Control of Horizontal Crackingin the Ends of PretensionedPrestressed Concrete Girders

by W. T. Marshall* and Alan H. Mattock**

SYNOPSIS

This paper describes a limited investigation of the stresses which occurin the ends of pretensioned prestressed concrete girders at the time oftransfer of prestress, and which can result in the formation of horizontalcracks in the ends of such girders. Also reported is a study of the stressesset up in vertical stirrup reinforcement near the ends of pretensionedprestressed girders when horizontal end cracking does occur. On the basisof experimental data obtained in this study, an equation is proposed forthe design of vertical stirrup reinforcement necessary to restrict the sizeof any horizontal end cracks which may occur in a pretensioned pre-stressed concrete girder.

IntroductionHorizontal cracks in the ends of

pretensioned prestressed concretegirders have been reported on num-erous occasions in recent years.These cracks have been observedmainly in beams having an I-shapedcross-section, but they have alsobeen noted in inverted tee and rec-tangular section girders.

A typical example of horizontalend cracking is shown in Fig. 1. Thecracks usually occur near the cen-troidal axis, and often close to thejunction of the web and the lowerflange in the case of I- and invertedtee-shaped girders. As would be ex-pected, this type of cracking occursmore commonly in I-shaped or in-verted tee-shaped members than in

*Regius Professor of Civil Engineeringat the University of Glasgow, and Visit-ing Engineer at the Portland CementAssociation Research and DevelopmentLaboratories, Skokie, Illinois.

**Principal Development Engineer, Struc-tural Development Section, Portland Ce-ment Association Research and Devel-opment Laboratories, Skokie, Illinois.

rectangular girders. This is due tothe smaller horizontal cross-sectionof the concrete available to resistthe tensile forces set up.

In a recent survey(') of prestressedconcrete highway bridges in theUnited States, horizontal end crack-ing was noted in 25 out of 41 pre-tensioned prestressed bridges ex-amined. This type of crackingoccured with the greatest frequencyin the case of girders having drapedstrands, where the strands were con-centrated in two groups at the ends.The provision of end blocks did notappear to prevent the formation ofthese cracks.

Horizontal end cracks occur as aresult of the high tensile stresses setup at the end face of girders be-tween the groups of strand. Themaximum tensile stress usually oc-curs near the centroidal axis, andthe cracks usually form in this lo-cation.

When an adequate amount ofvertical stirrup reinforcement is pro-vided in the end regions of preten-sioned prestressed girders, the

56 PCI Journal

Fig. 1Example of a Horizontal Crack in the endof a Pretensioned Prestressed Bridge Girder.(Crack marked to emphasize location)

Fig. 2Severe Horizontal Cracking in a Girder without Vertical Stirrup Re-inforcement.

October 1962 57

development of the horizontalcracks is restricted. In these cases,the cracks are fine; usually theirwidth is about one hundredth of aninch or less, and they penetrate onlya matter of inches into the end ofthe member. However, it should benoted that if vertical stirrup rein-forcement is not provided, the hori-zontal crack may widen and spreadalong the girder as may be seen inFig. 2. In the extreme case such acrack may split the girder from endto end. This behavior was observedduring the test program carried outat the PCA Laboratories, and alsooccurred a few times in plants be-fore vertical stirrups came into gen-eral use.

If sufficient vertical stirrup rein-forcement is provided, so that thehorizontal cracks are restricted toa few inches in length, and in widthto one hundredth of an inch or less,then these cracks would not be det-rimental to the performance ofgirders either at service load levelor at ultimate strength. The cracksare caused primarily by the con-centration of prestressing forces.They will not lengthen or widen asa result of the application of ad-ditional loads to the member, sincein a pretensioned prestressed con-crete girder the tension in the pre-stressing strand at the ends of thegirder does not appreciably increasewhen loads are applied to thegirder. If restricted as describedabove, the cracks would not begreater in width than those whichhave occurred in ordinary reinforcedconcrete members for over half acentury.

The field survey(') referred toearlier has shown that the provisionof end blocks does not ensure theabsence of horizontal cracking at

the ends of pretensioned prestressedconcrete girders. In addition, endblocks cannot restrict the growth ofa horizontal end crack once crack-ing has occurred, and they also addappreciably to the formwork costsfor a prestressed girder. It thereforeappears sound both from economicand structural considerations to omitend blocks from pretensioned pre-stressed concrete girders, and tosubstitute in their place an adequateamount of vertical stirrup reinforce-ment. The primary outcome of thisinvestigation is a proposal for de-sign criteria for this vertical stirrupreinforcement.

Previous Investigations

As far as can be ascertained, noexperimental investigations havebeen carried out previously whichhad as their primary purpose themeasurement of those stresses whichcause horizontal cracking in theends of pretensioned prestressedgirders. However, during an investi-gation of the transmission length ofwires in pretensioned prestressedconcrete, Base( 2 ) made measurementsof strains due to vertical tension inthe end zones of three girders.

Of particular interest are the datafrom Base's second and third gird-ers. Both these girders were of aninverted tee cross-section, 22 in.deep, 20 in. wide, and with a 3-in.thick web. The girders were pre-stressed by straight 0.20-in, diameterwires arranged in two groups, 62wires at the bottom of the sectionand 10 wires at the top. Ten 3/s-in.square twisted bars placed verticallyin the end nine inches of the secondgirder prevented the formation ofvisible cracks, although a maximumvertical tension strain of 0.00035 wasmeasured in the web at the end face.

58 PCI Journal

The vertical tension strains de-creased to zero at about 12 in. fromthe end face of the girder.

The third girder was providedwith eight 3/s-in. square twisted barsin the end 22 inches. In this case avisible horizontal crack occurred inthe lower part of the web, and themaximum vertical tension strain re-corded was in excess of 0.0015. Zerovertical strain did not occur untilabout 20 in. from the end face ofthe girder. The longitudinal strainsmeasured in the bottom flangereached a peak 12 in. from the endface of the girder in this case, asagainst a distance of about 48 in.in the case of the second girder,indicating a much shorter transferlength for the third girder than forthe second. The more extensivecracking of the third girder wastherefore probably due to a com-bination of the effects of the shortertransfer length and of the smaller

concentration of vertical stirrup re-inforcement close to the end of thegirder.

Base did not attempt to analyzethese results but commented asfollows: "Considerable work wouldbe necessary to produce and checkany theoretical method of forecast-ing the peak stress, but too great aconcentration of wires should ob-viously be avoided."

The results obtained by Base ap-pear to indicate that the maximumvertical tension stress probably de-pends on the arrangement of theprestressing wires at the end of thegirder, and more specifically thatdivision of the prestressing wiresinto concentrated groups leads tohigher vertical tension stresses. Themagnitude of these stresses is alsoapparently affected by the trans-mission length of the wire, shortertransmission lengths leading to high-er stresses. The results further sug-

8

111113/8_ I3/8"

10 Strands, 72 2 Strands21/4 Spacing

T3

rAl 8 3, 3 1/2" 1145,7, 89, 3 1/2i

22 1/2 9 1/2 42 B 4, 2 1/2' 46, 8, a 10, 2 1/2'

Jt1l3 3/4 gB5

B3 26 Strands,}10 STrands, 62 J 2 Spacing2 I Spacing

31/2 2 31/2 2I/2 2j)21/2

A[, 2, 3, & 4 A 5, 6, 7, 8, 9, a 10(1/4" Strand) (1/4" Strand )

Fig. 3Series A Girders, Al through A10.

October 1962 59

gent that in order to be effective inrestraining the growth of cracks,vertical stirrup reinforcement mustbe concentrated close to the endface of the girder.

Experimental InvestigationScope of Investigation

Test Series A concerned measure-ment of the concrete stresses in theend zones of pretensioned pre-stressed girders at the time of trans-fer of prestress to the concrete. Tenshort girders were tested; all hadbasically the same I-shaped cross-section and were prestressed by thesame size strand. The variables in-cluded in the test series were: webthickness, arrangement of prestres-sing strand, and surface condition ofthe strand.

Test Series B concerned measure-ment of the stresses set up at pre-stress transfer in the vertical stirrupreinforcement provided near theends of pretensioned prestressedgirders. From the results obtainedin these tests, it was sought to de-velop an empirical method for thedesign of end zone reinforcement.Twenty five girder specimens weretested, having two basic cross-sec-tions and containing two sizes ofvertical stirrup reinforcement. Theother variables included in the testswere the size and location of theprestressing strands. and the mag-nitude of the prestressing force.

Test GirdersAll test specimens were 10 ft long;

their cross-sections are shown inFigs. 3, 4, and 5. Since this investi-gation was concerned only with themeasurements of stresses in the endzones of pretensioned prestressedgirders at the time of transfer ofprestress, the length of the testgirders was governed by the re-quirement that the theoretical linear

distribution of prestress in the girdershould be achieved over at leastthe middle third of the girderlength. It was considered that ifthis condition was achieved, thenany further lengthening of the testspecimen would not significantlyaffect the stresses at the ends of thespecimens at transfer. Measurementof longitudinal strains along thelength of the specimens indicatedthat in every case the desired con-dition was achieved.

The stirrups used had two legsand were made of No. 2 or No. 3bars as indicated on Figs. 4 and 5.The cross piece was welded to thetwo legs at the points of intersec-tion. It should be noted that thestirrups had hooks pointing alongthe axis of the girder at the bottomof each leg to ensure satisfactoryanchorage. The stirrups were spacedas shown in Fig. 6.Materials

The prestressing steel was seven-wire, stress-relieved strand of 1/4, 3/a,or 1/2 in. diameter, having the prop-erties indicated in Table 1. Exceptfor the strand used in specimensA3, A4, A7, and A8, all strand wasfree of rust, and was cleaned of sur-face oil before tensioning. The 1/4-in.diameter strand used for specimensA3, A4, A7, and A8 was purposelyrusted by placing it in a moist cur-ing room for 8 days prior to use.

The stirrups were either No. 2deformed bar with a yield point of49.9 ksi, or No. 3 deformed bar witha yield point of 44.4 ksi. The No. 3deformed bar conformed to the re-quirements of ASTM A305 for de-formations, and the No. 2 bar usedwas similarly deformed.

The concrete used in the fabrica-tion of the specimens was made withType III Portland cement and 3/4-in.maximum size aggregate. The slump

60 PCI Journal

19 1/2"25"

6

4

E

3"5 1/4 3 at13/4

3^ ^32 Strands 4 Strands

1 B5,3 1/2"5' I I

-8384,5"SikrupsBI, p3 bar82, !2 bar J Stirrups83, 113bar

8435, p 2 bar

2

18 Strands2 J2'I ]22 ^16 Strands

3/8"J7 at 13/4 2 3/8 2 3/8 7 at 3/4 217 83,4,&5BI&2 (3/8' Strand)

13/8" Strand)

3 at3/4" --6 at 11/4

^H 15/83 I/4

12 8 Strands

^ 2 IIB 11 ands,

} 3/8 " Spacini._.-- 88, 31/2 1 I _. B I1, 3I/2

86 & 7, 5" 8930, 5"J l Stirrups J I Stirrups

86, 13 ba i 89, 13 barB7&8, q 2 bar 810 811 12 bar

l rl2 Strands 26 Strands}1J /2"Sp.6r,

51/4"3/8"

2 II/16 4 at - 4 at, 2 11/16"3/4/4 4 5/886,7,483/8" Strand) 89,10, 81111/4" Strand)

Im

2'}4 Strands^ r

814, 3 1/2"

-812813, 5"J L

Stirrupsi B 12, p 3 BarB 13 & 14, p 2 b a r

6 Strands

6 1/2" 2" 2^ 6 1/2B 12, 13, & 14

( 1/2" Strand )

Fig. 4Cross-sections, Girders Bl Through B14.

October 1962

61

1/2 I I 11/23 Ja"}8 Strands^L 2'.} 4 Strands21/4"\

1 B 15, 2 1/2 1 1 B 18, 2 1/2"

9 I/2' B 16 & 17, 3 I/2 B 19 & 20, 3 1/2'22 1/2Stirrups I StirrupsB 19, N3 bar

J l616, !3 batlB 15 & 17, 12 bar BIB & 20, p 2 bar

24 Strands,L3^/2'

2'1rl2 Strands

^2J I Spacing 1 1/2^^

27/8l' 1 I" 2 13/162 13/16 1 3/8 2 7/8'^ 3 at 1 3/4

915,16,&17 BIB, 19, &20(3/8' Strand(1/4 Strand)

2 121

I*)2'5trand5

3/6} 10 Strands,

I Spacing

B 21, 2 1/2" - B 24, 2 1/2"

B 22 8233 3 1/2" B 25, 3 1/2"StirrupsB22, ObarB21 823, $ 2 bar

I Stirrups) B 24 & 25, r2 bar

21 Strands2r61IOSpacngs'

I IJ/22,

4 1/2 3 I/2 I 3 I/2'B 21,22,823(1/2' Strand)

B24&25(1/4' Strand)

Fig. 5Cross-sections, Girders B15 Through 825.

62 PCI Journal

TABLE 1PROPERTIES OF PRESTRESSING STRAND

Nominal Strand Cross-sectional Stress at 1% UltimateDiameter, in. Area, sq. in. Extension, ksi Strength, ksi

1/4 0.0356 251 2803/s 0.0799 259 286r/a 0.1438 231 254

TABLE 2CONCRETE STRENGTHS AT TRANSFER OF PRESTRESS

CylinderGirder Strength, Girder

Number f' psi' Number

Al 5050 B3A2 4145 B4A3 4625 B5A4 4145 B6A5 4900 B7A6 4560 B8A7 5095 B9A8 4560 B10A9 4520 B11A10 4520 B12B1 4580 B13.B2 4580 B14

Cylinder CylinderStrength, Girder Strength,f', psi Number f', psi

7090 B15 40307090 B16 37157190 B17 37154855 B18 44504855 B19 45754995 B20 45754290 B21 40904290 B22 49754475 B23 49754415 B24 46604415 B25 46604200 ..

Series B Girders.

October 1962 63

was 2 in. for Series A and 3 in. forSeries B. The specimens were moistcured under plastic sheets at 70Ffor the first three days after casting,and subsequently were stored at70F and 50% relative humidity untiltransfer of prestress at an age ofseven days. The concrete cylinderstrengths at transfer are listed inTable 2. These concrete strengthsare in each case the average of three6 by 12-in, cylinders taken frombatches of concrete placed in thewebs of the test specimens. Thecylinders were cured alongside thegirder specimens.Fabrication and Test Procedure

The specimens were fabricatedand tested in sets of from one tothree girders. They were producedin a prestressing bed set up on thelaboratory test floor(') with a cleardistance of up to 38 ft between an-chorage blocks. The strands weretensioned individually using a cen-ter-hole ram with a 20-in. stroke.The tension in the strand was mea-sured by a load cell placed betweenthe hydraulic ram and the tempo-rary anchorage used to grip thestrand during the tensioning opera-tion. The strands were over-ten-sioned by an amount sufficient tocompensate for the loss in prestressdue to the "draw in" of the perma-nent anchorages. The prestress re-maining after permanent anchoragewas measured for five strands, usingload cells placed between the per-manent anchorages and the anchor-age cross head. By these methodsof control the initial prestress waskept very close to the chosen valueof 175,000 psi.

The day after prestressing, thestirrups (when used) and form-workwere positioned, and the concretewas cast. The girders were moistcured for three days, and the formswere then stripped. Two days later

SR-4 electrical resistance straingages and mechanical strain gagepoints were mounted on the girdersas shown in Fig. 7, or in a similarpattern. Seven days after casting theprestress was transferred to thespecimens by torch-cutting thestrand.

Readings of all gages were takenbefore cutting of the strand wascommenced, together with readingsfrom the load cells behind thestrand anchorages. The strandswere cut in predetermined groups,initially adjacent to each of the an-chorage blocks, and subsequentlybetween the specimens. Readings ofall gages and load cells were takenafter the cutting of each group ofstrands. In this way the prestressforces applied to the specimens andthe strains produced were meas-ured for each stage in the transferprocess.

Test Results

Discussion of Series A ResultsThe distribution of strains in the

ends of the girders without stirrupswas similar for all specimens. Figs.8 and 9 show typical examples ofthe distribution of vertical tensionstrains over the depth of the web atthe end of the girder, and along ahorizontal plane through the cen-troidal axis of the section. It can beseen that the maximum verticaltension stress existed only in a re-gion near mid-depth of the web andthat it died away rapidly on movinginto the girder from the end face.In all cases in which cracking didnot occur the vertical tension strainsbecame zero at a distance from theend face not greater than about onethird of the girder depth, and inmost cases at a distance of about onequarter the depth. These resultswere thought to indicate that ifvertical stirrup reinforcement were

64 PCI Journal

Fig. 7Typical Distributionof Strain Gages.

6 Whittemore gage points'centerstop and bottom of 10

2I/2 x x x

50"

I 3 t 21/2' 5'1 1/4" in

I all cases

I 2, I I ICI I^ I^

lo'

2

x x

I

r+ A-l2 SR4 resistance stramn gage

Fig. 8Distribution of Vertical TensionStrains Over the Depth of the Web atthe End of a Typical A-Series Girder, atVarious Stages of Cutting the Strands.

0 20 40 60 00 100 12Tensile Strain, micro in./in.

12(

101

8

6

E

4

or

Fig. 9Distribution of Vertical Tension 2Strain at the Level of the Central Axis inthe End Zone of a Typical A-Series Gird-er, at Various Stages of Cutting theStrands. H

E

Stage 5

4

Stage 3

2 4 b 0

Distance from End Face of Girder, in.

October 1962

65

to be used to control cracking re-sulting from these vertical tensionstresses, then in order to be mosteffective it should be concentratedas close to the end face of the gird-er as is practicable.

The values of the maximum ver-tical tension stress for each testspecimen at various stages of trans-fer were calculated from the meas-ured maximum vertical tensionstrains, using for the modulus ofelasticity (60,000VT) psi. An attemptwas made to correlate these meas-ured stresses with the maximumstresses calculated using an adapta-tion of Sievers'(4 ) theory for the dis-tribution of stress in the end zonesof post-tensioned prestressed gird-ers. The theoretical and measuredstresses were in reasonable agree-ment for the girders having equalgroups of strand in both top andbottom flanges. However, for thegirders in which the majority of thestrands were in the bottom flangethe analytical approach grossly un-der-estimated the maximum stresses.It is thought that this may be dueto the fact that a horizontal shearof considerable magnitude will existat the level of the centroid of thesection for this arrangement of pre-

stressing strand, and that the influ-ence of such a shear on the maxi-mum vertical tensile stress in theend zone was not taken into ac-count in the development of theexpression for vertical tensile stress.To resolve this matter completelywould require a somewhat extensiveexperimental investigation. At thisstage of the present investigation,however, it was decided to directattention to the stresses in stirrupreinforcement after cracking, ratherthan to continue the experimentalstudy of the stresses existing beforecracking.

Discussion of Series B Results

Twenty five specimens with endstirrups were tested in this series;seven of these did not crack, and infour girders the cracks occurred re-mote from the location of the straingages on the stirrups. Useful infor-mation was therefore obtained from14 of the 25 specimens. The behaviorand relevant characteristics of thetest specimens are recorded inTable 3.

Similarity of behavior was ob-served in 14 of the 18 girders whichcracked. In these girders the crackoccurred in the lower part of the

TABLE 3BEHAVIOR OF SERIES B GIRDERS

Percent of Percent ofStrand Strand at Strand Strand atGirder Diameter, Top of Girder Diameter, Top ofNumber in. Section Behavior* Number in. Section Behavior*B1 3/8 10 N B14 i/Z 40 CB2 3/8 10 N B15 i/4 25 NB3 3/8 20 C B16 1/4 25 CB4 3/8 20 C B17 1/4 25 CB5 3/8 20 C B18 3/8 25 C+B6 3/8 40 C+ B19 3/8 25 C -B7 3/8 40 C B20 3/8 25 NB8 3/8 40 C B21 i 25 NB9 1/4 40 C B22 i 25 NB10 1/4 40 C B23 25 NB11 i/4 40 C B24 50 CB12 rz 40 C B25 14 50 C+B13 i/z 40 C-L. cracking, C-j- cracking remote from gages on stirrups, N = no cracking.66 PCI Journal

web close to the centroidal axis ofthe section, and resulted in a dis-tribution of tension strains in thevertical stirrup reinforcement similarto those shown in Fig. 10. In theother four girders which cracked,the crack occurred in the upperpart of the web remote from thestrain gages on the stirrups; the dis-tribution of stirrup strains for thesefour girders is therefore not known.In all cases the width of the crackat the end face of the girder was be-tween 0.001 and 0.004 in. To thenaked eye the cracks appeared totravel from two to four inches intothe web of the girder from the endface.

The total tension force in the ver-tical stirrup reinforcement in eachgirder that cracked near the cen-troidal axis was calculated from themeasured stirrup strains, and islisted in Table 4, together with themeasured effective prestress forceafter transfer. The effective pre-stress was taken as the prestressjust before transfer, as measured bythe load cells at the strand ancho-rages, less the loss due to the mea-

sured longitudinal shortening of theconcrete at transfer. Also listed inTable 4 are the measured maximumstirrup stress, the length of the hori-zontal crack, and the ratio of thedepth of the girder to the transferlength of the prestress strand.

Since the cracks were very fine itwas almost impossible to determinetheir exact length by visual obser-vation. It was. therefore decided todetermine their length by referenceto the measured vertical stirrupstrains. If a value of (7 .5V f) ,) psi isassumed for the limiting tensilestress of the concrete, then the strainat the end of the crack will be givenby:

e^,= (7.51/f^)/E^

_ (7.5Vf^)/(60,000 f;.)

= 125 micro in./in.The point at which the verticaltension strain became 125 microin./in. was therefore taken to be thepoint to which the crack extended.Examples of the use of this criterionare shown in Fig. 10. It can also

TABLE 4-EFFECTIVE PRESTRESS FORCE AND TOTAL VERTICAL. STIRRUP FORCE AT TRANSFER

GirderNumber

EffectivePrestressForce, T,

kips

TotalStirrupForce, S

kips

ST

Size ofStirrup

Bar

MaximumStirrupStress,

ksi

Lengthof

Crack,in.

h

1,

B3 246 5.17 .0210 3 11.32 3.6 1.28B4 246 3.57 .0145 2 13.12 5.0 1.28B5 246 3.28 .0134 2 12.80 4.5 1.28B7 246 3.99 .0162 2 13.90 5.4 1.28B8 246 3.28 .0133 2 12.30 4.2 1.28B9 261 9.00 .0344 3 18.27 4.3 1.76B10 261 5.26 .0202 2 22.50 4.9 1.76B11 261 5.36 .0205 2 23.20 4.9 1.76B13 214 4.04 .0189 3 9.00 2.6 0.97B13 214 2.20 .0103 2 10.80 3.2 0.97B14 214 1.82 .0085 2 9.00 2.7 0.97B16 186 4.71 .0253 3 11.30 2.9 1.63B17 186 2.61 .0140 2 13.50 3.2 1.63B24 116 1.86 .0161 2 8.10 3.3 1.63

*h = girder depth; I t = strand transfer length calculated assuming an average transfer bond stressof 400 psi(-).

October 1962 67

be seen in Fig. 10 that within thenominal crack length the stirrupstrains vary in a linear manner, butthat beyond the crack length thereis considerable departure from thislinear variation. This behavior wasobserved in all the girders whichcracked and is considered to sup-port the criteria used to determinethe crack length.

It can be seen in Table 4 andFig. 10 that for girders which areidentical except for the stirrup rein-forcement, a higher total stirrupforce was measured in the No. 3bar stirrups than in the No. 2bar stirrups. This is apparently dueto the fact that the heavier stirrupsrestrained the opening of the crackto a greater degree and hence hadto supply a greater force. Compar-ison of results for girders identicalexcept for the stirrup reinforcementshows that the maximum stress inthe No. 2 bar stirrups was approxi-mately 20 per cent in excess of themaximum stress in the No. 3 barstirrups, and also that the averagelength of crack in girders reinforcedwith No. 2 bar stirrups was about 20per cent more than in girders rein-forced with No. 3 bar stirrups. It ap-pears that an increase of 120 percent in the cross-section of stirrupsresulted in a decrease in the lengthand width of the horizontal cracksof only about 17 per cent.

Since the behavior of the girderswith No. 2 bar stirrups was consid-ered satisfactory with respect tocracking, it was decided to use theresults from the tests of these gird-ers as a basis for the derivation of aformula for the amount of stirrupreinforcement necessary for satisfac-tory end zone performance in prac-tice.

The analytical study showed that,for a given cross-section, the magni-

tude of the vertical tensile stressesbefore cracking is a function of thestrand transfer length and the pre-stress force. The set of girders B6through B14 were tested to deter-mine whether these same parame-ters also influenced the tension inthe stirrup reinforcement aftercracking. The results obtained in thetests of these girders are plotted inFig. 11. It can be seen that, for therange of transfer lengths covered bythese tests, the ratio of the totalstirrup force to the prestressing forcecould be taken as inversely propor-tional to the strand transfer lengthfor a given cross-section and dis-tribution of prestressing strand. Thisrelationship is confirmed by the be-havior of both the girders with No.2 bar stirrups and the girders withNo. 3 bar stirrups.

The transfer length is a measureof the abruptness with which theprestress is transferred from thestrand to the concrete. The shorterthe transfer length, the more abrupt-ly is the prestress transferred to theconcrete and the higher are thevertical tension stresses in the endzone. The analytical study indicatedthat for the more general case ofgirders of different size, the relativeabruptness of transfer of prestress,expressed as a function of both thetransfer length and the depth of thegirder, is a more correct parameterto consider than the transfer lengthalone.

The set of girders B1 through B8were tested with the intention ofdetermining the influence on thestirrup stresses of the way in whichthe strands were distributed acrossthe section. Girders B1 and B2 didnot crack. Examination of the stir-rup forces measured in girders B3through B8 did not reveal a sig-nificant increase in stirrup forcewhen the amount of strand in the

68 PCI Journal

Girder B9#3 bar stirrups

800

600

400

c200

0U

E

47

x

_ X

Nominal crack xlength

1Ec = 125x 10-6

in 800 Girder B 10

C x #2 bar stirrups

Cvv7

600U)C

400

200 Nominal cracklength

=125x10-6t i

0 2 4 6 8 10Distance from End Face, in.

Fig. 10Typical Variation of Stirrup Strain with Dis-tance from End Face of Girder.

October 1962 69

.0,

(J)IH

01

.02

`oU-

U,.01

UUZ.U9.Ub.va

Inverse of Strand Transfer Length, (

Fig. 11Variation of Stirrup Force with Strand TransferLength-Girders B7 Through B14.

top of the section was increasedfrom 20 to 40 per cent of the total.

Ignoring therefore, for the timebeing, the influence of the mannerof distribution of the strand, theforce in the stirrups is seen to bea function of prestress force, strandtransfer length, and depth of section.We may write:

S :: Toh hz(1)If F and L are the fundamental

units of force and length, then thecorresponding dimensional equationis:

[F] = [F] o [L] u [L] .....(2)which is dimensionally consistent forx = 1 and z = y. Now the testdata plotted in Fig. 11 demonstratethat the tensile force in the stirrupsis inversely proportional to the trans-fer length for the range of transferlengths considered, hence y = 1and therefore z = 1, and we maywrite:

S =KT h O

It............ 3

where K is an empirical constant.In order to determine the value

of K, the results of tests of girderscontaining No. 2 bar stirrups havebeen plotted in Fig. 12 in terms ofthe ratio of the total stirrup forceto the effective prestress force, S/T,and of the ratio of the depth of thegirder to the transfer length of theprestressing strand h/lt. It can beseen that the relationship betweenSIT and h/lt can be idealized asthe straight line

S/T = 0.0106 (h/lt) ...... (4)that is, the value of K is 0.0106.Using this equation, the averageratio of S/T(test) to S/T (talc) is1.00, and the standard deviation is0.119.

Verification of the form of Eq. (4)is obtained from a consideration ofthe behaviour of two hypotheticalgirders which are similar in all re-spects. Let all dimensions of the firstgirder be N times the correspondingdimensions of the second girder.

i.e. hj = N h2; D 1 = N D2 etc.70 PCI Journal

The number of strands and stirrupswill be the same in each girder.

The theory of structural similitudeshows that if the two girders aremade from materials having thesame stress-strain relationships, andif concentrated loads P l and P2 areapplied to the first and second gird-ers, respectively, such that

Pi = N2 P2then the stresses at correspondinglocations within the girders will beequal.

Now the diameter of the pre-stressing strand in the first girder,Dl, is N times the diameter of thestrand in the second girder, D2.Therefore, if the strand in bothgirders is tensioned to the samestress, it follows that the prestressforce in the first girder, T1, is N2times the prestress force in the sec-ond girder, T2, and hence the corre-sponding stresses set up in the stir-rups of both girders will be equal.However, since the diameter of thestirrups in the first girder is N times

.025

the diameter of those in the . secondgirder, the total stirrup force, S 1 , inthe first girder will be N2 times thetotal stirrup force, S 2, in the secondgirder. Hence,

Sl _N2 S2 _ S2T1N2 T2T2

This result is also achieved by useof Eq. (4) if we make the reason-able assumption that the transferlength, it, is given by a constant, K',times the strand diameter, D, for sim-ilarly manufactured strand. Then:

S1 0.0106 hl = 0.0106 N h2Tl K'ND2

=0.0106 h2' = T2

Hence Eq. (4) yields results forgirders of different size which arein agreement with those obtainedby use of the theory of similitude.This result is seen as giving supportto the form of Eq. (4) even thoughthe effect of varying the depth ofgirder, h, was not investigated ex-

"II

.020 $e

.015 :

.010 + T 0.0106( * ) + r

in

.005

0'0.8 1.0 1.2 1.4 1.6 1.8

2.0

Depth of Girder/Transfer Length, ( fit )Fig. 12Relationship of Stirrup Force and the Ratio ofGirder Depth to Strand Transfer Length.

October 1962 71

perimentally.A further check was made on the

possible influence on the stirrupforces of the manner of distributionof the strand. In Fig. 13 the product(S/T)(lt/h) is plotted against thepercentage of strand in the top ofthe section for all girders reinforcedwith No. 2 bar stirrups. Also plottedis the average value of the product,0.0106. The values of the productappear to be scattered in a fairlyrandom manner about the averagevalue and the diagram shows nosignificant trend. It appears there-fore that Eq. (4) can be used tocalculate the total stirrup force re-gardless of the percentage of pre-stressing strand present at the topof the girder end section.

If f8 is the maximum allowablestress in the stirrups, then the aver-age stress will be very nearly f8/2,since the stirrup stresses can be as-sumed to vary linearly from a maxi-mum close to the end face of thegirder to zero near the end of thecrack. The total cross-sectional areaof stirrups necessary, A t, will thenbe given by Eq. (4) as

A (5)t =(f8/2)= 0.021 f8h ......... The amount of stirrup reinforce-

ment calculated using Eq. (5) shouldbe distributed uniformly over alength equal to one fifth of thegirder depth, measured from theend face of the girder. For mostefficient crack control the first stir-rup should be placed as close to theend face of the girder as possible,since surface crack width is to someextent controlled by concrete cov-ers. It is suggested that for designpurposes It may be assumed to be50 times the strand diameter.

It should be noted that crackingdue to restraint of shrinkage andthermal contraction of the concreteby formwork can occur in end zoneswhich would otherwise not crackdue to vertical tension stresses setup at transfer of prestress. However,once the horizontal cracks have beeninitiated, from whatever cause, thelocal stresses due to anchorage ofthe strands will tend to open them,and stirrup reinforcement is thennecessary. Since the amount of re-

.015

.010+-

Average = 0.0106

.005

015

20 25 30 35 40 45

50 55Percent of Strand at Top of Section

Fig. 13Variation of Stirrup Force with Manner of Distribution of Strand Acrossthe Section.

72 PCI Journal

inforcement involved is relativelysmall, it is strongly recommendedthat suitably designed stirrup rein-forcement should be placed in theend zones of all pretensioned pre-stressed concrete girders, even whenthe maximum vertical tension stressoccurring at the time of transfer isrelatively low.

Range of Applicabilityof Design Equation

Equation (5) is based on a sim-plified linear relationship betweenthe ratio of the stirrup force to theprestressing force, (S/T), and theratio of the depth of section to thestrand transfer length, (h/lt). Thissimplified linear relationship hasbeen justified experimentally onlyfor values of (h/lt) of up to about 2.It is considered that as (h/lt) in-creases beyond 2 the design equa-tion will tend to become conserva-tive, the degree of conservatismincreasing as (h/lt) increases. Thisis apparent if the limiting case of apost-tensioned girder is considered.In this case It is zero and Eq. (5)would yield infinity . for the cross-sectional area of the stirrups. Thecurve representing the true relation-ship between (S/T) and (h/l t) couldtherefore be expected to becomeasymptotic to the value of (S/T)corresponding to the case of a post-tensioned girder.

It should further be noted thatthe design equation has been de-veloped from measurements madeon the end zones of girders in whichthe strand was divided into twogroups located respectively in thetop and bottom of the section. Ifthe groups were placed closer to-gether, or if the strand were dis-tributed uniformly over the end faceof the girder, then it appears likelythat the stirrup forces set up wouldbe somewhat less than those occur-

ring in the case considered in thisexperimental investigation. The de-sign Eq. (5) will therefore tend tobe conservative for girders in whichthe groups of strand are spacedmore closely, or in which the strandis distributed uniformly over the endof the grider.

Design ExampleTo illustrate the use of design Eq.

(5), the case will be considered ofa Type IV AASHO-PCI standardbridge girder prestressed by 48 half-inch diameter strands, 16 of whichare draped. Each strand will be as-sumed to be tensioned to 25.2 kips.Depth of Type IV section, h = 54 in.Transfer length of %-in, strand,

it =50x 1/z= 25 in.Total prestress force,

T = 48 x 25.2 x = 1209.6 kipsMaximum reinforcement working

stress, f8 = 20 ksiArea of stirrups,

At = 0.021 T It0.021

1209.6 x 5420x25

= 2.74 sq. in.Use three No. 6 bar two-legged

stirrups in the end 101/a in. of thegirder, which yields At = 2.64 sq. in.

Additional stirrup reinforcementshould be provided in the remainderof the end zone of the girder. De-termination of the amount of thisreinforcement depends on shearforce due to load and is outside thescope of this investigation.

Concluding RemarksAs a result of field observations ( )

and of the observation of the be-havior of laboratory specimens it isconcluded that end blocks are notnecessary for the satisfactory per-formance of the end zones of pre-

October 1962 73

tensioned prestressed concrete gird-ers. A relatively small amount ofvertical stirrup reinforcement placedclose to the end face will ensuresatisfactory performance of the endzone of a pretensioned prestressedconcrete girder; any horizontalcracks which occur will be fine andshort and will not affect the serviceperformance of the girder.

It is considered desirable thatvertical stirrup reinforcement beplaced near the ends of all girders.Restraint of shrinkage and thermalcontraction of the web concrete byformwork can lead to cracking.Cracks originated in this mannerwill tend to open under the actionof the prestress forces, and henceadequate vertical stirrup reinforce-ment should always be provided.

Acknowledgments

The investigation reported hereinwas carried out in the StructuralLaboratory of the Portland CementAssociation, and contributions weremade by several members of thelaboratory staff. Particular credit isdue B. W. Fullhart and 0. A. Kur-vits, Senior Technicians, for theircontributions to the testing and in-strumentation work involved.

References

1. Fountain, R. S., "A Field Inspection ofPrestressed Concrete Bridges," to bepublished by the Portland Cement As-sociation.

2.. Base, G. D., "An Investigation of Trans-mission Length in Pretensioned Con-crete," Cement and Concrete Associa-tion Research Report No. 5, London,August 1958, 29 pp.

3. Hognestad, E., Hanson, N. W., Kriz,L. B., and Kurvits, 0. A., "Facilitiesand Test Methods of PCA StructuralLaboratory," Journal of the PCA Re-search and Development Laboratories,1, No. 1, 12-20, 40-44 (January 1959),1, No. 2, 30-37 (May 1959), 1, No. 3,35-41 (September 1959); PCA Devel-opment Department Bulletin D33.

4. Sievers, H., "t ber den Spannungszus-tand in Bereich der Ankerplatten vonSpanngliedern in vogespannten Stahl-betonkonstructionen," (Stress Conditionin the Vicinity of Anchorage Plates ofPrestressed Tendons in Prestressed Con-crete Structural Units) Ber Bauingeni-eur, Vol. 31, No. 4, Berlin, 1956, pp.134-135.

5. Hanson, N. W., and Kaar, P. H.,"Flexural Bond Tests of PretensionedPrestressed Beams," PCA DevelopmentDepartment Bulletin D28, reprintedfrom the Journal of the American Con-crete Institute, January 1959, Proceed-ings Vol. 55, pp. 783-802.

6. Clark, A. P., "Cracking in ReinforcedConcrete Flexural Members," Journalof the American Concrete Institute,April 1956, Proceedings Vol. 52, pp.851-862.

74 PCI Journal