Experimental Study and Analysis of a Full-Scale CFRP/CFCC ...

Table 1. Material properties of CFRP tendons/CFCC strands.Leadline CFCC 1 x 7 CFCC 1 x 37Properties (MCC 15 ) (Tokyo Rope 16 ) (Tokyo Rope 16 )Nominal diameter, in. (mm) 0.39 (10) 0.5 (12.5) 1.57 (40)Effective cross-sectional area, sq in. (mm 2 ) 0.111 (71.6) 0.118 (76.0) 1.17 (752.6)Guaranteed tensile strength, ksi (kN/mm 2 ) 328 (2.26) 271 (1.87) 205 (1.41)Specified tensile strength, ksi (kN/mm 2 ) 415 (2.86) 305 (2.10) 271 (1.87)Young’s modulus of elasticity, ksi (kN/mm 2 ) 21,320 (147) 19,865 (137) 18,419 (127)Elongation, percent 1.9 1.5 1.5Guaranteed breaking load, kips (kN) 36.4 (162) 31.9 (142) 240.5 (1070)Ultimate breaking load, kips (kN) 46 (204.7) 36 (160) 316.9 (1410)Table 2. Material properties of NEFMAC sheets.Modulus of elasticity, ksi (GPa) 12,540 (86.5)Ultimate strength, ksi (MPa) 217 (1500)Ultimate strain, percent 1.8Table 3. Material properties of precast concrete and concrete topping.Properties Precast concrete Concrete toppingModulus of elasticity, ksi (GPa) 5320 (36.7) 4580 (31.6)Strength, ksi (MPa) 7.81 (53.8) 5.7 (39.3)cated using soft materials. All Leadlinetendons were pretensionedusing series of conventional steelstrands, CFRP-tendon anchor systems,couplers, steel strand anchor chucks,and a hydraulic jack. Fig. 3 shows thepretensioned CFRP Leadline tendonsat a roller hold-down point.When the concrete achieved adequatestrength – 45.9 MPa (6.66 ksi) in48 hours – CFRP Leadline tendonswere released in the sequence shown inFig. 4. A typical draped tendon withhold-up and hold-down arrangementsis also illustrated in Fig. 4. A close-upview of pretensioned tendons, CFCCnon-prestressing strands, and flange reinforcementis provided in Fig. 5.Four externally draped post-tensioningCFCC strands were installed afterthe release of the pretensioning tendonsand form removal. Fig. 6 revealsthe post-tensioning strands as seenfrom below the beam. Post-tensioninganchorage details at Diaphragm D6are represented in Fig. 7. Each posttensioningstrand was tensioned usinga special apparatus (see Fig. 8) usingspecific sequencing.A special bearing system at DiaphragmsD3, D4, and D5 was designedand constructed to allow the design(anticipated) movement of theexternal CFCC strands due to trafficloads and temperature changes, and toaccommodate construction tolerances(see Fig. 6). This bearing system wascritical to provide an almost frictionlessbearing surface between the CFCCstrands and the diaphragm concrete.Fig. 3. Pretensioned CFRP Leadline tendons at roller hold-down location.plant, Prestressed Systems Inc., locatedin Windsor, Ontario, Canada.The sequence of fabrication is givenbelow:1. Installation of the CFCC andCFRP mild reinforcement, steel stirrups,CFRP prestressing Leadlinetendons, and other embedded itemssuch as hold-up and hold-down devices,vibrating wire, and electrical resistancestrain gauges in the formwork.2. Installation and stressing of thepretensioned CFRP tendons.3. Casting and curing of concrete.4. Release of pretensioned CFRPLeadline tendons after concreteachieved desired strength.5. Removal of the prestressed beamfrom the form.6. Installation of the longitudinalCFCC post-tensioning strands and applying60 percent of the total post-tensioningforce.Except for those in Row 10, allLeadline tendons were draped priorto pretensioning using hold-down andhold-up roller devices. These tendonholdingrollers were specially fabri-InstrumentationThe instrumentation installed in thetest beam was selected to achieve a setof measurement objectives. Thesemeasurements included the following:• Pretensioned load applied to CFRPLeadline tendons.• Transfer length of pretensionedCFRP Leadline tendons after release.• Beam camber/deflections duringfabrication/construction sequence.• Concrete strain distributions inbeam cross section and concrete topping.• Forces in CFCC post-tensioningstrands.122 PCI JOURNAL

Measurement ofPretensioning ForcesPretensioning forces were measuredusing a load cell installed between thestressing jack at the live end and thechuck anchor, and these forces wererecorded on a read-out device. A fewload cells were also positioned at thedead end of selected tendon lines betweenthe steel strand anchor chuckand the bulkhead for spot checkingand verification of the readings takenat the live end.In addition to these load cells, theelongation of the tendons and gaugepressure of the hydraulic pump wereused to verify the desired pretensioningforces. Once the jacking force,gauge pressure, and elongation hadbeen recorded, the stressing assemblywas transferred to the next tendon; thisprocedure was continued until all 60Leadline tendons were stressed.The total prestressing forces in eachCFRP tendon in Rows 1 through 5 andRows 6 through 10 were 82.3 and 86.7kN (18.5 and 19.5 kips), respectively,after seating losses, or about 40 and 42percent of the ultimate breaking loadof the Leadline tendons, respectively.Load cells located at the deadend were continuously monitored duringand after placement of the concreteto measure any changes in thepretensioning forces during concreteplacing and curing.Measurement of Transfer LengthTransfer length of the pretensioningtendons was measured using seven vibratingwire strain gauges with an effectivegauge length of 152 mm (6.0in.). Gauges were installed along oneweb at both ends of the beam duringfabrication. Strain gauges were installedend to end beginning at the endof the beam and extending over a distanceof approximately 1065 mm(42.0 in.). All seven gauges at eachbeam end were located at an elevationcoinciding with the centroid of thetotal designed pretensioning force, andstaggered about the mid-width centerlineof the web.Measurement of Early Age CamberA complete history of early age deflectionin the test beam was obtainedFig. 4. Release pattern for 60 pretensioned tendons.Fig. 5. Instrumentation and CFRP/CFCC reinforcing cage of one web (looking downfrom top).Fig. 6. Externallydraped post-tensioningCFCC strands (lookingfrom below DT beam).July-August 2003 123

Fig. 7. Post-tensioning anchorage details at Diaphragm D6.Fig. 8. Posttensioningapparatus for aCFCC externallydraped strand.using elevation reference points embeddedin the concrete surface. Elevationpoints were located along the topflange of the test beam at midspan, attwo quarter-span points, and near thebeam ends. Prior to prestressing tendonrelease, elevation measurementswere recorded for each point and usedas a reference for all subsequent measurements.After tendon release, measuredchanges in elevation relative tothe initial pre-release values were usedto calculate beam camber due to prestressingat the mid- and quarter-spanpoints.Total camber due to pretensioningforce and initial post-tensioning forcewas 20.1 mm (0.79 in.). Net camberafter the addition of the topping andthe final stage of post-tensioning was14.3 mm (0.56 in.). Note that a totalloss of 6.4 mm (0.25 in.) of midspancamber occurred due to changes in thesupport location and transportation ofthe beam from the fabrication plant tothe testing facility.The elevation of each referencepoint was determined using a precisionlevel and surveying rod with ahigh-resolution scale. The precisionlevel used for this application requiredan internal micrometer capable of resolvingelevation readings to the nearest0.030 mm (0.001 in.). The top surfaceof each embedded reference pointincorporated a machined indentationto match the pointed end of the surveyor’srod.Measurement of Concrete StrainsEmbedded electrical-resistancestrain gauges were installed for measuringstrain distributions along thedepth of cross sections at midspan andquarter-spans. A total of 30 straingauges were installed in the test beam.Of these 30 gauges, 21 were installedin the precast section at the fabricationplant, while the remaining nine gaugeswere installed in the cast-in-place concretetopping at the testing facility.Fig. 9. NEFMACreinforcing gridplaced over theflange top.Projected steel stirrupMeasurement ofPost-tensioning ForcesPrior to post-tensioning, all fourpost-tensioning strands were instrumentedwith load cells at one end (seeFig. 8). The load cells were installed124 PCI JOURNAL

etween the lock nut on the strand anchor/headand the bearing plate embeddedin the transverse DiaphragmD2 near the end of the beam (see Fig.2).Post-tensioning was applied to thefour CFCC strands in two separatestages (initial and final). The initialpost-tensioning (at the precast yard inWindsor, Canada) consisted of applying60 percent of the total desiredpost-tensioning force [449 kN (101kips)], equal to 32 percent of the ultimatebreaking load of CFCC post-tensioningstrands on the precast DTbeam. Final post-tensioning was appliedat the Construction TechnologyLaboratories, Inc. (CTL), testing facilityin Skokie, Illinois, after casting theconcrete topping. Final post-tensioningconsisted of the remaining 40 percentof the total post-tensioning force.Each stage of post-tensioning wasapplied in two parts. During the firstpart, approximately 50 percent [133.5kN (30.0 kips)] of the required forcewas applied by pulling the strandsfrom one end as shown in Fig. 8. Aftercompletion of the first part, the posttensioningsetup was moved to the oppositeend where the remaining 50 percentof the required force was applied.At the precast facility, the initialpost-tensioning was applied with thebeam supported at Diaphragms D2and D6 at intermediate transverse locations.The beam remained supportedin this manner until delivered to theCTL test facility. The beam supportconfiguration prevented the developmentof additional tensile strain due todead loads.At CTL, final post-tensioning wasapplied with the beam supported at theEnd Diaphragms D1 and D7 after theplacement of the concrete topping tosimulate the post-tensioning of the actualDT beams used in the BridgeStreet Bridge. The average measuredstrand force after the initial post-tensioningwas 268 kN (60.3 kips), whilethe average measured strand force immediatelyafter the final post-tensioningwas 449 kN (101 kips).CTL followed stressing proceduressimilar to those of the precast plant:the remaining 40 percent of the totalpost-tensioning force was appliedprior to the onset of structural testingFig. 10. Flexural load testing at the CTL testing facility.(described below). The average measuredforce in the CFCC strand priorto the test was 443 kN (99.6 kips),which was slightly less than that observedimmediately after the finalpost-tensioning.Casting of Concrete ToppingFinally, a 75 mm (3.0 in.) thick concretetopping was cast over the topflange of the precast DT beam. Inpreparation for casting, a CFRP NEF-MAC grid reinforcement (see Fig. 9)was installed over the top flange of thebeam. In addition, the nine remainingembedded concrete strain gauges requiredfor concrete strain distributionmeasurements were installed atmidspan and the two quarter-span locations.The NEFMAC reinforcement was(a) Test setup forflexural loading.(b) Supportcondition oftest beam.supplied in the form of grid sheets,each with a transverse dimension of2083 mm (82.0 in.). Each grid sheetincorporated longitudinal reinforcementelements spaced at 300 mm (12in.) and transverse elements spaced at100 mm (4 in.) intervals. The NEF-MAC reinforcement was supported on25 mm (1.0 in.) high plastic slab bolsters;these were epoxy-anchored tothe top surface of the beam flange.Each grid sheet was secured to theslab bolster using plastic ties.TEST SETUPThe flexural loading test setup andsupport condition for the beam areshown in Figs. 10a and 10b, respectively.The beam was loaded alongtwo lines to create a 3658 mm (12.0ft) wide constant moment region sym-July-August 2003 125

Output data from instrumentationwere monitored throughout the testand recorded at each loading increment.RESULTS AND DISCUSSIONFig. 11. Location of four-point loading system, showing location of displacementtransducers.Transfer LengthFig. 12 shows the concrete strainplotted against distance from the liveend of the beam. The strain valuesrefer to data recorded at various timeintervals (immediately, 5 minutes, 1hour, 26 hours, and 43 hours after release,and prior to initial post-tensioning).It can be seen from Fig. 12 thatthe transfer length (the distance fromthe beam end at which full effectiveprestress is transferred to the concrete)is approximately 381 mm (15.0 in.),compared to a theoretical value of 419mm (16.5 in.) [based on Eq. (4) inReference 11].Thus, experimental and calculatedresults for transfer length ofLeadline tendons are similar invalue. As indicated by the measureddata, concrete strains gradually increasedwith distance from the beamend until strains stabilized or becameuniform (at the location of the third vibrating-wirestrain gauge).Fig. 12. Concrete strain versus distance from beam live end.metrical about midspan (see Fig. 11).The loading lines were oriented orthogonallyto the longitudinal centerlineof the beam. Along each line, loadwas applied at two bearing points thatwere coincident with the beam webs.Roller supports in Fig. 10b were usedat each beam end to allow both longitudinalmovement and rotation underthe load.Load was applied using a series ofhydraulic jacks with load and extensioncapability sufficient to induceflexural failure. All loads applied tothe beam during the test were monitoredusing load cells. Beam displacementsat mid- and quarter-span locationswere monitored using twodisplacement transducers at each locationattached to the underside of thetwo webs.In addition to the applied loads anddeflections, output from the concretestrain gauges installed for measuringstrain distribution at midspan and twoquarter-span sections – and from thelongitudinal CFCC strand load cells –was monitored during the flexural test.Strain Distribution after Posttensioningand Service LoadThe distribution of concrete strainsalong the depth of the cross sections[at mid- and two quarter-span (Q-S)locations] after the final post-tensioningand due to service load is shown inFigs. 13a and 13b, respectively. Fig.13a indicates that after final post-tensioning,the strain at the top surface ofthe concrete topping (at midspan) issmall and tensile in nature, whereasthe strain throughout the depth ofcross section is in the compressionrange.High compressive strain developedat the bottom at each cross section location.The developed compressivestrain levels were desirable to eliminateany potential cracking problemsunder service load/traffic condition.The DT cross section dimensions, theplacement and arrangement of prestressingtendons, as well as post-ten-126 PCI JOURNAL

(a) After final post-tensioning(b) Due to service loadFig. 13. Strain measurements: (a) Average measured concrete strain along the depth of cross section after final post-tensioning;and (b) Strain due to service load.(a) At cracking load(b) At ultimate loadFig. 14. Average measured concrete strain: (a) At cracking load; and (b) At ultimate load.sioning strands, were selected so thatno cracking would occur under serviceloads.Superposition of the concrete strainreadings due to pretensioning andpost-tensioning shown in Fig. 13a andthe service load condition in Fig. 13bindicates that the net strain at the bottomand top surfaces of the beam iscompressive. This strain result is desirablein eliminating any potentialcracking problems in the CFRP prestressedconcrete beam under serviceloading.Strain Distribution at Cracking andUltimate LoadsAfter reaching the service loadlevel, incremental loading of the DTbeam continued to reach the appliedload corresponding to the developmentof the first concrete crack. To accomplishthis objective, the beam wasloaded in suitably small increments sothat careful observations could bemade of the bottom surface for beamstrains within the constant moment region.After application of each increment,loading was suspended to allowsufficient time to visually inspect forcracking.The total applied load correspondingto the development of the firstcrack was found to be 643.9 kN(144.7 kips). Figs. 14a and 14b showthe concrete strain distributions alongthe depth of cross sections at mid- andquarter-span (Q-S) locations at thecracking and ultimate load conditions.Note that the strain readings shown inFig. 14 are only due to the appliedloads and do not include the pretensioningand post-tensioning effects.Fig. 14 shows that the strain distributionsat the quarter-span sectionsare very close in value. In Fig. 14b,the ultimate strain at the top of concretetopping has reached a compressivestrain as high as 0.0025, whereasthe webs are in tension. Note that datafor web strain at the midspan sectionunder the ultimate load condition inFig. 14b are not presented due to failureof the corresponding strain gauge.Superposition of the concrete strainreadings due to pretensioning andpost-tensioning and the cracking loadconditions shown in Figs. 13a and 14aindicates that the net strain at the bottomof the beam is close to the strainvalue corresponding to the modulus ofrupture of concrete.July-August 2003 127

Fig. 15. Beam deflection response at midspan under ultimate load.The ultimate load applied during thetest was 2443.0 kN (549.1 kips). Averagemidspan deflections at the service,cracking, and ultimate load conditionswere 12.82, 18.67, and 342.3 mm(0.505, 0.74, and 13.48 in.), while thecorresponding forces in the unbondedpost-tensioning CFCC strands were450.0, 453.8, and 807.1 kN (101.1,102.0, and 181.4 kips), respectively.Fig. 16 indicates that the failure ofthe DT beam occurred in the constantmoment region along one side of themidspan Diaphragm D4 and that thefailure plane extended across thewidth of the beam. The ultimate failureof the DT beam was initiated bypartial separation between the bottomof the topping and top of the beamflange, which led to crushing of theconcrete topping. This was due to thedifference in the stiffness and strengthof the concrete topping and precastconcrete section.After the concrete topping failed, anattempt was made to further increasethe load. As Fig. 15 shows, however,the rupture of Leadline tendons ledto the beam collapse without any furtherincrease in the load. The forcelevels in the four CFCC post-tensioningstrands (see Fig. 17) nearly doubledduring the test, increasing fromapproximately 443 kN (99.6 kips) atthe onset of loading to 807 kN (181.4kips) at the ultimate load, yet none ofthe CFCC strands nor their anchorsruptured.Fig. 16. Closeupof beam failure.Externalunbonded CFCCpost-tensioningstrands did notrupture.ULTIMATE BEHAVIOR OFDT BEAMUltimate flexural strength wasreached when the beam was incrementallyloaded to a total load of 1303.9kN (293.1 kips) and then unloaded(see Fig. 15). This load representsabout 50 percent of the actual ultimatestrength. The loading and unloadingsequence was necessary to evaluatethe ductility of the beam. After the initialloading and unloading sequence,the beam was incrementally loaded toinduce flexural failure.ANALYSISAn analysis of experimental resultsobtained from the full-scale DT testbeam is presented in this paper. A theoreticalanalysis was also developed topredict the strength and failure modeof the DT test beam. The strains,stresses, and forces in strands, and deflectionof the beam were calculatedfor the service and ultimate loadingconditions. 14 Details of the theoreticalanalysis and calculations are presentedin Appendix A. 15-20CONCLUSIONSBased on the results of the test program,the following conclusions canbe drawn:128 PCI JOURNAL

Fig. 17. Force inCFCC posttensioningstrands,indicating responseunder ultimate loadand failure of DTbeam.1. The test results and constructionexperience gained from this evaluationhave been implemented in the designdocuments for the 12 DT beams requiredfor the construction of theBridge Street Bridge.2. The DT beam had significant reservestrength beyond the service load.The ultimate load and the crackingload of the DT beam were, respectively,approximately 5.3 and 1.4times the service load.3. The combined internal and externalprestressing forces induced the designedcompressive strain in the crosssection.4. The ultimate failure of the DTbeam was initiated by partial separationbetween the topping and the beamflange, which led to crushing of theconcrete topping followed by ruptureof the internal prestressing tendons.5. Significant cracking was observedprior to failure. However, none of theexternal unbonded CFCC post-tensioningstrands nor their anchors ruptured.6. Theoretical values and experimentalresults are very close, especiallyunder service load condition.The theoretical nominal moment capacityof the DT beam is about 0.9percent lower than the correspondingexperimental value, whereas the predictedforces in the unbonded posttensioningstrands at the service andultimate load conditions are 0.5 and6.1 percent lower than correspondingexperimental values.ACKNOWLEDGMENTSThe DT beam for this study wasconstructed by Prestressed SystemsInc. (PSI), Windsor, Ontario, Canada,and tested by Construction TechnologyLaboratories, Inc. (CTL), Skokie,Illinois. Hubbell, Roth & Clark(HRC), Consulting Engineers, BloomfieldHills, Michigan, provided the entiredesign, drawing details, and constructionadministration services. TheCity of Southfield and the FederalHighway Administration jointlyfunded the instrumentation and testingof the DT beam.The National Science Foundation,Division of Civil and Mechanical Systems,funded the research activities,testing of several one-third-scale models(conducted in the Structural TestingCenter at Lawrence TechnologicalUniversity), and the analytical evaluationof the full-scale testing.Mitsubishi Chemical FunctionalProducts, Inc., Tokyo Rope ManufacturingCo. Ltd., Autocon CompositesInc., Toronto, Canada, SumitomoCorporation of America, and MitsuiUSA Corporation also supported theongoing research investigation. Thetireless efforts of Drs. S. B. Singh andT. P. Murphy in these research andtesting investigations are highly appreciated.The authors also appreciate the technicalcomments of Dr. G. Tadros andthe many constructive comments ofthe PCI JOURNAL reviewers.July-August 2003 129

REFERENCES1. ACI Committee 440, “State-of-the-Art Report on Fiber ReinforcedPlastic Reinforcement for Concrete Structures,” ACI440R-96, American Concrete Institute, Farmington Hills, MI,1996, 153 pp.2. Rizkalla, S. H., “A New Generation of Civil EngineeringStructures and Bridges,” Proceedings of the Third InternationalSymposium on Nonmetallic (FRPRC) Reinforcement for ConcreteStructures, Sapporo, Japan, V. 1, October 1997, pp. 113-128.3. Dolan, C. W., “FRP Prestressing in the U.S.A.,” Concrete International,V. 21, No. 10, October 1999, pp. 21-24.4. Tadros, G., “Provisions for Using FRP in the Canadian HighwayBridge Design,” Concrete International, V. 22, No. 7,July 2000, pp. 42-47.5. Fam, A. Z., Rizkalla, S. H., and Tadros, G., “Behavior ofCFRP Prestressing and Shear Reinforcements of ConcreteHighway Bridges,” ACI Structural Journal, V. 94, No. 1, January-February1997, pp. 77-86.6. Shehata, E., Abdelrahman, A., Rizkalla, S. H., and Tadros, G.,“The New Generation,” Concrete International, V. 20, No. 6,June 1998, pp. 35-38.7. Grace, N. F., and Abdel-Sayed, G., “Ductility of PrestressedConcrete Bridges Using CFRP Strands,” Concrete International,V. 20, No. 6, June 1998, pp. 25-30.8. Grace, N. F., Navarre, F. C., Nacey, R. B., Bonus, W., andCollavino, L., “Design-Construction of Bridge Street Bridge –First CFRP Bridge in the United States,” PCI JOURNAL, V.47, No. 5, September-October 2002, pp. 20-35.9. Grace, N. F., and Abdel-Sayed, G., “Behavior of ExternallyDraped CFRP Tendons in Prestressed Concrete Bridges,” PCIJOURNAL, V. 43, No. 5, September-October 1998, pp. 88-101.10. Grace, N. F., “Response of Continuous CFRP Prestressed ConcreteBridges Under Static and Repeated Loadings,” PCIJOURNAL, V. 45, No. 6, November-December 2000, pp. 84-102.11. Grace, N. F., “Transfer Length of CFRP/CFCC Strands forDouble-T Girders,” PCI JOURNAL, V. 45, No. 5, September-October, 2000, pp. 110-126.12. Grace, N. F., Enomoto, T., and Yagi, K., “Behavior of CFCCand CFRP Leadline Prestressing System in Bridge Construction,”PCI JOURNAL, V. 47, No. 3, May-June 2002, pp. 90-103.13. Grace, N. F., Abdel-Sayed, G., Navarre, F. C., Nacey, R. B.,Bonus, W., and Collavino, L., “Full-Scale Test of PrestressedDouble-Tee Beam,” Concrete International, V. 25, No. 4,April 2003, pp. 52-58.14. Grace, N. F., and Singh, S. B., “Design Approach for CFRPPrestressed Concrete Bridge Beams,” ACI Structural Journal,V. 100, No. 3, May-June 2003, pp. 365-376.15. Mitsubishi Chemical Corporation (MCC), Leadline CarbonFiber Tendons/Bars, Product Manual, 1994.16. Tokyo Rope Mfg. Co. Ltd., Technical Data on CFCC, ProductManual, 1993.17. NEFMAC, Technical Data Collection, Autocon CompositesInc., Canada, 1996, pp. 1-10.18. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-02) and Commentary (318R-02),”American Concrete Institute, Farmington Hills, MI, p. 105.19. Chad, R. B., and Dolan, C. W., “Flexural Design of PrestressedConcrete Beams Using FRP Tendons,” PCI JOUR-NAL, V. 46, No. 2, March-April 2001, pp. 76-87.20. Naaman, A. E., and Alkhairi, F. M., “Stress at Ultimate in UnbondedPost-tensioning Tendons: Part 2 – Proposed Methodology,”ACI Structural Journal, V. 88, No. 6, November-December1991, pp. 683-692.130 PCI JOURNAL

APPENDIX A – ANALYSIS PROCEDUREThe following procedure illustrates the steps for predictingthe behavior of the full-scale pretensioned and post-tensionedDT test beam. The material properties of the tendons,strands, NEFMAC grids, and concrete are presented inTable 1. It should be noted that the term “tendon” is used forLeadline TM rods and “strand” is used for CFCC cablesthroughout the paper. It should also be noted that the crosssectionalarea of NEFMAC grids has not been included inthe analysis presented below:Step 1 — Calculate required moment capacity:Total dead load of beam = 142 kipsDead load /unit length, W d = 142/67 = 2.12 kip/ftDead load moment at midspan:MDWLd=82212 . × 67=8= 1188.8 kip-ftLet W be the total midspan load applied through 4-pointloading system. The distance between the center of eachsupport to the nearest load point = 27.5 ftDesign service live load = 104.3 kipsService live load moment at midspan,104. 3 × 27.5M L=2= 1434.13 kip-ftRequired moment capacity of the section,M required = 1.4M D + 1.7M L (ACI 318-02)= 4102.34 kip-ftStep 2 — Calculate balanced ratio, ρ b :fc′εcuρb= 085 . β 1(A1)f ε + ε −εwheref c ′ = strength of precast concrete = 7.81 ksi (see Table 3)ε fu = specified ultimate strain of bonded pretensioningtendons = 0.019 (see Table 1)f fu = specified strength of bonded pretensioning tendons= 415 ksi (see Table 1)β 1 = 0.85 – [(7810 – 4000)/1000] × 0.05 = 0.66(ACI 318-02)Based on experimental results, 25 percent loss in the prestressingforce in the bonded tendons is considered. The initialeffective strain in bottom pretensioning tendons, ε pbmi ,can be calculated as follows:19. 5 × 0.75ε pbmi=0. 111 × 21,320= 0.00622fucu fu pbmiBased on experimental observation, the ultimate strain inconcrete, ε cu , is taken as 0.0025. This value of strain is veryclose to the value of 0.003 recommended by ACI 318-02.The use of measured value of crushing strain in theoreticalevaluation will amount to an accurate comparison of the theoreticaland experimental values of the ultimate load carryingcapacity of the DT beam. Substituting the values in Eq.(A1) yields ρ b = 0.0017.Step 3 — Calculate cracking moment, M cr , andcracking load, P cr :Modulus of rupture of concrete,f r = 6.0 f c′= 530 psiVarious sectional properties of the DT beam section (seeFig. A1) are presented in Table A1.Total effective pretensioning force,F pre = 1139.3 × 0.75= 854.5 kipsTotal effective post-tensioning force, F post = 397.8 kipsStress at the bottom fiber of the section due to pretensioningforce,FpreFpreeb( σ c) b 1=− −(A2)A SpHere, A p and S bp are cross-sectional area and section modulus(with respect to bottom fiber) of precast section, respectively;e b is the eccentricity of resultant pretensioning forcewith respect to the centroid of the precast section. FromTable A1, A p = 1462 sq in., S bp =11,192 in. 3 , and e b = 13 in.Substituting values in Eq. (A2) results in (σ c ) b1 = –1.577 ksi.Since 60 percent of the total post-tensioning force wasused to prestress the precast DT concrete section and the remaining40 percent of the total post-tensioning force was appliedon the composite concrete section (DT beam plus concretetopping), then the stress at the bottom fiber of sectiondue to post-tensioning force, (σ c ) b2 , is:( σ c) b 2=06 . × Fpost06 . × Fposteup04 . × Fpost04× . Fpost× euc− −− −ApSbpAcSbc(A3)Here, e up and e uc are the eccentricity of unbonded posttensioningstrands with respect to the centroids of precastand composite sections, respectively. From Table A1, valuesof e up and e uc are 19.12 and 21.67 in., respectively. Substitutingthe values in Eq. (A3),(σ c ) b2 = –0.938 ksiThe cracking moment (M cr ) can be found from the followingexpression:(σ c ) b1 + (σ c ) b2 + M cr /S bc = f r (A4)Here, S bc is the section modulus of composite section withbpJuly-August 2003 131

Table A1. Section properties of DT beam cross section.Sectional properties Precast section Composite sectionCross-sectional area, sq in. 1462 1673Moment of inertia, in. 4 328,489 403,821Distance of centroid of the section from the bottom fiber of the section, in. 29.35 31.90Distance of centroid of the section from the top fiber of the section, in. 18.68 19.08Section modulus corresponding to the bottom fiber of section, in. 3 11,192 12,659Section modulus corresponding to the top fiber of section, in. 3 17,585 21,165Eccentricity of resultant pretensioned force, in. 13 N/AEccentricity of unbonded post-tensioning strands, in. 19.12 21.67Note: 1 in. = 25.4 mm; 1 sq in. = 645 mm 2 ; 1 in. 3 = 16387 mm 3 ; 1 in. 4 = 416231 mm 4 .N/A refers to a non-applicable value for calculating stresses and/or moment due to pretensioning forces because pretensioning forces were not applied to the composite section.respect to extreme tension fiber and is equal to 12,659 in. 3(see Table A1). Substituting the values in Eq. (A4),M cr = 3212.2 kip-ftThus, cracking load,Pcr( Mcr− MD)× 2=27.5= 147.2 kips > service load (104.3 kips)Percent difference between the theoretical and experimentalcracking loads:( 147. 2 − 144. 7)=× 100144.7= 17 . percentStep 4 — Compute flexural moment capacity:Ultimate bond reduction coefficient for externalstrands,Ω u= 54 . (A5)LuduwhereL u = horizontal distance between anchorages of unbondedpost-tensioning tendons = 51.4 ftd u = distance at midspan of external post-tensioningstrands from the extreme compression fiber =40.75 in.Substituting the values in Eq. (A5), we get Ω u = 0.36.Distance of the centroid of bottom bonded pretensioningtendons from the extreme compression fiber, d m = 47 in.Flange width, b = 83.46 in.Total effective pretensioning force, ∑F pi = 854.5 kipsTotal effective post-tensioning force, F pui = 397.8 kips* Equivalent tendon/strand means a tendon/strand located at centroid oftendons/strands and having cross-sectional area equal to the total cross-sectionalarea of corresponding tendons/strands.It should be noted that the actual reinforcement ratio of thesection is defined as the ratio of total weighted area of the tendonsand/or strands in the section to the effective concretearea. Here, weighting factor is defined as the ratio of the stressin a particular equivalent * tendon/strand to the specifiedstrength of bonded pretensioning tendons. The expression forthe reinforcement ratio is based on equilibrium of forces inconcrete and tendons and the strain compatibility conditions(see Fig. A1).Flexural stress in the equivalent bonded prestressing tendonat balanced condition, f pbb = 188.7 ksiStress in the equivalent non-prestressing strand of webs atbalanced condition, f pnbb = 152.4 ksiStress in the equivalent unbonded strand at balanced condition,f pub = 72.2 ksiStress in the equivalent non-prestressing tendons of flangeat balanced condition, f pnf = 39.94 ksiTotal cross-sectional area of bonded prestressing tendons,A pb = 6.66 sq in.Total cross-sectional area of non-prestressing strands in thewebs, A pn = 3.304 sq in.Total cross-sectional area of unbonded prestressing strands,A fu = 4.68 sq in.Total cross-sectional area of non-prestressing tendons in theflange, A pnf = 2.109 sq in.Reinforcement ratio,ρ =m∑ F + f A + f A + F + f A − f Aj = 1pi pbb pb=854. 5+ 188. 7× 6. 66 + 152. 4× 3. 304+ 397. 8+ 72. 2× 4. 68− 39. 94×2.10983.46× 47×415= 0.002 > 0.0017 (ρ b )pnbb pn pui pub fu pnfb pnfb × d × fmfu(A6)Since the section is over-reinforced, failure of the beam willoccur due to crushing of the concrete topping. The steps tocalculate the moment capacity of the DT beam are givenbelow:132 PCI JOURNAL

Fig. A1. Strain and stress distributions for DT beam at ultimate loading indicating compressive and tensile forces.a. Compute strain in tendons/strands:The strains in tendons/strands can be calculated from thestrain distribution as shown in Fig. A1. The ultimate failureof the DT beam was initiated by partial separation betweenthe topping and the beam flange, which led to crushing ofthe concrete at the strain level of 0.0025. Thus, strain in theconcrete at the extreme compression fiber, ε cu = 0.0025.Let c = depth to the neutral axis from the extreme compressionfiberStrain in bonded prestressing tendons,εpbj0. 0025× ( dj−c)= + εpbji(for j = 1, m) (A7)cStrain in unbonded post-tensioning strands,uε pu= Ωu+ 0.0046 (A8)Strain in non-prestressing strands,ε pnj0. 0025× ( d −c)c0. 0025× ( hj−c)= (A9)cStrain in non-prestressing tendons at flange top,0. 0025× ( c−d)ctε pnt= (A10)Strain in non-prestressing tendons at flange bottom,0. 0025× ( c−d)cbε pnb= (A11)In Eq. (A7), e pbji equals 0.0059 (for j = 1, 5) and 0.0062(for j = 6 to 10); d j (j =1 to 10) is the depth of an individuallayer, j, of the bonded pretensioning tendons from the extremecompression fiber. Here, d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , d 7 , d 8 ,d 9 , and d 10 are 21.8, 24.6, 27.4, 30.2, 33.0, 35.8, 38.6, 41.4,44.2, and 47.0 in., respectively; d u is the depth of unbondedpost-tensioning tendons from the extreme compression fiberand is equal to 40.75 in.In Eq. (A9), h j (j = 1 to 6) is the depth of an individuallayer, j, of the non-prestressing tendons provided in webs.Here, h 1 , h 2 , h 3 , h 4 , h 5 , and h 6 are 8.8, 16.4, 23.5, 31.8, 40.1,and 48.6 in., respectively; d t [see Eq. (A10)] and d b [see Eq.(A11)] are the depth of the top and bottom layers of non-prestressingtendons provided in the flange, respectively. Here,d t and d b are 4.5 and 6.9 in., respectively.b. Compute resultant forces in tendons/strands andconcrete:Total cross-sectional area of bonded prestressing tendonsin each row, A fb = 0.666 sq in.Total cross-sectional area of non-prestressing strands ineach row, A fn = 0.472 sq in. for Rows 1 to 5 and A fn = 0.944sq in. for Row 6.Total cross-sectional area of non-prestressing tendons atflange top, A fnt = 1.11 sq in.Total cross-sectional area of non-prestressing tendons atflange bottom, A fnb = 0.999 sq in.Using the stress-strain and force-stress relationships (i.e.,stress = strain × modulus of elasticity; and force = stress ×cross-sectional area of tendons of each layer), the resultantforces in each layer of prestressing and non-prestressing tendonscan be determined by the following expressions:Resultant force in pretensioning tendons,FFpbjpbj⎡35.( 5 dj− c)⎤= ⎢+ 83. 78 j⎣ c⎥ kips for = 1 to 5 (A12a)⎦⎡35.( 5 dj− c)⎤= ⎢+ 88. 03 j⎣ c⎥ kips for = 6 to 10 (A12b)⎦Resultant force in unbonded post-tensioning strands,Fpu =⎡77. 6 ( 40. 75 − c)⎣⎢ c+ 396. 54⎤kips (A13)⎦⎥July-August 2003 133

Table A2. Stresses in tendons of different layers at ultimate load condition of DT beam.Stress (ksi)Bonded Non-prestressing Unbonded Non-prestressing Non-prestressingpretensioning tendons located post-tensioning tendons located tendons located atLayer number tendons in webs tendons at flange top flange bottom1 205.6 0.40 145.5 25.8 11.22 222.7 43.63 239.8 84.04 256.9 131.25 274.0 178.56 297.5 226.8 N/A N/A N/A7 314.58 331.6N/A9 348.710 365.8Note: 1 ksi = 6.895 MPa.Resultant force in non-prestressing strands,23. 45( hj− c)=c46.( 9 hj− c)=cResultant force in non-prestressing tendons at flange top,Resultant force in non-prestressing tendons at flange bottom,FFpnb =Compressive force in concrete = C t + C f= 0.85f ′ c (E ct /E c )bh t + 0.85f ′ c b(0.66c – 2.95)= [1405.62 + 554.05 (0.66c – 2.95)] kips (A17)c. Compute the neutral axis depth c:From the equation of equilibrium,10pnt=⎡53. 3 ( c − 6. 9)⎤⎣⎢ c ⎦⎥∑ F + ∑ F + F = C(A18)pbjj = 1 j = 159. 2( c − 4. 5)cpnj35.5 23.45( 344 − 10 c)+ 859 . 05 + ( 217 . 8 − 7 c)+cc77.6( 40 . 75 − c) + 396 . 54c= 1405. 62 + 554. 05( 0. 66c− 2. 95)+59. 2( c − 4. 5) 53. 3( c − 69 . )+ccSolving for c gives c = 8.73 in.6kips for j = 1 to 5 (A14a)kips for j = 6 (A14b)pukipskips(A15)(A16)d. Compute stresses in tendons/strands:The value of strain in tendons/strands can be computedusing Eqs. (A7) through (A11), and then using the stressstrainrelationships, stresses in tendons/strands of differentlayers can be calculated.For example, stress in the tenth layer of bonded pretensioningtendons−=⎡0. 0025 ( 47 c)+ 0. 0062⎤⎣⎢⎦⎥ × 21,320c= 365.8 ksi < 415 ksi (OK)The calculated values of the stress in tendons of differentlayers are presented in Table A2.e. Compute the resultant force in tendons and concrete:Resultant forces in tendons of different layers can be calculatedby multiplying the corresponding tendons stress andequivalent cross-sectional area of tendons of a particularlayer. For example, resultant force in pretensioning tendonsof 10th layer is given by:F pb10 = 365.8 × 0.666= 243.6 kipsThe calculated resultant forces in tendons of different layersare presented in Table A3.From Eq. (A17), resultant force in concrete = C t + C f =2963.5 kipsTotal compressive force,C = C t + C f + F pnt + F pnb= 2963.5 + 28.6 + 11.2= 3003.2 kipsTotal resultant tension force,F = ∑ F + ∑ F + FR106pbjj = 1 j = 1pnj= 3004. 3 kips ≅ C (OK)pu134 PCI JOURNAL

Table A3. Resultant forces in tendons of different layers at ultimate load condition of DT beam.Force (kips)Bonded Non-prestressing Unbonded Non-prestressing Non-prestressingpretensioning tendons located post-tensioning tendons located tendons located atLayer number tendons in webs tendons at flange top flange bottom1 136.9 0.2 680.9 28.6 11.22 148.3 20.63 159.7 39.64 171.1 61.95 182.5 84.36 198.1 214.1 N/A N/A N/A7 209.58 220.8N/A9 232.210 243.6Note: 1 kip = 4.44 kN.f. Compute the ultimate moment capacity:Location of the center of gravity of the resultant tension forcemeasured from the extreme compression fiber, d = 37.61 in.Depth of Whitney’s equivalent stress block,a = β 1 c= 0.66 × 8.73= 5.76 in.Distance of the center of gravity of resultant compressionforce from the extreme compression fiber,d =295 .× + ×⎛ 281 .1405.6 1557. 9 2.95 +⎞2⎝ 2 ⎠ + 28. 3 × 4. 5 + 10. 6 × 69 .3003.2= 302 . in.Thus, nominal moment capacity,M n = F R (d – _ d)= 3004.3 × (37.61 – 3.02)= 8659.9 kip-ftDesign moment capacity,M u = 0.85 × 8659.9= 8210.9 kip-ft > 4102.34 kip-ft (M required ) (OK)g. Compare the theoretical and experimental values ofmoment capacity and post-tensioning tendon force:Experimental value of the moment capacity of DT beam =[(549 × 227.5)/2] + 1188.8 = 8739.0 kip-ftTheoretical nominal moment capacity of DT beam = 8659.9kip-ftPercent difference in experimental and theoretical momentcapacities,( 8739. 0 − 8659. 9)=× 1008739.0= 09 . percentExperimental value of post-tensioning force in eachpost-tensioning strand = 181.3 kipsTheoretical value of post-tensioning force in each tendon= 680.9/4 = 170.2 kipsPercent difference in experimental and theoretical valuesof post-tensioning forceh. Compute stresses due to service loads:Moment due to service loads, M = 1188.8 + 1434.13 =2622.93 kip-ft < 3203.9 kip-ft (cracking moment)Since moment due to service loads is less than thecracking moment, the section will remain uncracked at theservice load. The maximum stresses can be calculated asfollows:Maximum compressive stress in concrete at the extremecompression fiber,fct( 181. 3 − 170. 2)=× 100181.3= 61 . percentFpreFpreeb06 . × Fpost06 . × F= − + −ApStpApStp04 . × Fpost04 . × Fpost× e−ASSubstituting the values of sectional properties (seeTable A1) and other parametric values in Eq. (A19), weget f ct = 1.28 ksi < 4.69 ksi (0.6f c ′) (OK)Maximum concrete stress due to applied load1434.13 × 12=21,165= 081 . ksictcpostuceupM+Stc+(A19)July-August 2003 135

Experimental value of concrete stress = 1.0 ksiPercent difference in theoretical and experimentalvalues of concrete stressThis difference in analytical and experimentalstrain in concrete topping may be attributed to thechange in the strain condition because of the changingof the support condition of the beam during variousconstruction stages and due to different propertiesof the concrete topping and precast concrete.Stress in bottom prestressing tendons,f10 . − 081 .= × 10010 .= 19 percentEf M( d10− ytc)= E ε +EcIc= 21,320 × 0.0062 +pb10 f pb10i2622. 93 × 12 × ( 47. 0 −19. 08)4.0 ×40,3821= 132. 19 + 8.70= 140.89 ksi < 415 ksi (OK)Stress in bottom prestressing tendons due to appliedload1434. 13 × 12 × ( 47 −19. 08)= 40 . ×40,3821= 47 .6 ksi ≅ 4.73 ksi (experimental value)Stress in post-tensioning strands,Efp ( M − MD)efpu = Efpεucpui+ΩEcIc= 18, 419 × 0.0046 +( 2622. 93 − 1188. 8) × 12 × 21.67 2346 . ××403,8213= 86.86 ksi < 271 ksi (OK)Force in a CFCC strand at service load = 86.86 ×1.17 = 101.63 kipsExperimental value of post-tensioning force =101.12 kipsPercent difference in theoretical and experimentalvalues of post-tensioning force at service load101. 63 − 101.12=× 100101.12= 05 . percent(A20)i. Compute maximum deflection under serviceload (no cracking occurs under service load):M L = M – M D= 1434.13 kip-ftDistance between the support and nearest loadpoint, L 1 = 27.5 ftLongitudinal distance between load points, L 2 = 12 ftDeflection due to applied load,MLL ⎛ L ⎞ Lδ a= + ⎜ ⎟EcIc⎝ L ⎠+ ⎛ 22 ⎡⎝ ⎜⎞ ⎤182 2⎢ 40 .⎟ ⎥(A21)8L⎣⎢31 1 ⎠⎦⎥2× × × ⎡=+⎛ ⎞× ×⎝ ⎠ + ⎛ 21434. 13 12 ( 27. 5 12)8 12 12 ⎞ ⎤⎢ 40 .⎣⎝ ⎠ ⎥8 5320 40,3821 3 27. 5 27.5 ⎦= 0.502 in. ↓ ≅ 0.505 in. (experimental value)It should be noted that the precast DT beam was supported at DiaphragmsD2 and D6 prior to initial post-tensioning. The center-tocenterdistance between Diaphragms D1 and D2, d 12 , was 7.8 ft.Hence, midspan deflection of DT beam due to dead load is given byδdd=144 ⎡EIc p ⎣⎢δ =43WDL WDL L−⎛− d⎞12L384 12 ⎝ 2144 ⎡212 . × 67 212 . × 67−⎛⎢5320×328,489 ⎣ 384 12 ⎝= 001 . in. ↓Assuming that the loss of pretensioning forces up to the instantof initial post-tensioning be negligible,Effective pretensioning force = 1139.3 kipsDeflection due to pretensioning force at the instant of initial posttensioning1139. 3 × 13. 0 × ( 67 × 12)=8 × 5320 × 328,489= 069 . in. ↑Center-to-center distance between Diaphragms D2 and D6 =51.4 ftDeflection due initial post-tensioning06 . × 3978 . × 1912 . × ( 514 . × 12)=8 × 5320 × 32,8489= 012 . in. ↑Total deflection of the precast DT beam after the initial post-tensioning= 069 . + 012 . −001.= 080 . in. ↑Experimental value of deflection due to pretensioning and initialpost-tensioning = 0.79 in. ↑Percent difference in theoretical and experimental values of deflectionafter initial post-tensioning080 . − 079 .=079 .= 13 . percent⎠ + − ⎤5081. 85 79603.8in.⎦⎥4 326722−⎞⎠ + × − ⎤78 . 5081. 85 67 79603.8⎥⎦136 PCI JOURNAL

Measured loss in the deflection due to transportation andsupport condition change = 0.25 in. ↓Net deflection due to prestressing prior to final post-tensioning= 0.80 – 0.25= 055 . in. ↑Increase in deflection due to final post-tensioning04 . × 3978 . × 2167 . × ( 514 . × 12)=8 × 5320 × 40,3821= 008 . in. ↑Weight of the concrete topping150 × 83. 46 × 2.95=212 × 1000= 026 . kip / ft2Deflection due to concrete topping45 026 . × ( 67×12)=384 12 × 5320 × 403,821= 005 . in. ↓Net deflection at the time of final post-tensioning= 055 . + 008 . −005.= 058 . in. ↑Experimental value of the net deflection after final posttensioning= 0.56 in. ↑Percent difference in theoretical and experimental valuesof deflection after final post-tensioning058 . − 056 .=× 100056 .= 36 . percentAPPENDIX B — NOTATIONaA cA pA fbA fnA fntA fnbA fuA pbA pnA pnfbcCC t= depth of equivalent rectangular compressionblock, in.= cross-sectional area of composite DT beam, sqin.= cross-sectional area of precast DT beam, sq in.= cross-sectional area of bottom bonded prestressingtendons in each row, sq in.= cross-sectional area of non-prestressing tendonsin each row, sq in.= cross-sectional area of non-prestressing tendonsat flange top, sq in.= cross-sectional area of non-prestressing tendonsat flange bottom, sq in.= total cross-sectional area of unbonded post-tensioningtendons, sq in.= total cross-sectional area of bonded prestressingtendons, sq in.= total cross-sectional area of non-prestressing tendonsin the webs, sq in.= total cross-sectional area of non-prestressing tendonsin the flange, sq in.= width of compression face of member, in.= depth to the neutral axis from the extreme compressionfiber, in.= resultant compressive force, kips= resultant compression force in concrete topping,kipsC f = resultant compression force in flange of precastDT beam, kipsc.g.c = axis passing through the centroid of concretecross section of the DT beamc.g.p = axis passing through the center of gravity of theresultant pretensioned forced 12 = center-to-center distance between DiaphragmsD1 and D2, ftd = distance to center of gravity of the resultant tensionforce from the extreme compression fiber,in._dd jd md bd td u= distance to center of gravity of the resultant compressionforce from the extreme compressionfiber, in.= distance to centroid of bonded prestressing tendonsof an individual row from the extreme compressionfiber, in.= distance to centroid of bottom bonded prestressingtendons (mth row) from the extreme compressionfiber, in.= distance to centroid of non-prestressing tendonsat flange bottom from the extreme compressionfiber, in.= distance to centroid of non-prestressing tendonsat flange top from the extreme compressionfiber, in.= distance to centroid of unbonded post-tensioningJuly-August 2003 137

e be upe ucE cE ctE fE fnE fpf cf ctf ′ cf fuf funf fupf pbbf pbjf pbmf pbmif pnbf pnbbf pnfbf pnjf pnkf pntf puf rF piF preF postF pbjF pbmtendons from the extreme compression fiber, in.= eccentricity of resultant pretensioning force fromthe centroid of the precast concrete cross section,in.= eccentricity of unbonded post-tensioning tendonsfrom centroid of the precast concrete cross section,in.= eccentricity of unbonded post-tensioning tendonsfrom the centroid of composite section= modulus of elasticity of precast concrete, ksi= modulus of elasticity of concrete topping, ksi= modulus of elasticity of bonded tendon, ksi= modulus of elasticity of non-prestressing tendonsin webs, ksi= modulus of elasticity of unbonded tendon, ksi= stress in the concrete at extreme compressionfiber in very under-reinforced beam, ksi= stress in the concrete at extreme compressionfiber in over-reinforced beam, ksi= specified compressive strength of precast concrete,ksi= specified ultimate tensile strength of bonded prestressingtendons, ksi= specified ultimate tensile strength of non-prestressingtendons in webs, ksi= specified ultimate tensile strength of unbondedpost-tensioning tendons, ksi= flexural stress in the equivalent bonded prestressingtendons at balanced condition, ksi= total stress in bonded prestressing tendons of anindividual row, ksi= total stress in bottom prestressing tendons (mthrow), ksi= initial effective prestress in bottom bonded prestressingtendons (mth row), ksi= total stress in non-prestressing tendons at flangebottom, ksi= stress in the equivalent non-prestressing tendonin the webs, ksi= stress in the equivalent non-prestressing tendonin the flange, ksi= total stress in non-prestressing tendons of an individualrow in webs, ksi= total stress in non-prestressing tendons of bottomrow (kth row), ksi= total stress in non-prestressing tendons at flangetop, ksi= total stress in unbonded post-tensioning tendons,ksi= modulus of rupture of concrete, psi= initial effective prestress in bonded pretensioningtendons, kips= resultant effective pretensioning force in bondedtendons, kips= resultant effective post-tensioning force in unbondedtendons, kips= resultant tensile force in bonded prestressing tendonsof an individual row, kips= resultant tensile force in bonded bottom (mthrow) prestressing tendons, kipsF pnb = resultant compression force in non-prestressingtendons at flange bottom, kipsF pnj = resultant tensile force in non-prestressing tendonsof individual row in webs, kipsF pnk = resultant tensile force in non-prestressing tendonsof bottom row (kth row) in webs, kipsF pnt = resultant compression force in non-prestressingtendons at flange top, kipsF pu = resultant tensile force in unbonded post-tensioningtendons, kipsF pui = total initial effective post-tensioning force, kipsF R = resultant of the tensile forces in bonded and unbondedtendons, kipsh f = flange thickness of DT beam, in.h j = distance of centroid of non-prestressing tendonsof an individual row in webs, in.h k = distance of centroid of bottom non-prestressingtendons (kth row), in.h t = thickness of concrete topping, in.I c = moment of inertia of composite concrete crosssection, in. 4I p = moment of inertia of precast concrete cross section,in. 4k = number of rows of non-prestressing tendons inwebs of DT beamL = effective span of the beam, ftL u = horizontal distance between the ends of unbondedpost-tensioning strand, ftm = number of rows of bonded tendonsM = applied maximum moment due to service loads,kip-ftM cr = cracking moment capacity of section, kip-ftM u = design moment capacity of section, kip-ftM D = maximum bending moment due to dead load,kip-ftM L = maximum bending moment due to live load, kipftM n = nominal moment capacity of section, kip-ftM required = required moment capacity of the section, kip-ftp = number of materials used for tension reinforcementP cr = midspan load causing cracking of DT beam, kipsS bc = section modulus corresponding to the bottom extremefiber of composite section, in. 3S bp = section modulus corresponding to the bottom extremefiber of precast section, in. 3S tp = section modulus corresponding to the top extremefiber of precast section, in. 3S tc = section modulus corresponding to the top extremefiber of composite section, in. 3W = total midspan load, kipsW_d = self-weight of beam per unit length, kip/fty = distance of centroid of tension reinforcementfrom the extreme compression fiber, in.y bc = distance of centroid of composite section fromthe bottom fiber, in.y bp = distance of centroid of precast section from thebottom fiber, in.138 PCI JOURNAL

y tc = distance of centroid of composite cross sectionfrom the top fiber, in.β 1 = factor defined as the ratio of the equivalent rectangularstress block depth to the distance fromthe extreme compression fiber to the neutral axisε cu = ultimate compression strain in concrete (0.0025)ε fbj = flexural strain in the bonded prestressing tendonsof an individual rowε fbm = flexural strain in the bonded prestressing tendonsof bottom row (mth row)ε fu = ultimate tensile strain capacity of bonded prestressingtendonsε fun = ultimate tensile strain capacity of non-prestressingtendons in websε fup = ultimate tensile strain capacity of unbonded tendonsε pbj = total strain in bonded prestressing tendons of anindividual rowε pbji = initial strain in bonded prestressing tendons of anindividual rowε pbm = total strain in bonded prestressing tendons of mthrowε pbmi = initial strain in bonded prestressing tendons ofmth rowε pnb = total strain in non-prestressing tendons at flangebottomε pnj = total strain in non-prestressing tendons of an individualrow in websε pnkε pntε puε pui∆ε puα= total strain in non-prestressing tendons of bottomrow (kth row)= total strain in non-prestressing tendons at flangetop= total strain in unbonded post-tensioning tendons= initial strain in unbonded post-tensioning tendons= flexural strain in unbonded post-tensioning tendons= ratio of the effective post-tensioning force appliedto the precast DT beam to the total effectivepost-tensioning force(σ c ) b1 = stress at extreme tension fiber due to pretensioningforce, ksi(σ c ) b2 = stress at extreme tension fiber due to post-tensioningforce, ksiρρ bΩ uδδ aδ dδ p= tensile reinforcement ratio= balanced ratio= bond reduction coefficient at ultimate= maximum midspan deflection of the beam underservice loads, in.= maximum midspan deflection of the beam due toapplied load, in.= maximum midspan deflection of the beam due todead load, in.= maximum midspan deflection of the beam due toprestressing forces, in.July-August 2003 139