Theory of Shear Transfer Strength of Reinforced Concrete

ACI STRUCTURAL JOURNAL TECHNICAL PAPER Title no. 84-816

Theory of Shear Transfer Strength of Reinforced Concrete

by Thomas T. C. Hsu, S. T. Mau, and Bin Chen

A theory of shear transfer in initially uncracked concrete is pre-sented. The theory is based on the truss model and incorporates a softened compression stress-strain relation along the concrete s/ruls. For reinforced concrete specimens experiencing shear transfer across a plane, acrilical zone in the vicinity of the shear plane is identified. Within this zone, the stress distribution is assumed to be approxi-mately uniform after the formation of cracks. The governing equa-tions derived from the theory can then be applied to this crilical zone to obtain strain responses for the given stress conditions. The ulti-mate shear transfer strength is identified by tracing the complete shear stress-strain history using electronic computer. Comparison of theo-retical predictions to 32 test results reported in the literature gives good agreement.

The theory predicts that steel reinforcement parallel to the shear plane also contributes to the shear transfer strength, while the shear-friction concept in the current design codes recognizes only the con-tribution of steel reinforcement crossing the shear plane. Since the current design codes are based on test specimens with heavy rein-forcement parallel to the shear plane, they could be unconservative for the practical cases where only light reinforcemenl is provided parallel to the shear plane.

Keywords: building codes; failure mechanisms; friction; reinforced concrete; reinforcing steels; shear properties; shear strength; stress-strain relationships; structural analysis.

The problem of shear transfer across a plane in con-crete has been studied extensively in the past 15 years. Through experimental observations, it is established 1 that there are basically two kinds of distinctively dif-ferent behavior in shear transfer problems: shear trans-fer across an initially cracked plane, and shear transfer across an initially uncracked plane. The behavior in the former case is governed largely by the shear-slip char-acteristics of the cracked plane. Aggregate interlock, dowel action, and constraints in a direction normal to the shear plane affect the resistance to shear. 23.457 The final failure occurs along the existing crack [Fig. I (a)] with little or no additional cracks formed across the ex-isting crack, 8 except in cases with a high percentage of steel crossing the initial crack. 5. 7 For design purposes, the shear strength is predicted using an empirical for-mula9 based on the shear friction theory.s.to.tt

In contrast, shear failure across an initially un-cracked plane occurs after numerous cracks formed in a direction inclined to the shear plane [Fig. I (b)]. The final failure is usually due to the crushing of concrete

ACI Structural Journal I March-April 1987

in the compression struts formed approximately paral-lel to the direction of the cracks. 12. 16 The compression in the struts and the tension provided by the reinforcing bars across and parallel to the shear plane constitute a truss-like action. Although this truss-like action is well recognized, 18 direct application of the truss model would result in a much higher prediction of the shear strength. An attempt was made8 to bring the predicted shear strength in line with the measured strength by in-troducing shear stresses in the compression strut result-ing in a biaxial failure condition. 1718 However, this ad-ditional shear stress in the compressional strut consid-erably complicates the truss model theory.

The fundamental difficulty in predicting the shear transfer strength of initially uncracked concrete is in the uncertainty of the compressive strength of the strut. In a study of the behavior of reinforced concrete panels

Shear Plane

(a) Initially Cracked (b) Initially Uncracked

Fig. 1-Shear transfer test specimens

Received Apr. 7, 1986, and reviewed under Institute publication policies. Copyright 1987, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright propri-etors. Pertinent discussion will be published in the January-February 1988 ACI Structural Journal if received by Sept. I, 1987.

149

Thomas T. C. Hsu, FACJ, is a professor in the Civil Engineering Department at the University of Houston. Dr. Hsu is the author of many technical publi-cations, and in 1965 was the corecipient of ACt's Wason Medal for Materials Research. He is a member of ACT Committees 215, Fatigue; 358, Concrete Guideways; and joint ACI-ASCE Committees 343, Concrete Bridge Design; and 445, Shear and Torsion.

ACJ memberS. T. Mau is an associate professor in the Department of Civil Engineering, University of Houston. He received his BS and MS degrees from National Taiwan University and his PhD from Cornell University. He is a member of ACt Committee 446, Fracture Mechanics.

Bin Chen is a PhD candidate in the Department of Civil Engineering, Univer-sity of Houston. He received his MS degree from Tong Gi University, Shang-hai, China, in /98/.

Reinforced Concrete

Concrete

Concrete Reinforcement

d

Fig. 2-Truss model for reinforced concrete element

under predominately shear stresses, it was found that compressive strength of the diagonal struts formed af-ter the cracking of the concrete can be much lower than the standard cylinder strength. 19 20 This phenomenon has been called the softening of concrete. This soften-ing of the concrete struts is related to the tensile strain in a direction perpendicular to the struts. Using the softened stress-strain relation proposed in Reference 19, the shear strength and behavior of various reinforced concrete members had been predicted with good accu-racy. 21-2s

RESEARCH SIGNIFICANCE In this study, the softened truss model theory was

applied to the shear transfer problem and was found to be successful in predicting the shear transfer strength as well as the shear deformations of 32 initially uncracked specimens. Contrary to the well-known shear friction concept, the theory predicts that the ultimate failure is caused by the crushing of concrete in the compres-sional struts formed after cracking of concrete. Fur-thermore, the transverse reinforcement parallel to and in the vicinity of the shear plane also has an effect on the shear strength. Since the ACI shear friction provi-sions are based on test specimens with very high trans-

150

verse reinforcement percentages, the theory calls for additional test specimens with low transverse reinforce-ment percentages below 2 percent.

BASIC EQUATIONS The equilibrium, compatibility, and stress-strain

equations for a reinforced concrete element are first presented. Stresses and strains are considered positive in tension and are assumed uniformly distributed.

Stress transformation conditions (equilibrium) A concrete element is reinforced with longitudinal

bars in the /-direction and with transverse bars in the !-direction as shown in Fig. 2. It is subjected at its edges to the in-plane normal stresses a, and a, as well as the shear stresses r 11 After diagonal cracking, a series of diagonal compression struts is formed in the diagonal or d-direction, resulting in a truss-like action. It is assumed that the element takes only compressive stress ad in the direction of the compression struts, and ten-sion stress a, in the r-direction transverse to the compression struts. The shear stress rd, is assumed zero. The angle between the 1-t and d-r coordinate systems is designated as a. This angle is also the angle of inclina-tion of the compression struts with respect to the lon-gitudinal axis.

The stresses a1, a1, and Tit in the reinforced concrete element are resisted jointly by the concrete and the steel reinforcement. The stresses contributed by concrete are designated as a1"' a1"' and r,,, where the subscript c de-notes concrete. The concrete stresses in the two coor-dinate systems 1-t and d-r are transformed according to the usual stress transformation equations

(1)

(2)

Trrc = (ac - a,) sina COSO! (3)

These relationships can also be presented using Mohr's stress circle.

The steel reinforcement is assumed to contribute only normal stresses

p,, p,

j,,j,

a/s = Pt j, (4)

(5)

normal stress in steel in /- and !-direc-tions, respectively reinforcement ratio in /- and !-direc-tions, respectively steel stress in /- and !-directions, re-spectively

The total stress in a reinforced concrete element is the superposition of the concrete stresses, Eq. (1) to (3), and the reinforcement contribution, Eq. (4) and (5)


(6)

(7)

T11 = (ad - a,) sina cosa (8)

Strain transformation conditions (compatibility) Assuming that the strains are distributed uniformly

in the element, they can be transformed according to the following equations

(9)

(10)

"(1, = 2(Ed - E,) sina cosa (11)

where E1, E1 normal strains in the 1-t coordinate system

'Y1r shear strain in 1-t coordinate system normal strains in the d-r coordinate system (principal strains)

These conditions satisfy Mohr's strain circle.

Material laws The stress-strain relation in the direction of the

compression strut is represented by the following two equations suggested by Vecchio and Collins 19 and shown graphically in Fig. 3(a).

Ascending branch

Descending branch

a = _ J: l1 _ (EdiE 0 - 1/A)2l d A 2 - 1/A (12b)

where Ep = E/A is defined as the peak strain, with E0 taken as - 0.002. A is a coefficient to take care of the softening phenomenom and is expressed by

Substituting E1 and E, from Eq. (9) and (10) into Eq. (13), the expression for A can be simplified to

R A= d (14) where E, = E1 + E, - Ed. This relationship for E, can easily be observed from Mohr's strain circle . 26

The stress-strain relation in a direction perpendicular to the compression strut is shown in Fig. 3(b). Before the concrete is cracked, the ascending linear relation-ship is


O'"d

f I c

-T

Eq(12a) Eq(12b)

E =~ P >-.

(a) Compression Stress-Strain Relationship

Eq(15a)

Eq(15b)

(b) Tension Stress-Strain Relationship

Fig. 3-Softened stress-stress relationships for concrete

(15a)

where Ec = initial modulus of elasticity of concrete, taken to be - 2 fi IE,. This value is obtained by taking derivative of ad with respect to Ed in Eq. (l2a), and then taking Ed equal to zero.

For the descending branch after cracking the rela-tionship is assumed to be

if E, > E0 (15b)

where !a concrete cracking stress, taken as 4Jf: if 1: is expressed in psi. E,, = concrete cracking strain = fj E,. This expression is not the same as the equation given in Reference 19. In Reference 19, the term under the square root in the denominator is E, rather thanE, - E0 This correction makes the resulting Eq. (15b) consistent with the condition that a, = ;:, when E, = E". The effect of this modification is minor, since Ecr is usually much smaller than E,.

151

T R.

1 Critical Zone (b)

(a)

Fig. 4-Push-off test specimen (Reference 13) and crit-ical zone

;; = E,E, if E, ::::;; Ely (16a)

;; = ;;y if E, > Ely (16b)

fr = E,EI if E1 ::::;; E1y (17a)

fr = /ry if E1 > E1y (17b) where E, = Young's modulus of elasticity of the rein-forcement. fry, fry = yield stresses of the longitudinal and transverse reinforcement, respectively.

CONDENSATION OF EQUATIONS AND SOLUTION PROCEDURE

The preceding 11 equations, Eq. (6) through (12) and (14) through (17), contain 14 variables:;;, fr, a,, an r11 , E1, En "f1n ad, a, Ed, E, a, and A. For the shear transfer test specimens studied in this paper (described later), a, is specified. The other two stresses a1 and r 11 are related to the external applied load P1 by

where Ka

152

b h

(18)

(19)

coefficient describing the nonuniform distribution of stress a1 coefficient describing the nonuniform distribution of stress Tc thickness of test specimen (Fig. 4) width of test specimen in the longitudi-nal direction (Fig. 4) and length of shear plane in the transverse direction (Fig. 4)

Eliminating P1 from Eq. (18) and (19), but keeping Ka and Kr as variables

(20)

Denoting

(21)

and substituting K into Eq. (20), we obtain

(22)

where K = ratio of maximum transverse stress to max-imum shear stress. If the stresses are distributed uni-formly over the whole specimen, Ka and Kr equal to unity, and K = 1/h.

In this study, K will be assumed a known value in the critical zone. The justification for assuming a constant K shall be explained later. Using Eq. (22) and a speci-fied a,, the 14 variables listed previously are reduced to 12 unknowns. By selecting one of them as a known value, the remaining 11 unknowns can be solved by the 11 equations, Eq. (6) through (12) and Eq. (14) through (17). A solution procedure can be implemented by first reducing the number of equations.

It should be observed that the three stress-strain re-lationships of concrete, Eq. (12), (14), and (15), are given in the d-r axis, and are expressed in terms of six unknowns, ad, a,, Ed, E, a, and A. It is, therefore, pos-sible to transform the stresses and strains in the 1-t axis (!;, fr, E1 EJ onto the d-r axis, so that the equations can be reduced to five equations containing the six un-knowns. The two equations besides the three stress-strain relationships are derived as follows:

Substituting Eq. (16) into Eq. (6) and using Eq. (9) for E1

E1 < E1y a, = ad cos2a + a, sin2a + p,E,(Ed cos2a + E,sin2a) (23b)

Equating Eq. (7) and (8) and noting that a1 = Kr11 from Eq. (22)

(ad - a,) K sina cosa = ad sin2a + a, cos2a + P1/r (24)

Substituting Eq. (17) into Eq. (24) and using Eq. (10) for E1

(ad - a,)K sina cosa = ad sin2a + a, cos2a + P1/ry (25a)

(ad - a,)K sina cosa = ad sin2a + a, cos2a + p1Es(Ed sin2a + E, cos2a) (25b)


Note that Eq. (23) and (25) are expressed in terms of the six unknowns listed in the preceding paragraphs. By selecting a value for Ed, the other five unknowns, ad, a, E, a, and A, can be solved by Eq. (12), (14), (15), (23), and (25). Ed is selected because it is expected to vary monotonically as the load is increased. Once these six unknowns are obtained, the stress and strain in the 1-t axis (T,, E,, E,, "f1, j,, j,) can be easily calculated.

The iterative procedure to solve the five nonlinear si-multaneous algebraic equations is as follows:

1. Select a value for Ed. 2. Assume a value of a,. 3. Solve forE, from the stress-strain curves of Eq.

(15)

E, I 0.005 (I, [_ a,. a,

E =-, E,.

4. Find A from Eq. (14)

5. Find a" from Eq. (12)

6. Solve for a from Eq. (23)

then calculate sin2a, sina, and cosa. 7. Solve for a, from Eq. (25)

a, aAK sina cosa

aiK sina cosa - sin2a) - p,s(Edsin2a + E,COS2a)

K sina cosa + cos2a

8. If the calculated a, is close enough to the assumed a, value, a set of solution ad, a, E, a, and A has been obtained for the selected Ed value. Otherwise, a new a, is calculated by a bisection method and Steps 2 to 7 are repeated.


9. Select another value of Ed and repeat Steps 1 through 8. In this way, a set of solution for various Ed values can be obtained.

10. The value of Tit, E1, E, "f1,, j,, and j, can be calcu-lated from Eq. (8), (9), (10), (11), (16) and (17), respec-tively. The relationship of any two variables, such as T1, versus "(1,, can be plotted.

COMPARISON WITH TESTS The theory described in the previous steps will now

be applied to the shear transfer problem. A typical test specimen for shear transfer across a vertical shear plane is shown in Fig. 4. To apply the theory, it is necessary to know the ratios among the three in-plane stresses a,, a, and Tit. These stresses should also be uniformly dis-tributed over the region of interest. However, exami-nation of the test specimen in Fig. 4 shows that the stresses cannot be expected to distribute uniformly over the entire specimen, nor can they be expected to be uniform in the central test region. Before cracking, the shear stress along the shear plane T1, should be consid-erably larger near the two ends of the shear plane where the open slot disrupts the smooth geometry and intro-duces local stress concentration. For the same reason, the transverse normal stress in the direction of the load a, is larger near the two ends of the shear plane. The normal stress in the longintudinal direction a, is small and can be neglected.

After diagonal cracking, a cracked region is ob-served in the vicinity of the shear plane and eventually leads to failure. This cracked region will be called the critical zone and is the shaded area shown in Fig. 4.

A typical width of this zone was observed to be about 2 to 3 in. for a 10 in. wide specimenY Within this zone, the extensive cracking of the concrete had an effect of redistributing the shear stress and the transverse nor-mal stress more evenly along the shear plane. The cracking also reduced the stiffness in the zone as com-pared to that outside of the zone. This would cause a redistribution of the compression stress in the trans-verse direction to become more evenly distributed across sections perpendicular to the shear plane. Thus, within this critical zone the stresses might be assumed to be uniform and the theory developed in the previous sections could be used. More specifically, the shear stress T" was estimated as the average stress over the entire shear plane (i.e., Kr = 1); and the compressive stress in the transverse direction a, was estimated as the average stress over a cross-sectional plane perpendicu-lar to the shear plane (i.e., Ka = 1). The K ratio, there-fore, becomes 1/h as shown in Fig. 4. The normal stress in the longitudinal direction a, is assumed to be zero [Fig. 4(b)].

In determining the reinforcement ratio, the cross-sectional area of the longitudinal steel across the shear plane is divided by the area of the shear plane to obtain p1, and the area of the transverse steel is divided by the cross-sectional area of a plane perpendicular to the shear plane to obtain p,. As the cracks in the critical zone are constrained by the two rows of transverse

153

~

Ill .e 1--

ui VI w cc: 1-VI cc: c w :z: VI

1500

1000

500

SPECIMEN M2

NON-SOFTENED CONCRETE

SOFTENED CONCRETE

-x- TEST

6. YIELDING OF LONGIT. STEEL

0 PEAK STRESS IN CONCRETE

0 0.005 0.01 0.015 0.02 0.025

SHEAR STRAIN, )"

Fig. 5-Shear stress-shear strain curves for Specimen M2 (1 psi = 6.895 kPa)

;; .!: l-

ui Ill w

2000

1500

1!: 1000 Ill a:: c w :1: Ill

500

I I

I I

I I

//'-

f -----i(

- SPECIMEN M8 -

-0- NON-SOFTENED CONCRETE

-e- SOFTENED CONCRETE

-X- TEST

-6- YIELDING OF LONGIT. STEEL

-0- PEAK STRESS IN CONCRETE

0.002 0.004 0.006 0.008 0.01

SHEAR STRAIN, 'Y

Fig. 6-Shear stress-shear strain curves for Specimen M6 (1 psi = 6.895 kPa) 154

steel, and the transverse steel is almost uniformly de-ployed over the whole section, the use of the average steel ratio over the whole section for the transverse re-inforcement ratio in the critical zone is considered ap-propriate.

The initially uncracked shear transfer tests reported in the literature were studied by tracing the shear stress-shear strain history by the method as previously de-scribed. Fig. 5 and 6 show the shear stress versus shear strain curves for Specimens M2 and M6, respectively, obtained from References 13 and 16. For convenience, the starting point is taken at the zero stress state and successive tracing is done from uncracked state to cracked state of the concrete, even though the imposed ratio between the normal stress and shear stress is only applicable to the cracked state as explained previously.

Fig. 5 and 6 each provide three curves: one experi-mental and two theoretical. One theoretical curve is based on the softened compression stress-strain rela-tionship given by Eq. (12) and (14) and shown in Fig. 3(a), while the other one utilizes the nonsoftened compression stress-strain curve specified by the CEB-FIP Model CodeY The CEB-FIP curve has a para-bolic-rectangular shape. The ascending parabolic curve up to a strain of 0.002 is described by Eq. (12a), if the coefficient 'A is taken as unity, and the continuing hori-zontal branch terminates at a strain of 0.0035. Fig. 5 and 6 show that the theoretical curves using the soft-ened compression stress-strain curve agree very well with the experimental curve. In contrast, the theoreti-cal curves based on the nonsoftened compression stress-strain curve overestimates considerably the maximum stress as well as the strain at maximum stress.

It should be mentioned that the truss model theory is not intended for the prediction of behavior before cracking. Tests in Fig. 5 and 6 show quite reasonably that the specimens before cracking are considerably stiffer than those predicted. Only when the ultimate strength is approached can the predicted shear stresses and shear strains become valid .

Push-off tests The two specimens M2 and M6, discussed previously

and shown in Fig. 4, are subjected to the so-called push-off loading. A total of 20 push-off tests is re-ported in References 13, 16, and 28 for initially un-cracked specimens. The test results are compiled in Ta-ble 1, including the predicted shear stresses 7"'"''' the shear strains at peak stress 'Ymax,n and the longitudinal steel strains at peak stress E1 The shear stresses were computed from Tmax,c = p,lbl, assuming K = l. Also, assuming Ka = 1, then K = 1/h. For specimens No. 1 through 14, K = 10/10 = 1, and for specimens No. 15 through 20, K = 10/12 = 0.83. This means that the shear stresses are assumed to be uniformly distributed across the shear plane and the transverse stresses are uniformly distributed on the plane perpendicular to that.

A comparison of the calculated and experimental maximum shear stresses is given in Fig. 7. The agree-

ACI Structural Journal I March-April1987

ment between the calculated and the measured shear strengths is indeed very good. The mean value of the ratio of the measured shear strength to calculated shear strength is 1.054 and the standard deviation is 0.069. Among the 20 tests, 4 of them (1.1A, 1.1B, 6.1, and M.1) have only one row of longitudinal steel across the shear plane. Since the theory is applicable to cases with more evenly distributed reinforcements, appreciable er-ror in prediction is expected for these four cases. In fact, Table 1 shows that the test values in these cases are 10 to 20 percent higher than the predicted values. This indicates that other factors, such as aggregate in-terlocking and dowel action, which are not considered in the present theory, may become relatively important in these cases of low steel percentage. If these four cases are discounted from Table 1, the remaining 16 re-sults show excellent agreement between theory and test, with a mean value of 1.030 and a standard deviation of 0.048, for the ratio of experimental shear strength to calculated shear strength.

Fig. 7 also compares the experimental shear strength to the theoretical shear strength based on nonsoftened concrete. This theoretical prediction overestimates the actual strength by about 50 percent.

To see how sensitive predicted shear strength is to the assumption of the stress ratio K, results from different K values ranging from 0.5 to 2.0 are calculated for the first 14 cases in Table 1 and are shown in Table 2. The ratios of the experimental to calculated maximum shear stresses are plotted in Fig. 8 as a function of the K-ra-

; 2- 1500

"' "' w a: ...

2.0 ..

'l. ,( >< ., ., E E

"' "' !/) !/) 1.5 MEAN !/) !/) w w a: a: 1- 1-!/) !/) a: a: C( C( 1.0 w w ::r:: ::r:: !/) !/)

x x C( C( :::E :::E 0.5 a: < (,) w

0 0.5 1.0 1.5 2.0 2.5

K-RATIO

Fig. 8-Ejject ojK-ratio on calculated maximum shear stresses

iii _g. 1500 M6( P. = 0.0264) " ( ~ "' .. E

"' ~ I ui M2(P, = 0.0088) l Ul 1000 I w 1 I t a: 1- rr:

y Ul I a: oT C( w I max,c ::r:: 500 I *T Ul practical I I TEST max,t x regkn -1 :-sPECIMEN C( I :;

0.02 0.04 0.06 0.08 0.1

TRANSVERSE STEEL RATIO, p1

Fig. 9-Ejject of transverse steel ratio on calculated maximum shear stresses (1 psi = 6.895 kPa)

Table 2-Effect of stress ratio K (push-off) Theoretical prediction

K = 2.0 K = 1.0 K = 0.5 711/U\,I' -y,

'Y' -y, Number Specimen psi T X 10' s T X 10 ' s T X 10' s I I. IA 750 753 3.098 y 684 4.333 y 661 5.79 y

2 I.IB 844 768 2.857 y 696 4.700 y 672 5.76 y

3 1.2A 1000 1025 3.353 y 923 3.933 y 885 5.19 y

4 1.2B 980 1046 2.904 y 930 4.267 y 893 5.59 y

5 1.3A 1100 1163 4.131 N 1109 4.887 y 1063 4.90 y

6 1.3B 1070 1182 4.370 y 1098 4.246 y 1046 5.10 y

7 1.4A 1360 1395 3.950 N 1326 5.028 N 1289 5.50 y

8 1.4B 1280 1233 3.882 N 1173 4.836 N 1142 5.26 N

9 1.5A 1400 1457 3.972 N 1377 4.822 N 1334 5.29 N

10 1.5B 1384 1340 3.726 N 1268 4.711 N 1230 5.14 N

II 1.6A 1432 1451 3.596 N 1366 4.614 N 1321 5.07 N

12 1.6B 1420 1380 3.589 N 1300 4.355 N 1258 4.98 N

13 6.1 800 739 2.651 y 672 4.441 y 649 5.45 y

14 6.2 1240 1304 3.712 N 1235 4.672 N 1198 5.09 N r = r,,_,._, in psi, "' = "(,,_,,,. S = longitudinal steel; Y = yielded; N = not yielded. I psi = 6.895 kPa.

Table 3-Effect of transverse steel (push-off) j,, f,'' 7/J/(1\,,

Number Specimen p, psi psi psi 0.0025* 0.0055* 16 M2 0.0088

20 M6 0.0264 *Assumed p,. tp, in actual test specimen. I psi = 6.895 kPa.

52,700 3900 980 700

52,700 4120 1320 1174

insensitivity of the shear strength to the assumed K-ra-tio simply means that the shear strength is not sensitive to the unevenness of the compressive stress distribution. The choice of the simple expression K = 1/h would have the advantage of simplicity and would provide sufficient accuracy.

The sensitivity of shear strength to the assumption made in determining the amount of transverse steel is studied by comparing the results for two of the test

156

777

1209

7 011111 .,, psi 0.011''' 0.022* 0.0293* 0.044* 0.0587' 0.0733* 0.088*

829 877 895 918 932 942 950

1243 1273 1284 1297 1304 1310 1313

specimens, M2 and M6 (p, = 0.0088 and 0.0264). The transverse reinforcement ratio is assumed to vary from p, = 0.0025 to 0.088 for cases shown in Table 3. The effect of the transverse steel ratio on the maximum shear stress is plotted in Fig. 9. Fig. 9 shows that changing the reinforcement ratio from the actual 0.0587 to 0.0293 or from 0.0587 to 0.088 results in a change of shear strength by Jess than 4 percent. If, however, the transverse steel ratio is reduced from the test specimen


< w

8.0 .--------r------.---.------::l

6.0

4.0 ......

2.0

SPECIMEN 15-20

0 2.0 4.0 6.0 8.0

CALC. SHEAR STRAIN AT MAX. STRESS, Y (X10" 3 ) max,c

Fig. 10-Comparison of shear strains at maximum stress

value of 0.0587 to the minimum practical values of 0.0025, then the reduction of shear strength is as large as 25 percent. While the test specimens are heavily reinforced in the transverse direction to force failure at the shear plane, such heavy reinforcement parallel to the shear plane is generally not available in practical structures. Thus, design guidelines based solely on these tests may not be conservative.

Also compared in Table 1 are the shear strains at peak stress for the six specimens for which the test val-ues were reported from measured slip across the shear plane. A test shear $train is calculated by dividing the measured slip by the gage distance across the shear plane. A comparison of the calculated and experimen-tal shear strains at maximum stress is given in Fig. 10. The predicted shear strains at peak stress are in gener-ally good agreement with the test values. The scatter can be explained by the fact that the slip across the shear plane was measured at one level across the shear plane, while the predicted shear strain is an average value.

Push-off tests with imposed longitudinal tensile stress

A tensile stress is applied in the longitudinal direc-tion to a test specimen, 1415 as shown in Fig. 11. If the

Table 4-Effect of normal stress (push-off) feo J:.

Number Specimen p, p, psi psi E,, 1/f- a 21 EIU 0.0105 0.0360 52,700 4060 0.0013 0.5031 55.68

22 E4U O.QI05 0.0360 49,100 3860 0.0011 0.4236 60.00 23 E6U 0.0105 0.0360 50,800 4120 0.0007 0.3178 65.13

24 FlU 0.0157 0.0360 52,200 4035 0.0017 0.5865 52.17

25 F4U 0.0157 0.0360 53,200 4175 0.0013 0.5219 54.55

26 F6U 0.0157 0.0360 51,000 4245 0.0012 0.4411 58.86 1 psi = 6.895 kPa.


p = 12"

1: 2.75"

~

I [ ~ I ~ JJ I I 7 r--- h = 14" ---1

Fig. 11-Push-off test specimen with longitudinal ten-sion stress (Reference 14) (1 in. = 25.4 mm)

tensile stress is kept constant during the test, the shear strength is expected to be weakened. Test results are available to show this weakening effect. 14 The present theory can be applied easily to such cases by assigning a constant normal stress in the longitudinal direction. In these test specimens K = 12/14 = 0.86. Shown in Table 4 are the measured and calculated shear strengths for six tests, including four with imposed longitudinal tension. The agreement is acceptable but not as good as

Normal ,,, ,, Tou"/' TI!W\:''

T,U.\,1 "(,.,,,., "(.,.,,,, stress,

X JO' X 10' pSI psi Tmax,c xlO' xiO' Steel psi 2.469 0.456 1089 1004 1.0847 5.145 6.266 y 0

3.745 0.515 946 757 1.2497 5.595 1.800 y 200

5.178 0.563 607 545 1.1138 5.449 1.390 y 400

1.699 0.352 1369 1173 1.1671 5.279 N 0

2.125 0.437 1143 1087 1.0515 4.878 y 200

3.581 0.546 1066 877 1.2155 5.779 y 400

157

2.0 .... It >< >< IV IV E E

'"" '"" 1.5

ui ui (/) (/) w w a: a: 1- 1- (/) (/)

1.0 a: a: < 0.5

would be underestimated using an average value. This indeed is the result as shown in Fig. 15. The calculated maximum shear stresses are mostly greater than the test values.

DISCUSSION The proposed theory for shear transfer across anini-

tially uncracked plane in reinforced concrete is shown to be sound. Excellent agreement between predicted and measured shear strength is obtained for cases where the stress distribution in a critcal zone encompassing the shear plane is reasonably uniform after cracking of the concrete. It is shown that the predicted shear strength is not sensitive to the distribution of the compressive stress applied in the transverse direction, nor is it sen-sitive to the amount of transverse reinforcement in the specimen if a large amount of transverse reinforcement is used. However, if a specimen is lightly reinforced in the transverse direction near the shear plane, the reduc-tion of shear strength can be substantial when com-pared to the shear strengths of specimens with heavy transverse reinforcement reported in the literature.

According to the theory, the softening of concrete after cracking of the concrete plays an important role in determining the shear strength. Both the amount of longitudinal reinforcement across the shear plane and the transverse reinforcement near the shear plane are important factors determining the degree of softening of the concrete and therefore contribute to the shear resistance across the shear plane. The longitudinal re-inforcement has a more important role than that of the transverse reinforcement in shaping the shear resistance of the shear plane, as can be seen from both the test re-sults and theoretical predicitons.

The literature contains reports of only nine test spec-imens to measure the slip across the shear plane. De-spite the small number of tests and slip measurements available, comparison of predicted and test shear strains at peak stress shows general agreement. This is another indication that the proposed theory is sound.

Since its incorporation into the ACI Building Code in 1971, the shear friction theory has been widely applied in design practice. In this theory, it is assumed that the shear transfer strength of reinforced concrete is a func-tion of the longitudinal steel crossing the shear plane and the compression strength of concrete, but it is not a function of the transverse steel parallel to the shear plane. This study shows, however, that the transverse steel near the shear plane could have a significant ef-fect on the shear transfer strength. Since the shear fric-tion equations in the ACI Building Code were based on test specimens with a very high percentage of trans-verse steel near the shear plane, they could be uncon-servative when applied to the practical cases of shear transfer with a small percentage of transverse steel.

CONCLUSIONS 1. Shear strength across an initially uncracked plane

in reinforced concrete can be predicted with good ac-curacy by the proposed theory. This theory is based on


1500 / /

.. o,../

.e ~~/ / / ~ / / )( / /~o,.. .. E /

... / / '

1000 / /

vi / ,. til "/ ,/ w a: / 1- / .-til / / a: / / < / / w / / % / . til ..: 500

/ / / /

< / / ::E / / a: / / / / SPECIMEN No. 2 7-32 w / / c.. >< / / w /'/ /.V

AI"

0 500 1000 1500

CALC. MAX. SHEAR STRESS, T max,c (psi)

Fig. 15-Comparison of maximum shear stresses for pull-off tests (1 psi = 6.895 kPa)

the truss model and incorporates a softened compres-sion stress-strain relation along the concrete struts.

2. The current ACI Building Code design criterion for shear strength across a plane is based on the shear friction theory. Since the shear-friction theory assumes that the shear transfer strength is not a function of the reinforcement parallel to the shear plane, the ACI pro-visions are derived from test specimens that use heavy reinforcement parallel to and near the shear plane. The proposed truss model theory, however, indicates that this reinforcement could have a significant effect on the shear transfer strength. For the practical cases of de-sign with small percentages of reinforcement parallel to the shear plane, the code provisions for initially un-cracked concrete might be unconservative.

3. More tests are required for shear transfer speci-mens with light reinforcement of 0.2 to 2 percent par-allel to the shear plane. The critical zone of the future test specimens also should be carefully instrumented to ascertain the strain field.

ACKNOWLEDGMENTS The author wishes to thank Professor Alan Mattock for supplying

the detailed test data against which the proposed theory is compared. This work is partially supported by the Texas Advanced Research Program.

b d E, E, J: f,., ;; };, }; };, h

NOTATION thickness of test specimen a direction parallel to the direction of the compression strut initial Young's modulus of concrete Young's modulus of reinforcing bars cylinder compression strength of concrete cracking strength of concrete stress in the longitudinal steel reinforcement yielding stress of longitudinal steel reinforcement stress in the transverse steel reinforcement yielding stress of transverse steel reinforcement width of test specimen in the longitudinal direction (Fig. 4)

159

K

K.

K,

P, r

E,

E,, "A

a,

a,

a,, a,

ratio of normal stress in transverse direction to shear stress in the 1-t coordinate system ratio of normal stress in transverse direction in the critical zone to that of the average normal stress ratio of shear stress in the critical zone to that of the average shear stress over the shear plane the longitudinal direction (a direction perpendicular to the shear plane); length of shear plane in the transverse direction (Fig. 4). external applied load in the transverse direction direction perpendicular to the direction of the compression strut transverse direction; a direction parallel to the shear plane angle of inclination of the compression strut from the /-axis shear strain in the 1-t coordinate system tensile strain at which concrete cracks compression strain in the direction of the strut normal strain in the /-direction yield strain of longitudinal steel reinforcement compression strain at maximum stress in a uniaxial stress-strain curve of concrete cylinder peak strain of softened concrete defined as E,/A normal strain in the r-direction normal strain in the /-direction yield strain of transverse steel reinforcement coefficient for softening effect reinforcement ratio of the longitudinal steel reinforcement ratio of the transverse steel compressive stress in the concrete strut normal stress in the /-direction normal stress in concrete in the /-direction normal stress in the r-direction

a, normal stress in the /-direction a,. normal stress in concrete in the /-direction a" normal stress in steel in the /-direction T1, shear stress in the 1-t coordinate system T,,. shear stress in concrete in the 1-t coordinate system T,., = shear strength of test specimen

REFERENCES I. ACI-ASCE Committee 426, "The Shear Strength of Reinforced

Concrete Members," Proceedings, ASCE, V. 99, ST6, June 1973, pp. 1091-1187.

2. Dulacska, Helen, "Dowel Action of Reinforcement Crossing Cracks in Concrete," ACI JouRNAL, Proceedings V. 69, No. 12, Dec. 1972, pp. 754-757.

3. Paulay, T., and Loeber, P. J., "Shear Transfer by Aggregate Interlock," Shear in Reinforced Concrete, SP-42, American Con-crete Institute, Detroit, 1974, pp. 1-15.

4. Paulay, T.; Park, R.; and Phillips, M. H., "Horizontal Con-struction Joints in Cast-in-Place Reinforced Concrete," Shear in Reinforced Concrete, SP-42, American Concrete Institute, Detroit, 1974, pp. 599-616.

5. Walraven, J. C.; Vos, E.; and Reinhardt, H. W., "Experiments on Shear Transfer in Cracks in Concrete, Part I: Description of Re-sults," Report No. 5-79-3, Stevin Laboratory, Delft University of Technology, 1979, 89 pp.

6. Walraven, J. C., "Experiments on Shear Transfer in Cracks in Concrete, Part 2: Analysis of Results," Report No. 5-79-10, Stevin Laboratory, Delft University of Technology, 1979, 132 pp.

7. Walraven, J. C., Aggregate Interlock: A Theoretical and Exper-imental Analysis, Delft University Press, 1980, 197 pp.

8. Mattock, Alan H., and Hawkins, Neil M., "Shear Transfer in Reinforced Concrete-Recent Research," Journal, Prestressed Con-crete Institute, V. 17, No.2, Mar.-Apr. 1972, pp. 55-75.

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160

10. Birkeland, Philip W., and Birkeland, Halvard W., "Connec-tions in Precast Concrete Construction," ACI JouRNAL, Proceedings V. 63, No.3, Mar. 1966, pp. 345-368.

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12. Mattock, A. H., "Shear Transfer in Concrete Having Rein-forcement at an Angle to the Shear Plane," Shear in Reinforced Concrete, SP-42, American Concrete Institute, Detroit, 1974, pp. 17-42.

13. Mattock, Alan H., "Effect of Aggregate Type on Single Di-rection Shear Transfer Strength in Monolithic Concrete," Report No. SM74-2, Department of Civil Engineering, University of Washing-ton, Seattle, Aug. 1974, 72 pp.

14. Mattock, Alan H., "Effect of Moment and Tension Across the Shear Plane on Single Direction Shear Transfer Strength in Mono-lithic Concrete," Report No. SM74-3, Department of Civil Engineer-ing, University of Washington, Seattle, Oct. 1974, 103 pp.

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16. Mattock, Alan H.; Li, W. K.; and Wang, T. C., "Shear Transfer in Lightweight Reinforced Concrete," Journal, Prestressed Concrete Institute, V. 21, No. I, Jan.-Feb. 1976, pp. 20-39.

17. Kupfer, Helmut; Hilsdorf, Hubert K.; and Rusch, Hubert, "Behavior of Concrete Under Biaxial Stresses," ACI JOURNAL, Pro-ceedings V. 66, No.8, Aug. 1969, pp. 656-666.

18. Zia, Paul, "Torsional Strength of Prestressed Concrete Mem-bers," ACI JOURNAL, Proceedings V. 57, No. 10, Apr. 1961, pp. 1337-1359.

19. Vecchio, F., and Collins, M.P., "Stress-Strain Characteristics of Reinforced Concrete in Pure Shear," Final Report, IABSE Col-loquium on Advanced Mechanics of Reinforced Concrete (Delft, 1981), International Association for Bridge and Structural Engineer-ing, Ziirich, pp. 211-225.

20. Vecchio, F., and Collins, M. P ., "The Response of Reinforced Concrete to In-Plane Shear Normal Stresses," Publication No. 82-03, Department of Civil Engineering, University of Toronto, Mar. 1982, 332 pp.

21. Hsu, Thomas T. C., and Mo, Y. L., "Softening of Concrete in Torsional Members -Theory and Tests," ACI JouRNAL, Pro-ceedings V. 82, No.3, May-June 1985, pp. 290-303.

22. Hsu, Thomas T. C., and Mo, Y. L., "Softening of Concrete in Torsional Members - Design Recommendations," ACI JOURNAL, Proceedings V. 82, No.4, July-Aug. 1985, pp. 443-452.

23. Hsu, Thomas T. C., and Mo, Y. L., "Softening of Concrete in Torsional Members- Prestressed Concrete," ACI JoURNAL, Pro-ceedings V. 82, No.5, Sept.-Oct. 1985, pp. 603-615.

24. Hsu, Thomas T. C., and Mo, Y. L., "Softening of Concrete in Low-Rise Shear Walls," ACI JouRNAL, Proceedings V. 82, No.6, Nov.-Dec. 1985, pp. 883-889.

25. Mau, S. T., and Hsu, Thomas T. C., "Shear Design and Analysis of Low-Rise Structural Walls," ACI JouRNAL, Proceedings V. 83, No.2, Mar.-Apr. 1986, pp. 306-315.

26. Hsu, Thomas T. C., Torsion of Reinforced Concrete, Van Nostrand Reinhold Co., New York, 1984, 516 pp.

27. CEB-FIP Model Code for Concrete Structures, 3rd Edition, Comite Euro-International du Beton/Federation Internationale de Ia Precontrainte, Paris, 1978, 348 pp.

28. Hofbeck, J. A.; Ibrahim, I. 0.; and Mattock, Alan H., "Shear Transfer in Reinforced Concrete," ACI JouRNAL, Proceedings V. 66, No.2, Feb. 1969, pp. 119-128.

29. Chatterjee, P., "Shear Transfer in Reinforced Concrete," MS thesis, Department of Civil Engineering, University of Washington, Seattle, 1971, 48 pp.


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