SHEAR TRANSFER INREINFORCED CONCRETE-RECENT RESEARCHAlan H. MattockUniversity of WashingtonSeattle, Washington

Neil M. HawkinsUniversity of WashingtonSeattle, Washington

Shows how concrete strength, shear planecharacteristics, reinforcement, and direct stressaffect the shear transfer strength of reinforcedconcrete. Fundamental behavior of test specimensunder load is reported, and hypotheses to explain thebehavior are developed. It is concluded thatshear-friction provisions of ACI 318-71 give aconservative estimate of shear-transfer strengthbelow the stated limit of 800 psi. A design equationto develop higher shear transfer strengthis presented.

Test program

Shear transfer across a definite planemust frequently be considered 'in thedesign of precast concrete connec-tions t1 f 2r . A continuing study of thefactors affecting shear transfer strengthis in progress at the University ofWashington. Factors so far included inthe study are as follows:

1. The characteristics of the shearplane

2. The characteristics of the rein-forcement

3. The concrete strength

4. Direct stresses acting parallel andtransverse to the shear plane.

The influence of the first three fac-tors has been studied in tests( 3 ) of mon-olithically cast "push-off" specimens asseen in Fig. 1(a). Tests( 45 ) to study theinfluence of direct stresses acting paral-lel and transverse to the shear planewere made on the "pull-off" and mod-ified push-off specimens shown in Figs.1(b) and 1(c) respectively. In all cases,the shear transfer reinforcementcrosses the shear plane at right anglesand is securely anchored so that it candevelop its yield strength in tension.

PCI Journal / March-April 1972 55

Type AҟType BҟTy C

Pҟ Shear Transfer PRollers

it Iҟ̀

I I

II

-- —i— — --- Iҟ IҟACCICI

ShearPlane

SteelBracket

PҟPҟP(a)ҟ (b)ҟ (c)

Fig. 1. Shear transfer test specimens: (a) push-of}; (b) pull -off; (c) modified push-off

Additional reinforcement is providedaway from the shear plane, to preventfailures other than along the shearplane. The length and width of theshear planes were 10 x 5 in., 12 x 43/a

in., and 12 x 6 in. (approx. 25 x 13 cm,30x12 cm, and 30x15 cm) in thepush-off, pull-off and modified push-offspecimens respectively. When loadedconcentrically by a force P. the shearalong the shear plane is equal to P inthe push-off and pull-off specimens. Inthe modified push-off specimens, theconcentric force P produces a shearforce P cos 0 along the shear plane anda compressive normal force P sin 0across the shear plane. Six differentvalues of 0 were used to give differentratios of shear stress to transverse nor-

mal stress. The test program is sum-marized in Table 1.

The specimens were subjected tomonotonic loading to failure. In allcases, slip along the shear plane wasmeasured, and in some instances thelateral separation at the shear plane wasalso measured. Cracks were marked onthe faces of the specimens as they de-veloped. Detailed data for Series 1 to 6have already been published( 3 ). Thedata for Series 7 to 10 are summarizedin Tables 2 and 3. For convenience, theultimate shear strengths are expressedas average shear stresses vu, obtainedby dividing the ultimate shear forceV2, by the area of the shear plane bd(d is the length of the shear plane andb its width).

56

Table 1. Test program

Testseries Description

Specimentype

Numberof tests

1 Push-offҟtestsҟofҟinitiallyҟuncracked A 13specimens.. Reinforcementҟsizeҟcon-stant,ҟspacingҟvaries.ҟf°4000ҟpsi,f50 ksi.

2 Push -off tests of initially cracked spec- A 6imens.ҟReinforcementҟsizeҟconstant,spacing varies. f,ҟ4000 psi, f, o50 ksi.

3 Push -off tests of initially cracked spec- A 5imens.ҟReinforcementҟsizeҟvaries,spacing constant. f',,4000 psi, f;50ksi.

4 Push-off testsҟofҟinitiallyҟcracked A 5specimens. Higher strength reinforce-ment, f,,.-; 66 ksi.ҟReinforcement sizeconstant, spacing varies. f4000 psi.

5 Push-offҟtestsҟofҟinitiallyҟcracked A 5specimens.ҟLowҟstrengthҟconcrete,f2500 psi. Reinforcement size con-stant, spacing varies. f50 ksi.

6 Push -off tests of both initially cracked A 4and uncracked specimens. Dowel ac-tion destroyed by short rubber sleeveson reinforcement across shear plane.f'4000 psi, fy,50 ksi.

7 Pull-off testsҟofҟinitiallyҟuncracked B 6specimens.ҟReinforcementҟsizeҟandspacing varies. f,5000 psi, f 9 ,50 ksi.

8 Pull-off testsҟofҟinitiallyҟcracked B 6specimens.ҟReinforcementҟsizeҟandspacingҟvaries.ҟf,,5000ҟpsi,ҟf=50ksi.

9 Modifiedҟpush-offҟtestsҟofҟinitially C 6uncracked specimens.ҟReinforcementsizeҟconstant,ҟspacingҟvaries.ҟAngle9 varies (0, 15°, 30°, 45 0 ). f5500 psi,f=52 ksi.

10 Modifiedҟpush-offҟtestsҟofҟinitially C 10cracked specimens. Reinforcement sizeconstant, spacing varies. Angle 0 var-ies (0, 15 0 , 30 0 , 45°, 60 0 , 75°). f'4000and 6000 psi, f=52 ksi.

PCI Journal / March-April 1972 57

Table 2. Test data, Series 7 and 8

Specimennumber*

Reinforcementbar size

Number ofstirrups

(2 legs each)

Reinforcementyield point,

f9ksi

Concretestrength,

fpsi

pf,-,psi

v.,psi

7.1 #3 2 49.5 4850 384 8517.2 #3 3 49.5 5120 576 9087.3 #3 4 49.5 5050 768 9747.4 #2 2 56.0 5410 193 5677.5 #2 3 56.0 5070 289 6097.6 #2 5 56.0 5100 481 846

8.1 #3 2 49.5 4850 384 6978.2 #3 3 49.5 5120 576 8888.3 #3 4 49.5 5050 768 9258.4 #2 2 56.0 5410 193 5218.5 #2 3 56.0 5070 289 5728.6 #2 5 56.0 5100 481 746

*Specimens of Series 7 were initially uncracked; specimens of Series 8 werecracked along the shear plane before test.

Characteristics of the shear plane.Mast( 2 ) pointed out the need to consid-er the case where a crack may existalong the shear plane before shear isapplied. Such cracks occur for a varietyof reasons unrelated to shear, such astension forces caused by restrainedshrinkage or temperature deformationsor accidental dropping of a member.Certain shear transfer test specimenswere therefore cracked along the shearplane by the application of transverseline loads, before application of shearloading.

Slip was measurable from the begin-ning of the shear test for the initiallycracked specimens. However, no move-ment occurred in the initially uncrack-ed Type A specimens until diagonaltension cracks became visible at shearstresses of from 500 to 700 psi (35-49kgf/cm2). These cracks crossed theshear plane at an angle of from 40 to50 deg. They were each about 2 in.(5 cm) long, spaced 1 to 2 in. (2% to 5cm) apart along the shear plane. After

these cracks formed, there was a rela-tive longitudinal movement of the twohalves of the initially uncracked speci-mens. This was due to rotation of theshort concrete struts formed by the di-agonal tension cracks, when the sheartransfer reinforcement stretched. It wasfound that if a crack exists in the shearplane before the application of shear,then the slip at all stages of loading isgreater than when such a crack doesnot exist.

A crack in the shear plane reducesthe ultimate shear strength of under-reinforced specimens (Fig. 2). The de-crease is greater in the push-off speci-mens than in the pull-off specimens.The shear strength of the initiallycracked specimens is not directly pro-portional to the amount of reinforce-ment. Because of the observed weak-ening effect of a crack in the shearplane, most of the subsequent testswere made on initially cracked speci-mens, in order to obtain lower boundvalues of shear strength.

58

ti Table 3. Test data, Series 9 and 10

Specimennumber"

Angle 0,deg.

Number ofbars"'

Reinforcementyield

point, f,ksi

Concretestrength,

f,psi

pf,psi

o-,psi

pf3, +o- ,psi

v,psi

Failuretype"'

9.1 45 10 52.4 5500 800 2460 3260 2460 S9.2 30 12 52.2 5500 956 1480 2436 2560 S9.3 15 12 52.3 3940 976 406 1382 1515 S9.4 0 12 53.7 3940 985 0 985 1389 S9.5 30 8 51.0 6440 623 1655 2278 2870 S9.6 30 4 51.0 6440 312 1600 1912 2770 S

10.1 75 6 51.8 3450 475 3220 3695 862 C10.2 75 6 52.0 4390 476 3920 4396 1049 C10.3 60 8 51.8 3450 632 2780 3412 1610 C10.4 60 8 53.0 4390 648 3060 3708 1770 C10.5 45 10 52.7 4630 805 2265 3070 2265 S10.6 30 12 52.0 4630 954 1250 2204 2165 S10.7 15 12 52.4 4020 962 387 1349 1445 S10.8 0 12 53.7 4020 985 0 985 1115 S10.9 30 8 51.0 5800 623 1490 2113 2590 S10.10 30 4 51.0 5800 312 813 1125 1410 S

1. Specimens of Series 9 were initially uncracked; specimens of Series 10 were cracked along the shear plane before test.2. All reinforcing bars were No. 3's arranged in pairs crossing the shear plane.3. S = shear; C = compression.

ui

00 PUSH-OFF TESTSҟo.0-0-+- PULL-OFF TESTS

Series Iҟoҟ+ i0o Uncracked

/ҟSeries 7gooҟoҟd?ҟ+ҟ

Uncracked ҟi//0./ /ҟ o

300 ° / /ҟSeries 2/ҟi+ҟInitially crackedҟo

sooҟ+' /ҟin shear plane

tooҟ ,/Series 8/ҟInitially cracked

f^ 4000 pslWO j, fy - 50 ksiҟ

in shear plane

0 200 400 600 800 1000 1200 0 200 400 600 800

ҟ

pfy (psi)

Fig. 2. Variation of shear transfer strength reinforcement parameter pf y, with andwithout an initial crack along the shear plane

IҟIҟIҟIҟIҟIҟiPUSH-OFF TESTS

l00 ° -

Series 3ҟ °00 Bar size varies,ҟ+

spacing constant>ooҟ + 0 -

°ҟSeries 2300

-Bar size constant,

300 0 /ҟspacing variesҟ-

Foo /

ҟ

ҟ -f^-4000 psi, fy^50ksi

!00 -1ҟ Specimens initially cracked -

0 200 400 600 800 1000 1200 1400

p fy (psi)

Fig. 3. Effect of stirrup bar size and spacing on the sheartransfer strength of initially cracked push-off specimens

vu

(psi)

vu

(psi)

60

Characteristics of the reinforcement.The reinforcement parameter pf, , canbe changed by varying either the rein-forcement ratio p, the reinforcementyield strength f, or both. Also, for agiven shear plane the reinforcement ra-tio can be changed by changing thebar size and/or the bar spacing. InFig. 3, the results of tests of Series 2and 3 are compared to determinewhether the way in which the rein-forcement ratio is changed has any ef-fect on the relationship between ulti-mate shear strength and the reinforce-ment parameter pf,. In Series 2, p waschanged by varying the stirrup spacing,the bar size (No. 3) (9.5 mm) beingconstant. In Series 3, P was changed byvarying the bar size between '/s in.diam. and No. 5 (3.2 and 15.9 mm)while maintaining a constant spacingof 5 in. (12.7 cm). Fig. 3 shows that the

way in which p is changed does notaffect the relationship between shearstrength and the reinforcement para-meter pf,

In the tests so far discussed the sheartransfer reinforcement had a yieldstrength of about 50 ksi (3500 kgf/cm2). A group of push-off specimenswas therefore tested in which the rein-forcement had a yield strength of 66ksi (4640 kgf/cm2), to check whetherthe relationship between v 2 and pfywas independent of fy, and also to checkwhether it was possible to develop theyield strength of this higher strengthsteel. It was found that for given val-ues of pf, the specimens with 66 ksisteel had slightly higher shear strengthsthan the specimens reinforced with the50 ksi steel. This appears to indicatethat at ultimate strength the higherstrength steel stirrups developed a stress

PUSH-OFF TESTSI400-0

O

I2o0ҟSeries 2fc''- 4000 psi

1000ҟ 0ҟ+'

Vu800F^

(psi)

600F_0/ Series 5, f2500psi

fy = 50 ksi

Specimens initially cracked

0 200 400 600 800 1000 1200 1400

p fyҟ(psi)

Fig. 4. Effect of concrete strength on the shear transferstrength of initially cracked push -off specimens

PCI Journal / March-April 1972 61

greater than their yield point, i.e., strainhardening had occurred. This is quitepossible, as the yield plateau of thehigher strength reinforcement was con-siderably shorter than that of the in-termediate grade reinforcement. Ittherefore appears conservative to as-sume that the relationship between pf,,

and v,4 is the same for higher strengthreinforcement as for intermediate gradereinforcement, provided the yieldstrength does not exceed 66 ksi.

Concrete strength. The effect of varia-tion in concrete strength on the shearstrength of initially cracked push-offspecimens is illustrated in Fig. 4. Thespecimens of Series 2 and 5 were iden-tical in all respects except for concretestrength, Series 2 having 4000 psi (281kgf/cm2) concrete and Series 5 hav-ing 2500 psi (176 kgf/cm2) concrete.For values of pf, below about 600 psi(42 kgf/cm2) the concrete strengthdoes not appear to affect the sheartransfer strength. For higher values of

pfy the shear strength is lower for thelower strength concrete. The concretestrength therefore appears to set anupper limit value of pf., below whichthe relationship between v,, and pf„ es-tablished for 4000 psi concrete wouldhold for any strength of concrete equalto or greater than the strength beingconsidered, and above which the shearstrength increases at a lesser rate forthe concrete strength being considered.This change in behavior is discussedlater.

Direct stress parallel to the shear plane.In an earlier report( 3 ) a method wasproposed for the calculation of theshear transfer strength of initially un-cracked concrete. This was based onthe average shear and normal stressesacting on a concrete element in theshear plane, and made use of the fail-ure envelope for concrete proposed byZia( 6 ). This approach predicted the re-lationship between v,, and pf, , very

closely for the tests of initially un-cracked push-off specimens reportedhere, and also for tests of larger initial-ly uncracked composite push-off speci-mens reported by Anderson< 7 ). In thepush-off test, direct compressive stressesexist parallel to the shear plane, andthese were taken into account in thecalculation.

Using this method of calculation, ananalytical study was made of the influ-ence on shear transfer strength of directstress parallel to the shear plane. Fromthese calculations it appeared that if adirect tension stress existed parallel tothe shear plane, then the shear transferstrength would increase more slowly,as pfy was increased, than in the push-off test where a direct compressivestress exists parallel to the shear plane.This conclusion was disturbing from thedesigner's point of view, since in manypractical situations there is a direct ten-sile stress parallel to the shear plane. Itwas therefore decided to study thisproblem with pull-off tests using speci-mens of the type shown in Fig. 1(b).The shear is applied to the shear planeby a concentric tension force acting onthe specimen through steel bracketsbolted to longitudinal reinforcing barsembedded in the specimen on eitherside of the shear plane. In these speci-mens a direct tension stress exists paral-lel to the shear plane, the average in-tensity of which is about half the inten-sity of the applied shear stress. In thepush-off specimens tested previously, adirect compressive stress existed paral-lel to the shear plane, the average in-tensity of which was equal to that ofthe applied shear stress.

The ultimate shear strengths of thepull-off and push-off specimens arecompared in Fig. 5. For initially un-cracked specimens, the pull-off testsgave lower shear strengths than thepush-off tests, indicating that a directtension stress parallel to the shear plane

62

I UNCRACKEDҟINITIALLY CRACKED

400!-

200

1000

vu800

(psi)

600

400

200

OPush - off tests,Series

o .

'Pull-off tests,Series •

Push-off tests,Series 2 & 3 ^"o o _

0 +Z+0'

-+o PuII-off. tests,Series 8ҟ-

0

0ҟ200 400 600 800ҟ0ҟ200 400 600 800 1000

p fy (psi)

Fig. 5. Effect on shear transfer strength of direct stress acting parallel to the shearplane

is detrimental to shear transfer strengthin initially untracked concrete. How-ever, the reduction in shear strength ap-pears to be due to a reduction in thecohesion contribution of the concrete,and the rate of increase in v,, with in-crease in pfy is approximately the samein both the pull-off and push-off tests.This indicates that the method of calcu-lation proposed earlier( 3 ) is faulty andcannot be extrapolated to the case ofthe pull-off test.

For specimens cracked along theshear plane before being loaded inshear, the shear strengths of the push-off and the pull-off specmiens are es-sentially the same for any given valueof p f y . This is important practically,since it indicates that direct stresses

parallel to the shear plane may be ig-nored in design for shear transfer, if thedesign is based on the relationship be-tween v,, and p f, obtained in tests ofinitially cracked specimens.

Direct stress transverse to the shearplane. The effect of compressivestresses acting transverse to the shearplane was studied in Series 9 and 10.Modified push-off specimens were used,as shown in Fig. 1(c). The depth ofthe block-outs in the specimens wasadjusted so that the length of the shearplane joining their ends remained con-stant as the angle 0 varied. A system ofrollers on the top of the specimen per-mitted separations to develop, even forrelatively large applied loads. The spec-

PCI Journal / March-April 1972 63

Series 10Initially cracked,

2000ҟ /Modified push-off tests

^lҟ xConcrete failureҟXҟx •envelopeҟ̂•^"ҟ̂̀ -

1500ҟ Series 9vҟ ^•-O° ;^+ Uncracked,°ҟ̂' ° ҟModified push-off tests

(psi)ҟ/ҟ.Foo ^X

1000/ d'ҟ̂•X+

' ° ''+°ҟ,ҟUntracked,+ҟPush-off tests

,ҟInitially cracked,500

Push-off testsҟ f° 4000 psify = 50 ksi

500ҟ1000ҟ1500ҟ2000ҟ2500ҟ3000

Normal stressҟ(arNX + p fy)ҟ(psi)

Fig. 6. Effect on shear transfer strength of direct stress acting transverse to theshear plane

imens of Series 10 were initially crack-ed along the shear plane, while thoseof Series 9 were initially uncracked.

Failures were characterized by ashearing action along the shear planewhen angle 6 was 45 deg. or less, andby a crushing failure across the planefor 6 of 60 or 75 deg. The deformationsof the initially uncracked specimenswere extremely small until diagonaltension cracks developed across theshear plane at about 60 to 70 percent ofthe ultimate strength. As in the push-off specimens, these cracks formed at anangle of about 45 deg. to the shearplane. They were about 2 in. (5 cm)long, and between 1 and 2 in. (2 1/z to5 cm) apart. In specimens with angle

0 of 30 deg. or less, failure occurredwith a continuous crack propagatingthrough the diagonal tension cracks,along the shear plane. Deformations de-veloped rapidly after diagonal tensioncracking, at a rate which increased con-tinuously with increasing load, but de-creased as 6 increased. The slips atfailure were in excess of 0.03 in. (0.76mm) and the separations were largeenough to indicate yielding of the rein-forcement when 6 was 30 deg. or less.For the specimens with 0 equal to 45deg., separations did not develop rap-idly until immediately prior to failure.For the specimens with 0 of 30 deg.and having differing values of pf9, the

64

load-slip relationships were not influ-enced by the value of pfy until immedi-ately prior to failure,

Significant deformations of the pre-cracked specimens occurred from thecommencement of loading. The initialstiffnesses were almost identical for 0ranging from 45 to 75 deg. When 0 wasbetween 0 and 45 deg., the initial stiff-ness increased with both 8 and the val-ue of pf . When shearing failures oc-curred, the ultimate slips were similarto those observed in initially uncrackedspecimens. Separations began to devel-op rapidly at three-quarters of the ulti-mate load, for 0 between 0 and 30 deg.For. 0 equal to 45 deg., separations didnot develop until immediately prior tocollapse, while for angles 0 of 60 and75 deg. only contractions occurred.Separations at ultimate were as largeas 0.06 in. (1.52 mm).

The ultimate shear strengths of themodified push-off specimens which hadshearing type failures are compared inFig. 6 with results from the push-offtests of Series 1, 2 and 3. In this figurethe data from Series 9 and 10 are nor-malized to a concrete strength f of4100 psi (288 kgf/em2), the averageconcrete strength of the specimens inSeries 1 and 2. The values of appliednormal stress o-Nx and of v,., were mul-tiplied by the ratio 4100/f. . The totalnormal compressive stress across theshear plane is assumed to be equal too + pf,. Also shown in Fig.' 6 is afailure envelope for concrete with acylinder strength of 4100 psi. The in-trinsic shape of this failure envelopewas obtained from biaxial tests of con-crete reported by Kupfer, Hilsdorf andRusch( 8 ). The assumption that ONx.

may be added to pf, when estimating

vu can be seen to be conservative forall values of °N• Furthermore, undercertain conditions, the shear strengthcan be as large as the intrinsic strengthof the concrete. This occurred when

ON + pf was greater than 0.3 f f, andthe ratio of ONw to pfy was simultane-ously greater than 1.3. (An initiallycracked specimen having ON5 /pfy

equal to 2.6, but with ON, + pf v ofonly 0.2f ' developed a strength almostidentical with that of a simple push-offspecimen having pfy equal to 0.2f.)

Further investigations are needed todefine completely the effect on sheartransfer strength of the ratio of o- Nx topf, of direct tensile stresses actingtransverse to the shear plane, and ofapplying the shearing force after thedirect stress has been increased to itsmaximum value.

Hypotheses for behavior

Shear transfer behavior of initially un-cracked concrete with reinforcementnormal to the shear plane. Externalloads are assumed to cause a shearstress v along the shear plane and di-rect stresses 0Ny and o-Na parallel toand normal to the shear plane, respec-tively. As loading begins the concrete isuncracked; the transverse reinforce-ment is unstressed and thereforedoes not contribute an additional directstress across the shear plane.

Several short diagonal tension crackswill occur along the length of the shearplane and inclined to it at an angle awhen, under increasing shear, the' printcipal tensile stress in the concrete be-comes equal to the tensile strength ofthe concrete. The angle a will dependupon the particular combination of v;o-N and ^^„ existing at the time ofcracking. In push-off tests without,addi-tional externally applied direct stress

a is usually about 45 deg..When the shear load is further in-

creased a truss action develops, asshown in Fig. 7(a). Diagonal struts ofconcrete are formed by the y short, paral-lel diagonal tension cracks. When'4theshear acts on the truss, the struts tend to

PCI Journal / March-April 1972 65

Shear Plane

Appliedҟa ,shearҟ--V

N T IҟI T 'NCҟV

ExternallyҟI V'applied forceҟ,`y

VҟI C

N TI IT N

ReinforcementҟVforce

aҟC - CompressionAppliedҟin strutshear, V

V' - Shear(b)ҟT + N in strut

a x^

oy,xly'

x

Diagonalҟ / ^\tension cracksҟ y

(a)ҟ (c)

Fig. 7. Shear transfer in initially untracked concrete

rotate and so stress the transverse rein-forcement. Because the diagonal strutsare continuous with the concrete onboth sides of the shear plane, there willbe both compression and transverseshear in the strut. The applied shear istherefore resisted by the components ofthe strut compression and shear forcesacting parallel to the shear plane, asshown in Fig. 7(b).

The reinforcement crossing the shearplane will eventually develop its yieldstrength A„ffy, provided a failure ofthe concrete does not occur first. Fail-ure will finally occur when the con-crete struts fail under the combinedaction of compression and shear in thestruts, while the reinforcement contin-ues to develop its yield strength.

Consider an element of concretelying in the shear plane, at the middleof the thickness of a strut. With refer-ence to coordinates x' and y', thestresses acting on the element will beas shown in Fig. 7(c). They comprisea campression a-y acting parallel to thedirection of the diagonal tension cracks,and shear stresses r'5' oriented asshown. Because the faces of the strutformed by the diagonal tension cracksare unloaded free surfaces, o-x is zero.The pairs of values of a' and r,,, atfailure of the concrete can be obtainedfrom the failure envelope for the con-crete using the geometrical construc-tion shown in Fig. 8. A succession ofMohr circles is drawn tangent to thefailure envelope. The intersection of

66

any particular circle and the T axiswill define the point (o-', r''), sincea-„ is zero. The diametrically oppositepoint on the circle must therefore bethe point (o-v ., T''), where a,' andT y are a pair of stresses correspondingto failure of the concrete.

The state of stress in the element onthe shear plane can also be expressedas o-, o- and rxy with respect to theaxes x and y, normal and parallel to theshear plane, respectively. These stressescan be stated in terms of and TX yas follows:

(TX = cry,' sin2 a — 2Tx sin a cos a (1)

o-y = cry cos2 a + 2T' sin a cos a (2)

T ^.71 = —^ - sin a cos a+ r '!, (cos t a — sine a) (3)

If a =45 deg., then

o-y, (1a)^= 2 —Txv

a'v = 2 + Tx (2a)

— 0-̂ (3a)Tye -- 2

Since pairs of values of cry, and T5

corresponding to failure of the concretecan be obtained as shown in Fig. 8, itis possible to calculate values of max.,a-y and Txy which correspond to failureof the concrete.

Now at failure, o- is the direct stressacting across the shear plane as a re-sult of the shear transfer reinforcementbeing stressed to yield, plus any ex-ternally applied direct stress (TN;ü act-

Concrete failure envelope

Combinations of oy.and txlylcorresponding to failure.

o-

Fig. 8. Derivation of combinations of may, and Tx,, which cause failureof the concrete

PCI Journal / March-April 1972 67

r..

150

Vu

(psi)

100

Calculated; K = 1.0 , a 45°

/' ••ҟCalculated; K=0.84,x.=45°i

/•ҟTest data, Series I,

50 Initially uncracked

0 500 1000 1500 2000

pfy(psi)

Fig. 9. Comparison of calculated and test shear transfer strengths ofinitially uncracked push-off specimens

ing across the shear plane at failure,i.e.

Avd+ a-N.x — pfv + O N.x (4)^^— bd

Where A„f is the total cross-sectionalarea of shear transfer reinforcement,and f its yield strength. Also, -r, is theshear stress in the shear plane, at thecenter of a strut. We may then write,

v (5)vu — bd — KTxy

Hence if a=45 deg., then vu =—Ko-y /2 and p f + o- = (o /2 r'5 ') . The value of the coefficient Kwould be 1.0 if the shear stresses wereuniformly distributed across the strut,and could be as low as 0.67 if the shearstress distribution across the strut wereparabolic. When the external normal

stress o- is zero or small, the con-crete struts must rotate slightly in orderto strain the transverse reinforcement.This causes the ends of the diagonaltension cracks to propagate parallel tothe shear plane for a short distance.As a result, the shear stress in the strutis increased locally to a value higherthan the average shear stress v 14 basedon the total area of the shear plane.Failure then tends to shift from theshear plane •to a parallel plane con-taining the ends of the cracks, and oc-curs when the locally higher shearstress reaches a critical value. The di-agonal struts at each end of the shearplane may be incomplete, dependingon the exact location of the diagonaltension cracks. In an extreme case thiscould reduce the total effective cross-section resisting shear by an amount

68

equal to the width of the shear planemultiplied by the projected length onthe shear plane of a single diagonaltension crack. Both of the foregoingtypes of behavior can be expected toresult in K becoming less than 1.0.

Using Eqs. (1) and (3) it is thereforepossible to calculate pairs of values of(p f, + o-) and v2, corresponding toshear transfer failure, provided somevalue can be assigned to the coeffi-cient K. The actual distribution of shearstress across each strut will probablybe intermediate between the uniformand parabolic distributions. In Fig. 9the calculated relationship between v,and pf,,, corresponding to an averagevalue of 0.84 for K and the assumptionthat angle a is 45 deg., is seen to be inreasonably close agreement with resultsobtained in push-off tests of initiallyuntracked specimens in which w, waszero. Also shown are the relationshipbetween v,, and pf0 corresponding to avalue of 1.0 for K, and the failure en-velope used in the calculations. Theintrinsic shape of the failure envelopewas obtained from biaxial tests of con-crete reported by Kupfer, Hilsdorf andRusch( 8 ). Use was made of data re-ported for a concrete having a ratio oftensile to compressive strength corre-sponding to that of the concrete usedin the push-off specimens, (f1/f' —1/12.33).

The results of the modified push-offtests in which the ratio of o- ,x to p f„ isgreater than 1.3 agree reasonably wellwith calculations assuming K = 1.0. Inthis case the ends of the cracks do notpropagate parallel to the shear plane,and therefore the total cross section ofthe shear plane acts to resist the ap-plied shear.

In the pull-off specimens a directtensile stress existed parallel to theshear plane at the time of diagonal ten-sion cracking. The intensity of the di-rect tensile stress was half the intensity

of the shear stress. The diagonal tensioncrack angle a which corresponds to thisis 52 deg. This increase in the crackangle from 45 to 52 deg, would lead toa reduction in calculated strength ofabout 10 percent, other characteristicsremaining the same. The actual reduc-tion found in the pull-off tests wasgreater than this, indicating that the Kvalue is less in the pull-off specimenthan in the push-off specimen. The ex-tension of the ends of the diagonal ten-sion cracks parallel to the shear planewas more marked in the pull-off teststhan in the push-off tests. For a givenapplied shear, this would increase thelocal intensity of shear stress in thepull-off specimen. This increase wouldresult in failure at a lower averageshear stress, corresponding to a reduc-tion in the value of K.

Shear transfer behavior of initiallycracked concrete with reinforcementnormal to the shear plane. When aninitially cracked specimen is loaded inshear, slip will occur along the shearplane. The faces of the crack are roughand hence when slip occurs, the crackfaces are forced to separate. This sep-aration causes tension strains in the re-inforcement crossing the shear plane.The tension force so induced in the re-inforcement is balanced by an equalcompression force acting across thecrack. This compression force producesa frictional resistance to sliding be-tween the faces of the crack, thus op-posing the applied shear. The relativemovement of the concrete on oppositesides of the crack also subjects the indi-vidual reinforcing bars to a shearingaction. The resistance of the bars tothis shearing action, sometimes referredto as dowel action, also contributes tothe shearing resistance.

In an under-reinforced shear plane,the separation of the crack faces iseventually sufficient to strain the rein-

PCI Journal / March-April 1972 69

forcement to its yield point. At ulti-mate strength therefore, the compres-sion force across the crack is equal tothe yield strength of the reinforcementA,ffy. The frictional resistance to shearalong the crack is then equal to thisforce multiplied by the coefficient offriction for concrete. In addition to thefrictional resistance to shear, there isalso shear resistance due to the dowelaction of the reinforcement crossing thecrack in the shear plane, and the re-sistance to shearing off of asperitiesprojecting from the faces of the crack.It is hypothesized that the frictionalresistance to sliding and the reinforce-ment dowel effect are the principalcontributors to shear resistance. Thisview is supported by the fact that forvalues of pf greater than 200 psi (14kgf/cm2), the slope of the curve relat-ing v,, and pfy is equal to the coeffi-cient of friction between formed con-crete surfaces measured by Gaston andKriz ( 9 ) . Further, when the dowel actionwas destroyed in two initially crackedpush-off specimens (Series 6), the shearstrength dropped almost to that whichcould be provided by friction alone. Inthese specimens the reinforcement be-came kinked at ultimate, and hence acomponent of the reinforcement forceacted along the shear plane. It isthought that the excess strength ofthese specimens above the frictionalresistance was due to this kinking ef-fect.

The concrete strength does not ap-pear to affect the shear transfer strengthof an initially cracked under-reinforcedspecimen. This is consistent with theshear strength being primarily devel-oped by friction, since the coefficient offriction is independent of the concretestrength. The behavior hypothesis alsoexplains why the shear transfer strengthof initially cracked pull-off and push-off specimens are the same for the samevalue of the reinforcement parameter

p f ^. Direct stresses parallel to the shearplane will not affect either the friction-al resistance to sliding along the shearplane, or the dowel effect. Hence achange in this longitudinal direct stressfrom tension to compression does notaffect the shear transfer strength in thiscase.

In a heavily reinforced shear plane,or one subject to a substantial exter-nally applied normal compressive stress,it is possible for the theoretical shearresistance due to friction and dowel ef-fects to become greater than the shearwhich would cause failure in an initial-ly uncracked specimen having the samephysical characteristics. In such a case,the crack in the shear plane "locks up"and the behavior and ultimate strengththen become the same as for an initial-ly uncracked specimen. When this oc-curs, the shear strength becomes de-pendent upon the concrete strength,whereas before it was independent.This change in behavior corresponds tothe change in slope of the v,u/p f curvefor 2500 psi (176 kgf/cm2) concretein Fig. 4. In Fig. 2 it can also be seenthat at the highest values of pfy, thestrengths of both initially cracked andinitially uncracked specimens are thesame.

In a moderately to heavily reinforcedshear plane, diagonal tension cracksmay form at angle a to the shear plane,but failure still occurs by sliding alongthe crack in the shear plane at an ulti-mate shear strength less than that of thecorresponding initially uncracked speci-men.

Shear transfer in design

Section 11.15 of the ACI BuildingCode, ACI 31871( 0 ), allows designfor shear transfer to be based on the"shear-friction" hypothesis proposed byBirkeland(') and Mast( 2 ). This is a sim-plification for design purposes of the

70

1400

1200

1000

Vu800

(psi)600

400

200

Specimens initially crackedPush—off {O —ff = 4000psiҟO

• —f'e = 2500psi ҟOPull — off A — f = 5100psiҟO

0 O/

O 0/0ҟ•/o v„ = pfyr( p00 +0.5)

• •-D

p V„ = 800psiO

oppQҟLimit for ff = 2500psi, (0.2fc)

SHEAR FRICTION , p = 1.4

0ҟ _ AVf fyv°ҟb d N = pffy

0ҟ

200 400 600 800 1000 1200 1400

pfy (psi)

Fig. 10. Comparison of shear transfer strength calculated using theshear friction provisions of ACI 318-71 with measured strengths of

initially cracked push -off and pull-off specimens

hypothesis for the behavior of initiallycracked concrete described above. Inthe shear-friction approach, it is as-sumed that for some unspecified rea-son a crack exists in the shear plane.The shear resistance is then assumed tobe developed entirely by the frictionalresistance to sliding of one crack faceover the other, when acted upon by anormal force equal to the yield strengthof the reinforcement crossing the shearplane. A fictitiously high value of thecoefficient of friction p. is used to com-pensate for neglect of dowel action and

other factors. For a crack in monolithicconcrete, p. is taken as 1.4. For conser-vative calculation of strength, the sheartransfer strength is limited to 0.2f' or800 psi (56 kgf/cm2) whichever is theless. The shear-friction equation maybe written as

Avrf,v , = bd .µ =pfyl-. (6)

but not more than 0.2f' or 800 psi.In Fig. 10 the shear transfer strength

calculated according to Eq. (6) is indi-cated by unbroken lines and is com-

PCI Journal / March.-April 1.972 71

1400

1200

1000

Vu800

(psi)600

400

200

Specimens initially crackedҟ0

Push-off {O - fy = 4000psiҟ*Oҟ0• - Z = 250opsiPull-off A-fc' = 510opsiҟ0Modified {O - ff = 4000psiPush-off • -fc = 5800psiҟ0 ҟ0ҟlimit,

f,=40

0 0 0ҟ•A A

0• •

A0 A ! limit, fe' = 2500 psi

O ALQ Equation (7)

vu = 200psi + 0.8(pfy + o-Nx)but not more than 0.3fc,

minimum (p fy + oj,^) = 200psiI 1 I

( pfy t ONx )ҟ(psi)

Fig. 11. Comparison of shear transfer strength calculated using Eq. (7)with the measured strengths of initially cracked push-off, pull -off and

modified push-off specimens

pared with the measured strength ofall the initially cracked push-off andpull-off specimens tested in the pro-gram reported here. It can be seen. thatthe provisions of ACI 318-71 yield r aconservative estimate of the shear trans-fer strength of concrete cracked alongthe shear plane. However, it is alsoclear that shear stresses considerably inexcess of the arbitrary upper limit of800 psi (56 kgf/cm2) can be devel-oped if appropriate reinforcement isprovided and the concrete strength ishigh enough. Section 6.1.9 of the PCI

Design Handbook( 11 ) suggests that thisreinforcement may be designed usingEq. (6), providing that itt is multipliedby (300 /p f y + 0.50) when •p f 5 ex-ceeds 600 psi (42 kgf/cm2). This pro-posal is indicated by the broken line inFig. 10. It can be seen that this pro-posal is in accord with the trend of theexperimental data for concrete strengthsgreater than 2500 psi (176 kgf/cm2)

and for shear strengths less than about1300 psi (91 kgf/cm2). A deficiency inthis proposal is that no upper limit oneither pf,5 or v, is specified.

72

1200ҟ -CORBEL TESTS OF KRIZ a RATHS 0;

1000ҟ -

° •ҟvu 800psi• ' O

• •O i-pSlҟ°ҟ•ҟ•ҟr•ҟ,ҟimit, f

•ҟr= 2000psi

600ҟ0 00•* ҟ• a • ^^^`^ fc = 2000psi

•ҟAS'o ^J Equation (7)400 ©ҟa °ҟS®.0ҟ -limit, f"= 2000psi

g °o'° o° o ' Equation (6) • - Nu/Vu ° ^/2° ii® o °°^ҟN= 1.40 °-

200ҟ -I--ҟ̂̂ҟ= 3/4ҟ-OEҟ(Shear- (Shear-friction) o -ҟ..ҟ= vl

X - ҟ=5/4Circled ppints, corbels with stirrups

0ҟ100ҟ200ҟ300ҟ400ҟ500ҟ600ҟ700

(pfy + a-NX ) _ (pfy – N„/bd) psi

Fig. 12. Comparison of the shear strength of corbels calculated using Eqs. (6) and(7) with the measured strengths of tested corbels

An alternate approach, simpler to ap-ply in design, would be the use of thefollowing equation for shear transferacross a crack in monolithic concrete

v,, = 200 psi + 0.8 (pf5 + o-,,) (7)

with the restrictions that v,,, shall benot more than 0.3f' and (pfy + o)shallshall be not less than 200 psi (14 kgf/cm2). 0N, is the externally applied di-rect stress acting across the shearplane, taken as positive for a compres-sive stress and negative for a tensilestress. In Fig. 11, Eq. (7) is compared

with the measured strength of all theinitially cracked specimens tested in theprogram reported here and is seen tobe a lower bound to the data. Eq. (7)is slightly less conservative than Eq. (6)for (pfy + a-,,,) less than 333 psi (23kgf/cm2) and slightly more conserva-tive for values of (p f5 + ON) between333 and 572 psi (23 and 40 kgf/cm2),the limit of applicability of Eq. (6) ac-cording to ACI 318-71.

The results reported here validateEq. (7) for values of (p f y + ON) up to1400 psi (99 kgf/cm 2), when 0,, is

PCI Journal / March-April 1972 73

zero or compressive. The results re-ported by Kriz and Raths( 12 ) for corbelssubjected to shear and to tension forcesin the direction of the reinforcement,indicate that there are a wide varietyof conditions for which Eq. (7) is alsovalid for values of te,,, which are ten-sile. In Fig. 12 Eqs. (6) and (7) arecompared with data from Kriz' andRaths' corbel tests. In plotting Fig. 12,v,, was taken as the nominal shear stressat yield of the tension reinforcement,or at ultimate strength of the corbelif the yield of the tension reinforcementdid not occur. In accordance with Kriz'and Raths' findings and as required bySection 11.14 of ACI 318-71, the re-inforcement ratio p was taken as (A8 +Ah)/bd when shear only acted on thecorbel and as A 3 /bd when both shearV,, and tension N,, acted on a corbel.In this latter case, Jrx = N,1/bd. Itcan be seen that, providing the limita-tions placed on them are observed,both Eqs. (6) and (7) yield conserva-tive estimates of the ultimate strengthof corbels. (The corbel tests consideredincluded specimens for which the ratioof the tensile stress o- to the shearstress v,, varied from 0 to 1.25, andfor which the ratio of moment actingon the corbel to the shear times theeffective depth at the column face(a/d) varied from 0.11 to 0.62. Themaximum value of (pf, , — Nw/bd) was514 psi (36 kgf/cm2), and the maxi-mum value of pf, considered was 700psi (49 kgf/cm2).

Conclusions

Concerning design.

1. Within their range of applicabil-ity, the shear-friction provisions of ACI318-71 yield a conservative estimate ofthe shear transfer strength of reinforcedconcrete whether or not a crack existsin the shear plane.

2. Higher shear transfer strengthsthan the upper limit of 800 psi (56kgf/cm2) specified in ACI 318-71 canbe developed if appropriate reinforce-ment is provided and the concretestrength is adequate. Such reinforce-ment may be proportioned using Eq.(7).

Concerning fundamental behavior.1. A pre-existing crack along the

shear plane will both reduce the ulti-mate shear transfer strength and in-crease the slip at all levels of load.

2. Changes in strength, size, andspacing of reinforcement affect theshear transfer strength only insofar asthey change the value of the reinforce-ment parameter pf, for fy :66 ksi(4640 kgf/cm2).

3. In initially cracked concrete, theconcrete strength sets an upper limitvalue for pf9 below which the relation-ship between v, and pf, is indepen-dent of concrete strength. Above thisvalue of pf, the shear transfer strengthincreases at a much reduced rate forlower strength concrete and is equal tothat of similarly reinforced, initially un-cracked concrete.

4. Direct tension stresses parallel tothe shear plane reduce the shear trans-fer strength of initially uncracked con-crete, but do not reduce the shear trans-fer strength of concrete initially crackedin the shear plane.

5. An externally applied compres-sive stress acting transversely to theshear plane is additive to pfy in calcu-lations of the ultimate shear transferstrength of both initially cracked anduncracked concrete.

6. The shear transfer strength of ini-tially uncracked concrete is developedby a truss action after diagonal tensioncracking. Failure occurs when the in-clined concrete struts fail under a com-bination of shear and axial force.

74

7. The shear transfer strength of ini-tially cracked concrete with moderateamounts of reinforcement is developedprimarily by frictional resistance to slid-ing between the faces of the crack andby dowel action of the reinforcementcrossing the crack. When large amountsof reinforcement, or sufficient external-ly applied compression stresses normalto the shear plane are provided, thenthe crack in the shear plane "locks up"and shear transfer strength is developedas in initially untracked concrete.

Acknowledgments

This study was carried out in the Struc-tural Research Laboratory of the Uni-versity of Washington, Seattle, It wasmade possible by the support of donorsto the Structural Concrete ResearchFund at the University of Washington,and by the Bethlehem Steel Corpora-tion who donated reinforcing bars.

References1. Birkeland, P. W. and Birkeland,

H. W., "Connections in PrecastConcrete Construction," Journal of

the American Concrete Institute,Vol. 63, No. 3, March 1966, pp.345-368.

2. Mast, R. F., "Auxiliary Reinforce-ment in Concrete Connections,"Proceedings, ASCE, Vol. 94, ST6,June 1968, pp. 1485-1504.

3. Hofbeck, J. A., Ibrahim, I. O. andMattock, A. H., "Shear Transfer inReinforced Concrete," Journal of

the American Concrete Institute,Vol. 66, No. 2, Feb. 1969, pp. 119-128.

4. Chatterjee, P. K., "Shear Transferin Reinforced Concrete," MSCEThesis, University of Washington,Seattle, June 1971.

5. Vangsirirungruang, K., "Effect ofNormal Compressive Stresses onShear Transfer in Reinforced Con-crete," MSCE Thesis, University ofWashington, Seattle, July 1971.

6. Zia, P., "Torsional, Strength of Pre-stressed Concrete Members," Jour-nal of the American Concrete In-stitute, Vol. 57, No. 10, April 1961,pp. 1337-1359.

7. Anderson, A. R., "Composite De-signs in Precast and Cast-in-PlaceConcrete," Progressive Architec-ture, Vol. 41, No. 9, September1960, pp. 172-179.

8. Kupfer, H., Hilsdorf, H. K. andBusch, H., "Behavior of ConcreteUnder Biaxial Stresses," Journal of

the American Concrete Institute,Vol. 66, No. 8, Aug. 1969, pp. 656-666.

9. Gaston, J. R. and Kriz, L. B., "Con-nections in Precast Concrete Struc-tures—Scarf Joints," Journal of thePrestressed Concrete Institute, Vol.9, No. 3, June 1964, pp. 37-59.

10. "Building Code Requirements forReinforced Concrete (ACI 318-71)," American Concrete Institute,Detroit, Mich., 1971.

11. "PCI Design Handbook," Pre-stressed Concrete Institute, Chica-go, Ill., 1971.

12. Kriz, L. B. and Raths, C. H., "Con-nections in Precast Concrete Struc-tures—Strength of Corbels," Jour-nal of the Prestressed Concrete In-stitute, Vol. 10, No. 1, Feb. 1965,pp. 16-61.

Discussion of this paper is invited.Please forward your discussion to PCI Headquarters by July 1to permit publication in the July-August 1972 issue of the PCI JOURNAL.

PCI Journal / March-April -1972 75