Engineering Structures 150 (2017) 176–189

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Engineering Structures

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Comparison of curved prestressed concrete bridge population responsebetween area and spine modeling approaches toward efficient seismicvulnerability analysis

http://dx.doi.org/10.1016/j.engstruct.2017.07.0330141-0296/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: junwon.seo@sdstate.edu (J. Seo), luke.rogers@hdrinc.com

(L.P. Rogers).1 Current address: Bridge E.I.T, HDR, 701 Xenia Ave. South, Suite 600, Minneapolis,

MN 55416, United States.

Junwon Seo ⇑, Luke P. Rogers 1

Department of Civil and Environmental Engineering, South Dakota State University, United States

a r t i c l e i n f o

Article history:Received 23 September 2016Revised 13 June 2017Accepted 12 July 2017

Keywords:Curved precast-prestressed concrete bridgeSpine modelArea modelSeismic analysisFragilityEfficiencyGround motions

a b s t r a c t

This paper presents two finite element modeling approaches for seismic evaluation of curved precast-prestressed concrete (PSC) I-girder bridges and compares the results in a statistical and graphical manner.These approaches, including area and spine models, were applied to a simply-supported, curved PSC I-girder bridge under an ensemble of 3D synthetic ground motions. Along with performing non-linear timehistory (NLTH) analyses of the bridge to capture its separate seismic response, the efficiency of eachapproach was evaluated with respect to execution time and each was compared. This comparison revealsthat the seismic responses that were computed at low computational cost from the spine approach arereasonably analogous to those from the area model. Seismic fragility curves of a portfolio of curvedPSC bridges using the spine approach are also created to assess their vulnerability and then comparedto those gained from past studies. This comparison shows a reasonable agreement for the PSC portfolio.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Bridges are considered the most vulnerable elements in thenational transportation networks of the United States during anearthquake [5,9,2]. In case of bridges designed without sufficientseismic detailing, it has been reported from past earthquakes[7,19] that there has been significant damage on particular compo-nents of the bridges that were observed across regions in the Uni-ted States. Describing the probability of earthquake-inducedbridges experiencing different damage states has been vital to treatuncertainty related to their characteristics and incorporate ran-domness in seismic excitations. A fragility curve has been com-monly used to quantify the probability of bridge damage orfailure and vulnerability due to seismic loadings [33,20,23].

The majority of studies for seismic vulnerability assessments ofbridges have focused on regular bridges in the form of fragilitycurves [7,19,21,22,11,12,14]. However, there exist a significantnumber of existing bridges with irregular configurations nation-wide. Since irregular configuration factors in a bridge unfavorably

affect its seismic behavior and vulnerability [15], pertinent bridgedesign codes, such as the American Association of State Highwayand Transportation Officials (AASHTO) Load and Resistance FactorDesign (LRFD) Seismic Bridge Design [1] have dealt with require-ments associated with the irregularities. Curved precast-prestressed concrete (PSC) I-girder bridges that have beenregarded as one of the representative irregular bridge types havefrequently been constructed in both seismically active and moder-ate seismic regions as the complexity of traffic flow transitions andthe efficiency of transportation network increases.

As stated above, several seismic vulnerability studies [7,18], forstraight PSC I-girder bridges and their retrofits [21,22] to seismicexcitations have been performed. Specifically, Choi et al. [7] andNielson and DesRoches [18] examined seismic response of variousstraight PSC I-girder bridge configurations and types. It was foundthat simply supported straight PSC I-girder bridges exhibited seis-mic fragility and potential damage under moderate groundmotions. Padgett and DesRoches [23] proposed a methodologyfor the establishment of analytical fragility curves in part forstraight PSC I-girder bridges with different retrofit measures.Though these studies have successfully assessed seismic vulnera-bility of straight PSC I-girder bridges, these findings are not appli-cable to curved PSC I-girder bridges due to their unique featuresand system mechanisms under seismic excitations.

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 177

Recent studies [29,28,30–34,3] demonstrated that steel bridgeswith irregular configuration factors such as radius of curvaturehave more significant damage due to excessive torsional responsethan regular bridges. For example, Seo and Linzell [29–31] haveattempted to examine the seismic response and vulnerability ofcurved steel girder bridges using conventional three dimensional(3D) finite element models. These models required more complexcomputational solutions coupled with risk analysis for inherentrandomness in earthquakes in conjunction with uncertaintyrelated to irregular characteristics than straight ones. It wasdemonstrated that each curved steel bridge has distinctive charac-teristics, leading to different seismic responses affecting its vulner-ability. The significant finding from the work [30,31] was that thecurved steel bridges produced fairly higher seismic vulnerabilitiesthan those for straight ones due to their curvatures associated withthe other parameters. Based upon the literature review, unfortu-nately significant studies related to seismic evaluation and vulner-ability of curved PSC I-girder bridges have not been found. Tobetter understand seismic behavior and vulnerability of curvedPSC bridges at reasonably low computational cost using 3D finiteelement models, the need of developing a new seismic modelingtechnique for such bridge populations that enables its efficientseismic vulnerability assessment with variability in their bridgecharacteristics is of significance.

The current study focuses on developing a finite element anal-ysis modeling technique for seismic vulnerability assessment ofPSC I-girder bridges in a more efficient manner. To accomplishthe goal of this study, two 3D finite element modeling approaches,including a spine model and an area model, were developed andapplied to a curved two-lane, simply supported PSC I-girder bridge.The responses of both models of the bridge were compared statis-tically and graphically. The bridge was designed using currentAASHTO codes [1] and PCI (Precast/Prestressed Concrete Institute)practices [24]. Nominal materials and components for bridge con-struction were selected for use in the design. The 3D models wereloaded with an ensemble of synthetic ground motions having twohorizontal and one vertical acceleration components. These groundmotions varied in magnitude and intensity to make it possible torun a broad spectrum nonlinear time history analysis of seismicresponse. The effect of model approach on the seismic responseis also examined, along with comparing efficiency of the analysis.To validate the possibility of creating seismic fragility curves,component- and system-level fragility curves for a suite of 15curved simply supported PSC I-girder bridges that were alsodesigned following the AASHTO and PCI specifications were devel-oped using a spline model and compared to those developed inpast studies of straight ones.

2. Structural design of curved PSC girder I-girder bridge

A curved PSC I-girder bridgewas initially chosen and designed toexamine its seismic response and susceptibility and to explore thedifferences between area and spinemodels. The bridge design com-plied with the [1] LRFD Bridge Design Specifications [1] followingthe procedure outlined by the Precast/Prestressed Concrete Insti-tute Bridge Design Manual [24]. It was assumed that the bridge islocated in the central United States that would be considered Seis-mic Zone 1 where design considerations for seismic performanceare minimal. These requirements are described in the AASHTOSpecifications. The abutment type for these bridges is the seat-type abutment. Integral abutments are generally not favorable forcurved bridges [16]. Elastomeric pads each with two embeddedsteel dowelswere selected for use in both fixed and expansion bear-ings. The AASHTO I-girder type III was used based on establisheddesign requirements. Each girder is prestressed with strands of

tensile strength of 1860 MPa and is 13 mm in diameter. Bothstraight and harped strands are employed. The concrete cast inplace deck is 200 mm thick.

The bridge has the following attributes: the span was 23 m andwidth was 9.8 m so that four girders were deemed appropriate; thegirders used 24 straight tendons and 8 harped tendons; the radiusof curvature was 402 m; and an offset of 160 mmwas developed atthe mid-span centerline as a result of the curvature and spanlength. This with its design schematic can be illustrated in Fig. 1.The offset is the distance between the centerline and the arc chordof curvature at mid-span. Eq. (1) will determine the offset asfollows:

O ¼ L2

8Rð1Þ

where O, L, and R is the offset, span length, and radius of curvature,respectively. Concrete cast-in-place diaphragms were 160 mm thickand placed at the midspan and endspan. The seat type abutmentwas anchored by 8 cast-in-place concrete piles of 400 mm in diam-eter. A 75 mm expansion gap was located at each abutment-deckinterface.

3. Finite element modeling approach

Two modeling approaches, encompassing the area and spinemodels, were presented on the bridge type using the program CSI-Bridge [8]. It is important to note that area and spine modeling is ageneral term that can describe many different models. The modelsdescribed in this study were specific to the finite element modelingapproaches used for the seismic response and fragility investiga-tion of curved PSC bridges and were categorized into area andspine as appropriate. The spine modeling approach is one thatmodels a superstructure with beam elements, while the areamodel approach use plate elements or shell elements to idealizeit. The following subsections describe each approach for the super-structure and the modeling for the substructure with bearings forboth approaches.

3.1. Area model for superstructure

The area modeling approach is the more complex of the twoapproaches in the current study. The superstructure model is con-structed using shell and frame elements. Prestress tendons repre-senting the Gr. 270, 7-wire strands were modeled using frameelements located within each girder. Tendon prestress force is1100 MPa after all losses, including elastic shortening, steel relax-ation, creep, and shrinkage. Both harped and straight tendons weremodeled and placed in the correct position relative to the girderaccording to the structural design.

Each girder was represented by frame elements with the crosssection of a Type III girder that span between the supports. Theconcrete compressive strength of the girder is 48 MPa. Shear rein-forcement consists of #10 (#3 imperial) stirrups that were placedappropriately according to structural analysis. These frame ele-ments were assigned the correct cross section and material proper-ties of the girders specified during the designing process. Thebridge deck was idealized using shell elements. Maximum lengthof a particular shell element is 3.05 m. Increasing this lengthdecreases the number of joints and vice versa. The deck sectionshell elements represent a 200 mm slab thickness consisting of27 MPa concrete. Note that a shell elements are three or four-node area objects used to model membrane and plate-bendingbehavior and are useful for simulating bridge deck systems [8].

Diaphragms were also modeled using shell elements with cor-ner nodes at adjacent girder frame elements; thus, with four gird-

(a) Top view

(b) Elevation

(c) Cross-section at mid-span

(d) Girder Detail

At Midspan At Endspan

Fig. 1. Bridge plan and details.

178 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 179

ers for the bridge, there are three diaphragm shell elements, oneplaced within each of the spaces between girders. The position ofthe diaphragms was determined according to the designed bridgeand placed accordingly to connect adjacent girders. A single200 mm thick concrete diaphragm was used on this bridge and isplaced at the midspan. Fig. 2(a) shows a plan view of the areamodel of the superstructure, and Fig. 2(c) illustrates an elevationview of the girder with tendons in this model.

3.2. Spine model for superstructure

The spine model is the most common approach used in paststudies of seismic vulnerability of bridges [6,17,27]. This approachhas less complexity compared to the area one but at the cost ofhaving fewer joints to measure responses. The spine model forthe superstructure is primarily made up of frame elements. Theentire bridge deck and all of the girders are modeled using a singleframe element; this portion of the model is the most dissimilarfrom the area model. The spline modeling approach computesframe element properties such as stiffness by identifying the mate-rial properties, dimensions, and locations of girders and deck sec-tions. As a result of the superstructure modeling identification,the element has attributes such as axial stiffness, bending stiffness,torsional stiffness, and mass similar to that of the area model. Itshould be noted that the tendons were modeled in the same wayof the area model. This modeling approach greatly reduces analysistime and computational expense due to the fewer number ofjoints. A disadvantage of spine models is the limited analysis ofmembrane effects as compared to area models. Fig. 2(b) displaysthe plan view of spine model superstructure.

3.3. Model for substructure with bearings

Substructure components for both approaches were modeledusing links representing springs and frames available in the CSIprogram. A schematic of the bridge showing the bearing, founda-tion, and abutment can be seen in Fig. 2(c). The detailed substruc-ture model can be seen in Fig. 2(d). All the substructurecomponents with the bearings are the only elements that weremodeled in the same way between both area and spine models.

The bearings were represented by two separate spring elementslocated at the end of each girder. One spring was used for the elas-tomeric pad and one for the embedded steel dowels. Appropriatevalues for horizontal bearing stiffness were obtained from equa-tions presented in past studies [6,7,17,18]. The 25 mm thick elas-tomeric pad effective stiffness was determined to be 8.2 kN/mm.An initial stiffness for the steel dowels within the bearing is92 kN/mm. Stiffness for the vertical axis on the bearing allowedfor free movement upward and restricted movement below theresting position. This was modeled using gap elements availablein the program. This simulates no vertical restraint and allowsthe superstructure to lift-off from the bearing and abutment. Inthis way, actual conditions were replicated where the self-weightof the superstructure is the only significant resistance to verticallift-off from the support. Fixed bearings have a gap of 5 mm beforethe dowel is activated while the expansion bearings have a gap of25 mm before the dowel is activated. Alongside the bearings,impact elements were modeled using multi-linear springs to sim-ulate the impact of the superstructure with the abutment backwallduring seismic events. A 75 mm expansion gap separates the abut-ment from the superstructure. When this gap closes, impact occursand the spring is activated. Values for these springs were deter-mined using formulas from past research [17].

The foundation piles were modeled using spring elements. Stiff-ness was obtained from past studies for cast-in-place concretepiles [7,18,4]. An effective stiffness of 7 kN/mmwas adopted in this

study for the piles. Foundation springs under the abutment weremodeled accounting for both passive and active earth resistancevia use of multi-linear springs. This was achieved by definingtwo springs set in parallel, one for the passive pressure and theother for the piles. Passive resistance is encountered when theabutment pushes into the backfill. Passive resistance was calcu-lated using techniques from previous research [6,17]. The initialstiffness of 23 kN/mm for passive resistance was obtained. Passiveresistance increases as deformation increases; thus, a multi-linearspring was used to model this behavior. Active resistance is pro-duced by the foundation piles. In the transverse direction, the pileresistance is the only factor that should be accounted for. The effectof wing walls was ignored as in past studies [7,18].

The abutments were modeled using frame elements havingappropriate dimensions that were determined from the structuraldesign. Nominal material properties of the concrete were assignedto the abutment frame elements in the model. The compressivestrength of the concrete used in the abutment is 27 MPa. Main flex-ural reinforcement of the abutment consists of eight #22 bars(#7 bars imperial). Depth of the seat area of the abutment is910 mm. Height of the backwall is 1.37 m to accommodate theheight of the Type III girder and bearing.

4. Seismic analysis procedure

Seismic analysis procedure used for this study is detailed in thefollowing subsections. This consists of the selection of syntheticground motions and non-linear time history analysis (NLTH) forboth modeling approaches.

4.1. Synthetic ground motions

Since useful and widely varying real ground motion time histo-ries are not readily available, 300 synthetic ground motions acrossthe regions of interest, including the Central United States such asMemphis, were obtained from the University of New York at Buf-falo [10]. As shown in Fig. 3 for a Peak Ground Acceleration(PGA) histogram for the synthetic ground motion, the groundmotion pool has a broad spectrum of PGA values, and each includestransverse, longitudinal, and vertical acceleration components.

An approach of ground motion application used for this studywas decided based upon the recommendations in the past work[29–31]. It was recommended that the horizontal ground motions,including longitudinal and transverse components, be applied tocurved bridges. Note that past studies [7,35] used ground motionscontaining only the horizontally longitudinal and transverseground motions. The seismic fragility data of the bridges used inthe past work [6] was directly compared to those from this study.

Additionally, in the current study, it was preferred to incorpo-rate the vertical accelerations during the nonlinear time historyanalysis since curved bridges have varying geometry that couldpotentially exaggerate the effects of vertical ground motions. Nineground motions covering a wide array of PGAs were randomlyselected for use in this study. These ground motions had PGAranges of 0.1 g (minimum value), 0.5 g (median value), and 1.0 g(maximum value). Fig. 4 provides the spectral acceleration graphsrepresenting each components of the ground motions. Fig. 4(a)shows the spectral acceleration verse period for the longitudinal(x) acceleration where a maximum of 3.6 g, a minimum of 0.4 g,and a median of 1.75 g is shown. Fig. 4(b) displays the accelera-tions for the transverse (y) direction. Here, the maximum spectralacceleration is 3.8 g, a minimum of 0.45 g, and the median of 1.7 g.Lastly, in Fig. 4(c), the vertical (z) spectral accelerations are shownwhere the maximum is 1.95 g, a minimum of 0.5 g, and a median of1.6 g.

(a)

(b)

(c)

Beam-Column Frame Element

Typ. Girder line

X

Y

Forc

e (k

N)

Displacement (mm)

Fixed Steel Dowel

k = 93 kN/mm Forc

e(kN

)

Displacement (mm)

Expansion Steel Dowel

k = 93 kN/mm

Forc

e (k

N)

Displacement (mm)

Pile

k1 = 10.9 kN/mm

k2 = 2.1 kN/mm

Forc

e (k

N)

Displacement (mm)

Passive Soil Resistance

k1 = 23 kN/mmk2 = 3.9 kN/mm

k3 = 1.6 kN/mm

Forc

e (k

N)

Displacement (mm)

Elastomeric Pad

k = 5.4 kN/mm

Rigid LinkTendon ElementsDetail (d) and (e).

X

Z

Shell Element

X

Y

(d)

X

Z

Fig. 2. 3D finite element model details: (a) area model superstructure plan view; (b) spline model superstructure plan view; (c) elevation view with spring properties; and (d)model substructure detail.

180 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

(a)

(b)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

Spec

tral

Acc

eler

a�on

(g)

Period (s)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

Spec

tral

Acc

eler

a�on

(g)

Period (s)

(c)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

Spec

tral

Acc

eler

a�on

(g)

Period (s)

Fig. 4. Spectral acceleration of selected ground motions in different orthogonaldirections: (a) longitudinal direction; (b) transverse direction; and (c) verticaldirection.

Fig. 3. PGA histogram for the ground motions.

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 181

4.2. Non-linear time history analysis

Non-linear time history analysis was used to capture seismicresponses of the two modeling approaches applied to the bridge.Although this seismic analysis technique tends to be the mostcomputationally expensive, this analysis is considered one of themost reliable methodologies available in earthquake engineering[33]. As such, there have been many researchers that have used amethodology based on NLTH analysis to determine nonlinearbridge responses resulting from seismic excitations. The NLTHanalysis procedure is outlined in Fig. 5. This procedure is done anumber of times for each bridge, once for each ground motion timehistory. The two modeling approaches were considered and theground motions are paired with the bridge models and a non-linear time history analysis was performed. For each simulation,the peak seismic responses for key bridge components such asbearing deformation were collected. In addition to peak responses,the complete time history of loading and deformation of certaincomponents was extracted from the results to create hysteresis.

5. Seismic analysis results

The efficiency of each approach that was evaluated with respectto execution time is detailed herein. Seismic responses of criticalcomponents were captured NLTH analyses of the bridge betweenthe two approaches are also summarized in the following subsec-tions. Included in the critical components were the bearings, gird-ers, and abutments.

5.1. Analysis efficiency

The main advantage of the spine model over the area model isthe reduction in complexity. This analysis was performed withall six degrees of freedom (DOFs) enabled, although there is theoption to disable any number of them. As can be seen in Table 1,the number of joints in the spine model is 33% lower than in thearea model. This is the core factor that affects other factors andultimately determines the number of DOFs and equations of equi-librium. As noted, the reduction in frame elements by 46% from thearea to spine model is a direct result of the lack of shell elementswithin the spine model. Computational time is the result of theprogram creating and solving matrices based on the constraintsand DOFs. Due to the lower element and joint count, the numberof constraints is lower in the spine model. This also lowers thenumber of constraint equations by 24% and lowers the MasterDOF significantly by 76% from the area model to the spine model.Even after DOF reductions due to coupled constraints/restraints,the final Constraint Master DOF is 68% lower in the spine modelthan in the area model. The greatest benefit that can be seen when

using a spine model is the lower number of equilibrium equations.From the area model to the spine model, the reduction is on theorder of 48%. Similarly, the reduction in nonzero stiffness termsis around 46%.

The aforementioned reduction in number of joints, elements,constraints, and DOFs ultimately reduces computational time byreducing the number of equilibrium equations. This reduction istime on the order of 7% for this size of seismic analysis. Thecomputer used for this analysis utilized a 32-bit operating system,

Fig. 6. Bearing displacements: (a) vertical; (b) transv

Time (sec)

Abutment

Bearing

Girder

Time (sec)

Acc

eler

atio

n (g

)

Fig. 5. Non-linear time history analysis procedure.

Table 1Number of joints, elements, and DOFs for area versus spine models.

Area Spine PercentDifference (%)

Number of Joints 637 424 40Number of Joints with Restraints 48 48 0Number of Frame/Cable/Tendons 606 330 59Number of Shells 69 0 200Number of Link/Supports 216 219 1.3Number of Constraints 876 850 3.0Number of Coupled Constraint Equations 1096 836 27Constraint Master DOF before Reduction 1264 306 120Coupled Constraint/Restraint Master DOF 160 12 170Constraint Master DOF after Reduction 1104 294 120Total Number of Equilibrium Equations 2598 1344 64Number of Non-Zero Stiffness Terms 60,045 32,460 60Analysis Time (min:sec) 46:01 43:03 7

182 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

4 GB of ram, and a 3.6 GHz dual-core processor. These specificationsdirectly impact analysis speed which is timed by the program CSi-Bridge during an analysis run. The size of the analysis performed forthe designed bridge would be considered small in comparison totypical analysis with larger bridge models andmore groundmotiontime histories as in past studies of bridge fragility [6,17,27]. Due toincreased complexity of larger models, it can be expected that thetime savings when using the spine model over the area modelwould be significant for seismic vulnerability assessment of a num-ber of bridges with different structural characteristics.

5.2. Seismic responses of critical components

5.2.1. BearingsThe elastomeric bearings, both fixed and expansion types,

undergo displacements up to 33 mm as a result of the ground

erse; (c) longitudinal; and (d) percent difference.

Fig. 7. Displacements at midspan: (a) vertical; (b) transverse; (c) longitudinal; and (d) percent difference.

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 183

motions. These displacements are relative, meaning the differencebetween the bearing spring element ends that connect the girderto the abutment. In Fig. 6, it can be seen that the difference inresults between the two model approaches varies depending onthe direction of displacement. Fig. 6(a) displays the verticaldeformation where spine model reaches 2 mm while the areamodel reaches 1 mm. Fig. 6(b) shows the transverse displacements.These are the most similar across model approaches where themaximum reached is around 11 mm for both model approaches.Fig. 6(c) shows the longitudinal movements where the areamodel reaches a greater displacement at 33 mm verses the spinemodel at 25 mm. In sum, the percent differences are displayed inFig. 6(d). It can be seen that the greatest difference lies in thevertical direction at an average of about 90%, seconded by thelongitudinal direction at 40%. The transverse direction is the mostconsistent between models with a percent difference less than 5%.Discrepancies in bearing displacements between the approachesare caused by overlapped shell elements in the deck section ofthe area model.

5.2.2. GirderGlobal displacements were measured at the mid-span of

girder 2 on the centerline. Fig. 7 displays the results for allcomponent directions at the mid-span between the two modelapproaches. Fig. 7(a) shows the vertical displacements wherethe area model reaches a maximum of 7 mm compared to the6.5 mm of the spine model. Fig. 7(b) shows movement in thetransverse direction where the results are almost identicalbetween model approaches, a maximum of about 48 mm isreached. In Fig. 7(c), the longitudinal displacements are shown.The area model reaches 38 mm while the area model reaches30 mm. Finally, in Fig. 7(d) the percent differences are displayedshowing that the vertical and transverse directions are very con-sistent between modeling approaches with a percent differenceof less than 5%. The longitudinal average difference is around40%. Again, the discrepancy between models is due to the over-lapped shell elements in the deck causing a higher mass thanthe spine model.

5.2.3. AbutmentsThe abutment displacements displayed are the average of both

abutments. The difference between these was minor, less than 5%,so the average is an accurate representation. Fig. 8 shows theseresults for the abutments. Fig. 8(a) shows the vertical displace-ments where the spine model reaches 1.5 mm and the area modelreaches 0.6 mm. Fig. 8(b) shows transverse displacement, this isthe most similar between model approaches. A maximum of35 mm is reached for both models. In Fig. 8(c) the longitudinalresponse is shown, these values reach 3.5 mm for the area modeland 2.75 mm for the spine model. These values are low mainlydue to passive resistance provided by the abutment backfill.Fig. 8(d) shows the percent differences. Again, the transverse direc-tion is most similar between models with an average percent dif-ference of less than 15%. Differences in the longitudinal directionare about 90%. In the vertical direction, it is about 35%. As statedbefore, overlapped shell elements caused the discrepancy in abut-ment displacements between models.

5.2.4. Hysteric analysis of critical componentsFig. 9 shows hysteresis loops for the various components of the

bridge for both area and spine modeling approach. Included fromthe both approaches are the elastomeric pad, steel dowels, andpiles. In Fig. 9(a) the pad in the area model reaches a value of25 mm at 75 kN where as in the spine model the pad reaches15 mm and 75 kN. A difference of 50%. Fixed steel dowels shownin Fig. 9(b) are similar reaching the yielding force of 105 kN andthen fracturing, as indicated later in the hysteric loop by the exces-sive displacement with zero force. Fig. 9(c) shows the cast-in-placepile hysteresis from the spine model and the area model. The max-imum force is 140 kN at about 23 mm for the area model and21 mm at 125 kN for the spine model. The percent difference indisplacement is 9% and the difference in force is 11%.

5.2.5. Model comparison to past studiesPast bridge seismic analysis studies that incorporated the

straight single span PSC girder bridge inventory include Choi [6]and Nielson [17]. The material properties are consistent across

Fig. 8. Abutment displacements: (a) vertical; (b) transverse; (c) longitudinal; and (d) percent difference.

184 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

the studies and procedure for determining bearing, pile, and pas-sive action stiffnesses. The past studies used spine models, andthe results are compared to those resulting from the spine modelof the current study. The geometry and significant results are sum-marized in Table 2a and b comparing bridges in past studies to thebridge used herein. Note that the past studies summarized theresults of a typical bridge under a 0.6 g PGA ground motion foran example. The bridges used in the past work by Choi [6] andNielson [17] are comparable to the bridge studied herein, althoughthe Choi bridge has the longer span length compared to the currentstudy bridge. It can be seen that the mid-span displacement of thecurrent study bridge girder in the transverse direction is largerthan that of the girder resulting from the Nielson [17] study.

There has been a discussion as to whether or not the transversedirection is vulnerable to seismic loadings and results of studieshave been somewhat conflicting [6,17]. A contributing factor tothe high transverse abutment movement in the current study isthat there is no accounting for the wing wall resistance. Accountingfor this would lower these displacements. Also, longitudinal move-ment is slightly lower due to the dimensions of the elastomericbearing used on the bridge. Since there are four girders for the cur-rent study, each bearing is required to be larger, and therefore stif-fer than a bearing on a five girder bridge that was used for theNielson study [18].

6. Seismic fragility analysis

A suite of 15 curved PSC bridges having various radii of curva-ture is selected for evaluating their seismic vulnerability in theform of fragility curves. These bridges are modeled using the spinemodel approach and the procedures described above. All bridgesmentioned herein are single span with varying span length, width,and offset. A summary of the bridges included in this fragility studyis shown in Table 3. The various geometry and component charac-teristics are given. The 15 bridge configurations were obtainedusing the Central Composite Design (CCD) method based on three

parameters, each with three possible values. This procedure is out-lined by the past work [30]. The three parameters were bridge spanlength, bridge width, and offset. Three values each for the spanlength and bridge width are selected based on literature reviewand engineering judgment. The offset parameter is taken as therecommended value from the PCI [24]. This procedure was usedto attempt to include as many combinations as possible whilemaintaining a feasible amount for study.

6.1. Limit states

Limit states are needed to quantify damage during a NLTH anal-ysis and are used to create seismic fragility curves. These aredefined in terms of displacements for abutments and bearings,while a curvature or drift is used for columns. These states in thecurrent study follow damage state (i.e., slight, moderate, extensiveand complete) definitions developed by FEMA for the HAZUS-MHloss assessment package [36]. As in recent studies [25,30], the limitstates developed by Nielson [17] are implemented. These are acombination of physics based and survey based values. To blendthese values for each damage state, the Bayesian approach wasused which is outlined by Nielson [17]. The displacement valuesfor the limit states used in the current study are shown in Table 4.It should be noted that because the vertical displacement limitstates were included in the past work, fragility analyses with seis-mic response quantities resulting from vertical ground motionswere not unfortunately made in this study.

6.2. Probabilistic seismic demand models and fragility curvegeneration

To develop fragility curves, probabilistic seismic demandmodels(PSDMs)must be generated using displacement data from theNLTHanalysis. This assumes a lognormal distribution and helps correlatedemand to intensity measure [7]. The intensity measure is taken asa PGA for this study. A more detailed description is given in past

Fig. 9. Hysteresis loops under median PGA ground motions: (a) elastomeric pad; (b) steel fixed dowel; and (c) cast-in-place concrete piles.

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 185

work [6,17]. Once PSDMs have been created, the component fragi-lity curve is generated for each bridge component at each damagestate using Eq. (2), which have been used in various fragility studies[23,29,26,15,34]. This fragility generation approach has been con-sidered the most straightforward and common way to estimateseismic fragility behavior of structural components.

Pf ¼ Uln SD

SC

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2d þ b2

c

q0B@

1CA ð2Þ

where Pf is the exceedance probability, Sc is the median value of thestructural capacity defined for the damage state, bc is the dispersionor lognormal standard deviation of the structural capacity, Sd is theseismic demand in terms of a chosen ground motion intensityparameter, bd is the logarithmic standard deviation for the demandand Uð�Þ is the standard normal distribution function. Once compo-nent level fragility analysis is complete, the component-level fragi-lity data were used to create a system level fragility curve following

first-order reliability theory having upper and lower bounds. Thisrepresents the system-level probabilities of exceedance and is cal-culated by using Eq. (3) [7,13].

maxm

i¼1½PðFiÞ� 6 PðFsysÞ 6

Ymi¼1

½1� PðFiÞ� ð3Þ

where PðFsysÞ is the cumulative probability, PðFiÞ is the componentprobabilities. The fragility curve generation procedure is furtherdetailed in past studies [7,23,13].

6.3. Comparison of seismic fragility curves

A set of component fragility curves was generated for the singlespan curved PSC bridge class at different damage states followingEq. (2). Component level fragility curves for the current study aredisplayed in Fig. 10 for the fixed bearing, expansion bearing, abut-ment with fixed bearings, and abutment with expansion bearings.These component fragility curves were then compared to those of

Table 3Single span PSC bridge inventory.

Bridge Geometry Beam Abutment

Span (m) Offset (m) Curve Radius (m) Width (m) Beam Type No. of Beams Total Tendons No. Piles Back wall Height (m)

1 21.3 15.2 373.6 9.1 I-45 4 32 8 1.42 21.3 15.2 373.6 18.3 I-45 7 32 10 1.43 21.3 30.5 186.9 12.2 I-45 5 32 8 1.44 21.3 45.7 124.7 9.1 I-45 4 32 8 1.45 21.3 45.7 124.7 18.3 I-45 7 32 10 1.46 33.5 15.2 922.3 12.2 I-63 5 50 10 1.87 33.5 30.5 461.3 9.1 I-63 4 50 10 1.88 33.5 30.5 461.3 12.2 I-63 5 50 12 1.89 33.5 30.5 461.3 18.3 I-63 7 50 16 1.810 33.5 45.7 307.6 12.2 I-63 5 50 10 1.811 48.8 15.2 1951.3 9.1 I-72 7 68 14 2.012 48.8 15.2 1951.3 18.3 I-72 13 68 24 2.013 48.8 30.5 975.8 12.2 I-72 10 68 18 2.014 48.8 45.7 650.6 9.1 I-72 7 68 18 2.015 48.8 45.7 650.6 18.3 I-72 13 68 24 2.0

Table 4Limit states [17].

Components Slight Moderate Extensive Complete

Elastomeric Bearing-Long (mm) 28.9 104.2 136.1 186.6Elastomeric Bearing-Tran (mm) 28.8 90.9 142.2 195Abutment Trans/Long (mm) 9.8 37.9 77.2 NA

Table 2Comparison between past studies and current study: (a) bridge characteristics and (b) seismic responses for critical components.

Bridge Characteristic Choi [6] Nielson [17] Current Study

(a)Span (m) 34.4 18.3 21.3Width (m) 8.68 9 9.1No. Girders 5 5 4No. Piles/Abutment 31–74 8 8PGA (g) 0.6 0.65 0.5

Critical Component Seismic Response

Choi [6] Nielson [17] Current Study

(b)Bearing (mm) Long. 100 28 18

Trans. 100 5 7Vert. NA NA 1

Abutment (mm) Long. 3 <7 22Trans. 3 <7 3Vert. NA NA 0.6

Girder at mid-span (mm) Long. NA 30 22Trans. NA 7 31Vert. NA NA 7

186 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

past studies [6,17] using median PGA values. All component-levelmedian PGA values for the fixed bearing, expansion bearing, andabutment from the current study, the Choi study, and the Neilsonstudy are summarized in Table 5. It should be noted that medianPGA values above 4 g are considered insignificant because groundacceleration is unlikely to ever reach this level and are thereforerepresented by the value 99 [17]. The Neilson study and the cur-rent study incorporated several bridges into the single span class,while the Choi [6] study only included a single PSC I-girder bridge.The most notable finding displayed in Table 5 is that the abutmentsfor the current study show more fragility than in the other studies.The abutment has a media PGA for the slight damage state of 0.75 gin the transverse direction where the Neilson study shows 1.38 g.Likewise for slight damage, in the longitudinal direction for theabutment, the median PGA for the current study is on average

2.0 g where in the Neilson study the abutment active action med-ian PGA is 2.62 g.

From the component-level fragilities, the system level fragilitycurve shown in Fig. 11(c) was constructed using Eq. (3). The systemfragility curve for the current study is of the 15 bridges in the gen-erated bridge inventory. Similarly, Fig. 11(a) and 11(b) show thesingle span bridge fragility of the Choi study [7] and the single spanbridge class of the Neilson study [19], respectively. The slight dam-age state is the first damage level to be encountered by a bridgesystem, including bearing and abutment components. As seen inFig. 11(a), the slope of the slight damage fragility curve obtainedfrom the Choi [6] study is higher than those from the Nielsonand current study. This may be attributed to the fact that the Choibridge had the longer span length, resulting in large seismic dis-placements at bearings in the longitudinal and transverse direc-tions. This indicates that slight damage will occur most rapidlyover the 0.1–0.3 g range.

In Fig. 11(b) and (c), the slope of the slight damage curve is gen-tler, signifying a slower rate of damage proliferation. Likewise, thethree remaining damage states; moderate, extensive, and com-plete, show similarity between the Nielson [17] study and the cur-rent study in terms of fragility curve slope and peak probability.For the slight damage state, the probability of exceedance at1.0 g is around 0.95 for the current study and 0.90 for the Neilson

Fig. 10. Component fragility curves for: (a) fixed bearing; (b) expansion bearing; (c) abutment with fixed bearing; and (d) abutment with expansion bearing.

Table 5Median PGA values for bridge components.

Slight Moderate Extensive Complete

Component Median (g)

Choi [6] Fxd-Long 0.30 0.63 0.82 1.13Fxd-Trans 0.14 0.61 0.77 1.06Exp-Long 99 99 99 99

Neilson [17] Fxd-Long 0.66 2.10 2.67 3.56Fxd-Trans 99.00 99.00 99.00 99.00Exp-Long 0.62 2.11 2.72 3.68Exp-Trans 1.46 99.00 99.00 99.00Ab-Pass 99.00 99.00 99.00 99.00Ab-Act 2.62 99.00 99.00 99.00Ab-Tran 1.38 99.00 99.00 99.00

Current Study Fxd-Long 1.40 99.00 99.00 99.00Fxd-Trans 1.68 99.00 99.00 99.00Exp-Long 1.31 99.00 99.00 99.00Exp-Trans 1.68 99.00 99.00 99.00Ab-Fxd-Long 1.70 99.00 99.00 99.00Ab-Fxd-Trans 0.74 2.63 99.00 99.00Ab-Exp-Long 2.33 99.00 99.00 99.00Ab-Exp-Trans 0.75 2.66 99.00 99.00

Note: Fxd = Fixed Bearing; Exp = Expansion Bearing; Ab = Abutment; Long = Longi-tudinal; Trans = Transverse.

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 187

study. The Choi [6] study displays higher fragility than the othertwo studies mentioned as indicated by the more aggressive slopesand greater peak probabilities of the fragility curves for the moder-ate, extensive and complete damage levels.

The median PGA values taken from the fragility curves of Fig. 11for each study are displayed in Table 6. In the current study, underthe complete damage state, the median is 2.75 g, whereas the valueis 2.50 g for the Neilson study. This is a percent difference of 9%.When comparing the Neilson study with the current study, thepercent difference is 5% for the extensive damage state, 0.7% forthe moderate state, and 10% for the slight damage state. As shownin Fig. 11 and Table 6, the results correlate well between thebridges in the current study and the work of Neilson [17]. Consid-ering that single span bridges are less fragile than multi-spanbridges, [7,19] the effects of curvature are not as easily noticed ina fragility analysis.

As seen in Table 6, median values for the bridge in the Choi [6]study shows extremely low values indicating a fragile bridge. Thepercent differences between Choi and the current study are100%, 90%, 100%, and 104%, for the slight, moderate, extensive,and complete damage states, respectively. The results differ whencompared to the Choi study [7] mainly due to the differences inbridge geometry that were outlined in Table 2a. In addition, theChoi study only looked into seismic fragility of a single PSC bridge,whereas the current study investigated the bridge portfolio’sfragility.

7. Conclusions and future work

This paper proposed two modeling approaches, including thearea and spine modeling methods, for seismic fragility analysis ofcurved precast-prestressed concrete (PSC) bridge portfolios andcompared the results. These were developed based upon finite ele-

Fig. 11. System level fragility curve comparison for single span bridge class: (a)Choi [6]; (b) Neilson [17]; and (c) current study.

188 J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189

ment modeling techniques and were applied to a simply sup-ported, curved PSC I-girder bridge with radius of 402 m subjectedto a suite of horizontal and vertical ground motions. The focuswas to monitor changes in seismic responses and computational

Table 6System-level median PGA values for different damage states.

Slight (g) %dif Moderate (g)

Choi [6] 0.13 100.00 0.49 89.89

Nielson [17] 0.35 10.81 1.30 0.77

Current Study 0.39 1.29

%dif = percent difference between past study and current study.

efficiency between the two approaches along with the comparisonto past studies [6,17]. Component- and system-level fragilitycurves of a portfolio of single span, curved, PSC I-girder bridgesmodeled using the spine modeling approach were constructed.These curves were compared to those gained from the past studiesto identify similarities and verify the modeling approach that canbe efficiently used for seismic vulnerability analysis for such bridgeportfolios. Findings for this study are relevant to the presentedmodels and modeling techniques. It is important to realize thatthere are many finite element modeling methods available andthat these findings may not necessarily apply to all other methods.Significant findings from this study are as follows:

Comparison between the two approaches showed that the spinemodeling approach was considered more efficient to model theselected curved PSC bridges and simulate their seismic behaviorsthan the area model. Specifically, all the critical component seismicresponses obtained from the two approaches in the transversedirection were almost identical, though preliminary seismic resultsindicated a somewhat discrepancy between displacements in thevertical and longitudinal directions. However, it is the responsibil-ity of the curved PSC bridge design engineer to determine whichmodeling technique is appropriate for any given situation. Factorsaffecting this decision can include computational efficiency, impor-tance factor of the structure, seismic zone, level of detail desired,and structure complexity.

The efficiency comparison determined that the spine model hasseveral advantages over the area model. Specifically, a total timesaving of 7% was shown when using the spine model for a curvedPSC bridge. This would be significant for seismic vulnerability stud-ies for PSC bridge portfolios with vastly different configurations.

The component- and system-level fragility curves of the PSC I-girder bridge portfolio were compared to those from the past stud-ies. The fragility analysis turned out to be comparable among thestudies, and the comparison of these system fragilities solidifiesthe spine modeling approach for regional curved PSC bridges as apractical option.

A comprehensive parametric study to investigate the effect ofbridge characteristic parameters on seismic fragility of single-span curved PSC bridges should be further conducted. Because ver-tical seismic fragility of bridges has been in question for manyyears, the future parametric study should include vertical fragilityanalyses to provide evidence to support or deny that single-spancurved PSC bridges are vulnerable to vertical ground motions.Meanwhile, the use of area modeling approach in a fragility analy-sis considering the membrane effects and plate-bending behaviorsfor the curved superstructure system could be further investigatedto determine any increases in accuracy. Because this would becomputationally intensive, appropriate experimental design tech-niques are required. Finally, parameterized seismic fragility analy-sis enabling to account for uncertainty in configurationalparameters of curved PSC bridges should be investigated in anattempt to adequately assess their effects on structuralvulnerabilities.

%dif Extensive (g) %dif Complete (g) %dif

0.63 100.40 0.87 103.87

1.80 5.41 2.50 9.52

1.90 2.75

J. Seo, L.P. Rogers / Engineering Structures 150 (2017) 176–189 189

References

[1] AASHTO. LRFD bridge design specifications, Washington DC; 2012.[2] Abdelnaby A, Raji F, Yohannes A, Naimi A, Mishra S, Golias M. Impacts of the

1811–1812 earthquakes on existing transportation networks in Memphis area.In: 10NCEE, tenth U.S. national conference on earthquake engineering frontiersof earthquake engineering, 21–25 July, 2014 Anchorage, Alaska; 2014.

[3] Amirihormozaki E, Pekcan G, Itani A. Analytical fragility functions forhorizontally curved steel I-girder highway bridges. In: Seventh nationalseismic conference on bridges and highways, 20–22 May, 2013, Oakland,California; 2013.

[4] Caltrans. Caltrans seismic design criteria version 1.3, 1st ed. Sacramento, CA:California Department of Transportation; 2004.

[5] Central US Earthquake Consortium (CUSEC). Earthquake vulnerability oftransportation systems in the Central United States; 2000.

[6] Choi E. Seismic analysis and retrofit of mid-america bridges [PhDthesis]. Atlanta, Georgia: School of Civil and Environmental Engineering,Georgia Institute of Technology; 2002.

[7] Choi E, DesRoches R, Nielson B. Seismic fragility of typical bridges in moderateseismic zones. Eng Struct 2004;26(2):187–99.

[8] Computers and Structures Inc., 2014. CSIBridge v16; 2014.[9] DesRoches R, Delemont M. Seismic retrofit of simply supported bridges using

shape memory alloys. Eng Struct 2002;24:325–32.[10] Engineering Seismology Laboratory. SUNY at Buffalo. Buffalo, New York; 2005.

<http://civil.eng.buffalo.edu/engseislab/index.htm> [accessed March 2014].[11] Ghosh J, Padgett JE. Aging considerations in the development of time-

dependent seismic fragility curves. J Struct Eng 2010;136(12):1497–511.[12] Ghosh J, Rokneddin K, Padgett JE, Dueñas-Osorio L. Seismic reliability

assessment of aging highway bridge networks with field instrumentationdata and correlated failures. I: methodology. J Earthquake Spectra 2014;30(2):795–817.

[13] Han R, Li Y, van de Lindt J. Seismic risk of base isolated non-ductile reinforcedconcrete buildings considering uncertainties and mainshock–aftershocksequences. Struct Saf 2014;50:39–56.

[14] Jeon J-S, Shafieezadeh A, Lee DH, Choi E, DesRoches R. Damage assessment ofolder highway bridges subjected to three-dimensional ground motions:characterization of shear-axial force interaction on seismic fragilities. EngStruct 2015;87:47–57.

[15] Jeon J-S, DesRoches R, Kim Taesik, Choi E. Geometric parameters affectingseismic fragilities of curved multi-frame concrete box girder bridges withintegral abutments. Eng Struct 2016;122:121–43.

[16] Minnesota Department of Transportation (MNDOT), LRFD bridge designmanual. Minnesota Department of Transportation 3485 Hadley AvenueNorth Mail Stop 610 Oakdale, MN 55128-3307; 2014.

[17] Nielson B. Analytical fragility curves for highway bridges in moderate seismiczones [Ph.D. thesis]. Atlanta, Georgia: School of Civil and EnvironmentalEngineering, Georgia Institute of Technology; 2005.

[18] Nielson B, DesRoches R. Seismic performance assessment of simply supportedand continuous multispan concrete girder highway bridges. J Bridge Eng2007;2(5):611–20.

[19] Nielson B, DesRoches R. Analytical fragility curves for typical highway bridgeclasses in the Central and Southeastern United States. Earthquake Spectra2007;23(3):615–33.

[20] Nielson BG, DesRoches R. Seismic fragility methodology for highway bridgesusing a component level approach. Earthquake Eng Struct Dynam2007;36:823–39.

[21] Padgett J. Seismic vulnerability assessment of retrofitted bridges usingprobabilistic methods [Ph.D. thesis]. Atlanta: Georgia Institute ofTechnology; 2007.

[22] Padgett J, DesRoches R. Bridge functionality relationships for improved seismicrisk assessment of transportation networks. Earthquake Spectra2007;23:115–30.

[23] Padgett J, DesRoches R. Methodology for the development of analyticalfragility curves for retrofitted bridges. Earthquake Eng Struct Dynam 2008;37(8):1157–74.

[24] Precast/Prestressed Concrete Institute (PCI). Bridge design manual. 3rded. Chicago, IL: First Release; 2011.

[25] Ramanathan K, DesRoches R, Padgett JE. Analytical fragility curves formultispan continuous steel girder bridges in moderate seismic zones. TranspRes Rec : J Transp Res Board 2010;2202:173–82.

[26] Ramanathan K, DesRoches R, Padgett JE. A comparison of pre- and post-seismicdesign considerations in moderate seismic zones through the fragilityassessment of multispan bridge classes. Eng Struct 2012;45:559–73.

[27] Seo J. Seismic vulnerability assessment of a family of horizontally curved steelbridges using response surface metamodels [Ph.D. thesis]. University Park,Pennsylvania: Department of Civil and Environmental Engineering, ThePennsylvania State University; 2009.

[28] Seo J. Statistical determination of significant curved I-girder bridge seismicresponse parameters. Earthquake Eng Eng Vibr 2013;12(2):251–60.

[29] Seo J, Linzell D. Horizontally curved steel bridge seismic vulnerabilityassessment. Eng Struct 2012;34:21–32.

[30] Seo J, Linzell D. Use of response surface metamodels to generate system levelfragilities for existing curved steel bridges. Eng Struct 2013;52:642–53.

[31] Seo J, Linzell D. Nonlinear seismic response and parametric examination ofhorizontally curved steel bridges using 3D computational models. J Bridge Eng2013;18(3):220–31.

[32] Seo J, Park H. Probabilistic seismic restoration cost estimation fortransportation infrastructure portfolios with an emphasis on curved steel I-girder bridges. Struct Saf 2017;65:27–34.

[33] Shinozuka M, Feng MariaQ, Kim HK, Kim SHM. Nonlinear static procedure forfragility curve development. J Eng Mech 2000;126(12):1287–96.

[34] Torres E, Seo J, Rogers L. Probabilistic structural performance evaluation ofconcrete slab bridge system subjected to scour and earthquake. ACI Spec Publ2017;316:73–94.

[35] Wilson T, Mahmoud H, Chen S. Seismic performance of skewed and curvedreinforced concrete bridges in mountainous states. Eng Struct2014;70:158–67.

[36] FEMA. MH MR4 technical manual: Multi-hazard loss estimation methodologyearthquake model. Dept. of Homeland Security Emergency Preparedness andResponse Directorate. Washington, DC: FEMA Mitigation Div.; 2003.