This paper presents a study on the fabrication,instrumentation, and flexural testing of a full-scaledouble-tee (DT) beam, prestressed using bondedpretensioned CFRP Leadline tendons andunbonded carbon fiber composite cable (CFCC)post-tensioning strands. The beam was designedto simulate the performance of the DT beams usedfor the construction of the three-span Bridge StreetBridge, the first vehicular concrete bridge everbuilt in the United States that uses CFRP materialas the principal structural reinforcement. Testingfocused on measurement of strain distributionsalong the length and depth of the beam, transferlength, camber/deflection, cracking load, forces inpost-tensioning strands, ultimate load-carryingcapacity, and mode of failure. In addition, ananalysis approach is presented to theoreticallyevaluate the response of the tested beam. It wasobserved that the ultimate failure of the beam wasinitiated by partial separation between the toppingand the beam flange, which led to the crushing ofthe concrete topping followed by rupture ofbottom tendons. The tested beam was found tohave significant reserve strength beyond theservice load. Theoretical calculations are similarin value to the corresponding experimental results– especially under the service load condition.

Experimental Study and Analysisof a Full-Scale CFRP/CFCCDouble-Tee Bridge Beam

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Nabil F. Grace, Ph.D., P.E. Professor and ChairmanDepartment of Civil EngineeringLawrence Technological UniversitySouthfield, Michigan

Tsuyoshi EnomotoEngineer

Bridge and Structural Cable DepartmentEngineering Division

Tokyo Rope Mfg. Co., Ltd.Tokyo, Japan

George Abdel-Sayed, Dr. Eng., P. Eng.Professor EmeritusDepartment of Civil EngineeringUniversity of WindsorWindsor, Ontario, Canada

Kensuke YagiManager, Engineering

Construction Materials DivisionMitsubishi Chemical Functional

Products, Inc.Tokyo, Japan

Loris Collavino, P.E.PresidentHollowcore IncorporatedPrestressed Systems, Inc.Detroit, Michigan

Concrete bridges prestressed and reinforced usingcarbon fiber reinforced polymer (CFRP) materialsare being used worldwide.1 However, the concrete

bridges constructed using CFRP tendons are few in num-ber.1-7 The most recent example of this new technology isthe Bridge Street Bridge,8 the first CFRP prestressed con-crete bridge built in the United States using CFRPLeadline tendons* and carbon fiber composite cable(CFCC) strands.†

Extensive research, funded by the National Science

July-August 2003 121

* Leadline tendons manufactured by Mitsubishi Chemical Functional Products, Inc., Japan.

† CFCC strands manufactured by Tokyo Rope Manufacturing Co., Ltd., Japan.

‡ NEFMAC grid reinforcement manufactured by Autocon Composites, Inc., Ontario, Canada.

Foundation, addressing the behaviorof this new bridge system, was con-ducted on one-third scale bridge mod-els in the Structural Testing Center atLawrence Technological University(LTU) in Southfield, Michigan.7,9-13

The results of this research formed thedesign basis that led to the double-tee(DT) beam tests and the award-win-ning Bridge Street Bridge design.

The objective of the full-scale test-ing presented here is to experimentallydetermine the various response param-eters for the DT beam – a design al-most identical to that used in theBridge Street Bridge. The parametersinclude strains, deflections, forces inpost-tensioning strands at service load,cracking load, and ultimate load con-ditions. Full-scale testing was requiredto verify the assumptions made in theanalysis and design approach pre-sented in Appendix A.

FABRICATION ANDINSTRUMENTATION

The cross section of the DT testbeam is shown in Fig. 1, while Fig. 2shows the plan and elevation of thebeam with the location of the exter-nally draped CFCC post-tensioningstrands. The concrete cross sectionconsists of the DT precast section and75 mm (3.0 in.) thick concrete top-ping. Total depth of the precast sectionis 1220 mm (48.0 in.) with a flangethickness of 150 mm (5.90 in.). Totalwidth of flange is 2120 mm (83.5 in.).

Reinforcement consists of 10 rowsof 10 mm (0.39 in.) diameter bondedprestressing CFRP Leadline tendonsand 6 rows of 12.5 mm (0.5 in.) non-prestressing CFCC strands in eachweb of the DT beam. The vertical dis-tance between adjacent rows of theprestressing tendons is 70 mm (2.8in.). The cross section also consists offour externally draped 40 mm (1.57in.) diameter post-tensioning CFCCstrands and nineteen 10 mm (0.39 in.)diameter non-prestressing rods placedlongitudinally in the flange.

Fig. 1. Cross section of DT test beam at midspan.

In addition, the flange is reinforcedwith two layers of transverse 10 mm(0.39 in.) diameter Leadline tendons.The post-tensioning strands are longi-tudinally draped between DiaphragmsD2 and D6 and located at a depth of1035 mm (40.8 in.) from the top sur-face of concrete topping at midspan.As shown in Fig. 2, the DT beam con-sists of seven diaphragms (D1 throughD7). The total overall length of the DTbeam is 20.88 m (68.5 ft), and the ef-fective span is 20.40 mm (67.0 ft).The mechanical properties of the

CFRP Leadline tendons and CFCCstrands are given in Table 1, while theproperties of the NEFMAC‡ reinforc-ing grid (used in the concrete topping)and the concrete are presented in Ta-bles 2 and 3, respectively.

Fabrication

The DT test beam was cast using asingle pan form incorporating the twostems, top flange, and seven transverseintegral diaphragms. All fabricationactivities took place at the precast

Fig. 2. External post-tensioning arrangement for CFCC strands, showing location ofthe seven diaphragms.

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plant, Prestressed Systems Inc., lo-cated in Windsor, Ontario, Canada.

The sequence of fabrication is givenbelow:

1. Installation of the CFCC andCFRP mild reinforcement, steel stir-rups, CFRP prestressing Leadlinetendons, and other embedded itemssuch as hold-up and hold-down de-vices, vibrating wire, and electrical re-sistance strain 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 ap-plying 60 percent of the total post-ten-sioning force.

Except for those in Row 10, allLeadline tendons were draped priorto pretensioning using hold-down andhold-up roller devices. These tendon-holding rollers were specially fabri-

Table 1. Material properties of CFRP tendons/CFCC strands.Leadline CFCC 1 x 7 CFCC 1 x 37

Properties (MCC15) (Tokyo Rope16) (Tokyo Rope16)

Nominal diameter, in. (mm) 0.39 (10) 0.5 (12.5) 1.57 (40)

Effective cross-sectional area, sq in. (mm2) 0.111 (71.6) 0.118 (76.0) 1.17 (752.6)

Guaranteed tensile strength, ksi (kN/mm2) 328 (2.26) 271 (1.87) 205 (1.41)

Specified tensile strength, ksi (kN/mm2) 415 (2.86) 305 (2.10) 271 (1.87)

Young’s modulus of elasticity, ksi (kN/mm2) 21,320 (147) 19,865 (137) 18,419 (127)

Elongation, percent 1.9 1.5 1.5

Guaranteed 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 1.8

Table 3. Material properties of precast concrete and concrete topping.Properties Precast concrete Concrete topping

Modulus 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 Lead-line tendons 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 ten-dons at a roller hold-down point.

When the concrete achieved ade-quate strength – 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 re-inforcement is provided in Fig. 5.

Four externally draped post-tension-ing CFCC strands were installed afterthe release of the pretensioning ten-dons and form removal. Fig. 6 revealsthe post-tensioning strands as seenfrom below the beam. Post-tensioninganchorage details at Diaphragm D6are represented in Fig. 7. Each post-tensioning strand was tensioned usinga special apparatus (see Fig. 8) usingspecific sequencing.

A special bearing system at Di-aphragms D3, D4, and D5 was de-signed and constructed to allow the de-sign (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 friction-less bearing surface between the CFCCstrands and the diaphragm concrete.

Instrumentation

The 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 re-lease.

• Beam camber/deflections duringfabrication/construction sequence.

• Concrete strain distributions inbeam cross section and concrete top-ping.

• Forces in CFCC post-tensioningstrands.

Fig. 3. Pretensioned CFRP Leadline tendons at roller hold-down location.

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Measurement of Pretensioning Forces

Pretensioning 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 be-tween the 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 pretension-ing forces. 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, respec-tively. Load cells located at the deadend were continuously monitored dur-ing and after placement of the con-crete to measure any changes in thepretensioning forces during concreteplacing and curing.

Measurement of Transfer Length

Transfer length of the pretensioningtendons was measured using seven vi-brating wire strain gauges with an ef-fective gauge length of 152 mm (6.0in.). Gauges were installed along oneweb at both ends of the beam duringfabrication. Strain gauges were in-stalled end to end beginning at the endof the beam and extending over a dis-tance of 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 center-line of the web.

Measurement of Early Age Camber

A complete history of early age de-flection in the test beam was obtained

Fig. 4. Release pattern for 60 pretensioned tendons.

Fig. 6. Externallydraped post-tensioningCFCC strands (lookingfrom below DT beam).

Fig. 5. Instrumentation and CFRP/CFCC reinforcing cage of one web (looking downfrom top).

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using elevation reference points em-bedded in the concrete surface. Eleva-tion points were located along the topflange of the test beam at midspan, attwo quarter-span points, and near thebeam ends. Prior to prestressing ten-don release, elevation measurementswere recorded for each point and usedas a reference for all subsequent mea-surements. After tendon release, mea-sured changes in elevation relative tothe initial pre-release values were usedto calculate beam camber due to pre-stressing at 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 preci-sion level and surveying rod with ahigh-resolution scale. The precisionlevel used for this application requiredan internal micrometer capable of re-solving elevation readings to the near-est 0.030 mm (0.001 in.). The top sur-face of each embedded reference pointincorporated a machined indentationto match the pointed end of the sur-veyor’s rod.

Measurement of Concrete Strains

Embedded electrical-resistancestrain gauges were installed for mea-suring strain 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 con-crete topping at the testing facility.

Measurement of Post-tensioning Forces

Prior to post-tensioning, all fourpost-tensioning strands were instru-mented with load cells at one end (seeFig. 8). The load cells were installed

Projected steel stirrup

Fig. 9. NEFMACreinforcing gridplaced over the

flange top.

Fig. 8. Post-tensioning

apparatus for aCFCC externally

draped strand.

Fig. 7. Post-tensioning anchorage details at Diaphragm D6.

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between the lock nut on the strand an-chor/head and the bearing plate em-bedded in 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 apply-ing 60 percent of the total desiredpost-tensioning force [449 kN (101kips)], equal to 32 percent of the ulti-mate breaking load of CFCC post-ten-sioning strands on the precast DTbeam. Final post-tensioning was ap-plied at the Construction TechnologyLaboratories, Inc. (CTL), testing facil-ity in Skokie, Illinois, after casting theconcrete topping. Final post-tension-ing consisted of the remaining 40 per-cent of 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 post-tensioning setup was moved to the op-posite end where the remaining 50 per-cent of the required force was applied.

At the precast facility, the initialpost-tensioning was applied with thebeam supported at Diaphragms D2and D6 at intermediate transverse lo-cations. The beam remained supportedin this manner until delivered to theCTL test facility. The beam supportconfiguration prevented the develop-ment of 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 ac-tual DT beams used in the BridgeStreet Bridge. The average measuredstrand force after the initial post-ten-sioning was 268 kN (60.3 kips), whilethe average measured strand force im-mediately after the final post-tension-ing was 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 testing

(described below). The average mea-sured force in the CFCC strand priorto the test was 443 kN (99.6 kips),which was slightly less than that ob-served immediately after the finalpost-tensioning.

Casting of Concrete Topping

Finally, a 75 mm (3.0 in.) thick con-crete topping 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 re-quired for concrete strain distributionmeasurements were installed atmidspan and the two quarter-span lo-cations.

The NEFMAC reinforcement was

supplied in the form of grid sheets,each with a transverse dimension of2083 mm (82.0 in.). Each grid sheetincorporated longitudinal reinforce-ment elements 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 bol-sters; 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 and

support condition for the beam areshown in Figs. 10a and 10b, respec-tively. The beam was loaded alongtwo lines to create a 3658 mm (12.0ft) wide constant moment region sym-

(a) Test setup forflexural loading.

Fig. 10. Flexural load testing at the CTL testing facility.

(b) Supportcondition of test beam.

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metrical about midspan (see Fig. 11).The loading lines were oriented or-thogonally to the longitudinal center-line of 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 longi-tudinal movement and rotation underthe load.

Load was applied using a series ofhydraulic jacks with load and exten-sion capability sufficient to induceflexural failure. All loads applied to

the beam during the test were moni-tored using load cells. Beam displace-ments at mid- and quarter-span loca-tions were monitored using twodisplacement transducers at each loca-tion attached 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.

Output data from instrumentationwere monitored throughout the testand recorded at each loading incre-ment.

RESULTS AND DISCUSSION

Transfer Length

Fig. 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 re-lease, and prior to initial post-tension-ing). 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 in-creased with distance from the beamend until strains stabilized or becameuniform (at the location of the third vi-brating-wire strain gauge).

Strain Distribution after Post-tensioning and Service Load

The distribution of concrete strainsalong the depth of the cross sections[at mid- and two quarter-span (Q-S)locations] after the final post-tension-ing and due to service load is shown inFigs. 13a and 13b, respectively. Fig.13a indicates that after final post-ten-sioning, 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 lo-cation. The developed compressivestrain levels were desirable to elimi-nate any potential cracking problemsunder service load/traffic condition.The DT cross section dimensions, theplacement and arrangement of pre-stressing tendons, as well as post-ten-

Fig. 12. Concrete strain versus distance from beam live end.

Fig. 11. Location of four-point loading system, showing location of displacementtransducers.

July-August 2003 127

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 bot-tom and top surfaces of the beam iscompressive. This strain result is de-sirable in eliminating any potentialcracking problems in the CFRP pre-stressed concrete beam under serviceloading.

Strain Distribution at Cracking andUltimate Loads

After reaching the service loadlevel, incremental loading of the DTbeam continued to reach the applied

load corresponding to the develop-ment of the first concrete crack. To ac-complish this objective, the beam wasloaded in suitably small increments sothat careful observations could bemade of the bottom surface for beamstrains within the constant moment re-gion. After application of each incre-ment, loading was suspended to allowsufficient time to visually inspect forcracking.

The total applied load correspond-ing to 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 applied

loads and do not include the preten-sioning and post-tensioning effects.

Fig. 14 shows that the strain distri-butions at the quarter-span sectionsare very close in value. In Fig. 14b,the ultimate strain at the top of con-crete topping has reached a compres-sive strain 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 fail-ure of 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 bot-tom of the beam is close to the strainvalue corresponding to the modulus ofrupture of concrete.

Fig. 14. Average measured concrete strain: (a) At cracking load; and (b) At ultimate load.

Fig. 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

(a) After final post-tensioning

(b) At ultimate load

(b) Due to service load

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ULTIMATE BEHAVIOR OFDT BEAM

Ultimate flexural strength wasreached when the beam was incremen-tally loaded to a total load of 1303.9kN (293.1 kips) and then unloaded(see Fig. 15). This load represents

The ultimate load applied during thetest was 2443.0 kN (549.1 kips). Aver-age midspan 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 fail-ure of 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 fur-ther increase in the load. The forcelevels in the four CFCC post-tension-ing strands (see Fig. 17) nearly dou-bled during 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.

ANALYSISAn analysis of experimental results

obtained from the full-scale DT testbeam is presented in this paper. A the-oretical analysis was also developed topredict the strength and failure modeof the DT test beam. The strains,stresses, and forces in strands, and de-flection of the beam were calculatedfor the service and ultimate loadingconditions.14 Details of the theoreticalanalysis and calculations are presentedin Appendix A.15-20

CONCLUSIONSBased on the results of the test pro-

gram, the following conclusions canbe drawn:

Fig. 15. Beam deflection response at midspan under ultimate load.

Fig. 16. Closeupof beam failure.

Externalunbonded CFCC

post-tensioningstrands did not

rupture.

about 50 percent of the actual ultimatestrength. The loading and unloadingsequence was necessary to evaluatethe ductility of the beam. After the ini-tial loading and unloading sequence,the beam was incrementally loaded toinduce flexural failure.

July-August 2003 129

1. The test results and constructionexperience gained from this evaluationhave been implemented in the designdocuments for the 12 DT beams re-quired for the construction of theBridge Street Bridge.

2. The DT beam had significant re-serve strength beyond the service load.The ultimate load and the crackingload of the DT beam were, respec-tively, approximately 5.3 and 1.4times the service load.

3. The combined internal and exter-nal prestressing forces induced the de-signed compressive strain in the crosssection.

4. The ultimate failure of the DTbeam was initiated by partial separa-tion between 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-tension-ing strands nor their anchors ruptured.

6. Theoretical values and experi-mental results are very close, espe-cially under service load condition.The theoretical nominal moment ca-pacity of the DT beam is about 0.9percent lower than the correspondingexperimental value, whereas the pre-dicted forces in the unbonded post-tensioning strands at the service andultimate load conditions are 0.5 and6.1 percent lower than correspondingexperimental values.

ACKNOWLEDGMENTSThe DT beam for this study was

constructed by Prestressed SystemsInc. (PSI), Windsor, Ontario, Canada,and tested by Construction Technol-ogy Laboratories, Inc. (CTL), Skokie,Illinois. Hubbell, Roth & Clark(HRC), Consulting Engineers, Bloom-field Hills, Michigan, provided the en-tire design, drawing details, and con-struction administration services. TheCity of Southfield and the Federal

Highway Administration jointlyfunded the instrumentation and testingof the DT beam.

The National Science Foundation,Division of Civil and Mechanical Sys-tems, funded the research activities,testing of several one-third-scale mod-els (conducted in the Structural Test-ing Center at Lawrence TechnologicalUniversity), and the analytical evalua-tion of the full-scale testing.

Mitsubishi Chemical FunctionalProducts, Inc., Tokyo Rope Manufac-turing Co. 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 ap-preciated.

The authors also appreciate the tech-nical comments of Dr. G. Tadros andthe many constructive comments ofthe PCI JOURNAL reviewers.

Fig. 17. Force inCFCC post-tensioning strands,indicating responseunder ultimate loadand failure of DTbeam.

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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 Struc-tural Concrete (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 Pre-stressed Concrete 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 Un-bonded Post-tensioning Tendons: Part 2 – Proposed Methodol-ogy,” ACI Structural Journal, V. 88, No. 6, November-De-cember 1991, pp. 683-692.

REFERENCES

July-August 2003 131

APPENDIX A – ANALYSIS PROCEDURE

The following procedure illustrates the steps for predict-ing the behavior of the full-scale pretensioned and post-ten-sioned DT 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 forLeadlineTM rods and “strand” is used for CFCC cablesthroughout the paper. It should also be noted that the cross-sectional area 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, Wd = 142/67 = 2.12 kip/ftDead load moment at midspan:

= 1188.8 kip-ft

Let 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 ft

Design service live load = 104.3 kips Service live load moment at midspan,

= 1434.13 kip-ft

Required moment capacity of the section, Mrequired = 1.4MD + 1.7ML (ACI 318-02)

= 4102.34 kip-ft

Step 2 — Calculate balanced ratio, ρb:

wheref ′c = strength of precast concrete = 7.81 ksi (see Table 3)εfu = specified ultimate strain of bonded pretensioning

tendons = 0.019 (see Table 1)ffu = specified strength of bonded pretensioning tendons

= 415 ksi (see Table 1)β1 = 0.85 – [(7810 – 4000)/1000] × 0.05 = 0.66

(ACI 318-02)Based on experimental results, 25 percent loss in the pre-

stressing force in the bonded tendons is considered. The ini-tial effective strain in bottom pretensioning tendons, εpbmi,can be calculated as follows:

ε pbmi = ××

=

19 5 0 75

0 111 21 320

0 0062

. .

. ,

.

ρ β εε ε εb

c

fu

cu

cu fu pbmi

f

f= ′

+ −0 85 1. (A1)

ML = ×104 3 27 5

2

. .

MW L

Dd=

= ×

2

2

8

2 12 67

8

.

Based on experimental observation, the ultimate strain inconcrete, εcu, 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 the-oretical and experimental values of the ultimate load carry-ing capacity of the DT beam. Substituting the values in Eq.(A1) yields ρb = 0.0017.

Step 3 — Calculate cracking moment, Mcr, andcracking load, Pcr:

Modulus of rupture of concrete, fr = 6.0

= 530 psiVarious sectional properties of the DT beam section (see

Fig. A1) are presented in Table A1.Total effective pretensioning force, Fpre = 1139.3 × 0.75

= 854.5 kips Total effective post-tensioning force, Fpost = 397.8 kipsStress at the bottom fiber of the section due to pretension-

ing force,

Here, Ap and Sbp are cross-sectional area and section mod-ulus (with respect to bottom fiber) of precast section, respec-tively; eb is the eccentricity of resultant pretensioning forcewith respect to the centroid of the precast section. FromTable A1, Ap = 1462 sq in., Sbp =11,192 in.3, and eb = 13 in.Substituting values in Eq. (A2) results in (σc)b1 = –1.577 ksi.

Since 60 percent of the total post-tensioning force wasused to prestress the precast DT concrete section and the re-maining 40 percent of the total post-tensioning force was ap-plied on the composite concrete section (DT beam plus con-crete topping), then the stress at the bottom fiber of sectiondue to post-tensioning force, (σc)b2, is:

Here, eup and euc are the eccentricity of unbonded post-tensioning strands with respect to the centroids of precastand composite sections, respectively. From Table A1, valuesof eup and euc are 19.12 and 21.67 in., respectively. Substi-tuting the values in Eq. (A3),

(σc)b2 = –0.938 ksi

The cracking moment (Mcr) can be found from the follow-ing expression:

(σc)b1 + (σc)b2 + Mcr/Sbc = fr (A4)

Here, Sbc is the section modulus of composite section with

( )

(A3)

2σ c b

post

p

post up

bp

post

c

post uc

bc

F

A

F e

S

F

A

F e

S

=

−×

−×

−×

−× ×0 6 0 6 0 4 0 4. . . .

( ) (A2)1σ c bpre

p

pre b

bp

F

A

F e

S= − −

′fc

132 PCI JOURNAL

respect to extreme tension fiber and is equal to 12,659 in.3

(see Table A1). Substituting the values in Eq. (A4), Mcr = 3212.2 kip-ft

Thus, cracking load,

Percent difference between the theoretical and experi-mental cracking loads:

Step 4 — Compute flexural moment capacity:

Ultimate bond reduction coefficient for externalstrands,

where Lu = horizontal distance between anchorages of un-

bonded post-tensioning tendons = 51.4 ftdu = distance at midspan of external post-tensioning

strands from the extreme compression fiber =40.75 in.

Substituting the values in Eq. (A5), we get Ωu = 0.36. Distance of the centroid of bottom bonded pretensioning

tendons from the extreme compression fiber, dm = 47 in.Flange width, b = 83.46 in.Total effective pretensioning force, ∑Fpi = 854.5 kipsTotal effective post-tensioning force, Fpui = 397.8 kips

Ωuu

u

L

d

= 5 4. (A5)

= − ×

=

( . . )

..

147 2 144 7

144 7100

1 7 percent

PM M

crcr D= − ×

=

( )

..

2

27 5147 2 kips > service load (104.3 kips)

It should be noted that the actual reinforcement ratio of thesection is defined as the ratio of total weighted area of the ten-dons and/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, fpbb = 188.7 ksi

Stress in the equivalent non-prestressing strand of webs atbalanced condition, fpnbb = 152.4 ksi

Stress in the equivalent unbonded strand at balanced condi-tion, fpub = 72.2 ksi

Stress in the equivalent non-prestressing tendons of flangeat balanced condition, fpnf = 39.94 ksi

Total cross-sectional area of bonded prestressing tendons,Apb = 6.66 sq in.

Total cross-sectional area of non-prestressing strands in thewebs, Apn = 3.304 sq in.

Total cross-sectional area of unbonded prestressing strands,Afu = 4.68 sq in.

Total cross-sectional area of non-prestressing tendons in theflange, Apnf = 2.109 sq in.

Reinforcement ratio,

= 0.002 > 0.0017 (ρb)

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:

=+ × + × + + × − ×

× ×854 5 188 7 6 66 152 4 3 304 397 8 72 2 4 68 39 94 2 109

83 46 47 415

. . . . . . . . . .

.

ρ =

+ +∑ + + −

× ×=

F f A f A F f A f A

b d f

pi pbb pbj

m

pnbb pn pui pub fu pnfb pnf

m fu

1

(A6)

* Equivalent tendon/strand means a tendon/strand located at centroid of tendons/strands and having cross-sectional area equal to the total cross-sectional area of corresponding tendons/strands.

Table A1. Section properties of DT beam cross section.

Sectional properties Precast section Composite section

Cross-sectional area, sq in. 1462 1673

Moment of inertia, in.4 328,489 403,821

Distance of centroid of the section from the bottom fiber of the section, in. 29.35 31.90

Distance of centroid of the section from the top fiber of the section, in. 18.68 19.08

Section modulus corresponding to the bottom fiber of section, in.3 11,192 12,659

Section modulus corresponding to the top fiber of section, in.3 17,585 21,165

Eccentricity of resultant pretensioned force, in. 13 N/A

Eccentricity of unbonded post-tensioning strands, in. 19.12 21.67

Note: 1 in. = 25.4 mm; 1 sq in. = 645 mm2; 1 in.3 = 16387 mm3; 1 in.4 = 416231 mm4.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.

July-August 2003 133

a. Compute strain in tendons/strands:The strains in tendons/strands can be calculated from the

strain 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, εcu = 0.0025.

Let c = depth to the neutral axis from the extreme com-pression fiber

Strain in bonded prestressing tendons,

Strain in unbonded post-tensioning strands,

Strain in non-prestressing strands,

Strain in non-prestressing tendons at flange top,

Strain in non-prestressing tendons at flange bottom,

In Eq. (A7), epbji equals 0.0059 (for j = 1, 5) and 0.0062(for j = 6 to 10); dj (j =1 to 10) is the depth of an individuallayer, j, of the bonded pretensioning tendons from the ex-treme compression fiber. Here, d1, d2, d3, d4, d5, d6, d7, d8,d9, and d10 are 21.8, 24.6, 27.4, 30.2, 33.0, 35.8, 38.6, 41.4,

ε pnbbc d

c= (A11)

0 0025. ( )× −

ε pnttc d

c= (A10)

0 0025. ( )× −

ε pnjjh c

c= (A9)

0 0025. ( )× −

ε puu

u

d c

c= (A8)

0 00250 0046

. ( ).

× − +Ω

ε εpbjj

pbji

d c

cj m= (for = 1, ) (A7)

0 0025. ( )× −+

44.2, and 47.0 in., respectively; du is the depth of unbondedpost-tensioning tendons from the extreme compression fiberand is equal to 40.75 in.

In Eq. (A9), hj (j = 1 to 6) is the depth of an individuallayer, j, of the non-prestressing tendons provided in webs.Here, h1, h2, h3, h4, h5, and h6 are 8.8, 16.4, 23.5, 31.8, 40.1,and 48.6 in., respectively; dt [see Eq. (A10)] and db [see Eq.(A11)] are the depth of the top and bottom layers of non-pre-stressing tendons provided in the flange, respectively. Here,dt and db 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, Afb = 0.666 sq in.

Total cross-sectional area of non-prestressing strands ineach row, Afn = 0.472 sq in. for Rows 1 to 5 and Afn = 0.944sq in. for Row 6.

Total cross-sectional area of non-prestressing tendons atflange top, Afnt = 1.11 sq in.

Total cross-sectional area of non-prestressing tendons atflange bottom, Afnb = 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 ten-dons can be determined by the following expressions:

Resultant force in pretensioning tendons,

Resultant force in unbonded post-tensioning strands,

Fc

cpu = − +

77 6 40 75396 54

. ( . ). kips (A13)

Fd c

cjpbj

j=−

+

=35 5

88 03. ( )

. kips for 6 to 10 (A12b)

Fd c

cjpbj

j=−

+

=35 5

83 78. ( )

. kips for 1 to 5 (A12a)

Fig. A1. Strain and stress distributions for DT beam at ultimate loading indicating compressive and tensile forces.

134 PCI JOURNAL

Resultant force in non-prestressing strands,

Resultant force in non-prestressing tendons at flange top,

Resultant force in non-prestressing tendons at flange bot-tom,

Compressive force in concrete = Ct + Cf

= 0.85f ′c(Ect/Ec)bht + 0.85f ′cb(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,

Solving for c gives c = 8.73 in.

d. Compute stresses in tendons/strands: The value of strain in tendons/strands can be computed

using Eqs. (A7) through (A11), and then using the stress-strain relationships, stresses in tendons/strands of differentlayers can be calculated.

For example, stress in the tenth layer of bonded preten-sioning tendons

The calculated values of the stress in tendons of differentlayers are presented in Table A2.

e. Compute the resultant force in tendons and con-crete:

Resultant forces in tendons of different layers can be cal-culated by 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:

Fpb10 = 365.8 × 0.666 = 243.6 kips

The calculated resultant forces in tendons of different lay-ers are presented in Table A3.

From Eq. (A17), resultant force in concrete = Ct + Cf =2963.5 kips

Total compressive force,

C = Ct + Cf + Fpnt + Fpnb

= 2963.5 + 28.6 + 11.2 = 3003.2 kips

Total resultant tension force,

F F F F

C

R pbjj

pnjj

pu= ∑ + ∑ +

= ≅= =1

10

1

6

3004 3. kips (OK)

= − +

×

=

0 0025 470 0062 21 320

365 8

. ( ). ,

.

c

c

ksi < 415 ksi (OK)

35 5344 10 859 05

23 45217 8 7

77 640 75 396 54

1405 62 554 05 0 66 2 95

59 2 4 5 53 3 6 9

..

.( . )

.( . ) .

. . ( . . )

. ( . ) . ( . )

cc

cc

cc

c

c

c

c

c

−( ) + + − +

− +

= + − +− + −

F F F Cpbjj

pnjj

pu= =∑ ∑

1

10

1

6

+ + = (A18)

Fc

cpnb = −

53 3 6 9. ( . ) kips (A16)

Fc

cpnt = −59 2 4 5. ( . ) kips (A15)

=−46 9. ( )h c

cjj kips for = 6 (A14b)

=−23 45. ( )h c

cjj kips for = 1 to 5 (A14a)

Stress (ksi)

Bonded Non-prestressing Unbonded Non-prestressing Non-prestressing

pretensioning tendons located post-tensioning tendons located tendons located at

Layer number tendons in webs tendons at flange top flange bottom

1 205.6 0.40 145.5 25.8 11.2

2 222.7 43.6

3 239.8 84.0

4 256.9 131.2

5 274.0 178.5

6 297.5 226.8 N/A N/A N/A

7 314.5

8 331.6N/A

9 348.7

10 365.8

Table A2. Stresses in tendons of different layers at ultimate load condition of DT beam.

Note: 1 ksi = 6.895 MPa.

July-August 2003 135

f. Compute the ultimate moment capacity:Location of the center of gravity of the resultant tension force

measured from the extreme compression fiber, d = 37.61 in.Depth of Whitney’s equivalent stress block, a = β1c

= 0.66 × 8.73 = 5.76 in.

Distance of the center of gravity of resultant compressionforce from the extreme compression fiber,

Thus, nominal moment capacity, Mn = FR(d –

_d)

= 3004.3 × (37.61 – 3.02) = 8659.9 kip-ft

Design moment capacity, Mu = 0.85 × 8659.9

= 8210.9 kip-ft > 4102.34 kip-ft (Mrequired) (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-ft

Theoretical nominal moment capacity of DT beam = 8659.9kip-ft

Percent difference in experimental and theoretical momentcapacities,

= − ×

=

( . . )

..

8739 0 8659 9

8739 0100

0 9 percent

d =

× + × +

+ × + ×

=

1405 62 95

21557 9 2 95

2 81

228 3 4 5 10 6 6 9

3003 23 02

..

. ..

. . . .

.. in.

Force (kips)

Bonded Non-prestressing Unbonded Non-prestressing Non-prestressing

pretensioning tendons located post-tensioning tendons located tendons located at

Layer number tendons in webs tendons at flange top flange bottom

1 136.9 0.2 680.9 28.6 11.2

2 148.3 20.6

3 159.7 39.6

4 171.1 61.9

5 182.5 84.3

6 198.1 214.1 N/A N/A N/A

7 209.5

8 220.8N/A

9 232.2

10 243.6

Table A3. Resultant forces in tendons of different layers at ultimate load condition of DT beam.

Note: 1 kip = 4.44 kN.

Experimental value of post-tensioning force in eachpost-tensioning strand = 181.3 kips

Theoretical value of post-tensioning force in each ten-don = 680.9/4 = 170.2 kips

Percent difference in experimental and theoretical val-ues of post-tensioning force

h. 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 the

cracking 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,

Substituting the values of sectional properties (seeTable A1) and other parametric values in Eq. (A19), weget fct = 1.28 ksi < 4.69 ksi (0.6f ′c) (OK)

Maximum concrete stress due to applied load

= ×

=

1434 13 12

21 165

0 81

.

,

. ksi

fF

A

F e

S

F

A

F e

S

F

A

F e

S

M

S

ctpre

p

pre b

tp

post

p

post up

tp

post

c

post uc

tc tc

= − +×

−×

+

×−

× ×+

0 6 0 6

0 4 0 4

. .

. . (A19)

= − ×

=

( . . )

..

181 3 170 2

181 3100

6 1 percent

136 PCI JOURNAL

Experimental value of concrete stress = 1.0 ksi Percent difference in theoretical and experimental

values of concrete stress

This difference in analytical and experimentalstrain in concrete topping may be attributed to thechange in the strain condition because of the chang-ing of the support condition of the beam during vari-ous construction stages and due to different propertiesof the concrete topping and precast concrete.

Stress in bottom prestressing tendons,

Stress in bottom prestressing tendons due to ap-plied load

Stress in post-tensioning strands,

Force in a CFCC strand at service load = 86.86 ×1.17 = 101.63 kips

Experimental value of post-tensioning force =101.12 kips

Percent difference in theoretical and experimentalvalues of post-tensioning force at service load

i. Compute maximum deflection under serviceload (no cracking occurs under service load):

ML = M – MD

= 1434.13 kip-ftDistance between the support and nearest load

point, L1 = 27.5 ft

= − ×

=

101 63 101 12

101 12100

0 5

. .

.. percent

f EE

E

M M e

Ipu fp puifp

c

D

c

uc= + −

= × +

× − × × ×

=

ε ( )

, .

.( . . ) .

,

.

Ω

18 419 0 0046

3 462622 93 1188 8 12 21 67

403 821

2

3

86 86

ksi < 271 ksi (OK)

= × × × −

= ≅

4 01434 13 12 47 19 08

40 3821

4 7

.. ( . )

,

. 6 ksi 4.73 ksi (experimental value)

f EE

E

M d y

Ipb f pb if

c

tc

c

10 1010

2622 93 12 47 0 19 08

40 3821

132 19 8 70

= + −

= ×

× × × −

==

ε ( )

. ( . . )

,

(A20)

21,320 0.0062 +

4.0

. + .

140.89 ksi < 415 ksi (OK)

= − ×

=

1 0 0 81

1 0100

19

. .

. percent

Longitudinal distance between load points, L2 = 12 ftDeflection due to applied load,

It should be noted that the precast DT beam was supported at Di-aphragms D2 and D6 prior to initial post-tensioning. The center-to-center distance between Diaphragms D1 and D2, d12, was 7.8 ft.Hence, midspan deflection of DT beam due to dead load is given by

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 post-

tensioning

Center-to-center distance between Diaphragms D2 and D6 =51.4 ft

Deflection due initial post-tensioning

Total deflection of the precast DT beam after the initial post-ten-sioning

Experimental value of deflection due to pretensioning and initialpost-tensioning = 0.79 in. ↑

Percent difference in theoretical and experimental values of de-flection after initial post-tensioning

= −

=

0 80 0 79

0 791 3

. .

.. percent

= + −

= ↑

0 69 0 12 0 01

0 80

. . .

. in.

= × × × ×× ×

= ↑

0 6 397 8 19 12 51 4 12

8 5320 32 8489

0 12

2. . . ( . )

,

. in.

= × × ×× ×

= ↑

1139 3 13 0 67 12

8 5320 328 489

0 69

2. . ( )

,

. in.

δ

δ

d

c p

D D

d

E I

W L W L Ld L

=

− −

+ −

=

= ↓

×× − × −

+ × −

144

384 12 25081 85 79603 8

0 01

4

12

3

4 3144

5320 328 4892 12 67

384

2 12 67

12

67

27 8 5081 85 67 79603 8

. .

.

,. .

. . .

in.

in.

δ aL

c c

M L

E I

L

L

L

L= +

+

= × × ×× ×

+

+

= ↓ ≅

12

2

1

2

1

2

2 2

8

8

34 0

1434 13 12 27 5 12

8 5320 40 3821

8

34 0

12

27 5

12

27 5

0 502

.

. ( . )

,.

. .

.

(A21)

in. 0.505 in. (experimental value)

July-August 2003 137

Measured loss in the deflection due to transportation andsupport condition change = 0.25 in. ↓

Net deflection due to prestressing prior to final post-ten-sioning

= 0.80 – 0.25

Increase in deflection due to final post-tensioning

Weight of the concrete topping

Deflection due to concrete topping

Net deflection at the time of final post-tensioning

Experimental value of the net deflection after final post-tensioning = 0.56 in. ↑

Percent difference in theoretical and experimental valuesof deflection after final post-tensioning

= − ×

=

0 58 0 56

0 56100

3 6

. .

.. percent

= + −

= ↑

0 55 0 08 0 05

0 58

. . .

. in.

= × ×× ×

= ↓

5

384

0 26 67 12

12 5320 403 821

0 05

4. ( )

,

. in.

= × ××

=

150 83 46 2 95

12 10000 26

2

. .

. kip / ft

= × × × ×× ×

= ↑

0 4 397 8 21 67 51 4 12

8 5320 40 3821

0 08

2. . . ( . )

,

. in.

= ↑0 55. in.

a = depth of equivalent rectangular compressionblock, in.

Ac = cross-sectional area of composite DT beam, sqin.

Ap = cross-sectional area of precast DT beam, sq in.Afb = cross-sectional area of bottom bonded prestress-

ing tendons in each row, sq in.Afn = cross-sectional area of non-prestressing tendons

in each row, sq in.Afnt = cross-sectional area of non-prestressing tendons

at flange top, sq in.Afnb = cross-sectional area of non-prestressing tendons

at flange bottom, sq in.Afu = total cross-sectional area of unbonded post-ten-

sioning tendons, sq in.Apb = total cross-sectional area of bonded prestressing

tendons, sq in.Apn = total cross-sectional area of non-prestressing ten-

dons in the webs, sq in.Apnf = total cross-sectional area of non-prestressing ten-

dons in the flange, sq in.b = width of compression face of member, in.c = depth to the neutral axis from the extreme com-

pression fiber, in.C = resultant compressive force, kipsCt = resultant compression force in concrete topping,

kips

Cf = resultant compression force in flange of precastDT beam, kips

c.g.c = axis passing through the centroid of concretecross section of the DT beam

c.g.p = axis passing through the center of gravity of theresultant pretensioned force

d12 = center-to-center distance between DiaphragmsD1 and D2, ft

d = distance to center of gravity of the resultant ten-sion force from the extreme compression fiber,in._

d = distance to center of gravity of the resultant com-pression force from the extreme compressionfiber, in.

dj = distance to centroid of bonded prestressing ten-dons of an individual row from the extreme com-pression fiber, in.

dm = distance to centroid of bottom bonded prestress-ing tendons (mth row) from the extreme com-pression fiber, in.

db = distance to centroid of non-prestressing tendonsat flange bottom from the extreme compressionfiber, in.

dt = distance to centroid of non-prestressing tendonsat flange top from the extreme compressionfiber, in.

du = distance to centroid of unbonded post-tensioning

APPENDIX B — NOTATION

138 PCI JOURNAL

tendons from the extreme compression fiber, in.eb = eccentricity of resultant pretensioning force from

the centroid of the precast concrete cross section,in.

eup = eccentricity of unbonded post-tensioning tendonsfrom centroid of the precast concrete cross sec-tion, in.

euc = eccentricity of unbonded post-tensioning tendonsfrom the centroid of composite section

Ec = modulus of elasticity of precast concrete, ksiEct = modulus of elasticity of concrete topping, ksiEf = modulus of elasticity of bonded tendon, ksiEfn = modulus of elasticity of non-prestressing tendons

in webs, ksiEfp = modulus of elasticity of unbonded tendon, ksifc = stress in the concrete at extreme compression

fiber in very under-reinforced beam, ksifct = stress in the concrete at extreme compression

fiber in over-reinforced beam, ksif ′c = specified compressive strength of precast con-

crete, ksiffu = specified ultimate tensile strength of bonded pre-

stressing tendons, ksiffun = specified ultimate tensile strength of non-pre-

stressing tendons in webs, ksiffup = specified ultimate tensile strength of unbonded

post-tensioning tendons, ksifpbb = flexural stress in the equivalent bonded pre-

stressing tendons at balanced condition, ksifpbj = total stress in bonded prestressing tendons of an

individual row, ksifpbm = total stress in bottom prestressing tendons (mth

row), ksifpbmi = initial effective prestress in bottom bonded pre-

stressing tendons (mth row), ksifpnb = total stress in non-prestressing tendons at flange

bottom, ksifpnbb = stress in the equivalent non-prestressing tendon

in the webs, ksifpnfb = stress in the equivalent non-prestressing tendon

in the flange, ksi fpnj = total stress in non-prestressing tendons of an in-

dividual row in webs, ksifpnk = total stress in non-prestressing tendons of bottom

row (kth row), ksifpnt = total stress in non-prestressing tendons at flange

top, ksifpu = total stress in unbonded post-tensioning tendons,

ksifr = modulus of rupture of concrete, psiFpi = initial effective prestress in bonded pretensioning

tendons, kipsFpre = resultant effective pretensioning force in bonded

tendons, kipsFpost = resultant effective post-tensioning force in un-

bonded tendons, kipsFpbj = resultant tensile force in bonded prestressing ten-

dons of an individual row, kipsFpbm = resultant tensile force in bonded bottom (mth

row) prestressing tendons, kipsFpnb = resultant compression force in non-prestressing

tendons at flange bottom, kipsFpnj = resultant tensile force in non-prestressing ten-

dons of individual row in webs, kipsFpnk = resultant tensile force in non-prestressing ten-

dons of bottom row (kth row) in webs, kipsFpnt = resultant compression force in non-prestressing

tendons at flange top, kipsFpu = resultant tensile force in unbonded post-tension-

ing tendons, kipsFpui = total initial effective post-tensioning force, kips FR = resultant of the tensile forces in bonded and un-

bonded tendons, kipshf = flange thickness of DT beam, in.hj = distance of centroid of non-prestressing tendons

of an individual row in webs, in.hk = distance of centroid of bottom non-prestressing

tendons (kth row), in.ht = thickness of concrete topping, in.Ic = moment of inertia of composite concrete cross

section, in.4

Ip = moment of inertia of precast concrete cross sec-tion, in.4

k = number of rows of non-prestressing tendons inwebs of DT beam

L = effective span of the beam, ftLu = horizontal distance between the ends of un-

bonded post-tensioning strand, ftm = number of rows of bonded tendonsM = applied maximum moment due to service loads,

kip-ftMcr = cracking moment capacity of section, kip-ftMu = design moment capacity of section, kip-ftMD = maximum bending moment due to dead load,

kip-ftML = maximum bending moment due to live load, kip-

ftMn = nominal moment capacity of section, kip-ftMrequired = required moment capacity of the section, kip-ftp = number of materials used for tension reinforce-

ment Pcr = midspan load causing cracking of DT beam, kipsSbc = section modulus corresponding to the bottom ex-

treme fiber of composite section, in.3

Sbp = section modulus corresponding to the bottom ex-treme fiber of precast section, in.3

Stp = section modulus corresponding to the top ex-treme fiber of precast section, in.3

Stc = section modulus corresponding to the top ex-treme fiber of composite section, in.3

W = total midspan load, kipsWd = self-weight of beam per unit length, kip/ft_y = distance of centroid of tension reinforcement

from the extreme compression fiber, in.ybc = distance of centroid of composite section from

the bottom fiber, in.ybp = distance of centroid of precast section from the

bottom fiber, in.

July-August 2003 139

ytc = distance of centroid of composite cross sectionfrom the top fiber, in.

β1 = factor defined as the ratio of the equivalent rect-angular stress block depth to the distance fromthe extreme compression fiber to the neutral axis

εcu = ultimate compression strain in concrete (0.0025)εfbj = flexural strain in the bonded prestressing tendons

of an individual rowεfbm = flexural strain in the bonded prestressing tendons

of bottom row (mth row)εfu = ultimate tensile strain capacity of bonded pre-

stressing tendonsεfun = ultimate tensile strain capacity of non-prestress-

ing tendons in websεfup = ultimate tensile strain capacity of unbonded ten-

donsεpbj = total strain in bonded prestressing tendons of an

individual rowεpbji = initial strain in bonded prestressing tendons of an

individual rowεpbm = total strain in bonded prestressing tendons of mth

rowεpbmi = initial strain in bonded prestressing tendons of

mth rowεpnb = total strain in non-prestressing tendons at flange

bottomεpnj = total strain in non-prestressing tendons of an in-

dividual row in webs

εpnk = total strain in non-prestressing tendons of bottomrow (kth row)

εpnt = total strain in non-prestressing tendons at flangetop

εpu = total strain in unbonded post-tensioning tendonsεpui = initial strain in unbonded post-tensioning ten-

dons ∆εpu = flexural strain in unbonded post-tensioning ten-

donsα = ratio of the effective post-tensioning force ap-

plied to the precast DT beam to the total effec-tive post-tensioning force

(σc)b1 = stress at extreme tension fiber due to pretension-ing force, ksi

(σc)b2 = stress at extreme tension fiber due to post-ten-sioning force, ksi

ρ = tensile reinforcement ratio ρb = balanced ratioΩu = bond reduction coefficient at ultimateδ = maximum midspan deflection of the beam under

service loads, in.δa = maximum midspan deflection of the beam due to

applied load, in.δd = maximum midspan deflection of the beam due to

dead load, in.δp = maximum midspan deflection of the beam due to

prestressing forces, in.