seismic response of skewed rc box-girder bridges

8/10/2019 Seismic response of skewed RC box-girder bridges

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Vol.7, No.4 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION December, 2008

Earthq Eng & Eng Vib(2008) 7:415-426 DOI: 10.1007/s11803-008-1007-4

Seismic response of skewed RC box-girder bridges

Ahmed Abdel-Mohtiand Gokhan Pekcan

Civil and Environmental Engineering Department, University of Nevada Reno

Abstract: It is critical to ensure the functionality of highway bridges after earthquakes to provide access to importantfacilities. Since the 1971 San Fernando earthquake, there has been a better understanding of the seismic performance of

bridges. Nonetheless, there are no detailed guidelines addressing the performance of skewed highway bridges. Several

parameters affect the response of skewed highway bridges under both service and seismic loads which makes their behavior

complex. Therefore, there is a need for more research to study the effect of skew angle and other related factors on the

performance of highway bridges. This paper examines the seismic performance of a three-span continuous concrete box

girder bridge with skew angles from 0 to 60 degrees, analytically. Finite element (FE) and simplified beam-stick (BS) models

of the bridge were developed using SAP2000. Different types of analysis were considered on both models such as: nonlinear

static pushover, and linear and nonlinear time history analyses. A comparison was conducted between FE and BS, different

skew angles, abutment support conditions, and time history and pushover analysis. It is shown that the BS model has the

capability to capture the coupling due to skew and the significant modes for moderate skew angles. Boundary conditions and

pushover load profile are determined to have a major effect on pushover analysis. Pushover analysis may be used to predict

the maximum deformation and hinge formation adequately.

Keywords:skew bridge; seismic response; pushover analysis; time history analysis; modeling

Correspondence to: Gokhan Pekcan, Dept. of Civil and

Environmental Engineering, University of Nevada Reno, NV

89557

Tel: (775) 784-4512; Fax: (775) 784-1390

E-mail: [email protected] Assistant; Assistant Professor

Supported by: In part by the California Department of

Transportation Under Caltrans Contract No. 59A0503, and the

Dept. of Civil and Environmental Engineering (UNR)

Received: October 3, 2008; Accepted: November 3, 2008

1 Introduction

Many advances have been made in developingdesign codes and guidelines for static and dynamicanalyses of regular or straight highway bridges.However, there remains significant uncertainty withregard to the structural system response of skewedhighway bridges as it is reflected by the lack ofdetailed procedures in current guidelines. In fact, asevidenced by past seismic events (i.e., 1994 Northridge Gavin Canyon Undercrossing and 1971 San Fernando

Foothill Boulevard Undercrossing), skewed highwaybridges are particularly vulnerable to severe damage dueto seismic loads. Even though a number of studies havebeen conducted over the last three decades to investigatethe response characteristics of skewed highway bridgesunder static and dynamic loading, research findingshave not been sufficiently comprehensive to addressglobal system characteristics. Due to the fact that the

current seismic design guidelines do not provide explicitprocedures, a significantly large number of bridges areat risk with consequential threat to loss of function, lifesafety, and economy. Many of the existing bridges maybe prone to earthquake induced damage and may requiresubstantial retrofit measures to achieve the desiredseismic performance and post-earthquake serviceability.Researchers and practicing design engineers need to fullyunderstand the overall system response characteristicsof skewed highway bridges for the proper detailing ofsystem components.

It is generally agreed that bridges with skew anglesgreater than 30 degrees exhibit complex responsecharacteristics under seismic loads. Several studies haveinvestigated the effects of skew angle on the response ofhighway bridges (Maleki, 2002; Bjornsson et al., 1997;Saiidi and Orie, 1991; Maragakis, 1984). Saiidi andOrie (1991) noted the skew effects and suggested thatsimplified models and methods of analysis would result insufficiently accurate predictions of seismic response forbridges with skew angles of less than 15 degrees. On theother hand, Maleki (2002) concluded that slab-on-girderbridges with skew angles up to 30 degrees and spansup to 65 feet have comparable response characteristics

to straight bridges, and therefore, simplified modelingtechniques such as rigid deck modeling can be used inmany instances. Bjornsson et al. (1997) conducted anextensive parametric study of two-span skew bridgesmodeled with rigid deck assumption. In this study,


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416 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.7

maximum relative abutment displacement (MRAD)was considered as a critical quantity associated withfailures due to unseating. MRAD was found to beinfluenced strongly by the impact between the deck

and the abutments. Critical skew angle as a function ofspan length and width was introduced to maximize therotational impulse due to impact and was found to bebetween 45 and 60 degrees.

In evaluating and comparing results across variousanalytical studies, one must consider the underlyingassumptions and idealizations implemented in theanalytical treatment of the skew highway bridges. Thesemay pertain to material modeling, inelastic (hysteretic)response characteristics of components, as well asconditions at the boundaries, soil-structure interaction,component geometry (i.e., idealized beam-stick versus

full finite element), superstructure (i.e., rigid versusflexible), seismic mass (i.e., distributed versus lumped),etc. For instance, Meng and Lui (2000) suggested thatthe effects of modeling boundary conditions properlymay outweigh the effects of skew angle on the overalldynamic response characteristics of a bridge. In fact,differences in assumptions can create inconsistencies inresults as seen in the analysis of the Foothill BoulevardUndercrossing, which sustained severe damage duringthe San Fernando earthquake. A study conducted byWakefield et al. (1991) concluded that the failure wascontrolled by rigid-body motion, which agreed with aprevious study conducted by Maragakis (1984). But, astudy conducted by Ghobarah and Tso (1974) explainedthat the failure was induced by flexural and torsionalmotion. Ghobarah and Tso (1974) assumed the deckwas fixed at the abutments, while Wakefield et al. (1991)assumed free translation of the deck at the abutments.For the present study, a benchmark bridge is selected andthe effect of various parameters interacting with the skewangle is studied. Also, the adequacy of using simplifiedbeam-stick (BS) models and pushover analysis is testedversus FE model and time history analysis.

2 Benchmark bridge

To facilitate the objective of this study, ahighway bridge was chosen from Federal HighwayAdministrations (FHWA) Seismic Design of BridgesSeries (Design Example No.4). The bridge is acontinuous three-span box-girder bridge with 97.536 m(320 ft) total length, spans of 30.48, 36.576, and 30.48m (100, 120, and 100 ft), and 30o skew angle (Fig. 1;FHWA, 1996). The superstructure is a cast-in-placeconcrete box-girder with two interior webs and has awidth-to-span ratio (W/L) of 0.43 for the end spans and0.358 for the middle span. The intermediate bents have a

cap beam integral with the box-girder and two reinforcedconcrete circular columns. Reinforced concrete columnsof the bents are 1.219 m (4 ft) in diameter supported onspread footings. The longitudinal reinforcing steel ratioof the column is approximately 3% and the volumetric

steel ratio of the spirals is approximately 0.8%. Also, theaxial load ratio of the column is 14%. The abutmentsare seat-type with elastomeric bearings under theweb of each box girder. In the longitudinal direction,

movement of the superstructure is limited by the gapbetween the superstructure and the abutment back-wall.In the transverse direction, interior shear keys preventthe movement. This bridge was designed to be builtin the USA in a zone with an acceleration coefficientof 0.3g following 1995 AASHTO guidelines. Seismicperformance of highway bridges is expected to beaffected by a combination of various parameters suchas: skew angle, boundary conditions, superstructureflexibility, stiffness and mass eccentricity, in-spanhinges and restrainers, width-to-span ratio, and directionof strong motion components with respect to orientation

of bridge bent and abutments. This study investigatesthe effect of some of these parameters summarizes as:skew angle, abutment gap, shear keys, and direction ofearthquake motion.

3 Modeling of bridges

To facilitate a comparative parametric study of theseismic response of skewed highway bridges, a numberof detailed three-dimensional (3D) finite element (FE)as well as beam-stick (BS) models were developed(Fig. 2) using SAP2000 (2005). In all of the models,

the superstructure was assumed to be linear-elastic, andall of the nonlinearity was assumed to take place in thesubstructure elements including bents, internal shear-keys, bearings, and abutment gap (gap opening andclosing). The benchmark bridge was altered to producemodels with different skew angles, but with the sameoverall dimensions. Skew angles of 0, 30, 45, and 60degree were of interest.

For FE models, FE mesh was used to model deck,soffit, girders, and diaphragms. On the other hand, bentcolumns and footing were modeled using 3-D frameelements. For the nonlinear analyses, nonlinearity is

assumed to take place in the form of localized plastichinges at the top and bottom of columns. The behaviorof nonlinear [uncoupled] axial and moment hingesis characterized by the axial force-displacement andmoment-rotation relationship, respectively. It shouldbe noted that the coupling between moments developedabout the x-axis and y-axis was not considered inmodeling. Footing-soil interaction was modeled usinglinear translational and rotational springs at base of thefootings (Fig. 2). Nonlinear link elements were used tomodel bearings, abutment gap, and shear keys.

The bearings were designed based on a shearmodulus of elasticity (G) of 1.03 MPa (150 psi), andbearing dimensions of 0.127 m (5 in.) height and 0.165m2(256 in.2) of area that supports the bridge. The initialstiffness (k

o) is calculated to be 1.33 kN/mm (7.68 kip/

in.). The failure limit state is assumed to occur at 100%shear strain (127 mm (5.0 in.) displacement) and the


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No.4 Ahmed Abdel-Mohti et al.: Seismic response of skewed RC box-girder bridges 417

corresponding lateral load is 169 kN (38.4 kips).Gap link elements were used to model the abutment

gap which its stiffness activates only when the gap closes.On the other hand, the multi-linear plastic link elements

were used to simulate the response of internal shearkeys. Capacities of internal shear keys were estimatedbased on comprehensive study done by Megally et al.(2002). Three shear keys, one at center of each cell, werealigned along the skew angle of the bridge (Fig. 3).

For simplified BS models, the same procedure ofmodeling was followed. Nonetheless, superstructure

was represented by a single beam element having theequivalent properties of the entire deck (Table 1). Also,the bent cap was modeled using a 3D frame element witha high inertia to facilitate the force distribution to the

columns. Additional mass was assigned at abutments,mid-spans, and bent caps to account for the additionalweight of the diaphragms. The abutments are modeledby lumping the bearing, shear key, and gap link elementsfrom the FE model into individual nonlinear linkelements respectively used in BS Model (Fig. 3).

(b) Plan and elevation views

Fig. 1 Benchmark bridge geometry (*1ft = 304.8 mm and 1in. = 25.4 mm)

(a) Bent elevation

(Looking parallel to bents)

Elevation

Brg abut A 320'-0''

100'-0'' 120'-0'' 100'-0''

Brg abut B

Bent 1 Bent 2

30 Skew

(TYP)

43'-0

''

26'-0

''

Plan

BrgabutA

BrgabutB

Bent1

Bent2

Exp ExpFixPin

FixPin

43'-0'' 12'-0'' 5'-6'' 5'-6'' 12'-0''

4'-0''4'-0''

Bent

11'-3 1/8'' 11'-3 1/8''

4'-0'' Column (TYP)

Concrete hinge

(TYP)

1'-6''

6'-0''

20'-0''

3

'-6''

2'-0''

(TYP)

Column Column

14'-0''14'-0''Square FTG (TYP)

1'-0'' (TYP)

Section

8''


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4 Analysis of skew highway bridges

As mentioned earlier, the main objective of the studyis to examine the effect of various modeling assumptionson the seismic response of skewed highway bridges.Table 2 presents the analytical matrix of the study.The analysis includes modal, pushover, and linear andnonlinear time history analyses. Also, a comparison wasconducted between the simplified BS and FE models toassess the accuracy of BS models to capture response of

models with different skews. Another comparison wasdone between results of nonlinear pushover analysesversus linear and nonlinear time history analyses. Notethat analyses were conducted for two extreme cases ofboundary conditions Case I with shear keys and Case

II with no shear keys. Both pushover analysis and timehistory analyses results are discussed in more detail inthe following paragraphs.

4.1 Modal analysis

Modal analyses were conducted for each skew and fortwo abutment support conditions (Case I and Case II) todetermine the vibration modes of the FE and BS modelsdiscussed previously. Note that structural dynamic

characteristics of bridges are expected to be capturedmore accurately by the FE models. Nonetheless, basedon the comparisons between the FE and BS models,conclusions can be drawn with respect to the level ofaccuracy of approximations due to simplified BS models

(a) FE model (b) BS model

Fig. 3 Abutment support detail of bridge models

Fig. 2 FE and BS model details

(a) Finite element (FE) model (b) Beam-stick (BS) model

Column 1

(C1)

Skewangle

Skewangle

Column 3(C3)

Column 2(C2)

Z

Y X

Column 4(C4)

Nonlinear hinges(all columns)

Translational androtational springs

(all columns)

Z

Y X

Skewangle

Translational androtational springs

(all columns)

Nonlinear hinges(all columns)

Skewangle

Rigid links Footings

Skew

angle

B

B & GB & G

B & GB & G

BB

B

SKSK

SK

ZY XBBearing element

GGap elementSKShear key element

Skew

angle

B & SK

B & G

BBearing elementGGap elementSKShear key element

ZY X

Table 1 Section properties for beam-stick model

Superstructure Bent cap beam Bent column

Area (m2) 6.76 2.51 1.17

Ix- torsion (m4) 10.16 863 0.22

Iy(m4) 3.46 863 0.11

Iz(m4) 83.69 863 0.11

Density (kg/m3) 2915 2402 2402


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in general.Note that the comparisons for Case II only are

presented in Table 3. For Case II, it was observed thatthe principal longitudinal, transverse, vertical, and

coupled vibration periods predicted by both modelsare generally in good agreement. The largest percentdifference is approximately 11% if 60o skew bridge isexcluded. The modal periods start to deviate and thelargest percent difference approaches to approximately35%. However, for Case I, it was found that the principallongitudinal, transverse, vertical, and coupled vibrationperiods predicted by both models are in good agreementwith 13% relative difference as well. The largest percentdifference is approximately 6% if 60o skew bridge isexcluded. Similarly, the modal periods start to deviateand the largest percent of difference becomes 33% for

larger skew angles.

4.2 Time history analysis

Linear and nonlinear time history analysis wereconducted on FE models with different skew anglesusing seven ground motions in order to investigate

nonlinear response; effect of skew angle on seismicperformance, and effect of direction of strong componentof ground motion with respect to longitudinal andtransverse directions of the bridge models. Also, time

history analysis results were used to measure theaccuracy of pushover analysis. Since the hystereticbehavior and failure mode of the shear keys could not bemodeled in SAP2000, two separate cases were studied:one with rigid restraints at the shear key locations(Case I), and one without restraints at these locations(Case II). Time history analyses were conducted forthese two extreme cases (Cases I and II) and differentdirection of components of ground motions.4.2.1. Selection of ground motions

Design acceleration response spectrum (ARS) isassumed to be that of Caltrans with M

W of 6.5, PGA

of 0.3g, and soil type B. Four ground motions wereselected from the PEER strong motion data base (http://peer.berkeley.edu/smcat) among the historicallyrecorded motions withM

Wof 6.0 or larger and epicenter

distance of 20 km or less. These four motions wereKobe, Landers, Sylmar, and Rinaldi. All of the fourground motions were scaled to have PGA of 0.3g. The

Table 3 Percent differences between FE and BS periods for case II %

Mode desc. 0oskew Mode desc. 30oskew Mode desc. 45oskew Mode desc. 60oskew

Trans./Coup.

(1,1)*0.24

Coupled

(1,1)2.04

Coupled

(1,1)5.03

Trans./Coup.

(1,1)3.47

Longitudinal

(2,2)3.36

Coupled

(2,2)8.06

Coupled

(2,2)10.6

Long./Coup.

(2,2)0.58

Transverse

(4,4)3.11

Transverse

(4,4)2.86

Transverse

(3,4)5.26

Transverse

(3,3)9.19

Vertical

(6,6)5.31

Vertical

(6,6)1.62

Vertical

(6,6)2.68

Vertical

(4,4)22.8

Vertical (12,9) 4.67Vertical

(12,9)1.89

Vertical

(8,9)34.7

Vertical

(6,6)26.1

Vertical(12,12)

34.3 Vertical(10,9)

26.1

Vertical

(12,12)23.5

* The numbers refer to the corresponding FE and BS mode numbers (FE, BS).

Table 2 Analytical matrix

Analysis type Analysis cases Parameters

Modal analysis Modal With shear keys (Case I) Without shear keys (Case II)

Skew: 0, 30, 45, 60 Skew: 0, 30, 45, 60Pushover analysis Transverse mode,

Multimode,

Uniform load

With shear keys (Case I) Without shear keys (Case II)

Skew: 0, 30, 45, 60 Skew: 0, 30, 45, 60

Time history analysis-

Nonlinear and linear

Kobe Restrained (Case I) Unrestrained (Case II)

Landers

Sylmar

Rinaldi

Strong motion in transverse

direction / Weak motion in

longitudinal direction (ST)

Strong motion in Longitudinal

direction / Weak motion in

transverse direction (SL)

Simqke 1

Simqke 2

Skew: 0, 30, 45, 60 Skew: 0, 30, 45, 60


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remaining three ground motions were artificial groundmotions generated using SIMQKE (1999) to matchthe target 5%-damped design ARS. Figure 4 presentsa comparison of the scaled ground motions along withartificially generated motions and Caltrans ARS. Theaverage of the selected seven motions matches thedemand curve, especially within 0.4 to 0.7 seconds,which is of major interest as periods of models fallwithin this range. All four ground motions consistedof two components with different intensities, thereforetwo analysis cases were considered; the two cases

are designated as ST and SL. For the ST, the strongercomponent was applied in the transverse direction whilethe opposite was true for the SL case. On the other hand,SIMQKE motions were applied equally in both directions.

4.3 Pushover analysis

Nonlinear static pushover analyses are performed onBS and FE models developed with various skew anglesunder consideration (0, 30, 45, and 60). The objectiveswere to study: (1) the accuracy of BS models versus FEmodels in capturing the overall nonlinear behavior of thebridges through pushover analysis, (2) the applicability

of pushover analyses for relatively large skew angles,and (3) the effect of the pushover load profile.

Pushover analysis is anticipated to be sensitive to theorientation of shear keys and the pushover load profile.Therefore, three pushover load profiles are considered.These profiles are: (1) first dominant transversemode (Mode 1); (2) combination of transverse and alongitudinal modes (Mode 1+2); and (3) uniform load.Pushover capacities in terms of load and displacementare reported for all of the examined parameters as well ashinge formation sequence in columns (C1, C2, C3, andC4), which are shown in Fig. 2(a).

4.3.1. FE versus BS models resultsFor straight bridges, FE and BS models predictedcomparable load and displacement capacities when Mode1 or Mode 1+2 were applied to Case I. A slight differencewas observed between the results of both models when

uniform load profile was used for the pushover analysis.For both load profiles Mode 1 and Mode 1+2, sequenceof hinge formation varied between FE and BS models.Yielding of columns initiated at the bottom then at the

top in the FE model, and the opposite was the casefor the BS model. This was due to the difference inmodeling of the deck diaphragm between the models,which is expected to affect the load distribution and pathfrom deck to columns. However, the same sequence ofhinge formation of columns was observed between themodels when the uniform load profile was applied,although the FE model predicted larger pushovercapacities and displacement. On the other hand, a goodagreement was observed between pushover capacitiespredicted by FE and BS models for all load profilesfor Case II. The same sequence of hinge formation

was observed in Mode 1 and the uniform load profiles.For 30o skew, pushover loads predicted by both

models were in a good agreement in the first transversemode and uniform load profile for Case I. The sequenceof hinge formations was predicted consistently with bothmodes and with similar values of pushover capacitiesand displacements regardless of the load profile.Furthermore, application of the uniform load profileled to consistent results between the FE and BS modelsto predict pushover curve and sequence of yielding forCase II. This shows the adequacy of the uniform loadprofile to predict pushover capacity of bridges relyingon simplified BS models for highway bridges withmoderate skew angles.

For 45o skew, similar behavior was observed ascompared to the 30o skew. However, yielding loadand displacement values started to deviate betweenthe FE and BS models when subjected to the firsttransverse mode for Case I. For Case II, applicationof the uniform load profile remains the first choice toachieve consistency between the models (BS and FE).The transverse load profile led to the prediction of largerpushover capacities and displacements, whereas smallercapacities and displacements were predicted with themulti-mode profile in FE models as compared to the BS

models.For 60o skew, when FE and BS models were

subjected to transverse mode and uniform load profiles,similar pushover capacities were predicted for Case I.Application of the multi-mode profile predicted smallercapacity in the FE model. For Case II, the uniform loadprofile still resulted in a reasonable agreement betweencapacities between the FE and BS models, although theBS model predicted larger displacements. Application ofthe first transverse mode profile led to similar pushovercurves with larger pushover capacities predicted by the FEmodel. A fair agreement was noticed between the predict

sequence of hinge formation between the models whensubjected to transverse mode and uniform load profiles.4.3.2. Effect of skew angle

By considering each of the three load profiles,different conclusions can be drawn. Application of the

1.2

1.0

0.8

0.6

0.4

0.2

0.0

PSA(g)

0 1 2 3 4 5

Period (s)

Caltrans ARS (0.3g)Kobe EWLanders EWSylmar EWRinaldi EWSimqke 1Sinqke 2Simqke 3Average

Fig. 4 ARS spectra scaled to 0.3 PGA


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transverse mode leads to comparable hinge formationstages and first yield displacement as skew varies, exceptfor 45oskew where larger first yield displacement waspredicted for Case I. Also, major variation in predicted

pushover capacity was observed as a function of skew.For Case II, no certain yield displacement pattern wasfollowed. Also, similar pushover capacity was observeddue to the application of the transverse mode for all skewsexcept 45oskew, which predicted slightly larger capacity.

Application of the combined mode led to analmost identical pattern for Case I. Pushover load anddisplacement are similar regardless of the skew angle,while capacity of the bridge with no zero skew waspredicted to be smaller. For Case II, pushover capacitydecreases with increasing skew angle. It was observedthat 45ohas the smallest pushover capacity but it is close

to that of 60o

skew model.Application of the uniform load profile led to lower

pushover capacity and displacement as skew increases,

contradicting with results of the other load profiles forCase I (Fig. 5). For Case II, it predicted smaller pushovercapacities than in the case of the transverse mode profile(Fig. 6).

4.3.3. Effect of shear keyFigure 7 summarizes the effectiveness of shear key

for all load profiles. Application of the transverse modeshows that effectiveness of the shear keys contributionto pushover capacity is independent of skew angle. Asexpected, pushover capacity of Case I is larger thanCase II due to the absence of the shear key, whichessentially resists lateral loads. Nonetheless, the locationof the first yield and ultimate capacity remain the sameamong models with different skews. On the other hand,application of the combined mode suggested that theshear key does not provide a major additional capacity

as the skew changes. The application of uniform loadshows that the effectiveness of the shear key is dependenton skew angles; it reduces as skew angle increases. For

Uniform load (finite element)

Skew angle Stage Displ. (mm) Base shear (kN) Description

0 I 17 10399 Yield at bottom of C1, C2, C3, C4

II 61 22163 Yield at top of C1, C2, C3, C4

30 I 16 8646 Yield at bottom of C3

II 30 13552 Yield at bottom of C1, C2, C4 and at top of C2, C3, C4

III 48 16102 Yield at top of C1

IV 95 16905 Failure at top of C3


II 16 6801 Yield at bottom of C1, C2, C4

III 31 9694 Yield at top of C1, C2, C3, C4

IV 114 11471 Failure at top of C3

60 I 13 5131 Yield at bottom of C3 and C4

II 15 5568 Yield at bottom of C1 and C2

III 29 7756 Yield at top of C1, C3, C4

IV 52 8998 Yield at top of C2

V 130 8922 Failure at top of C3 and C4

Fig. 5 Uniform load pushover curves at various skew angles: Case I shear keys effective

30

25

20

15

10

5

0

Base

shear(103kN)

0 30 60 90 120 150

Displacement (mm)

0 Skew30 Skew45 Skew60 Skew


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moderate skew (< 30o), larger pushover capacity and

smaller displacement is predicted by Case I; while forlarger skew (> 45o), it suggests that the shear keys donot have a significant effect on pushover capacity anddisplacement.

4.4. Comparison between time history and pushover

analyses

In order to investigate the accuracy of pushoveranalyses in predicting the lateral capacity of skewedhighway bridges, individual maximum responsesfrom nonlinear time history analyses are plotted onthe pushover curves. These data are plotted for the FEmodels only (Figs. 8 and 9). The maximum resultantbase shear and maximum displacement at the center ofthe midspan (control node) in the transverse directionare plotted for each of the seven ground motions. Twoabutment support conditions are considered: Case I

(shear key/restrained) and Case II (without shear key/

not restrained) with two cases of ground motions. Thefirst case, ST, applies the stronger component in thetransverse direction of the bridge while the second case,SL, is the opposite. It should be noted that additionaltime history analysis was done using Kobe and Sylmarmotions scaled to 0.5g for Case I and ST case andcompared against pushover curves.

For Case I, the time history results compare wellfor the 0 degree skew. Both ST and SL cases followthe Mode 1+2 and Uniform Load pushover curves verywell; although, the base shear capacity is slightly overpredicted by the Uniform Load pushover curve. Thetime history results tend to cluster around the first hingeformation on the pushover curves. The first transversemode pushover curve predicts the closest response for0.5g ground motions. For the 30 degree skew, all thepredicted pushover curves compare well to the timehistory results for the ST and SL cases; although, the

Fig. 6 Uniform load pushover curves at various skew angles: Case II shear keys failed

12

10

8

6

4

2

0

Baseshear(103k

N)

0 30 60 90 120 150

Displacement (mm)

0 Skew30 Skew45 Skew60 Skew

Uniform load (finite element)

Skew angle Stage Displ. (mm) Base shear (kN) Description

0 I 13 5241 Yield at bottom of C1, C2, C3, C4

II 23 6886 Yield at top of C1, C2, C3, C4

III 144 8674 Failure at top of C1, C2, C3, C4

IV 144 3647 Failure at bottom of C1, C2, C3, C4


II 26 7944 Yield at bottom of C3 and C4 and at top of C1, C2, C3, C4

III 135 10344 Failure at top of C1, C2, C3, C4


II 47 9065 Yield at bottom of C3 and C4 and at top of C1, C3, C4

III 75 10664 Yield at top of C2


II 17 5612 Yield at bottom of C1, C3, C4

III 30 7510 Yield at top of C1, C3, C4

IV 35 7947 Yield at top of C2

V 132 8883 Failure at top of C4


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Fig. 7 Effect of shear keys on pushover curves at various skew angles

Transverse mode Mode 1+2 Uniform load

30

20

10

0

30

20

10

0

30

20

10

0

30

20

10

0

Baseshear(103kN)

Baseshear(103kN)

Baseshear(103k

N)

Baseshear(10

3kN)

0 35 70 105 140 0 35 70 105 140 0 35 70 105 140Displacement (mm)

0 Skew

30 Skew

45 Skew

60 Skew

Case I (w/shear keys)

Case II (w/o shear keys)


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Fig. 8 Comparison of time history results and pushover capacities for 30skew, ST direction, Case I - shear keys effective

30 skew stages

Finite element

Stage Description

Mode 2 I Yield at top of C1 and C2

II Yield at top of C3 and C4 and at bottom of C1, C2, C3, C4

Mode 1+2 I Yield at top of C2

II Yield at top of C1, C3, C4 and at bottom of C1, C2, C3, C4

III Failure at bottom of C2 and C3

Uniform load I Yield at bottom of C3

II Yield at bottom of C1, C2, C4 and at top of C2, C3, C4

III Yield at top of C1

IV Failure at top of C3

Kobe

Kobe (0.5g)

Landers

Sylmar

Sylmar (0.5g)

Rinaldi

Simqke 1

Simqke 2

Simqke 3

# Pushover stages

Fig. 9 Comparison of time history results and pushover capacities for 60skew, ST direction, Case I - shear keys effective

Kobe

Kobe (0.5g)

Landers

Sylmar

Sylmar (0.5g)

Rinaldi

Simqke 1

Simqke 2

Simqke 3

# Pushover stages

60 skew stages

Finite element

Stage Description

Mode 1 I Yield at bottom of C1, C2, C3, C4

II Yield at top of C1, C2, C3, C4

Mode 1+2 I Yield at bottom of C3

II Yield at bottom of C1, C2, C4 and at top of C2 and C3

III Yield at top of C1 and C4

IV Failure at top of C3

Uniform load I Yield at bottom of C3 and C4

II Yield at bottom of C1 and C2

III Yield at top of C1, C3, C4IV Yield at top of C2

V Failure at top of C3 and C4

30

20

10

0

Baseshear(103kN)

0 40 80 120

Displacement (mm)

0 40 80 120

Displacement (mm)0 40 80 120

Displacement (mm)

Mode 2 Transverse mode Mode 1+2 Uniform load

Baseshear(103kN)

Baseshear(103kN)

30

20

10

0

30

20

10

0

30

20

10

0

Baseshear(10

3kN)

0 40 80 120

Displacement (mm)

0 40 80 120

Displacement (mm)0 40 80 120

Displacement (mm)

Mode 2 Transverse mode Mode 1+2 Uniform load30

20

10

0

Baseshear(10

3kN)

Baseshear(10

3kN)

30

20

10

0


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Mode 1+2 pushover curve underpredicts the responsefrom the Kobe and Sylmar motions for the ST case(Fig. 8). The Kobe, Landers, and Sylmar motions arecloser to the fully yielded mechanism predicted by

the pushover curves while the Rinaldi and SIMQKEmotions are closer to the first hinge formation. Onceagain, the first transverse mode profile gives the betterprediction for ground motions beyond the design level,but the Uniform Load also gives satisfactory results. Forthe 45 degree skew and ST case, the three SIMQKE andLanders motions fall below the pushover curves whilethe Kobe, Rinaldi, and Sylmar motions suggest that thepushover curves predicts smaller maximum base shear,except for the first transverse mode profile. The timehistory responses are clustered around the fully yieldedmechanism for Mode 1+2 and the Uniform Load. For

0.5g ground motions, the first transverse mode pushovercurve accurately predicts the maximum response fromthe time history analyses. For the SL case, the pushovercurves over predict the maximum response due tothe three SIMQKE motions. In both Mode 1+2 andUniform Load, the maximum responses from the timehistory analyses are clustered around the fully yieldedmechanism. For the 60 degree skew shown in Fig. 9,the time history results compare well with the Mode 1+2pushover curve for both cases. The first transverse modepredicts a larger capacity and the 0.5g ground motionresults do not compare well with any of the pushovercurves; although, the results are conservative with respectto the first transverse mode pushover curve. For the STand SL cases, the maximum responses are clusteredin between the partial and fully yielded mechanisms.

For Case II, for the ST and SL cases, the firsttransverse mode and Uniform Load pushover curvesunderpredict the response from all seven groundmotions for the 0 degree skew. The Mode 1+2 pushovercurve compares well with the time history results. Allthe motions cluster near the fully yielded mechanismfor both cases. For the 30 degree skew, the pushovercurves and time history results compare well for the STand SL cases; although the first transverse mode and

Uniform Load pushover curves slightly underpredict thetime history responses, especially for the SL case. Onceagain, the motions are clustered around the fully yieldedmechanism. For the 45 degree skew, the seven groundmotions are more scattered, especially the Kobe andLanders motions. The scatter for the ST case followsthe first transverse mode and Uniform Load pushovercurves exactly. The Kobe and Landers motions clusteraround the fully yielded mechanism while the maximumresponses due to the rest of the motions fall near thepartially yielded system. For the SL case, all threepushover curves slightly underpredict the time history

responses. For the 60 degree skew, the three pushovercurves once again slightly underpredict each groundmotion response for the ST and SL cases. All of theground motions are clustered near the fully yieldedmechanism for both cases.

5 Conclusions

As mentioned earlier, the behavior of skewedhighway bridges is complex and modeling assumptions

affect the predicted seismic performance. In this study,various parameters were studied such as: skew angle,shear key effect, direction of components of groundmotions, as well as the adequacy of the pushover methodto estimate the lateral capacity of skew bridges. Thefollowing conclusions are drawn:

(1) Predicted modal properties with the FE and BSmodels were comparable; the BS model was successfulin capturing the modal coupling due to the skew andthe significant modes needed for further analysis.Nonetheless, FE models should be considered whendealing with very large skew angles (> 30) in order to

capture the higher mode effects.(2) Boundary conditions (hence modeling

assumptions) have a significant effect on pushoveranalyses and choice of pushover load profile.

(3) For Case I and II, the Uniform Load profile wasthe most consistent in predicting similar sequencesof hinge formation between the FE and BS models.Ultimately, the BS model, when compared to theFE model, accurately captured the overall nonlinearbehavior of the bridge when using the Uniform Loadprofile.

(4) Based on the pushover analyses, each load profileconsidered suggested a different conclusion regardingthe effectiveness of the shear keys; while the UniformLoad profile results suggested that the effectiveness ofthe shear keys reduces with skew angle, the Multimodeprofile results suggested that the shear keys do notnecessarily improved the pushover capacity, and theTransverse Mode profile results suggested that shearkeys are effective in providing additional capacity that isfully effective despite the skew angle.

(5) The direction of the two horizontal componentsof the strong motions relative to the longitudinal andtransverse directions did not have any significant effecton the overall response.

(6) The pushover curves considered in this studypredicted the maximum deformation response and thesequence of hinge formations reasonably accuratelywhen compared to response predictions from individualnonlinear time history analyses.

(7) For larger skew angles, the results from thenonlinear time history analyses agree with the observedyield mechanism in the columns from the pushoveranalyses.

References

American Association of State Highway andTransportation Officials (AASHTO) (1998), LRFDBridge Design Specifications, 2nd Edition, WashingtonD.C.


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seismic response of skewed rc box-girder bridges

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