Distribution of Compressive Stresses in Transversely Posttensioned Concrete Bridge Decks J.Paul Smith-Pardo,M.ASCE1;Julio A.Ramirez,M.ASCE2;and Randall W.Poston,F.ASCE3Abstract:A parametric study was carried out in order to understand the salient aspects affecting the distribution of compressive stresses in transversely posttensioned concrete bridge decks.Alternativefinite element modeling techniques and alternative software were considered and the corresponding analytical results were compared with the experimental results from previous investigations.It was found that the distribution of compressive stresses is mainly affected by the support conditions of the girders and the axial stiffness of the diaphragms.DOI:10.1061/͑ASCE͒1084-0702͑2006͒11:1͑64͒CE Database subject headings:Finite elements;Model tests;Superstructures;Stress distribution;Bridge decks;Bridges,concrete.Research SignificanceDurability problems in reinforced concrete bridge decks are an ongoing concern for Departments of Transportation͑DOTs͒in the United States.Corrosion of reinforcement is considered the most significant source of deterioration in concrete decks in the United States.Limiting the service load stresses below the cracking level should improve the long-term performance of the deck.An earlier study investigating the use of transverse posttensioning as a mean of delaying the initiation of reinforcement corrosion in concretedecks was conducted at the University of Texas,Austin,Tex.͑Almustafa1983;Mora1983;Poston1984;Ralls1984͒.From this research,design specifications were proposed for the useof transverse posttensioning as a mechanism to control crackingin the longitudinal direction of the deck.The specifications werepublished in a special report of the Precast/Prestressed ConcreteInstitute(PCI)Journal͑Poston et al.1989͒as proposals to modify the existing American Association of State Highway and Transportation Officials͑AASHTO1983͒provisions. However,major limitations of the design specifications from the Texas study were the exclusion of the effect of the position of the diaphragms,and the effect of the support conditions for the girders on the distribution of transverse stresses induced from posttensioning.Moreover,the provisions were derived based on the results from afinite element analysis program with limitations on the number of elements.In this paper,the problem of modeling transversely posttensioned deck-on-girder superstructures usingfinite element͑FE͒analysis is critically addressed by comparing the results from alternative techniques suggested in the literature against experimental results.In addi-tion,the variables not considered in the Texas study are taken into account and incorporated into an extensive parametric study whose results can be used to specify a scheme of transverse posttensioning of decks for crack control in the design. IntroductionThe fundamental question to be answered is:how are transverse stresses distributed on the top surface of a concrete bridge deck that is subjected to a uniform transverse posttensioning applied at the edges.It is also desirable to determine the parameters signifi-cantly affecting such distribution of transverse stresses because some insight can be gained about the required magnitude and distribution of applied posttensioning to reach the condition of low or no tensile stresses on the deck under superimposed service loads.Modeling of Deck-on-Girder SuperstructuresAlternatives for modeling deck-on-girder superstructures range from closed-form solutions of particular cases͑Westergaard 1930;Duberg et al.1960;Cao and Shing1999͒to numerical solutions͑Chen et al.1957͒,and somewhat more recently to modeling techniques.Whereas in two-dimensional͑2D͒FE models͑Mabsout et al.1999͒the elements representing the deck,the girders,and the diaphragms are located in the same plane,in three-dimensional͑3D͒FE models͑Bishara and Elmir 1990;Bishara et al.1993;Mabsout et al.1997;Chan and Chan 1999͒,these elements are located in different planes and are connected through rigid links.The most notorious deficiency of the2D models in relation to3D FE models is that the eccentricity1Engineer,Berger/Abam Engineers Inc.,333019th Ave.South, Suite300,Federal Way,WA98003;formerly,Instructor,School of Civil Engineering,Purdue Univ.,West Lafayette,IN47906͑corresponding author͒.E-mail:paul.smith@2Professor,School of Civil Engineering,Purdue Univ.,West Lafayette,IN47906;formerly,Assistant Head;E-mail:ramirez@ 3Fellow,Prin.Whitlock Dalrymple Poston Associates;402W.7th St., Austin,TX78701.E-mail:rposton@Note.Discussion open until June1,2006.Separate discussions must be submitted for individual papers.To extend the closing date by one month,a written request must befiled with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November11,2003;approved on March16,2005.ThisFour distinct3D FE analysis techniques appropriate for the modeling of deck-on-girder superstructures are found in the literature and are described as follows:1.Type I:Girders and diaphragms are modeled as frameelements rigidly connected to the deck.The deck is modeled with shell elements.2.Type II:Top and bottomflanges and webs of girders aremodeled as frame elements.Diaphragms and deck are mod-eled as shell elements.Elements are connected through rigid links.3.Type III:Topflanges of girders and diaphragms are modeledas frame elements rigidly connected to the deck.Webs of girders and deck are modeled using shell elements.4.Type IV:Girders and diaphragms are modeled with solidbrick elements͑three translations at every node͒,and the deck is modeled with shell elements.Evaluation of Alternative Finite Element Models Using Experimental ResultsTo evaluate the accuracy of each modeling alternative with respect to the calculated distribution of stresses induced from edge transverse posttensioning of the deck,a test specimen from the Texas study͑Mora1983;Ralls1984;Poston et al.1989͒was considered.The geometric characteristics of the test speci-men are shown in Fig.1.The applied transverse posttensioning at the edges was about 4.1MPa͑0.6ksi͒and consisted of 6.3mm seven-wire strand tendons͓1,750MPa͑250ksi͔͒spaced at140mm,except in549mm wide regions over the diaphragms where the spacing was halved and thus the applied transverse posttensioning was about8.2MPa͑1.2ksi͒.The higher transverse stress applied over the diaphragms was intended to overcome deck.Tendons were straight and their resultant force was assumed concentric with respect to the centerline of the concrete slab.Only one half of the span of the deck was transversely posttensioned in the Texas specimen.Two diaphragm arrange-ments were tested:thefirst,which will be referred to as the all-diaphragm case,included both interior and end diaphragms; and the second,which will be referred to as the end-diaphragm case,included end diaphragms only.Stresses in the experimental program were inferred from strain gage readings assuming a constant modulus of elasticity for concrete͑Almustafa1983; Ralls1984͒.The computer program SAP2000was used to represent the test superstructure using modeling alternatives2D and3D Types I–III.The computer program ANSYS5.7was used for modeling alternative Type IV.The calculated distribution of transverse stresses,using three-dimensional Type I FE modeling, is shown in Fig.2with contour lines for the half of the deck that was transversely posttensioned.Experimental values for the stresses on the top surface of the deck are indicated in the same figure with dots.The comparison of experimental and calculated stresses is summarized in Table1for the different FE modeling techniques.Error is defined as the difference between the calcu-lated and measured value divided by the measured value at a given location.Positive errors correspond to overestimations of the experimental values.The mean error is computed based on the absolute values of individual errors.The maximum over-estimations and underestimations are also indicated in Table1. Minimum overestimations and underestimations were almost zero ͑less than4%͒for all modeling alternatives and thus are not reported.The values presented in Table1suggest that that there is no significant difference in the estimation of induced transverse stresses when using any of the FE modeling alternatives evalu-Fig.1.Test structure from Texas studythe all-diaphragm and for the end-diaphragm cases.The results show that the analytical models overestimate the measured values at certain locations but underestimate at other locations.Table 1also indicates that modeling the eccentricity of the posttensioning does not have a significant effect on the distribution of transverse stresses.Parametric StudyAn extensive parametric study was conducted as part of the research project carried out at Purdue Univ.͑Smith-Pardo and Ramirez 2002͒.The base structure for the parametric study,depicted in Fig.3,is similar to the prototype in the Texas study ͑Poston et al.1989͒.It consisted of a deck-on-girder superstructure having a span length of 23,200mm ͑76ft ͒and a deck width of 17,700mm ͑58ft ͒.The superstructure had seven girders,two interior diaphragms,and two exterior diaphragms.Girders were C-type standard Texas Department of Highways and Public Transportation ͓f c Јϭ35MPa ͑5ksi ͔͒at 7,400mm ͑24.3ft ͒and 7,700mm ͑25.3ft ͒.The deck ͓f c Јϭ35MPa ͑5ksi ͔͒was 210mm ͑8.25in.͒thick.Each girder was supported on unreinforced elastomeric bearing pads at both ends.Each bearing was modeled as a roller with two mutually perpendicular springs representing the shear stiffness of the elastomeric pads.One of the springs is oriented in the longitudinal ͑traffic ͒direction and the other in the direction perpendicular to the traffic ͑parallel to the diaphragms ͒.The interaction between the two springs is neglected in the analyses.Elastomeric bearing pads had b ϭ305mm ͑12in.͒ϫw ϭ508mm ͑20in.͒cross sections and a thickness h ϭ102mm ͑4in.͒.The shear modulus was assumed as G ϭ1.2MPa ͑170psi ͒.The corresponding shear ͑spring ͒stiffness is calculated as K s =Gbw /h ϭ1.8kN/mm ͑10kip/in.͒in each direction.The expression for K s is easily derived by assuming small deformation of the elastomeric bearings in shear only ͑Poston 1984͒.Three-dimensional Type I FE modeling was used for the analyses.The choice of this modeling technique was arbitrary as the evaluated modeling alternatives produced similar results.Unlike the Texas study ͑in which the applied stresses in zones of the deck above the diaphragms were twice those at other zones ͒,all the structures defined in the parametric study were subjected to a uniform unit compressive stress applied along the edges of the deck.The choice of a uniform distribution of applied stresses is convenient because it avoids introducing variables such as the width and the magnitude of the applied stresses in zones of the deck above the diaphragms.For the parametric study transverse stresses were evaluated along four types of lines on the surface of the deck defined as follows ͑Fig.3͒:1.Strip 1͑S1͒:located on the top surface of the deck along enddiaphragms;2.Strip 1͑S2͒:located on the top surface of the deck halfwaybetween an end diaphragm and adjacent interior diaphragms;Table 1.Error ͑%͒in Calculated Transverse Stresses on Top Surface of Deck Relative to Experimental Results from Texas Specimens Errora2D 3D Type I 3D Type II 3D Type III3D Type IV ͑a ͒All diaphragm caseMean b1614141412Maximumunderestimation −40−38−38−42−36Maximum overestimation 2730323223͑b ͒End-diaphragm case Mean1514141413Maximumunderestimation−26−28−28−28−28Maximum overestimation 3131292928aError ϭ͑calculated stress-measured stress ͒/measured stress ϫ100%.Measured stress ϭstrain gage reading ϫelastic modulus of deck’s concrete.bMean error ϭaverage ͑absolute value of individual errors ͒.Fig.2.Calculated and measured stresses for Texas structure4.Strip 4͑S4͒:located on the top surface of the deck between consecutive interior diaphragms that are closer to the end of the deck.Along a particular line,transverse stresses were calculated every 610mm ͑2ft ͒as this corresponded to the grid size in the finite element model.For each line,the minimum and the mean stresses were reported.The minimum transverse stress is of major importance from the point of view of design,because its magnitude should be sufficient to prevent longitudinal crack-ing of the deck under service loading.In the discussion that follows,transverse stress refers to that oriented perpendicular to the direction of the traffic ͑i.e.,perpendicular to the direction of the girders ͒.Contours of Induced Transverse Stresses for Base Case StructureThe resulting distribution of transverse stresses on the top surface of the deck for the base case structure is shown in Fig.4with contour lines.Because of symmetry,only one quarter of the deck is shown.From Fig.4it is important to notice that the magnitude of the induced compressive stress near the end of the deck is significantly smaller than that in zones near midspan.Although the space limitation for this paper makes it impossible to show individual contour maps for each case considered in the parametric study,it should be mentioned that the topology of the distribution of transverse stresses is similar to the one shown in Fig.4in all cases while the magnitudes change significantly.Effect of Number of DiaphragmsThe base case illustrated in Fig.3was consecutively modified by changing the number of interior diaphragms in the superstructure.Fig.3.Base structure for parametricstudyFig.5shows the variation of the calculated minimum and average transverse stresses at different strips as a function of the number of diaphragms.It can be seen that there is not a signifi-cant change in the distribution of transverse stresses with the number of interior diaphragms.In Strips2–4the average and the minimum stresses are similar to each other.This suggests that there is a uniform distribution of induced stresses in regions of the deck away from the ends regardless of the number of diaphragms. The minimum and average stresses along any of the strips do not significantly decrease as the number of diaphragms is increased. This suggests that,for the domain of the foregoing parametric study,the restraining effect of each individual diaphragm can be considered localized͑i.e.,independent from the restraining effect of other diaphragms͒.This is a relevant observation that may allow prescribing different levels of posttensioning at different diaphragm regions for design purposes.It is also observed from Fig.5that the minimum and the average top transverse stress in regions of the deck above the end diaphragm͑S1͒are signifi-cantly smaller than the corresponding stresses in regions above interior diaphragms͑S3͒.This can be explained by noticing that:͑1͒the restraining effect imposed by the supports of the girders is more significant near the ends of the deck;and͑2͒the effective width of the deck tributary to end diaphragms is smaller than that for interior diaphragms,thus the resultant applied eccentric loadis higher͑and so are the stresses on the deck͒for the compositeinterior diaphragms.Effect of Boundary Conditions of GirdersThe base case illustrated in Fig.3was modified by changingthe level of restraint against displacement in the longitudinal andtransverse direction at the ends of each girder.Such supportconditions at the ends of each girder were simulated with a rollerand two mutually perpendicular springs having elastic constantsequal to K s͑the base case structure had K sϭ1.8kN/mm͒.The springs were again oriented parallel and perpendicular to thedirection of the traffirge values of K s are relevant becausethey approximate the support condition for integral bent bridges.The minimum and average stresses as a function of K s atdifferent strips are shown in Fig. 6.It can be seen that thedegree offixity at the ends of the girders significantly affects thedistribution of induced transverse stresses,especially near the endof the deck.More restrained support conditions for the girderswould be qualitatively equivalent to a stiffer diaphragm at the endof the deck.Such an equivalent element will attract a significantportion of the applied transverse stresses,and thereby willreduce Fig.5.Effect of number ofdiaphragmsthe induced compressive stresses in the deck.It is important to notice from Fig.6that the minimum and average stresses at different strips are approximately constant for values of K sϽ20kN/mm͑thus,any size of neoprene pad for which K sϽ20kN/mm would have the same effect͒then decrease inthe range20kN/mmϽK sϽ10,000kN/mm andfinally remain constant for K sϾ10,000kN/mm.Effect of Reinforced Concrete Diaphragm SizeThe base case illustrated in Fig.3was modified by changing the cross-sectional area of all the diaphragms and keeping the same aspect ratio͑height/widthϭ2.0͒.Concrete diaphragms were mod-eled as elastic beam elements having a young modulus, E cϭ27,600MPa͑4,000ksi͒.The results from thefinite element analyses are summarized in Fig.7.It is noticed that the minimum and average stresses at Strip1significantly depend on the size of the diaphragms.The restraining effect of the diaphragms,however,is considerably less significant at Strips2–4.This can be explained by the following two opposite effects:1.As the cross-sectional area of the diaphragm is increased therestrained effect increases and the induced compressive stress decreases;and2.As the height of the diaphragm is increased,the eccentricityof the applied transverse load on the composite diaphragm–deck section is increased thusflexural stresses on the top flange are higher.The superposition of effects͑1͒and͑2͒to a similar extent must be responsible for the relatively minor change in the induced stress in Strips2–4as the diaphragm size is increased.For Strip 1,on the other hand,the effect͑1͒is more significant than the effect͑2͒because the tributary area of deck for the end diaphragms is smaller.Effect of Position of End DiaphragmThe base case is modified by changing the position in the longi-tudinal͑traffic͒direction of the end diaphragm with respect to the end of the deck.For the base case structure͑Fig.3͒,in particular, end diaphragms were located at305mm from the respective ends of the deck.Fig.8summarizes the results from the analysesin Fig.7.Effect of diaphragmsizeterms of the minimum and average stresses at different strips. It is observed that the transverse stresses at the end regions of the deck͑S1͒increase as these elements are placed farther from the ends of the deck.This can be explained in part by considering the diaphragms as composite with the deck.As the end dia-phragms are placed away from the edge of the deck their tributary area of deck increases and so does the resultant compressive force ͑for an applied unit stress͒.Notice that induced transverse stresses in other regions of the deck remain basically unchanged for different positions of the end diaphragm.Summary and ConclusionsThe primary objective of this research was to evaluate how different parameters͑geometry and physical parameters͒affect the distribution of transverse compressive stresses in concrete decks under transverse posttensioning.The experimental results from previous studies were used to test the values calculated from alternative modeling techniques and/or alternativefinite element programs.Following the validation and selection of a convenient alternative modeling technique,a parametric study was carried out in order to identify the most relevant variables.A base case structure was defined and then subjected to transverse unit forces representing the posttensioning.The following can be concluded: 1.Different FE modeling schemes for a particular deck-on-girder structure subjected to transverse stresses only͑test model of the Texas study͒were found to produce results reasonably similar to the experimental values.The average ratio of calculated to experimental transverse stresses on the top surface of the deck is at most1.16for two different tests.2.The degree of restraint at the end of the girders has asignificant effect on the distribution of transverse stresses on the top of the deck.The induced compressive stresses at different regions of the deck remains almost constant for all the possible͑neoprene pad or others͒bearings for which K sϽ20kN/mm.Support conditions for which K sϾ10,000kN/mm are considered asfixed for practical purposes and might correspond to the condition of an integral end bridge.3.The restraining effect of every diaphragm can be approxi-mately taken as independent of the effects of other dia-phragms in the superstructure.Prescribing different levels of posttensioning at different regions near the diaphragm is a reasonable approach for design purposes.In particular, higher transverse forces are needed at regions near the end diaphragms as compared to those needed at regions near the interior diaphragms.4.The restraining effect of the diaphragm can be explainedby considering theflexural action of the applied transverse posttensioning on a composite diaphragm–deck section. 5.Superimposing the induced transverse stresses from deadand live loads with the induced stresses from transverse post-tensioning may lead to a feasible alternative to limit cracking on the top surface of the deck and thus mitigate the onset of corrosion of the reinforcement.AcknowledgmentsThis study was part of the Joint Transportation Research Project ͑JTRP͒entitled“An Investigation on Transversely Prestressed Concrete Bridge Decks.”The project was funded by the Indiana Department of Transportation͑INDOT͒.Special thanks are extended to Scott Snowbolds and Gregory Klevitzki from the research division of INDOT,and to Keith Hoernschemeyer from the Federal Highway Administration͑FHWA͒for their continuous feedback during the course of the study.The views,conclusions, and suggestions from this study represent those of the writers and not necessarily those of the sponsor agency.ReferencesAlmustafa,R.A.͑1983͒.“The analysis of transverse prestressing effects in bridge decks.”PhD dissertation,Univ.of Texas at Austin,Austin, Tex.American Association of State Highway and Transportation Officials ͑AASHTO͒.͑1983͒.Standard specifications for highway bridges, 13th Ed.,Washington,D.C.Bishara,A.G.,and Elmir,W.E.͑1990͒.“Interaction between cross frames and girders.”J.Struct.Eng.,116͑5͒,1319–1333. 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