameters to variations in bridge geometry and ground motion characteristics, it is shown that bridges
with larger abutment skew angles bear a higher probability of collapse due to excessive rotations, and that
the shear keys can play a major role in reducing deck rotations and thus the probability of collapse.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Bridges with skew-angled abutments, also denoted henceforth
as ‘‘skewed bridges,’’ are constructed to accommodate geometry
constraints resulting from alignment of a waterway or roadway
crossing that occurs at an angle that is different from 90°. In the
present study, the focus is on the seismic response of short skewed
bridges with ‘‘seat-type’’ abutments, which are very common in
construction practice in California. Observations from past earthquakes [1–3] suggest that there are significant differences between
the response of straight and skewed bridges. A manifestation of
this difference is the inherent tendency of the decks of skewed
bridges to rotate about their vertical axes under seismic excitation,
which can lead to unseating from abutments, and ultimately, to
collapse. The primary cause of this excessive rotation is the eccentric passive resistance of the abutment backfill.
Excessive deck rotations can lead to loss of bridge functionality
and significant downtime for repairs, or loss of stability and casualties. Site investigation of the Foothill Boulevard Undercrossing located in California, 34.25N and 118.5W, (which has a skew angle of
approximately 60°) showed that it rotated in its horizontal plane,
resulting in a permanent offset of approximately 7.5 cm (i.e.,
0.9 103 radians of deck rotation) in the direction of increasing
skew angle during the 1971 San Fernando Earthquake [1]. Recon⇑ Corresponding author. Tel.: +1 949 824 9866; fax: +1 949 824 2117.
E-mail address: [email protected] (F. Zareian).
0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.engstruct.2012.06.013
naissance report of the 2010 Chile Earthquake [2,3] states that
skewed bridges in affected regions rotated, mainly about their center of stiffness, and that those with weak exterior shear keys suffered higher levels of damage due to transverse unseating.
Development of analytical/numerical models that can capture
the peculiar collapse mechanisms of skewed bridges under seismic
excitation, and can accurately quantify their damages have been a
subject of research for quite some time. Ghobarah and Tso [4] used
a spine-line model to represent bridge deck, and columns; they
concluded that the bridge collapse was caused by coupled flexural-torsional motions of the bridge deck or by the excessive compression demands that resulted in column failures. Using simplified
beam models, Maragakis and Jennings [5] concluded that the angle
of the skew and the impact between the deck and abutment govern
the response of skewed bridges. Wakefield et al. [6] conjectured
that if the deck is not rigidly connected to the abutments, dynamic
response of the bridge will be dominated by planar rigid body rotations of the deck rather than coupled flexural and torsional
deformations.
A more recent study by Meng and Lui [7] proposed that the seismic response of a bridge is strongly influenced by the column
boundary conditions and skew angle. In a subsequent study [8],
they used a dual-beam stick model to represent the bridge deck,
and showed that in-plane deck rotations are mostly due to abutment reactions. Using nonlinear static and dynamic analyses,
Abdel-Mohti and Pekcan [9] investigated the seismic performance
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
of a three-span continuous RC box-girder bridge for abutment
skew angles spanning between 0° and 60°. They used detailed finite element as well as simplified beam-stick models, and concluded that simplified beam-stick models can capture the
coupled lateral–torsional response of the skewed bridges for moderate skew angles.
An approximate method for dynamic analyses of skewed
bridges with continuous rigid decks was proposed by Kalantari
and Amjadian [10]. They developed a three degree-of-freedom
analytical model to determine the natural frequencies, mode
shapes, and internal forces for short skewed bridges. Dimitrakopoulos [11] investigated the seismic response of short skewed bridges
with pounding deck-abutment joints, and proposed a non-smooth
rigid-body approach to analyze their response. Dimitrakopoulos
concluded that the tendency of skewed bridges to rotate after
deck-abutment collisions is a factor of the skew angle, deck geometry, and the friction between the deck and abutment.
Although the aforementioned studies resulted in a better
understanding of seismic behavior of skewed bridges, a broader
and quantified understanding of the spectrum of their potential responses and collapse mechanisms under seismic loading is needed.
In this study, a modeling technique, which is computationally efficient yet adequately detailed, is described. Through extensive
parametric studies—conducted using nonlinear time-history analyses of the said models—the trends in the seismic response of
skewed bridges were also investigated by considering variations
in their configuration, as well as ground motion characteristics.
The varied configuration parameters and attributes comprise global torsional resistance, skew angle, column height, and span
arrangements. The applied ground motions are those that were recorded on rock or soil, as well as those that contained pronounced
velocity pulses. Seismic response parameters that are examined include maximum planar deck rotations and maximum column-bent
drift ratios.
2. Methodology
Three bridges from the vast bridge stock in California are selected and named ‘‘seed bridges.’’ For parametric seismic response-sensitivity analyses, variations for each type of seed
bridges are developed. These bridges are recently designed and
built (ca. 2000) at regions with high seismicity. The first bridge is
the Jack Tone Road Overcrossing (denoted as Bridge A) with two
spans supported on a single column. The second bridge is the La
Veta Avenue Overcrossing (denoted as Bridge B) with two spans
supported on a two-column bent. As such, it has a larger global torsional stiffness than Bridge A. The third bridge is the Jack Tone Road
Overhead, (denoted as Bridge C) with three spans and two threecolumn bents.
Table 1 summarizes key geometric properties of the seed
bridges; and their detailed descriptions may be found in [12]. They
have configurations that are frequently encountered in California.
They have prestressed RC box-girder superstructures, and seat-
Table 1
General geometric properties of the seed bridges.
Parameters
Bridge A
Bridge B
Bridge C
Span lengths (m)
Deck width (m)
Deck depth (m)
Deck centroid (m)
Columns per bent
Column height (m)
Column diameter (m)
33 + 34
8.3
1.40
0.75
1
6
1.7
47 + 44
23
1.90
1.04
2
6.7
1.7
47.5 + 44 + 36
23.5
1.92
1
3
7.5
1.7
type abutments. Seat-type abutments (Fig. 1) feature exterior
shear keys that are used to counter possible transverse deck movements. They are proportioned and detailed to act as fuses that will
break off under the design earthquake [13,14]. In the longitudinal
direction, a lightly reinforced backwall holds an engineered backfill
in place; it is designed to break off and allow the mobilization of
backfill, which, in turn, generates passive resistance [13].
In order to investigate the trends in key response parameters, a
matrix of model bridges are generated by using the three bridges
(A–C) as seeds and by creating reasonable variations in their configuration and geometric attributes (for preliminary work, see,
for example [12,15]). The seed bridges primarily differ in their global torsional resistance. The bridge matrix contains multiple analytical bridge models, each of which representing one particular
realization of skew angle, span arrangement, and column height.
The ranges and definitions of the varied parameters are shown
on Table 2.
2.1. Ground motion characteristics
Response-sensitivities of the model bridges need to be probed
using a diverse set ground motions that are representative of those
recorded in California. To that end, three sets of ground motions
were selected from the ‘‘EQ Library,’’ which was developed as part
of the PEER Transportation Research Program [16]. These sets are denoted as ‘‘Soil-site,’’ ‘‘Rock-site,’’ and ‘‘Pulse-like.’’ Each set comprises forty un-scaled three-component ground motions. The EQ
Library motions do not represent motions that are specific to the
sites of the seed bridges considered in this study, nor do they represent a target hazard. This characteristic of selected motions provides a unique opportunity to observe the general trends within
the seismic responses as function of ground motion intensity
measures.
Selected motions have a variety of spectral shapes, durations,
and directivity periods. For the first two sets—i.e., soil-site and
rock-site sets—the mean and variance of the natural logarithm of
spectral acceleration match the generic Mw = 7 and R = 10 km scenario for California [16]. The pulse-like ground motions contain
strong velocity pulses of varying periods in their strike-normal
components. Fig. 2 displays the response spectra of both the
strike-normal (SN) and the strike-parallel (SP) components of the
three ground motion sets. An effort is also made to investigate
the response-sensitivity with respect to the angle of incidence by
varying the incidence angle from 0° to 150°, with 30° increments
(Table 2).
2.2. Bridge matrix
The definitions of the parameters varied in the response-sensitivity and their ranges are provided in Table 2. There are three
bridge types (A–C), two column-height variations (original column
height of Colorig, and extended column height of Colext =
1.5 Colorig), two span arrangements of symmetrical (i.e., equal
span length) and asymmetrical (i.e., ratio of span length equals to
1.2), five abutment skew angles (0°, 15°, 30°, 45°, and 60°), three
types of ground motions (i.e., ‘‘Soil-site,’’ ‘‘Rock-site,’’ and ‘‘Pulselike’’) each of which include a suit of 40 three-dimensional acceleration records, and six incidence angles (0°, 30°, 60°, 90°, 120°, and
150°). Given these variations, 14,400 nonlinear time-history analyses had to be carried out for each bridge.
2.3. Collapse criteria
The simplified modeling technique and structural analysis software adopted in this study are not capable of simulating every conceivable collapse mechanism. This is particularly true for
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Bearing pad
Stemwall
Shear key
Backwall
Wingwall
(a)
(b)
Fig. 1. Configuration of typical seat-type abutment: (a) skewed abutment, (b) straight abutment.
Table 2
Definition of bridge geometry parameters and ground motions used in the sensitivity
study.
Sensitivity parameters
Geometric characteristics
Abutment skew angle (°)
Span arrangement
Column height
Ground motion characteristics
Ground motion type
Angle of incidence (°)
and/or validated modeling techniques are adopted whenever possible [22]. Details are provided in what follows.
Variation range
3.1. Superstructure modeling
0, 15, 30, 45, 60
Symmetrical (span ratio = 1.0)
Asymmetrical (span ratio = 1.2)
Colorig = Original column size
Colext = 1.5 Colorig
Soil-site
Rock-site
Pulse-like
0, 30, 60, 90, 120, 150
sequential collapse scenarios. With that proviso, the analysis output is post-processed in order to identify collapse cases using predefined collapse criteria. Non-simulated collapse is defined using
the following criteria: Collapse occurs if one of the following criteria is met:
Column-bent maximum drift ratio is greater than 8%.
Deck displacement relative to the abutment in the longitudinal
unseating direction is greater than the seat length (of each seed
bridge).
It was shown by Hutchinson et al. [17] that if a column maximum drift ratio is less than 8%, then the maximum residual drift
ratio of that column can be considered to be less than 1%—a number that provides an indication of the bridge serviceability after an
earthquake [18].
3. Three-dimensional modeling of skewed bridges
OpenSees structural analysis platform [19] was used to carry
out the time-history analyses. This platform provides an adequate
element and material response library for earthquake engineering
applications, and enables scripted execution of repetitive nonlinear
time-history analyses through which the model parameters and input ground motions can be systematically varied. A representative
bridge model used in the simulations is displayed in Fig. 3. The
model comprises seat-type abutments, shear keys, expansion
joints, column-bents, and the superstructure. Modeling assumptions are known to have significant effects on the dynamic response characteristics of short bridges [20,21]. Therefore, verified
The bridge deck and the cap-beam form the bridge superstructure, which are modeled using three-dimensional spine-line elements that trace the bridge’s alignment. These elements were
designated as elastic elements and were assigned with uncracked
section properties (i.e., prestresed concrete section properties).
The cap-beam was given arbitrarily (105) high torsional and inplane flexural stiffness values, because the cap-beam and the deck
are integrally constructed in the seed bridges. Superstructure elements were endowed with both translational and rotational mass
in order to capture the dynamic response accurately.
3.2. Abutment modeling
The abutment system comprises backwall, backfill, wingwalls,
shear keys, bearing pads, and usually a pile group that supports
the seat. Among these, only the longitudinal responses of the backfill (passive pressure) and the expansion joint, the transverse responses of the shear keys, and the vertical responses of the
bearing pads and the stemwall are considered explicitly in the models. The rest are omitted, because their contributions to the overall
response, and their effects on failure modes are deemed to be
insignificant.
Experimental data on lateral passive response of abutment
backfills are limited [23–26]. To the authors’ knowledge, there
are no data sets from controlled experiments on backfill response
for skewed abutments. Therefore, the model intuitively devised
and adopted here for response of skewed abutments is not experimentally validated. That said, the longitudinal response is modeled by using five nonlinear springs in series with gap elements
as shown in Fig. 4. The nonlinear springs and the gap elements represent the passive backfill response and the expansion joint,
respectively. The strength and initial stiffness of the soil springs
are obtained from recommendations provided in the Caltrans
SDC [13], which, in turn, were derived from large-scale abutment
testing [24,26].
For all abutment skew angles, the direction of the backfill passive pressure is assumed to be perpendicular to the backwall. Consequently, the forces tangential to the backwall—which would be
due to friction between the wall and the soil—are omitted here.
This is a conservative assumption, as those frictional forces would
always work to reduce deck rotation. The backfill springs are
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Sa (g)
Individual GM Spectrum-SN
Median Spectrum-SN
10
10
Individual GM Spectrum-SP
Median Spectrum-SP
0
10
-1
10
-1
10
0
10
1
Period (sec)
10
0
-1
10
-1
10
Sa (g)
10
10
-1
-1
10
0
10
1
Period (sec)
10
0
-1
10
-1
10
Sa (g)
10
10
-1
10
10
0
Period (sec)
10
1
Individual GM Spectrum-SP
Median Spectrum-SP
0
-1
0
Period (sec)
(b)
Individual GM Spectrum-SN
Median Spectrum-SN
10
1
Individual GM Spectrum-SP
Median Spectrum-SP
0
10
10
Period (sec)
(a)
Individual GM Spectrum-SN
Median Spectrum-SN
10
0
10
1
10
0
-1
10
-1
10
0
10
1
Period (sec)
(c)
Fig. 2. Response spectra of strike-normal (SN) and strike-parallel (SP) components of ground motion types: (a) ‘‘Soil-site,’’ (b) ‘‘Rock-site,’’ (c) ‘‘Pulse-like’’.
equally spaced and attached to a rigid bar representing the deck.
Properties of the five abutment nonlinear hyperbolic springs are
slightly different from each other, depending on their relative location to the obtuse angle between the backwall and the longitudinal/traffic direction (cf., point OBT in Fig. 4). The stiffness and
strength of these springs are assumed to increase linearly, as function of abutment skew angle and distance from point OBT. It is assumed that the volume of engineered backfill soil that can be
mobilized per unit wall-width in the event of backwall breakage
is larger, from point OBT towards the acute side (cf., point ACU in
Fig. 4). It is further assumed that this variation is linear in the simplest—and therefore the most reasonable—that can be adopted
without direct guidance from experimental observations.1 Finally,
it is postulated that the maximum stiffness/strength variation occurs
1
Incidentally, the abutment modeling assumptions are similar to those adopted in
[27] who used three-dimensional finite element simulations of a skew abutment [28]
to develop their simplified models.
for the largest skew angle (60°), and it is equal to 30%. Therefore, the
stiffness/strength variation factor b for a given the skew angle a can
be computed as in,
b ¼ 0:3
tan a
tan 60
ð1Þ
Multiple analyses conducted with different values of the maximum
variation between OBT and ACU springs (i.e., 30% in Eq. (1)) indicate
that the results are not highly sensitive to variations in b (less than
2% difference in the median of deck rotation of Bridge A when b is
varied from 0% to 60%). As such, the aforementioned modeling
assumptions are a reasonable (and arguably non-controversial)
extension to previous attempts [27,28].
The backwall and the shear keys are sacrificial elements that are
designed to break off to protect the bridge foundation against higher dynamic forces [13,21]. The shear key response is modeled with
a tri-linear backbone curve, which mimics the behavior observed
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Fig. 3. Generic model (a° skew) used for nonlinear time-history analyses.
Displ. (cm)
OBT
0
2
4
6
8
10
0
150
Elastic
Superstructure
α
Force (KN)
Rigid Beam
300
450
600
750
900
ACU
(a)
(b)
Fig. 4. Backfill soil springs: (a) configuration diagram, (b) backbone curve.
Bridge "A"
Bridge "B"
Bridge "C"
Force (KN)
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
Deformation (cm)
Fig. 5. Shear key for the seed bridges depicted in scale (right), and their force–deformation backbone curves (left).
in full-scale tests [29]. Fig. 5 shows these backbone curves for the
three seed bridges. The transverse stiffnesses of bearing pads are
omitted (i.e., they were considered to behave very flexible in the
transverse direction). Vertical response is modeled by two parallel
springs as shown in Fig. 6. The first spring represents the flexible
portion of elastomeric bearing pad in vertical direction (K1),
whereas the second represents vertical stiffness of the stemwall
and abutment embankment (K2).
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
bridges. The modeling technique used in this study is similar to
the approach employed by Cruz and Saiidi [32] that was verified
by a recent large-scale test. They simulated a large-scale four-span
bridge test in the University of Nevada, Reno, and showed a comparable correlation between seismic demands derived from analytical models and the experimental data.
F
K
2
K
1
4. Trend observations
D
Δ
Fig. 6. Vertical force–displacement backbone curve of the abutment (K1: the
elastomeric bearing pad stiffness, and K2: the stemwall and abutment embankment
stiffness).
3.3. Column-bent modeling
Progression of column yielding and damage is expected under
strong ground motions, and thus nonlinear fiber-based force-based
beam finite elements are used to represent the columns (Fig. 7). In order to achieve a more realistic representation of their responses, the
beam finite elements representing them are endowed with the ability to respond inelastically at every quadrature point. All fiber sections are assigned with the UniaxialMaterial model tag of
OpenSees [19]. Three different constitutive rules are used simultaneously within a cross-section: (i) confined concrete, (ii) unconfined
concrete, and (iii) steel rebar (Fig. 7). A forced-based element—viz.,
NonlinearBeamColumn—is used, which enables a more accurate
accounting of the moment distribution within the element [30]. A
single force-based element with 10 quadrature points is used per column; and this is usually deemed to provide adequate accuracy [19].
In order to model the portion of the column-bent embedded in the
superstructure, a rigid element is attached to the top of the nonlinear
beam-column element. The length of this rigid element is set equal to
the distance between the centroid of the soffit-flange of the superstructure box-girder and the column top.
Assumption on column base boundary condition can significantly affect estimates of bridge seismic demands obtained from
nonlinear time-history analysis. Based on information gathered
from structural drawings and common engineering practice, a
fixed base connection for a single-column bridge (‘‘Bridge A’’)
and a pinned base connection for multi-column bridges (‘‘Bridges
B and C’’) are assigned in bridge models used in this study. Including soil-pile (or foundation)-structure interaction increases the
flexibility of the substructure, which results in longer period of
vibration and larger estimates for deformation demands [31].
However, this research is focused on identifying trends in seismic
demands of skewed bridges due to variation in bridge geometric
parameters rather than accurate estimates of seismic demands.
Nevertheless, focus is maintained on accurate modeling of seattype abutment, which has a larger effect on behavior of skewed
The seed bridge models and their variations were systematically analyzed using the selected ground motion sets under multiple angles of incidence. In total, 43,200 time-history analyses were
conducted, and the results were scrutinized to identify dependencies, correlations and trends. To this end, two seismic response
parameters were obtained, which were the maximum planar deck
rotation (hrot), and the maximum total column drift ratio (hcol). Detailed investigation of simulated data suggested the necessity for
differentiating results into several groups based on three distinct
regimes of skew bridge behavior: bridge-collapse (according to
the collapse criteria discussed in Section 2.3), bridge survival with
shear key failure, and bridge survival with shear key survival.
In what follows, the effects of ground motion characteristics on
bridge response are discussed; and the use of an efficient ground
motion Intensity Measure, IM, to express IM-EDP curves is proposed (EDP denotes Engineering Demand Parameter). Because the
ground motion sets used in this study are not tuned towards a
target hazard level, it is necessary to interpret the results using
IM-EDP curves that are conditioned on no-collapse (henceforth
denoted simply as IM-EDP curves), and the probability of collapse
(given IM). Trends and dependencies between IM-EDP curves and
EDPs other than hrot and hcol may be found in [33].
4.1. Effects of ground motion characteristics and the selection of an
efficient intensity measure
Seismic response parameters of skewed bridges are significantly sensitive to the characteristics of applied seismic excitation;
higher values of response parameters and incidences of collapse
are observed for skewed bridges that experience pulse-like ground
motions. Fig. 8 displays the variation of hrot and hcol for no-collapse
cases as a function of ground motion peak acceleration PGAres
(maximum of the resultant of two SN and SP components) for all
ground motion sets, and for five variations of the abutment skew
angle. This figure is generated only for Bridge A with its original
column height and by assuming symmetric spans; but other variations in bridge geometrical properties display similar trends for
hrot and hcol (see [33]). In each plot, EDPs are discriminated into
two categories: based on whether the shear keys survived (circles
in Fig. 8) or lost resistance (squares) during seismic excitation.
These results clearly indicate that the probability of shear key failure is higher when the bridge experiences a pulse-like motion
(higher concentration of black squares) for any abutment skew
Rigid Element
Unconfined
Concrete
Confined
Concrete
A
A
Steel Rebar
Section A-A
Fig. 7. Column modeling scheme.
NonlinearBeamColumn
Element
143
Pulse-Like
PGAres (g)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
1
Soil-Site
0.1
Rock-Site
1
0.1 1
0.1
10
0.1 1
0.1
10
1
0.1 1
Shear key-survive
0.1 1
0.1
10
0.1 1
0.1
10
1
0.1 1
0.1
10
1
0.1 1
0.1
10
1
0.1
10
60°
1
1
0.1
10
1
0.1
0.1 1
1
0.1 1
45°
1
0.1
10
1
0.1
30°
0.1
10
10
0.1 1
10
0.1 1
10
1
0.1 1
0.1
10
1
0.1 1
0.1 1
1
0.1 1
0.1
10
Shear key-failure
θ rot
(radx10-3)
Pulse-Like
PGAres (g)
(a)
00°
15°
1
Soil-Site
0.1
Rock-Site
1
1
10
1
0.1
0.1
1
10
0.1
1
10
Shear key-survive
10
0.1
0.1
1
10
0.1
1
10
10
0.1
0.1
1
1
10
1
1
10
1
1
60°
1
1
1
1
45°
1
1
1
0.1
30°
0.1
10
0.1
1
10
1
10
1
1
10
1
1
0.1
0.1
1
1
10
0.1
1
10
θ col ( %)
Shear key-failure
(b)
Fig. 8. Sensitivity of response of Bridge A with original column height and symmetric spans to type of seismic excitation and abutment skew angle: (a) PGAres–hrot, (b) PGAres–hcol.
angle. Given the significance of pulse-like ground motions in
imposing high seismic demands compared to other types of ground
motions, the trend observations will henceforth be confined only
to those analyses carried out using the pulse-like ground motion
set. Results from the soil- and rock-site ground motion sets may
be found in [33].
We opted to use the peak resultant ground velocity (PGVres) as
the ground motion intensity measure for generating IM-EDP curves
in this study. PGVres is computed as the maximum of the SN and SP
components’ resultant velocity history. The efficiency of different
intensity measures were studied by monitoring the variations in
IM-EDP plots (similar to Fig. 8 but with different IMs) as well as dispersion of collapse fragility curves considering a wide range of IMs
as collapse-potential indicators. Other IMs that were considered
include spectral acceleration at the first mode period of strikenormal component, denoted as SaSN(T1), peak ground acceleration
of strike-normal component (PGASN), peak ground velocity of
strike-normal component (PGVSN), and peak resultant ground
acceleration (PGAres). Fig. 9 displays a sample collapse fragility
curve derived for Bridge A with original column height, symmetric
span, and 60° abutment skew angle, subjected to the pulse-like
ground motion set. A logistic regression was applied to the 240
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Prob. of Collapse
1
Collapse indicator
Lognormal fit
0.75
0.5
0.25
0
100
200
300
PGVres (cm/s)
Fig. 9. Development of the collapse fragility curve for Bridge A (symmetric span,
original column height, 60° abutment skew angle, pulse-like GM).
data points (resulting from 40 ground motion records and 6 angles
of incidence) for generating this collapse fragility curve. Data
points shown in Fig. 9 are pairs of PGVres and an indicator parameter, which is equal to ‘‘1’’ if the ground motion caused collapse, and
‘‘0’’ otherwise. Using this approach, the collapse fragility curves of
Colext-Symt.
Disper.
Colorig-Asymt.
Disper.
Colorig-Symt.
Disper.
Bridge A
Bridge B
Bridge C
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
15
30
45
60
0
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
15
30
45
60
0
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
15
30
45
60
0
Skew Angle
0
15
30
45
60
0
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
Skew Angle
SaSN (T1)
15
30
0
45
60
0
PGA res
30
45
60
45
60
45
60
45
60
Skew Angle
0
15
30
Skew Angle
0
15
30
Sk
0
Skew Angle
PGA SN
15
0.5
0.5
0
Colext-Asymt.
Disper.
all bridges in bridge matrix were obtained along with the median
and dispersion values of fitted lognormal distributions.
Implication of PGVres being a superior IM compared to other IMs
may be deduced from Fig. 10, in which the dispersions of various candidate collapse-potential indicator IMs are presented. Fig. 10 displays
these collapse-potential IM dispersions for three bridges types (A–C)
and four combinations of column height and span arrangements—
namely, original column height with symmetric span (Colorig-Symt.),
original column height with asymmetric span (Colorig-Asymt.), extended column height with symmetric span (Colext-Symt.), and extended column height with asymmetric span (Colext-Asymt). In
each plot, the dispersion as a function of assumed abutment skew angle is shown. These results clearly indicate that the dispersion of collapse fragility curves obtained using PGVres as the collapse-potential
indicator is consistently smaller on average than those of other IMs
for Bridges B and C. A similar trend is observed for Bridge A; however,
PGAres exhibits a smaller dispersion compared to PGVres for cases with
extended column height. It is useful to acknowledge that no single IM
(i.e., no truncated/reduced model or parameter of the ground motion) by itself can capture all aspects of seismic hazard; nevertheless,
given the observed superiority of PGVres for most cases, it was used
for identifying collapse-potential and for obtaining the IM-EDP
curves throughout the rest of this study.
15
A
l
30
Skew Angle
PGVSN
PGVres
Fig. 10. Dispersions in collapse fragility curves obtained for the bridge matrix using the pulse-like ground motion set, and logistic regressions of data for various IMs.
145
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
4.2. Trends in the statistical parameters of the collapse fragility curves
Probability of collapse, given the level of ground motion intensity, is sensitive to the type of seismic excitation, abutment skew
angle, bridge column height, and span arrangement. Fragility
curves of all bridges in the bridge matrix were developed using
PGVres as the collapse-potential indicator, and using logistic regression as described in Section 4.1. Fig. 11 displays the median collapse capacity and the associated dispersion for all bridges
considered obtained by using the pulse-like ground motion set.
The rock- and soil-site ground motion sets had significantly fewer
instances of collapse, so these results are omitted here [33].
The effect of abutment skew angle on collapse-potential of
bridges is found to be a function of bridge type. For a single-column
two-span bridge (i.e., Bridge A) the collapse-potential is small at
zero skew angle (note the high median collapse-potential at zero
skew angle for Bridge A in Fig. 11). The median collapse-potential
drops for higher skew angles, exhibiting a higher probability of collapse at the same level of ground motion intensity. Bridge C, in contrast, does not show a similar correlation between the abutment
skew angle and the collapse-potential. This phenomenon can arguably be attributed to the type of failure resulting from the columnbase boundary conditions. Because Bridges A and C were modeled
with fixed- and pinned-base boundary conditions, respectively,
there is a high moment demand at the top of the columns of Bridge
C, regardless of the abutment skew angle, which can rapidly exhaust column load-resistance capacity and lead to collapse. In contrast, the single column in Bridge A with a fixed-base can better
handle the moment demand at zero skew angle. However, for larger
Col orig-Symt.
Bridge A
Col orig-Asymt.
The strength of shear key has a large influence on the bridge deck
rotation. For larger skew angles, probability of shear key failure—
hence, the probability of having a large deck rotation—increases.
Fig. 12a–c show the variation of maximum deck rotation, hrot, for
bridges A, B, and C, respectively, as a function of PGVres obtained from
the pulse-like ground motion set. These figures display the sensitivity of hrot to five variations in abutment skew angle, and to various
combinations of span configuration and column height. In each plot,
the cases where shear keys survived (circles in Fig. 12) or have lost
resistance (squares) during seismic excitation are marked. Data
points representing ground motions for which the bridge collapsed
are not shown. Fig. 12 also reveals that higher hrot is expected once
a shear key fails. Moreover, span arrangements appear to have less
effect compared to column height on the deck rotation demand.
Bridge B
Bridge C
200
200
150
150
150
100
100
100
50
50
50
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
Col ext -Symt.
4.3. Trends in the deck rotation values
200
0
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
Col ext -Asymt.
abutment skew angles, the bridge becomes susceptible to deck
rotation, which may increase its collapse-potential.
Certain combinations of abutment skew angle, column height,
and span arrangement can create a condition at which the bridge
is more susceptible to deck rotation; and therefore, probability of
collapse is increased, and vice versa. Fig. 11 shows this phenomenon for Bridge B: the median collapse-potential is decreased for
skew angles between 0° and 15°, once original column heights
and symmetric span is considered. However, for same column size
and asymmetric span, the median collapse capacity is increased.
Similar trends can be seen for Bridge A.
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
0
0
15
30
Skew Angle
45
60
0
15
30
Skew Angle
45
60
0
0
15
30
45
60
0
15
30
45
60
0
15
30
45
60
0
15
30
45
60
Skew Angle
Fig. 11. Median and dispersion of collapse capacity, expressed in terms of PGVres for the bridge matrix obtained using the pulse-like ground motion set.
PGVres (cm/s)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
146
00°
15°
100
30°
100
10
100
10
0.1 1
0.1 1
10
100
10
100
0.1 1
10
100
10
100
0.1 1
10
100
10
10
0.1 1
10
10
10
0.1 1
10
10
0.1 1
10
100
10
10
0.1 1
100
100
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
10
10
Shear key-survive
10
100
10
0.1 1
10
10
0.1 1
10
10
10
0.1 1
100
100
10
0.1 1
10
10
10
100
10
0.1 1
100
10
0.1 1
60°
100
10
10
100
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
(a)
00°
15°
100
30°
100
10
100
10
0.1 1
0.1 1
10
100
10
100
0.1 1
10
100
10
100
0.1 1
10
100
10
10
Shear key-survive
10
10
10
10
10
10
0.1 1
10
10
0.1 1
10
100
10
0.1 1
0.1 1
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
100
10
0.1 1
10
10
0.1 1
10
10
10
0.1 1
100
100
10
0.1 1
10
10
10
100
10
0.1 1
100
10
0.1 1
60°
100
10
10
100
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
(b)
Fig. 12. PGVres–hrot conditioned on no-collapse plots obtained for the bridge matrix using the pulse-like ground motion set: (a) Bridge A, (b) Bridge B, (c) Bridge C.
The incidence of shear key failure is higher for Bridge A (a twospan single-column-bent bridge) than Bridges B or C; and its seismic response is affected more by the abutment behavior that those
of other bridges. In Fig. 12, the two cases of shear key survival and
failure have seemingly divided the IM-EDP plots into two distinct
regions. However, in Bridge B, these regions overlap somewhat,
suggesting that the incidences of collapse in Bridge B are not dominated by excessive deck rotations, and that excessive translations
play a significant role.
Deck rotations can be detected even for symmetrical bridges
that have 0° skew (Fig. 12). These rotations are caused by the columns that become asymmetrically damaged (concrete losing
strength in tension, etc.) during the course of strong motions.
Bridge columns and shear keys experience loads exceeding their
yield capacity, and behave nonlinearly. This destroys the initial
symmetry in geometry and boundary conditions, and leads to the
initiation of deck rotations and in some cases, ultimately, to
collapse.
147
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
100
30°
100
10
100
10
0.1 1
10
100
10
10
100
10
100
10
10
100
10
100
10
10
Shear key-survive
10
10
10
10
10
10
0.1 1
10
10
0.1 1
10
100
10
0.1 1
0.1 1
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
10
0.1 1
100
100
10
0.1 1
10
10
0.1 1
100
10
0.1 1
100
10
0.1 1
60°
100
10
0.1 1
100
10
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
(c)
Fig. 12 (continued)
4.4. Trends in column drift ratios
For a single-column bridge, the column drift ratio is a response
parameter that depends on the translation of the bridge deck in
two orthogonal directions along the ground motion record is applied. However, in multi-column-bent skewed bridges, the rotation
of the deck results in an additional translation of the column-top;
and this can potentially increase the maximum column drift ratio
for these bridges compared to their zero-skew counterparts.
Fig. 13 clearly displays this effect where PGVres and hcol data pairs
are shown for the pulse-like ground motion set, only for cases
where collapse has not occurred. The first row of plots in
Fig. 13a, which is for Bridge A with symmetric spans and original
column heights, reveals that hcol is not sensitive to abutment skew
angle. However, first row of Fig. 13b, which is for Bridge B, indicates that hcol increases with the skew angle. Similar trends may
be observed with a one-to-one comparison between the plots in
Figs. 12 and 13; the effect of deck rotation in increasing the column
drift ratio for higher skew angles in Bridge B (multi-column-bent)
or lack of it for Bridge A (single-column-bent) is evident.
hcol for Bridge C has a similar behavior to that for Bridge A, despite the fact that Bridge C is a multi-column-bent three-span
bridge (Fig. 13c). This behavior may be attributed to the torsional
rigidity of Bridge C, in which the two multi-column-bents provide
a relatively high stiffness against transverse rotation. However, given the modeling assumptions made in this study whereby the column bases in a multi-column-bent are modeled with pinned
connections, there is sudden drop in stiffness once the moment
capacity of columns exceeds yield moment capacity, at which
point, the bridge is likely to experience collapse.
Two less catastrophic damage levels of hcol, concrete cover spalling and buckling of longitudinal bars, are also studied. Berry and
Eberhard [34] suggested that the onset of cover spalling can represent the first flexural damage state to concrete columns and the
onset of longitudinal bar buckling indicates the significant (or
functionary) damage state to concrete columns. Based on the
regression analysis of experimental data, they proposed two formulations for the predefined damage states. Table 3 represents
the corresponding numerical evaluation for hcol, the two damage
states. In Fig. 13, three distinct regions are specified by two dotted
lines. The blue line (dark gray in black and white print) represents
hcol-spall, and the red line (light gray in black and white print) indicates hcol-bb. For all bridges, the shear key failure status plays a significant role in damage state evaluation of the column-bent drift
ratio.
5. Conclusions
In this study, three actual concrete box-girder bridges (of California) with seat-type abutments were used as seed models; and
a matrix of bridge models were created by varying their abutment
skew angles, column heights and span arrangements. These models were subjected to three sets of forty ground motion records
through nonlinear time-history analyses. These ground motions
were not selected for a target hazard level; but rather they are representative of the types of expected seismic excitation in California. As such, response trends were studied through IM-EDP and
collapse fragility curves.
The results indicated that ground motions with high velocity
pulses induce higher seismic demands, at least for the bridges
studied. Collapse-potential appeared to have an inverse relation
with the abutment skew angle. Particularly, the shear key failure
status has a major effect on induced demands in single-column
bridges. This indicator has a significant influence on the planar
rotation of the deck; i.e., larger deck rotations are expected for
when shear keys fail. The global torsional stiffness, abutment skew,
and column elevation of skewed bridges also significantly affect
the seismic response of skew bridges. On the other hand, seismic
response of a skewed bridge is not sensitive to the span arrangement, at least for the configurations considered in this study.
The results also indicated that under the same seismic excitation, demands for bridges with skewed abutments, such as deck
PGVres (cm/s)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
148
00°
15°
100
30°
100
10
100
10
1
1
10
100
10
100
10
100
10
100
10
1
1
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
10
Shear key-survive
10
10
10
1
10
1
1
100
100
10
1
100
10
10
100
10
10
10
1
10
1
10
1
100
10
1
100
10
10
100
10
100
10
1
10
1
60°
100
10
10
100
45°
10
1
10
1
10
θ col (%)
Shear key-failure
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
(a)
00°
15°
100
30°
100
10
100
10
1
1
10
100
10
100
10
100
10
100
10
1
1
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
10
Shear key-survive
10
10
10
1
10
1
1
100
100
10
1
100
10
10
100
10
10
10
1
10
1
10
1
100
10
1
100
10
10
100
10
100
10
1
10
1
60°
100
10
10
100
45°
10
1
10
1
10
θ col (%)
Shear key-failure
(b)
Fig. 13. PGVres–hcol conditioned on no-collapse plots obtained for the bridge matrix using the pulse-like ground motion set: (a) Bridge A, (b) Bridge B, (c) Bridge C.
Table 3
Column-bent drift ratios for two damage states.
Column-bent damage states
Bridge A
Bridge B
Bridge C
Concrete spalling, hcol-spall (%)
Long bar buckling, hcol-bb (%)
1.8
5.0
2.1
5.2
2.4
7.9
rotation and column drift ratio, are higher than those for bridges
with straight abutments. It was also clear that the shear key
strength has a significant influence on the overall bridge behavior,
and shear keys can limit deck rotations and suppress the potential
amplification in column drift ratios resulting from deck rotation.
Modeling assumptions adopted in this study were based on earlier component-scale, verification and validation studies with certain exceptions. First among these are modeling assumptions for
the passive response of the abutment backfill. While purely
numerical studies and validated models for straight abutments
suggest that this is a reasonable model, experimental studies are
149
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
100
30°
100
10
100
10
1
10
100
10
10
100
10
100
10
10
100
10
100
10
Shear key-survive
10
10
10
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
1
10
10
1
10
1
1
100
100
100
10
1
10
10
1
10
1
10
1
100
100
10
1
10
10
1
100
10
1
100
10
1
60°
100
10
1
100
10
45°
10
1
10
1
10
θcol (%)
Shear key-failure
(c)
Fig. 13 (continued)
clearly needed in this area. Similarly, component-scale tests and
high-fidelity numerical simulations should be conducted to narrow
the uncertainties associated with assigning the column base
boundary conditions, which may significantly enhance seismic demand estimation.
Acknowledgements
This study is based on work supported by the PEER Transportation Research Program under Grant No. UCLA-45782. This financial
support is gratefully acknowledged. Any opinions, findings, and
conclusions or recommendations expressed in this paper are those
of the authors and do not necessarily reflect the views of sponsors.
The authors thankfully acknowledge the valuable technical discussion held with Dr. Anoosh Shamsabadi of California Department of
Transportation, and Dr. Majid Sarraf of Parsons Corp. throughout
this study.
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Seismic behavior of reinforced concrete bridges with skew-angled
seat-type abutments
Peyman Kaviani a, Farzin Zareian a,⇑, Ertugrul Taciroglu b
a
b
Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USA
Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USA
a r t i c l e
i n f o
Article history:
Received 30 July 2011
Revised 10 April 2012
Accepted 5 June 2012
Available online 20 July 2012
Keywords:
Skewed bridges
Performance-based earthquake engineering
Shear key
Seat-type abutment
a b s t r a c t
This study focuses on identifying trends in seismic behavior of reinforced concrete bridges with seat-type
abutments under earthquake loading, especially with respect to abutment skew angle. To that end, a
detailed approach for modeling skew-angled seat-type abutments is proposed; and a comprehensive variety of bridge configurations are considered. Specifically, three short bridges located in California are
selected as seed bridges, from which different models are spawned by varying key bridge structural
parameters such as column-bent height, symmetry of span arrangement, and abutment skew angle.
Through extensive nonlinear time-history analyses conducted using three suites of ground motions, it
is demonstrated that demand parameters for skew-abutment bridges, such as deck rotation and column
drift ratio, are higher than those for straight bridges. By investigating the sensitivity of various response
parameters to variations in bridge geometry and ground motion characteristics, it is shown that bridges
with larger abutment skew angles bear a higher probability of collapse due to excessive rotations, and that
the shear keys can play a major role in reducing deck rotations and thus the probability of collapse.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Bridges with skew-angled abutments, also denoted henceforth
as ‘‘skewed bridges,’’ are constructed to accommodate geometry
constraints resulting from alignment of a waterway or roadway
crossing that occurs at an angle that is different from 90°. In the
present study, the focus is on the seismic response of short skewed
bridges with ‘‘seat-type’’ abutments, which are very common in
construction practice in California. Observations from past earthquakes [1–3] suggest that there are significant differences between
the response of straight and skewed bridges. A manifestation of
this difference is the inherent tendency of the decks of skewed
bridges to rotate about their vertical axes under seismic excitation,
which can lead to unseating from abutments, and ultimately, to
collapse. The primary cause of this excessive rotation is the eccentric passive resistance of the abutment backfill.
Excessive deck rotations can lead to loss of bridge functionality
and significant downtime for repairs, or loss of stability and casualties. Site investigation of the Foothill Boulevard Undercrossing located in California, 34.25N and 118.5W, (which has a skew angle of
approximately 60°) showed that it rotated in its horizontal plane,
resulting in a permanent offset of approximately 7.5 cm (i.e.,
0.9 103 radians of deck rotation) in the direction of increasing
skew angle during the 1971 San Fernando Earthquake [1]. Recon⇑ Corresponding author. Tel.: +1 949 824 9866; fax: +1 949 824 2117.
E-mail address: [email protected] (F. Zareian).
0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.engstruct.2012.06.013
naissance report of the 2010 Chile Earthquake [2,3] states that
skewed bridges in affected regions rotated, mainly about their center of stiffness, and that those with weak exterior shear keys suffered higher levels of damage due to transverse unseating.
Development of analytical/numerical models that can capture
the peculiar collapse mechanisms of skewed bridges under seismic
excitation, and can accurately quantify their damages have been a
subject of research for quite some time. Ghobarah and Tso [4] used
a spine-line model to represent bridge deck, and columns; they
concluded that the bridge collapse was caused by coupled flexural-torsional motions of the bridge deck or by the excessive compression demands that resulted in column failures. Using simplified
beam models, Maragakis and Jennings [5] concluded that the angle
of the skew and the impact between the deck and abutment govern
the response of skewed bridges. Wakefield et al. [6] conjectured
that if the deck is not rigidly connected to the abutments, dynamic
response of the bridge will be dominated by planar rigid body rotations of the deck rather than coupled flexural and torsional
deformations.
A more recent study by Meng and Lui [7] proposed that the seismic response of a bridge is strongly influenced by the column
boundary conditions and skew angle. In a subsequent study [8],
they used a dual-beam stick model to represent the bridge deck,
and showed that in-plane deck rotations are mostly due to abutment reactions. Using nonlinear static and dynamic analyses,
Abdel-Mohti and Pekcan [9] investigated the seismic performance
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
of a three-span continuous RC box-girder bridge for abutment
skew angles spanning between 0° and 60°. They used detailed finite element as well as simplified beam-stick models, and concluded that simplified beam-stick models can capture the
coupled lateral–torsional response of the skewed bridges for moderate skew angles.
An approximate method for dynamic analyses of skewed
bridges with continuous rigid decks was proposed by Kalantari
and Amjadian [10]. They developed a three degree-of-freedom
analytical model to determine the natural frequencies, mode
shapes, and internal forces for short skewed bridges. Dimitrakopoulos [11] investigated the seismic response of short skewed bridges
with pounding deck-abutment joints, and proposed a non-smooth
rigid-body approach to analyze their response. Dimitrakopoulos
concluded that the tendency of skewed bridges to rotate after
deck-abutment collisions is a factor of the skew angle, deck geometry, and the friction between the deck and abutment.
Although the aforementioned studies resulted in a better
understanding of seismic behavior of skewed bridges, a broader
and quantified understanding of the spectrum of their potential responses and collapse mechanisms under seismic loading is needed.
In this study, a modeling technique, which is computationally efficient yet adequately detailed, is described. Through extensive
parametric studies—conducted using nonlinear time-history analyses of the said models—the trends in the seismic response of
skewed bridges were also investigated by considering variations
in their configuration, as well as ground motion characteristics.
The varied configuration parameters and attributes comprise global torsional resistance, skew angle, column height, and span
arrangements. The applied ground motions are those that were recorded on rock or soil, as well as those that contained pronounced
velocity pulses. Seismic response parameters that are examined include maximum planar deck rotations and maximum column-bent
drift ratios.
2. Methodology
Three bridges from the vast bridge stock in California are selected and named ‘‘seed bridges.’’ For parametric seismic response-sensitivity analyses, variations for each type of seed
bridges are developed. These bridges are recently designed and
built (ca. 2000) at regions with high seismicity. The first bridge is
the Jack Tone Road Overcrossing (denoted as Bridge A) with two
spans supported on a single column. The second bridge is the La
Veta Avenue Overcrossing (denoted as Bridge B) with two spans
supported on a two-column bent. As such, it has a larger global torsional stiffness than Bridge A. The third bridge is the Jack Tone Road
Overhead, (denoted as Bridge C) with three spans and two threecolumn bents.
Table 1 summarizes key geometric properties of the seed
bridges; and their detailed descriptions may be found in [12]. They
have configurations that are frequently encountered in California.
They have prestressed RC box-girder superstructures, and seat-
Table 1
General geometric properties of the seed bridges.
Parameters
Bridge A
Bridge B
Bridge C
Span lengths (m)
Deck width (m)
Deck depth (m)
Deck centroid (m)
Columns per bent
Column height (m)
Column diameter (m)
33 + 34
8.3
1.40
0.75
1
6
1.7
47 + 44
23
1.90
1.04
2
6.7
1.7
47.5 + 44 + 36
23.5
1.92
1
3
7.5
1.7
type abutments. Seat-type abutments (Fig. 1) feature exterior
shear keys that are used to counter possible transverse deck movements. They are proportioned and detailed to act as fuses that will
break off under the design earthquake [13,14]. In the longitudinal
direction, a lightly reinforced backwall holds an engineered backfill
in place; it is designed to break off and allow the mobilization of
backfill, which, in turn, generates passive resistance [13].
In order to investigate the trends in key response parameters, a
matrix of model bridges are generated by using the three bridges
(A–C) as seeds and by creating reasonable variations in their configuration and geometric attributes (for preliminary work, see,
for example [12,15]). The seed bridges primarily differ in their global torsional resistance. The bridge matrix contains multiple analytical bridge models, each of which representing one particular
realization of skew angle, span arrangement, and column height.
The ranges and definitions of the varied parameters are shown
on Table 2.
2.1. Ground motion characteristics
Response-sensitivities of the model bridges need to be probed
using a diverse set ground motions that are representative of those
recorded in California. To that end, three sets of ground motions
were selected from the ‘‘EQ Library,’’ which was developed as part
of the PEER Transportation Research Program [16]. These sets are denoted as ‘‘Soil-site,’’ ‘‘Rock-site,’’ and ‘‘Pulse-like.’’ Each set comprises forty un-scaled three-component ground motions. The EQ
Library motions do not represent motions that are specific to the
sites of the seed bridges considered in this study, nor do they represent a target hazard. This characteristic of selected motions provides a unique opportunity to observe the general trends within
the seismic responses as function of ground motion intensity
measures.
Selected motions have a variety of spectral shapes, durations,
and directivity periods. For the first two sets—i.e., soil-site and
rock-site sets—the mean and variance of the natural logarithm of
spectral acceleration match the generic Mw = 7 and R = 10 km scenario for California [16]. The pulse-like ground motions contain
strong velocity pulses of varying periods in their strike-normal
components. Fig. 2 displays the response spectra of both the
strike-normal (SN) and the strike-parallel (SP) components of the
three ground motion sets. An effort is also made to investigate
the response-sensitivity with respect to the angle of incidence by
varying the incidence angle from 0° to 150°, with 30° increments
(Table 2).
2.2. Bridge matrix
The definitions of the parameters varied in the response-sensitivity and their ranges are provided in Table 2. There are three
bridge types (A–C), two column-height variations (original column
height of Colorig, and extended column height of Colext =
1.5 Colorig), two span arrangements of symmetrical (i.e., equal
span length) and asymmetrical (i.e., ratio of span length equals to
1.2), five abutment skew angles (0°, 15°, 30°, 45°, and 60°), three
types of ground motions (i.e., ‘‘Soil-site,’’ ‘‘Rock-site,’’ and ‘‘Pulselike’’) each of which include a suit of 40 three-dimensional acceleration records, and six incidence angles (0°, 30°, 60°, 90°, 120°, and
150°). Given these variations, 14,400 nonlinear time-history analyses had to be carried out for each bridge.
2.3. Collapse criteria
The simplified modeling technique and structural analysis software adopted in this study are not capable of simulating every conceivable collapse mechanism. This is particularly true for
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Bearing pad
Stemwall
Shear key
Backwall
Wingwall
(a)
(b)
Fig. 1. Configuration of typical seat-type abutment: (a) skewed abutment, (b) straight abutment.
Table 2
Definition of bridge geometry parameters and ground motions used in the sensitivity
study.
Sensitivity parameters
Geometric characteristics
Abutment skew angle (°)
Span arrangement
Column height
Ground motion characteristics
Ground motion type
Angle of incidence (°)
and/or validated modeling techniques are adopted whenever possible [22]. Details are provided in what follows.
Variation range
3.1. Superstructure modeling
0, 15, 30, 45, 60
Symmetrical (span ratio = 1.0)
Asymmetrical (span ratio = 1.2)
Colorig = Original column size
Colext = 1.5 Colorig
Soil-site
Rock-site
Pulse-like
0, 30, 60, 90, 120, 150
sequential collapse scenarios. With that proviso, the analysis output is post-processed in order to identify collapse cases using predefined collapse criteria. Non-simulated collapse is defined using
the following criteria: Collapse occurs if one of the following criteria is met:
Column-bent maximum drift ratio is greater than 8%.
Deck displacement relative to the abutment in the longitudinal
unseating direction is greater than the seat length (of each seed
bridge).
It was shown by Hutchinson et al. [17] that if a column maximum drift ratio is less than 8%, then the maximum residual drift
ratio of that column can be considered to be less than 1%—a number that provides an indication of the bridge serviceability after an
earthquake [18].
3. Three-dimensional modeling of skewed bridges
OpenSees structural analysis platform [19] was used to carry
out the time-history analyses. This platform provides an adequate
element and material response library for earthquake engineering
applications, and enables scripted execution of repetitive nonlinear
time-history analyses through which the model parameters and input ground motions can be systematically varied. A representative
bridge model used in the simulations is displayed in Fig. 3. The
model comprises seat-type abutments, shear keys, expansion
joints, column-bents, and the superstructure. Modeling assumptions are known to have significant effects on the dynamic response characteristics of short bridges [20,21]. Therefore, verified
The bridge deck and the cap-beam form the bridge superstructure, which are modeled using three-dimensional spine-line elements that trace the bridge’s alignment. These elements were
designated as elastic elements and were assigned with uncracked
section properties (i.e., prestresed concrete section properties).
The cap-beam was given arbitrarily (105) high torsional and inplane flexural stiffness values, because the cap-beam and the deck
are integrally constructed in the seed bridges. Superstructure elements were endowed with both translational and rotational mass
in order to capture the dynamic response accurately.
3.2. Abutment modeling
The abutment system comprises backwall, backfill, wingwalls,
shear keys, bearing pads, and usually a pile group that supports
the seat. Among these, only the longitudinal responses of the backfill (passive pressure) and the expansion joint, the transverse responses of the shear keys, and the vertical responses of the
bearing pads and the stemwall are considered explicitly in the models. The rest are omitted, because their contributions to the overall
response, and their effects on failure modes are deemed to be
insignificant.
Experimental data on lateral passive response of abutment
backfills are limited [23–26]. To the authors’ knowledge, there
are no data sets from controlled experiments on backfill response
for skewed abutments. Therefore, the model intuitively devised
and adopted here for response of skewed abutments is not experimentally validated. That said, the longitudinal response is modeled by using five nonlinear springs in series with gap elements
as shown in Fig. 4. The nonlinear springs and the gap elements represent the passive backfill response and the expansion joint,
respectively. The strength and initial stiffness of the soil springs
are obtained from recommendations provided in the Caltrans
SDC [13], which, in turn, were derived from large-scale abutment
testing [24,26].
For all abutment skew angles, the direction of the backfill passive pressure is assumed to be perpendicular to the backwall. Consequently, the forces tangential to the backwall—which would be
due to friction between the wall and the soil—are omitted here.
This is a conservative assumption, as those frictional forces would
always work to reduce deck rotation. The backfill springs are
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Sa (g)
Individual GM Spectrum-SN
Median Spectrum-SN
10
10
Individual GM Spectrum-SP
Median Spectrum-SP
0
10
-1
10
-1
10
0
10
1
Period (sec)
10
0
-1
10
-1
10
Sa (g)
10
10
-1
-1
10
0
10
1
Period (sec)
10
0
-1
10
-1
10
Sa (g)
10
10
-1
10
10
0
Period (sec)
10
1
Individual GM Spectrum-SP
Median Spectrum-SP
0
-1
0
Period (sec)
(b)
Individual GM Spectrum-SN
Median Spectrum-SN
10
1
Individual GM Spectrum-SP
Median Spectrum-SP
0
10
10
Period (sec)
(a)
Individual GM Spectrum-SN
Median Spectrum-SN
10
0
10
1
10
0
-1
10
-1
10
0
10
1
Period (sec)
(c)
Fig. 2. Response spectra of strike-normal (SN) and strike-parallel (SP) components of ground motion types: (a) ‘‘Soil-site,’’ (b) ‘‘Rock-site,’’ (c) ‘‘Pulse-like’’.
equally spaced and attached to a rigid bar representing the deck.
Properties of the five abutment nonlinear hyperbolic springs are
slightly different from each other, depending on their relative location to the obtuse angle between the backwall and the longitudinal/traffic direction (cf., point OBT in Fig. 4). The stiffness and
strength of these springs are assumed to increase linearly, as function of abutment skew angle and distance from point OBT. It is assumed that the volume of engineered backfill soil that can be
mobilized per unit wall-width in the event of backwall breakage
is larger, from point OBT towards the acute side (cf., point ACU in
Fig. 4). It is further assumed that this variation is linear in the simplest—and therefore the most reasonable—that can be adopted
without direct guidance from experimental observations.1 Finally,
it is postulated that the maximum stiffness/strength variation occurs
1
Incidentally, the abutment modeling assumptions are similar to those adopted in
[27] who used three-dimensional finite element simulations of a skew abutment [28]
to develop their simplified models.
for the largest skew angle (60°), and it is equal to 30%. Therefore, the
stiffness/strength variation factor b for a given the skew angle a can
be computed as in,
b ¼ 0:3
tan a
tan 60
ð1Þ
Multiple analyses conducted with different values of the maximum
variation between OBT and ACU springs (i.e., 30% in Eq. (1)) indicate
that the results are not highly sensitive to variations in b (less than
2% difference in the median of deck rotation of Bridge A when b is
varied from 0% to 60%). As such, the aforementioned modeling
assumptions are a reasonable (and arguably non-controversial)
extension to previous attempts [27,28].
The backwall and the shear keys are sacrificial elements that are
designed to break off to protect the bridge foundation against higher dynamic forces [13,21]. The shear key response is modeled with
a tri-linear backbone curve, which mimics the behavior observed
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Fig. 3. Generic model (a° skew) used for nonlinear time-history analyses.
Displ. (cm)
OBT
0
2
4
6
8
10
0
150
Elastic
Superstructure
α
Force (KN)
Rigid Beam
300
450
600
750
900
ACU
(a)
(b)
Fig. 4. Backfill soil springs: (a) configuration diagram, (b) backbone curve.
Bridge "A"
Bridge "B"
Bridge "C"
Force (KN)
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
Deformation (cm)
Fig. 5. Shear key for the seed bridges depicted in scale (right), and their force–deformation backbone curves (left).
in full-scale tests [29]. Fig. 5 shows these backbone curves for the
three seed bridges. The transverse stiffnesses of bearing pads are
omitted (i.e., they were considered to behave very flexible in the
transverse direction). Vertical response is modeled by two parallel
springs as shown in Fig. 6. The first spring represents the flexible
portion of elastomeric bearing pad in vertical direction (K1),
whereas the second represents vertical stiffness of the stemwall
and abutment embankment (K2).
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
bridges. The modeling technique used in this study is similar to
the approach employed by Cruz and Saiidi [32] that was verified
by a recent large-scale test. They simulated a large-scale four-span
bridge test in the University of Nevada, Reno, and showed a comparable correlation between seismic demands derived from analytical models and the experimental data.
F
K
2
K
1
4. Trend observations
D
Δ
Fig. 6. Vertical force–displacement backbone curve of the abutment (K1: the
elastomeric bearing pad stiffness, and K2: the stemwall and abutment embankment
stiffness).
3.3. Column-bent modeling
Progression of column yielding and damage is expected under
strong ground motions, and thus nonlinear fiber-based force-based
beam finite elements are used to represent the columns (Fig. 7). In order to achieve a more realistic representation of their responses, the
beam finite elements representing them are endowed with the ability to respond inelastically at every quadrature point. All fiber sections are assigned with the UniaxialMaterial model tag of
OpenSees [19]. Three different constitutive rules are used simultaneously within a cross-section: (i) confined concrete, (ii) unconfined
concrete, and (iii) steel rebar (Fig. 7). A forced-based element—viz.,
NonlinearBeamColumn—is used, which enables a more accurate
accounting of the moment distribution within the element [30]. A
single force-based element with 10 quadrature points is used per column; and this is usually deemed to provide adequate accuracy [19].
In order to model the portion of the column-bent embedded in the
superstructure, a rigid element is attached to the top of the nonlinear
beam-column element. The length of this rigid element is set equal to
the distance between the centroid of the soffit-flange of the superstructure box-girder and the column top.
Assumption on column base boundary condition can significantly affect estimates of bridge seismic demands obtained from
nonlinear time-history analysis. Based on information gathered
from structural drawings and common engineering practice, a
fixed base connection for a single-column bridge (‘‘Bridge A’’)
and a pinned base connection for multi-column bridges (‘‘Bridges
B and C’’) are assigned in bridge models used in this study. Including soil-pile (or foundation)-structure interaction increases the
flexibility of the substructure, which results in longer period of
vibration and larger estimates for deformation demands [31].
However, this research is focused on identifying trends in seismic
demands of skewed bridges due to variation in bridge geometric
parameters rather than accurate estimates of seismic demands.
Nevertheless, focus is maintained on accurate modeling of seattype abutment, which has a larger effect on behavior of skewed
The seed bridge models and their variations were systematically analyzed using the selected ground motion sets under multiple angles of incidence. In total, 43,200 time-history analyses were
conducted, and the results were scrutinized to identify dependencies, correlations and trends. To this end, two seismic response
parameters were obtained, which were the maximum planar deck
rotation (hrot), and the maximum total column drift ratio (hcol). Detailed investigation of simulated data suggested the necessity for
differentiating results into several groups based on three distinct
regimes of skew bridge behavior: bridge-collapse (according to
the collapse criteria discussed in Section 2.3), bridge survival with
shear key failure, and bridge survival with shear key survival.
In what follows, the effects of ground motion characteristics on
bridge response are discussed; and the use of an efficient ground
motion Intensity Measure, IM, to express IM-EDP curves is proposed (EDP denotes Engineering Demand Parameter). Because the
ground motion sets used in this study are not tuned towards a
target hazard level, it is necessary to interpret the results using
IM-EDP curves that are conditioned on no-collapse (henceforth
denoted simply as IM-EDP curves), and the probability of collapse
(given IM). Trends and dependencies between IM-EDP curves and
EDPs other than hrot and hcol may be found in [33].
4.1. Effects of ground motion characteristics and the selection of an
efficient intensity measure
Seismic response parameters of skewed bridges are significantly sensitive to the characteristics of applied seismic excitation;
higher values of response parameters and incidences of collapse
are observed for skewed bridges that experience pulse-like ground
motions. Fig. 8 displays the variation of hrot and hcol for no-collapse
cases as a function of ground motion peak acceleration PGAres
(maximum of the resultant of two SN and SP components) for all
ground motion sets, and for five variations of the abutment skew
angle. This figure is generated only for Bridge A with its original
column height and by assuming symmetric spans; but other variations in bridge geometrical properties display similar trends for
hrot and hcol (see [33]). In each plot, EDPs are discriminated into
two categories: based on whether the shear keys survived (circles
in Fig. 8) or lost resistance (squares) during seismic excitation.
These results clearly indicate that the probability of shear key failure is higher when the bridge experiences a pulse-like motion
(higher concentration of black squares) for any abutment skew
Rigid Element
Unconfined
Concrete
Confined
Concrete
A
A
Steel Rebar
Section A-A
Fig. 7. Column modeling scheme.
NonlinearBeamColumn
Element
143
Pulse-Like
PGAres (g)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
1
Soil-Site
0.1
Rock-Site
1
0.1 1
0.1
10
0.1 1
0.1
10
1
0.1 1
Shear key-survive
0.1 1
0.1
10
0.1 1
0.1
10
1
0.1 1
0.1
10
1
0.1 1
0.1
10
1
0.1
10
60°
1
1
0.1
10
1
0.1
0.1 1
1
0.1 1
45°
1
0.1
10
1
0.1
30°
0.1
10
10
0.1 1
10
0.1 1
10
1
0.1 1
0.1
10
1
0.1 1
0.1 1
1
0.1 1
0.1
10
Shear key-failure
θ rot
(radx10-3)
Pulse-Like
PGAres (g)
(a)
00°
15°
1
Soil-Site
0.1
Rock-Site
1
1
10
1
0.1
0.1
1
10
0.1
1
10
Shear key-survive
10
0.1
0.1
1
10
0.1
1
10
10
0.1
0.1
1
1
10
1
1
10
1
1
60°
1
1
1
1
45°
1
1
1
0.1
30°
0.1
10
0.1
1
10
1
10
1
1
10
1
1
0.1
0.1
1
1
10
0.1
1
10
θ col ( %)
Shear key-failure
(b)
Fig. 8. Sensitivity of response of Bridge A with original column height and symmetric spans to type of seismic excitation and abutment skew angle: (a) PGAres–hrot, (b) PGAres–hcol.
angle. Given the significance of pulse-like ground motions in
imposing high seismic demands compared to other types of ground
motions, the trend observations will henceforth be confined only
to those analyses carried out using the pulse-like ground motion
set. Results from the soil- and rock-site ground motion sets may
be found in [33].
We opted to use the peak resultant ground velocity (PGVres) as
the ground motion intensity measure for generating IM-EDP curves
in this study. PGVres is computed as the maximum of the SN and SP
components’ resultant velocity history. The efficiency of different
intensity measures were studied by monitoring the variations in
IM-EDP plots (similar to Fig. 8 but with different IMs) as well as dispersion of collapse fragility curves considering a wide range of IMs
as collapse-potential indicators. Other IMs that were considered
include spectral acceleration at the first mode period of strikenormal component, denoted as SaSN(T1), peak ground acceleration
of strike-normal component (PGASN), peak ground velocity of
strike-normal component (PGVSN), and peak resultant ground
acceleration (PGAres). Fig. 9 displays a sample collapse fragility
curve derived for Bridge A with original column height, symmetric
span, and 60° abutment skew angle, subjected to the pulse-like
ground motion set. A logistic regression was applied to the 240
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P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Prob. of Collapse
1
Collapse indicator
Lognormal fit
0.75
0.5
0.25
0
100
200
300
PGVres (cm/s)
Fig. 9. Development of the collapse fragility curve for Bridge A (symmetric span,
original column height, 60° abutment skew angle, pulse-like GM).
data points (resulting from 40 ground motion records and 6 angles
of incidence) for generating this collapse fragility curve. Data
points shown in Fig. 9 are pairs of PGVres and an indicator parameter, which is equal to ‘‘1’’ if the ground motion caused collapse, and
‘‘0’’ otherwise. Using this approach, the collapse fragility curves of
Colext-Symt.
Disper.
Colorig-Asymt.
Disper.
Colorig-Symt.
Disper.
Bridge A
Bridge B
Bridge C
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
15
30
45
60
0
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
15
30
45
60
0
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
15
30
45
60
0
Skew Angle
0
15
30
45
60
0
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0
15
30
45
60
0
Skew Angle
SaSN (T1)
15
30
0
45
60
0
PGA res
30
45
60
45
60
45
60
45
60
Skew Angle
0
15
30
Skew Angle
0
15
30
Sk
0
Skew Angle
PGA SN
15
0.5
0.5
0
Colext-Asymt.
Disper.
all bridges in bridge matrix were obtained along with the median
and dispersion values of fitted lognormal distributions.
Implication of PGVres being a superior IM compared to other IMs
may be deduced from Fig. 10, in which the dispersions of various candidate collapse-potential indicator IMs are presented. Fig. 10 displays
these collapse-potential IM dispersions for three bridges types (A–C)
and four combinations of column height and span arrangements—
namely, original column height with symmetric span (Colorig-Symt.),
original column height with asymmetric span (Colorig-Asymt.), extended column height with symmetric span (Colext-Symt.), and extended column height with asymmetric span (Colext-Asymt). In
each plot, the dispersion as a function of assumed abutment skew angle is shown. These results clearly indicate that the dispersion of collapse fragility curves obtained using PGVres as the collapse-potential
indicator is consistently smaller on average than those of other IMs
for Bridges B and C. A similar trend is observed for Bridge A; however,
PGAres exhibits a smaller dispersion compared to PGVres for cases with
extended column height. It is useful to acknowledge that no single IM
(i.e., no truncated/reduced model or parameter of the ground motion) by itself can capture all aspects of seismic hazard; nevertheless,
given the observed superiority of PGVres for most cases, it was used
for identifying collapse-potential and for obtaining the IM-EDP
curves throughout the rest of this study.
15
A
l
30
Skew Angle
PGVSN
PGVres
Fig. 10. Dispersions in collapse fragility curves obtained for the bridge matrix using the pulse-like ground motion set, and logistic regressions of data for various IMs.
145
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
4.2. Trends in the statistical parameters of the collapse fragility curves
Probability of collapse, given the level of ground motion intensity, is sensitive to the type of seismic excitation, abutment skew
angle, bridge column height, and span arrangement. Fragility
curves of all bridges in the bridge matrix were developed using
PGVres as the collapse-potential indicator, and using logistic regression as described in Section 4.1. Fig. 11 displays the median collapse capacity and the associated dispersion for all bridges
considered obtained by using the pulse-like ground motion set.
The rock- and soil-site ground motion sets had significantly fewer
instances of collapse, so these results are omitted here [33].
The effect of abutment skew angle on collapse-potential of
bridges is found to be a function of bridge type. For a single-column
two-span bridge (i.e., Bridge A) the collapse-potential is small at
zero skew angle (note the high median collapse-potential at zero
skew angle for Bridge A in Fig. 11). The median collapse-potential
drops for higher skew angles, exhibiting a higher probability of collapse at the same level of ground motion intensity. Bridge C, in contrast, does not show a similar correlation between the abutment
skew angle and the collapse-potential. This phenomenon can arguably be attributed to the type of failure resulting from the columnbase boundary conditions. Because Bridges A and C were modeled
with fixed- and pinned-base boundary conditions, respectively,
there is a high moment demand at the top of the columns of Bridge
C, regardless of the abutment skew angle, which can rapidly exhaust column load-resistance capacity and lead to collapse. In contrast, the single column in Bridge A with a fixed-base can better
handle the moment demand at zero skew angle. However, for larger
Col orig-Symt.
Bridge A
Col orig-Asymt.
The strength of shear key has a large influence on the bridge deck
rotation. For larger skew angles, probability of shear key failure—
hence, the probability of having a large deck rotation—increases.
Fig. 12a–c show the variation of maximum deck rotation, hrot, for
bridges A, B, and C, respectively, as a function of PGVres obtained from
the pulse-like ground motion set. These figures display the sensitivity of hrot to five variations in abutment skew angle, and to various
combinations of span configuration and column height. In each plot,
the cases where shear keys survived (circles in Fig. 12) or have lost
resistance (squares) during seismic excitation are marked. Data
points representing ground motions for which the bridge collapsed
are not shown. Fig. 12 also reveals that higher hrot is expected once
a shear key fails. Moreover, span arrangements appear to have less
effect compared to column height on the deck rotation demand.
Bridge B
Bridge C
200
200
150
150
150
100
100
100
50
50
50
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
Col ext -Symt.
4.3. Trends in the deck rotation values
200
0
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
Col ext -Asymt.
abutment skew angles, the bridge becomes susceptible to deck
rotation, which may increase its collapse-potential.
Certain combinations of abutment skew angle, column height,
and span arrangement can create a condition at which the bridge
is more susceptible to deck rotation; and therefore, probability of
collapse is increased, and vice versa. Fig. 11 shows this phenomenon for Bridge B: the median collapse-potential is decreased for
skew angles between 0° and 15°, once original column heights
and symmetric span is considered. However, for same column size
and asymmetric span, the median collapse capacity is increased.
Similar trends can be seen for Bridge A.
0
15
30
45
60
0
0
15
30
45
60
0
200
200
200
150
150
150
100
100
100
50
50
50
0
0
0
15
30
Skew Angle
45
60
0
15
30
Skew Angle
45
60
0
0
15
30
45
60
0
15
30
45
60
0
15
30
45
60
0
15
30
45
60
Skew Angle
Fig. 11. Median and dispersion of collapse capacity, expressed in terms of PGVres for the bridge matrix obtained using the pulse-like ground motion set.
PGVres (cm/s)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
146
00°
15°
100
30°
100
10
100
10
0.1 1
0.1 1
10
100
10
100
0.1 1
10
100
10
100
0.1 1
10
100
10
10
0.1 1
10
10
10
0.1 1
10
10
0.1 1
10
100
10
10
0.1 1
100
100
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
10
10
Shear key-survive
10
100
10
0.1 1
10
10
0.1 1
10
10
10
0.1 1
100
100
10
0.1 1
10
10
10
100
10
0.1 1
100
10
0.1 1
60°
100
10
10
100
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
(a)
00°
15°
100
30°
100
10
100
10
0.1 1
0.1 1
10
100
10
100
0.1 1
10
100
10
100
0.1 1
10
100
10
10
Shear key-survive
10
10
10
10
10
10
0.1 1
10
10
0.1 1
10
100
10
0.1 1
0.1 1
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
100
10
0.1 1
10
10
0.1 1
10
10
10
0.1 1
100
100
10
0.1 1
10
10
10
100
10
0.1 1
100
10
0.1 1
60°
100
10
10
100
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
(b)
Fig. 12. PGVres–hrot conditioned on no-collapse plots obtained for the bridge matrix using the pulse-like ground motion set: (a) Bridge A, (b) Bridge B, (c) Bridge C.
The incidence of shear key failure is higher for Bridge A (a twospan single-column-bent bridge) than Bridges B or C; and its seismic response is affected more by the abutment behavior that those
of other bridges. In Fig. 12, the two cases of shear key survival and
failure have seemingly divided the IM-EDP plots into two distinct
regions. However, in Bridge B, these regions overlap somewhat,
suggesting that the incidences of collapse in Bridge B are not dominated by excessive deck rotations, and that excessive translations
play a significant role.
Deck rotations can be detected even for symmetrical bridges
that have 0° skew (Fig. 12). These rotations are caused by the columns that become asymmetrically damaged (concrete losing
strength in tension, etc.) during the course of strong motions.
Bridge columns and shear keys experience loads exceeding their
yield capacity, and behave nonlinearly. This destroys the initial
symmetry in geometry and boundary conditions, and leads to the
initiation of deck rotations and in some cases, ultimately, to
collapse.
147
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
100
30°
100
10
100
10
0.1 1
10
100
10
10
100
10
100
10
10
100
10
100
10
10
Shear key-survive
10
10
10
10
10
10
0.1 1
10
10
0.1 1
10
100
10
0.1 1
0.1 1
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
0.1 1
100
100
100
10
0.1 1
10
10
0.1 1
10
0.1 1
10
0.1 1
100
100
10
0.1 1
10
10
0.1 1
100
10
0.1 1
100
10
0.1 1
60°
100
10
0.1 1
100
10
45°
10
0.1 1
10
0.1 1
10
θ rot
(radx10 -3)
Shear key-failure
(c)
Fig. 12 (continued)
4.4. Trends in column drift ratios
For a single-column bridge, the column drift ratio is a response
parameter that depends on the translation of the bridge deck in
two orthogonal directions along the ground motion record is applied. However, in multi-column-bent skewed bridges, the rotation
of the deck results in an additional translation of the column-top;
and this can potentially increase the maximum column drift ratio
for these bridges compared to their zero-skew counterparts.
Fig. 13 clearly displays this effect where PGVres and hcol data pairs
are shown for the pulse-like ground motion set, only for cases
where collapse has not occurred. The first row of plots in
Fig. 13a, which is for Bridge A with symmetric spans and original
column heights, reveals that hcol is not sensitive to abutment skew
angle. However, first row of Fig. 13b, which is for Bridge B, indicates that hcol increases with the skew angle. Similar trends may
be observed with a one-to-one comparison between the plots in
Figs. 12 and 13; the effect of deck rotation in increasing the column
drift ratio for higher skew angles in Bridge B (multi-column-bent)
or lack of it for Bridge A (single-column-bent) is evident.
hcol for Bridge C has a similar behavior to that for Bridge A, despite the fact that Bridge C is a multi-column-bent three-span
bridge (Fig. 13c). This behavior may be attributed to the torsional
rigidity of Bridge C, in which the two multi-column-bents provide
a relatively high stiffness against transverse rotation. However, given the modeling assumptions made in this study whereby the column bases in a multi-column-bent are modeled with pinned
connections, there is sudden drop in stiffness once the moment
capacity of columns exceeds yield moment capacity, at which
point, the bridge is likely to experience collapse.
Two less catastrophic damage levels of hcol, concrete cover spalling and buckling of longitudinal bars, are also studied. Berry and
Eberhard [34] suggested that the onset of cover spalling can represent the first flexural damage state to concrete columns and the
onset of longitudinal bar buckling indicates the significant (or
functionary) damage state to concrete columns. Based on the
regression analysis of experimental data, they proposed two formulations for the predefined damage states. Table 3 represents
the corresponding numerical evaluation for hcol, the two damage
states. In Fig. 13, three distinct regions are specified by two dotted
lines. The blue line (dark gray in black and white print) represents
hcol-spall, and the red line (light gray in black and white print) indicates hcol-bb. For all bridges, the shear key failure status plays a significant role in damage state evaluation of the column-bent drift
ratio.
5. Conclusions
In this study, three actual concrete box-girder bridges (of California) with seat-type abutments were used as seed models; and
a matrix of bridge models were created by varying their abutment
skew angles, column heights and span arrangements. These models were subjected to three sets of forty ground motion records
through nonlinear time-history analyses. These ground motions
were not selected for a target hazard level; but rather they are representative of the types of expected seismic excitation in California. As such, response trends were studied through IM-EDP and
collapse fragility curves.
The results indicated that ground motions with high velocity
pulses induce higher seismic demands, at least for the bridges
studied. Collapse-potential appeared to have an inverse relation
with the abutment skew angle. Particularly, the shear key failure
status has a major effect on induced demands in single-column
bridges. This indicator has a significant influence on the planar
rotation of the deck; i.e., larger deck rotations are expected for
when shear keys fail. The global torsional stiffness, abutment skew,
and column elevation of skewed bridges also significantly affect
the seismic response of skew bridges. On the other hand, seismic
response of a skewed bridge is not sensitive to the span arrangement, at least for the configurations considered in this study.
The results also indicated that under the same seismic excitation, demands for bridges with skewed abutments, such as deck
PGVres (cm/s)
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
148
00°
15°
100
30°
100
10
100
10
1
1
10
100
10
100
10
100
10
100
10
1
1
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
10
Shear key-survive
10
10
10
1
10
1
1
100
100
10
1
100
10
10
100
10
10
10
1
10
1
10
1
100
10
1
100
10
10
100
10
100
10
1
10
1
60°
100
10
10
100
45°
10
1
10
1
10
θ col (%)
Shear key-failure
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
(a)
00°
15°
100
30°
100
10
100
10
1
1
10
100
10
100
10
100
10
100
10
1
1
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
10
Shear key-survive
10
10
10
1
10
1
1
100
100
10
1
100
10
10
100
10
10
10
1
10
1
10
1
100
10
1
100
10
10
100
10
100
10
1
10
1
60°
100
10
10
100
45°
10
1
10
1
10
θ col (%)
Shear key-failure
(b)
Fig. 13. PGVres–hcol conditioned on no-collapse plots obtained for the bridge matrix using the pulse-like ground motion set: (a) Bridge A, (b) Bridge B, (c) Bridge C.
Table 3
Column-bent drift ratios for two damage states.
Column-bent damage states
Bridge A
Bridge B
Bridge C
Concrete spalling, hcol-spall (%)
Long bar buckling, hcol-bb (%)
1.8
5.0
2.1
5.2
2.4
7.9
rotation and column drift ratio, are higher than those for bridges
with straight abutments. It was also clear that the shear key
strength has a significant influence on the overall bridge behavior,
and shear keys can limit deck rotations and suppress the potential
amplification in column drift ratios resulting from deck rotation.
Modeling assumptions adopted in this study were based on earlier component-scale, verification and validation studies with certain exceptions. First among these are modeling assumptions for
the passive response of the abutment backfill. While purely
numerical studies and validated models for straight abutments
suggest that this is a reasonable model, experimental studies are
149
PGVres (cm/s)
Colext-Asymt. Colext-Symt. Colorig-Asymt. Colorig-Symt.
P. Kaviani et al. / Engineering Structures 45 (2012) 137–150
00°
15°
100
30°
100
10
100
10
1
10
100
10
10
100
10
100
10
10
100
10
100
10
Shear key-survive
10
10
10
10
10
10
10
1
10
10
1
10
100
10
1
1
100
100
10
1
10
10
1
10
1
1
100
100
100
10
1
10
10
1
10
1
10
1
100
100
10
1
10
10
1
100
10
1
100
10
1
60°
100
10
1
100
10
45°
10
1
10
1
10
θcol (%)
Shear key-failure
(c)
Fig. 13 (continued)
clearly needed in this area. Similarly, component-scale tests and
high-fidelity numerical simulations should be conducted to narrow
the uncertainties associated with assigning the column base
boundary conditions, which may significantly enhance seismic demand estimation.
Acknowledgements
This study is based on work supported by the PEER Transportation Research Program under Grant No. UCLA-45782. This financial
support is gratefully acknowledged. Any opinions, findings, and
conclusions or recommendations expressed in this paper are those
of the authors and do not necessarily reflect the views of sponsors.
The authors thankfully acknowledge the valuable technical discussion held with Dr. Anoosh Shamsabadi of California Department of
Transportation, and Dr. Majid Sarraf of Parsons Corp. throughout
this study.
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