experimental study of prestressed concrete under combined torsion, bending and shear

7/27/2019 Experimental Study of Prestressed Concrete Under Combined Torsion, Bending and Shear

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EXPERIMENTAL STUDY OFPRESTRESSED CONCRETE UNDERCOMBINED TORSION,BENDING, AND SHEARArthur E. McMullenAssociate Professor of Civil EngineeringDepartment of Civil EngineeringUniversity of CalgaryCalgary, AlbertaCanada

H. Roger WoodheadDesign Engineer,McKenzie, Snowball, Skalbaniaand Associates Ltd.Vancouver, British ColumbiaCanada

Test results are presented for 26 rectangulareccentrically pretensioned concrete beams subjectedto various combinations of torsion, bending, andshear.To investigate the effect of the presence of stirrups,18 beams were provided with closed rectangularstirrups and 8 contained no stirrups.Other variables investigated were the average levelof prestress, the stirrup spacing, the T/(Vb) ratio,and the T/M ratio.Torque-shear and torque-moment interactionequations are proposed for both beams withoutstirrups and beams with stirrups.

Considerable progress has recently been for the design of reinforced concretemade in understanding the effect of subjected to torsion. However, the ACItorsion on reinforced concrete and as Torsion Committee was unable toa result the most recent edition of the propose similar design recommenda-ACI Building Code' contains provisions tions for prestressed concrete since in-

PCI Journal/September-October 19735


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I.75" t31/a' x 9 1/," (79.4 x 232 mm)CLOSED STIRRUPS (exceptSERIES I)

6-1/2(12.7mm) DIA. 7WIRE STRANDS

CROSS SECTION IN TEST LENGTH52"(1320mm) I TEST LENGTH=56"(1420mm) 36(914mm)

-10"(254mm) 72"( 1.83 m)36' (9I4mm) D INDICATES LOCATION OF DEFLECTION GAGET INDICATES LOCATION OF TWISTMETE SERIES I)LONGITUDINAL SECTIONFig. 1. Details of typical test specimen

adequate test data were available.2The investigation reported herein is anattempt to provide more test data andforms part of a larger study.3

EXPERIMENTAL PROGRAMTwenty-six rectangular, eccentrically

pretensioned beams were subjected tovarious combinations of torsion, bend-ing, and shear. Three of the specimenswere tested in purr torsion and two inbending and shear alone. The speci-mens were divided into four series:

Series I consisted of eight specimenswithout stirrups, tested under variousloading combinations so that the in-teraction of torsion, bending, and shearcould be observed.

Series II contained five specimenswith closed rectangular stirrups and

with the same average level of pre-stress o-, as Series I (0.1 f,'). Theeffect of stirrups on the interactioncould thus be observed.Series III contained six beams withthe same size and spacing of stirrupsas Series II but with o-, = 0.2 ft,'. Thusthe effect of o-p on the interactioncould be examined.Series V consisted of seven beamssimilar to those of Series III exceptthat they were tested at a constant tor-sion to shear ratio [T/(Vb) = 1.5] .The quantity of stirrups was varied toobserve the effect on the strength incombined loading.All beams had a nominal cross sec-tion of 6 x 12 in. (152 x 305 mm) andwere 12 ft (3.66 mm) in length. Theywere pretensioned eccentrically by six1/2in. (12.7 mm) diameter seven-wire

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Table 1. Details of test specimens

f "` f f,t Qn, ^p e , Transverse s tee lSizes f:y s,Be am psi psi psi fo ' in. ksi in .(N/mm 2 ) (N/mm 2 ) (N/mm 2 ) (mm) (N/mm2) (mm)- 1 6850(47.2) 494(3.41) 660(4.55) 0.096 0.65(16.5) -- -1 - 2 6850(47.2) 494(3.41) 660(4.55) 0.096 1.48(37.6) -- -1 - 3 7170(49.5) 538(3.71) 800(5.51) 0.111 1.50(38.1) -- -1 - 4 7170(49.5) 538(3.71) 700(4.83) 0.098 1.46(37.1) -- -1 - 5 5760(39.7) 549(3.79) 775(5.35) 0.134 1.19(31.0) -- -1 - 6 5760(39.7) 549(3.79) 750(5.17) 0.130 1.53(38.8) -- -- 7 7400(51.1) 572(3.94) 680(4.69) 0.092 1.28(32.5) -- -- 8 7400(51.1) 572(3.94) 680(4.69) 0.092 1.15(29.2) -- -

II-1 5340(368) 554(3.82) 700(4.83) 0.131 1.79(45.4) No.2 26(179) 4(101)1 1 - 2 5340(36.8) 554(3.82) 700(4.83) 0.131 1.68(42.7) No.2 26(179) 4(101)1 1 - 3 5780(39.9) 633(4.37) 790(5.45) 0.136 1.52(38.6) No.2 26(179) 4(101)1 1 - 4 5780(39.9) 633(4.37) 675(4.66) 0.117 1.41(35.8) No.2 26(179) 4(101)1 1 - 5 5520(38.1) 555(3.83) 730(5.04) 0.132 1.61(40.9) No. 2 26(179) 4(101)1 1 1 - 1 6530(45.0) 570(3.93) 1220(8.41) 0.187 1.54(39.1) No. 2 33(228) 4(101)1 1 1 - 2 6530(45.0) 570(3.93) 1080(7.45) 0.166 1.56(39.6) No. 2 33(228) 4(101)1 1 1 - 3 6060(41.8) 506(3.49 1100(7.59) 0.182 1.64(41.7) No.2 33(228) 4(101)1 1 1 - 4 6060(41.8) 506(3.49) 1075(7.42) 0.178 1.71(43.4) No.2 33(228) 4(101)1 1 1 - 5 6580(45.4) 534(3.68) 1175(8.10) 0.176 1.75(44.4) No. 2 33(228) 4(101)1 1 1 - 6 6580(45.4) 534(3.68) 1125(7.75) 0.171 1.86(47.2) No.2 33(228) 4(101)

V -1 5730(39.5) 546(3.77) 1350(9.31) 0.235 1.82(46.1) No.2 33(228) 2(51)V -2 5730(39.5) 546(3.77) 1200(8.27) 0.209 1.66(42.2) No. 2 33(228) 6(152)V -3 6720(46.3) 566(3.90) 1250(8.62) 0.186 1.83(46.5) No.2 33(228) 8(203)V -4 6720(46.3) 566(3.90) 1270(8.76) 0.189 1.78(45.2) No. 2 33(228) 10(254)V -5 5520(38.1) 555(3.83) 1250(8.62) 0.227 1.66(42.2) No. 2 61(421) 2(51)V -6 5640(38.9) 592(4.08) 1180(8.14) 0.209 1.58(40.1) No.3 55(379) 8(203)V -7 5640(38.9) 592(4.08) 1260(8.68) 0.224 1.66(42.2) No.3 55(379) 2(51)* Cylinder compressive strength f 0 ' is average value from five 6 x 12 in. (152 x 304 mm)cyl inders.Modulus of rupture f,. is average value from five 6 x 6 x 24 in. (152 x 152 x 608 mm)prisms.#No. 2 (6 .3 m m d iam e te r ) bars were p lain and No. 3 (9 .5 m m d iam e te r ) were de forme d.

prestressing strands. The top strandswere stressed to a lower level than thebottom strands. All specimens con-tained additional stirrups outside thetest length to ensure that failure oc-curred within the test length. Detailsof the specimens are furnished in Table1 and Fig. 1. The stress-strain curve ofthe prestressing strand is shown in Fig.2.

The specimens were cast in formslocated inside a prestressing frame(Fig.3). The strands were tensionedby a hydraulic jack; the force in each

strand being monitored by a load cell.To measure the prestress applied tothe beam, an 8 in. (204 mm) demecgage was used. The concrete for eachpair of beams was mixed in five batchesin a drum mixer. Each mix had atarget 28-day compressive strength of6000 psi (41.4 N/mm2 ) and consistedof the following approximate propor-tions by weight:Cement: Fine aggregate: Coarse aggre-gate 1:2.4:3.8.Water-cement ratio: 0.5.

The amount of water was adjusted



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40 15 03613 0

S211 0

!84000 p6 5012

ULTIMATE TENSILE STRESS94 8 1 k i8 , LOAD AT I% ELONGATION=37.9 kips 30(168101)MODU LUS OF ELASTICITY =274alOeps i4 (18.9s104N/mm21 100.2, 0.4 0.6 U.6 1.0 I.cPERCENT ELONGATION)NCHES/ INCH 10-2Fig. 2. Stress-strain curve for prestress-ing strand

so that a slump of about 11/2 in. (38mm) was obtained. One standard 6 x12 in. (152 x 304 mm) cylinder andone 6 x 6 x 24 in. (152 x 15 2 x 608 mm )prism were cast from each mix forcontrol specimens. The cylinders weretested in compression, and modulus ofrupture tests under third-point loadingwere carried out on the prisms.

Fig. 3. Prestressing bed

Electrical resistance strain gages w ereused to measure the strains in thestirrups. Four gages were attached toa stirrup, one on each leg. Deflectionswere measured at the locations shownin Fig. 1. Each deflection gage con-sisted of a scale hanging from a balljoint fixed to each side of the beam.The readings were taken with a geo-

Fig. 4. General view of test setup

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SF 0.

TEST LENGTH

NORTH(F IXED HEAD) I SOUTHT-. P I T

3 L32PL9

1 2L P3B . M . D .

Fig. 5. Schematic diagram of test setup

BehaviorBeams without stirrupsThe speci-

mens without stirrups tested under lowT/(Vb) ratios failed in an explosiveand sudden manner in shear-compres-sion. At failure the compression zonewas usually completely destroyed andthe top strands arched upwards. Atypical example is Specimen I-I [T/(Vb) = 0] shown in Fig. 6.

As the proportion of torsion was in-creased the failure became less violent.The cracks still commenced on the hot-

tom face and they spread faster on theeast face, on which torsional and flex-ural shear stresses were additive, thanon the west face. On the east face theycurved towards the load point althoughthey were almost vertical on the westface. On both faces the cracks weremore closely spaced near the loadpoint.Specimens tested under a high T/(Vb) ratio [T/(Vb) 2.92] displayeda torsion failure with the cracks onthree faces forming a spiral, the ends

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Top

West

Bottom

East

Fig. 6. Crack pattern for Specimen I-1of which were joined by a line ofcrushing on the other (side) face. Fewcracks appeared prior to failure whichwas sudden but not violent.

It was found that as the proportionof torsion increased, the angle betweenthe crack and the beam axis decreased,the distance from the load point to the

compression zone increased and theratio of the torque at which the firstcracks (flexural or torsional) were ob-served, to ultimate torque, T,,, in-creased.

Beams with stirrupsThe behaviorof these beams in the early stages of atest was similar to that of the beams

Top

West

Bottom

East

Fig. 7. Crack pattern for Specimen 111-3



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Top

West

Bottom

East

Fig. 8. Crack pattern for Specimen 111-4

without stirrups. In the later stages thebehavior was modified by the presenceof the stirrups in that the violence offailure was reduced and at high T/(Vb)ratios the specimens displayed in-creased strength and ductility aftercracking.

The beams tested under low T/(Vb)ratios failed when crushing occurred onthe top surface near the load point. Asthe T/(Vb) ratio increased there wasevidence of crushing on one of theside faces as well as on the top face.The crushing on the top face becameinclined to the beam axis and movedaway from the load. A typical exampleis Specimen III-3 [T/(Vb) = 1.5]shown in Fig. 7. At failure of thespecimens tested under predominantlytorsional loading, a spiral crack formedon three faces and was joined by aline of crushing on the fourth (side)face. A typical torsion failure is illus-trated in Fig. 8 for Specimen III-4[T/(Vb) = cc ] . As observed for thespecimens without stirrups, the ratioT c ,/T,, increased and the angle of crackdecreased as T/(Vb) increased.

Comparison of the crack patterns forSeries II and III indicated that whenthe level of prestress was increased thefailure became more destructive andthe inclination of the cracks to thebeam axis decreased, especially at highT/(Vb) ratios. The crack patterns forthe beams in Series V showed that asthe quantity of stirrups decreased theviolence of failure increased.Deform ation characteristicsThe torque-twist and load-deflectioncurves for the beams in Series I (nostirrups) are given in Figs. 9 and 10,respectively. The torque-twist curveswere generally smooth and not discon-tinuous at the cracking torque as isthe case with similar unprestressedspecimens. The initial portion of thetorque-twist curve was almost linearfor beams tested under high T/(Vb)ratios.

After cracking occurred, the slope ofthe torque-twist curve gradually de-creased until near failure it was a lm-stzero. The load-deflection curves forSeries I were also linear until flexural

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12 0

TU)az.60wDvX030

I-8

I-3I-2

r2zU05r0I-

0TWIST/UNIT LENGTH- RAD./IN.xl0 -6 (RAD/MMx 10-6)

45(1.77)Fig. 9. Torque-twist curves for Series I

30

a-20

Q0JJQ10

U

ri

0QO O60UF -

30 >

i-V . CENTRAL D EF LECTION- IN . x 10 -2 (MM)15(3.8)

Fig. 10. Load-deflection curves for Series I

cracking commenced. After cracking,the slope of the curves gradually de-creased but not as markedly as that ofthe torque-twist curves. In fact, at largeT/(Vb) ratios there was little change inslope after cracking.Torque-twist and load-deflection

curves for typical beams with stirrups(Series III) are shown in Figs. 11 and12, respectively. The curves are gen-erally similar to those of the beamswithout stirrups except that the beamsloaded under high T/(Vb) ratios dis-played increased torsional ductility

PCI Journal/September-October 1973 3


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REINFORCEMENT FACTOR KN MI01V-5V I V-7

V-3

V-4

0REINFORCEMENT FACTOR x 1 y l A t f t y IN KIPS

s

Fig. 13. Variation of M,, with reinforcement factor for specimens in Series V

750

700C' ,

z650

600

550

SPECIMEN REINFORCEMENT FACTORINCH KIP KNMV-4 4.88 0 .55V-3 6.11 0 .69V-2 8.14 0.92111-3 1 2 . 2 1.38V-6 24.0 2 . 7 1V I 24.4 2.76V-5 4 5 . 1 5.10V-7 95.8 10.82

75

2z7

6 0

vary in any consistent manner withthe quantity of stirrups, increased asthe level of prestress increased, and de-creased as T/(Vb) decreased. It wasmostly dependent on the loading ratio,T/(Vb); a result which confirms theobservations of Henry and Z ia.4

The initial slope of the load-deflec-tion curves was found to be inde-pendent of the quantity of stirrupsand the level of prestress but it in-creased slightly as T/(Vb) decreased.The slope of the load-deflection curvesafter cracking was found to increase asthe level of prestress increased. Thedeflection at the ultimate load increasedas T/(Vb) decreased and as the levelof prestress decreased.Reinforcemen t strains

The stirrup strains were small andsometimes even compressive in theearly stages of a test. They did not in-crease markedly at the cracking loadbut increased just prior to the ultimateload. The stirrup strain appeared to bemainly dependent on the location ofthe stirrup relative to the failure sur-

face. In general, if a stirrup that wasprovided with a gage was intersectedby the failure surface, it indicated theyield strain at or immediately after theultimate load.Since the magnitude of the stirrupstrain was so dependent on the loca-tion of the failure surface relative tothe stirrup, it is difficult to draw anyfirm conclusions regarding yield of thestirrups. For the same torque level, thestirrup strains in Series II were gen-erally higher than those in Series III.In Series V. the strains tended to de-crease as the quantity of stirrups wasincreased. Specimen V-5 had gages onfour different stirrups and the resultsindicate that the strains decreased asthe distance from the load increased.Since the torque and shear force wereconstant along the test length, it wouldappear that the distance from the fail-ure crack affected the strains as theinclined failure crack occurred close tothe load point.For Specimen IlI-4, tested in puretorsion, the gage on the east leg ofthe stirrup indicated yield subsequent



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00I-

0.

0.2.4.6.8. 0Mu / Muo(a ) TORSIONMOMENT0.2.4.6.8.0

vu! Vuo(b) TORSION SHEARFig. 14. Nondimensional interaction diagrams for specimens

without stirrups

m

0 .8

0 .60

0.4

0 .2

1 0 7

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to failure even though the east faceof the specimen contained the com-pression zone. Gangarao and Zia 5 madea similar observation for sev eral of theirbeams and concluded that it indicatedthat the compression zone was small.However, a better explanation may bethat the tensile strain acting parallel tothe line of crushing was large enoughto cause yield of the stirrups near thatface.Effect of stirrups on strength incombined loading

The specimens in Series V containedvarying amounts of stirrups and weretested at a constant T/(Vb) ratio of1.5. Fig. 13 illustrates the variation ofultimate moment, M,,, with the rein-forcement factor,x1y1Awherex l and y l are the smaller and largercenter-to-center dimensions of a stirruprespectively and A t is the area of oneleg of a stirrup.

The value of M,, and not Tisplotted since the beams were found tofail in the bending mode. 3 Below a re-inforcement factor of 25 in.-kips (2.82kNm), except for Specimen V-6, M,,,increased as the reinforcement factorincreased. However, above this valuethere was little increase in M und itcan thus be concluded that SpecimensV-1, V-5, and V-7 were over-reinforced.It should be noted that the longi-tudinal prestressing steel was identicalfor all specimens in Series V and there-fore the over-reinforcement must havebeen caused by the stirrups. SpecimenV-6 may have failed prematurely be-cause of the large stirrup spacing andit is interesting to note that Gangaraoand Zia5 suggested that the maximumvalue of stirrup spacing should be one-half the larger stirrup dimension.

As stated above, the violence of fail-ure was found to increase as the rein-forcement factor decreased. Since thefailure of Specimens V-2, V-3, and V-4was noticably more violent than theothers it would appear that to avoid

an overly destructive failure in thistype of section the reinforcement fac-tor should be greater than 10 in.-kips(1.13 kNm).Interaction of torsion, bending,and shear

The interaction of torsion, bending,and shear is a much more complexproblem than that of torsion and bend-ing alone. It is essentially a three-di-mensional problem and an interactionsurface must be developed to fullypredict the behavior. However, for con-venience it is usual to present experi-mental data on two-dimensional inter-action diagrams.

The torsion-moment interaction dia-gram is the projection on the torsion-moment plane of the intersection of theinteraction surface and a plane definedby the loading ratio. Similarly, thetorsion-shear interaction diagram is thisintersection projected on the torsion-shear plane.Since the "pure" shear strength of abeam in the absence of bending mo-ment cannot be determined experi-mentally it must be defined. In this in-vestigation, V,, 0 is defined as the shearforce acting at failure of a specimentested in bending and shear alone.Thus, for beams which contained ade-quate shear reinforcement, such asSeries II and III, V00 is the shear forcethat was acting when the beam attainedits full flexural capacity. For Series I,however, V00 is less than this valuesince Specimen I-1 failed before itsfull flexural capacity was reached.

Figs. 14 and 15 show the interactiondiagrams in nondimensional form. Itcan be seen that the addition of stir-rups significantly affected the shape ofthese interaction diagrams, especiallyin the region corresponding to highT/(Vb) ratios.

For the beams in Series II (o-, 0.1fe ' ), the torsional strength was not re-duced below the pure torsional strengthuntil a bending moment and shear



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2T01T u o

1 .0

0 .8

0.6

0 .4

0 .2

.0

0 .8

o 0.6

0 .4

0 .2

aII2(Tu

0.2.4.6.8. 0Mu/Muo

(a) TORSIONMOMENT

0.2.4.6.8.0Vu/Vuo

(b) TORSIONSHEAR

Fig. 15. Nondimensional interaction diagrams for specimenswith stirrups

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force of more than 25 percent of theultimate capacities were simultaneouslyapplied. An increase in the level of pre-stress increased the strength in com-bined loading, especially when T/(Vb)was large.For Series III (a-, 0.2 f0') the pres-ence of a small amount of bending andshear actually increased the torsionalcapacity above the pure torsionalstrength. In addition, the bending ca-pacity was equal to M,u oven when atorque equal to 30 percent of T ,u owassimultaneously applied.

For the specimens without stirrupsthat were tested in this investigationthe torsion-moment interaction can beapproximated by the relation:

MMu+Tu _11)u oand the torsion-shear interaction by:

V.+[T_ '=12)V.uoSince no tests in combined loadingon beams of this type have previouslybeen reported it is not possible to com-pare these relations with results ob-

tained by other investigators.For the specimens with stirrups thatwere tested in this investigation, the

interaction can be conservatively ex-pressed by the following equations:

[,a]2+[Tuo]213)and C]'+2V10 Tuo](4)Eq. (3) is identical to that proposedby Henry and Zia4 but Eq. (4) cannotbe compared to their torsion-shear in-teraction equation since they adopteda different definition for V , u o .In Eqs. (1) to (4) the experimentalvalues of T ,, o ,M o ,nd V,, , o have beenused. If these values are not availablethey must be calculated. Current meth-

ods of calculating these values can befound in References 3, 4, and 5.CONCLUSIONS

1. The specimens that were testedunder predominantly flexural loadsfailed when crushing occurred on thetop surface and for the beams withoutstirrups this failure was explosive. Asthe proportion of torsion was increasedthe failure became less violent, thecompression zone on the top face be-came inclined to the beam axis andgradually moved from the top face to aside face (and also away from the loadpoint).2. For the beams without stirrupsthe torsion-shear-moment interactioncan be approximated by Eqs. (1) and(2) while for the beams with stirrupsEqs. (3) and (4) conservatively predictthe interaction.3. The conclusion of Gangarao andZia 5 that the maximum stirrup spacingshould be yl /2 was substantiated.4. The torque twist curves were gen-erally smooth and not discontinuous atthe cracking torque. The initial portionof a torque-twist curve was almostlinear and the initial torsional stiffnesscould be approximated by the elastictorsional stiffness. The value of thepost-crack ing torsional stiffness wa s de-pendent mainly on the loading ratio.The reduction in the flexural stiffnessafter cracking was not as large as thatof the torsional stiffness.5. The strains in a stirrup were large-ly dependent on the location of thestirrup relative to the failure surface.The strains were low initially and didnot increase markedly at the crackingtorque but did increase rapidly justprior to failure.

ACKNOWLEDGMENTSThe research on which this paper isbased was carried out in the Depart-



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ment of Civil Engineering at The Uni-versity of Calgary. Financial assistancewas provided by the National ResearchCouncil of Canada and is gratefullyacknowledged.

APPENDIX-NOTATIONA tarea of a single leg of a stirrupb =width of a rectangular beameeccentricity of prestressf0 = concrete cylinder compressivestrengthf, =modulus of rupture of concreteft,=yield stress of stirrupL =ength of spanM= bending momentMe ,. = cracking momentM,, =ultimate bending capacity of

beam under combined loadingM ,u 0= ultimate bending capacity of

beam under pure flexureP = vertical load on beam in testsspacing of stirrupsT = twisting momentT o ,.torque at which first cracks wereobservedTu= ultimate torsional capacity of

beam under combined loadingT ,u 0 ultimate torsional capacity of

beam in pure torsionVar shear force at crackingVu ultimate shear of a beam under

combined loadingVhear force acting at failure ofa specimen tested in momentand shear alonex l = shorter center-to-center dimen-sion of a closed rectangular

stirrupy l = larger center-to-center dimen-sion of a closed rectangularstirrupo-, = average level of prestress onsection

REFERENCES1. "Building Code Requirements forReinforced Concrete (ACI 31 8-71 ),"American Concrete Institute, De-

troit, 1971, 144 pp.2. Hsu, T. T. C., and Kemp, E. L.,"Background and Practical Applica-tion of Tentative Design Criteria

for Torsion," ACI Journal, Proceed-ings Vol. 66, No. 1, January 1969,pp. 12-23.3. Woodhead, H. R., and McMullen,A. E., "A Study of Prestressed Con-crete Under Combined Loading,"Research Report No. CE 72-43, De-partment of Civil Engineering, Uni-versity of Calgary, 1972.4. Henry, R. L., and Zia, P., "Behaviorof Rectangular Prestressed ConcreteBeams Under Combined Torsion,Bending and Shear," Department ofCivil Engineering, North CarolinaState University at Raleigh, April1971.

5. Gangaroa, H. V. S., and Zia, P.,"Rectangular Prestressed ConcreteBeams Under Combined Bendingand Torsion," Department of CivilEngineering, North Carolina StateUniversity at Raleigh, April 1970_

Discussion of this paper is invited,Please forward your discussions to PCI Headquartersby February 1, 1974, to permit publication in theMarch-April 1974 PCI JOURNAL.

experimental study of prestressed concrete under combined torsion, bending and shear

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