Nonlinear Continuity Analysisof Precast, Prestressed ConcreteGirders with Cast-in-PlaceDecks and DiaphragmsAmir Mirmiran, Ph.D., P.E.

ProfessorDepartment of Civil EngineeringNorth Carolina State University

Raleigh, North Carolina

Siddharth KulkarniStructural EngineerBechtel CorporationHouston, Texas

Reid Castrodale, Ph.D., P.E.Associate & Senior EngineerRalph Whitehead Associates

Charlotte, North Carolina

Richard Miller, Ph.D., RE.Associate ProfessorDepartment of Civil andEnvironmental EngineeringUniversity of CincinnatiCincinnati, Ohio

Makarand Hastak, Ph.D.Assistant Professor

School of Civil EngineeringPurdue University

West Lafayette, Indiana

An analytical study was carried out to determine

whether and how much the performance of

continuity connections for precast, prestressed

concrete girders with cast-in-place decks is

affected by positive moment reinforcement in

continuity diaphragms. A flexibility-based

analytical tool is developed that predicts

time-dependent restraint moments and the

effectiveness of the continuity connection under

service live loads. The model considers the

different nonlinear stress-strain responses of the

continuity diaphragm and the girder/deck

composite sections, and the change in the

stiffness of the structure under time-dependent

effects. The study confirms previous findings

that total midspan moments are virtually

independent of the amount of positive moment

reinforcement provided. This, however, does not

mean that positive moment connections are

unnecessary. Cracking of the diaphragm in the

absence of such connections significantly

reduces the effectiveness of continuity for

service live loads and may raise durability

concerns. Based on the analytical results to date,

a minimum amount of positive moment

reinforcement is recommended to avoid a

significant loss of continuity and to control

cracking of the diaphragm under service loads.

60 PCI JOURNAL

Continuity in prestressed concrete bridges has several benefits. It provides redundancy foroverload conditions or extreme events,enhances the riding surface for vehicles, and improves the durability ofthe bridge by eliminating joints at thesupports. It also improves structuralbehavior, which may reduce construction costs by increasing span lengthsor girder spacing.

Continuity of precast girders can beachieved by providing continuous reinforcement in the deck over the piersand a concrete diaphragm between theends of the girders at interior supports.This type of connection has been usedsuccessfully in several states for manyyears. Fig. 1 shows the sequence ofconstruction. The girders act as simplespan members for dead loads, beforethe continuity connection is cast. Oncethe continuity diaphragm and deck arecast, the composite girder/deck sectionwill carry live loads and superimposeddead loads as a continuous structure.

Time-dependent effects such ascreep, shrinkage and thermal gradientcause restraint moments in the continuity connection. The continuity connection is subject to negative and positivemoments. Reinforcement is providedin the deck over the support to resistnegative moment. Positive moment reinforcement extends from the ends ofthe girders into the diaphragm.

In the last 40 years, this type of connection has been the subject of severalstudies. A brief overview of each investigation follows:

1. Around 1960, the Portland Cement Association (PCA) carried outlaboratory and analytical studies onthe continuity of precast girders with acast-in-place deck and a continuity diaphragm with two types of positivemoment connections, i.e., straight barswelded to a structural angle, andhooked bars.’-6 The welded connections were found to be adequate, whilethe hooked connections fractured prematurely in fatigue because of a verytight (non-standard) bend radius.

Continuity behavior was evaluatedfor two two-span specimens, one withno positive moment connection andthe other using the hooked bar detail.A continuity connection for bothspecimens was cast at the girder age

of 28 days. The specimen with nopositive moment connection crackedand lost much of its rigidity at thecenter support to provide continuityfor live loads. The degree of continuity as a ratio of the elastic momentwas as low as 18 percent at the timeof crack closure, but had risen to 56percent at service load.6 However,positive moment cracking did not affect the ultimate capacity of the negative moment connection.

The specimen with hooked barsshowed no cracking in the diaphragm,but might have cracked if the test duration had been long enough. The continuity moment in this specimen wasreduced to about 80 percent of theelastic moment. These studies resultedin a design method7 to estimate restraint moments due to creep andshrinkage. It also recommended limiting the stress in bent bars in the diaphragm to 60 percent of yieldstrength under live loads and time-dependent effects.

2. In the late 1970s, as part of theMissouri Cooperative Highway Research Program, the University ofMissouri-Columbia studied the feasibility of extending strands into thecontinuity diaphragm to develop apositive moment connection.8-”Threestrand configurations were tested forthe connection, i.e., straight, frayed(untwisted) and 90-degree bent. Thebent strands provided the best anchorage with half the slip of the other two

types. Based on full-scale tests, a design method was proposed, which limited the stress in the strand to 15 percent of its ultimate strength to avoidfatigue failure. The study also suggested that a continuity diaphragmcould be cast before the deck.

3. In the late l980s, under a National Cooperative Highway ResearchProgram, the Construction Technology Laboratories (CTL) performed analytical studies on this type of continuous bridge and developed a programBridgeRM to predict time-dependentrestraint moments.’2The study indicated that positive moments generatedby time-dependent effects may crackthe continuity connection. It alsoshowed that with positive moment reinforcement in the form of bent bars,the connection is more resistant tocracking, but restraint moments will inturn be larger. Therefore, both reinforced and unreinforced connectionsmay be expected to crack eventually.The study showed that the totalmidspan moments are virtually independent of the positive moment reinforcement provided in the continuitydiaphragm. Finally, the study concluded that positive moment connections are difficult, time-consuming andcostly to install, and have no structuralbenefit.

4. In the mid-l990s, the IndianaDOT sponsored a study at PurdueUniversity to evaluate restraint moments in precast, prestressed concrete

Precast PrestressedConcrete Girders

(a) Erect Simple Span Precast Prestressed Concrete Girders

Negative Moment ReinforcementPositive Moment Reinforcement

Deck

Fig. 1. Continuity connection in precast, prestressed concrete bridges.

September-October 2001 61

bridges.’3”4Six two-span bridges weretested, three with AASHTO Type Igirders,’3 one with a 27 in. (686 mm)box girder,’3 and two with 6 in. (153mm) form panels.’4Bent strands wereembedded into the continuity diaphragm for all specimens. The studyshowed that neither the CU methodnor the PCA method could provide anaccurate prediction of the restraint moments. Therefore, an empirical designmethod was proposed to take into account the length and relative stiffnessof the diaphragm.’4

5. Studies at the University of Nebraska, Omaha, in the mid-1990s, explored different methods of establishing continuity using the NebraskaNU-Type girders.’5”6It was concludedthat the construction sequence has asignificant effect on the positive restraint moments. It was recommendedthat if the continuity diaphragm is castbefore the deck, the deck should becast within 230 days to prevent cracking. It was also suggested that, if thediaphragm is cast first, a negative mo

ment connection should be providedbetween the girders to prevent cracking and spalling at the joint betweenthe diaphragm and the deck. The studies further indicated that, if the deckand diaphragm are cast at the sametime, an unbonded joint should be provided between the diaphragm andgirder to allow end rotation of thegirder under the weight of the deck.

6. In England, Clark and Sugie,’7inthe late 1990s, studied the positive andnegative moment connections of precast girders. They suggested that, instead of calculating the effect of creepand shrinkage, the connection shouldbe designed for a positive moment of550 kip-ft (750 kN-m) for spans in the65 to 120 ft (20 to 36 m) range wherethe beams are at least 42 in. (1.1 m)deep. For smaller beams, they suggested designing for a positive moment of 440 kip-ft (600 kN-m).

7. Rabbat and Aswad,’8 in the early1990s, reviewed standard details forcontinuity diaphragms used in Tennessee and elsewhere, which had been

developed based on the PCA method.7They showed that these details providepositive moment reinforcement equivalent to a range of 1.19 to 1.70 timesMcr, where Mc,. is the cracking moment of the diaphragm concrete withthe cross-sectional shape of the composite girder/deck section. Therefore,as a simple approach, they proposedusing 1 .2Mcr as the minimum positivemoment reinforcement in the continuity diaphragm. No analysis for restraint moments due to time-dependent effects would then be required.

It is clear from the above studiesthat, while there is little question onthe design of negative moment reinforcement for continuity, there are twoimportant issues regarding the positivemoment connection:

(a) How much, if any, positive moment reinforcement should be provided in the continuity diaphragm?;and

(b) If provided, how much restraintmoment is developed from time-dependent effects, and how effective is

Fig. 2. Typical moment-curvature diagrams for girder/deck and diaphragm sections.

Girder/Deck at Mid-SpanGirder/Deck at Support

-o.o-o.o-o- Continuity Diaphragm with 2.4 M,Positive Moment Reinforcement

I I H— Continuity Diaphragm with 1.2 McrPositive Moment Reinforcement

• Continuity Diaphragm with 0.6 MerPositive Moment Reinforcement

Continuity Diaphragm with 0.1 McrPositive Moment Reinforcement

Curvature (1W3 rad/m)

-40 -30 -20 -10 0 10 20 30 404000! • • • I • • • I • • • I • •

______

5000

3000! 4000

20003000

2000

.t” 1000.OOOOOOOOOOO

Iii I 11111111111111 I 11111111 III 1000

•

E 0 ‘ . ““1111”II I”. iiiI.IIUflhIHIItIt 0 E0

-1000-1000

IIIIrIIIIiIiiiiiiii.. —2000

-2000-3000

-3000

-1000

AASHTO Type III Girder

20-1/2 in. ‘I’ Strands(18 Straight, 2 Draped)

8 in. Deck with 7.04 sq. in.Negative Moment Reinforcement

-750 -500 -250

_I_ I I I I I

0 250

Curvature (106 rad/in)

500 750

-4000

1000

62 PCI JOURNAL

the connection in maintaining the continuity for service loads?

ANALYTICALMODEUNG

In order to evaluate the above issues, a flexibility-based analyticaltool has been developed to predictthe time-dependent response of precast, prestressed concrete girdersmade continuous. The study takesinto account the creep and shrinkage effects, prestress losses, age atloading, and construction sequence.The nonlinear stress-strain responseof materials and the change in thestiffness of the members caused bytime-dependent effects are alsoconsidered.

This analytical tool can be used toassess restraint moments as a functionof positive and negative reinforcementin the continuity connection, as well asthe length of the diaphragm. It canalso be used to evaluate how effectively the connection maintains conti

nuity, should it crack under time-dependent effects. The different components of the model are describedbelow:

Time-Dependent EffectsTime-dependent effects in pre

stressed concrete bridges includecreep and shrinkage of concrete,relaxation of steel, and the increase ofconcrete strength over time. Thermaleffects may also cause restraint moments in the bridge, but these were notconsidered in this study.

Creep of concrete results from thesustained load of prestressing and thedead weight of the bridge. Simple-span girders are free to deform with norestraint. However, if the girders aremade continuous, creep deformationswill be restrained at the continuityconnections.

The creep effect from prestressing ispartially counteracted by the effect ofdead loads. Creep due to prestressmakes the girders camber up andcauses positive restraint moments.

Creep due to dead loads, on the otherhand, results in negative restraint moments. The relative size of the two opposing effects depends on several factors, including the age at whichcontinuity is established.

Shrinkage is the reduction of concrete volume due to loss of moisture.The most significant effect ofshrinkage in continuous bridges isthe differential shrinkage that occursbecause of the difference in type andage of girder and deck concrete.19Since girders are cast before thedeck, most of the girder shrinkagemay have already occurred beforethe deck concrete is placed. Thegirder will, therefore, act to restrainthe shrinkage of the deck concrete.Differential shrinkage between theprecast girder and the deck typicallycauses a downward deflection (sag).If the girders are made continuousby a cast-in-place continuity diaphragm, their end rotations will berestrained, causing negative moments in the diaphragm.

Fig. 3. Comparison of the CTL method12 with present study using uniform linear response.

0 5 10

Time from Transfer of Prestress (Years)

15 20

800

I

600

400

0

200

-200

-400

0 1000 2000 3000 4000 5000 6000 7000 8000

Time from Transfer of Prestress (Days)

-600

-800

September-October 2001 63

The incremental change in theshrinkage moment at each time step isgiven by:

= AESfEdAd(e+

for deck age 28 days

AM= Ar5EdAd ( t

EA1+ di d

EgAg

for deck age > 28 days

(1)

(2)

where z1 is the incremental differential shrinkage strain, Ad and Ag arethe cross-sectional areas of deck andgirder, respectively, e is the distancefrom the centroid of the compositesection to the bottom of the deck, t isthe slab thickness, and EdI and Eg arethe modulus of elasticity of the deckconcrete (at time t1) and the girder, respectively.

The deck reinforcement can furtherrestrain shrinkage of the deck. This isaccounted for by the Dischinger effectfactor,’2’13 which is multiplied by deckshrinkage strain at each time step.

In the present study, both creep andshrinkage strains are estimated usingthe ACI 20920 method, including correction factors for relative humidity,

volume-to-surface ratio, and age atloading (for creep). The present studyalso uses the general equation fromACT 209 to estimate the increase inconcrete strength with time.2°

Mattock5 suggested that the creepand shrinkage restraint momentsshould be multiplied by a creep effectfactor of (1 — e) and (1 — ejl4), respectively, where 4) is the creep coefficient at each time step. The ultimatecreep coefficient for concrete typescommon to prestressed concretebridges is between 1.5 and 2.5.20 Thecreep effect factors computed usingthis range of creep coefficients varyin a much narrower range of 0.78 to0.92 for creep and 0.52 to 0.37 forshrinkage.

Relaxation is the loss of stress insteel held at a relatively high constantstrain for a period of time. Relaxationcontinues for an indefinite period, although its rate decreases with time.Losses due to steel relaxation beforetransfer of prestress and elastic shortening of the girder at transfer are calculated using appropriate coefficientsfor stress-relieved and low-relaxationstrands.

The present study follows the procedure of the PCI Committee2’for estimating prestress losses, but uses the

correction factors of the ACT 20920 forloading age, relative humidity and volume-to-surface ratio. Prestress lossesare calculated from the time of transfer using a time-step analysis. Foreach time step, the losses due to creep,shrinkage and relaxation are subtracted from the prestress at the previous time step to calculate the prestressfor the current time step.

The creep loss is calculated by multiplying the creep coefficient of thematerials by the modular ratio between prestressing steel and concrete.The shrinkage loss for each time stepis obtained by multiplying the changein shrinkage strain by the modulus ofelasticity of steel. The loss due to relaxation is obtained using the strandstress from the preceding time step.The additional stress in the strands dueto dead load of the deck is taken intoaccount after the deck is cast. This increase in strand stress is equal to theadditional concrete stress at the levelof the centroid of strands multipliedby the modular ratio.

Moment-Curvature Analysis

To carry out the flexibility analysis,the moment-curvature relationship ofthe girder/deck section and the continuity diaphragm section are required.

Fig. 4.Cross section of the

PCA test girders.5 39.00”

13.00”

26 strands 1/4”dia.(1’ on center vertical

64 PCI JOURNAL

The program RESPONSE, developedby Collins and Mitchell,22 was usedfor this purpose.

The program RESPONSE developsthe entire moment-curvature relationship of a prestressed or reinforcedconcrete section subjected to moment,axial load and shear. It has a numberof modeling options, including up tofive different types of concrete withdifferent stress-strain curves, up tofive different types of mild reinforcingsteel with bilinear or trilinear stress-strain curves, and up to five differenttypes of prestressing steel using themodified Ramberg-Osgood functionfor the stress-strain response.

Typical moment-curvature diagrams are shown in Fig. 2 for threedifferent sections along a bridgecomprising AASHTO Type III girders with straight and draped strands.The thick and thin lines in the figureshow the moment-curvature relationships for the midspan and end sections, respectively, where the strandsare in their lowest and highest positions, respectively.

The moment-curvature relationshipsfor the conventionally reinforced continuity diaphragm section are shownfor four different amounts of positivemoment reinforcement equivalent to0.1, 0.6, 1.2, and 2.4 times Mer. Thereinforcement in the positive momentconnection is assumed to be fully effective across the diaphragm. This as-

sumption will be evaluated in the experimental phase of this study. In allfour cases, the same amount of reinforcement is provided in the deck toresist negative design moments. Modeling of the effective width of the diaphragm is discussed in more detaillater in this paper. No tension stiffening was included in the analysis.

Flexibility Method forTime-Dependent Analysis

Both the PCA method7 and the CTLmethod12 (BridgeRM) use momentdistribution factors for calculating therestraint moments. This assumes constant stiffness along the bridge. However, as shown in Fig. 2, the stiffnessalong the bridge (and especially overthe support) is not uniform mainly dueto the difference in concrete strengthand the amount and type of reinforcement. Also, nonlinear behavior such ascracking in the continuity diaphragmor the deck will further reduce thestiffness at those locations along thebridge under live loads and time-dependent effects. These changes instiffness along the bridge will result inmoment redistribution.

The present study addresses theseissues by adopting a flexibility-basedanalysis, which comprises two modules, i.e., a time-dependent analysisfor any given time period, and a liveload continuity analysis at any timeafter continuity is established.

The analysis is performed incrementally over the specified time period.Each span and diaphragm is dividedinto a number of segments. The moments due to the differential shrinkageand creep effect of prestressing are assumed constant in the span and zero inthe diaphragm. The moment due to thecreep effect of dead load is parabolicin the span and zero in the diaphragm.

The flexibility analysis is carried outusing the interior support reaction asredundant by making the structure determinate. The total moment at eachsection is then calculated by addingthe moments due to time-dependenteffects (i.e., creep from prestress anddead load, and differential shrinkage),the moment due to dead load of thecomposite section in the span, and themoment due to the assumed redundantreactions at the interior supports.

Once the total moment is established at each section, the programfinds the corresponding curvaturefrom the appropriate moment-curvature relationship. Curvatures for sections between the end of the precastgirder and the hold-downs for drapedstrands are found by interpolating between the responses of the end section and midspan of the girder. Fromthe total moment and total curvature,the moment and curvature due todead loads are subtracted to arrive atthe moments and curvatures due totime-dependent effects. This proce

Fig. 5. Positive moment connection in PCA5 Girder 3/4.

Ends of Rec8st Grders

rN[

C. — ------1-

-

- .3-ord ba th hook endsentedded in the ends of precast girders

ELEVA11ON

No.4 -Bar Cap(,ragn Reinfornnt

SEC11ON

September-October 2001 65

Table 1. Data used for the PCA tests.5’24 dure is carried out at different see-

Material Data measured by PCA5 Data assumed by Suttikan

Prestressing steel = 175 kips (778 kN) Modified Ramberg-OsgoodStress-strain function for

f,,,. = 280 ksi (1930 MPa) stress-relieved strands

f = 254 ksi (1751MPa)

E= 28,700 ksi (198000 MPa)

Negative moment f3, = 48.5 ksi (334 MPa)reinforcement E, = 29,500 ksi (203400 MPa) and

-- elastic-perfectly-plastic behaviorPositive moment f = 50 ksi (345 MPa)

reinforcement

Moist cured, Type III cementGirder concrete (5.2 bags per cubic yard)

f2s = 5.45 ksi (37.6 MPa) 150 lb/ft3 (2403 kg/m3 ) unit weight-—-------

-------———-—----——— E, = 33 w’5f’°5Deck and continuity Moist cured, Type III cement ACT 209 time and loading age factorsdiaphragm concrete (5.2 bags per cubic yard) High strength Stress-strain curve

.f-28 = 4.82 ksi (33.4 MPa)

Relative humidity — 70 percent

Ultimate creepcoefficient in girder — 2.197and deck concrete

Ultimate shrinkage strain — 667 3 micro strainsin girder concrete

Ultimate shrinkage — 640.4 micro-strains

strain in deck concrete

Fig. 6.Time-dependent

variation of centersupport reaction for

PCA5Girder 3/4.

tions along the span and the di

aphragm to obtain the curvature dia

gram for each time step.

The deflection at the interior support

is then calculated by the moment-area

method using the curvature diagram.

The program iterates on the interior

support reaction to eliminate the de

flection at the support. The moment

thus obtained in the diaphragm section

is the time-dependent restraint mo

ment at the interior support.

Flexibility Method forLive Load Continuity Analysis

In order to determine the degree of

continuity for live loads, a second mod

ule was developed in conjunction with

the time-dependent analysis. At a speci

fied age, live load is applied in the form

of two equal point loads acting at the

center of each span of a two-span

bridge. This type of loading was se

lected to allow a comparison with the

tests of PCA5 and others. The live loadis then normalized with respect to equivalent service live loads, which would

generate the same moments caused by

the HS-20 loading with appropriate im

pact and distribution factors.23

3

2

PCA lests5

Suttikan’s PBEAM Program24

Mattock’s Effective Modulus Method5

Mattock’s Rate of Creep Method5

— Present Study

‘

•ll

“C

I

CsC6DI5i

Cs

CsCs

C.’Cs

0

—1’

6C

Cs.Cs

riD

CsC.’c

CsCsC

C.’CSC.’

-2

0

-4

150 300 450

Time from Removal of Deck Formwork (Days)

600

-9

750

66 PCI JOURNAL

The same flexibility-based analysisis used as described in the previoussection, except that live load momentsand curvatures are now added to thecorresponding dead load and time-dependent moments and curvaturesfrom the last step of the time-dependent analysis. The live loads are applied incrementally, until the maximum corresponding curvature in agirder/deck section along the bridge orin the diaphragm section is exceeded,triggering failure. The program calculates live load moments at interiorsupports and midspan, and comparesthem with the respective theoreticalelastic moments based on a full continuity assumption.

A Continuity Index is defined andcomputed as the ratio of live load moment at center support (or midspan)obtained from the nonlinear analysisto the corresponding elastic moment atthat location assuming full continuity.An index of one represents full continuity. The index is less than one at theinterior support, and greater than oneat midspan. This indicates moment redistribution as the relative stiffness ofthe diaphragm and span change underpositive and negative moments. Thecontinuity indices are found for various levels of live load up to failure.

Verification

To verify the accuracy of the proposed method of analysis, its resultswere first compared to linear elasticmodels such as the CTL method12(BridgeRM) by ignoring the effects ofdifferent stiffness of the diaphragmand the girder/deck sections and theircracking. For this verification, thesame uniform linear moment-curvature relationship was assumed for allsections. A two-span bridge was considered with 65 ft (19.8 m) longAASHTO Type III girders at 8 ft (2.4m) spacing, a 2 ft (0.6 m) long continuity diaphragm, and an 8 in. (203mm) thick cast-in-place compositeconcrete deck.

Negative reinforcement in the deckand prestressing in the girder were designed based on the AASHTO Standard Specifications.23Eighteen straightstrands were used with their effectivecentroid at 3.9 in. (99 mm) from thebottom of the girders. Two draped

strands were placed at 43 in. (1.1 m)from the bottom of the girders at theend section, and at 5 in. (127 mm)from the bottom at the hold-downs.

The ultimate creep coefficient ofconcrete was assumed to be 2.3, andthe ultimate shrinkage strain was takenas 600 micro-strains for concrete inboth the girder and the deck. Fig. 3confirms that the proposed method ofanalysis asymptotically approaches theCTL method,12 if uniform linear elastic response were incorporated alongthe bridge.

Additional verifications were madeagainst the PCA tests5 for the twohalf-scale I-girder specimens. Each

girder had a span length of 33 ft (10m), and was prestressed with 28 — 1/4

in. (6.4 mm) diameter seven-wirestress-relieved straight strands at aninitial prestressing force of 175 kips(778 kN). Fig. 4 shows the specimencross section.

One specimen (Girder 3/4) incorporated a positive moment connectionusing four No. 3 bars with hookedends extending from the girders (seeFig. 5). As shown in the figure, thebars overlapped in the diaphragm. Theother specimen (Girder 1/2) had nopositive reinforcement in the diaphragm. The girders were cast a dayafter the strands were tensioned, and

Total Live Load (kN)

S

I

Cl)I

a

aaS

a0

Total Live Load (kips)

Fig. 7. Variation of center support reaction due to live loads for PCA5 Girder 3/4.

September-October 2001 67

the prestressing force was releasednine days after tensioning.

The girders were loaded with 800 lb(3.56 kN) concrete blocks at 3 ft (0.9m) spacing to compensate for thelesser weight of the half-scale model.For both specimens, the deck was castalong with the 3 in. (76 mm) continuity diaphragm when the girders were28 days old. The deck formwork wasremoved at the girder age of 35 days.Both long-term measurements and intermittent load tests were carried out.

Moment-curvature relationships forthe girder/deck and diaphragm sections were developed using the RESPONSE22 program. The diaphragmsection was modeled as a T-sectionwith its web width equal to the widthof the bottom flange of the girder.Table 1 summarizes the measured5andassumed data for the PCA tests.

Fig. 6 shows test data for the centersupport reaction due to time-dependent effects up to 680 days from removal of deck formwork. For comparison purposes, the same figure also

shows analytical results from the present study, the program PBEAM,24rate of creep method5 and the effective modulus method.5 It can be seenthat the nonlinear flexibility analysisdeveloped in the present study provides a much closer agreement withtest data compared to the other analytical models.

The present study slightly overestimates the effect of differential shrinkage in the early ages because the creepproperties of the girder and deck wereassumed to be the same. The studyalso predicts an initial uplift force assoon as the deck formwork is removed, because continuity was assumed to initiate when the diaphragmwas cast. However, differential shrinkage was mainly developed after thedeck forms were removed.

Continuity behavior of Girder 3/4was tested at different ages of the girders, using a point load in the middle ofeach span. Fig. 7 shows the PCA test5data at the girder ages of 45, 422, 530,and 684 days. The present study and

the elastic theory moments based onthe full continuity assumption are alsoplotted for comparison. A good agreement with test data is noted.

The experiments and the presentstudy both indicate that the diaphragmdoes not provide full continuity evenat an early age. Furthermore, the effective continuity decreases over time.Note, however, that the present studyunderestimates the continuity at laterages. This is attributed in part to thefriction of the support bearing, whichis neglected by the analysis, as discussed below for Girder 1/2 with nopositive reinforcement.

The measured time-dependent variation of the center support reaction forGirder 1/2 is compared to results fromthe present study in Fig. 8. The testsshowed that the diaphragm cracked at330 days after removal of the form-work.5 The present study also predictscracking of the diaphragm, but at anearlier age as marked on the figure.This can be explained with referenceto the inset in Fig. 8.

Fig. 8. Time-dependent variation of center support reaction for PCA5Girder 1/2.

9.5

2.5

2

l.5

0

(12

4-

c-)0.5

0

-0.5

7.5

5.5

3.5

0

aC’)I4-aU4-Cs

a0

Cs

1.5

-0.5

0 150 300 450 600 750

-2.5

Time from Removal of Deck Formwork (Days)

-4.5

68 PCI JOURNAL

The positive restraint moment develops a tensile force in the bottom ofthe diaphragm. When this tensile forceexceeds the tensile strength (i.e., modulus of rupture) of the unreinforcedconcrete, the diaphragm would crack.Once it cracks, the restraint momentwill remain constant until the ultimatecurvature of the diaphragm is reached.

In the experiment, however, a steelbearing plate was provided under thediaphragm and the ends of both girders. The friction between the plate andthe concrete provided additional resistance to the tensile force, and mighthave delayed the cracking. The present study does not consider this friction force, and therefore, predictscracking at an earlier stage and with alower uplift reaction.

In addition to the PCA tests,5 thepresent study compared favorablyagainst test results from Purdue University.’3 The comparisons, presentedin Reference 25, are not repeated herefor brevity. However, additional verifications are planned for the ongoing

full-scale experiments at the University of Cincinnati.

PARAMETRIC STUDYA parametric study was performed

on the same two-span bridge withAASHTO Type III girders that wasdescribed in the previous section forthe comparison to the CTL method. 12

The study considered the amount ofpositive moment reinforcement in thecontinuity diaphragm, the effectivewidth of the diaphragm, girder age atcontinuity, and girder age at the liveload test.

Time-DependentRestraint Moments

The restraint moments were calculated for four different amounts ofpositive moment reinforcement corresponding to 0.1, 0.6, 1.2, and 2.4 timesMe,., where reinforcement corresponding to Mc,. is 2.0 sq in. (1290 mm2).The negative reinforcement in thedeck was 7.04 sq in. (4542 mm2) for

all cases. Fig. 9 shows the effect ofpositive moment reinforcement on thetime-dependent restraint moments atthe center support. The figure insetshows the effect of age at continuity.

For all cases, the restraint momentsare initially negative due to differential shrinkage. The negative restraintmoments are virtually independent ofpositive moment reinforcement. Whencontinuity is established at the age of28 days, as creep effects dominate andtend to cause uplift in the center support, the effect of positive moment reinforcement becomes apparent. For avery small amount of positive momentreinforcement (0. lMcr), the diaphragmcracks and can sustain only a small restraint moment due to its reduced flexural stiffness.

Additional positive moment reinforcement stiffens the diaphragm andresults in increased restraint moments.The increase, however, is smaller aspositive moment reinforcement exceeds 1 .2M. As shown in the Fig. 9inset, if establishment of continuity is

200

600

400

E

E 0

C

-200

-400

-600

0 5

Time from Transfer of Prestress (Years)

20

7000 8000

800

600

400

:°°I-200

-400

-600

-800

10 15

0 1000 2000 3000 4000 5000 6000

Fig. 9. Effect of positive reinforcement on time-dependent restraint moment.

September-October 2001 69

Total Live Load (kN)

Fig. 10. Typical variations of live load moments at midspan and center support.

Fig. 11. Typical variations of continuity index at center support.

0

3000

300

2000

600 900 1200 1500

1000

4000

S© 0

0

-1000

3000

2000

-2000

-Equivalent Service - -

Live Load for CenterSupport Moment

S

1000

-1000

-30001

0.1 Mcr Positive Moment Reinforcement I Softening of deck IContinuity at 28 Days 1 at center support

Live Load Test at 1000 Days

0

-2000

100 200

-3000

Total Live Load (kips)

300 400

-4000

0

2

Normalized Live Load for Mid-Span

32

L5

—

Cc)

0.5

0

lEquivalent Service Live Loadfor Center Support Moment

0 1 2

0.1 Mcr Positive Moment ReinforcementContinuity at 28 Days

Live Load Test at 1000 Days

3

Normalized Live Load for Center Support

4 5

70 PCI JOURNAL

delayed, restraint moments may neverbecome positive.

It is important to determine howmuch of the diaphragm width is effectively engaged in providing continuity.The diaphragm section for the abovecase studies was modeled as a T-section with an effective web equal to thebottom flange of the girder. The otherextreme case is when the diaphragm ismodeled as a rectangular section withan effective width equal to that of thecomposite slab.

Despite considerable change in thecross section, the moment-curvaturefrom the RESPONSE22 programshowed only a slight increase in thenegative moment response, but nochange in the positive moment response. The time-dependent analysisfurther confirmed that the difference inthe restraint moments for different diaphragm widths was quite negligible.Therefore, the diaphragm was modeledas a T-section, as described above.

Continuity Analysis

Continuity analysis was performedfor four different amounts of positive

1.00

0.75

a

1: 0.50c) 0.25

0.00

moment reinforcement (0.1, 0.6, 1.2,and 2.4 times Mcr) and two differentgirder ages when continuity is established (28 or 90 days after prestresstransfer). In each case, the live loadcontinuity test was carried out at different girder ages to consider a widespectrum of time-dependent restraintmoments that may exist in the continuity diaphragm. The restraint moments may be positive or negative depending on when continuity isestablished and when the live load testis conducted.

Fig. 10 shows a typical plot of liveload moments at midspan and centersupport for a positive moment reinforcement of 0. lMc,., with continuityestablished at 28 days, and the liveload test conducted at 1000 days. Themoments at midspan and the centersupport are shown as positive and negative, respectively. The elastic analysis, also shown in Fig. 10, overestimates the support moment, butunderestimates the midspan moment.The equivalent service live loads forthe support and midspan moments arealso marked in the figure.

Fig. 11 shows continuity indices forthe above case. Live loads are normalized with respect to the equivalent service live loads. Initially, a very lowlevel of continuity exists due to cracksin the bottom of the diaphragm. Sincelive loads place the bottom of the diaphragm in compression, the cracksclose and more continuity develops.

A further increase in the continuityis achieved at higher load levels, whenthe girder cracks at midspan, thus increasing the relative stiffness of the diaphragm. With softening of the deckover the support (i.e., yielding of deckreinforcement), continuity is onceagain reduced, until failure occurs atmidspan or the support.

Fig. 12 shows the variation of centersupport continuity index for positivemoment reinforcement of 0. lMcr anddifferent ages at the load test. At laterages, the support continuity index decreases, indicating that the section resists less moment for a particular levelof live load. At the girder age of 3500days, the diaphragm has alreadycracked under the effect of positive restraint moment.

0 1 3

Normalized Live Load

Fig. 12. Continuity indices for positive moment reinforcement of 0.1 Mcr (deck cast at 28 days).

4 5

September-October 2001 71

Fig. 13. Continuity indices for positive moment reinforcement of 1 .2M, (deck cast at 28 days).

Fig. 14. Continuity indices when deck is cast at 90 days.

1.00

0.75ta—

aaaCj0.50

(30.25

0.00

0 1 2 3 4 5Normalized Live Load

1.000.1 Mcr to 2.4 Mcr Positive Moment Reinforcement

Continuity at 90 Days

0.75ta

aaaC

0.50

0.25

0.00

Age at Load Test

0

400 Days

1000 Days

2

—.-—3500 Days

3

Normalized Live Load

4 5

72 PCI JOURNAL

Fig. 15. Effect of positive moment reinforcement on continuity index (60 days after prestress transfer).

Fig. 16. Effect of positive moment reinforcement on continuity index (400 days after prestress transfer).

September-October 2001

1.00

ServiceLive Load

0.75

Continuity at 28 DaysLoad Test at 60 Days

C—

CC

C0c-)I0

CriD

C

c) 0.25

0.50

Positive MomentReinforcement

0.00

0

—0.1 Mcr

0.6 Mcr

1.2Mcr

—2.4 Mcr

2 3

Normalized Live Load

4 5

1.00

0.75

C

CC

CC1: 0.50

0.25

0.00

0 1 2 3 4 5

Normalized Live Load

73

Fig. 17. Effect of positive moment reinforcement on continuity index (1000 days after prestress transfer).

Fig. 18. Effect of positive moment reinforcement on continuity index (3500 days after prestress transfer).

1.00

0.75

a—

aaaC

0.50

C_) 0.25

0.00

0 1 2 3 4 5

Normalized Live Load

1 .00

0.75

a—

aaC1: 0.50

C..) 0.25

0.00

0 1 2 3 4 5

Normalized Live Load

74 PCI JOURNAL

When live loads are applied, crackshave to close before the diaphragmcan resist the live load moments.Therefore, the continuity index is initially close to zero. On the other hand,for the load test at the girder age of 60days, the restraint moment in the diaphragm is negative, meaning that thebottom of the diaphragm is still incompression.

Fig. 13 depicts the variation of thecenter support continuity index for apositive moment reinforcement of1.2Mr, showing the same general pattern. However, the variation of thecontinuity index is much smaller forlarger amounts of positive moment reinforcement. The initial range of variation of support continuity index is between 0.01 and 0.68 for 0.lMcr,whereas it is between 0.61 and 0.68for 1.2Mcr, and between 0.66 and 0.68for 2.4Mcr.

Fig. 14 shows the center supportcontinuity index with continuity established at the age of 90 days. In thiscase, the curves for different amountsof positive moment reinforcement be-

tween 0. lM, and2.4Mcr are identical.This is because of the predominant effect of differential shrinkage, whichkeeps the restraint moments negativethroughout the life of the bridge.

The continuity index in this case islarger than that for continuity established at 28 days. The increase in thecontinuity index at higher levels oflive load is attributed to the softeningof the girders under positive momentin the midspan. Therefore, the higherrelative stiffness of the diaphragmleads to an increase in the continuityindex prior to failure.

Figs. 15 to 18 show the support continuity index for various levels of positive moment reinforcements and loading ages. For the load test at 60 days,the continuity index is virtually independent of the amount of positive moment reinforcement, mainly becauserestraint moments at this time are negative. The effect of positive momentreinforcement, however, becomesmore pronounced as the time-dependent effects increase over time (i.e.,greater age at load test).

Fig. 19 captures this concept as aplot of center support continuity indexversus age at load test for equivalentservice live load. When continuity isestablished at 28 days, the effect ofgirder age at load test (i.e., time-dependent effects) on the continuity behavioris significant for small amounts of positive moment reinforcement, but diminishes for larger amounts of positivemoment reinforcement. When continuity is established at 90 days, time-dependent effects on continuity are insignificant, irrespective of the amountof positive moment reinforcement.

In summary, it is clear that positivemoment cracking lowers the continuity index. Therefore, delaying the establishment of continuity and providing adequate positive momentreinforcement both improve the effectiveness of the continuity connection. However, the support continuityindex always remains less than one,regardless of the amount of positivemoment reinforcement, age at continuity, age at load test, and the levelof live load.

Fig. 19. Support continuity indices at service load for different amounts of positive moment reinforcement and continuity age.

1.00

C

0.75 ....“.

If)

0.50

0.25Cl)

0.00 • .

0 1000

0.1 M1, Continuity at 28 Days€ 0.6 M1, Continuity at 28 Days

1.2 Mcr, Continuity at 28 Days

• 2.4 Mcr, Continuity at 28 Days0.1 to 2.4 M1, Continuity at 90 Days

2000

Age at Live Load Test (Days from Prestress Transfer)

3000 4000

September-October 2001 75

This is due to the relative stiffnessand cracking behavior of the supportsection with respect to the midspan section. The support section is designedonly for negative live load moment,whereas the midspan section is designed for positive live load and deadload moments. Moreover, the supportsection is only reinforced with mildsteel, whereas the midspan section isprestressed, delaying the cracking.

With its stiffness being lower thanthe midspan section, the support section will always receive less live loadmoment than computed from an elasticanalysis. However, the design examplein Appendix B shows that the effect ofthis reduction may not be very large.

Total Mdspan Moments

The total midspan moments can becalculated by adding the dead load,time-dependent and live load moments. Fig. 20 shows the restraint, liveload, and total midspan moments as afunction of age at load test for two different amounts of positive moment reinforcement. The figure is derived fora continuity age of 28 days, but plots

for continuity at other ages are similar.The figure clearly shows that total

midspan moment at service live loadis independent of the amount of positive moment reinforcement or the ageat load test. When a very smallamount of positive reinforcement isprovided in the continuity diaphragm,it cracks and prevents further increasein its restraint moment. The decreasedstiffness results in reduced continuity.On the other hand, a larger positivemoment reinforcement leads to higherrestraint moments. This confirms asimilar finding by the CTL study.’2

CONCLUSIONSA flexibility-based model was de

veloped to study the time-dependentbehavior of continuity connections forprecast, prestressed concrete girderswith a cast-in-place deck. The modelconsiders the different nonlinearstress-strain responses of the diaphragm and the girder/deck, and thechange in their stiffness under time-dependent effects and loads. The studyhas led to the following conclusions:

1. Time-dependent restraint moments and continuity for live loads arehighly dependent on the girder agewhen continuity is established. If girders are older when continuity is established, the predominant effect is differential shrinkage, which may preventthe development of positive restraintmoment or uplift at the center support.

2. Positive moment reinforcement inthe continuity diaphragm has a significant effect on restraint moments, whencontinuity is established at early ages.The analysis indicates that the continuity diaphragm may crack if no positivemoment reinforcement is provided.With a sufficient amount of positivemoment reinforcement, the diaphragmcrack will be limited but higher positive restraint moments will develop.

3. The continuity behavior of thebridge is generally better when continuity is established at later girder ages.In such cases, the continuity behavioris also independent of the amount ofpositive moment reinforcement provided in the diaphragm.

4. Because of the relative stiffnessand cracking behavior of the continu

Fig. 20. Effect of positive moment reinforcement on midspan moments.

2500

E0

continuity at 28 Days

Iit (0.lMcr and

l.2Mcr

ILweL0adMomen2Mç

2000

2000

1750

____________

1500

1250

1000

________

750----

- boo

500

500

250

0

0

E

1500E0

1000 2000

Age at Live Load Test (Days from Prestress Transfer)

3000

0

4000

76 PCI JOURNAL

ity diaphragm with respect to themidspan section, full continuity is notexpected in this type of bridge, andlinear elastic analysis may not be conservative for calculating live load moments at midspan.

5. Although total midspan momentsare virtually independent of theamount of positive moment reinforcement, positive moment connectionsare recommended to address durability and structural integrity. A minimum amount of positive moment reinforcement equivalent to 1.2Mcr issuggested, because (a) additional reinforcement above 1 does not appreciably improve continuity for liveloads; (b) it conforms to the minimumreinforcement requirements in theAASHTO Standard Specifications;23and (c) it correlates well with standarddetails that have been used in severalstates. 18

6. The width of the continuity diaphragm does not have a significant effect on the restraint moments, and canbe modeled as a T-section with its webequal to the width of the bottom flangeof the prestressed concrete girder.

WHERE DO WE GOFROM HERE?

While the present study shows thatonly partial continuity may exist inbridges made continuous for live

loads, the performance of thesebridges has generally been quite acceptable. In fact, some bridge engineers believe that the primary value ofcontinuity design may not lie in themoment carrying capacity of the connection, but rather the structural integrity and the improved durability ofthe bridge as a result of eliminatingthe joints.

Practicing engineers should considerproviding the minimum reinforcementthat prevents excessive cracking of thediaphragm section without overcrowding the connection. They shouldalso be cognizant of the girder age atthe time of establishing continuity toavoid excessive time-dependent effects. Based on findings of earlierstudies, some engineers have addressed this issue by specifying a minimum girder age when continuity isestablished.

Additional work is also needed inthese areas:

1. Cracking of continuity diaphragms in some states has been attributed to thermal effects. Therefore,the effect of thermal gradients on restraint moments should be considered.The model presented in this paper canbe extended to accommodate thermalgradients as another component in theflexibility-based analysis.

2. Additional parametric studiesshould be performed to determine the

effect of a wider range of girder types,span lengths, girder spacings and otherdesign variables.

3. Details and design parametersshould be considered to reduce congestion of reinforcement in the continuity diaphragm and in the ends of thegirders, while maintaining the minimum reinforcement for durability andperformance considerations.

Work is currently under way by theauthors on a number of specimenswith bent strands or bent bars as positive moment reinforcement with orwithout girder embedment into thecontinuity diaphragm. Final designrecommendations will be made at theconclusion of the experimental phaseof this study.

ACKNOWLEDGMENTSThis study is sponsored by the Na

tional Cooperative Highway ResearchProgram (NCHRP) Project 12-53, forwhich the authors are grateful.

The authors sincerely thank the reviewers of the PCI JOURNAL fortheir valuable comments, which havecertainly enhanced the paper.

The opinions and findings expressedherein, however, are those of the authors alone, and not necessarily theviews or positions of the AASHTO,the NCHRP, or the NCHRP ProjectPanel.

September-October 2001 77

REFERENCES

1. Kaar, P. H., Kriz, L. B., and Hognestad, E., “Precast-Prestressed Concrete Bridges: 1. Pilot Tests of Continuous Girders,” Journal of the PCA Research and Development Laboratories, V. 2, No. 2, May 1960, pp. 2 1-37.

2. Hanson, N. W., “Precast-Prestressed Concrete Bridges: 2. Horizontal Shear Connections,” Journal of the PCA Research andDevelopment Laboratories, V. 2, No. 2, May 1960, pp. 38-58.

3. Mattock, A. H., and Kaar, P. H., “Precast-Prestressed ConcreteBridges: 3. Further Tests of Continuous Girders,” Journal ofthe PCA Research and Development Laboratories, V. 2, No. 3,September 1960, pp. 5 1-78.

4. Mattock, A. H., and Kaar, P. H., “Precast-Prestressed ConcreteBridges: 4. Shear Tests of Continuous Girders,” Journal of thePCA Research and Development Laboratories, V. 3, No. 1,January 1961, pp. 19-46.

5. Mattock, A. H., “Precast-Prestressed Concrete Bridges: 5. Creepand Shrinkage Studies,” Journal of the PCA Research an4 Development Laboratories, V. 3, No. 2, May 1961, pp. 32-66.

6. Mattock, A. H., and Kaar, P. H., “Precast-Prestressed ConcreteBridges: 6. Test of Half-Scale Highway Bridge ContinuousOver Two Spans,” Journal of the PCA Research and Development Laboratories, V. 3, No. 3, September 1961, pp. 30-70.

7. Freyermuth, C. L., “Design of Continuous Highway Bridgeswith Precast, Prestressed Concrete Girders,” PCI JOURNAL,V. 14, No. 2, April 1969, pp. 14-39.

8. Salmons, J. R., and McCrate, T. E., “Bond of Untensioned Prestressing Strand,” interim Report 73-5A, Missouri CooperativeHighway Research Program, Missouri State Highway Department, Columbia, MO, August 1973, 108 pp.

9. Salmons, J. R., and May, G. W., “Strand Reinforcing for EndConnections of Pretensioned I-Beam Bridges,” interim Report73-5B, Missouri Cooperative Highway Research Program,Missouri State Highway Department, Columbia, MO, May1974, 142 pp.

10. Salmons, J. R., “End Connections of Pretensioned I-BeamBridges,” Final Report 73-SC, Missouri Cooperative HighwayResearch Program, Missouri State Highway Department,Columbia, MO, November 1974, 51 pp.

11. Salmons, J. R., “Behavior of Untensioned-Bonded PrestressingStrand,” Final Report 77-1, Missouri Cooperative HighwayResearch Program, Missouri State Highway Department,Columbia, MO, June 1980, 73 pp.

12. Oesterle, R. G., Gilkin, J. D., and Larson, S. C., “Design ofPrecast-Prestressed Bridge Girders Made Continuous,”NCHRP Report 322, Transportation Research Board, NationalResearch Council, Washington, DC, November 1989, 97 pp.

13. Abdalla, 0. A., Ramirez, J. A., and Lee, R. H., “StrandDebonding in Pretensioned Beams - Precast Prestressed Concrete Bridge Girders with Debonded Strands - Continuity Issues,” Report FHWA/INDOT/JHRP-92-24, Indiana Department of Transportation and Purdue University, West Lafayette,IN, June 1993, 235 pp.

14. Peterman, R. J., and Ramirez, J. A., “Restraint Moments inBridges with Full-Span Prestressed Concrete Form Panels,”PCI JOURNAL, V. 43, No. 1, January-February 1998, pp. 54-73.

15. Tadros, M. K., Ficenec, J. A., Einea, A., and Holdsworth, S.,“A New Technique to Create Continuity in Prestressed Concrete Members,” PCI JOURNAL, V. 38, No. 5, September-October 1993, pp. 30-37.

16. Ma, Z., Huo, X., Tadros, M. K., and Baishya, M., “RestraintMoments in Precast/Prestressed Concrete ContinuousBridges,” PCI JOURNAL, V. 43, No. 6, November-December1998, pp. 40-56.

17. Clark, L. A., and Sugie, I., “Serviceability Limit State Aspectsof Continuous Bridges Using Precast Concrete Beams,” TheStructural Engineer, Institution of Structural Engineers, London, U.K., V.75, No. 11, June 1997, pp. 185-190.

18. Rabbat, B. G., and Aswad, A., “Design of Precast PrestressedGirders Made Continuous,” Report to PCI Committee onBridges, October 1992, 4 pp.

19. Birkeland, H. W., “Differential Shrinkage in CompositeBeams,” ACI Journal, V. 56, 1960, pp. 1123-1136.

20. ACI Committee 209, “Prediction of Creep, Shrinkage andTemperature Effects in Concrete Structures,” in Designingfor Creep and Shrinkage in Concrete Structures, SP-76,American Concrete Institute, Farmington Hills, MI, 1998,47 pp.

21. PCI Committee on Prestress Losses, “Recommendations forEstimating Prestress Losses,” PCI JOURNAL, V. 20, No. 4,July-August 1975, pp. 44-75.

22. Collins, M. P., and Mitchell D., Prestressed Concrete Structures, Prentice Hall, New York, NY, 1997.

23. AASHTO, Standard Specifications for Highway Bridges, 16thEdition, Washington, DC, 1996.

24. Suttikan, C., “A Generalized Solution for Time-Dependent Response and Strength of Noncomposite and Composite Prestressed Concrete Beams,” Ph.D. Thesis, University of Texas,Austin, TX, 1978.

25. Kulkarni, S., “Time-Dependent Analysis of Precast, Prestressed Girders Made Continuous,” M.S. Thesis, Universityof Cincinnati, Cincinnati, OH, August 2000.

APPENDIX A - NOTATION

Ad = cross-sectional areas of deck= cross-sectional area of girder

e2’ = distance from centroid of composite section to bottom of deck

E = modulus of elasticity of concrete

Ed = modulus of elasticity of deck concrete (at time t)Eg = modulus of elasticity of girder concreteE5 = modulus of elasticity of steel

f = compressive strength of concretef-28 = 28-day compressive strength of concrete

= yield strength of mild steel reinforcement

= yield strength of prestressing strandsultimate strength of prestressing strands

Finitiat = initial jacking forceMcr = cracking moment of continuity diaphragm concrete

with girder composite sectiont = slab thickness

= time stepw unit weight of concrete

= incremental differential shrinkage strain

= incremental change in shrinkage moment= creep coefficient for concrete

78 PCI JOURNAL

APPENDIX B - DESIGN EXAMPLE

This design example will demonstrate the effect of the partial continuity discussed in the paper on the design of a bridge comprised of precast,prestressed concrete girders made continuous for live load.

The structure used for the verification section of this paper will be usedfor the design example. The bridge hastwo 65 ft (19.8 m) spans of AASHTOType III girders at 8 ft (2.4 m) spacing. The length of the continuity diaphragm is 2 ft (0.6 m). An 8 in. (203mm) thick composite concrete deckcompletes the bridge.

Two sets of design conditions willbe considered:

Case 1: Continuity established atgirder age of 28 days.

Case 2: Continuity established atgirder age of 90 days.

For both cases, a positive momentconnection with reinforcement equivalent to 1.2Mcr is used in the continuity diaphragm. This amount of reinforcement is selected to control

cracking at the continuity diaphragmand to maintain a greater degree ofcontinuity (i.e., increased continuityindex), if cracking of the continuitydiaphragm occurs.

From the data presented in thepaper, the bridge of Case 1 woulddevelop large positive moments atthe continuity connection at laterages. Therefore, the amount of positive moment reinforcement has asignificant effect on the continuitybehavior of the bridge. The bridge ofCase 2 would not develop a positivemoment at the connection, so itscontinuity behavior would not be affected by the amount of positive moment reinforcement.

The continuity index at the centersupport for both cases can be obtained from Fig. 19. The values forthe two cases are:

Case 1: Continuity index at centersupport = 65 percent.

Case 2: Continuity index at centersupport =78 percent.

The live loading used for this example will be the simplified configuration used elsewhere in the paper,namely, equal point loads at midspanof both spans. For this loading, themoment diagram for live load for asimple span (continuity index at center support = 0 percent) and for fullcontinuity (continuity index at centersupport = 100 percent) are shown inFig. B 1 for one span of the bridge.The effective live load moment is alsoshown, using continuity indices forthe center span of 65 and 78 percentfor Cases 1 and 2. respectively.

The moments at the center supportand midspan are listed in Table B 1.From these moments, the continuityindices at midspan are computed,and listed in the table. The continuity index for a specific location isthe ratio of a given moment to themoment for the fully continuouscondition.

From the continuity indices atmidspan, it can be seen that, for Case

Fig. BI. Effective continuity for two-span bridge with equal point loads in each span.

1100

E

0

1000

500

-500

-1000

0

750

400S

I50

-300 ©

-650

0.0 0.2 0.4 0.6 0.8 1.0

-1000

Fraction of Span Length (One Span of Two-Span Bridge Shown)

-1350

September-October 2001 79

1, even though the negative moment

developed at the center support is only

65 percent of the moment for a fullycontinuous bridge (a 35 percent reduction), the rnidspan moment only increases 21 percent. For the more favorable condition of Case 2, where thenegative moment developed at thecenter support is 78 percent of the moment for a fully continuous bridge (a22 percent reduction), the midspanmoment increases 13 percent.

Preliminary studies indicate thatthe effect of partial continuity onthe moment envelope may be evensmaller than the values shownabove. This may be a significantpart of the reason that, even though

the analysis presented in this paperindicates that the continuity connections for girders made continuousare only partially effective in estab

lishing continuity, bridges designed

neglecting this partial continuity

have generally performed well overthe years.

Tab’e Bi. Continuity indices and moments at center support and midspan.

Center support Midspan

Negative moment Continuity Positive moment ContinuityBridge type kip-ft (kN-m) index kip-ft (kN-m) index

I -Simple span : 0 (0) 0 percent 960 (1302) 160 percent

Fully continuous 720 (976) 100 percent 600 (814) 100 percent

Partially Case 1 468 (635) 65 percent 726 (984) 121 percentcontinuous - -

________________________

—

_________________________

Case 2 562 (762) 78 percent 679 (921) 113 percent

80 PCI JOURNAL