Engineering Structures 70 (2014) 158–167

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Engineering Structures

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Seismic performance of skewed and curved reinforced concrete bridgesin mountainous states

http://dx.doi.org/10.1016/j.engstruct.2014.03.0390141-0296/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: tswilson@rams.colostate.edu (T. Wilson), hussam.mahmoud@

colostate.edu (H. Mahmoud), suren.chen@colostate.edu (S. Chen).

Thomas Wilson, Hussam Mahmoud ⇑, Suren ChenDept. of Civil and Environmental Engineering, Colorado State Univ., 1372 Campus Delivery, Fort Collins, CO 80523, United States

a r t i c l e i n f o

Article history:Received 11 November 2013Revised 30 March 2014Accepted 31 March 2014

Keywords:Nonlinear analysisSeismic effectsGround motionSkewedCurvedRC bridges

a b s t r a c t

A number of skewed and curved highway bridges have experienced damage or collapse due to seismicevents, and has most recently been observed during the Chile earthquake in 2010. In the Mountain Westregion, bridges integrating skew and curvature are becoming an increasingly prominent component ofmodern highway transportation systems due to their ability to accommodate geometric restrictionsimposed by existing highway components. There is however very little information available on the com-bined effects of skew and curvature on the seismic performance of Reinforced Concrete (RC) bridges. Acomprehensive performance analysis is performed on eight bridge configurations of various degrees ofskew and curvature with low-to-moderate seismic excitations which are characteristic of the MountainWest region. Nonlinear time-history analysis is carried out on each bridge configuration using detailedfinite element (FE) models. The results show a considerable impact on the seismic performance due tothe effects of skew and curvature, with stacking effects observed in the combined geometries. Insightson the complexities of curvature, skew, loading direction and support condition are also made, whichmay lend themselves to more informed design decisions for practicing engineers in the future.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction Jennings [4] studied the dynamic behavior of skewed bridges, fol-

Explicit knowledge of the behavioral response of skewed andcurved highway bridges to seismic events is essential to designingsafe transportation systems. Typically, geometrically complexbridges will exhibit a more complex seismic response as comparedto regular, straight bridges.

1.1. Seismic performance of skewed bridges

Although skewed bridges offer many benefits to transportationdesign, the offset angle of the superstructure has in the past led toseismic-induced failure, particularly due to excessive deflections ofthe superstructure. The collapse of the Rio Bananito bridge in CostaRica [1], the Gavin Canyon undercrossing in Northridge [2], andAmerico Vespucio/Miraflores bridge in Chile [3], are a few exam-ples of where skew has contributed to seismic induced failures ofRC bridges. Failures were typically characterized by in plane rota-tion of the bridge span about the acute angle of the abutment orpier; often supported by a bearing or seat type support wheretranslational and rotational restraint is minimal. Maragakis and

lowed by Wakefield et al. [5] who investigated the failure of theFoothill Boulevard Undercrossing during the San Fernando earth-quake in 1971. Failure was characterized by unseating and columndamage, attributed to rigid-body motion in the superstructure.Pushover analysis conducted by Bignell et al. [6] showed that theskew angle of the bridge can significantly reduce the comparativeultimate capacity during an earthquake by up to 2/3 in the longitu-dinal direction. More recently in a study by Saadeghvaziri et al. [7],it was found that seismic-induced impact between spans ofskewed bridges can impose large shear stresses on bearings. Whilethe behavior discussed is observed for higher angles of skew; stud-ies have indicated that bridges with skew angles below 30 degreestend away from higher mode effects, and can be analyzed asstraight or represented without using complex FE models [8,9].

1.2. Seismic performance of curved bridges

Curved bridges are susceptible to a similar asymmetrical failuremode as skewed bridges. The effect of curvature on the seismicresponse of highway bridges has been examined in many studies,although it has been predominantly concentrated on steel bridges[10–15]. Early vulnerabilities in curved bridges were identifiedthrough the failure of the I 5/14 Freeway Overpass during the1971 San Fernando earthquake and subsequent analyses conducted

T. Wilson et al. / Engineering Structures 70 (2014) 158–167 159

by Williams and Godden [16]. Large excitations were induced in thesuperstructure of the I 5/14 Freeway causing sections to displace outof shear keys and triggering bending failures in columns bases. Wil-liams and Godden studied the behavior of multispan curved bridgesunder seismic excitation, and discovered that the inclusion ofexpansion joints can cause extensive damage through repetitiveimpacting in translational and torsional modes. More recently, Gal-indo et al. [13] investigated the effect of four different radii of curva-ture on the seismic performance of steel I-girder bridges withbearing supports. The study showed that the relative degree of cur-vature has a significant effect on the bridges response and that forshorter bridge radii there is increased vulnerability to joint residualdamage and pounding effects. Unseating was also observed to stemfrom large rotations in the superstructure causing the deck to rotateoff the bearing supports. This is consistent with the findings of Heet al. [17], who attributed curvature as the primary parameteraffecting seismic load levels at bearing supports and critical crossframe members. From a modeling standpoint, Mwafy and Elnashai[11] assessed the impact of conventional design assumptions onthe capacity estimates of curved steel bridges. Modeling of joints,including friction levels in bearing supports, were found to affectthe bridges performance and that simplified conservative designdecisions can in cases lead to a non-conservative representation ofthe scenario.

1.3. Seismic performance of curved and skewed bridges

Numerical studies that assess the seismic performance of bothskewed and curved bridge geometries were not found in the liter-ature review conducted. Existing research conducted on skew andcurvature independently suggests common vulnerabilities. Forexample, both bridge configurations appear to be susceptible todeck unseating, tangential joint damage, pounding effects as wellas large in-plane displacements and rotations of the superstruc-ture. Ultimately, detailed analyses on the seismic performance ofcurved and skewed bridges are needed to identify specific behav-ioral vulnerabilities that can lead to improved structural design.

For low seismic hazard regions such as the Mountain West,knowledge of typical vulnerabilities to bridges and earthquakeresistant design practices are limited. In the event that an earth-quake does occur, it is important to identify vulnerabilities in typi-cal RC bridges in advance, and create a knowledge base forimprovement of future bridge designs. In the present study, a para-metric analysis is conducted to evaluate the seismic performance ofskewed and curved, three-span RC bridges characteristic of theMountain West region. Detailed FE models are developed and ana-lyzed for eight bridge configurations of various degrees of skew andcurvature with consistent structural and geometric components.The bridges examined follow typical design for the region, includingthe adoption of continuous deck design and integral abutments.Nonlinear time-history analysis is conducted on each bridge config-uration under seven sets of earthquake records scaled to a site loca-tion in Denver, Colorado. The research is targeted at gaining a betterunderstanding of the global behavior of various bridge configura-tions during a low-to-moderate seismic event. To better aid designengineers in making informed design decisions, the effects of earth-quake input loading direction and abutment support condition,including integral and bearing supports are also examined.

2. 3-D finite element modeling and seismic analysis

2.1. Structural components

The bridges investigated in this study vary in degree of skewand radii of curvature; yet each bridge is constructed with the

same structural components. The bridge superstructures (Fig. 1a)are composed of a 205 mm deep concrete slab deck supported byeight 1.73 m deep, parallel pre-stressed concrete I-girders. Thegirders are reinforced longitudinally near the top, and prestressednear the bottom of each section, with stirrup bracings at 45 cmintervals. End spans are embedded into pier caps creating integral,fixed connections (Fig. 1b). The rectangular pier caps supportingthe superstructure are 1.54 m in depth, and are each supportedby an interior and exterior pier-column (Fig. 1c). Each column isreinforced with longitudinal rebar, and transverse stirrups thatrun in alternating directions at a spacing of 41.5 cm. Integral abut-ments support the superstructure and encase contiguous I-girders,and are reinforced with rebar tied into the deck.

2.2. Development of the finite element models

The structural performance of the bridges selected for this studyare evaluated using 3-D FE models constructed in SAP2000 [18].The bridge deck is modeled using thin shell elements that spanintermediate nodes of the girders and are further meshed intoquadrants. Frame elements capture axial, shear, and bendingdeformations, and are used to model the abutments, girders, bentcaps, and pier-columns. Concrete confinement of the pier-columnsis based on the model developed by Mander et al. [19]. Prestressingtendons are modeled as equivalent loads (after losses) and followthe geometry of tendons at each girder. Girders are connected toshell elements by use of fully constrained rigid links. The columnsare fixed at the soil foundations in all rotational and translationaldegrees of freedom (DOF). In order to account for inelastic columnbehavior in the substructure, plastic hinges are implemented at thetop and bottom of the pier-columns (Fig 1b). The plastic hinges inthe columns utilize a lumped plasticity model that is based on aforce-biaxial moment (PMM) interaction. The PMM interactionhinges account for axial force fluctuations and bending in orthog-onal directions of the targeted member, as well as degradationbehavior and ductility estimation.

The abutment–girder connections are modeled using rigid linksthat are characteristic of the integral fixity between the abutmentand girder. The abutment is considered to have fixity from the sur-rounding soil and pile foundation in all DOF except the longitudinal.The backing soil behind the abutment is represented by the use of amulti-linear, longitudinal, compressive spring (Fig. 1b) followingthe Caltrans design procedures for backing soil behind an integralabutment [20]. The stiffness is based on passive earth pressure testsand force deflection results derived from large-scale abutmentexperiments [21] with equations refined by Shamsabadi et al. [22].

2.3. Ground motion selection and scaling for mountainous states

Seven sets of earthquake records are selected in accordancewith the AASHTO Guide Specifications for LRFD Bridge Design fromthe Pacific Earthquake Engineering Research (PEER) Center strongmotion database [23,24]. To simulate typical earthquake groundmotion for states in the Mountain West region, Denver, Coloradois chosen as a prototype site location. A stiff soil profile for Denveris selected, and a design response spectrum is developed using theUSGS database and AASHTO Guide Spec. Strong motion records arechosen based on a moment magnitude range between Mw 6.5 and7.0, a stiff soil condition with shear wave velocity range of 300–550 m/s, and a 20–30 km range for the Joyner–Boore distance tothe fault (Rjb). The characteristics of the selected ground motionsare listed in Table 1. Fig. 2 shows the response spectra for the faultnormal component of the selected records, and the design responsespectrum developed for the site condition. The scaling factor iscomputed for the fault normal and parallel directions by matchingthe AASHTO design response spectra to the average of the seven

(a)

(b) (c)Fig. 1. (a) Plan view of bridge – radius 1370 m, skew 30�, (b) elevation view and (c) bent cross-section.

Table 1Earthquake characteristics.

Record # Event Year Station Mag. (Mw) Significant duration (5–95%, s) Rjb (km) Vs30 (m/s)

1 San Fernando 1971 LA – Hollywood Stor FF 6.61 11.9 22.9 464.22 Imperial Valley 1979 Calipatria Fire Station 6.53 25.1 23.2 301.83 Superstition Hills 1987 Wildlife Liquef. Array 6.54 29.1 24.0 304.34 Irpinia, Italy 1980 Mercato San. Severino 6.9 28.4 29.8 513.35 Loma Prieta 1989 Agnews State Hospital 6.93 18.4 24.3 351.66 Northridge 1994 LA – Baldwin Hills 6.69 17.6 23.5 435.77 Kobe, Japan 1995 Kakogawa 6.9 17.6 22.5 457.6

0 0.5 1 1.5 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

2

Period (sec)

Spec

tral

Acc

eler

atio

n (g

)

San Fernando, CAImperial Valley, CASuperstition Hills, CAIrpania, ItalyLoma Prieta, CANorthridge, CAKobe, JapanAverage DRS

Fig. 2. Earthquake and AASHTO design response spectrum.

160 T. Wilson et al. / Engineering Structures 70 (2014) 158–167

earthquake response spectrums at the fundamental period of thebridge structure [23].

2.4. Nonlinear time-history analysis procedures

Nonlinear time-history analysis is conducted using the directintegration method; material and geometric nonlinearities are

included. Fixed Rayleigh damping coefficients are used that repre-sent 2% damping in the first and second modes. The method oftime integration follows the Hilber–Hughes–Taylor method withalpha, beta and gamma coefficients at 0, 0.25, and 0.5, respectively.The integration time step is kept at 0.01 s and a standard iterationconvergence tolerance of 0.0001 is used following a sensitivityanalysis. Two orthogonal components of the ground motion set

T. Wilson et al. / Engineering Structures 70 (2014) 158–167 161

are employed simultaneously in each analysis. The fault normalcomponent of the ground motion is employed in the global longi-tudinal direction, while 40% of the fault parallel component isapplied in the global transverse direction. In a study conductedby Bisadi et al. [25], this method has been shown to produce thelowest probability of underestimation of seismic demand, althoughit may be uneconomical for design purposes.

3. Configuration of bridges in parametric study and modalanalysis results

3.1. Part I – Baseline bridge model configuration (radius: 1730 m;skew: 30)

Part one of this study examines a single skewed and curvedbridge configuration and the impact of typical design decisionssuch as the abutment support condition, and the directional com-ponents of the loading (Table 2). The geometric specifications meetdimensional requirements utilized by the Colorado Department ofTransportation bridge design office and detailing is taken directlyfrom a highway bridge on I-25 in Northern Colorado.

3.2. Part II – Geometric configurations of bridges

Part two examines eight RC bridges of varying curvature andskew, constructed from the baseline model for parametric compar-ison. The bridge configurations were developed with extensiveguidance and input from the Colorado Department of Transporta-tion and are representative of a range of typical skewed and curvedbridges in Colorado (Table 3). Each bridge consists of three spans,with two identical side spans and a middle span kept at consistentlengths of 22.1 m and 29.5 m, respectively. Characteristics thatwould otherwise affect the structural response such as the membermaterial properties, deck width, mean pier height, and membercross-section are kept consistent. Characteristics such as theslanted length of piers and abutments are subject to deviationsin accordance with variations in skew, and curvature and otherrealistic design considerations.

Table 2Bridge components for baseline comparison.

Scenario Skew (degrees) Curvatureradius (m)

Component

1 30 1370 Baseline model2 30 1370 Reversed directional loading3 30 1370 Bearing support

Table 3Bridge configurations for parametric study: Part II.

Bridge # Skew (degrees) Curvatureradius (m)

Super elevation (degrees)

1a 0 0 02 30 0 03 45 0 04 0 1370 45 0 910 66b 30 1370 47 45 1370 48 30 910 6

a Benchmark model.b Baseline model.

3.3. Modal analysis

Modal analysis is conducted on each of the bridge configura-tions introduced above, for which the primary translational modesare extracted. Ritz vectors are utilized for determining modeshapes, and target dynamic participation ratios of 99% are imposedin the local longitudinal and transverse directions. Depictions ofthe mode shapes and tabulated summaries of the modes withhighest participation ratios in the primary translational directionsare shown in Table 4.

The benchmark model (#1 in Table 4), which contains no skewor curvature, induces a longitudinal fundamental mode of vibra-tion with a period 0.21 s (Fig. 3a). Transverse and vertical modesfollow with lesser periods and contain negligible rotational partic-ipation. The curved bridge configurations (#4 and #5), yield com-parable mode shapes and translational participation ratios to thebenchmark bridge. Curved bridges differ however by introducingtorsional rotation into the mode shapes, as seen for example inthe fundamental longitudinal mode that incurs twist about thevertical DOF. Higher contributions from rotational DOF are alsoobserved in the mode shapes for transverse and vertical directions,as seen in Table 4. In skewed bridges (#2 and #3), participation insecondary translational directions of the first two modes isobserved. This is reflected in the fundamental longitudinal modesshape, where torsional vibration occurs about the primary axis(Fig. 3b). In the skewed and curved bridges, independent geometriceffects are superposed. The alternate translational participationassociated with skew is observed in combination with increasedrotational participation associated with curvature.

4. Parametric study results – Baseline model

4.1. Evaluation criteria

The earthquake time-histories previously described are appliedto the skewed and curved bridge configurations shown in Table 3.The demand imposed by seismic loading is compared to the mem-ber section capacity using demand-deformation relationships forframe members. Demand/capacity (D/C) ratios are generated forcolumn sections using axial force–uniaxial moment relationships,and evaluated using an axial force, biaxial moment surface interac-tion shown in Fig. 4.

4.2. Time-history analysis results of the baseline model

Lateral displacement of the pier-columns is limited, and reachespeak drift ratios of 0.18% and 0.037% in the longitudinal and trans-verse directions, respectively (Fig. 5a). Coupling effects betweendiagonally opposite pier-columns are also observed as an effectof the planar rotation generated from skew. At the abutments,the resistance of the soil ranges from no observed resistance in par-ticular load cases to 4913.8 kN across the back wall. Although therelative deformation is limited in the superstructure, a significantdemand develops across the pier-columns (Table 5). Large actionsare imposed on the interior front and back exterior columns, andare coupled in magnitude. The longitudinal shear induced by theresponse of the bridge reaches 92.3% of the nominal capacity, whilethe longitudinal moment reaches 69.2% of its capacity. In additionto the unidirectional analysis, the demand on the critical interiorcolumn for the largest earthquake excitation (San Fernando) isplotted against the triaxial surface capacity (Fig. 5b). The sectioncut of the triaxial surface shows that the demand exceeds the sec-tion capacity in several instances of the earthquake, and that sub-sequent damage may be expected.

Table 4Modal participation factors of bridges.

Bridge # Skew (deg.) Curv. radius (m) Mode Period (s) UX (kN s2) UY (kN s2) UZ (kN s2) RX (kN m s2) RY (kN m s2) RZ (kN m s2)

1a 0 0 1 0.21 44 0 0 0 �14 02 0.13 0 �37 0 30 0 03 0.11 0 0 �35 0 2 3

2 30 0 1 0.19 44 �14 0 �25 8 02 0.11 �16 �36 0 36 �34 03 0.10 0 0 33 0 0 �48

3 45 0 1 0.19 40 �17 0 �27 1 942 0.12 20 30 0 �20 98 �1623 0.10 0 0 �31 0 �174 75

4 0 1370 1 0.20 43 0 0 54 4108 �587272 0.13 1 35 0 �2799 68 �20583 0.10 0 �1 33 45014 1211 7

5 0 910 1 0.21 44 �1 0 120 4259 �404712 0.13 �1 �31 1 3466 �30 17463 0.10 0 3 27 25081 905 �34

6b 30 1370 1 0.20 40 �15 1 2577 3836 �538752 0.12 �17 �31 0 2511 �1636 239393 0.10 1 2 �29 �40015 �1015 �745

7 45 1370 1 0.19 38 �18 2 5033 3691 �508202 0.12 �22 �28 0 2250 �2177 305643 0.10 0 �1 33 44737 1281 403

8 30 910 1 0.20 40 �17 1 2910 3938 �367782 0.13 �19 �33 1 4447 �1848 190873 0.11 0 2 32 29548 1308 �531

a Benchmark model.b Baseline model.

Longitudinal(global)

Vertical(global)

Transverse(global)

Fig. 3. Fundamental mode shapes.

−5,000−2,500

02,500

5,000

−4

−2

0

2

4

x 104

0

5000

10000

15000

20000

Longitudinal Moment(kN−m)

Transverse Moment (kN−m)

Axi

al F

orce

(kN

)

Fig. 4. Axial force, biaxial moment interaction surface for the pier-column section.

162 T. Wilson et al. / Engineering Structures 70 (2014) 158–167

4.3. Effect of earthquake input direction

In the analysis of the baseline model above, the earthquakeinput referred to as the Primarily Longitudinal Combination

(PLC), is applied fully (100%) in the global longitudinal and partially(40%) in the global transverse direction. In order to make a surfacecomparison on the effect of ground motion direction, a new seis-mic input referred to as the Primarily Transverse Combination(PTC) is defined. The new combination consists of a full (100%) loadcontribution to the global transverse direction and a partial (40%)contribution to the longitudinal direction.

In comparison to the analysis using the PLC, the drift ratios atthe pier cap for the PTC are reduced in the longitudinal and trans-verse directions. Comparatively, maximum D/C ratios developed inthe columns for the PTC are also on average 55.5% smaller. This ispredominantly attributed to the asymmetrical strength and rigid-ity of the column sections in the transverse direction, and theadded stiffness derived from the transverse fixity at the abutments.The abutment reactions are also lower in all DOF for the PTC overthe PLC.

4.4. Effect of support condition

Preliminary analyses show that a small variation in supportcondition can significantly affect the bridge excitation and subse-quent distribution of actions. In order to study the effect of impos-ing an entirely different support condition, the integral abutmentsare substituted with bearing pads and reinforced with shear keys.The support connection is modeled by restraining translation in

-2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 3000-1000

-500

0

500

1000

1500

2000

2500

3000

3500

4000

Longitudinal Moment (kN-m)

Axi

al F

orce

(kN

)

Surface Capacity

D/C=<1

D/C>1

0.04

0.06

0.08

0.1

0.12

0.14

Time-History Record

SanFern

ando

, CA

Impe

rial V

alley

, CA

Supers

tition

Hill

s, CA

Irpan

ia, It

alyLom

a Prie

ta, C

ANor

thridg

e, CA

Kobe,

Japa

n

Dri

ft R

atio

(%

)

(a)

(b)

Fig. 5. (a) Peak drift ratios for the baseline bridge (global) and (b) triaxial pier-column capacity with demand- for Loma Prieta time-history.

Table 5Maximum demand on bridge pier – at location 1 for the baseline bridge (local coordinates).

Axial force (kN) Shear (long.) (kN) Shear (trans.) (kN) Uniaxial moment (long.) (kN m) Uniaxial moment (trans.) (kN m)

Maximum demand 3872.6 1957.0 706.4 2371.3 9290.3Demand/capacity 0.076 0.923 0.289 0.692 0.465

T. Wilson et al. / Engineering Structures 70 (2014) 158–167 163

the vertical and transverse directions and rotation about the verti-cal and longitudinal axes. In addition, a three inch gap spring isused to represent the longitudinal expansion typically allowedbetween the superstructure and abutment. This is a typical designdetail employed by the DOT and has been used in several existingColorado bridges.

The results from the analyses show that the use of a bearingtype connection releases the structure in the longitudinal direc-tion. This causes an increase in the structural period to 0.394 s,while maintaining the original longitudinal mode shape. A reduc-tion in structural demand is observed across the bridge, readilyapparent at the pier-columns. At the abutments, the bearing sup-ports develop a reduced transverse force and moment demandby 12.3% and 33.6%, respectively. At the bent cap, actions devel-oped across the 6 DOF are 53.7% lower than the bridge model sup-ported by integral abutments. This may infer that there aresignificant advantages to employing bearing type supports overintegral abutments for curved and skewed bridges. On the otherhand, designers should make appropriate considerations to

potential pounding effects, as well as the strength of transversesupport components such as shear keys, despite it not being a con-cern for this specific bridge configuration and loading scenario.

5. Parametric study results – Curved and skewed bridgeconfigurations

5.1. Critical drift ratios at the top of interior columns

The bridges assessed are composed of non-slender reinforcedconcrete members with integral connections that restrain rotationand segmented deformation. Subsequently, large deformations arenot easily incurred without significant damage to members or con-nections (Fig. 6). The displaced response of the bridge across thedifferent configurations would likely be more prevalent in lessrigid bridges with more slender members and flexible supports.Effects from curvature on the translational response to groundmotion are not observed. Planar rotation of the superstructure is

R0 S0 R0 S30 R0 S45 R4500 S0 R3000 S0 R4500 S30 R4500 S45 R3000 S300

0.05

0.1

0.15

0.2

0.25

Dri

ft R

atio

(%

)

LongitudinalTransverse

Fig. 6. Longitudinal and transverse drift ratio at pier.

164 T. Wilson et al. / Engineering Structures 70 (2014) 158–167

observed in skewed bridges, which is characterized by an increasein translational drift and decrease in longitudinal drift at pier-columns.

5.2. Resistance behind integral abutment

The non-skewed bridge configurations induce primarily longi-tudinal vibration and therefore, yield the highest resistance fromsoil springs (Fig. 7). Skewed bridges induce more planar rotationalmotion about the center of the superstructure. This causes sub-stantially higher forces on the transverse supports of the abut-ments, and less stress longitudinally on the backing soil.Alternative abutment configurations where initial transverse resis-tances are comparable, such as shear keys or wing walls, would beheavily loaded in the scenarios involving skewed geometries andmay exceed expected capacities.

5.3. Shearing forces in the pier-columns

The shear capacity of the pier-column sections is heavily depen-dent on axial, moment and shear demand, thus a comparison of thedemand and capacity over the time history is made (Fig. 8a). In bothskewed bridges, the skew angle causes a reduction in the transverse

0

1000

2000

3000

4000

R0 S0

R0 S30

R0 S45

R4500

S0

R3000

S0

R4500

S30

R4500

S45

R3000

S30

For

ce (

kN)

Longitudinal

0

1000

2000

3000

4000

5000

6000

R0 S0

R0 S30

R0 S45

R4500

S0

R3000

S0

R4500

S30

R4500

S45

R3000

S30

For

ce (

kN)

Transverse

(a)

(b)

Fig. 7. Reaction force at abutment – (global).

column shear, and a substantial increase in longitudinal shear(Fig. 8b). The longitudinal shear in the 45 degree skewed bridge isthe highest at 2892 kN, and exceeds the capacity of the section atthe two boxed intervals shown in Fig. 8a. Both the longitudinaland transverse shear forces observed in the pier-columns of curvedbridges decrease for higher radii of curvature. The longitudinalshear D/C ratios observed in all the skewed bridge configurationsare close to or exceed unity. In the combined geometries, the shear-ing forces are comparatively higher in the transverse directions andvaried in the longitudinal direction. Significant damage due to lon-gitudinal shear is expected and should be a consideration in designfor high angles of skew, and particularly for geometries that containa combination of skew and curvature.

5.4. Axial force and moment demand on pier-columns

In comparison to the benchmark model (of no skew or curva-ture) the bridge configurations with skew and curvature yieldhigher D/C ratios in the substructure. In all model sets, the distri-bution throughout the pier-columns shows the highest concentra-tion of demand at the bases of pier-columns.

Imposing skew on the bridge structure directs bending in thesubstructure away from the weak axis of the columns and alsocauses cross coupling of actions between adjacent columns. Inthe longitudinal (weak) axis of the pier-columns: the typically crit-ical longitudinal moment decreases proportional to the skew angle(Fig. 9a). It is accompanied by an amplification of the moment inthe strong axis, which becomes critical in the 45 degree skew con-figuration. In the curved bridge configurations, although equivalentdeformation is observed, longitudinal moment increases propor-tional to higher degrees of curvature. In combined geometries,interaction between skew and curvature leads to a stacking effectwhere higher moments are observed in both the longitudinal andtransverse directions, and approach nominal capacities.

To evaluate capacity of the pier-columns for axial compressionwith orthogonal bending moments, the column demand is evalu-ated with respect to the PMM interaction capacity. For each bridgeconfiguration the most critical time-point is selected from the timehistory, and the resulting demand is plotted against the triaxialsurface capacity (Fig. 9b). The demand exceeds the capacity ofthe pier-column section in 3 out of 8 bridge configurations. Dam-age is predicted in the pier-columns of the curved bridge configu-ration of lowest radii (910 m) due to larger imposed longitudinalbending-moments. In the combined curved and skewed bridgeconfigurations, higher longitudinal moments (caused by curvature)combined with higher transverse moment (caused by skew) lead toexceedance of capacity in both 910 m and 1730 m curved, 30-degree skew bridges.

The analyses performed also yield critical locations for high seis-mic demand and locations where damage may occur (Fig. 10). Forthe curved bridge configurations, the two interior columns consis-tently display higher D/C ratios as compared to exterior columns,and the separation is proportionally larger for higher degrees ofcurvature. For skewed bridge configurations, coupling of actions

0 5 10 15 20

0

1000

2000

3000

4000

5000

6000

7000

Time (secs)

Shea

r Fo

rce

(kN

)

Nominal Shear Capacity

Shear Demand

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Shea

r D

eman

d/C

apac

ity

R0 S0

R0 S30

R0 S45

R4500

S0

R3000

S0

R4500

S30

R4500

S45

R3000

S30

Longitudinal (Local)Transverse (Local)

(a)

(b)

Fig. 8. (a) Pier shear force/nominal capacity – Northridge time-history and (b) critical shear D/C ratios.

0

0.2

0.4

0.6

0.8

1

Mom

ent D

eman

d/C

apac

ity

R0 S0

R0 S30

R0 S45

R4500

S0

R3000

S0

R4500

S30

R4500

S45

R3000

S30

Longitudinal (Local)

Transverse (Local)

-3000 -2000 -1000 0 1000 2000 3000

0

500

1000

1500

2000

2500

3000

3500

4000

Longitudinal Moment (kN-m)

Axi

al F

orce

(ki

ps)

Surface Capacity

D/C>1

D/C=<1

(a)

(b)

Fig. 9. (a) Section analysis – unidirectional moment demand/nominal section capacity at critical pier and (b) triaxial D/C ratios of pier-columns in various bridgeconfigurations.

T. Wilson et al. / Engineering Structures 70 (2014) 158–167 165

in diagonally opposite columns is observed from effects of planartorsion in the superstructure. The unequal distribution of demandbetween pier-columns is also more apparent for higher degrees ofskew. The behavior observed independently in curved and skewedbridges is proportionately observed in combined geometries. Both

skew and curvature have in common the interior front column loca-tion as a focus point for high concentrations of demand. This loca-tion yields higher shear forces as well as higher axial andmoment demands, which in two of the three combined geometriesexceeds the nominal capacity.

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

R0 S0 R0 S30 R0 45 R4500 S0 R3000 S0 R4500 S30 R4500 S45 R3000 S30

Dem

and/

Cap

acit

y (N

orm

aliz

ed)

Column A

Column C

Column B

Column D

Column B

Column A Column C

Column D

Fig. 10. Normalized triaxial demand ratios at critical pier-columns.

166 T. Wilson et al. / Engineering Structures 70 (2014) 158–167

6. Conclusions

Numerical studies that evaluate the combined effects of both cur-vature and skew on the seismic performance of bridges were notfound in the review of existing studies. The present study examinesthe seismic behavior of typical, local reinforced concrete bridgesincorporating continuous deck design, integral abutments, and stiffbacking soils. It also includes an examination of the effects of skewand curvature in various geometric configurations. Although the pro-totype bridge was selected for Denver Colorado, the findings may aiddesign and analysis of RC bridges with similar characteristics in otherlow-to-moderate seismic regions. Detailed nonlinear time-historyanalyses were conducted on each bridge configuration subjected tomultiple seismic excitations. The following conclusions are drawnbased on the numerical simulation results:

� Seismic analyses of the bridges in this study yields limiteddeformation due to the integral pier connection and abutmentsupports. Despite the limited response, the actions developedin the pier-columns of the substructure were found to exceednominal section capacities in several instances of dynamictime-history loading.� Ground motion applied primarily in the lengthwise direction

(100/40) of the bridge was confirmed to control the analysesfor the geometries considered. Loading in the transverse axes(40/100) of the baseline bridge yielded a 55.5% reduction inthe column D/C ratios, lower deformations at pier caps, andlower resistance forces at supports.� In a comparative study between integral abutments and bearing

supports, integral abutments were found to induce higherdemand on the columns and generate larger deformations.Adoption of a bearing support in the bridge increased the periodof vibration, subsequently reducing bridge excitation and low-ering demand on the structural components. Although, it wasnot the case in this study, considerations should be made toknown vulnerabilities. These may include pounding effects atexpansion joints of curved bridges and unseating from planarrotation of the superstructure resulting from skew.� The skewed bridge configurations analyzed induced planar

rotation in the superstructure resulting in coupled demandbetween diagonally opposite columns. It also directed seismicdemand away from principal axes, resulting in an increase inlongitudinal shear and transverse moment demand in thepier-columns. The effects of skew were observed to be propor-tional to the skew angle and an established difference was read-ily visible at the supposed near conservative skew angle of30 degrees.

� Curved bridge models induced higher moment demand in theweak axis of the substructure and overall lower shear demandin comparison to the other configurations. Curvature also intro-duced higher moment demand on the interior pier-columns,which increased with lower radii. The interior columns experi-enced the highest longitudinal moment demand out of the modelset and exceeded the nominal capacity in the PMM analyses.� The bridges analyzed incorporating both skew and curvature

exhibited stacking effects proportional to the influence of eachgeometric parameter. The result is higher observed D/C ratios inthe columns and larger deformations in the superstructure, whichin some cases results in damage to the substructure. This wouldsuggest that for bridges involving both skew and curvature, amore rigorous design and analysis approach should be consideredthat accounts for geometric effects under seismic loads.� With respect to structural modeling, the bridges analyzed sug-

gest that for high levels of skew (exceeding 30 deg.), certainranges of curvature (<1370 m), and for combinations of thetwo geometries, a more rigorous modeling approach may beconsidered to capture complex interactions.� The analysis was performed for a low seismic region; however

the earthquake loading induced a demand on the bridge config-urations that exceeded the nominal capacities in the substruc-ture. Although bridges in low seismic regions are generallynot considered for seismic analysis or design, bridges with com-plex geometric configurations including skew and curvaturemay need specific attention as demonstrated in the analysesof select configurations in this study.

Acknowledgement

The research study was supported by the Colorado Departmentof Transportation, which assisted in providing important informa-tion and details on current bridge design in the region.

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