Strength and Behavior ofPrestressed Concrete Members

With Unbonded Tendons

by Ned H. Burns and David M. Pierce*

SYNOPSIS

Experimental test results are presented for 15 prestressed concrete mem-bers with unbonded tendons, utilizing two shapes of cross sections. Ribbedcross sectional shape was used for beams tested as indeterminate two-spancontinuous members and as determinate double cantilever members simu-lating the negative moment region of multiple span members. An I-shapedsection was used for eight simply supported beams, four of which were sim-ple beams with an overhang on one end. Web reinforcement was used foronly four of the 15 beams tested.

Bonded reinforcement was included in all cases to distribute cracking inthe concrete and to contribute to the ultimate moment capacity at criticalsections. Development of cracking, ultimate load capacity, and the load-deflection response were of interest. Some members failed in flexure, whileothers failed in shear.

The test results are compared with values computed following the 1963ACI Building Code requirements. Both flexural and shear strengths at ulti-mate are underestimated by the ACI Code. The trends from comparisons ofmeasured to predicted values are discussed in the paper for beams havingthe two types of cross sections, and overall trends are presented. The con-tribution of the bonded deformed bars to strength and behavior of memberswith unbonded tendons is discussed.

INTRODUCTION

The topic of this experimentalstudy is but one example of a situa-tion where the "art" of prestressedconcrete has somewhat preceded thebasic "science". All 15 prestressedconcrete beams tested in this investi-

°Associate Professor of Civil Engineeringand Research Assistant, The University ofTexas, Austin, Texas.

gation at The University of Texaswere post-tensioned with thestressed tendons being left un-bonded. In addition to the un-bonded prestressing tendons, allbeams also contained bonded, un-stressed steel reinforcement. Thisparticular combination of prestress-ing and bonded reinforcement isbeing used rather extensively in con-

,October, 1967 15

struction; but the research informa-tion behind design assumptions forthis structural system is extremelysparse when compared with thelarge volume of tests reported onfully bonded members.

There does exist a fundamentaldifference in the behavior underload of a beam containing bondedprestressing tendons and one thatcontains unbonded tendons. Forpurposes of this paper, the term"tendon" may apply equally well toprestressing elements composed ofwires, strands, or bars. The tendonsfor all of the tests reported here werecomposed of four 1/4-in, diameterwires.

In a beam that contains bondedtendons the strain that exists at anypoint in a tendon is a function of theinitial tendon strain and the externalmoment carried by the beam at thesection in question. Thus, for bondedtendons the maximum tendon stresswill occur at the point of maximumexternal moment. The changes instrain occurring in the concrete atthe level of the prestressing tendonsfor bonded beams may be assumedalso to occur in the tendons.

The change in strain in an un-bonded tendon is a function of theexternal moment at all points alongthe beam. Any changes in force oc-curring along the length of the ten-don must be due to friction forceswhich develop. The limiting case foran unbonded beam would be that ofno friction forces, with the resultthat tendon stress over the entirelength between anchorages wouldbe uniform. Slip between the tendonand the concrete tends to reduce thetendon stress, compared with whatone might observe in a bonded mem-ber at the point of maximum mo-ment. The force distribution is gov-erned by the initial tendon stress,applied loading, tendon profile, and

coefficients of friction.For identical conditions with the

only variable being the bonded orunbonded tendon, the difference inresponse to load can be quite notice-able. The beam with the bonded ten-don will develop many small cracksspaced close together while thebeam with only the unbonded ten-don will tend to develop a few largecracks in the vicinity of points wheremaximum moments occur.

Since beams containing unbondedtendons are known to develop somerather wide cracks, which may beundesirable, the addition of un-stressed reinforcement can be usedto help control this cracking. Actual-ly, the addition of unstressed bondedreinforcement serves two main pur-poses. First, the additional reinforce-ment tends to distribute the cracks,making the beam crack much like abeam containing bonded tendons.Second, the additional unstressed re-inforcement contributes to the ulti-mate moment capacity of the sec-tion.

From the standpoint of design forflexure, the 1963 ACI Building Codeis careful to distinguish betweenbeams containing bonded tendonsand those containing unbonded ten-dons. It is interesting to note, how-ever, that no similar distinction ex-ists when design for shear is beingconsidered. The key assumptionsmade by the ACI Code in ultimatestrength design for flexure and shearof prestressed concrete beams aresummarized in Table I.

One major objective of the seriesof tests reported in this paper was toexamine the validity of the currentACI Code in predicting the strengthof beams having unbonded tendons.Predictions for both flexural andshear strength by ACI Code expres-sions were compared with those ob-served in the laboratory tests. The

16 PCI Journal

Table I—Design of Prestressed Concrete Beams (ACI 318-63)

TYPEMEMBER

ULTIMATEFLEXURE

ULTIMATESHEAR (SAME)

BONDED fsu = fs I 1 -0.5 pfs / Vu" is smaller of two

estimates based on

type of shear crack-

UNBONDED fsU ' fse t 15,000 psi ing developed.

distribution of cracking and addi-tional strength accomplished by theuse of bonded unstressed reinforce-ment was also of major interest inthese tests.

DESCRIPTION OF TEST PROGRAMThe total program of testing re-

ported in this paper represents fourindependent series of tests. Two se-ries used the ribbed cross sectionand two used the I-shaped cross sec-tion shown in Fig. 1. All beams wereof normal weight concrete with fof approximately 5000 psi. A briefdescription of the four differenttypes of beams used is presentedbelow.

Double-Cantilever Ribbed BeamsThis series of tests contained four

beams which were supported and

loaded as shown in Fig. 2. Betweenthe supports the cross section be-came a solid diaphragm, simulatingthe restraint offered by a crossingbeam in the pan-joist method of con-struction. This type specimen actsmuch like the negative moment re-gion between points of inflection ofa continuous beam. The distancefrom the edge of the diaphragm tothe load point varied and is pre-sented in Table II. Different ratiosof moment to shear at the face of thediaphragm were produced by adjust-ing the length of cantilever.

All beams contained two un-bonded tendons. Each of these twotendons (one in each rib) was com-posed of four /4-in. diameter wires.All tendons had the same drape pat-tern and essentially the same initial

Deforme 6 x 6 - +IE 10 mesh Deformedbars ,bars

Tendon8' 4-1/4"wires -1/2° 3" 5 14°

Tendons^' Deformed bar 4- 1 /4wires

Deformed 3../2" I/2" bars

7-1/2 F3+ 43436"

RIBBED SECTION I- SHAPED SECTION

Fig. 1—Cross Sections for Test Beams

October, 1967 17

P/c P/2

t_—vARES_----..H 18°-—VARIES°. °. I 92

DOUBLE CANTILEVER

RIBBED CROSS SECTION

P/2 P/2

6- 6"—I8" H-6- 6

15^-0"— I5-0"

CONTINUOUSCONTINUOUS BEAM

Fig, 2—Loading Arrangement for Ribbed Beams

prestress force. The major variablefor this series of tests was theamount of unstressed bonded rein-forcement. A complete description ofthe four double-cantilever beams ispresented in Table II.

Continuous Ribbed Beams

This group contained three 31 ft.10 in. long beams loaded and sup-ported as shown in Fig. 2. Thesebeams also had a solid diaphragmbetween the interior supports. Thesymmetrical loads were applied 6 ft.

6 in. from the edge of the diaphragmfor the three beams. All beams con-tained two tendons, each composedof four 1/4-in, diameter wires. Thedrape patterns of the tendons wereall identical: eccentricity from thebottom fiber was 8 in. at the end,3ahs in. at the low point 7 ft. 8 in.from the end, and 8 in. at the edgeof the diaphragm.

The major variable for this seriesof tests was the amount of unstressedbonded reinforcement. Table IIIcontains a complete description of

Table II—Double-Cantilever Ribbed Beamst

Distance from Total Bonded Total Bonded TotalBeam Diaphragm Top Bottom Prestress f'

to Load Reinforcement ° Reinforcementin. kips psi

B-1 39 4—No. 2 2—No. 2 64.2 4850B-2 30 4—No. 2 2—No. 2 56.6 4940

CB-3 37 4—No. 2 4—No. 3 49.5 5170B-4 30 4—No. 3 2—No. 5 57.4 4680

"All members contained 6 x 6—No. 10 welded wire mesh in the slab, as shown inFig. 1, in addition to the tabulated reinforcement. f, = 58 ksi for all bars andf^ = 70 ksi for No. 10 wire.18 PCI Journal.

Table Ill—Two-Span Ribbed Beams

Positive Moment Negative Moment TotalTotal Bonded Total Bonded Total Bonded Total Bonded Pre-

Beam Top Bottom Top Bottom stress f^Reinforcement Reinforcement Reinforcement° Reinforcement kips psi

CB-1 None 2—No. 3 4—No. 2 4—No. 3 50.6 4620CB-2 None 2—No. 4 7—No. 3 4—No. 4 51.1 4640CB-4 Same as CB-2 plus stirrups of No. 10 wire at 7 in. o.c. 48.3 5050

°All members contained 6 x 6—No. 10 welded wire mesh in the slab, as shown inFig, 1, in addition to the tabulated reinforcement. f,, = 55 ksi for No. 2 and No. 3bars, f = 44 ksi for No. 4 bars, f, = 70 ksi for No. 10 bars.

the continuous ribbed beam series.

I-Shaped Simple Beams

There were four beams simplysupported in this group of speci-mens, two with 9 ft. spans and twowith 12 ft. 6 in. spans The positionof the load points for all beams inthis series was constant at 18 in. ei-ther side of the centerline, as shownin Fig. 3. These beams retained thesame cross section throughout theirentire length.

The prestressing reinforcementwas again provided by two tendonswhich were composed of four ^-in.

diameter wires. The symmetricalparabolic drape pattern of the toptendon had an eccentricity of 93/4 in.from the bottom fiber at the endsand 41/4 in. at the centerline of thebeam. The bottom tendon followed aflatter symmetrical drape with theeccentricity from the bottom fiberbeing only 3% in. at the ends and2'/4 in. at centerline.

The major variables were the ini-tial amount of prestressing force andthe amount of unstressed bonded re-inforcement. A complete summary ofthe properties for this series of beamsis given in Table IV.

P/2 P/2

SIMPLE BEAM I CROSS SECTION

P/3 2P/3

BEAM WITH LOADED OVERHANG

Fig. 3—Loading Arrangement for I-Shaped Beams

October, 1967 19

Table IV—Simple Span I-Shaped Beams

Total Bonded TotalBeam Span Length Bottom Prestress

Reinforcement° kips psi

AB-1 9 ft. 0 in. 2—No. 3 63.8 4910AB-2 9 ft. 0 in. 2—No. 3 28.8 3980AB-3 12 ft. 6 in. 2—No. 3 59.9 4950AB-4 12 ft. 6 in. 2—No. 6 59.0 4880

*No top reinforcement was provided in these simple beams. f,, = 45 ksi for No. 3bars, f,, = 55 ksi for No. 6 bars.

I-Shaped Simple Beams with Over-hang

This group of four beams spanned12 ft. with a 3 ft. loaded overhang atone end as shown in Fig. 3. Allbeams had the same cross section,tendon profile, and loading arrange-ment. The beams contained two un-bonded tendons, each composed offour 1/4-in. diameter wires. The ten-don profile for the centroid of thetwo tendons (Fig. 3) had an eccen-tricity at the load point on the over-hang of 7/8 in. above the centroid ofthe concrete I-section. The eccen-tricity increased to a maximum of

41/8 in. above the centroid of con-crete at the support; then begins areversal until at a point 7 ft. 6 in.from the support a maximum eccen-tricity of 41/s in. below the centroidof concrete is reached. From thispoint the eccentricity decreased un-til at the end of the beam the eccen-tricity was 7/s in. above the centroidof concrete.

The major variables in this seriesof tests were the amount of prestressforce, amount of web reinforcement,and additional bonded unstressedlongitudinal reinforcement. A com-plete description of this series ofbeams is given in Table V.

Table V—I-Shaped Simple Beams With Overhang

Beam

Total BondedTop

Reinforcement°and Location

Total BondedBottom

Reinforcement

WebReinforcement

(U-Stirrups)

PrestressForcekips

f^

psi

BB-1 4—No. 3 4-3/s-in.dia. None 65.9 5720overhang to PI 7-wire strand

BB-2 4—No. 3 4-3/s-in.dia. No. 2 bars @ 6 in. 65.9 5470total length 7-wire strand

BB-3 2—No. 4 2-1h-in. dia. No. 2 bars @ 6 in. 34.9 6500total length 7-wire strand

BB-4 4—No. 3 4-a/s-in.dia. No. 10 gauge wire 65.0 5050total length 7-wire strand @ 6 in.

1—No. 2 1—No. 2 bartotal length

= 54 ksi for No. 2 bars, f, = 55 ksi for No. 3 bars, f5 = 44 ksi for No. 4 bars.

20 PCI Journal

DISCUSSION OF RESULTS

The present study was conductedas four series of tests utilizing twodifferent cross-sectional shapes asdescribed above. Some of the signifi-cant results from each of the fourseries will be discussed below beforetaking an overall look at the conclu-sions which grew out of the entirestudy.

Double-Cantilever Ribbed Beams

Flexural failures were observedfor each of the four beams in thisseries. None of these members con-tained web reinforcement, and theexpected test result following thecurrent ACI Code would have beena shear failure. Table VI gives the

ratios of observed-to-predicted flex-ure and shear strengths.

Fig. 4 shows the comparative load-deflection curves for two companionbeams in this series. The form ofthese two curves is representative ofall of the ribbed double-cantileverbeams. Beam B-2 was essentiallyidentical to Beam B-4 in materials,level of prestress, and loading. Notethat the additional bonded deformedbar reinforcement in Beam B-4 in-creased its moment capacity com-pared with Beam B-2. The bondedreinforcing steel acting at its yieldstress was considered in the predic-tion of ultimate moment capacity forthese beams, but observed flexuralstrength was consistently more thanpredicted (Table VIE).

Table VI-Summary of Results

Flexural ShearBeam Prestress Strength Ratio Strength Ratio

Designation Force Observed to Observed to Failure Modekips Predicted Predicted

Double-Cantilever Ribbed

13-1 64.2 1.13 1.16 FlexureB-2 56.6 1.19 1.39 Flexure

CB-3 49.5 1.24 1.47 FlexureB-4 57.4 1.18 1.71 Flexure

Two-Span Continuous Ribbed

CB-1 50.6 1.38 1.73 FlexureCB-2 51.1 1.22 1.92 ShearCB-4 48.3 1.15 1.48 Flexure

I-Shaped Simple Beams

AB-1 63.8 1.51 1.74 Flexure-ShearA13-2 28.8 1.80 1.82 Flexure-ShearAB-3 59.9 1.23 1.52 FlexureAB-4 59.0 1.24 2.14 Shear

I-Shaped Beams with Overhang

BB-1 65.9 0.84 1.48 ShearBB-2 65.9 1..00 1.27 Flexure-ShearBB-3 34.9 1.12 1.24 FlexureBB-4 65.0 0.97 1.71 Shear

October, 1967 21

5E

4f

400,a-

00 32

°w 24

aal6

8

B-4

B-2

P/2 P/230 11 3011

_____ _____ 39 L 1-J 391' _____0.3 0.4 0.5 0.6

•• • •

Failure of these members wasrather sudden, although some tracesof concrete crushing were noted inthe ribs prior to collapse. A diagonaltension-type crack formed at col-lapse, but this appeared to be sec-ondary with flexure as the primaryfailure mode.

Two-Span Ribbed Beams

The cross section for each of thesethree beams was identical to thatused in the double-cantilever beams(Fig. 1). Applied loading (Fig. 2)produced both positive and negativepeak moments in these indetermi-

nate two-span beams. Ultimate loadfor flexure was predicted on the ba-sis of simplified limit analysis assum-ing "plastic hinging" at this com-puted Mir for each of the twosections where maximum momentoccurred. Analysis following the ACICode shear strength provisions wasalso made. Only Beam CB-4 con-tained stirrups and their contribu-tion to ultimate shear strength wasconsidered in the analysis.

Fig. 5 shows a comparison of load-deflection curves for the two com-panion beams in this series whichhad no web reinforcement. Beam

DEFLECTION - INCHES

Fig. 4—Influence of Additional Bonded Reinforcement on Load-Deflection Re.sponse

22 PCI Journal

50

40CB-2

a-

CB-IYOa0J

w

a-Ja-

0

P/2 6,-6„ 6,-6„ P/2

III]i'-0 15-0„

rn0 0.5 1.0 1.5 2.0 2.5 3.0

3.5

DEFLECTION - INCHES

Fig. 5—Comparative Load-Deflection Response for Ribbed Continuous Beams

CB-1 failed in flexure with consid-erable crushing of the concrete inthe ribs at the point of peak negativemoment. The predicted shear failuredid not occur and the ultimate loadexceeded the flexural strength pre-diction by 38% and shear strengthprediction by 73%. Distribution ofcracking (Fig. 6) shows clearly the"hinge" positions, and the effective-ness of the bonded bars in distribu-tion of cracks is also indicated.

Beam CB-2 in Fig. 5 failed inshear with complete collapse of onespan. This beam differs from BeamCB-1 only in the amount of bondedreinforcement. Added reinforcementat both positive and negative mo-ment sections to boost the flexuralstrength was quite effective;strength was increased above that ofthe companion beam (Fig. 5) andcracks were very well distributed, asshown in Fig. 6. The higher shear

accompanied by the larger momentat the section of peak negative mo-ment produced a shear-compressiontype of failure which was suddenand complete. The load was 22%above the predicted flexural strengthand 92% above the predicted shearstrength at failure. Note the reducedductility for Beam CB-2 in Fig. 5compared to the companion beam.

Addition of approximately theminimum amount of web reinforce-ment under the ACI Code for BeamCB-4 produced a flexural failurewith a load-deflection responsemuch like that of Beam CB-1 in Fig.5. Analysis would have predicted ashear failure for Beam CB-4 and theACI Code minimum web reinforce-ment would not have been com-puted to be sufficient to produce aflexural failure. In all beams of thisseries the observed shear strengthwas underestimated to a larger ex-

October, 1967 23

tent than was the case for flexuralstrength as indicated by the ratios ofTable VI.

I-Shaped Simple Beams

Fig. 7 shows a comparison of load-deflection curves for two companionbeams in this series of I-shaped sim-ple beam tests. The only differencebetween the beams was the level ofprestress force initially provided forthe unbonded tendons. Reducing theprestress force by about one-half be-tween Beam AB-1 and Beam AB-2greatly changed the strength and be-havior. Predicted strength followingACI Code expressions takes into ac-count this difference, but the ratiosin Table VI indicate that both flex-ure and shear strength were still wellabove the predicted values.

Failures of both Beam AB-1 andBeam AB-2 was the result of exten-sion of diagonal cracking in the un-reinforced thin web of the I-section.The failure was not associated withflexural cracking as is noted in thecrack patterns of Fig. 8. Also note

that cracks were quite well distrib-uted.

Only one beam in this series failedin flexure; Beam AB-3 failed at 23%above the predicted ultimate load.All the beams failing in a manner re-flecting that shear cracking was aprimary part of failure sustainedloads well above the predicted shearstrength level as is noted in the ra-tios of Table VI.

I-Shaped Simple Beams with Over-hang

The loading (Fig. 3) applied tothe four beams of this series pro-duced simulated continuity; but thefact that the moments were determi-nate meant that no redistribution ofmoment between positive and nega-tive moment sections could occur.These beams were loaded to pro-duce constant shear over the lengthof the beam and all four of thebeams would have been predicted tofail in shear. As indicated by the ob-served-to-predicted strength ratiosof Table VI, only two of the beams

P/2 P/2

PATTERN OF CRACKING FOR SPECIMEN CB-I

P/2 P/2

r i/I "i k' f

PATTERN OF CRACKING FOR SPECIMEN CB-2

Fig. 6—Patterns of Cracking for Continuous Beams

24 PCI Journal

70

60

U)

Y 500

3400

a30

a20

10

0

ii_LI_z

AB-I FSe =63.8k

AB-2 FSe=28.8k

P/2 P/23'

0 .2 .4 .6 .8 1.0 1.2 1.4

DEFLECTION - INCHESFig. 7—Influence of Amount of Prestressing on Companion I.Shaped SimpleBeams

CRUSHEDATFAILUREt 77 ti

CRACK PATTERN FOR BEAM AB-I

CRUSHEDATFAILURE

C RACK PATTERN FOR BEAM AB-2Fig. 8—Crack Patterns for I-Shaped Simple Beams

October, 1967 25

80

70

60

50

40

w30

a

lLs]

BB-2FSe=64.4k

1._X

BB-3FSe=34.9k

i

1 P/3 2P/3

I7'_6'^ 4IlI

/-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0

actually did experience shear fail-ures, with one classified as a shear-compression type failure.

The bonded reinforcement con-sisted of 7-wire prestressing strandsat the failure section of all fourbeams. Large strains must occur inthese initially unstressed strands fortheir predicted ultimate steel stressto be reached. The larger strains areassociated with more severe crack-ing and, apparently, slightly less re-serve strength than would be ex-pected for the deformed bars whichwere used in the other three seriesof tests. But the two beams failing inshear (BB-1 and BB-4) carried 48%and 71% more than the predictedload assuming shear failures.

Fig. 9 shows a comparison of two

load-deflection curves for companionbeams BB-2 and BB-3 in this series.The beam with lower initial level ofprestress developed first flexuralcracking at earlier load. However,the form of the two companion load-deflection curves was quite similaras was the case with beams havingthe same cross section loaded as sim-ple spans (Fig. 7). The predictedflexural strength was realized forboth beams, but smaller reserve wasnoted for Beam BB-2, which had thehigher level of initial prestress force.

The indication is clear that use ofunstressed bonded strands, togetherwith stressed unbonded tendons cangive a very satisfactory member. Fig.10 shows how well the cracks weredistributed in the same two beams

DEFLECTION - INCHESFig. 9-Load-Deflection Response of Companion I-Shaped Simple Beams With Overhang

26 PCI Journal

P/3

2P/3P.I.

CRACK PATTERN OF BEAM BB -2

P/3

2P/3P.I.

CRACK PATTERN OF BEAM BB -3

Fig. 10—Crack Patterns for I-Shaped Simple Beams With Overhang

compared in Fig. 9. Ultimate loadfor Beam BB-2 was higher and morecracking in the web had developedat failure. Beam BB-2 exhibited bothtypes of cracking which the ACICode attempts to predict: shearcracks in the thin web near the pointof inflection and inclined flexuralcracks near the points of peak mo-ment.

The variation of web reinforce-ment within this series resulted inreasonable trends in strength forcompanion beams. Fig. 11 shows thecomparison of load-deflection curvesfor Beam BB-1 (with no stirrups)and Beam BB-2 (with sufficient stir-rups to develop predicted flexuralcapacity) . The striking change inductility is clearly noted in the com-parison of these two beams whichwere essentially identical except forthe web reinforcement.

CONCLUSIONS

From the overall results of all fourseries of tests described above, sev-eral conclusions can be reached.Two different cross sections and fourdifferent patterns of loading utilizedin this study would make more validthe consistent trends in results froma very limited series of tests.

1. Combination of stressed, un-bonded tendons with un-stressed bonded reinforcementproduced a very satisfactorystructural system from thestandpoint of both strength andbehavior.

2. 1963 ACI Building Code provi-sions are conservative for bothflexure and shear strength pre-dictions at ultimate. Moretrends for overly conservativepredictions were observed in

October, 1967 27

ii iIiBB-2+2 STIRRUPS at 6°FSe =64.4k

BB -INO STIRRUPSFs = 65.9k

P/3 2P/3

3•_0•• 7l-6° ¢ 4'-6"

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0DEFLECTION - INCHES

Fig. 11—Effect of Stirrups on Strength and Ductility of I-Shaped Simple Beams With Overhang

80

70

60U)a-

Y 50

0J 4C0wJ

30

20

I0

•

shear strength than in flexuralstrength.

3. Use of bonded 7-wire strandmay give a very adequate struc-tural system; but larger steelstrains required to develop thepotential strength of the strandsmay be associated with moresevere concrete cracking andless reserve strength than withdeformed bar reinforcement.

4. Continuous beams with suffi-cient strength to prevent shearfailure (even if no web rein-forcement is used) can develop"plastic hinging" at points ofpeak moment before reachingultimate load capacity.

ACKNOWLEDGMENTS

The work reported in this paperwas made possible by financial sup-port from The Prescon Corporation,Corpus Christi, Texas. All prestress-ing wires and anchorage hardwarewere furnished by Prescon and agrant by Prescon to The Universityof Texas at Austin provided the nec-essary funds to support the study.Four Master's theses, in which thetest results are reported in moredetail, have been used as the basisfor this paper. Mr. Oscar Atkins,Mr. Duane Brandt, Mr. Shan-TorngChen, and Mr. Harry Jones are re-sponsible for the four different seriesof tests and the authors acknowledge

28 PCI Journal.

their work which has been broughttogether here.

The helpful assistance given byMr. Eugene Dabney of The PresconCorporation throughout the entirecourse of this study was of greathelp and is acknowledged by theauthors.

REFERENCES

1. Atkins, Oscar L., "Behavior of I-ShapedPrestressed Concrete Beams with Un-bonded Tendons," unpublished Master's

thesis, The University of Texas, Austin,Texas, January 1967.

2. Brandt, Duane V., Jr., "Behavior ofPrestressed Concrete Members Rein-forced with Unbonded Tendons andBonded Unstressed Tendons," unpub-Iished Master's thesis, The University ofTexas, Austin, Texas, August 1967.

3. Chen, Shan-Torng, "Behavior of RibbedPrestressed Concrete Beams with Un-bonded Tendons," unpublished Master'sthesis, The University of Texas, Austin,Texas, January 1967.

4. Jones, Harry L., "Behavior of Pre-stressed Concrete Members with Un-bonded Tendons," unpublished Master'sthesis, The University of Texas, Austin,Texas, January 1967.

Discussion of this paper is invited. Please forward your discussion to PCI Headquartersbefore January 1 to permit publication in the April 1968 issue of the PCI JOURNAL.

October, 1967 29