the behavior of reinforced concrete corbels - pci.org journal/1976/march... · plicability of this...

The Behavior ofReinforced ConcreteCorbelsAlan H. MattockProfessor of Civil Engineering andHead, Division of Structures & MechanicsUniversity of WashingtonSeattle, Washington

K. C. Chen*Structural EngineerURS Consulting Engineering CompanySeattle, Washington

K. Soongswang*Civil EngineerSoutheast Asia Technology CompanyBangkok, Thailand

*Former Graduate Student, Department of Civil Engineering,University of Washington, Seattle, Washington.

52

Review of current designprovisions for corbels

C orbels (or brackets) projecting fromthe faces of reinforced concrete

columns are used extensively in pre-cast concrete construction to supportprimary beams and girders.

The design of corbels is governed bythe provisions of Section 11.14 of ACI318-71. 1 Under these provisions, corbeldesign may either be based on therather complicated empirical Eqs. (11-28) and (11-29), which are valid fora/d < 1.0, or if a/d is limited to 0.5,a simpler method of design based onthe shear-friction provisions of Section11.15 may be used.

However, when the shear-frictionprovisions are used for corbel design,the limitations on quantity and spacingof reinforcement specified in Section11.14 must still be observed. These lim-itations derive from the recommenda-tions of Kriz and Raths.2

ACI 318-71 requires that unless spe-cial measures are taken to avoid hori-zontal tension forces, all corbels shallbe designed for an appropriate horizon-tal force N,, in addition to the verticalload V,,, and that Nu/Vu shall not betaken as less than 0.2.

In this case the reinforcement ratio pis limited by Section 11.14 to

max. p = A$/bd = 0.13f',/ffthat is:

max. pf, , = 0.13j'Therefore, using shear-friction:

max. v u = p0.13 f'0 + o-N.)The normal stress o- is negative for

tension. Therefore, when a tension forceN,, acts:

N1, /O^Nx =— vn)V (

V.

that is:

max. V,,= /L [0.13f', --. (max. v,,)1U

max. V--+ju ( Vu =0.13f'0

SynopsisAn experimental study is re-ported of the behavior of re-inforced concrete corbelssubjected to both verticaland horizontal loads.Twenty-eight corbel speci-mens were tested, of whichtwenty-six contained hori=zontal stirrup reinforcement.The variables included in thestudy were the shear span toeffective depth ratio, the ra-tio of the vertical load to thehorizontal load, the amountsof main tensile and stirrupreinforcement, and the typeof aggregate.Criteria for the design ofhorizontal stirrup reinforce-ment are developed.It is shown that, subject toprovision of the recommend-ed amount of stirrup rein-forcement, the useful ulti-mate strength of corbels canbe taken to be the lesser of(a) the shear strength of thecorbel-column interface, cal-culated using either theshear-friction provisions ofSection 11.15 of ACI 318-71or the modified shear-frictionequation, and;(b) the vertical load corres-ponding to the developmentof the flexural ultimatestrength of the corbel-col-umn interface.In the next issue a follow-uppaper will propose specificcode provisions for design-ing corbels together withnumerical examples.

PC] JOURNAL/March-April 1976 53

therefore:0.13f,

max. vu = 1 Nu

µ + V"For corbels monolithic with their

supporting column, for which µ = 1.4,the maximum allowable shear stress dueto the limitation p not greater than0.13f' 0 will vary from 0.142f' whenNa/V,, has its minimum value of 0.2,to 0.076f' 0 when NU/VU is equal tounity,* i.e., v y (max.) according to Sec-tion 11.14 varies from 71 percent downto 38 percent of the vu (max.) allowedunder the shear-friction provisions ofSection 11.15.

The requirement of Section 11.14that horizontal stirrups or ties with anarea Ah not less than 0.50A,, be provid-ed, is designed to prevent a prematurediagonal tension or "diagonal splitting"failure of the corbel. This requirementis also based on Kriz and Raths' recom-mendation.2

The provision leads to heavy stirruprequirements when a large N,,, acts withV. However, the data obtained fromtests of corbels with stirrups subject tocombined loading, on which Kriz andRaths based their recommendation, wasrather limited, involving only five speci-mens with arbitrarily proportioned stir-rup reinforcement.

In addition to the quantity of stirrupsprovided, the following were also variedin these five specimens—the shear spanto depth ratio, the effective depth, theN„/Va ratio, and the concrete strength.(The reinforcement ratio p = A8/bdwas the same in all the specimens withstirrups.)

°No upper limit to N,,/V,, is specified in Sec-tion 11.14 of ACI 318-71, but Eq. (11-28) inSection 11.14 does not yield sensible results if avalue greater than 1.0 is substituted for N,,/V„in the equation.'The reinforcement ratio p used in Eq. (11-28)

is based on the area of the main tension rein-forcement only (p = A,/bd).

The results obtained in these fivetests were also the basis for a conclusionthat no consistent increase in shear ca-pacity was obtained in a corbel subjectto combined loading, when stirrupswere provided. The requirement in Sec-tion 11.14 of ACI 318-71, that the maintension reinforcement only be consid-eredt when calculating the strength ofa corbel subject to combined loading,stems from this conclusion of Kriz andRaths.

Purpose of this studyThe study reported in this paper wasdirected toward the extension of theshear-friction method of corbel designto corbels having shear span to depthratios a/d of up to 1.0, subject to com-binations of vertical and horizontalloads such that NU/VU < 1.0.

It has already been shown 3 that thesimultaneous action of a moment equalto the flexural ultimate strength of acracked section does not reduce theshear which can be transferred acrossthe crack, and that the arbitrary limi-tation of the use of the shear-friction de-sign method to cases when a/d < 0.5is unwarranted.

On the basis of the work reportedearlier, 3 it was tentatively proposedthat, providing a premature diagonaltension failure of the corbel is prevent-ed by the provision of an appropriateminimum amount of stirrup reinforce-ment, the ultimate strength of a corbelcan be calculated by taking it to be thelesser of

(a) The shear strength of the corbelinterface, calculated using the shear-friction provisions of Section 11.15 ofACI 318-71; and

(b) The vertical load correspondingto the development of the flexural ulti-mate strength of the corbel-column in-terface, using the provisions of Section10.2 of ACI 318-71.

The experimental study reported here

54

was designed to check the range of ap-plicability of this approach to corbeldesign and to develop rational require-ments

A$ for stirrups in corbels.

Preliminary considerationsConsider the corbel shown in Fig. la,acted on by a shear V and a horizontalforce N,w. It is assumed that the maintension reinforcement AS and the hori-zontal stirrups A,4 can develop their'yield strengths, fv and fvv, respectively.

The maximum values of the reactiveforces acting at the interface betweenthe corbel and the column will then beas shown in Fig. 1b, if shear resistancealong the interface is calculated usingthe shear-friction hypothesis.

For the corbel to remain in equilib-rium the following requirements mustbe satisfied:

1. 1F„=0

i.e., V„ < pC (1)

2. 1F,=0i.e., N,, < AAJ, + fAnfvv - C (2)

3. 1M=0i.e., V„a + N„(h — d + id)

çt.Ajvjd + 4A,f„vj1 d (3)Now

idd — Y, (4Ajv + fA,,f^,p^—N,,= 0.85f', b

=j'd —1/2 (6.85f

PAhfa;v

,,b )

where d would be the internal leverarm if only flexural reinforcement ofstrength (4Aj,, — N„) was acting in thesection, that is:

j'd=d — Acv —N,^( _______

Substituting this value for jd in Eq.(3) and simplifying, we obtain:V„a+N„(h— d+ jd) <4A8fvj'd+

0.85f^'

c b^hfvv {j^a — ^2 ^A f^ )}

Fig. 1a. A typical corbel.

We may conservatively neglect thecontribution of the stirrup force OA 7, f,,vto the resistance moment and write:

V3a +N„(h—d+id)<,OAjv1d (4)Therefore:

V,,a + N1/h — d) N,, jdAS — -f-

4fv1d 4f, dNow jd/j' d is slightly less than 1.0.

Therefore, it is conservative to write:

A ^. V,,a + N,%h — d) N8

Ofv ► d OfvThat is:

A, A1 +At

u

OAsfy —< - Tia Interfacejd r04hvy—^

pill, `t

pC

Fig. 1b. The corbel as a"free body.”

Ah

(5)

(6)

Nu

d h

PCi JOURNAL/March-April 1976 55

where

Af = area of reinforcement necessaryto resist the applied moment[Vua-+ N,1(h — d)]

At = area of reinforcement necessaryto resist the horizontal force N,

Eliminating C between Eqs. (1) and(2), and simplifying, we obtain:

Ah' f v ( + Asfr) (7)

If fv = we may write:

V N

A^ ^Mfv + A8or

An. Avf+At—A8 (8)where A01 is the area of reinforcementnecessary to carry the shear V,, calcu-lated using the shear-friction Eq. (11-30) of ACI 318-71.

Substituting for A8 in Eq. (8) fromEq (6) we have:

A1,, A,,, —At (9)It is possible that Eq. (9) could yield

a very small or zero value for A,,,. How-ever, this would not be admissible, sinceit is presupposed that the amount ofhorizontal stirrups present in the corbelis sufficient to prevent a prematurediagonal tension failure of the corbel. Asuitable minimum amount of stirrupsmust therefore be specified to ensurethis.

In this study, it was decided to startwith the hypothesis that the minimumamount of stirrups should be that re-quired to carry the difference in shearbetween the ultimate shear the corbelwas to carry, and the shear at which itwould fail in diagonal tension if no stir-rups were provided. It was thereforenecessary to determine a suitable wayof calculating the shear strength of acorbel without stirrups.

After reviewing the literature, it wasdecided to examine the applicability tocorbels of the following equation devel-

oped by Zsutty4 for the shear strengthof directly loaded reinforced concretebeams without stirrups and havingshear span to depth ratios of 2.5 or less:

d li3 10vu2 = 60 f P a )(2.

a/d ) ( )

Eq. (10) was developed from thefollowing equation for the shearstrength of slender beams (a/d> 2.5),previously proposed by Zsutty:5

d leav at = 60 f'0p

d (11)

a/

A study was made of the applicabilityof Eq. (10) to those vertically loadedcorbels tested by Kriz and Raths2 whichwere reported to have failed by "diago-nal splitting" (i.e., diagonal tension).

The initial calculations showed thatEq. (10) grossly overestimates theshear at diagonal tension failure inmembers with such small values of a/d.A direct comparison was thereforemade of the measured shear stresses at"diagonal splitting" failure, v,,, and the"slender beam" shear strengths, v,,, cal-culated using Eq. (11).

This is shown in Fig. 2. By so doing,it was hoped to determine whetherZsutty's "arching factor," 2.5/(a/d), inEq. (10) could be modified in a simplemanner, or limits be placed on its ap-plicability, so that Eq. (10) could bemade applicable to corbels with smalla/d values.

It can be seen that reasonable agree-ment can be obtained between meas-ured and calculated stresses if an upperlimit of 2.5 is placed upon the factor[2.5/(a/d)].

The upper limit controls when a/dis 1.0 or less.

It was therefore decided that in thefirst corbel specimens to be tested inthis study, the minimum horizontal webreinforcement would be made equal to

Pnfvv (min.) = Vu — V,,2 (12)

56

0ivu = 60(ff.p. d )! (2.5) 00

900

800Vu

(psi)700

0

0/0 • - aid < 0.30

0-,0.30<a/d<0.50/so 0 - 0.50 < a/d 1.00

500

400150 200 250 300 350 400

Vul = 60(fc.p.a ' (psi)

Fig. 2. Comparison of measured and calculated shear stressesat "diagonal splitting" failure in tests of corbels by Kriz

and Raths. 2 (Corbels subject to vertical load only.)

600

wherepry = Ai,/bd

= design ultimate shear stressd iia

v,,0 = 60 (f op a) K (13)

where K = 2.5/(a/d)but not greater than 2.5

For aid 1.0, Eq. (13) becomes:d its

vuz= 150 (fop a) (14)

Hence:

d 1/3pnfvv (min.) = v,, — 150 (ra p a )

(15)

Experimental Study

A total of 28 corbel specimens weretested, 26 of which contained horizon-tal stirrup reinforcement. The test pro-gram was developed as it progressedand is summarized in Table 1.

Series A and B were tested to studyminimum stirrup requirements in cor-bels designed to carry vertical loadsonly. Series C, D and E were tested tocheck the design method proposedwhen applied to corbels made of sandand gravel concrete, subject to bothvertical and horizontal loads. In SeriesC and E, the design ultimate shear

PCI JOURNAL/March-April 1976 57

Table 1. Summary of test program.

Series

No. ofSpecimens

Type ofConcrete

Horizontal

Stirrups

Design

N/VuU

Design

vu(psi)

A 2Sand &

No 0 800 Gravel

8 4Sand &

Yes 0 800 Gravel

C 4Sand &

GravelYes 0.75 800

D 3Sand &

Yes 1.0 565Gravel

c 4Sand &

Yes 1.0 800 Gravel

F 4 All-lightweight Yes 1.0 800

G 1 All-lightweight Yes 0 800

H 5Sand &

Yes 1.0 1200 Gravel

J 1 All-lightweight Yes 1.0 560

stress was made equal to 0.2f' = 800psi the maximum allowable accordingto the shear-friction provisions of Sec-tion 11.15 of ACI 318. In Series D andE, the design Nu/V,,, was made equalto the proposed maximum value of 1.0for this method of design. Within eachof these tests series, a/d was varied,having values up to and including 1.0.

Series F, G and J were tested tocheck the design method proposedwhen applied to corbels made of all-lightweight concrete, subject to bothvertical and horizontal loads. In theseseries also, the limiting values of vu,Nu/Vu , and a/d were explored.

Series H was tested to examine thepossibility of using the following pre-viously proposed7 equation for theshear transfer strength of sand andgravel concrete, in place of the shear-friction equation when designing -cor-bels as proposed.

v,, = 0.8 pfy + 400 psi (16)but not greater than 0.3f'c.

The specimens were designed for vuequal to the maximum value of 0.3f,simultaneously with NU/V,,, having itsmaximum value of 1.0 and a/d havingvalues up to and including 0.68.

Grade 60 reinforcement was used inSeries E through J, to ensure that any

conclusions reached would be valid forcorbels reinforced with this higherstrength steel.

Test specimensEach specimen consisted of a 38-in.length of column with two corbels pro-jecting from the column in a symmetri-cal fashion, as shown in Fig. 3. Themain tension reinforcement consisted ofparallel straight bars, anchored by shortbars of equal diameter welded acrosstheir ends. The closed horizontal stir-rups were #2 deformed bars and wereuniformly distributed in the two-thirdsof the effective depth nearest the maintension reinforcement.

Steel bearing plates (4 x 1 x 6 in.)were welded to the main tension rein-forcement. Series A, B, and G speci-mens had plain bearing plates. The re-mainder of the specimens had groovedbearing plates, as shown in Fig. 3, to fa-cilitate applying the horizontal force tothe corbels.

The amount and strength of rein-forcement used in each specimen areshown in Tables 2 and 3. The designcompressive strength of the concrete attest was 4000 psi for all specimens. Theactual concrete strengths at test aregiven in Tables 2 and 3.

58

10 or13 12 6,8,10 or13

Size and numberof bars varies.2 2

2,4, 6 I 1

i or 9 4^^

anchor bar.to

U) #2 bar closedstirrups, number& spacing varies.

#2 bar#2 bar ties at 6in. crs.

4 #4, 60 grade bars.

All dimensionsin inches.

Elevation

____ ii:itiiPlan, Series A,B,C,D

_______ _______

_ L

Plan, Series E,F,G,H,J

Fig. 3. Typical corbel test specimens.

Design of test specimensThe test specimens were designed asproposed above, except that in the caseof Series C and D, the term N,u(h-d)was omitted when calculating A$ usingEq. (5). This was done because it wasinitially assumed that the unconserva-tism of this action would be cancelledby the conservatism of ignoring the re-sistance moment contribution of the

horizontal stirrups. However, when thetest results from Series C and D werestudied, this assumption was found tobe incorrect. The term Nu,(h-d) wastherefore included when calculating A8in the design of the specimens of SeriesE through J. The capacity reductionfactor was taken equal to 1.0 in thesedesign calculations.

Series A—The specimens of Series A

PCI JOURNAL/March -April 1976 59

were intended to provide a reference asto the effectiveness of the horizontalstirrups provided in the specimens ofSeries B. The specimens of Series Awere therefore made as nearly as pos-sible identical to the correspondingspecimens of Series B, except that thehorizontal stirrups were omitted.

Series B-All the specimens of SeriesB were designed to have an ultimateshear stress of 0.2f, i.e., 800 psi. Thefirst three specimens of the series wereprovided with minimum stirrup rein-forcement as proposed above, that is:

d itsp7,f„v (min.) = vu - 150 f p a 1

(15)

The value of phf„y (min.) calcuatedusing Eq. (15) was found to be almost

a constant quantity, being 186, 183,and 179 psi for Specimens B1, B2, andB3, respectively. This is very nearlyequal to %(A„ff/bd) = 190 psi inthis case where vu = 800 psi.

This is apparently due to the occur-rence of the product '(p • d/a) in theequation for v,,,2 . When the shear span"a" is increased, then for a constantV,,, M,, increases and "p" increases al-most in proportion to the increases in"a." Hence the product (p . d/a) re-mains approximately constant andtherefore v,,2 remains approximatelyconstant.

For smaller values of v,4, p would de-crease in proportion to v,4 but v,,2 woulddecrease less rapidly because it is pro-portional to p l /s . The quantity (vu

-v,2) would therefore decrease more rap-

Table 2. Details of corbel specimens (Series A, B, C & D).

Spec. a

Main Reinforcement Stirru s

As ConcreteAs fy Ah fey

No. d Bars 2 2 P bdStrength

(in. ) (ksi) (in. ) (ksi) fc (psi)

A2 0.67 2 #5 0.62 46.6 0 - 0.0116 3675

A3 1.01+1 #3 0.99 53.6

0 - 0.0186 3850

61 0.44 2 #4 0.40 48.5 0.20 65.0 0.0075 3630

B2 0.67 2 #5 0.62 46.5 0.20 67.0 0.0116 3450

B3 101. +^ #3 0.9954.1

0.20 64.0 0.0186 3760

B3A 1.01+1 #3 0.99 50.9

0.40 65.5 0.0186 4165

C1 0.45 +^ #4 1.08 48.^ 0.20 60.0 0.0203 4010

C2 0.68 3 #6 1.32 50.4 D.20 67.4 0.0248 3715

C2A 0.682 #6

+ 1

.

19 50.0 0.20 65.6 0.0223 3705

C3 1.022 #7

1.6450.0

0.40 65.5 0.0310 4385+1 #6 50.9

D1 0.45 2 #6 0.88 50.2 0.10 70.0 0.0165 3910

D2 0.682 #6

+1 #41.08

47.70.20 64.0 0.0203 3805

D3 1.01 3 #6 1.32 48.4 0.25 67.6 0.0248 3700

60

idly than would (vu/3) and hence, theminimum Ah, by this hypothesis wouldbe less than A01/3. At this point, it ap-peared that a simple and conservativerule for minimum stirrup reinforcementmight be that it should be not less thanAvf/3.

Specimens B1 and B2 performed in asatisfactory manner, but Specimen B3developed an ultimate strength of only600 psi. The characteristics of the speci-mens as constructed were re-examined.It was found that because the actualstrength of the larger diameter barsused for Af was less than the value of50 ksi assumed in the design and thestrength of the #2 bars used for the

horizontal stirrups was more than 50ksi, the yield strength of the stirrupsprovided in Specimen B2 was veryclose to Af f,/2. ,/2.

A fourth specimen, B3A, was there-fore designed in which Af was made thesame as in Specimen B3, but a largernumber of stirrups was provided so thattheir total yield strength Ahf, was asnearly as possible equal to Atf^/2. Thisspecimen behaved in a satisfactorymanner.

It therefore appeared that it might beappropriate to require that the closedstirrups or ties parallel to the main ten-sion reinforcement shall have a totalyield strength at least equal to half the

Table 3. Details of corbel specimens (Series E, F, G, H & J).

Spec. a

Main Reinforcement Stirrup_sA

P bd

Concrete

StrengthAs fY Ah 6VyNo. d Bars

(in. 2 ) (ksi) (in. 2 ) (ksi)fc (psi)

El 0.22 # 6 0.84 59.1 0.15 64.0 0.0189 4030+2

E2 0.45 #6+^ 0.93 67.0 0.15 67.0 0.0210 4450

E3 0.68 1.08 0.25 65.0 0.243 4220+1 #4 64.0

E4 1.01 3 #6 1.32 62.5 0.35 67.0 0.0297 4055

F2 0.45 0.93 0.15 68.2 0.0210 3715+1 #2 65.0

F3 0.68 6#+^ 1.08 0.25 64.0 0.0243 373064.064

F4 1.01 3 #6 1.32 63.2 0.35 68.2 0.0297 4035

F4A 1.01 3 #6 1.32 63.4 0.35 68.0 0.0297 3715

G4 0.99#6

+2 0.30 67.4 0.0147 37502 65.5

H1 0.23 #2 0.93+1 fi8.

0.20

64.0 0.0210 39202

H2 0.45 +^ #4 1.08 0.25 64.0 0.0243 393064.0.

H3 0.68 3 #6 1.32 63.2 0.35 67.0 0.0297 3855

H3A 0.68 3 #6 1.32 64.1 0.35 68.6 0.0297 3960

H3B 0682 #6

1.32 0.35 68.0 0.0297 3820 .+1 #6 64.8

J4 1.01 2 #6 0.88 64.8 0.25 64.0 0.0198 3645


yield strength of the reinforcement re-quired for flexure, or one-third the yieldstrength of the reinforcement requiredto resist shear (calculated using theshear-friction provisions of ACI 318-71), whichever is the greater.

If the yield point of the stirrups andthe main tension reinforcement is thesame, the above requirement reduces toAh (min.) = .A/2 or Av f/3, whichever isthe greater.

Series C-The corbels of this serieswere designed to be companion speci-mens to Corbels B1, B2, and B3A. TheSeries C corbels were provided with ad-ditional main tension reinforcement, sothat the strength of the main tensionreinforcement in the Series C corbel wasgreater than that provided in its com-panion Series B corbel by an amountequal to the horizontal force N. that itwas proposed to apply to the corbelconcurrently with the shear.

The horizontal stirrup reinforcementwas made the same as that provided inthe companion Series B corbels. That is,Ahf, was made as nearly as possibleequal to the greatest of the values givenby 1/s or 1/z Affy.

This was done to check whether thisamount of stirrup reinforcement is ade-quate when the maximum allowablehorizontal force acts on the corbel con-currently with the shear. These corbelsbehaved satisfactorily. Therefore, thisamount of stirrup reinforcement wasprovided in all subsequent test speci-mens.

Series D and E—The specimens ofboth these series were designed to carryan ultimate horizontal force equal to theultimate shear force. However, the cor-bels of Series E were made of a reducedwidth, so that a design ultimate shearstress of 800 psi could be attained whenusing available hydraulic rams to applythe horizontal force.

Series F, G, and J—The specimens ofSeries F were of all-lightweight con-

crete and were designed to have thesame ultimate shear strength and ulti-mate outward horizontal force as thecorresponding specimens of Series E.However, in this case the shear-frictioncalculation was modified as proposedpreviously,6 ^u being multiplied by thecoefficient 0.75 for all-lightweight con-crete contained in Section 11.3.2 of ACI318-71.

Specimen F2 (a/d = 0.45) behavedsatisfactorily, yielding a t 825 psi.However, Specimen F4 (aid = 1.01),which was tested next, yielded a v,,of only 540 psi, or 0.13f'. '. The char-acteristics of the specimen as construct-ed were re-examined, but there was noimmediately apparent reason for thepoor behavior. A duplicate specimenF4A was fabricated and tested. It be-haved the same as Specimen F4, yield-ing a v, of 0.14J',. A specimen G4, withthe same a/d as Specimen F4, was de-signed to carry an ultimate shear stressof 0.2f' with zero horizontal load. Itwas subjected to vertical load only, todetermine whether the presence of thehorizontal load N. had influenced themaximum shear stress attained in Cor-bels F4 and F4A.

The ultimate shear stress developedin Specimen G4 was also 0.14f',, indi-cating that the unexpected low value ofv74 in Specimens F4 and F4A did notresult from the action of N,,,. Neitherthe main tension reinforcement nor thestirrup reinforcement developed theiryield strengths in Corbels F4, F4A, andG4. These corbels appeared to fail byshear-compression of the concrete in abeam type shear failure.

In push-off tests of all-lightweightconcrete previously reported 6 (i.e., a/d= 0), it was possible to develop a maxi-mum shear transfer stress of 02f',. Itis apparently necessary to define the up-per limit of v,, for structural light-weight concrete corbels in terms of a/d.It was decided to check whether a shearstress of 0.14f' could still be devel-

62

oped at a/d = 1.0 if the main rein-forcement yielded.

Specimen J4 (a/d = 1.0) was de-signed to have vu = 0.14f'^, N,,/V,4 =1.0 and a moment capacity (based onthe reinforcement yield strength) corre-sponding to the design ultimate shear.This corbel developed an ultimate shearstress of 0.13f' 0 and the reinforcementjust reached its yield point at failure.

Series H—The sand and gravel con-crete corbels of this series were de-signed to have an ultimate shearstrength of 53.4 kips with N,, /V, =0.68. This corresponds to the upperlimit shear stress of 0.3f', specifiedwhen using Eq. (16) for shear design,instead of the shear-friction equation.

Materials and fabricationThe sand and gravel concrete was madefrom Type III portland cement, sand,and 3/4-in, maximum size gravel, in theproportions 1.0:2.77:3.44 by weight.

The all-lightweight concrete was madefrom the same coated aggregate used inthe study6 of shear transfer strength.The concrete was made from Type Iportland cement, coarse and fine light-weight aggregate in the proportions1.0:1.48:2.10. In both concretes, waterwas added sufficient to produce a 3-in.slump. Approximately 6 percent of airwas entrained in the concrete. The sandand gravel concrete was moist cured 1day and then cured in the air of the lab-oratory until test at age 4 days.

The Iightweight concrete was moistcured 7 days and then cured in air un-til test at 28 days. (This is the standard-ized curing procedure for tests of struc-tural lightweight concrete.)

The deformed reinforcing bars usedfor the main tension reinforcement andfor the column reinforcement, con-formed to ASTM Specification A615.The #2 bars used for stirrup reinforce-ment had deformations similar to thelarger bars conforming to ASTM Speci-fication A615.

Fig. 4. Arrangement for test ofcorbel specimens.

A stiff metal jig was used to hold thebearing plates in alignment with oneanother and the correct distance apart,while they were being welded to themain tension reinforcement. In theform, the bearing plates were screwedto metal supports to ensure their cor-rect alignment and location.

Testing arrangements andinstrumentationFor convenience, the specimens weretested in an inverted position, as maybe seen in Fig. 4. The specimens of Se-ries A, B, and G were supported direct-ly on the plain bearing plates, throughfree rollers resting on the tops of thetwo legs of the U-shaped supportingframe. The vertical load was applied bya Baldwin hydraulic testing machinethrough an SR4 gage load cell locatedconcentrically on the top of the column.

The specimens of Series C, D, E, F,H, and J were also supported on freerollers; but through an intermediatebearing plate having a projection on its


upper face which mated with the 1/z in.deep groove in the bearing plate weldedto the corbel reinforcement. Short stubaxles projected from each end of theintermediate bearing plates, the centersof the stub axles being at the level ofthe horizontal face of the corbel.

The horizontal force was applied tothe corbels through these stub axlesby two 20-kip capacity hydraulic rams,arranged one in front and one behindthe specimen, as may be seen in Fig. 4.The horizontal force was monitored bytwo links instrumented with SR4 gagesso as to act as transducers. The verticalload was applied by the testing machinein the same way as for Series A and B.

The average deflection of the corbelsat their loading points was measuredusing a linear differential transformer,attached to the column at its centerline.Resistance strain gages were used tomeasure the strain in the main tensionreinforcement at the interfaces betweenthe column and the corbels in all tests.

In Series F, G, H, and J the strain inthe stirrup reinforcement was measuredat points where the stirrups crossed aline joining the center of the bearingplate and the intersection of the slopingface of the corbel and the column face.The load cell, transducer links, differen-tial transformer, and strain gages weremonitored continuously during the testsby a Sanborn strip chart recorder.

In Series E through J, the maximumwidth of crack was measured at an ar-bitrary "service load" of 0.55 V,, (de-sign) and 0.5Nu.

Testing procedureThe specimens of Series A and B weresubjected to an incrementally increasingload until failure occurred.

The specimens of Series C and Dwere first subjected to the horizontalforce N. This horizontal force was thenmaintained constant while the verticalload was increased incrementally, untilfailure occurred. It was considered that

applying the full ultimate horizontalload before applying the vertical loadwas the most severe loading conditionto which a corbel could be subjected,and that the measure of strength so ob-tained would therefore be conservative.

In the remaining tests, it was decidedto try and obtain some measure of ser-viceability as well as strength, by mea-suring crack widths at service load. Itwas arbitrarily decided to consider thatthe dead load shear VD and the liveload shear VL were equal. Then, ifth = 0.85, we have VD = VI = 0.275 V,,.

Also, since ACI 318-71 requires thatthe horizontal force be treated as alive load, the service horizontal forceN = 0.5N. The following loading se-quence was used:

(1) N = 0, V increased incremental-ly to VD = 0.275 V,.

(2) V = VD, N increased incremen-tally to 0.5 N.

(3) N = 0.5Nu, V increased incre-mentally to VD + VL = 0.55 V.

(4) V = Vp + VL, N increased incre-mentally to Nu.

(5) N = Nu, V increased incremen-tally until failure occurred.

The maximum crack widths at V =VD + VL and N = 0.5 N,, were re-corded, this being regarded as theservice load.

In all the tests the cracks weremarked at each increment of loading.

Test Results

Specimen behaviorSpecimens subject to vertical load

only—The first cracks to form were flex-ure cracks, which propagated from theintersection of the column face and thehorizontal face of the corbel. Thesecracks penetrated about halfwaythrough the depth of the corbel atabout half the ultimate shear, at whichtime diagonal tension cracks appearedin the corbels.

These cracks were aligned roughly

64

along a line running from the intersec-tion of the sloping face of the corbeland the column face, to a point betweenthe inner edge of the bearing plate andthe center of the bearing plate. Theinitial length of these cracks was aboutone-third the effective depth. As the ap-plied shear was further increased, thediagonal tension cracks increased inlength, at first rapidly, then much moreslowly as the ultimate load was ap-proached.

In the case of the corbels without stir-rups, essentially only one diagonal ten-sion crack formed in each corbel and avery sudden failure occurred when theconcrete at the head of these crackssheared through. This type of diagonaltension failure was referred to by Krizand Raths2 as a "diagonal splitting"failure.

In the case of the corbels with hori-zontal stirrups, additional inclinedcracks occurred, usually starting as flex-ure cracks at the horizontal face of thecorbel. The failure of Corbel B1 wasclassified as a flexural failure, since itwas characterized by wide opening ofthe flexural cracks, while the diagonaltension cracks remained fine.

The failures of the remaining corbelsof Series B were classified as "beamshear" type failures, i.e., a shear failureof the type which occurs in a reinforcedconcrete beam with stirrup reinforce-ment. In this case, the flexure cracks re-mained fine and failure was character-ized by widening of one or more diago-nal tension cracks and the shear-com-pression failure of the concrete near theintersection of the sloping corbel faceand the column face. Failure was quiteabrupt, but less brittle and with morewarning than in the case of the diago-nal tension failures of the corbels with-out stirrups. In all cases, the deflectionswere very small and provided no warn-ing of failure.

Specimens subject to both verticaland horizontal Ioads—Two types of

crack were caused by horizontal force:,(1) More or less vertical cracks due

to the direct tension stresses producedin the concrete by the horizontal force,and

(2) Cracks approximately alignedwith the main tension reinforcementand evidently caused by the splittingaction of the bar deformations due tothe high bond stresses caused by thetransfer of some of the horizontal forcefrom the main tension reinforcement tothe surrounding concrete.

The first cracks to form due to verti-cal loads were flexure cracks, whicheither occurred independently or as ex-tensions of the more or less verticalcracks caused by the horizontal force.Diagonal tension cracks initiated at be-tween 40 and 75 percent of the ulti-mate load, the average diagonal tensioncracking shear being 58 percent of ulti-mate. In some cases, they propagatedfrom the end of a vertical crack causedby the horizontal force, but in othercases, they initiated independently andcut across existing vertical crackscaused by the horizontal forces.

A typical example of the develop-ment of cracks is shown in Fig. 5. Thedashed lines indicate cracks which oc-curred when the horizontal load was in-creased. Full lines are cracks which oc-curred when the vertical load was in-creased.

Corbel Specimen D1 failed in flexurewith wide opening of the flexure crackadjacent to the column face as the maintension reinforcement yielded, the di-agonal tension cracks remaining fine.The failures of all the other specimenssubject to both vertical and horizontalforces (except Specimen E4) were clas-sified as "beam shear" type failures andwere characterized by widening of thediagonal tension cracks at failure andshear compression failure of the con-crete adjacent to the intersection of thesloping face of the corbel and thecolumn face.

PC! JOURNAL/March-April 1976 65

Cracking at Service Load

Cracking at Ultimate

Cracking during application of vertical loadCracking during application of horizontal load - --

Fig. 5. Typical development of cracking(Series E through H).

However, as may be seen from Ta-bles 4 through 8, in most cases themain tension reinforcement yielded be-fore failure occurred. In the case ofSpecimen E4, a brittle fracture of thereinforcement occurred as the reinforce-ment yielded. This was caused by em-brittlement of the reinforcement due tothe welding of the bearing plates ontothe reinforcement. Subsequent speci-mens were annealed after welding toeliminate this problem.

The maximum width of crack at ser-vice load in the specimens of Series 1 . ,G, H, and J was less than 0.01 in. in allcases. The actual widths are shown inTables 6 through 8. Complete recordsof deflection, reinforcement strain andcracking have been reported else-where. 7 ' 8

Ultimate strengthThe loads acting at failure, i.e., the

horizontal force N,, and the ultimate

66

shear force V,u (test) are shown in Ta-bles 4 through 8, together with thenominal ultimate shear stress vu (test)V,, (test)/bd and the shear forceV. (test) which was acting at yield ofthe main reinforcement. The type offailure exhibited by each specimen isalso indicated in these tables as follows:

D.T. = diagonal tension failureF = flexural failureB.S. = "beam shear" type failure

Discussion of test resultsMinimum stirrup reinforcement-The

brittle and complete diagonal tensionfailure of the Series A corbels without

stirrups, provides additional support forKriz and Raths' contention that all cor-bels should be reinforced with horizon-tal stirrups in addition to the main ten-sion reinforcement, in order to elimi-nate the possibility of this type of fail-ure occurring.

Except for Specimen B3, the remain-ing corbels all had horizontal stirrups,the yield strength of which was half theyield strength of that part of the maintension reinforcement not resisting thehorizontal force Nu, i.e.,

Ahfv1 = r/2(A$ ff - Nu)

In all these corbels which were made

Table 4. Test data-Series A & B (all forces in kips).

Specimen No. A2(1) A3(1) B1 82 83A

Nu 0 0 0 0 0

Vu (test) 35.6 28.0 47.0 38.9 42.1

vu(test)(psi) 664 526 870 725 791

vu/fC, 0.18 0.14 0.24 0.21 0.19

V(test) * * 39.7 + (2)

Vu(test) 35.6 28.0 39.7 38.9 42.1

Vu (S.F.) 39.4 41.0 39.2 37.0 42.6

Vu (Mod.S.F.) 44.5 61.5 47.5 54.9 66.6

V(flex) 39.2 44.6 41.1 39.0 44.1

V1 (calc) 39.2 41.0 39.2 37.0 42.6

Vu(test)0.91 0.68 1.20 1.05 0.99

Vl ca c

V(test)0.91 0.63 1.01 1.05 0.99

Vl calc

V2 (calc) 39.2 44.6 41.1 39.0 44.1

Vu(test)0.91 0.63 1.14 1.00 0.95

V2 ca c

V(test)0.91 0.63 0.97 1.00 0.95

V2 calc

FailureType

D.T. D.T. F. B.S. B.S.

* 'Did not yield + Strain at failure not known

(1) No stirrup reinforcement

(2) Main reinforcement very close to yield at failure

"D.T." denotes a diagonal tension failure

"F" denotes a flexural failure

'B.S." denotes a beam-shear type failure

1.

2.

3.

4.

5.

6.

7.

8.

9.10.

11.

12.

13.

14.

15.

16.


rnwTable 5. Test data-Series C & D (all forces in kips).

Specimen No. C1 C2 C2A 03 Dl 02 03

Nu 32.0 34.0 30.0 30.5 30.0 30.0 30.0

Vu (test) 44,0 40.0 40.5 37.6 28.0 34.0 32.8

vu (test)(psi) 826 751 760 706 525 638 616

vu/f 0.21 0.20 0.21 0.16 0.13 0.17 0.17

V(test) 38.0 + (1) * (2) 19.0 24.2 25.0

V(tent) 38.0 40.0 - 37.6 19.0 24.2 25.0

V5 (S.F.) 42.6 39.6 39.5 42.3 29.7 40.5 39.4

Vu (Mod.S.F.) 48.2 58.1 54.7 69.5 38.2 48.7 57.9

V(flex) 36.2 37.1 33.2 40.1 21.8 24.2 26.3

V 1 (calc) 36.2 37.1 33.2 40.1 21.8 24.2 26.3

Vu(test)1.21 1.08 1.22 0.94 1.28 1.40 1.25

V ltalc

V'(test)1.05 1.09 - 0.94 0.87 1.00 0.95

V 1talc

V2 (calc) 36,2 37.1 33.2 40.1 21.8 24.2 26.3

Vu(test)

V2 talc

1.21 1.08 1.22 0.94 1.28 1.40 1.25

V'(test)

V 2 talc

]90 1.08 - 0.94 0.87 1.00 0.95

Failure

TypeB.S. B.S. B.S. B.S. F. B.S. B.S.

* Did not yield + Strain at failure not known

(1) Yielded, but Vy not known

(2) Main reinforcement very close to yield at failure

"B.S." denotes a beam-shear type failure

F. denotes a flexural failure

Table 6. Test data-Series E (all forces in kips)Specimen No. El I E2 E3 E4

i1 32.5 34.4 35.7 35.1

Vu (test) 55.0 46.0 48.5 35.5

vu (test)(psi) 1240 1035 1095 800

vu/f^ 0.31 0.23 0.26 0.20

Vy (test) 50.5 35.0 37.5 33.2

VU(test) 50.5 35.0 37.5 33.2

Vu (S.F.) 36.0 35.5 35.5 35.5

V0 (Mod.S.F.) 42.1 42.8 56.1 53.9

V(flex) 69.9 34.8 36.0 35.1

V 1 (calc) 36.0 34.8 35.5 35.1

Vu(test)1.53 1.32 1.37 1.01

U 1talc

V(test)1.40 1.01 1.05 0.95

V 1talc

V2 (calc) 42.1 34.8 36.0 35.1

Vu(test)1.31 1.32 1.35 1.01

V2 talc

V(test)1.20 1.01 1.04 0.95

V2 talc

Failure

TypeB.S. B.S. B.S. (1)

Service Load

Crack Width (in)0.002 0.003 0.003 0.003

(1) Brittle fracture of main tension reinforcement


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

1.

2.

3.

4.

5.

6.

7.

B.

9.

10.

11.

12.

13.

14.

15.

16.

C)C-0C

13Z

ZZNO

CD

•.

2.

3.

4.

5.

6.

7.

8.

9:

10.

•1.

12.

13.

14.

15.

16.

17.

Table 7. Test data-Series F, G & J (all forces in kips)Specimen No. F2 F3 F4 F4A 54 34

Nu 35.6 35.7 35.4 35.5 0 25.0

V5 (test) 36.5 24.0 24.0 23.5 24.0 21.5

vu (test)(psi) 820 540 540 530 530 485

vu/fc 0.22 0.15 0.13 0.14 0.14 0.13

V(test) 28.0 * * * * 21.5

VU(test) 28.0 24.0 24.0 23.5 24.0 21.5

Vu (S.F.) 33.0 33.1 35.5 33.0 34.0 22.6(1)

V5 (Mod.S.F.) 33.0 33.1 35.5 33.0 34.0 22.6(1)

V(flex) 36.2 35.4 35.4 33.0 34.0 24.8

V1 (calc) 33.0 33.1 35.4 33.0 34.0 22.6

V5 (test) 1.11 0.73 0.63 0.71 0.71 0.95

V 1 (ca c)

V(test)0.35 0.73 0.68 0.71 0.71 0.87

V 1calc

V2 (calc) 33.0 33.1 35.4 33.0 34.0 22.6

V5(test)

V2 calc1.11 0.73 0.68 0.71 0.71 0.87

V(test)0.85 0.73 0.68 0.71 0.71 0.87

V2 talc

Failure

TypeB.S. B.S. B.S. B.S. B.S. B.S.

Service Load

Crack Width(in)0.004 0.005 0.008 0.008 0.008 0.005

Table 8. Test data-Series H (all forces in kips).

Specimen No. 91 H2 H3 H3A H3B

1u 35.3 35.7 35.6 36.0 35.6

Vu (test) 67.0 50.0 47.4 39.6 46.1

v u (test)(psi) 1510 1125 1065 890 1040

v/fcU

0.39 0.29 0.28 0.23 0.27

V(test) 44.9 44.5 42.5 39.6 45.7

V(test) 44.9 44.5 42.5 39.6 45.7

Vu (S.F.) 34.8 34.9 34.2 35.2 35.9

Vu (Mod.S.F.) 48.1 52.3 51.3 5'.7 50.9

V(flex) 82.0 53.1 52.4 53.5 52.4

V 1 (calc) 34.3 34.9 34.2 35.2 33.9

Vu(test)1.93 1.43 1.38 1.13 1.36

V 1 (ca lc)

VU(test)1.29 1.28 1.24 1.13 1.35

V lcalc

V2 (calc) 43.1 52.3 51.3 52.7 50.9

Vu(test)1.39 0.96 0.92 0.75 0.91

V2 talc

VU(test)0.93 0.85 0.S3 0.75 0.90

V2 calc

Failure

TypeB.S. B.S. B.S. B.S. B.S.

Service Load

Crack Width (in)0 0.002 0.003 0.004 0.004

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13:

14.

15.

16.

17.

* Did not yield "B.S." denotes a beam-shear type failure

0) (1) vu limited to 0.14fc in this case

CD


of sand and gravel concrete, and forwhich reinforcement strains were ob-tained, it is known that the stress in themain tension reinforcement at failurewas equal to or very close to the yieldpoint of the steel.

Furthermore, the failure was muchless abrupt than in the case of the cor-bels without stirrups. (The behavior ofthe all-lightweight concrete corbels willbe discussed later.)

It is considered that this behavior in-dicates that the amount of horizontalstirrup reinforcement provided in thesecorbels is adequate to eliminate prema-ture diagonal tension failures and topermit the potential strength of themain tension reinforcement to be devel-oped.

Providing the yield point of the stir-rup reinforcement is at least equal tothat of the main tension reinforcement,the following appears to be an appro-priate "minimum stirrup reinforcement"requirement:

"Closed stirrups or ties parallel to themain tension reinforcement, having atotal cross-sectional area A h not lessthan 0.50 (A$ — N,u/ shall be uni-formly distributed within two-thirds ofthe effective depth adjacent to themain tension reinforcement."

This amount of stirrup reinforcementwill prevent premature diagonal tensionfailure of the corbel and will permit theyield strength of the main tension rein-forcement to be developed. However,the failure of the corbel may be either aflexural failure or a beam shear typefailure after yield of the flexural rein-forcement. Either of these modes offailure is considered acceptable, sincethe full strength of the main tensile re-inforcement is developed.

Ultimate strength of sand and gravelconcrete corbels—Since Specimens A2and A3 without stirrups failed in diago-nal tension, it is appropriate to comparetheir strengths with that predicted by

Eq. (14). This equation predicts ulti-mate shear strengths of 32.2 kips and32.4 kips, respectively, for SpecimensA2 and A3. These compare reasonablywell with the test values of 35.6 and28.0 kips, considering the scatter inher-ent in the diagonal tension failurestrength of reinforced concrete beams.

From the viewpoint of design prac-tice, the ultimate strength results ofmost interest are those of the specimenswhich satisfied the minimum stirrup re-inforcement requirement proposedabove. In Tables 4 through 8, the mea-sured ultimate shear strengths of thesecorbels have been compared with theultimate shear strength calculated invarious ways:

(a) Using the shear-friction provi-sions of Section 11.15 of ACI 318-71,but setting the capacity reduction factor0 equal to unity. The term Atiff2 ,y wastaken as equal to

(AJ — N,) + AhfvyThis calculated shear strength is re-

ferred to as V,,, (S.F.). (See Line 7 inTables 4 through 8.) In the case of theall-lightweight concrete corbels, . thevalue of p. was multiplied by the coeffi-cient 0.75 as previously proposed.

(b) Using the modified shear-fric-tion equations previously proposed.3'6

(i) For sand and gravel concretevu = 0.8(pf, + opt,,) + 400 psi (16)

but not greater than 0.3f0.(ii) For all-lightweight concrete

v = 0.8(p f, + o) + 200 psi (17)but not greater than 0.2f', nor 800 psi.where (pf„ + ON) was taken as

(ASf^ + Ajfvv — Nu)/bdThis calculated shear strength is re-

ferred to as V,^ (Mod. S.F.) (see Line 8of Tables 4 through 8).

The ultimate vertical load corre-sponding to flexural failure, V(flex), isgiven by

V(flex) = [M Nu(h — d)] /a (18)The moment of resistance of the cor-

bel-column interface plane, Mu, was

70

based on a flexural reinforcementstrength equal to (Aj — N.u), and useof the equivalent rectangular stress dis-tribution, as defined in Section 10.2 ofACI 318-71. Any contribution of thehorizontal stirrup reinforcement to flex-ural strength was neglected (see Line 9of Tables 4 through 8).

V1 (calc) shown on Line 10 of thetables is the lesser of Vu(S.F.) andV(flex). This method of calculating theshear capacity of a corbel correspondsto the design procedure proposed earli-er in this paper.

Vo(calc) shown on Line 13 of the ta-bles is the lesser of V,,(Mod. S.F.) andV(flex). This method of calculating theshear capacity of a corbel correspondsto the design procedure proposed earli-er, modified by using the modifiedshear-friction relationship in place ofthe shear-friction provisions of ACI318-71.

It can be seen from Line 11 of the ta-bles that V1(calc) is a generally conser-vative estimate of the ultimate shearstrength, the average value ofV(test)/V1(calc) for Series B, C, D,and E being 1.20, and for Series H be-ing 1.45.

On Line 5 of the tables is given V,the vertical load at which the main ten-sion reinforcement yielded. It can beseen that in a number of cases, yield ofthe reinforcement occurred at a verti-cal load considerably less than the ulti-mate load. This behavior tends to begreatest for low values of a/d and forsmaller reinforcement ratios, such asSeries D. This behavior was also point-ed out by Kriz and Raths.2

The load factors specified in ACI318-71 are intended to provide a safetymargin against excessive cracking dueto yield of the reinforcement, as well asto provide safety against collapse. Theflexural ultimate strength of a rein-forced concrete beam calculated accord-ing to Section 10.2 of ACI 318-71, cor-

responds very closely to the moment atyield of the flexural tension reinforce-ment in a reinforced concrete beam ofusual proportions.

To provide the same safety marginagainst wide cracking in corbels due toyield of the main tension reinforcementwhen using the load factors of ACI 318-71, as is provided in the case of ordi-nary reinforced concrete beams usingthe same load factors, it is proposedthat the "useful ultimate strength," V',,be taken as the vertical load at yield ofthe main tension reinforcement or theultimate load when the main tension re-inforcement does not yield.

The measured useful ultimatestrengths V. (test) of the corbels test-ed is given on Line 6 of Tables 4through 8. It can be seen from Line12 of the tables that Vl (calc) is a goodpredictor of the useful ultimate strengthV'u, the average value of V'u(test)/Vl(calc) for Series B, C, D, and E being1.03 and for Series H being 1.26.

These results indicate that the pro-posed method of corbel design will en-sure both adequate strength and ade-quate serviceability.

The values of V,/test)/V2(calc)shown on Line 14 of the tables indi-cate that V2(calc) is also a good pre-dictor of the ultimate strength Vu(test),although slightly less conservative thanV1(calc). The average value of Vu(test)/Vo(calc) is 1.18 for Series B, C, D,and E and 0.99 for Series H. The re-sult obtained for Series H is the moresignificant since this series of corbelswas designed for maximum shear stressallowed by the modified shear-frictionequation, i.e., 0.3f',. With the excep-tion of Corbel H3A, this shear stresswas exceeded or closely approached.

(The reason for the low value of ul-timate strength given by Corbel H3A isnot known. As far as could be deter-mined, its physical properties were al-most identical with those of Specimens


H3 and H3B, which yielded higher ul-timate strengths. The yield strength Vi,,

was also low for this specimen.)On Line 15 of the tables are given

values of V'u(test)/V2(calc). The av-erage value of V'u(test)/V2(calc) is1.00 for Series B, C, D, and E and 0.85for Series H. The average for Series His low because the values of V5 (test) areall low relative to V(flex).

This probably indicates that at highultimate shear stresses such as 0.3f',the interaction between shear and di-rect stress in the flexural compressionzone reduces the average normal stressat failure to a value significantly lessthan that corresponding to the parame-ters of the equivalent rectangular, stressdistribution specified in Section 10.2.7of ACI 318-71.

This would lead to a reduction in theinternal lever arm of the interface planebetween the corbel and the column,with a consequent reduction in the mo-ment at yield of the reinforcement.

The safety factor against yield of thereinforcement will therefore be less ifthe modified shear-friction equation isused in design, and the ultimate shearstress is made equal to the maximum al-lowed, i.e., 0.3f'0.

However, it can be seen from Line 17of Tables 6 and 8 that the average val-ue of the maximum crack width at ser-vice load was the same for Series H andE, even though the shear stress at ser-vice load was 50 percent higher in Se-ries H than in Series E. In both cases,the average maximum crack width atservice load was very small, being ap-proximately 0.003 in.

Because the crack widths in the Se-ries H corbels were so small at serviceload and the average value ofV(test)/V2(calc) for Series H wasequal to 0.99, it is considered that themodified shear-friction equation, Eq.(16), could be used for shear design inthe corbel design procedure proposed,

in place of the shear-friction provisionsof ACI 318-71.

Ultimate strength of lightweight con-crete corbels-Data obtained in the testsof the all-lightweight concrete corbelsof Series F, G, and J are shown in Ta-ble 7. As already mentioned in the dis-cussion of the design of the specimensof Series F, it was found that the maxi-mum shear stress attainable decreasedas a/d increased, the failure at largervalues of a/d occurring by shear com-pression of the concrete before the yieldstrength of both the main tension rein-forcement and the stirrup reinforcementcould be developed.

The variation of the useful ultimateshear stress with the shear span todepth ratio a/d is shown in Fig. 6. Thevalue shown at a/d equal to zero is thevalue of 0.2f' previously developed inpush-off tests of all-lightweight con-crete.6

It is proposed that in the shear de-sign of all-lightweight concrete corbels,(using either the shear-friction equationor the modified shear friction equation),the maximum ultimate shear stress v,,

(max.) be limited to

a ,v,1(max.) _ (0.2 — 0.07 a f ^

(19)

but not more than (800 - 280 d l psi.

It can be seen in Fig. 6 that uuse ofthis equation will result in a close esti-mate of the maximum useful shear stressobtainable in an all-lightweight con-crete corbel. This limiting value wouldreplace the limiting value of "0.2f' nor800 psi" previously proposed for all-lightweight concrete when using eitherthe shear-friction equation or the modi-fied shear-friction equation, Eq. (17).

In Table 9, a revised comparison ismade of the measured and calculatedstrengths of the all-lightweight concrete

72

0.25

^

= (0.2 - 0.07 °-d ) fc

o

Push-offtestsfs)

O - Series FA - 64

+- J4

0.20

0.15

bdV^f

0.10

0.05

0 0.2 0.4 0.6 0.8 1.0

o/d

Fig. 6. Variation of maximum obtainable useful ultimateshear stress in all-lightweight concrete corbels, with the

shear span to depth ratio, a/d.

Table 9. Revised comparison of measured andcalculated ultimate strengths of lightweight concrete

corbels (all forces in kips).

Specimen No. F2 F3 F4 F4A 94 J4

Vu (test) 36.5 24.0 24.0 23.5 24.0 21.5

V(test) 28.0 * * * * 21.5

V(test) 28.0 24.0 24.0 23.5 24.0 21.6

Vu (S.F.) 27.9 25.3 23.2 21.3 22.2 20.9

Vu (Nod.S.F.) 27.9 25.3 23.2 21.3 22.2 20.9

V(flex) 36.2 35.4 35.4 33.0 34.0 24.8

V(calc) (1) 27.9 25.3 23.2 21.3 22.2 20.9

Vu(test)1.31 0.95 1.03 1.10 1.08 1.03

V calc

V-(test)1.00 0.95 1.03 1.10 1.08 1.03

V talc

* Did not yield

(1) For all these specimens, V 1 (calc) = V2 (calc) = V(calc). This is

because the calculated shear strengths V u (S.F.) and Vu(Nod.S.F.)

are both always governed by the maximum allowable shear stress as

given by Eq.(19).


corbels, using Eq. (19) as the upperlimit to the ultimate shear stress calcu-lated using both the shear-friction andmodified shear-friction equations. It canbe seen that use of Eq. (19) leads to aclose estimate of the useful ultimatestrength of all the all-lightweight con-crete corbels tested.

The shear strength of an all-light-weight concrete corbel is less than thatof a sand and gravel concrete corbelwith identical dimensions, reinforce-ment, and concrete compressivestrength, as may be seen from Series Eand F. The reason for this difference inbehavior is probably the difference insmoothness of the faces of the diagonaltension cracks which form in the cor-bcls.

The crack faces in the sand andgravel concrete are very rough. This isbecause the bond strength between thecement paste and the aggregate par-ticles is less than the tensile strength ofthe aggregate particles, so a crack gen-erally follows the interface of the ag-gregate particles and the cement paste.

The crack faces in the lightweightconcrete were smoother than in thesand and gravel concrete. This is be-cause the bond strength between thecement paste and the lightweight aggre-gate particles is greater than the tensilestrength of the lightweight aggregate,so a crack due to tension passes throughthe lightweight aggregate particles, andthe crack faces are relatively smooth.

Because of the smoothness of thecrack faces, it is probable that little ornone of the inclined compression forcecould be transferred across the principaldiagonal tension crack in the all-light-weight concrete corbels. Consequently,all of the diagonal compression forcewould have to pass through the com-pression zone above the tip of thediagonal tension crack in the all-light-weight concrete corbels.

However, in the sand and gravel

concrete corbels, some of the diagonalcompression force was apparently trans-ferred across the principal diagonal ten-sion crack by aggregate interlock ef-fects. (Evidence of this was local com-pression spalling adjacent to the crackfaces at ultimate.)

Consequently, only part of the diago-nal compression force would have topass through the compression zoneabove the tip of the principal diagonaltension crack. For a given appliedshear, the stresses in the compressionzone above the tip of the principal di-agonal tension crack would therefore begreater in the lightweight concrete cor-bel than in the sand and gravel concretecorbel.

Conversely, stresses in the compres-sion zone high enough to initiate fail-ure would occur at a lower appliedshear in the lightweight concrete cor-bels than in the sand and gravel cor-bels.

No tests were made of sanded light-weight concrete corbels, due to limita-tions of time and resources. It is prob-able, however, that the maximum ulti-mate shear stress developable for sand-ed-lightweight concrete would be simi-lar to that for all-lightweight concrete,because of similarity in shear transferbehavior previously demonstrated.6

The following equation has previous-ly6 been proposed for the shear trans-fer strength of sanded lightweight con-crete:

v,, = 0.8(pf, + o-N^) + 250 psi (20)but not more than 0.2f' nor 1000 psi.

By analogy with the case of all-light-weight concrete, it is proposed thatwhen Eq. (20) is being used in the de-sign of corbels, the upper limit to vushould be changed to

"but not more than (0.2 — 0.0-f'..

nor (1000 — 350 \)p

74

Conclusions for Design

On the basis of the study reported here, necessary to resist shear, whichever isthe following conclusions are drawn: the greater.

1. The design of corbels using theload factors in Section 9.3 of ACI 318-71 should be based on the "useful ulti-mate strength," in order that an ade-quate safety margin against wide crack-irig shall be maintained.

(In evaluating test data, the "usefulultimate strength" is defined as the ver-tical load at yield of the main tensionreinforcement, or the vertical load atfailure if yield of the main tension re-inforcement does not occur.)

2. Subject to the provision of mini-mum horizontal stirrup reinforcementaccording to Conclusion 3 below, the"useful ultimate strength" of corbelssubjected to a combination of verticaland horizontal loads can be calculatedwith satisfactory accuracy by taking itto be the lesser of

(a) The shear strength of the corbel-column interface, calculated usingeither the shear-friction provisions ofSection 11.15 of ACI 318-71° or themodified shear-friction equa tion ; * and

(b) The vertical load correspondingto the development of the flexural ulti-mate strength of the corbel-column in-terface, taking into account the addi-tional moment Nu(h-d), imposed on thecorbel by the horizontal tension forceN,,,, and using the provisions of Section10.2 of ACI 318-71.

3. A minimum amount of horizontalstirrup reinforcement must be providedin corbels to eliminate the possibility ofa premature diagonal tension failure.The yield strength of this stirrup rein-forcement, Ahf5, should be not less thanone-half the yield strength of that partof the main tension reinforcement nec-essary to resist moment, or one-thirdthe yield strength of the reinforcement

4. The use of the shear-friction pro-visions of Section 11.15 of ACI 318-71in the design of corbels, can be extend-ed beyond the limits currently set inSection 11.14 of ACI 318-71. An alter-nate Section 11.14 is proposed in Refer-ence 9.

5. In the design of all-lightweightconcrete corbels for shear, using eitherthe shear-friction or modified shear-fric-tion equation, the ultimate shear stressvu should not exceed:

(0.2 -0.07a)f

nor (800 — 280 a) psi

6. In the design of sanded-light-weight concrete corbels, using eithershear-friction or modified shear-friction,the ultimate shear stress v,, should notexceed:

(0.2- 0.07a)f;/a

nor (1000 — 350 psi

Acknowledgments

This study was carried out in the Struc-tural Research Laboratory of the Uni-versity of Washington. It was jointly sup-ported by the National Science Founda-tion, through Grant No. GK-33842X, andby the Prestressed Concrete Institute,through its PCI Graduate Fellowship pro-gram. Lightweight aggregate was suppliedby a member of the Expanded Shale, Clayand Slate Institute.

*Modified for lightweight concrete as proposedhere and" in Reference 6.


References

1. ACI Committee 318, "Building CodeRequirements for Reinforced Concrete(ACI 318-71)," American Concrete In-stitute, 1971.

2. Kriz, L. B., and Raths, C. H., "Connec-tions in Precast Concrete Structures—Strength of Corbels," PCI JOURNAL,V. 10, No. 1, February, 1965, pp. 16-61.

3. Mattock, A. H., Johal, L. P., and Chow,C. H., "Shear Transfer in ReinforcedConcrete with Moment or Tension Act-ing Across the Shear Plane," PCIJOURNAL, V. 20, No. 4, July-August,1975, pp. 76-93.

4. Zsutty, T. C., "Shear Strength Predic-tion for Separate Categories of SimpleBeam Tests," ACI Journal, V. 68,No. 2, February, 1971, pp. 138-143.

5. Zsutty, T. C., "Beam Shear StrengthPredicted by Analysis of ExistingData," ACI Journal, V. 65, No. 11,November, 1968, pp. 943-951.

6. Mattock, A. H., Li, W. K., and Wang,T. C., "Shear Transfer in LightweightReinforced Concrete," PCI JOURNAL,V. 21, No. 1, January-February, 1976,pp. 20-39.

7. Mattock, A. H., "Effect of Moment andTension Across the Shear Plane onSingle Direction Shear TransferStrength in Monolithic Concrete," R3-port SM74-3, Department of Civil En-gineering, University of Washington,Seattle, Washington, October, 1974.

8. Soongswang, K., "A Study of theStrength of Reinforced Concrete Cor-bels," MSCE thesis, University ofWashington, Seattle, Washington, Au-gust, 1975.

9. Mattock, A. H., "Design Proposals forReinforced Concrete Corbels." (To bepublished in May-June, 1976, PCIJOURNAL.)

Discussion of this paper is invited.Please forward your discussion toPCI Headquarters by August 1, 1976.

Coming in the Next IssueThe May-June, 1976, PCI JOURNAL willcontain a follow-up paper by Dr. Alan H.Mattock on "Design Proposals for ReinforcedConcrete Corbels."This paper will present a model code clauseand design procedures for corbels. Includedalso will be design examples for both normalweight and lightweight concrete corbels, usingboth the shear-friction and modifiedshear-friction approaches to shear transferdesign of the corbel-column interface.

76

Appendix—Notation

A,,. = area of shear plane, sq in.Ar = area of reinforcement necessary for

flexure, sq in.Ah = total area of stirrup reinforcement

parallel to main tension reinforce-ment, sq in.

A, = area of main tension reinforcement,sq in.

A t = area of reinforcement necessary toresist horizontal tension force N,sq in.

A,, = area of shear-friction reinforcement,sq in.

a = shear span; distance between a con-centrated load and face of support,in.

b = width of compression face of mem-ber, in.

C = resultant concrete compression forcein flexural compression zone

d = distance from extreme compressionfiber to centroid of tension rein-forcement, in.

f' = compressive strength of concretemeasured on 6 X 12-in, cylinders,psiyield point stress of stirrup rein-forcement, psi

f = yield point stress of reinforcement,psi

h = overall depth of corbel at columnface

jd = distance from centroid of maintension reinforcement to center ofaction of resultant concrete com-pression force C, in.

j,d = distance from centroid of horizon-

tal stirrup reinforcement to centerof action of resultant concretecompression force C, in.

j'd = distance from centroid of flexuraltension reinforcement to center ofaction of resultant concrete com-pression force if only flexural re-inforcement of strength (A,f5—N,)were acting in section, in.

Mu = ultimate moment of resistance ofcorbel-column interface plane, in.-kips

N.. = design ultimate tensile force oncorbel acting simultaneously withV.,,, kips

V(flex) = [M,,, — N,, (h-d)] /aV,, = ultimate shear strength, kipsV',, = useful ultimate shear strength,

kips=V,, or V. if main tension rein-

forcement does not yieldV, = shear acting at yield of main tension

reinforcement, kipsv,.. = nominal ultimate shear stress, psi

= V.ulçbbd= coefficient of friction used in shear-

friction calculationsp = A,,/A,, in modified shear-friction

equationp = A.,/bd in corbelsph = A/bdoNS = externally applied normal stress

acting across shear plane, psi (posi-tive if compression, negative if ten-sion)

0 = capacity reduction factor, as de-fined in Section 9.2 of ACI 318-71


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