I. INTRODUCTION I. Introduction Reinforced concrete, like other structural ma- terials, has been the subject of extensive experi- mental and analytical research and the past sixty years have witnessed a steady advance in knowl- edge of the behavior of reinforced concrete members under static loads. With the aid of numerous tests, a rather complete understanding has been obtained of the ultimate strength of such members in pure flexure and under pure axial compression. In addition, there have been developed theories for members subjected to combined flexure and axial compression. However, no such extensive informa- tion is available for members subjected to com- binations of flexure and shear, or of flexure, com- pression and shear. In previous research, major emphasis has been placed on the evaluation of the contribution of web reinforcement, and the shear strength of a rein- forced concrete member has been interpreted in the light of a truss analogy. Experimental evidence, however, has forced certain modifications of the original truss analogy equation. The contribution of the beam itself, without the benefit of web reinforcement, has been taken into consideration. Furthermore, it has been found that the moment- shear ratio affects the ultimate strength in shear. These modifications, suggested by different authors, have retained essentially the truss analogy relation but have added new terms to account for effects other than that of web reinforcement. All the modified equations, however, have been derived experimentally for each given series of test speci- mens and have usually failed to give good correla- tion with other test data, outside the range of test variables for which the equations were derived. Current design specifications have apparently been based on certain minimum values obtained from tests. Although these specifications yield satisfactory or even conservative values in most practical cases, test specimens have been reported which failed in shear at a lower load than that given by the usual "safe working stresses." This indicates a definite need for a better understanding of the phenomenon of shear failure and for a more reliable set of design rules. 2. Object and Scope of Investigation The object of this investigation was to review and correlate the results of previous research in the field of shear and diagonal tension, to determine the modes and characteristics of shear failure of rein- forced concrete beams, and to establish a general expression for the shear strength of reinforced concrete beams under different loading conditions. The investigation was limited to members subjected to combinations of shear and flexure only. More than one thousand tests of beams having a wide range of physical properties and subjected to different types of loading were studied. A basi- cally new empirical equation was derived for the shear strength of simple-span rectangular beams without web reinforcement and under one or two symmetrical concentrated loads. It is shown herein that the basic equation can be interpreted with the aid of the conventional theory of compression fail- ures of reinforced concrete beams. This equation was first presented in a previous technical report.("* The basic empirical equation was extended to include beams with web reinforcement, and the amount of web reinforcement required to prevent shear failures was determined. Furthermore, the same equation was modified to apply to simple- span T-beams and restrained beams under sym- metrical concentrated loads. It was found also that the basic equation could be used to determine the shear strength of a reinforced concrete beam under uniform load and, possibly, under any type of distributed loading. 3. Acknowledgment The studies reported herein were made as a part of a research program to establish by analysis and by studies of the available test data criteria for the structural design of reinforced concrete box culverts. The work was carried out in the Structural Re- search Laboratory of the Department of Civil Engi- neering in the Engineering Experiment Station of * Superscripts in parentheses refer to corresponding entries in the Bibliography. ILLINOIS ENGINEERING EXPERIMENT STATION the University of Illinois in cooperation with Ohio River Division Laboratories, Corps of Engineers, U.S. Army, under Contract DA-33-017-eng-222. This bulletin is based upon a thesis by A. Laupa submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the Uni- versity of Illinois in 1953. The thesis was written under the direction of N. M. Newmark and C. P. Siess. 4. Notation The following notation is used: a = distance from end support to concentrated load in simple-span beams a= angle of inclination of web reinforcement with respect to axis of beam A = given by Eq. 45 A = compressive area of concrete as determined by "straight line" theory A, = area of web reinforcement b= width of rectangular beam or width of flange of T-beam b'= width of web of T-beam C=internal compressive force in concrete; (also various numerical coefficients as de- fined in text) d = distance from centroid of tension reinforce- ment to compression face of beam D= total depth of beam e = thickness of flange of T-beam o,= ultimate compressive strain in concrete, taken as 0.004 : = strain in steel at yield point Ec= modulus of elasticity of concrete E,= modulus of elasticity of reinforcing steel f =a distance as shown on Fig. 13 fc = compressive stress in extreme fiber of con- crete, given by straight line theory f' = compressive strength of 6- by 12-in. con- crete cylinders fe'= compressive strength of concrete cubes fr = modulus of rupture f,= stress in tension reinforcement f = stress in compression reinforcement f,= yield stress of tension reinforcement f,'= yield stress of compression reinforcement f, = stress in web reinforcement fA = yield stress in web reinforcement F = total force in web reinforcement, see Fig. 4 F= shape factor of T-beams, given by Eq. 34 g=a distance as shown on Fig. 13 h=a distance as shown on Fig. 13 Icr = moment of inertia of "straight line" cracked transformed section, either rectangular or T-section IR = moment of inertia of uncracked rectangu- lar section having the same width as the flange of an otherwise similar T-section IT = moment of inertia of uncracked T-section jd= internal moment arm kd = depth of compression zone of concrete as determined by "straight line" theory k,d = depth of compression zone of concrete at shear failure C ki= k3f,,bd' a parameter which determines k3f' kmbd the magnitude of the compressive force C. It is the ratio of the average compressive stress to the maximum compressive stress in concrete k2.= fraction of the depth of compression zone which determines the position of the com- pressive force C in concrete k3 = ratio of maximum compressive strength of concrete in beam to compressive strength of standard test cylinders K = (sin a + cos a) sin a L=span length of test beam L'= total length of test beam M= bending moment M,= shear-compression moment of beam with- out web reinforcement, given by Eqs. 18, 35, 44 M,= shear-compression moment of beam with web reinforcement, given by Eq. 28 E, n=- E= elastic modular ratio, taken as 10,000 5+ ± n' = = plastic modular ratio A. =-- , where A,= area of tension reinforce- ment P= bd,' where A,'= area of compression rein- forcement po = given by Eq. 43 p = given by Eq. 47 P = total load on beam P, =load which corresponds to M, Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS P, = load which corresponds to M,w q= f" = reinforcing index qcr = value of q which determines the boundary between initial flexural failure by crushing of concrete and by yielding of reinforce- ment, given by Eq. 32 r = A for rectangular beams bs sin a w- for T-beams b's sin a s = spacing of web reinforcement along axis of beam td= distance between centroids of tension and compression reinforcement V v= nominal shearing stress in concrete, bjd V V ,bkd or b as defined in text bkd = nominal shearing stress at ultimate loabD v.= nominal shearing stress at ultimate load ve=nominal shearing stress at ultimate load for shear-proper, given by Eq. 48 V= shearing force Vo = critical magnitude of shear needed to cause diagonal tension failures in the transition region of a/d between shear-compression and flexural failures-see Section 18 and Fig. 26 x= clear distance between two load blocks- see Fig. 27-a; also other distances as de- fined in text The following notation was used in designating modes of failure in the tables: B= bond C = flexural compression Cr = crushing at hooks DT= reported diagonal tension failures; most beams failed in shear, a few in bond as marked in the tables S = shear T = flexural tension T - S = tension with shear-type final collapse II. SIMPLE-SPAN RECTANGULAR BEAMS WITHOUT WEB REINFORCEMENT AND UNDER ONE OR TWO SYMMETRICAL CONCENTRATED LOADS 5. Review of Earlier Research In previous research, shear failures have been treated conventionally as failures in diagonal ten- sion. Since the real value of diagonal tension stress was generally difficult to determine, the unit shear- ing stress V v = bjd(1) was considered as a measure of diagonal tension. The effect of web reinforcement was taken into account by considering a beam acting as a truss, in which the top chord was formed by the com- pression zone of the concrete, the bottom chord by the longitudinal reinforcement, the tension web members by the web reinforcement, and the com- pression web members by the concrete in the web of the beam. From these assumptions the following equation was derived to represent stress in the web reinforcement: f = (2) where the value of K depended on the angle of in- clination of web reinforcement. It was realized that Eqs. 1 and 2 were approxi- mate in nature, and thus empirical data were used to correlate the real behavior of test beams with the above theoretical considerations. It was ob- served that measured stresses in the web reinforce- ment were, in general, considerably less than predicted by Eq. 2; this discrepancy was attributed to the fact that a portion of the total shear was carried by the concrete. In 1927 Richart(2) modi- fied Eq. 2 from the truss analogy in the following manner: v = C + Krf, where the constant C was found to vary from 90 to 200 psi, and it was stated that C probably depended "upon the percentage of web reinforcement used and also on the quality of the concrete." More complete conclusions regarding the con- tribution of the concrete to resist shear had been reached by Talbot some twenty years earlier. In 1909, Talbot3") reported that for beams without web reinforcement the ultimate nominal shearing stress v increases as the quality of concrete in- creases, as the amount of longitudinal reinforce- ment increases, and as the span length L decreases. These conclusions, however, were apparently dis- regarded by most later investigators. Only in rela- tively recent years have new attempts been made to evaluate in quantitative terms the contribution of the various elements of a beam to its strength in shear. In 1945 Moretto(4) presented the following equation for the shearing strength of a simply supported beam: v = Krfy, + 0.10 f,' + 5000 p This was essentially an extension of Eq. 3 sug- gested by Richart. In 1951 Clark(5) reported the following formula: v = 2500 -\/7 + 0.12 f' (d/a) + 7000 p (5) This equation was the first to account quantita- tively for all of the variables listed by Talbot in 1909 as influencing the shearing strength of rein- forced concrete beams. In a previous report(1) attempts were made to correlate the results of previous research and to investigate the validity of Eqs. 4 and 5 as well as other empirical equations in the following form: v = Krf,ý + Cif' + C2p (d/a) v = Krf. + [Cif/' + Cap] (d/a) All these attempts to relate the nominal shearing strength of simple-span reinforced concrete beams to a function consisting of the truss analogy term Krf,, and linear terms of fe' and p failed to give good correlation with test results. Thus, all of the Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS empirical equations which were derived for a cer- tain range of test variables were found to be not applicable outside that particular range. Since the introduction of the concept of the truss analogy over 50 years ago, major emphasis has been placed, in general, on the evaluation of the contribution of web reinforcement to shear strength. The contribution of the beam itself, with- out the benefit of any web reinforcement, has remained a relatively unknown quantity. Further- more, any uncertainties with regard to the contri- bution of web reinforcement have reflected directly on the contribution of the beam itself, thus render- ing both questionable. The first problem, therefore, is the evaluation of shear strength of a beam without web reinforce- ment. In the following section a general expression for the shearing strength of such beams is derived. 6. Derivation of Basic Empirical Equation After the formation of a diagonal tension crack, a reinforced concrete beam which does not fail in tension will fail either in the compression zone of the concrete or in bond. Although the cause of these two types of failure is different, their appearance is often very nearly the same. When a beam fails by the destruction of the compression zone, the shear force which was previously carried by the concrete is transferred to the level of the longi- tudinal reinforcement. This leads to splitting of the concrete along the reinforcing bars. When a beam fails in bond, however, slipping of the longitudinal reinforcement produces cracking of the concrete along the bars and effectively reduces the bending resistance of the section. This causes a concentrated angle change at the end of the diagonal crack and leads to a premature destruction of the compression zone of the concrete. Since the above phenomena take place simultaneously, it has often been difficult to determine the real cause of failure. In the early tests in which plain bars without end anchorage were used as tension reinforcement, bond failures were frequently considered as diagonal tension, that is, shear failures. In many of the more recent experimental investigations, however, the possi- bility of bond failure has been eliminated by the use of special types of end anchorage in addition to deformed bars having good bond characteristics. Splitting along the longitudinal reinforcement has still been observed and sometimes even considered as a primary mode of failure. This phenomenon, however, is believed to be secondary to the failure of the beam by destruction of the compression zone. Failure by destruction of the compression zone takes place, in general, under a concentrated load, at the section of maximum moment and maximum shear. The real cause of failure has not been gener- ally understood. It has been suggested that this failure is the result of the principal stresses, com- pressive or tensile, or of the maximum shearing stress. As has been mentioned, the conventional theory, treating shear failures as diagonal tension failures, considered the nominal shearing stress v as a measure of diagonal tension. Previous research has indicated that v is a function of the following variables: v= b- = F p, f', d ' Krfw All empirical equations suggested by different in- vestigators, however, have failed to give good cor- relation with all of the available test results. Furthermore, the conventional theory pictures the nominal shearing stress v as being distributed over the entire cross-section of a beam, uniform from the level of tension reinforcement to the neutral axis. The formation of a diagonal crack, however, radi- cally changes the state of stress in a reinforced concrete beam. Since there can be no transfer of stress across a crack, the nominal shearing stress cannot possibly be the criterion of shear failure which occurs at loads greater than that causing first crack, and the state of stress in the uncracked com- pression zone should be investigated in order to de- termine the probable cause of final failure. A basic equation for the shear strength of a simple-span rectangular beam without any form of web reinforcement and under one or two symmetri- cal concentrated loads was derived by considering the state of stress in the compression zone of the concrete. It was first assumed that the total shear- ing force V is resisted solely by the compression area of the concrete. For beams without compres- sive reinforcement the area of the compression zone is given by kdb, where the quantity k,d refers to the depth of the compression zone at shear failure. Thus the average shearing stress is given by v = V/kdb. It was further assumed that the ulti- mate shearing unit stress v. was a function of fe'. Test results have shown that the shear capacity of the compression zone decreases as the moment- shear ratio M/V increases. Since the ratio M/V ILLINOIS ENGINEERING EXPERIMENT STATION equals a for the beams considered, this effect has usually been taken into consideration by the d/a- ratio, and there seems to be a linear relationship between this ration and the shear capacity of the beam. Since both the horizontal compressive stresses and the vertical shearing stresses are as- sumed to be resisted by the same compressive area, it seems more reasonable to consider the shear- compressive force ratio V/C rather than the M/V- ratio as influencing the ultimate load in shear. For the type of beam under consideration it can be written that V/C = jd/a. Thus the ultimate shear- ing stress v, can be expressed as follows: V _ d v- kdb- jd F, (fZ') (9) If both sides of Eq. 9 are multiplied by the factor aks/dfc', and the ratio Fi (fe') /f,' considered as a new function F(fe'), Eq. 9 can be rewritten as: Va M bd2f-- = k,jF(f/') or bdf = kjF(f/') (10) Equation 10 is in a form which suggests that the criterion for shear failure is a limiting moment rather than a limiting shearing stress. There is some supporting evidence for this observation in previous test results. Beams with no web reinforcement tested by Clark(5' had the d/a-ratio as the only variable; all these beams failed at a nearly constant moment, although the total shear force at failure depended upon the location of the loads on the beams. In 1906, Moritz(6) reported a series of tests on small mortor beams with the d/a-ratio as the only variable, and his results again show that the ultimate moment was nearly the same for all posi- tions of loads. Thus the so-called shear or diagonal tension failures seem to be failures in compression, the criterion of failure being a limiting average compressive stress or a limiting total compressive force in the compression zone of the concrete. This type of failure differs from flexural compression failures only because the compressive area is re- duced in depth as a result of diagonal tension cracking. In Eq. 10 there are two main unknowns: the depth of the compression zone, kd, and the limit- ing average compressive stress, related to F(fc'). The quantity j can be considered as a constant, since it does not vary over a large range. The depth of the compression zone can be de- termined accurately for flexural failures, both in tension and in compression, by considering statical equilibrium and the strain relations involved. For shear failures, however, no theoretical relationship between the extent of diagonal tension cracking and the physical properties of the beam has been found. Consequently, th6 depth of the compression zone must be determined empirically. From previous in- vestigations it can be shown qualitatively that k, is a function of If' and p. Furthermore, this func- tion must be a complex one, since different empiri- cal equations considering v as a linear function of fc' and p have failed to agree with test results. In order to facilitate the empirical evaluation of k,, it was deemed advantageous to consider the ratio ks/k rather than k, alone. The value of k as de- termined by the straight line theory is also a function of f,' and p. It was felt that there might be some similarity between the functions repre- senting k, and k, so that the ratio k8/k might be easier to evaluate than k, alone. It was considered that if the ratio ks/k is either a constant or a func- tion of fC', Eq. 10 can be written as M bd2f,' = k F(f.') and the unknown function F(fc') can be directly from available test data. If this done, the ratio k,/k must also depend Eq. 11 must be rewritten as M -bd2f = k F(f.', p) (11) evaluated cannot be on p and (11a) Equation 11 was derived for beams without compression reinforcement. For beams with both tension and compression reinforcement, Eq. 11 must be modified to take into account the added effect of the compression reinforcement. If it is assumed that a beam fails before the compression reinforce- ment yields, an expression for the limiting moment of shear failure can be derived by considering that the presence of compression reinforcement increases the compression area of the transformed section by an amount equal to np'bd, the steel area trans- formed to concrete; thus: Ac = bkd + np'bd = bd (k + np') (12) This modified compression area leads to the follow- ing equation which corresponds to Eq. 11 for beams without compression reinforcement: M , = (k + up') F(f,') (13) Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS The quantity k refers to the theoretical depth of the compression zone as ordinarily determined from transformed areas. For beams with tension reinforcement only, the numerical value of k is obtained from the well-known equation k = (pn)2 + 2pn - pn (14) For beams with both tension and compression re- inforcement the following equation can be derived: k = V [n (p + p')]2 + 2n (p + p' - p't) -n (p + p') (15) where td is the distance between the centers of the tension and compression reinforcement. In all subsequent calculations the value of the modular ratio n used in the above equations was determined by Jensen's formula(7) 5 + 10,000 (16) n=5+ (16) f'1 which has been found to give reliable results. 7. Test Data In order to determine the unknown function .F(f,') in Eqs. 11 and 13, experimental results of previous research were analyzed. Attention was directed first only to simple-span rectangular beams without web reinforcement and subjected to one or two symmetrical concentrated loads. All known tests of such beams were included in the analysis except those of very early beams for which there was some doubt about the compressive strength of the concrete used. A total of 125 beams from 15 different investi- gations were considered. These beams were tested over a period of 43 years and had a wide variation in their physical properties. Table 1 lists the differ- ent investigations, giving their entry numbers in the Bibliography and the numbers of the tables in which they are analyzed. This table also sum- marizes the range of test variables for the different groups of beams. There were 111 of these beams which failed in shear, 7 of them, however, yielding before failure. The remaining 14 beams failed in bond, although their mode of failure was reported as diagonal tension. These beams are discussed later in this section. Thirty beams were provided with both tension and compression reinforcement; all other beams were reinforced in tension only. The test results for the different groups of beams are analyzed in Tables 2 through 16. Both the physical properties as reported by the investi- gators and the calculated quantities are given for each individual beam. All dimensions are given in inches and the compressive strength of concrete in pounds per square inch. In most cases, concrete strength was determined from tests on 6- by 12-in. standard cylinders. In a few cases, tests either on cubes or on modulus of rupture beams were em- ployed; these cases are noted in the tables and the concrete strength is reduced to that of a standard cylinder by the formulas: f,' = 0.75 fe,,' for cubes and f/ = 6.7 f, for modulus of rupture beams. In order to evaluate the function F (fc'), the quantity M/bd2fC'(k + np') was calculated for each beam. For beams without compression rein- forcement the term (k + np') reduces to k. In Table 1 Range of Test Variables for Simple-Span Rectangular Beams Without Web Reinforcement and Under One or Two Symmetrical Concentrated Loads Test Series Richart Series 1910 Series 1911 Series 1913 Series 1917 Series 1922 Richart and Jensen Thompson, Hub- bard, and Fehrer Moretto Clark M.I.T. Gaston Laupa Moody Series A Series B Series III Total Entry in. Bibl. (2) (8) (9) (4) (5) (10) (11) (1) (12) Table No. No. No. of of Beams S,T-S Fail. psi 2030-2670 1490-2350 2180 4770 3696-4522 2230-4760 2570 3550-4640 3120-3800 3130-4880 4020-4750 2140-4690 880-4570 1770-5970 2500-3620 880-5970 p p' b d a a/d L No. of Loads 1.23-1.92 1.65-1.94 1.47 2.74-3.71 2.33 2.80 2.50 3.98 0.98 1.40-3.14 1.38-1.90 0.93-4.11 % % in. in. in. 8 10 24 8 10 24 8 15 40 8.1 10 48 8 21 36 8 21 32 8 12 20 0.50 5.5 8 p'=p 4-6.25 6 ... . (6 0.80-2.37 .... 1.90 . 2.72-4.25 p'=0.5p 0.80-4.11 18.25 15.37 7 10.58 10.5 32 18-36 30 36 48-51 2.4 2.4 2.67 4.8 1.71 1.52 1.67 1.75 1.17-2.34 4.28 3.40 4.48-4.79 7 10.5 31.5 3 6 10.56 36 3.41 7 21 32 1.52 1.17-4.8 ads ILLINOIS ENGINEERING EXPERIMENT STATION Table 2 Tests by Richart, Series 1910. Simple-Span Rectangular Beams Without Web Reinforcement Beam P.' psi 280.1 2670 280.2 2320 280.3 2030 Reference: (2) Dimensions: b=8; d=10; a - 24; a/d=2.4; L=72; L'= 78 Loading: 2 equal loads at X-points Reinforcement: Plain round bars; fI,= 38,500 psi for Beam 280.3; not given for others Concrete Strength: Tests on 6-in. cubes; reduced to cyl. strength by /' -0.75 f.' Age at Test: Around 60 days Reported Ca p Reinf. Anch. Ps.t Mode k Mt__ Bars of bd2f'k % No., Size kips Fail. 1.23 5-%" None 23.8 DT 0.380 0.352 1.92 5-Y/s 18.8 DT 21-0 DT 0.378 0.322 0.456 0.340 Iculated Ratio Mode Mt-t of M. Fail. 0.78 B 0.69 B 0.71 B Table 3 Tests by Richart, Series 1911. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (2) Dimensions: b= 8; d= 10; D= 12; a= 24; L=72; L'f-78 Loading: 2 equal loads at h-points Reinforcement: Plain round bars; f,= 34,200 psi for Beam 293.3; not given for others Concrete Strength: Tests on 6- by 8- by 40-in. control beams; reduced to cyl. strength by f/'- 6.7 f, Age at Test: Around 60 days Beam f/ psi 291.1 1690 291.2 291.3 294.1 1490 294.2 294.3 293.4 2350 293.5 293.6 293.1 2040 293.2 293.3 286.1 1660 286.2 286.3 286.5 2160 286.6 286.7 * Nuts tightened t Nuts not tightened Reported p Reinf. Anch. Pe.st Mode Bars of % No., Size kips Fail. 1.65 3-'/ Hooks 25.3 DT 22.5 DT 27.7 DT 15-in. 25.0 DT over- 20.2 DT S " hang 24.7 DT Nuts* 27.4 DT and 34.5 DT,T Plates 19.3 DT Nutst 20.0 DT and 21.4 DT Plates 24.8 DT None 18.0 DT 17.6 DT 22.5 DT 1.94 5-% " 17.4 DT 18.5 DT 22.1 DT 0 0 0 0 0 0 Calculated k Mt_ Ratio Mode bdf,'k M__t of M. Fail. .446 0.503 1.02 8 0.448 0.91 8 0.551 1.12 8 .457 0.551 1.10 S 0.445 0.89 S 0.544 1.08 S .421 0.415 0.89 S 0.523 1.13 S 0.293 0.63 B .431 0.341 0.71 B 0.365 0.76 B 0.423 0.88 B .448 0.363 0.73 B 0.355 0.72 B 0.454 0.92 B .452 0.267 0.56 B 0.284 0.60 B 0.340 0.72 B Table 4 Tests by Richart, Series 1913. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (2) Dimensions: b=8; d=15; D= 17; a=40; a/d=2.67; L=120; L'=126 Loading: 2 equal loads at %-points Reinforcement: %-in. plain round bars; I,= 36,300 psi Concrete Strength: Tests on 6-in. cubes; reduced to cyl. strength by f/,'=0.75 ft.' Age at Test: 225 days Mode of Fail. Reported f,' p Reinf. Anch. Ptws Bars psi % No., Size kips 2180 1.47 4-Y' Hooks 24.9 Beam 16B20.1 16B20.2 16B1.1 16B1.2 16B2.1 1BTt2 0 16B20.1 3210 16B20.2 3210 16B1.1 2450 16B1.2 2670 16B2.1 2450 16B2.2 2450 Calculated k Mtu, Ratio Mode bdf'k Mt.1t of M. Fail. 0.409 0.311 0.66 B Table 5 Tests by Richart, Series 1917. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (2) Dimensions: b=8.1; d=10; D-12; a-48; a/d=4.8; L=114; L'=120 Loading: 2 equal loads Reinforcement: Plain round bars; f,-45,700 psi for %-in. bars; f -40,600 psi for %-in. bars Age at Test: About 60 days Reported Calculated p Reinf. Anch. Pt.st Mode k 'MIf- Ratio Mode Bars of bd2f'k Mt.1 of % No., Size kips Fail. M. Fail. ANALYZED WITH ACTUAL CONCRETE STRENGTH IN COMPRESSION ZONE 3.71 5-%. None 31.0 DT 0.523 0.368 1.04 S " " 29.6 DT 0.523 0.352 0.99 S 3.69 Hooks 32.0 DT 0.522 0.381 1.07 S " 28.8 DT 0.522 0.343 0.97 S 2.74 5-i " 26.6 DT 0.472 0.350 0.99 8 " 29.5 DT 0.472 0.388 1.09 8 ANALYZED WITH CONCRETE STRENGTH USED IN LOWER PORTIONS OF BEAMS .... ... ..... .... ... 0.531 0.540 1.28 .... ... ..... .... ... 0.531 0.516 1.21 .... ... ..... .... ... 0.550 0.701 1.52 .... ... ..... .... ... 0.543 0.586 1.30 .... ... ..... .... ... 0.499 0.660 1.43 .... ... ..... .... ... 0.499 0.732 1.59 Beam 301.1 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 6 Tests by Richart, Series 1922. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (2) Dimensions: b = 8 d - 21; D= 24; a= 36; a/d = 1.71; L= 108; L'- 120 Loading: 2 equal loads at %-points Reinforcement: 1-in. corrugated round bars; f/,52,400 psi Age at Test: About 60 days Reported Calculated p Reinf. Anch. Pt-t Mode k Mt-t Ratio Mode Bars of bdf,'k M- of % No., Size kips Fail. M. Fail. 2.33 4-1 %8 None 149.4 B,DT 0.441 0.424 1.09 8 148.0 B,DT 0.446 0.458 1.13 S Hooks 165.5 B,DT 0.435 0.429 1.17 8 " 126.0 B,DT 0.437 0.339 0.91 8 Table 7 Tests by Richart and Jensen, 1931. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (8) Includes only those beams which were made of concrete with natural sand and gravel aggregates Dimensions: b=8; d=21; D=24; a=32; a/d=1.52; L=96; L'=108 Loading: 2 equal loads at h-points Reinforcement: 1-in. plain round bars; f/= 37,600 psi Age at Test: 28 days (moist cured 28 days) Reported Calculated p Reinf. Anch. Ps.t Mode k Mget Mt Bars of bde'k M % No., Size kips Fail. 2.8 6-1' Hooks 142.9 DT 0.463 0.294 0 159.7 DT 0.463 0.339 0 151.8 DT 0.467 0.344 0 134.1 DT 0.473 0.333 0 105.8 DT 0.510 0.422 0. atio Mode rte of f. Fail. .83 S .94 S .91 S .84 8 90 8 89 8 Table 8 Tests by Thompson, Hubbard, and Fehrer, 1938. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (9) Dimensions: b=8; d= 12; a= 20; a/d= 1.67; L=60; L'=74 for Series I; L'= 86 for Series II Loading: 2 equal loads of h-points Reinforcement: Four %-in. round old-style deformed bars; f, = 36,000 psi Concrete Strength: The average value of f/' reported for all beams Age at Test: 28 days Reported Calculated Beam l' p Reinf. Anch. Ptus Mode k Mt.. Ratio Mode Bars of bdWf.'k M1..* of psi % No., Size kips Fail. M, Fail. I B-1 2570 2.5 4-%' Hooks 84.0 DT 0.482 0.589 1.30 S I B-2 " " " " 88.0 DT 0.616 1.36 S I B-3 " 86.0 DT 0.603 1.33 S II K-1 13-in. 88.0 DT 0.616 1.36 8 II K-2 " over- 84.0 DT 0.589 1.30 S hang Note: These beams were apparently tested without rollers at the beam supports. This may have restrained the horizontal movement of the beams during loading. The ultimate loads of the beams are unusually high and correspond to their flexural capacities. Table 9 Tests by Moretto, 1945. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (4) Dimensions: b=5.5; d=18.25; D=21; a=32; a/d-1.75; L=96; L'-120 Loading: 2 equal loads of h-points Tension Reinforcement: Four 1-in. sq. deformed bars; f,=48,000 psi Compression Reinforcement: Two h-in. sq. deformed bars End Anchorage: Hooks Age at Test: 28 days Reported Calculated Beam f/ p p' t Pt..t Mode k k+np' Mt.& Ratio Mode of bdf,'(k+np') Mt- of psi % % kips Fail. M. Fail. 1N1 3550 3.98 0.50 0.932 70.0 DT 0.516 0.556 0.310 0.76 8 1N2 3620 " 88.0 DT 0.514 0.553 0.383 0.94 S 2N1 4340 " 78.5 DT 0.502 0.538 0.293 0.78 8 2N2 4640 " " " 90.5 DT 0.502 0.537 0.318 0.88 S Beam 221.1 221.2 222.1 222.2 Beam 1 2 3 4 5 6 11 6.5 DT 0.498 0.403 0. ILLINOIS ENGINEERING EXPERIMENT STATION Table 10 Tests by Clark, 1951. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (5) Dimensions: b=8; d=15.37; D=18; L=72 Loading: 2 equal loads at various positions Reinforcement: 2 No. 7 deformed bars; f,= 53,710 psi End Anchorage: %- by 8-in. steel plates j-in. thick welded to the end of bars Age of Test: 28 to 30 days; beams kept moist until the day prior to testing fReported Calculated Beam f/ p a a/d Ph.k psi % in. kips AO-1 3120 0.98 36 2.34 40.0 2 3770 " 48.5 3 3435 " " 53.5 BO-1 3420 0.98 30 1.95 54.4 2 3468 " 42.4 3 3410 " " " 57.6 CO-1 3580 0.98 24 1.56 78.4 2 3405 " " " 79.9 3 3420 " " " 75.1 DO-1 3750 0.98 18 1.17 99.6 2 3800 116.9 3 3765 " 100.4 Mode k MteM of bdW'k Fail. DT DT T DT DT DT DT T DT DT T DT 0.329 0.320 0.324 0.324 0.323 0.324 0.322 0.324 0.324 0.320 0.320 0.320 0.370 0.382 0.457 0.388 0.299 0.412 0.430 0.458 0.428 0.394 0.457 0.395 Ratio Mode Mt-, of M. Fail. 0.86 S 0.96 S 1.10 T-S 0.93 S 0.72 S 0.99 8 1.05 T-S 1.10 T-S 1.03 8 0.98 S 1.15 T-S 0.99 S Table 11 Tests at M.I.T., 1951. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (10) Dimensions: b= variable; d= 7; D=8; a=30; a/d= 4.28; L= 60; L'=65 Loading: One load at midspan Reinforcement: Type of bars not given; f,=52,220 psi for /-in., f,= 48,370 psi for h-in.; f,= 46,240 psi for %-in. bars End Anchorage: Not given Age at Test: 8 days Reported Calculated Beam f' p p' Reinf. t b Ps,,. Mode k k+np' Mtso Ratio Mode Bars of bdf/'(k+np') MA.* of psi % % No., Size in. kips Fail. M. Fail. T-2b 3580 1.40 1.40 2-W" 0.857 4 10.0 S 0.327 0.436 0.392 0.96 S c " " " " 10.0 8 ." 0.392 0.96 S T-3a 3470 3.14 3.14 2-%'" 10.5 S 0.405 0.652 0.355 0.86 S b " " " " 7.0 S " " 0.237 0.57 8 c " " " 8.5 S " " 0.288 0.70 S T-5a 3460 2.18 2.18 2-%" 9.5 S 0.372 0.544 0.387 0.94 S b " " " " 2 10.1 S " 0.412 1.00 S c " " " " 10.3 S " 0.417 1.01 8 T-6b 3130 1.40 1.40 2-%" 7.6 S 0.331 0.446 0.417 0.97 8 c " " 8.2 S " " 0.450 1.05 S T-11b 4190 1.40 1.40 2-%" 6.25 12.0 S 0.321 0.425 0.330 0.87 S T-12a 4880 2.18 2.18 2-%5 5.75 15.8 8 0.360 0.514 0.334 0.95 S b " 15.0 S " 0.318 0.91 8 c " 14.3 8 " " 0.303 0.87 S Table 12 Tests by Gaston, 1952. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (11) Dimensions: b=6; d=10.58; D=12; a=36; a/d=3.40; L=108; L'=120 Loading: 2 equal loads at %-points Reinforcement: Deformed bars Age at Test: Around 30 days Reported Beam /' p Reinf. f, Anch. Mt5t Mode Bars of psi % No., Size ksi kip-in. Fail. T2Ma 4320 1.38 2-No. 6 47.7 None 332.3 S T2Mb 4020 " " 48.3 Hooks 351.7 S T2Mc 4470 1.90 2-No. 7 46.8 None 450.2 8 Calculated k Mitt Ratio Mode bdf'k Mt** of M. Fail. 0.359 0.319 0.85 S 0.363 0.359 0.92 8 0.405 0.377 1.02 S Table 13 Tests by Laupa, 1953. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (1) Dimensions: b= 6; D = 12; L= 108; L= 120; distance a given from center of end support to edge of column stub Loading: One load at center of 108-in. span, applied through 6- by 12-in. column stub, 6 in. high Reinforcement: Deformed bars End Anchorage: None, straight bars Age at Test: Around 28 days Reported Calculated Beam /' p Reinf. f, d a a/d Pt.t Mode k MAtt Ratio Mode Bars of bdf.'k M-" of psi % No., Size ksi in. in. kips Fail. M, Fail. S-2 3900 2.08 3-No. 6 41.2 10.58 48 4.54 19.1 S 0.415 0.421 1.07 S S-3 4690 2.52 2-No. 8 59.4 10.44 4.60 23.9 S 0.446 0.419 1.17 S S-4 4470 3.21 2-No. 9 44.8 10.37 4.63 25.0 8 0.478 0.435 1.18 S S-5 4330 4.11 2-No. 10 45.7 10.31 4.66 22.4 S 0.531 0.367 0.98 S S-11 2140 1.90 2-No. 7 47.5 10.51 4.57 15.2 8 0.450 0.571 1.20 S S-13 3800 4.11 2-No. 10 44.1 10.31 4.66 22.4 8 0.528 0.420 1.05 8 S-1 3940 1.46 3-No. 5 44.6 10.65 51 4.79 16.8 T-S 0.361 0.443 1.13 T-S S-9 2140 0.93 3-No. 4 44.3 10.72 48 4.48 11.5 T-S 0.344 0.543 1.15 T-S S-10 2280 1.39 2-No. 6 41.8 10.58 " 4.54 15.4 T-S 0.396 0.608 1.30 T-S Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 14 Tests by Moody, Series A, 1953. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (12) Dimensions: b=7; d= 10.3-10.8; D=12.0; a=31.5; a/d= 2.92-3.06; L= 63; L'=75 Loading: One load at midspan Reinforcement: Intermediate grade deformed bars End Anchorage: None, straight bars Age at Test: About 28 days Reported Ca d p Reinf. psi in. % 4400 10.30 2.17 4500 10.50 2.15 4500 10.55 2.22 4570 10.63 2.37 3065 10.50 1.62 3125 10.55 1.63 2785 10.63 1.60 2430 10.69 1.66 920 10.55 0.81 880 10.70 0.83 1000 10.75 0.80 980 10.80 0.82 Bars No., Size 1-11 2-8 2-7; 1-6 4-6 1-8; 2-4 2-7 2-6; 1-5 4-5 1-7 2-5 3-4 2-4: 2-3 Mode k of Fail. 0.426 0.423 0.428 0.437 0.401 0.401 0.404 0.419 0.395 0.403 0.384 0.391 Table 15 Tests by Moody, Series B, 1953. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (12) Dimensions: b=6; d=10.56; D=12; a=36; a/d=3.41; L=108; L'=120 Loading: 2 equal loads at %-points Reinforcement: 2 No. 7 intermediate grade deformed bars End Anchorage: None, straight bars Age at Test: About 28 days Reported p lculated Mt..s bdV'k 0.306 0.322 0.357 0.319 0.420 0.435 0.442 0.483 0.501 0.608 0.577 0.569 Calculated Pts.t Mode k Mte Ratio Mode of bd2f,'k Mt__ of kips Fail. M. Fail. 26.0 S 0.397 0.334 1.01 S 16.0 S 0.441 0.408 0.89 S 23.5 8 0.414 0.413 1.03 S 19.8 S 0.447 0.540 1.15 S 23.4 S 0.395 0.362 0.98 S 15.8 S 0.445 0.421 0.90 S 23.0 S 0.395 0.354 0.96 S 14.0 S 0.465 0.462 0.94 S 24.0 S 0.393 0.278 0.92 S 22.0 S 0.418 0.412 1.00 S 27.0 S 0.395 0.336 1.05 S 21.2 S 0.424 0.464 1.06 S 25.0 S 0.396 0.313 0.97 S 19.4 S 0.421 0.382 0.90 S 23.0 S 0.397 0.291 0.89 S 17.0 S 0.446 0.437 0.94 8 Table 16 Tests by Moody, Series III, 1953. Simple-Span Rectangular Beams Without Web Reinforcement Reference: (12) Dimensions: b=7; d= 21; D f24; a= 32; a/d= 1.52; L= 96; L'= 120 Loading: 2 equal loads at h-points Tension Reinforcement: Four deformed bars Compression Reinforcement: Two deformed bars; t=0.91 Age at Test: 28 days Reported Calculated Beam f/ p p' f, Anch. P..es Mode k k+np' Mtat Ratio Mode of bdf.(k+snp') M" of psi % % ksi kips Fail. M. Fail. 24a 2580 2.72 1.36 45.7 Hooks 133 S 0.432 0.552 0.484 1.07 8 b 2990 " " " 136 S 0.424 0.538 0.439 1.01 8 25a 3530 3.46 1.73 45.4 120 8 0.456 0.582 0.303 0.74 S b 2500 " " 130 S 0.455 0.610 0.442 0.97 8 26a 3140 4.25 2.13 43.8 189 S 0.485 0.659 0.473 1.10 S b 2990 " " 178 8 0.488 0.665 0.464 1.07 S 27a 3100 2.72 1.36 45.7 None 156 8 0.433 0.545 0.479 1.11 S b 3320 " " " 160 S 0.429 0.538 0.464 1.10 S 28a 3380 3.46 1.73 45.4 " 136 S 0.458 0.596 0.350 0.84 8 b 3250 " .... " 153 S 0.519 0.519 0.470 1.11 S 29a 3150 4.25 2.13 43.8 175 S 0.485 0.659 0.437 1.02 8 b 3620 " " " 196 S 0.480 0.645 0.435 1.07 8 Ratio M.. 0.82 0.88 0.97 0.88 0.97 1.01 0.99 1.05 0.95 1.14 1.10 1.08 Beam 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 f/ ILLINOIS ENGINEERING EXPERIMENT STATION M te bd 'f(A Fig. 1. Evaluation of Eq. 17 for Simple-Span Fig. 1 the above quantity is plotted against fe'. It is seen that the concrete strength varies from about 1000 to 6000 psi. Within these limits the function F(fc') can be approximated by a linear equation: F(fe') = 0.57 - 4.5' (17) 10, where f,' is the compressive strength of a standard cylinder in pounds per square inch. Substitution of Eq. 17 into Eq. 13 yields an equation for moment, subsequently called the shear-compression moment, at which a simple-span reinforced concrete beam without web reinforcement and under one or two symmetrical concentrated loads fails in shear: = (k + np') (0.57 - 4.5 f(18) Five beams analyzed in Table 8V) show unusu- ally high ratios of Mtest to M,. Moreover, the ulti- mate loads corresponded to the flexural capacity of the beams although there was no indication of Rectangular Beams Without Web Reinforcement flexural failure in either the measured deflections or the crack patterns reported. The peculiar be- havior of these beams may be due to the fact that apparently no rollers were provided at the beam supports. This may have restrained the longitudinal movement of the beams and contributed to their high shear strengths. Data for these beams are not shown in Fig. 1. The agreement between test results and Eq. 18 is believed to be satisfactory. The average ratio of M/MS for the 106 beams which failed in shear and are shown in Fig. 1 is 0.986; the standard devi- ation is 0.119. The group of beams which were loaded through a column stub at midspan(1) failed at a somewhat higher load than that given by Eq. 18. Whether the apparent increase in shear strength was caused by the column stub or by the use of a single concentrated load could not be de- termined from the available data. Six beams from four different investigations failed at a considerably lower load than predicted by Eq. 18. However, all Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS these beams had companion specimens which failed in close agreement with the predicted values. In this connection, it must be pointed out that all compression failures are sensitive to the com- pressive strength of the concrete at the section of failure. The compressive strength reported for a test beam is the average strength obtained from control cylinders. Since even control cylinders can vary widely in strength, it is not reasonable to ex- pect the concrete strength to be uniform throughout a test beam. If the concrete strength at the section of failure happens to be different from the average strength of the control cylinders, the test beam may fail at a load different from the predicted load. It is believed that much of the scatter in test results can be attributed to variation of the concrete strength from the average value. This is especially true in those cases for which only the average con- crete strength was reported for the entire test series or for a group of companion specimens. Further- more, it may be of significance that the beams were tested over a period of almost a half of a century, and that the beams were both made and cured under greatly different conditions. No systematic difference can be detected be- tween beams reinforced in tension only and beams reinforced both in tension and compression. If the A MA latter group of beams is considered separately, the average ratio of M/Ms for thirty such beams is 0.940 and the standard deviation 0.14. It is inter- esting to note, however, that five of the six beams which fell considerably lower than the predicted values were provided with compression reinforce- ment. This explains also why the average ratio for this type of beam is somewhat lower than that for all beams combined; if these beams are excluded, the average ratio is 0.986 and the standard devia- tion 0.084. Equation 18 was based on assumptions made in deriving Eq. 11; that is, the ratio k,/k is a function of fc' alone and does not depend on p. In order to check this assumption and to investigate whether Eq. 11a might not represent better the moment at shear failure, the ratios M/M, are plotted against p in Fig. 2. Although the steel percentages used in the test beams vary over a large range of values, no consistent relationship can be detected between the ratio M/M, and p. Consequently, the ratio k,/k does not seem to be influenced by p and Eq. 18 is therefore assumed to be valid for beams with any amount of longitudinal reinforcement. Series 1917 of the beams tested by Richart (Table 5) provides further data for a study of the mechanism of shear failure. These beams were pro- Fig. 2. Effect of Longitudinal Steel Percentage on Ratio of Measured to Computed Shear-Moment. Simple-Span Rectangular Beams Without Web Reinforcement ILLINOIS ENGINEERING EXPERIMENT STATION vided with a 4-in. thick layer of high-strength con- crete at the top of each beam "as a precaution against premature failure of the beam by crushing of the concrete." Table 5 gives an analysis of these beams using the reported values of the concrete strength both in the compression zone and in the lower portions of the beams. It is seen that the use of the actual concrete strength for the compression zone gives very good agreement with Eq. 18, whereas the use of the concrete strength for the lower portions of the beams results in differences of as much as 59 percent between measured and predicted values. Thus it is clearly evident that the load at failure is controlled solely by the strength of the compression zone of the concrete. The strength of the remaining part of the concrete sec- tion does not greatly influence the shear strength of a beam. Fourteen beams, although reported as diagonal tension failures, failed in bond. These were beams tested by Richart; three from Series 1910 (Table 2), ten from Series 1911 (Table 3), and one from Series 1913 (Table 4). A total of 18 beams of Series 1911 were without web reinforcement. These beams were very nearly the same in every respect except for the end anchorage of the tension reinforcement. All beams with the longitudinal steel well-anchored either by conventional hooks, by overhang, or by an end plate tightened against the end of the beams, failed in shear at a load in good agreement with Eq. 18. All other beams, however, either with unanchored straight bars or with end plates not tightened, failed at a much lower load; this sug- gested bond failures. Some typical beams of this group were checked for their bond strength by a procedure suggested by Mylrea(13). Mylrea gives an empirical relationship between the length of em- bedment of a plain round bar in a simple-span beam and the cumulative bond stress the bar can develop before bond failure. By using as the length of embedment the distance from the end of the bar to the 45-deg diagonal crack, the cumulative bond stress as given by Mylrea agreed closely with the steel stress calculated from the load at failure. This indicates that the ultimate bond resistance was reached and that the beams failed in bond before developing their ultimate shear capacity. The three beams of Series 1910 with unanchored straight bars also failed in bond. The only beam of Series 1913 for which concrete strength was reported was rein- forced with hooked plain bars. However, it failed at a low load, and a photograph at failure indicated a possible bond failure. 8. Theoretical Interpretation of Basic Empirical Equation a. Beams Reinforced in Tension Only. Equa- tion 18 where the quantity np' reduces to zero for beams without compression reinforcement can be interpreted in the light of the conventional theory of compression failures of reinforced concrete beams. The only modification is in the depth of the compression zone. The following stress block is assumed: C = kk 3c'ksbd k) T = Cd (1 - k2k,) = kjk3fs'kbd' (1 - k2k,) M bd2fJ = kikak, (1 - k2k,) For beams failing in flexure, the parameter kAks is the ratio between the average stress in the con- crete of a beam and the strength of a standard 6- by 12-in. test cylinder in axial compression. This parameter has been evaluated experimentally by previous investigators. In Fig. 3 the values of kak3 as obtained by Gaston("1 and Billet(14 have been plotted against fe'. There is considerable scatter in the measured values as would be expected in an investigation of this kind. A reasonable approxima- tion, however, can be obtained by a linear relation- ship between kiks and f,'. When f,' is within the limits of 2000 and 6000 psi, kxk3 can be approxi- mated as follows: 10.8f3' 0 4.5f' kik3 = 1.37 - 10, = 2.4 0.57 10) Substitution of this function into Eq. 19 and the use of k2 = 0.45 as in the case of beams failing in flexure gives: iff -., =2.4 0.57- 'I ) k, (1-0.45k,.) Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS 2.0 1.6 1.2 k,k, 0.8 0.4 0 Fig. 3. Relation Between Equation 21 is based on two conditions of static equilibrium, C = T and M = Cd(1 - k2ks), and on a fully developed concrete stress-block having a value of kks3 as given by Eq. 20. Consequently, Eq. 21 constitutes a general expression for the ultimate moment of a rectangular beam reinforced in tension only and is valid for any mode of failure, provided that the properties of a concrete stress- block at the instant of failure are represented by Eq. 20. The depth of the compression zone of concrete, ksd, remains the only unknown in Eq. 21. This dis- tance can be determined with the aid of strain re- lations for beams which fail in flexure. Test obser- vations have shown that a beam fails in flexure when the concrete crushes at a limiting strain of about 0.004 and that the strain distribution over the depth of the beam remains practically linear up to the final failure. This information and the known stress-strain relationship of the reinforcing steel permit the calculation of the parameter k8 from the condition of static equilibrium of the internal forces of a section. After the value of k, has been determined, the ultimate moment can be calculated from Eq. 21. If, however, a linear distribution of strain over the depth of the beam is assumed for shear failures, the ultimate concrete strain of about 0.004 and the fact that the steel strain must be below yielding, S[(psi) kik, and Concrete Strength say below 0.0015, result in a value of k, at failure greater than 0.7. This is incompatible with test observations which show that the diagonal cracks extend higher than the flexural cracks at failure. Since the depth of the compression zone rarely ex- ceeds about 0.3 - 0.4d for flexural failures, the value of k, must be much below 0.7 for shear fail- ures. This would be possible if the ultimate con- crete strain were, say, 10 times smaller than 0.004, but such a small strain is not possible at the loca- tion of a diagonal crack. Actually, the presence of the diagonal crack disrupts the normal distribution of the steel strain along the tension reinforcement. Since there can be no transfer of stress across the diagonal crack, consideration of moment shows that the steel stress must be the same both at a vertical section through the upper end of the crack and at the intersection of the reinforcing bars and the crack. Thus, the steel strain must be practically uniform over this distance. Furthermore, in order to preserve the continuity of the beam, the total elon- gation of steel between these two sections must have a geometrically corresponding, although not numerically equal, total shortening of the top con- crete fiber over a much shorter distance at the location of the diagonal crack. This requires a con- centration of concrete strain in that region. Conse- quently, the strain distribution over the depth of the beam cannot be linear at the section of failure. ILLINOIS ENGINEERING EXPERIMENT STATION Since the concrete strain must be concentrated at the location of the diagonal crack, it is likely that the ultimate concrete strain is still about 0.004, as in the case of flexural failures, and that the concrete stress-block is fully developed. However, the actual distribution of strain is unknown and cannot be determined from the data available at the present time. Consequently, no theoretical re- lationship can be written for the depth of the com- pression zone at shear failure. In order to interpret test results and to determine a general expression for shear-compression failures, either the value of k,, or the magnitude of the steel stress, or some relationship between the average strains in the re- inforcement and in the concrete must be determined empirically. In this investigation it was chosen to evaluate k, empirically. Equation 18 was obtained to represent the shear-compression capacity of rec- tangular simple-span beams under one or two con- centrated loads. A comparison between Eqs. 18 and 21 reveals that both these equations have the same form. Equating the two yields a relationship between k, and k: k, = k (22) = 2.4 (1 - 0.45 k,) from which: k, = 1.11 - V 1.23 - 0.926 k (23) Since k remains usually within 0.2 and 0.5, Eq. 22 shows that k, is practically a constant fraction of k, the depth of the compression zone computed by the "straight line" theory. This finding shows why the previous attempt to use k as a measure of k, gave satisfactory results. However, since the relationship between the two was determined em- pirically, it can only be speculated why these two quantities are related. Zwoyer used in his investigation(29) an empiri- cal relationship between the average values of the concrete strain on the top surface of the beam and at the level of the reinforcing steel. In addition, the parameter kks3 was determined from data ob- tained in tests of prestressed concrete beams and the same value was used subsequently for ordinary reinforced concrete beams. The average ultimate strain in the concrete was found to be 0.00385. Moody used the parameter kiks3 as obtained from flexural failures and evaluated the magnitude of the steel stress from test results.(12) Two different ex- pressions were obtained for the steel stress; one for simple-span beams and another for restrained rec- tangular reinforced concrete beams under sym- metrical concentrated loads. b. Beams Reinforced in Both Tension and Compression. Equation 19 was derived for beams without compression reinforcement. For beams re- inforced in both tension and in compression it can be modified as follows: M = khk3fc'k.bd2 (1 - k2k,) + f,' p'bd2t (24) where td is the distance between the centers of the tension and compression reinforcements, f/' is the stress in the compression reinforcement, and p' is the ratio of compression reinforcement. Since the ultimate strain in the concrete is ap- proximately 0.0040 and the yield strain for rein- forcing bars is usually less than 0.0017, yielding of the compression reinforcement precedes crushing of the concrete in most flexural compression fail- ures. For shear compression failures, however, di- agonal cracks extend higher than the vertical cracks caused by flexural tension. It is conceivable that a beam can fail in shear either before or after the compression reinforcement yields. Expressions for the ultimate shear moment for both of these cases are derived in the following paragraphs, and the validity of these equations is determined with the help of experimental data. If it is first assumed that the compression rein- forcement has reached its yield stress f,' at shear failure and that k, is still given by k, = 2.4 (1- k2k,)' 2.-4 (-k2k.)' Eq. 24 for maximum shear-moment can be writ- ten as: Ma-= k (0.57 - 4.5f/) + n'p't bd2f, ' k 10 ) npt Since this equation assumes that the compression reinforcement has yielded while the tension rein- forcement is still elastic, the elastic modular ratio n is to be used for the tension reinforcement and the plastic modular ratio n' = f'//f' for the com- pression reinforcement when computing the quan- tity k. An expression for the maximum shear moment for the second case, a beam failing in shear before the compression reinforcement yields, was derived previously: bdf- = (k + np') (0.57 - 4.5f) I Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS In this expression the elastic modular ratio n is used for both tension and compression reinforce- ment in computing the quantity k. Equations 25 and 18, based on different assump- tions, are greatly different. Equation 25 gives a much higher ultimate moment than Eq. 18. In the analysis of previous test data, thirty of the 106 beams under consideration were provided with compression reinforcement. If these beams are con- sidered as having failed after the compression re- inforcement had yielded, the internal resisting moment given by the compression reinforcement acting at its yield stress is almost as large as and in several cases even larger than the total external moment. Thus, it must be concluded that these beams failed in shear before the compression rein- forcement yielded. Furthermore, since the thirty beams with compression reinforcement gave good agreement with Eq. 18, this equation can be used to take the effect of compression reinforcement into consideration. According to Eq. 18 the shear strength of a beam with compression reinforcement is but little greater than that of the same beam without; p' decreases the value of k while adding the term np', so that the quantity (k + np') is but little greater than the value of k for a beam with- out compression reinforcement. 9. Properties and Limitations of Basic Empirical Equation The basic empirical equation was derived for simple-span rectangular beams without web rein- forcement and under one or two symmetrical con- centrated loads. Different variables have the following effect on Eq. 18: a. Ratio of a/d. Equation 18 considers shear failures as compression failures. The load at failure is determined by a limiting shear-compression mo- ment. In that sense, the ratio a/d loses its usual meaning; that is, as affecting the shearing strength of a beam. The quantity a relates the magnitude of the applied load to the moment at failure, M = Va, and the effective depth d affects both the lever arm of the internal moment and the area of the compression zone. For the beams analyzed, the ratio a/d varied from 1.17 to 4.80. This variation did not seem to have any effect on the agreement between the test results and the predicted values. It is conceivable, however, that as the ratio a/d increases and the relative magnitude of the shearing stresses decreases, a beam will either fail in shear at a higher load than that given by Eq. 18 or, for still higher values of a/d will fail in flexure. This phenomenon is discussed further in Section 18. Con- versely, as the ratio a/d decreases to a very small value, it is expected that the mode of failure changes from shear-compression to shear-proper. This question is discussed in more detail in Sec- tion 19. b. Tensile Reinforcement. The amount of ten- sile reinforcement affects the size of the compres- sion zone of the concrete. It was found empirically that the moment at failure could be related to k and that the actual depth of the compression area was practically a constant proportion of k, or k, = c 2.4 (1 - k2k.) c. Concrete Strength. The shear strength of a beam is directly proportional to the following func- tion of fe': f,'(0.57 - 4.5 fc'/10)k. It is seen that as f,/ increases, both the quantity (0.57 - 4.5 fc'/ 10) which represents the effect of kfks, and the value of k decrease. Thus the shear strength is not a linear function of fe'. As an example, for a beam with one percent tension reinforcement, an increase of f,' from 2500 to 5000 psi increases the shear strength 36 percent. d. Compression Reinforcement. The contribu- tion of compression reinforcement to the shear strength is rather small and can be included in the analysis by considering p' in computing both the elastic k and the transformed concrete area. This procedure led to Eq. 18. e. Column Stub. Beams which had a column stub cast integrally with the beam at midspan failed consistently at slightly higher loads than beams without a column stub. These increases in strength were somewhat larger for lower values of concrete strength than for higher values of concrete strength. III. SIMPLE-SPAN RECTANGULAR BEAMS WITH WEB REINFORCEMENT AND UNDER ONE OR TWO SYMMETRICAL CONCENTRATED LOADS 10. General Considerations In Chapter II a rather simple expression was derived for the shear strength of a simple-span beam without web reinforcement. Here an attempt is made to extend that expression to beams which are provided with web reinforcement. The contribution of web reinforcement to the shear strength can be pictured in different ways. As has been mentioned, the conventional theory originally assumed that all shear was carried by the web reinforcement. Later modifications of the concept of truss analogy, prompted by experimental evidence, allocated a certain proportion of the shear to be resisted by the concrete. Essentially, even the modified expressions for the shear strength implied that the contribution of web reinforcement was determined by the properties of the web reinforce- ment alone, as expressed by the term Krfy,, and not influenced by the shear strength of the beam without web reinforcement. Another approach to the effect of web reinforce- ment is to consider that its contribution is de- termined by both the properties of the web reinforcement and the shear strength of the beam itself. The two alternatives are examined in more detail in the following paragraphs. Test observations show that, in general, web reinforcement which crosses the main diagonal crack yields before the beam fails in shear. Figure 4a shows a simple-span beam shortly before shear failure. For convenience, only the main diagonal crack is shown, although in reality numerous cracks appear as the beam is being loaded. Figure 4b shows the portion of the beam to the left of the crack as a free-body diagram, and Fig. 4c shows the approximate locations of the internal forces at the assumed 45-deg diagonal crack. The force F is the resultant of all stirrup forces crossing the crack. It has been projected down to the level of the tension reinforcement and divided into horizontal and vertical components. The other symbols have their usual meanings. One possible assumption is that the contribution of web reinforcement is independent of the shear strength of the same beam without web reinforce- ment. If this is true, then it must be possible to determine the increase of the shear capacity of the beam solely from the amount and physical prop- erties of the web reinforcement. The following cal- culation attempts this: (1/s) (cot a + 1) jd = number of stirrups crossed by crack (1/s) (cot a + 1) jd Afy,- = F = total tension force in stirrups (1/2) cos 2a (jd)2 brfy = moment given by stirrups acting at their yield stress, about point A The moment due to all forces about A is then: Va = Cjd - (1/2) b(jd)2 cos 2a rfJ According to this equation the total internal resisting moment is made up of two parts; the web reinforcement resists directly a part of the applied moment, the remainder being resisted by the com- pressive force C. The direct contribution of the web reinforcement is influenced by the angle of inclina- tion of the stirrups. For vertical stirrups, cos 2a = - 1, and the moment of the stirrup forces is added to Cid. As the angle a decreases, the direct contribu- tion of the web reinforcement decreases also. At a = 45 deg, this contribution is zero. For a less than 45 deg, cos 2a reverses its sign; this indicates that the direct contribution is detrimental to the shear strength of the beam since the moment of the stirrup forces is subtracted from Cjd. The remain- ing part of the internal resisting moment is pro- vided by Cjd. For vertical stirrups the horizontal component of the stirrup force F reduces to zero. Consequently, the term Cjd is equal to that of the beam without web reinforcement, given by Eq. 18. As the value of a decreases, the horizontal com- ponent of F increases and, consequently, the value of C increases. This increases the part of the in- ternal resisting moment provided by Cjd. 24 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS The above assumptions regarding the effect of web reinforcement can easily be checked for verti- cal stirrups where the magnitude of C is determined by Eq. 18. For this case the preceding expression for Va can be rewritten as follows: Va = (1/2)b(jd)2 rfv, + bd2 f' k (0.57 - 4.5 fc'/10") or M = M, + (1/2) b (jd)2rf In this equation all quantities can be determined and the validity of the equation can be checked against test results. This was done for Clark's and Moretto's beams with vertical stirrups. It was found, however, that the increase of the shear ca- pacity of the beams was much greater than the direct contribution of the web reinforcement as given by the above equation. Furthermore, the difference between the two was consistently larger than could be accounted for by inaccuracies in the assumed locations of the internal forces, e.g., as Fig. 4. Internal Forces at Section of Diagonal Crack given by the 45-deg crack in Fig. 4c. Consequently, it was concluded that the foregoing assumptions regarding the effect of web reinforcement were not valid. The above reasoning served one useful purpose. It showed not only that the shear strength is af- fected by the internal forces in the stirrups but also that the presence of web reinforcement changes the location of the neutral axis. Web reinforcement hinders the development of diagonal cracks; thus a larger compression area is available to resist the compressive stresses in the concrete. The combined effects of web reinforcement on the shear-compres- sion capacity are (1) to contribute directly a por- tion of the internal resisting moment which can be either beneficial or detrimental, depending on the angle of inclination of the web bars, (2) to provide a larger ultimate compression force through a larger compression area in the concrete, and (3) to de- crease the moment arm of the larger compression force through lowering the neutral axis of the beam. It is also conceivable that the presence of web re- inforcement restricts the concentration of the com- pressive concrete strain in the region of the main diagonal crack. In estimating the total effect of web reinforce- ment, only the direct contribution of stirrup forces can be determined rationally. However, even this contribution depends on the assumed angle of in- clination between the main diagonal crack and the axis of beam. The other two effects of web rein- forcement cannot be determined rationally. More- over, there is no theoretical basis for estimating the effect of stirrups on restricting the concentra- tion of concrete strain in the region of the diagonal crack. For these reasons it was considered desirable to express the total effect of web reinforcement empirically rather than to attempt to separate the different effects. This is done in the following sec- tion by assuming that the shear strength of a beam with web reinforcement is greater than that of the same beam without web reinforcement by an amount that is a function of the strength of the unreinforced beam and the amount and yield strength of the web reinforcement provided. 11. Stirrups as Web Reinforcement The findings of the previous section suggest that the shear strength of a beam with a reinforced web is affected not only by the amount and properties of web reinforcement but also by the shear strength of the beam itself. Since the most important func- ILLINOIS ENGINEERING EXPERIMENT STATION tion of web reinforcement appears to be its resist- ance to the extension and widening of diagonal cracks, it is logical to assume that a given amount of web reinforcement will increase the shear strength of a beam in proportion to that of the same beam without web reinforcement. Further- more, test results show that in most cases web reinforcement yields before the beam fails in shear, the latter indicating that both the amount of web reinforcement and its yield strength influence the load at failure. All available test data on simple-span beams with stirrups as web reinforcement were analyzed in the light of the above assumptions. A total of 179 beams from 11 different investigations were included; 87 of them failed in shear, 91 in flexure, and one additional beam failed because of insuffi- cient anchorage of stirrups. Different groups of beams are analyzed in Tables 18 through 28; Table 17 summarizes the range of test variables. In addi- tion to shear failures, it was found advantageous to consider also beams which failed in flexure. Several empirical expressions for the shear strength of such beams were investigated. The most consistent results were obtained by plotting the ratio P/P, against the quantity rfw, where P is the measured load and P, the load corresponding to the shear capacity of the same beam without web rein- forcement, Eq. 18. Figure 5 shows such a plot for the 87 beams which failed in shear. Satisfactory agreement with test results was obtained with the following linear equation: PW/P, = 1 + 1 Table 17 Range of Test Variables for Simple-Span Rectangular Beams With Stirrups and Under One or Two Symmetrical Concentrated Loads Test Series Richart(l) Series 1910 Series 1913 Series 1922 Slater, Lord, Zipprodt(W) Slater, Lyse(1) Table No. No. No. No. of of of Beams Shear Flex. Fail. Fail. f' d a/d psi in. 2030-3570 1380-2180 3689-4124 3000-5960 1210-5060 10 15 21 32.75 16.9 4.1- 12.2 npson, Hub- 23 3 3 .. 2570 12 rd, FehrerO) ston, Cox<") 24 20 10 10 3190 12 etto('> 25 40 26 14 2320-5060 18.25 19.50 k() 26 50 43 7 2000-6900 15.37 12.37 on(n" 27 9 .. 9 2120-5900 9.23- 10.72 dy(1) 28 2 2 .. 3250;3680 21 179 87* 91 * One additional beam failed because of insufficient anchorage of stirrups. t Assumed values. Table 18 Tests by Richart, Series 1910. Simple-Span Rectangular Beams With Stirrups Reference: (2) Dimensions: b=8; d=10; a=24; a/d=2.4; L-72; L'=78 Loading: 2 equal loads at %-points Tension Reinforcement: Monolith, ovoid, and corrugated bars Concrete Strength: Tests on 6-in. cubes; reduced to cyl. strength by f/,= 0.75 f.' Age at Test: From 60 to 70 days Beam // Tension p Reinf. psi No., Size % 282.1 2420 2-%" 1.40 2 3570 monolith 3 2410 281.1 2670 3-1iM' 1.56 2 2320 ovoid 3 2030 BOTH ST 281.5 2570 2-41s« 1.48 6 2570 and 1-%' 7 2030 ovoid 284.1 2420 4-%' 1.50 2 2560 corr. 3 2410 284.5 2570 4-%' 1 50 6 2900 corr. 7 2030 * Bent-up bars not included in web reinforces f, Web Reinf. ksi STIRRUPS AS WEB %' round 37.7 loops 40.0 Ms6 round loops IRRUPS AND BENT-UP BAB 37.6 iMe" round loops and 63.3 fie' round stirr. and 2%- "' sq. stirr. and 64.8 2~%" a r A. rf., deg % ksi psi 3 REINFORCEMENT 45 0.35 54.5 191 90 0.52 93.3 485 A USED AS WEB REINFORCEMENT 90 0.34* 99.4 339* 45 0.25* 63.7 159* and " " 90 45 0.56* 55.6 311* and " 5 " 90 Mode of Fail. T T T T T T T DT,B T T T DT T T T p' a Thor ba John More Clar Gast Mood Tota % 1.40-1.56 1.47 2.33 2.33-2.50 2.1-4.7 2.50 0.39-0.87 .3.98 1.86 1.63-3.42 0.62-7.22 4.25 deg 45; 90 45 90 90 90; 20 90 90 90; 67.5; 45 90 90 90 % 0.35;0.52 0.17-1.39 1.38-1.40 0.23-0.88 0.42-0.85 0.36 0.10 0.28-1.12 0.34-1.22 0.28-1.83 0.52;0.95 ksi 54.5; 93.3 40t 39.6-42.9 70 73.4 38.2 45t 46.0-55.0 48.0 45t 44.0; 47.3 P. Eq. 18 kips 29.3 36.1 29.3 32.4 29.8 27.4 31.0 30.9 26.9 30.1 31.1 30.0 31.1 33.4 27.0 Ratio P. 1.09 0.89 1.15 1.24 1.22 1.34 1.32 1.22 1.49 1.76 1.58 1.58 1.75 1.50 1.88 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 19 Tests by Richart, Series 1913. Simple-Span Rectangular Beams With Stirrups Reference: (2) Dimensions: b= 8; d=15; D= 17; a= 40; a/d=2.67; L - 120; L'= 126 Loading: 2 equal loads at 4-pointa Tension Reinforcement: Four %-in. plain round bars; p=0.0147; f,=36,300 psi End Anchorage: Hooks Web Reinforcement: Y-, %-, and M-in. plain round bars; f,= 40,000 psi assumed Concrete Strength: Tests on 6-in. cubes; reduced to cyl. strength by f// =0.75 f4' Age at Test: From 80 to 238 days a r rf,. Pgt. Mode P. of Eq. 18 deg % psi kips Fail. kips 45 0.17 68 38.7 DT 37.8 0.35 140 39.5 T 34.8 0.35 140 40.2 T 32.0 0.78 312 38.0 T 33.6 0.78 312 40.0 T 28.6 0.78 312 40.0 T 33.4 0.78 312 35.7 T 34.8 1.39 556 40.7 T 33.0 0.82 328 44.0 T 36.2 Table 20 Tests by Richart, Series 1922. Simple-Span Rectangular Beams With Stirrups Reference: (2) Dimensions: b=8; d=21; D=24; a=36; a/d=1.71; L-108; L'=120 Loading: 2 equal loads at s-points Tension Reinforcement: Four 1%-in. corrugated round bars; p=0.0233; /,f52,400 psi End Anchorage: Hooks Web Reinforcement: Plain round vertical stirrups Age at Test: Around 60 days Beam 303.1 304.1 2 305.1 2 306.1 2 307.2 308.1 Beam 223.1 2 224.1 2 225.1 2 229.1 2 BENT-UP BARS AS WEB REINFORCEMENT 1%Y 0.96 52.4 503 223.4 T 1 nt-up " " " 211.7 T 1 Table 21 Tests by Slater, Lord, and Zipprodt, 1926. Simple-Span Rectangular Beams With Stirrups Reference: (15) Dimensions: a=57; L= 114; L'= 128; D= 36 (18 for Beam 61) Loading: One load at midspan Flexural Reinforcement: Equal tension and compression reinforcement; 1l-in. round plain bars; f,=about 55,000 psi; some bars, not known which, had much lower yield strength End Anchorage: Hooks Web Reinforcement: %- and %-in. plain round vertical bars; f.= about 70,000 psi Age at Test: About 60 days d a/d p=p' t Stirr. Size in. % in. 32.75 1.74 2.50 0.901 3 2.48 2.48 16.9 3.37 2.33 0.867 % where P,, is the shear strength of a beam with stirrups, P,, that of the same beam without web reinforcement, and fy, is expressed in pounds per square inch. It is seen that most beams fall within ± 15 per- cent of the value predicted by Eq. 26. Only 7 beams failed at considerably lower load. All these beams had a very small a/d-ratio, and for two of them, tested by Moody,("2) it was reported that the stress in the stirrups was but 83 and 67 percent of their yield strength. It is likely that these beams did not fail in shear-compression but in shear-proper. This mode of failure is discussed in more detail in Sec- tion 19. r r/,. Pt«t % psi kips 0.82 574 496.2 0.81 567 496.2 0.88 616 540.0 0.23 161 121.3 Mode of Fail. T T T DT Ratio Pt..s P. 1.02 1.13 1.26 1.13 1.40 1.20 1.02 1.23 1.22 P. ;q. 18 kips 36.6 30.2 36.4 32.0 32.2 35.4 34.1 37.6 p kRatio Eq. 18 PML lkips P. 415.6 1.19 347.6 1.43 429.6 1.26 99.2 1.22 The average ratio between the load at failure and that given by Eq. 26 is 1.017 for the 80 beams which failed in shear-compression; the standard deviation is 0.089. This agreement is somewhat better than that obtained previously for beams without web reinforcement. As a further check on Eq. 26, the ratio P/P, is plotted against rf,,, in Fig. 6 for beams which failed in flexure, either in tension or in compression. It is well known that beams which have been tested to obtain information about their shear strength have frequently failed in tension. Some of these beams, however, were rather close to their shear capacity at failure, as indicated by well-developed a r A. rf,, Pitt Mode of E in. % ksi psi kips Fail. STIRRUPS AS WEB REINFORCEMENT 4 1.38 42.9 592 212.5 T 1 S. 216.4 T 1 7 1.40 40.1 561 218.5 T 1 .. 216.0 T 1 11 1.39 39.6 550 227.3 T 1 S091 2 T 1 2- Be Beam /,' psi 43 4880 48 3000 50 5960 61 3600 Ratio F-* P.. 0.90 Ratio P. 1.56 1.66 1.60 1.64 1.72 1.63 1.67 1.54 Ratio P.. 0.92 ILLINOIS ENGINEERING EXPERIMENT STATION Table 22 Tests by Slater and Lyse, 1930. Simple-Span Rectangular Beams With Stirrups Beam f/ psi IA 1210 B 1520 C 1450 2A 2530 B 2940 C 2910 3A 4020 B 4200 C 4000 4A 4670 B 4660 C 5060 6A 2490 B 2600 C 2670 7A 2800 B 2860 C 3200 8A 3020 B 2650 C 2600 9A 3120 B 2670 C 2900 10A 3040 B 2750 C 2660 10A-A 3730 B 3900 Reference: (16) Dimensions: a=36; L=114; L'=132; a/d-2.95 to 8.78 Loading: Two equal loads Tension Reinforcement: Rail-steel bars; f, = from 59,300 to 63,000 psi End Anchorage: Hooks Web Reinforcement: %-in. stirrups; f,w=73,400 psi Age at Test: 28 days d p a r rf,, Krf,. Pwie, in. % 10.2 2.1 10.3 2.8 3.7 10.1 4.7 14.2 3 0 12.2 2.8 8.0 3.1 5.9 3.2 4.1 3.0 4.0 * Stirrups too short. psi psi kips 627 33.0 32.0 36.2 47.4 40.0 46.8 58.6 64.8 66.6 74.5 71.1 79.0 92.5 106.6 92.0 619 69.3 63.9 71.3 309 135 25.8 31.7 33.4 317 139 15.3 18.6 16.6 315 138 6.8 7.7 6.2 9.8 10.3 " 9.0 Mode of Fail. C C C C C C C C C C C C DT* C C C C C C DT C C C C C C C C C C P. Eq. 18 kips 15.3 17.9 17.3 27.9 30.5 30.3 39.1 39.8 39.0 43.0 42.9 43.9 53.7 55.3 56.1 42.1 42.6 45.4 19.1 17.7 17.5 10.4 9.5 10.0 4.9 4.6 4.5 6.0 6.1 6.0 Ratio P. 2.15 1.79 2.09 1.70 1.31 1.54 1.50 1.63 1.71 1.73 1.66 1.80 1.73 1.92 1.64 1.65 1.50 1.57 1.35 1.79 1.91 1.47 1.96 1.66 1.37 1.65 1.37 1.63 1.68 1.49 Table 23 Tests by Thompson, Hubbard, and Fehrer, 1938. Simple-Span Rectangular Beams With Stirrups Reference: (9) Dimensions: b=8; d= 12; a=20; a/d= 1.67; L=60; L'= 74 Loading: 2 equal loads at 3-points Tension Reinforcement: Four %-in. round old-style deformed bars; f,=36,000 psi End Anchorage: Hooks Web Reinforcement: %-in. round vertical stirrups at 3.5 in.; r =0.0036; fI. =38,200 psi Age at Test: 28 days Beam f/ p rf,, Pt.t Mode P. Ratio Ratio of Eq. 18 Pt__ Ptt psi % psi kips Fail. kips P. P.. I C-I 2570 2.5 136 97.0 S 65.0 1.49 1.17 2 " 88.0 S 1.35 1.06 3 " 98.0 S 1.51 1.18 Note: These beams, like those in Table 8, were tested without rollers at the beam supports. Their ultimate loads are significantly above their com- puted flexural strengths. Table 24 Tests by Johnston and Cox, 1939. Simple-Span Rectangular Beams With Stirrups Reference: (17) Dimensions: b=12, d - 12; D= 13.3; a=36; a/d=3.00; L -108; L'= 120 Loading: Two equal loads at %-points Tension Reinforcement: Hard grade deformed and sq. twisted bars End Anchorage: Hooked Concrete Strength: Average concrete strength reported Web Reinforcement: Vertical Y-in. deformed stirrups at 8 in.; inter, grade; /, -45,000 psi assumed Aee at Test: 28 days %451 0.451 f, rf,. Pt, Mode P. Ratio Ratio of Eq. 18 P.- __ ksi psi kips Fail. kips P. P.. 62.2 47 30.6 T 30.9 0.99 0.90 . . 30.9 T " 1.00 0.91 59.2 28.7 T 29.0 0.99 0.90 " 28.4 T " 0.98 0.90 60.3 28.3 T 29.6 0.96 0.88 " 28.2 T " 0.95 0.87 59.2 45.2 DT 38.9 1.16 1.06 45.3 T " 1.16 1.06 60.3 44.5 T 40.2 1.11 1.01 45.2 DT " 1.12 1.02 63.2 52.8 T 39.9 1.32 1.21 .. 52.9 DT " 1.32 1.21 58.6 46.0 DT 39.7 1.16 1.06 " 45.5 DT " 1.15 1.05 58.4 54.2 DT 40.7 1.33 1.22 54.3 T " 1.33 1.22 61.8 47.1 DT 39.5 1.19 1.09 45.7 DT " 1.16 1.06 64.4 42.8 DT 38.9 1.10 1.01 " 45.5 DT " 1.17 1.07 Beam Al I II A2 I II A3 I II Bl I II B2 I II B3 I II I1I I T1 I II T2 I II T3 I II Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 25 Tests by Moretto, 1945. Simple-Span Rectangular Beams With Stirrups Reference: (4) Dimensions: b= 5.5; d = 18.25; D= 21; a= 32; a/d= 1.75; L= 96; L'= 120 Loading: 2 equal loads at h-points Tension Reinforcement: Four 1-in. square deformed bars; p =0.0398; f,= 48,000 psi End Anchorage: Hooks Compression Reinforcement: Two %-in. square deformed bars; t=0.932; p'= 0.005 Web Reinforcement: 4-in. plain bars, %-in. and h-in. deformed bars; s=6.5 in. Age of Test: 28 days r % 0.28 ", 0.615 11 1.1 " Beam IV Y 1 2 2V Y4 1 2 11 1 2 21 Y 1 2 1D Y 1 2 2D Y 1 2 IV % 1 2 2V % 1 2 11 % 1 2 21 Ya 1 2 LD % 1 2 2D % 1 2 lV / 1 2 2V~ 1 2 II ' 1 2 21 ) 1 2 1D q 1 2 2D M 1 2 laV Y 1 2 laV Y 1 2 0.28 0.615 in.; a= 32 46.0 52.0 ' rf,, Pes..t kips 154 116.4 116.8 135.8 134.4 126.9 115.0 144.0 120.0 115.3 121.8 142.3 138.8 295 142.9 150.8 148.3 139.0 162.5 155.5 171.0 165.0 132.0 127.5 139.9 147.4 568 157.0 157.0 188.8 184.0 177.5 178.0 196.3 196.1 165.0 145.0 171.5 180.2 SERIES IA in.; a/d= 1.64; p= 0.0186; 129 105.2 107.0 320 115.6 119.1 Mode of Fail. DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT DT C C C C C C C C C C C C p'=0.0047; t= 0.936 DT DT T,DT T,DT diagonal cracks. Furthermore, some beams tested to provide data on their flexural strength can also be utilized to obtain information about their shear capacity. Figure 6 is used with the following cri- terion in mind: a beam will fail either in flexure or in shear, whichever capacity is reached first. Beams which failed in shear were used to derive Eq. 26 for their shear strength. Beams which failed in flexure, however, must fall below the line repre- senting their strength in shear in Fig. 6. If they fall above, Eq. 26 cannot be correct; if it is correct, the beams should have failed in shear rather than in flexure since their shear capacity was smaller than their flexural capacity. Figure 6 indicates that Eq. 26 is a reliable expression for shear-compression strength; all beams with a few exceptions fall below the line representing this equation. The flexural capacity of the beams was reached at different ratios of P/P,, Eq. 26 being the limit. Only 4 of the 91 beams fall substantially above this limit. Two of these beams were tested by Johnston and Cox17 ) and only the average concrete strength was reported for 20 beams; it is likely, therefore, that the actual value of fc' for the individual beams was greater than the average and that P, for this strength would be in- creased sufficiently to bring the ratio P/Ps into agreement with other test results. Two other beams in this category were tested by Slater and Lyse.(16 One of the beams had one companion specimen which failed in shear and another which failed at a much lower load. Both companion specimens of the other beam failed at a considerably lower load. Figures 5 and 6 can also be used to determine the relative effectiveness of different angles of in- clination and the yield strength of stirrups. Most of the beams considered in the analysis had vertical stirrups; there were, however, beams with stirrups inclined at 67.5, 45 and 20 deg. The effect of differ- ent angles of inclination was taken into considera- tion in plotting Figs. 5 and 6 by computing the b=5.5 in.; d= 19.5 P. Eq. 18 kips 96.4 85.3 103.5 104.9 97.8 92.8 104.4 102.7 90.0 92.9 90.0 92.5 81.2 91.2 100.0 97.6 95.4 90.0 99.9 99.2 83.9 81.0 95.2 93.0 94.8 92.9 105.6 102.6 85.8 86.3 98.3 100.8 86.2 72.9 89.5 93.8 82.5 80.0 79.7 78.6 Beam A/ p a psi % in. Al-1 3575 3.10 36 2 3430 3 3395 4 3590 B1-1 3388 3.10 30 2 3680 3 3435 4 3380 5 3570 B2-1 3370 3.10 30 2 3820 3 3615 B6-1 6110 3.10 30 C1-1 3720 2.07 24 2 3820 3 3475 4 4210 C2-1 3430 2.07 24 2 3625 3 3500 4 3910 C3-1 2040 2.07 24 2 2000 3 2020 C4-1 3550 3.10 24 C6-2 6560 3.10 24 3 6480 4 6900 D1-1 3800 1.63 18 2 3790 3 3560 D2-1 3480 1.63 18 2 3755 3 3595 4 3550 D3-1 4090 2.44 18 D4-1 3350 1.63 18 D1-6 4010 3.42 24 7 4060 8 4030 E1-2 4375 3.42 25 D2-6 4280 3.42 30 7 4120 8 3790 D4-1 3970 3.42 30 2 3720 3 3200 D5-1 4020 3.42 30 2 4210 3 3930 * Considered tension failure. ILLINOIS ENGINEERING EXPERIMENT STATION Table 26 Tests by Clark, 1951. Simple-Span Rectangular Beams With Stirrups Reference: (5) Loading: 2 equal symmetrical loads at various positions on beam Tension Reinforcement: Deformed bars End Anchorage: A- by 8-in. plates j-in. thick welded to the end of bars Web Reinforcement: %-in. vertical deformed bars; f,_= 48,020 psi Age at Test: 28 to 30 days; beams kept moist until the day prior to testing a/d a r rf,. Pt,,t Mode P. Ratio of Eq. 18 Pt«t in. % psi kips Fail. kips P, 8- by 18-in. Beams; Span=72 in.; d= 15.37; f,=46,500 psi 2.54 7.2 0.38 182 100.0 DT 75.9 1.32 94.0 DT 74.5 1.26 100.0 DT 74.0 1.35 110.0 DT 76.0 1.44 1.95 7.5 0.37 178 125.4 DT 88.9 1.41 115.4 DT 92.3 1.25 128.1 DT 89.2 1.43 120.6 DT 88.6 1.36 108.6 DT 91.0 1.19 1.95 3.75 0.73 351 135.4 DT 88.4 1.53 144.9 DT 94.0 1.54 150.6 DT 91.5 1.64 1.95 7.5 0.37 178 170.6 DT 106.3 1.60 1.56 8.0 0.34 163 124.9 DT 100.9 1.24 139.9 DT 101.9 1.37 110.6 DT 97.6 1.13 128.6 DT 106.3 1.21 1.56 4.0 0.69 331 130.4 DT 97.0 1.34 . 4 135.4 DT 99.6 1.36 145.6 T 97.8 1.49 129.6 DT 100.5 1.29 1.56 8.0 0.34 163 100.6 DT 71.6 1.40 90.1 DT 70.9 1.27 84.6 DT 71.3 1.19 1.56 8.0 0.34 163 139.1 DT 113.5 1.23 1.56 8.0 0.34 163 190.6 DT 132.2 1.44 195.6 DT 132.1 1.48 192.7 DT 130.4 1.48 8- by 18-in. Beams; Span=72 in.; d= 15.37 in.; f,= 48,630 psi 1.17 6.0 0.46 221 135.4 DT 124.1 1.09 160.4 T 121.0 1.32 115.4 DT 120.4 0.96 1.17 4.5 0.61 293 130.4 DT 119.0 1.10 . 4 140.4 DT 123.4 1.14 150.4 T 120.9 1.24 150.6 T 120.3 1.25 1.17 3.0 0.92 442 177.6 DT 148.5 1.20 1.17 2.25 1.22 586 140.4 DT 116.8 1.20 6- by 15-in. Beams; Span 96 in.; d= 12.37; f, 46,500 psi 1.94 8.0 0.46 221 78,6 DT 60.3 1.30 80.6 DT 60.5 1.33 83.6 DT 60.4 1.38 6- by 15-in. Beam; Span= 115 in.; d= 12.37 in.; f,= 46,500 psi 2.02 5.0 0.73 351 99.7 DT 60.0 1.66 6- by 15-in. Beams; Span= 120 in.; d - 12.37 in.; f,= 46,500 psi 2.43 6.0 0.61 293 75.7 DT 49.5 1.53 70.7 DT 48.8 1.45 75.7 DT 46.9 1.61 2.43 7.5 0.49 235 75.7 DT 46.0 1.58 70.7 DT 46.6 1.52 74.2 DT 43.2 1.72 2.43 10.0 0.37 178 65.7 DT 48.2 1.36 70.7 DT 49.3 1.44 70.7 DT 47.8 1.48 Table 27 Tests by Gaston, 1952. Simple-Span Rectangular Beams With Stirrups Reference: (11) Dimensions: b = 6; D= 12; a= 36; a/d= 3.36 to 3.90; L= 108; L'= 120 Loading: 2 equal loads at A-points Tension Reinforcement: Deformed bars End Anchorage: None, straight bars Web Reinforcement: Y- and %-in. vertical deformed stirrups; fw= 45,000 psi assumed Age at Test: Around 30 days Beam // d p f, r rf,. MtAt Mode M. Ratio of Eq. 18 Mt__t psi in. % ksi % psi kip-ft Fail. kip-ft M. T1Lb 2520 10.72 0.62 46.0 0.28 126 20.2 T 18.7 1.08 T2La 2120 10.65 0.97 40.4 0.42 189 24.2 T 20.0 1.21 T4Lb 2810 10.44 2.52 43.3 0.92 414 47.8 T 32.4 1.48 T5L 2500 10.37 3.22 40.2 0.92 414 53.9 T 32.3 1.67 T11L 2900 9.23 7.22 45.3 1.83 824 67.6 C 35.4 1.91 T1Ha 5880 10.58 1.38 44.2 1.05 473 35.1 T 34.9 1.01 T2H 5400 10.44 2.52 45.6 1.05 473 53.9 T 42.3 1.27 T3H 5920 9.52 4.20 43.2 1.83 824 67.7 T 42.4 1.60 T5H 5900 9.23 7.22 40.6 1.83 824 86.3 T 46.8 1.85 Ratio Pt-t P.. 0.97 0.92 0.99 1.06 1.04 0.92 1.05 1.00 0.88 0.90 0.90 0.96 1.18 0.94 1.03 0.85 0.91 0.81 0.82 0.90 0.78 1.06 0.96 0.90 0.93 1.09 1.12 1.12 0.76 0.92 0.67 0.69 0.72 0.78 0.79 0.64 0.55 0.90 0.92 0.96 0.98 0.96 0.91 1.02 1.07 1.03 1.17 1.00 1.06 1.09 Ratio Pt 0.83 0.78 0.84 0.91 0.88 0.79 0.89 0.84 0.75 0.95 0.98 1.03 1.06 0.95 1.05 0.85 0.96 1.00* 1.03* 1.11* 0.98* 0.87 0.78 0.73 0.86 0.94 0.97 0.95 0.91 1.08* 0.78 0.89 0.94 1.01* 1.02* 0.83 0.96 0.81 0.82 0.86 1.04 0.96 0.90 0.98 0.97 0.92 1.02 0.84 0.90 0.91 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 28 Tests by Moody, Series III, 1953. Simple-Span Rectangular Beams With Stirrups Reference: (12) Dimensions: b -7; d= 21; D= 24; a=32; a/d= 1.52; L= 96; L'= 120 Loading: 2 equal loads at 4-points Flexural Reinforcement: No. 11 deformed bars; f,= 43,800 psi; t=0.914 End Anchorage: Hooks Web Reinforcement: Vertical stirrups Age at Test: 28 days Beam f/ p p' Web s Reinf. psi % % in. 30 3680 4.25 2.13 No. 3 6 31 3250 " No. 4 r f. % ksi 0.52 47.3 0.95 44.0 r/fw Pte.t Mode P. of Eq. 18 psi kips Fail. kips 246 215 S 179.5 418 228 S 169.8 Ratio Ratio Pte.t Pte.t P, P.. 1.20 0.80 1.34 0.73 ratio of web reinforcement from the conventional expression: A, (27) r bs sin a (27) The conventional theory considers that the param- eter Krf,,, is the measure of shear strength. Since the concept of the truss analogy is disregarded by the present analysis, there is no justification for employing the quantity K. Furthermore, while the variation in K is rather small for a between 45 and 90 deg, for smaller values of a the coefficient K de- creases rapidly. For beams of Slater and Lyse Fig. 5. Effect of Web Reinforcement on Shear Strength. Simple-Span Rectangular Beams with Stirrups which had stirrups inclined at 20 deg, K is equal to 0.44. The use of this low value of K would shift these beams considerably to the left in Fig. 6. Consequently, the beams would lie above the shear strength line. Since the beams failed in flexure, the use of Krfy, rather than rfJ, is not justified. The yield strength of the web reinforcement varied from about 44,000 to 73,400 psi for beams which failed in shear and from about 40,000 to 93,000 psi for beams which failed in flexure. The majority of the beams, however, had their yield strength between 45,000 and 55,000 psi. This varia- tion is perhaps not large enough to bring out the effect of yield strength. However, the beams re- ported by Slater and Lyse were reinforced with stirrups of relatively high yield strength, f,,, = 73,400 psi. If the ratio P/P, were plotted against r alone, these beams would again fall above the shear strength line determined from other test results. This indicates that the quantity rfw is a more cor- rect measure of shear strength than the ratio r alone. It seems reasonable to believe that stirrups with higher yield strength offer greater resistance to the extension and widening of the diagonal cracks than stirrups of low yield strength. 12. Bent-Up Bars as Web Reinforcement Relatively few simple-span beams with bent-up bars as web reinforcement have been tested to determine their strength in shear. The only source of experimental data is the beams tested by Richart,(2 but practically all these beams failed in tension. Series 1917 included 32 beams with hooked bent-up bars. The variables were the amount, angle of inclination, and spacing of the web bars. The main body of the beams was made of concrete from 2450 to 3770 psi; at the top center of each beam, however, there was a 4-in. deep zone of higher strength concrete, f/' = 4770 psi. The beams were tested twice: they were first loaded to yielding P, ILLINOIS ENGINEERING EXPERIMENT STATION 2.4 2.2 2.C 1.6 1.6 Plest Ps 1.4 1.2 /.C 0.6 Fig. 6. Simple-Span Rectangular Beams Fail with loads placed 48 in. from the end supports, and they were then retested with loads 36 in. from the supports. All beams failed in tension. In order to obtain some indirect information about the shear strength of these beams, some of the beams with the smallest ratio of web reinforce- ment are analyzed in Table 29. The shear capacity of the beams was calculated by using the steel percentage p at the critical section, that is, under the concentrated load, to determine the value of k. Their P/P,-ratios are plotted against rf,, in Fig. 7. This figure shows that four beams with rf,, equal to 210 psi were very close to shear failures, pro- vided that Eq. 26 holds true for beams reinforced with bent-up bars. Photographs taken of these beams after failure show well-developed diagonal cracks. In all probability the beams were very close to their shear capacity. Two beams of Series 1922 were also provided with bent-up bars as web reinforcement. These beams are analyzed in Table 20 and shown in Fig. ing in Flexure. Beams Reinforced with Stirrups 7. Both beams failed in tension and, as seen in the figure, lie below their strength in shear as given by Eq. 26. Three beams of Series 1911 had one longitudinal bar bent up at a rather small angle in order to reinforce the entire shear span, 24 in. long. These beams are reported to have failed in diagonal ten- sion. Table 30 analyzes the beams by using s = a in Eq. 27 to calculate their ratio of web reinforce- ment. Undoubtedly, this procedure is approximate, and these beams fall somewhat low in Fig. 7. How- ever, a sketch of one of the beams after failure shows extensive cracking at the end hooks of the reinforcement and indicates a possible failure in the anchorage. With the help of Fig. 7 and more numerous tests on T-beams which are analyzed later, it was concluded that the contribution of bent-up bars to the shear strength of a beam is the same as that of stirrups. Consequently, Eq. 26 can be used in both cases. Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 29 Tests by Richart, Series 1917. Simple-Span Rectangular Beams With Bent-Up Bars Reference: (2) Dimensions: b=8.1; d=10; D=12; a=48; a/d=4.8; L=114; L'=120 Loading: 2 equal loads Tension Reinforcement: Plain round bars, hooked; f,= 37,500 to 45,700 psi Web Reinforcement: Bent-up bars, hooked Concrete Strength: f// = 4770 psi for a zone 4-in. deep and 54-in. long at top center of each beam; f/= 3040 to 3770 for the remainder Age at Test: About 60 days a r rf,« deg % psi 45 0.56 210 45 0.56 45 0.80 320 45 0.80 370 45 3.28 1490 28 1.96 900 45 1.29 590 11 " 11 * Distance from load to first bent-up bar. Some additional information about the effective- ness of bent-up bars is available from the tests of Series 1917. One of the variables investigated was the distance from the load point to the first bent-up P -7 Pt«t Mode of kips Fail. 40.2 T 40.0 T 42.2 T 40.8 T 40.8 T 40.0 T 41.9 T 41.5 T 36.3 T 31.5 T 45.5 T, DT 40.9 T 37.7 T 41.1 T P. Eq. 18 kips 30.3 30.3 30.3 30.3 30.1 30.1 30.3 30.3 30.0 30.3 30.2 30.2 30.2 30.2 bar. This distance varied from 9.6 to 16.8 in., or up to 1.68 times the effective depth of the beams. The analysis of some of these beams was included in Table 29. It is seen that even these beams failed rfFm (psi) Fig. 7. Simple-Span Rectangular Beams Failing in Flexure. Beams Reinforced With Bent-Up Bars Beam 16B6.1 6.2 16B7.1 7.2 16B8. 1 8.2 16B9.1 9.2 16B10. 1 10.2 16B18.1 18.2 16B19.1 19.2 ILLINOIS ENGINEERING EXPERIMENT STATION in tension, although a considerable part of the shear span in the immediate region of maximum moment was without any direct web reinforcement. The highest P/P,-ratio at failure was 1.51. Conse- quently, well-anchored bent-up bars, although not covering the entire shear span, appear to be bene- ficial in resisting the development of diagonal cracks. This phenomenon was also observed for Table 30 Tests by Richart, Series 1911 Simple-Span Rectangular Beams With Bent-Up Bars Reference: (2) Dimensions: b 8; d0; D - 12; a-24; a/d-2.4; L -72; L'=78 Loading: 2 equal loads at A-points Tension Reinforcement: Three t-in. plain round bars, hooked; p = 0.0165; f,=about 38,000 psi Web Reinforcement: One o-in. round bar bent up; a about 27 deg Concrete Strength: Tests on 6- by 8- by 40-in. control beams; reduced to cylinder strength by f/' = 6.7 f, Age at Test: Around 60 days Beam f/ r* rf,. Pt.t Mode P. Ratio Ratio of Eq. 18 Pte..t Pte.t psi % psi kips Fail. kips p 292.1 1760 0.50 190 30.7 DT 25.5 1.20 0.87 2 " " " 28.9 DT 25.5 1.14 0.83 3 29.8 DT 25.5 1.17 0.85 *r computed as r= A sin 270. beams of Series 1910 which had vertical and diagonal stirrups supplemented by bent-up bars. It is seen in Table 18 that the addition of only one layer of bent-up bars, not covering the entire shear span, increased the shear strength of the beams sufficiently to permit a tension failure. 13. Maximum Useful Amount of Web Reinforcement Excluding bond failure, a reinforced concrete beam can fail either in flexure or in shear. Flexural failures can be initiated either by yielding of ten- sion reinforcement or by crushing of concrete on the top of the beam, depending on the physical properties of the beam. Since the flexural capacity of a beam can be determined accurately, the pur- pose of this analysis is to find the amount of web reinforcement necessary to force a beam to fail in flexure rather than in shear. Expressions for the shear capacity of a simple- span rectangular beam under one or two sym- metrical concentrated loads were derived previ- ously. Equation 26 can be rewritten as: M 1 + 2rf (28) M8 = 10, where Ms is given by Eq. 18. Expressions for the flexural capacity of a beam are taken from a previous technical report.(") The ultimate flexural moment is given as: Mf = pf. k2 pf. bd f,' f' kk f, ' ) (29) When a beam fails in tension, the yield stress f, is substituted for f, in Eq. 29. For compression fail- ures, the steel stress f, is below its yield strength; it can be determined from the following equation: = _ E .ekk3f/ (1 ~uE)2 1 f. = \ - p- -2 EUE) - uE (30) Whether the stress in the tension reinforcement at failure is below or at its yield stress is determined by the following criterion. The reinforcing index q is defined as: q = , c (31) The critical value of q is given by kik3 qcr= - 1 Cy (32) If q > qcr, the steel stress at failure is below its yield stress and the beam fails in compression. If q = Qcr, the beam fails by crushing of concrete as soon as the tension reinforcement yields. If q < qer, the steel stress at failure is either at or above its yield stress and the beam fails initially by yielding of the tension reinforcement. Another critical value of q can be utilized to determine whether or not the steel stress reaches work hardening at failure; this, however, is an unnecessary refinement in the pres- ent analysis. The following numerical values are used in the above equations: k2 = 0.45 kik3 = 2.4 (0.57 - 4.5f,) 10, (20) Eu = 0.004 E, = 30,000,000 psi The behavior of beams with different values of the reinforcing index is shown by Fig. 8. To facili- tate the presentation of expressions for shear strength, the quantity M/bd2f/ for the ultimate moment is plotted against the parameter p/f,' Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS rather than q. The curves are drawn for f, =45,000 psi, fc' = 3,000 psi, and fy,, = 45,000 psi. If the beams have sufficient web reinforcement to fail in flexure, the ultimate moment is determined by curves 1 and 2. For p/fi' < (P/fc') cr the beams fail in tension according to curve 1, obtained from Eq. 29 by substituting f, =fI. At (p/fc')r = 1.69 X 10-" in.2/lb, computed from Eq. 32, the mode of failure changes from tension to compression. For p/fc > (p/fc') c, the ultimate moment is given by curve 2, computed by Eq. 29 with steel stresses ob- tained from Eq. 30. If, however, no web reinforcement is provided, the maximum load is governed by curves 1 and 3. Curve 3 represents the shear strength of a beam without web reinforcement, given by Eq. 18. The intersection between these two curves determines the transition between tension and shear failures. When some web reinforcement is provided, the shear strength increases according to Eq. 28 and the transition between the two types of failures takes place at a larger value of p/fc. Curve 4 shows this for r = 0.005. If it is desired that the beams fail in flexure for any value of p/If', the shear strength must be Fig. 8. Relation Between Strength in Shear and Flexure as Function of Reinforcement Percentage greater than the flexural strength for the entire range of ultimate moment, curve 1 for tension and curve 2 for compression failures. It is seen that a shear strength curve 5 passing through the inter- section between curves 1 and 2 satisfies this con- dition. Computations based on the value of of per, which corresponds to qcr obtained from Eq. 32, show that the corresponding ratio of web reinforce- ment is 0.011 for the variables under consideration. f" (ps/I Fig. 9. Maximum Useful Amount of Web Reinforcement as Function of Concrete Strength and Yield Strength of Reinforcement Thus r = 0.011 corresponds to the maximum useful amount of web reinforcement for the values of fC', /y, and fyw used in the above example. This limit was calculated for other combinations of f//, If, and fyw and is shown in Fig. 9 graphically. The maximum useful amount of web reinforce- ment does not depend on the percentage of tension reinforcement. It forces a beam of any amount of tension reinforcement to fail in flexure, either in tension or in compression. However, for any value of p except that at the transition between tension and compression failures, this maximum useful amount is more than sufficient to insure flexural failures; see curve 5 of Fig. 8. In practice, most beams are designed to fail in tension if loaded to destruction. These beams would fall considerably to the left of the transition point and, consequently, would require much less web reinforcement to pre- vent shear failures. Table 31 shows an analysis of simple-span beams designed according to the present ACI Code balanced design requirements and loaded with concentrated loads. This analysis considers rec- p/f' (10sin?1// ILLINOIS ENGINEERING EXPERIMENT STATION Table 31 Amount of Web Reinforcement Required to Prevent Shear Failures in Rectangular Beams Normal ACI Beams Without Compression Reinforcement p//f. 10-5 in.'/lb 0.46 0.45 0.45 0.46 n Eq. 16 10.0 9.0 8.3 7.7 k M. Eq. 14 bdf// Eq. 18 0.345 0.166 0.351 0.160 0.376 0.164 0.398 0.160 // q Mr Ratio r/,. r(%) bdf/3 Mf/Al. psi computed for fA.(ksi) = Eq. 31 Eq. 29 Eq. 26 40 45 50 f,=40,000 psi 2000 0.182 0.169 1.02 11 0.03 0.02 0.02 2500 0.181 0.167 1.04 22 0.06 0.05 0.04 3000 0.181 0.167 2 11 0.03 0.02 0.02 3750 0.184 0.168 1.05 25 0.06 0.06 0.05 fy = 45,000 psi 2000 0.205 0.188 1.14 69 0.17 0.15 0.14 2500 0.203 0.186 1.16 81 0.20 0.18 0.16 3000 0.204 0.186 1.14 69 0.17 0.15 0.14 3750 0.207 0.187 1.17 84 0.21 0.19 0.17 f, = 50,000 psi 2000 0.228 0.207 1.25 126 0.32 0.28 0.25 2500 0.226 0.205 1.28 140 0.35 0.31 0.28 3000 0.227 0.205 1.25 126 0.32 0.28 0.25 3750 0.229 0.205 1.28 141 0.35 0.31 0.28 * Steel percentages as given by ACI Code balanced design requirements for f.= 20,000 psi and f. = 0.45 f/. tangular beams reinforced in tension only with the steel percentages corresponding to allowable stresses of f, equal to 20,000 psi and fc equal to 0.45 fe'. The amount of web reinforcement neces- sary to prevent shear failures has been calculated with the aid of Eq. 28 for several values of f, and fy,. It is seen that as f, increases and fy decreases, the amount of web reinforcement necessary to en- sure flexural failures increases. For f, equal to 50,- 000 psi and f/w equal to 40,000 psi, about 0.35 percent web reinforcement is required, while for both f, and fe, equal to 45,000 psi, about 0.20 per- cent will be sufficient. IV. SIMPLE-SPAN T-BEAMS UNDER ONE OR TWO SYMMETRICAL CONCENTRATED LOADS 14. T-Beams Without Web Reinforcement The basic empirical equation 18 was derived for simple-span rectangular beams. It has been shown that this equation can be interpreted by means of the conventional theory of compression failures, as modified by diagonal tension cracking, and that the failure criterion is the ultimate compressive strain in the concrete. The above concept of shear failures as shear- compression failures was extended to include T- beams. Since the moment-rotation relationship of a T-beam differs from that of a rectangular beam, a correction must be made to take into considera- tion the effect of the shape of the beam on the com- pressive strain in the concrete. But since the distribution of the concrete strain had not been de- termined previously, the exact form of the shape factor cannot be established. If a linear strain dis- tribution is assumed, strain in any fiber is given by M = El y where y is the distance from the neutral axis to the fiber under consideration. Comparing a T-section with a rectangular section of the same width as the flange in the T-section, the following relationship can be written if the ultimate strain in the concrete is the same in both cases: MT = MR IcTYR (33) where the subscripts R and T refer to rectangular and T-sections, respectively, Ic refers to the mo- ment of inertia of a section transformed to concrete, and y refers to the distance from the neutral axis to the top fiber in the concrete, all quantities taken at the instant of failure. If the strain distribution were linear and all quantities could be determined, the above expression would give the relationship between shear moments of a T-section and a rec- tangular section of the same width. However, the formation of a diagonal crack produces a non- linear strain distribution. The stress in the tension reinforcement is approximately uniform from the lower end of the crack to a vertical section through the uper end of the crack. These conditions affect also the distribution of concrete strain at the top of the beam, causing a certain concentration of strain at the end of the diagonal crack. Furthermore, since the section cracks progressively as load is applied, the exact values of I and y cannot be determined. Consequently, Eq. 33 cannot be applicable. An approximate shape factor was derived by assuming that the effect of shape of a beam is de- termined primarily by its moment of inertia. In an uncracked state, the moment of inertia of a T-beam is considerably smaller than that of a similar rec- tangular beam. After extensive cracking, the value of I of a section transformed to concrete is very nearly the same in both cases. At the instant of failure, the relationship between the two is un- known; it was approximated by the ratio of the average values of I of the uncracked and the fully cracked state. Thus, the shape factor takes the fol- lowing form: IT + Icr IR + Icr where IR and IT refer to the uncracked rectangular and T-sections, respectively, and Icr refers to the "straight line" cracked transformed section of either a rectangular or a T-section since both have very nearly the same moment of inertia. The above shape factor makes it possible to modify Eq. 18 for rectangular beams so that it applies to T-beams. The compressive area Ac of a T-section as determined by the conventional "straight line" theory is substituted for bkd and the equation is rewritten as follows: dF M. 0.57 104.5 AodfF, 10s The validity of Eqs. 34 and 35 must be deter- mined with the help of test results. All available data on T-beams under one or two symmetrical ILLINOIS ENGINEERING EXPERIMENT STATION concentrated loads were analyzed; the range of test variables is summarized in Table 32 and the physi- cal properties and calculated quantities of indi- vidual beams are given in Tables 33 through 39. All units are given in inches and pounds. The width of the flange is indicated by b, that of the web by b', and the thickness of the flange by e. Other symbols have their usual meaning. Some beams were rein- forced with straight unanchored bars and failed in bond; these beams are not included in the analysis. Beams without web reinforcement are consid- ered first. Ferguson and Thompson(20, 21) have re- ported tests on beams of a number of different shapes. Some of the beams were provided with shoulders; that is, the width of the upper part of the web was greater than that of the lower part. These beams are analyzed in Tables 36 and 37, and the quantity M/Acdf/'F is plotted against f/' in Fig 10. It is seen that in most cases Eq. 35 gives reasonable agreement with the test results. How- Table 32 Range of Test Variables for Simple-Span T-Beams. Under Two Symmetrical Table No. No. No. of of Beams Shear Fail. I' A. Reinf. in Flange Concentrated Loads b b d psi in.' in. BEAMS WITHOUT WEB REINFORCEMENT 2510-2690 3.9 None 19.7 1700 1.56 Yes 42 3570;3610 3.91 None 20 2540-3500 0.88 None 19; 22 3960-6580 1.58 None 17 BEAMS WITH WEB REINFORCEMENT 2650 3.9 None 19.7 2580 3.9 None 19.7 1700 2.34;3.51 Yes 42 3799-4346 3.91 None 20 1370-1540 6.77 Yes 53.2 a/d F, in. in. in. 13.9 10.9 21 4.5-7 8.25 13.9 13.4 10.0-10.9 21 21.3 3.9 4.25 6 1.5;2.13 1.5 3.9 3.9 4.25 6 3.9 2.83 3.30 1.71 4.0-6.22 3.39 2.83 2.94 3.3-3.6 1.71 2.77 0.815 0.55 0.76 0.58-0.65 0.65-0.75 0.82 0.80 0.59;0.62 0.76 0.63 Table 33 Tests by Bach and Graf, Heft 10, 1911. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (18) Dimensions: b=19.7; b'=7.9; Df-15.7; d=13.9; e=3.9; a=39.4; a/d=2.83; L-118.1; L'=133.9 Loading: 2 equal loads at %-points Tension Reinforcement: Two 1.57-in. plain round bars; A.=,3.90 in.'; f,= 43,600 psi Anchorage: Hooks Web Reinforcement: Plain round vertical stirrups Reinforcement in Flange: None Concrete Strength: Average f/.'3530 psi; '//0.75 .f' =2650 psi; variation from -8 to +12 percent Age at Test: About 45 days Number of Beams: 3 companion specimens in each group; 2 in groups c and d BEAMS WITHOUT WEB REINFORCEMENT Group Size 8 W.R. in. in. b 0.51 3.35 c 0.51 5.51 d 0.28 5.51 8 0.39 7.87 9 0.28 7.87 10 0.20 7.87 11 0.39 5.91 12 0.28 5.91 13 0.20 5.91 15 0.39 3.94 16 0.28 3.94 17 0.20 3.94 18 0.20 1.97 14 0.39 5.91 19 0.28 5.91 20 0.28 5.91 21 .79by .08 5.91 22 .79by .08 5.91 23 0.28 5.91 Ps..t A. Fe Muse Ratio 0.815 Adf'Ft 0.418 0.354 0.398 0.408 0.412 BEAMS WITH WEB REINFORCEMENT f,. rf,, Pt.t F, P, Eq. 35 ksi psi kips kips 37.8 580 94.4 0.82 60.6 38.6 370 86.0 40.2 109 77.2 41.0 160 80.0 " 43.8 83 72.0 48.2 48 65.8 41.0 213 82.9 43.8 114 79.4 48.2 63 72.4 41.0 324 94.1 43.8 171 88.2 48.2 96 80.1 48.2 188 89.3 41.0 213 82.2 43.8 114 71.9 43.8 114 74.4 57.5 305 87.5 52.2 141 77.2 43.8 114 67.6 M. 0.93 0.79 0.88 0.89 0.92 Ratio Ratio P-1 P-t1 P. P,, 1.56 0.73 1.42 0.82 1.27 1.04 1.32 1.00 1.19 1.03 1.08 0.99 1.37 0.96 1.31 1.06 1.19 1.05 1.55 0.94 1.45 1.08 1.32 1.10 1.47 1.07 1.36 0.96 1.19 0.97 1.23 1.00 1.44 0.90 1.27 0.99 1.12 0.92 Mode of Fail. S,B S,B S,B S,B S,B Mode of Fail. B,S? B,S? S S S S S 8 S S S S S S 8 S S Test Series Bach, Graf Heft 10(1) Braune Myers(") Richart Ser. 1922(2) Thompson Ferguson(20) Ferguson Thompson(21) Bach, Graf Heft 100)' Heft 12(22) Braune Myers(u) Richart Ser. 1922(2) Graf Heft 67(<) Beam e-330 331 7-441 442 444 kips 57.3 48.5 52.9 52.9 57.3 r % 1.56 0.95 0.27 0.39 0.19 0.10 0.52 0.26 0.13 0.79 0.39 0.20 0.39 0.52 0.26 0.26 0.53 0.39 0.26 I Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS ever, two series with the largest number of beams, Series A and B, indicate consistently lower shear strengths than those given by Eq. 35. This discrep- ancy could mean either that the shape factor given by Eq. 34 is fundamentally incorrect or that there are some other considerations besides the effect of the moment of inertia which determine the com- pressive strain in the concrete. It is noticed that the beams reported by Ferguson and Thompson have, in general, very wide and thin flanges. It is known that in such beams parts of the flanges at some distance from the web do not resist their full share of the bending moment. This phenomenon is dis- cussed in some detail for an elastic medium by Timoshenko.241 It can be seen in Fig. 10 and Tables 36 and 37 that beams which fall the greatest percentage below the predicted values have very large d/e and b/b'-ratios; that is, the depth of the beam is large relative to the thickness of the flange and the width of the flange is large relative to that of the web. The beams of Series A, B, C, and D had the same d/e-ratio, 5.5, while b/b'-ratio was equal to 4.25 for Series A and 2.43 for Series D. The two beams of Series D failed at a load rather close to the predicted values, 85 and 93 percent, respec- tively, while the beams of Series A reached only about 70 percent of their predicted capacity at fail- ure. The beams of Series B and C were provided with shoulders of the same width as the web width of the beams of Series D. The depth of the shoulder, e", was 4 in. for Series B and 7 in. for Series C. The addition of shoulders reduces the unsupported width of the flange and in that sense, should have the same effect as a decrease in the b/b'-ratio. Table 34 Tests by Braune and Myers, 1917. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (19) Dimensions: b=42; b'=6; D=12; e=4.25; a=36; L=108; L'- 120 Loading: Two equal loads at M-points Tension Reinforcement: 5%-in. square twisted bars; f,= 40,320 and 66,350 psi for the two bars tested Reinforcement in Flange: Four %-in. round long. bars; H-in. square transverse bars at 8 in.; all beams except I Web Reinforcement: h-in. square plain bars and bent-up bars Concrete Strength: Average f,,'= 2270 psi; f'= 0.75 fe'= 1700 psi Age at Test: 90 davys BEAMS WITHOUT WEB REINFORCEMENT Beam p* d a/d Ptest Mode of % in. kips Fail. I-1 0.34 10.9 3.30 33.4 S 2 " " 28.8 S BEAMS WITH WEB REINFORCEMENT Beam pt d a/d Web a r rf,. Reinf. % in. deg. % psi II-1 0.51 10.9 3.30 Stirr. 90 0.52 350 2 11" " " I '? III-1 0.51 10.9 3.30 3B-up ? ? 2 " " Bars IV-1 0.91 10.0 3.60 Stirr. 90 2.36 1580 2 "" +B-up 45 V-1 0.91 10.0 3.60 Stirr. 90 2.82 1870 2 " 1 " +B-up 45 * Bars not hooked. t Bars hooked. Ft M 'tF Asdfe'F, Ptýt kips 92.0 86.0 97.4 95.4 129.6 139.2 139.2 139.2 Mode of Fail. T T T T T T T T 0.562 0.484 F* P. Eq. 35 kips 0.59 39.1 0.59 39.1 0.62 43.7 0.62 43.7 Ratio M. 1.14 0.98 Ratio Pte. P. 2.36 2.20 2.49 2.44 2.97 3.19 3.19 3.19 Table 35 Tests by Richart, Series 1922. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (2) Dimensions: b=20; b'=8; D=f24; d=21; e=6; a= 36; a/d=1.71; L= 108; L'=120 Loading: 2 equal loads at W-points Tension Reinforcement: Four 1%-in. corrugated round bars; p= 0.0093; f,= 52,400 psi Anchorage: Hooks Web Reinforcement: Plain round vertical stirrups Reinforcement in Flange: None Age at Test: About 60 days BEAMS WITHOUT WEB REINFORCEMENT Pts. Mode A, Fs of kips Fail. in.2 180.3 DT 125.0 0.76 167.2 DT 125.4 BEAMS WITH WEB REINFORCEMENT s r /,, r/,» Pte.s in. % ksi psi kips 4 1.38 42.9 592 259.5 . ... 245.5 7 1.40 40.1 561 258.5 265.8 11 1.39 39.6 550 261.4 257.2 Mt-t A4df,'Ft 0.451 0.421 Mode of Fail. T T T T T T Ratio Mode M. " of M, Fail. 1.10 S 1.03 S Fs p, Ratio Eq. 35 ___ kips P. 172.7 1.50 178.1 1.38 167.5 1.54 178.3 1.49 173.0 1.51 Beam 2210.1 2210.2 Beam 226.1 2 227.1 2 228.1 2 71 4.8 1.47 ILLINOIS ENGINEERING EXPERIMENT STATION Although the beams of Series B failed at but slightly higher loads than those of Series A, the two beams of Series C reached 75 and 84 percent, re- spectively, of their predicted strength at failure. These results show that for the same value of d/e, the agreement between the measured and calculated loads improves as the ratio b/b' decreases, and thus addition of shoulders does have partially the same effect as that of decreasing the b/b'-ratio. Further- more, deeper shoulders have a greater effect on the increase of the shear strength than shallower shoul- Beam N-1 2 3 G-4 5 6 L-1 2 3 HB-2 5 8 KB-1 4 7 ders. This is apparently related to the formation and propagation of cracks in the tension zone of the concrete. However, since the shape factor of Eq. 34 was primarily intended for ordinary T- beams without shoulders, it is not expected that it would apply equally well for more complex shapes of T-beams. The remaining beams had rather large b/b'-ratios, varying from 4.47 to 5.18, while the d/e-ratio varied from 2.11 to 4.67. All these beams except those of Series N failed at a load in good agreement with Eq. 35. The beams of Series N Table 36 Tests by Thompson and Ferguson, 1950. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (20) Dimensions: a=28; L=84; L'=96 Loading: Two equal loads at %-points Tension Reinforcement: Two %-in. round deformed bars, inter, grade; A,= 0.88 in.2 End Anchorage: Welded anchorage plate Web Reinforcement: None Reinforcement in Flange: None Shoulders: Width = b; depth from top of beam= e'= 0 Age at Test: 28 days Series H-B and K-B: Beams with B-tile considered in analysis; comp. strength of B-tile= 4160 psi; %-in. layer of tile included in the overall dimensions of beams Mode of Failure: All beams failed in shear f.' Pt-t A, F, M__ Adfc'F, psi kips in.' b=19; b'=4.25; b"=0; d=7; D=7.5; e=1.5; e"=0; a/d=4.0; d/e=4.67; b/b'=4.47 3000 10.68 30.67 0.65 0.357 2990 10.76 30.67 0.361 2540 9.66 30.97 0.378 b= 22; b'= 4.25; bV= 0; d=4.5; D= 5.5; e= -1.5; e'=0; a/d-6.22; d/e =3.00; b/b'= 5.18 3320 6.30 30.99 0.58 0.326 3150 7.10 31.28 0.383 3170 7.90 31.19 0.425 b=19; b'=4.25; b"=7; d=6.25; D=7.5; e= 1.5; e= 3.5; a/d=4.48; d/e=4.17; b/b'= 4.47 3150 12.30 30.95 0.61 0.463 3280 13.40 30.81 0.487 3220 12.30 30.88 0.454 b=22;b'= 4.25; Vb=0; d=4.5; D=5.5; e=2.13; e"=0; a/d=6.22; d/e=2.11; b/b'=5.18 3270 9.14 31.09 0.60 0.470 3150 9.14 31.19 " 0.487 3020 8.90 31.48 0.489 b=19; b'= 4.25; b"= 8.25; d= 6.25; D=7.5; e=2.13; e'=4.13; a/d= 4.48; d/e=2.93; b/b'= 4.47 3340 13.78 34.44 0.62 0.435 3350 12.25 34.44 0.385 3500 14.76 34.20 0.447 Ratio Mtet M, 0.82 0.83 0.83 0.78 0.90 1.00 1.08 1.15 1.07 1.11 1.13 1.12 1.03 0.92 1.08 Table 37 Tests by Ferguson and Thompson, 1953. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (21) Dimensions: b=17; d=8.25; D=9.5; e=1.5; a= 28; a/d=3.39; L=64; L'=72 Loading: Two equal loads at J-points Tension Reinforcement: Two No. 8 deformed bars, rail steel; A.= 1.58 in.' End Anchorage: Welded steel block at each end Web Reinforcement: None Reinforcement in Flange: None Age at Test: Around 28 days Shoulders: Width = b", depth from top of beam = e Mode of Failure: All beams failed in shear I' b' b" el d/e b/b' Ptes. A, Ft Mt Beam A 1 2 3 4 5 6 D 1 2 B 1 2 3 4 5 C 1 2 kips 13.06 12.12 15.12 14.22 15.22 16.00 21.90 23.40 15.94 14.20 17.72 19.72 17.22 19.74 17.44 in.' 31.38 31.54 31.10 31.10 30.74 30.58 35.16 35.37 34.81 35.02 34.60 34.39 34.46 35.02 35.02 est Ratio M. 0.254 0.68 0.256 0.65 0.252 0.74 0.238 0.70 0.198 0.72 0.246 0.78 0.312 0.85 0.350 0.93 0.231 0.69 0.218 0.62 0.231 0.75 0.238 0.83 0.220 0.73 0.295 0.84 0.261 0.75 Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS had the largest d/e-ratio, 4.67, and failed at a somewhat lower load than that predicted. In con- clusion, the above findings suggest that the shape factor of Eq. 34 is applicable whenever beams with abnormally high d/e' and b/b'-ratios are excluded. In the above comparison, the beams of Series H-B and K-B were of composite tile-concrete con- struction. One 5/s-in. thickness of tile was included in the overall dimensions of the beams in calcu- lating the shear strength of the beams. These beams were made with B-type tile which had but slightly higher compressive strength than that of the con- crete used. It was seen that these beams failed at about the predicted load. Beams made with tiles of higher concrete strength, not included in the analyses, failed at a somewhat higher load; the high strength tiles seemed to have acted as a form of web reinforcement in increasing the load at failure. Figure 11 shows the above beams, except Series A and B, together with results from other investi- gations. The beams of Series A and B were ex- cluded because of the simultaneous high ratios of d/e and b/b', and as discussed previously, the shoulders of the beams of Series B were not deep enough to increase their shear strength. It is seen Table 38 Tests by Bach and Graf, Heft 12, 1911. Simple-Span T-Beams With Bent-Up Bars Under Two Symmetrical Concentrated Loads Group Pt.es Ratio Ptest kips P, 25 76.0 1.34 29 92.6 1.63 50 81.9 1.44 31 83.8 1.48 34 86.7 1.53 33 92.6 1.63 47 105.8 1.86 45 86.9 1.53 46 95.2 1.68 36 101.4 1.79 38 109.1 1.92 48 100.7 1.77 49 107.3 1.89 43 90.0 1.58 44 99.9 1.76 40 100.3 1.77 42 104.3 1.84 * Cr = crushing at hooks. Reference: (22) Dimensions: b= 19.7; b'V=7.9; D=15.7; d =around 13.4; e=3.9; a= 39.4; a/d=2.94; L=118.1; L'=133.9 Loading: 2 equal loads at %-points Tension Reinforcement: From 4 to 7 plain round bars; A,=about 3.9 in.2; f,= about 47,000 psi Anchorage: Only hooked bars included Web Reinforcement: Bent-up bars Reinforcement in Flange: None Concrete Strength: Average f.' =3440 psi; /' =0.75; f1' = 2580 psi; variation from -8.3 to +7.0 percent Age at Test: Around 45 days Number of Beams: Three companion specimens in each group Calculated Quantities: Ft=0.80; P.=56.8 kips (from Eq. 35) No. of B-up Layers 1 2 3 5 a No. and Area of Bent-up Bars deg No.-sq in. 18 2-1.91 45 3-1.79 3-1.77 45 2-1.78;2-1.78 2-1.78;2-1.78 2-1.78;2-1.78 2-1.78;2-1.78 30 2-1.78;2-1.78 2-1.78;2-1.78 45 1-0.89;2-0.88;2-0.88 1-0.89;2-0.88;1-0.88 1-0.89;2-0.88;2-0.88 1-0.89;2-0.88;1-0.95 30 1-0.89;2-0.88;2-0.88 1-0.89;2-0.88;1-0.95 45 Five times 1-0.54 Five times 1-0.54 No. and Area Straight Bars No.-sq in. 2-1.92 3-2.10 3-2.00 1-0.35 1-0.35 1-0.35 1-0.35 1-0.35 1-0.35 1-1.25 2-1.25 1-1.25 2-1.18 1-1.25 2-1.18 1-1.25 2-1.25 Spacing Between Bends, From Load (in.) 0 10.8 14.8 2.0-25.6 0-27.6 2.0-25.6 3.9-17.3 2.0-10.8 2.0-21.7 0-12.8-9.8 0-12.8-8.5 3.9-10.8-7.9 3.9-10.8-6.5 2.0-14.4-9.9 2.0-14.0-10.0 1.2-8.3-8.1-5.9-8.5 1.0-8.5-8.1-5.9-8.5 Beam P' psi III 6-1 1540 2 III 7-1 2 III 8-1 2 Table 39 Tests by Graf, Heft 67, 1931. Simple-Span T-Beams Under Two Symmetrical Concentrated Loads Reference: (23) Dimensions: b=53.2; b'= 9.9; D= 23.6; d=21.3; e=3.9; a=59.1; a/d=2.77; L= 177.2; L'= 205 Loading: 2 equal loads at H-points Tension Reinforcement: Ten 0.866-in. plain round bars; A.= 6.77 in.2; f,= 46,000 psi; all bars hooked Reinforcement in Flange: Four 0.28-in. long. plain round bars; 0.28-in. transverse bars at 4.5 in., under loads at 2.5 in.; f, = 48,000 psi Web Reinforcement: Five long. bars bent up at 45 deg, s=about 10.2 in.; 0.28-in. vert. stirrups at 7.1 in. Concrete Strength: Tests on 7.9-in. cubes; reduced to cyl. strength by f,'=0.75 f..' Age at Test: Around 30 days Type of r rf,. Pt.es Mode F, P. Ratio Ratio Ratio Bent-up of Eq. 35 Pt Ptt Ptt Bars % psi kips Fail. kips P. P.. PI 148370 c 1370 7 III 9-1 1410 k. 2* * Cr = Crushing at hooks. 1.33 638 231 T 220 T 170 SCr* 165 SCr 209 S,Cr 176 S,Cr 209 S,Cr 182 S,Cr 0.63 85 2.72 1.20 1.11 85 2.59 1.14 82 2.07 0.91 82 2.01 0.88 76 2.75 1.20 0.94 76 2.32 1.02 ' 78 2.68 1.18 0.96 78 2.33 1.02 r r,. % psi 1.15 540 1.15 540 1.15 540 1.50 710 3.52 1650 1.90 890 1.24 580 1.24 580 1.08 510 1.08 510 1.37 640 1.34 630 1.02 480 1.02 480 Mode* of Fail. Cr Cr Cr Cr Cr Cr T Cr Cr T, Cr T T, Cr T Cr Cr T, Cr T ILLINOIS ENGINEERING EXPERIMENT STATION 0.6 0.5 0.4 0.3 UMest 0.2 0.I 0 c, (psi) Fig. 10. Tests by Ferguson and Thompson. Si that when beams with abnormally large d/e and b/b'-ratios are excluded, Eq. 35 gives satisfactory agreement with test results. In some beams of Bach and Graf, there is some doubt about the primary mode of failure; heavy cracking at the end hooks of of the tension reinforcement indicated possible anchorage failure. This might explain why one of these beams is somewhat low. Beams of Richart and of Braune and Myers show good agreement with Eq. 35. Although the beams of Braune and Myers had a very high b/b'-ratio, 7.0, the d/e-ratio was rather small, 2.56, and the beams failed according to Eq. 35. It is concluded that the shear strength of simple-span T-beams without web reinforcement as normally used in construction can be predicted by Eq. 35 where the shape factor is computed by Eq. 34. Beams with abnormally high d/e' and b/b'- ratios are outside the scope of Eq. 35, their shear strength is lower because the effective width of such flanges is reduced. No attempt was made, however, to determine an expression for the effective flange width. Moreover, T-beams of such dimensions are not permitted by the present ACI Code require- imple-Span T-Beams Without Web Reinforcement ments for isolated beams.* In the following section, it is shown that the use of transverse reinforcement in the flange effectively counteracts the reduction in the effective width of the flange and thereby in- creases the scope of Eq. 35. This phenomenon was also observed for the beams of Braune and Myers in the present comparison. 15. T-Beams With Web Reinforcement T-beams considered in this section are analyzed in Tables 33, 34, 35, 38, and 39. A summary of the test variables is included in Table 32. The ratio P/P,, where P, was obtained from Eq. 35, was cal- culated for each beam. This ratio is plotted against the parameter rf,, in Fig. 12 for beams which failed in shear. The ratio of web reinforcement was com- puted with respect to the width of the web. Heft 10 by Bach and Graf reports tests on 81 beams.0s8 The beams were tested in 28 groups, 25 groups of three and 3 groups of two companion specimens. All beams were reinforced with two ten- sion bars. One group of 3 beams had 2.82 sq in. of tension reinforcement and failed in tension. The *Building Code Requirements for Reinforced Concrete (ACI 318-51), American Concrete Institute, Detroit, 1951. Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS remaining beams were provided with about 3.9 sq in. of tension steel and failed either in shear or in bond. Beams with straight unhooked bars failed at a lower load than similar beams with hooked bars, apparently in bond. Beams with hooked bars failed mostly in shear; these beams are analyzed in Table 33. Figure 12 shows that most of the beams give good correlation with Eq. 26, originally derived for rectangular beams. Only two groups of beams failed at a somewhat lower load than that predicted. However, photographs of beams after failure show rather extensive cracking at the end hooks of the tension reinforcement. Beams with larger amounts of web reinforcement resisted a higher load at failure and showed more marked cracking. It is possible that the two groups with the highest amount of web reinforcement failed in bond through excessive bending of the anchorage hooks. Heft 67 by Graf reports tests on 8 T-beams under two symmetrical concentrated loads.(23) These beams were provided with transverse rein- forcement in the flanges, and although the flanges were rather thin and wide, no reduction was noticed 0.6 o0 0. Mte A .df 0. '4 3 st Ft 2 n 0.57- /05 Z £ * in the effective flange width. All beams were rein- forced with the same amount of web reinforcement; the only variable was the arrangement of bent-up bars. Four different groups of two beams were in- vestigated; the test results are given in Table 39. Beams of Group 6 were reinforced with regular bent-up bars, the horizontal part of the bends being carried over the transverse reinforcement in the flanges. This arrangement of web reinforcement was the most effective one; the beams failed in tension, and as seen in Fig. 12, the load at failure was about 20 percent higher than that predicted for shear. Beams of Group 8 were reinforced with "brought- back" bent-up bars; all longitudinal bars were first taken to the end of the beam, bent up there and then bent down at the desired spacing to serve as web reinforcement. The beams failed in shear at a load slightly higher than the predicted load; some crushing of concrete was observed at the end hooks of the "brought-back" bars. Beams of Group 9 were provided with conventional bent-up bars ex- cept that the bends had no horizontal extension at the top of the beam. This type of web reinforce- ment was about as effective as that of Group 8. /000 2000 300 f (ps) 4000 5000 Fig. 17. Failure Moment as Function of Concrete Strength. Simple-Span T-Beams Without Web Reinforcement bU600 0 00 +15% 0 0 0 0 0 0 0 -+ 15% ----o ^ * - 0 -15% * Bach and Graf, Heft /0 * Braune and Myers o Richart, Series 1922 o Ferguson and Thompson except Series A and B ILLINOIS ENGINEERING EXPERIMENT STATION 2.8 2.6 2.4 2.2 2.0 Ptest 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 500 600 700 Tension failure / +15 % 2rf, 1/+ // 03 * Bach and Gra/ Heft /0, 3 beams each o Graf, Heft 67, 2 beams each Pw/P. = 1 ±+ ,10 10- (26) Fig. 12. Effect of Web Reinforcement on Strength of Simple-Span T-Beams Failing in Shear Beams of Group 7 were reinforced with loose, "floating" type of inclined bars, hooked on both ends. These beams failed at a somewhat lower load than that predicted, indicating that this type of web reinforcement was not fully effective. T-beams tested by Richart were provided with vertical stirrups.(2) As seen in Table 35, all these beams failed in tension. Beams tested by Braune and Myers had both vertical stirrups and bent-up where P, is determined from Eq. 35 and r from the following equation: As b's sin a (27a) For bent-up bars, there is some danger of a prema- ture failure because of cracking and crushing of concrete at the hooks. This failure can be prevented by using sufficiently large hooks and, especially, by using transverse reinforcement in the flanges of the beam. 0 /00 200 300 400 rfy, (psi) bars as web reinforcement."(9 These beams failed in tension also; see Table 34. However, the web re- inforcement was sufficiently effective to permit high ratios of P/P, at failure, the highest ratio being 3.19. Beams reported by Bach and Graf in Heft 12 were reinforced with bent-up bars.(22) A total of 87 beams were tested; the tension reinforcement con- sisted of from 4 to 7 bars of the same total area, and no transverse reinforcement was used in the flanges. All beams with unhooked longitudinal bars failed at a lower load than similar beams with hooked bars, evidently in bond. Beams with hooked longitudinal and bent-up bars are analyzed in Table 38. It is seen that despite the large ratios of web reinforcement only a few beams failed in ten- sion. Some other beams might have had yielding of the lower layer of the tension reinforcement at fail- ure. In most beams, failure was initiated by exces- sive cracking and crushing of the concrete at the hooks. The most effective arrangements of bent-up bars can be found from Table 38. It was concluded that the shear strength of simple-span T-beams with web reinforcement can be determined by the same expression as that for rectangular beams: V. RESTRAINED BEAMS UNDER SYMMETRICAL CONCENTRATED LOADS 16. Modes of Failure* Simple-span beams under concentrated loads fail at the location of an applied load, at the sec- tion of maximum shear and maximum moment. Shear stresses combined with flexural stresses are instrumental in producing a main diagonal crack; after this crack has formed, the beam fails in compression. In restrained beams, shear and moment con- ditions permit, in general, the formation of three main diagonal cracks as shown in Fig. 13. The A P, I 2/ 1 'p, - p? ID 4h- 17Z Possible Location of Main Diagonal Cracks -r P Shear Diagram Moment Diogram Fig. 13. Restrained Beam Under Symmetrical Concenlrated Loads beam may fail at any of these three cracks, depend- ing on the magnitude of shear and moment at the section under consideration and on the arrange- ment of both longitudinal and web reinforcement. Although the static moment is the same on both sides of section A, the magnitude of shear can be different in spans f and g. The crack at the section of greater shear forms first, and for small shear ratios it is even conceivable that the beam fails at that section before the other crack has formed. Al- * The modes of failure of restrained beams, and particularly the for- mation of two main diagonal cracks in the region in which the moment changes sign, were first described by E. Hognestad in an unpublished report, "Shear Failures in Concrete Beams," Department of Theoretical and Applied Mechanics, University of Illinois, 1951. though span g has constant shear, the moments can be different at sections A and B. Depending mainly on the relative magnitudes of moment, either one or two cracks form. The crack at the larger moment develops first and the beam fails, in general, at that crack. Various modes of failure are discussed below. Special emphasis is placed on the arrangement of reinforcement. It is assumed that span f has suffi- cient reinforcement so that the beam fails in span g. a. Continuous Top and Bottom Reinforcement. A free-body diagram for this arrangement of longi- tudinal reinforcement is shown in Fig. 14. It is as- sumed first that only one diagonal crack forms before failure. Figure 14 shows crack 2 and assumes that shear is resisted exclusively by the compres- A B Fig. 14. Continuous Top and Bottom Reinforcements. Restrained Beam With No Bond Failure sion area of the concrete. The top longitudinal reinforcement is in tension at crack 2 and in com- pression at section B. If there is no possibility for bond failure between these two sections, e.g., if span g is long relative to the effective depth of the beam and bars of good bond characteristics are used, a shear failure similar to that in simple-span beams is expected to take place. Thus, Eq. 18 can be em- ployed directly to determine the shear strength of such beams without web reinforcement and Eqs. 18 and 28 to determine that of such beams with web reinforcement. If two cracks are present and bond failure does not occur between them, the mode of failure is unchanged and the shear capacity of the beam can be determined by the same equations at the section of maximum moment. I I I I f" ILLINOIS ENGINEERING EXPERIMENT STATION If, however, bond is destroyed between the re- inforcing bars and the concrete in the middle por- tion of span g, Eqs. 18 and 28 no longer represent the shear strength of the beam. Bond failures are likely to take place when span g is relatively short. Then only a small distance separates the diagonal crack from either section A or B, and a change in stress from tension to compression in the reinforce- ment must take place over this length. If a bond failure results from the high bond stresses in this region, both the top and bottom reinforcing bars are in tension as shown in Fig. 15 for one crack and in Fig. 16 for two cracks at failure. For simplicity, it is assumed that the whole tensile force, TA or TB, is carried through the middle portion of the beam. This redistribution of internal forces is very un- favorable to the shear capacity, and the beam fails at a much lower load than it would if no "com- pressive" reinforcement was provided. An approximate expression for the shear strength of a beam with both top and bottom re- inforcement in tension can be derived as follows: C = kik3kdbfc' C = Ta + TB 7, t V T, Section B Fig. 15. Continuous Top and Bottom Reinforcement. Bond Destroyed in Restrained Beam With One Crack Fig. 16. Continuous Top and Bottom Reinforcement. Bond Destroyed in Restrained Beam With Two Cracks It is further assumed that the factor k, as derived for simple-span beams and given by Eq. 22 remains valid for restrained beams. The quantity k is de- termined again by the "straight line" theory. This can be done as follows: C = (1/2) bkdfc TB = pbdn k f M. = Cd (1 - k2k,) - TAtd (37) From Eqs. 36, 39, 40, and 41: Equation 37 determines the moment at shear failure. However, there are two unknowns, k, and TA, which must be evaluated before the shear mo- ment M, can be expressed quantitatively. If the tensile force TA is determined by assuming that the moment arm is the same for both sections A and B, the following relationship can be written: k = V/ (pn)2 + 2pon - pon where p= p(1+ MA (43) The shear moment as given by Eq. 37 can now be rewritten as: TA/TB = MA/MB (38) From Eqs. 36 and 38: where C TA =M - MB MX "+1 (39) = Ak (0.57 4.5f' bd2f- ' =5 10, ) A = 1 - (M + 1) (1 - kwk,) (44) (45) Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS k is given by Eq. 42 k2 is taken as 0.45 k, is given by Eq. 23. Equation 44 determines the shear strength of a restrained beam which fails at section B after bond has been destroyed from this section to crack 2 so that both the top and bottom reinforcing bars are in tension. This equation can be used for any sec- tion provided that the subscript B refers to the section under consideration and the subscript A to the adjacent section from which the tensile force TA is carried through to the section B. It was de- rived by assuming that the longitudinal reinforce- ment was continuous throughout the entire length of the beam and that the whole tensile force at one section was carried through to the other section. This is a conservative estimate, since it is likely that in some cases a part of the tensile force is re- sisted by partial bond which can exist even though the reinforcing bar might be slipping in the entire region from section A to B. If the actual ratio be- tween the top and bottom tensile forces can be de- termined for the section at failure, the actual ratio Ts/TA should be substituted for MB/MA in Eqs. 43 and 45. The first crack in span g will form at the section of maximum moment. If a beam fails in shear after only this crack has formed as shown in Fig. 15, redistribution of the internal forces due to bond failure has taken place at section B, whereas the bottom longitudinal reinforcement is still in com- pression at section A. Although both the top and bottom reinforcing bars are in tension at section B, diagonal cracking has not reduced the compression area, and the beam cannot fail in shear at that sec- tion. Consequently, section A is the critical section, and the shear strength of the beam is determined by Eq. 18 at the section of maximum moment. If two cracks are present at failure and full redistri- bution of the internal forces has taken place as shown in Fig. 16, Eq. 44 is applicable at section A as well as at section B. The shear strength of the beam is determined by Eq. 44 at the section of maximum moment. Under certain conditions it is conceivable that despite the formation of two cracks only partial bond failure and redistribution of the internal forces has taken place. This may be the case if, for example, the moment at section A is much greater than the moment at section B. Then the bond stresses are much higher in the top reinforcement than in the bottom reinforcement, and local bond failure may take place only along the top longitudinal bars. The shear capacity of the beam is given by Eq. 44 at section B and by Eq. 18 at section A, the section of maximum moment. The beam fails at the section of the small- est shear strength. However, since the conditions for partial redistribution of the internal forces cannot be determined in advance, it is more conser- vative to assume full bond failure and full redistri- bution whenever two cracks are present at failure. The validity of Eq. 44 is checked against test results in Section 17. b. Straight Bars Cut Off Beyond the Theoreti- cal Point of Contraflexure. A diagram for this ar- rangement of longitudinal reinforcement is shown in Fig. 17. When the length of embedment, both x and y, is sufficient to prevent a bond failure, it is expected that the shear strength of a restrained beam can be determined by Eqs. 18 and 28. How- ever, when the length of anchorage is small or re- inforcing bars of poor bond characteristics are used, the failure may be a sudden stripping out of the reinforcement and a complete destruction of the beam. Failures of this type have been reported by Richart and Larson(25' and by Moody.(12) Figure 18 shows a sketch of a beam in this category after failure. c. Beams With All Bars Bent Up. Figure 19 shows this arrangement of longitudinal reinforce- ment which appears to be an effective one since it prevents any possibility of bond failures and uses the bent-up bars as web reinforcement. When the bars are bent up at some distance from the support, P,+ . cs - n r -- dP o Fig. 17. Straight Bars Cut Off Beyond Point of Contraflexure. Restrained Beam Fig. 18. Stripping Type of Bond Failure. Restrained Beam ILLINOIS ENGINEERING EXPERIMENT STATION it seems advisable to use a few stirrups between the first bend and the load point. The shear strength of such beams is determined by Eqs. 18 and 28. While such an arrangement of reinforce- ment is very effective, care must be taken with the design and fabrication of bends. Richart and Lar- son 25) observed frequent crushing of the concrete at the bends after yielding of reinforcement. d. Beams With Both Bent-Up and Straight Longitudinal Bars. A diagram of such a beam is shown in Fig. 20. This type is similar to that dis- cussed in paragraph c. When bond failures are pre- vented, shear capacity is given by Eqs. 18 and 28. I' I I I II I I I II Ii 1 P - g Fig. 19. Restrained Beam With All Bars Bent-Up Pr i Fig. 20. Restrained Beam With Bent-Up and Straight Bars When, however, numerous bars are left straight, a premature bond failure similar to that discussed in paragraph b is possible. 17. Test Data on Restrained Beams The only tests on restrained beams reported in the literature are those by Richart and Larson(25) and by Moody.<12) These tests are analyzed and the validity of previously derived equations checked in the following paragraphs. a. Tests Reported by Richart and Larson. Richart and Larson reported tests on 59 beams, 17 in Series 1911 and 42 in Series 1917. Beams of Series 1911 failed either in tension or in bond, and the concrete strength was not recorded for all beams. Thus, very little information is available about the shear strength of these beams, and they are not included in the present analysis. Beams of Series 1917 were designed to investi- gate the effect of various arrangements of bent-up bars in span g. The type of beam is shown in Fig. 21, and Table 40 gives the arrangement of reinforcement for each individual beam. All beams Shear I I P/4 I M-M, 8P Momenl Fig. 21. Typical Restrained Beam of Richart and Larsen had eight 5/8-in. round plain bars over the sup- port A. The overhanging portions of the beam, span f, were heavily reinforced so as to produce failures in span g. Most beams of Series 1917 failed in tension. There are, however, a few beams which throw some light on the modes of shear failure as discussed under paragraphs b, c, and d in the previous section. Beams 380 represent beams with straight longi- tudinal bars cut off beyond the point of contra- flexure. From the description and photographs of failure it appears that these beams failed in bond by stripping off the concrete above the bars at failure as shown in Fig. 17. This premature bond failure cannot be predicted by any of the shear strength equations of this report; it is a matter of bond characteristics of the reinforcing bars. The rest of the beams were of the types dis- cussed under paragraphs c and d in Section 16, with some or all of the longitudinal bars bent down in span g. Beams 388, 389, and 400, which had four of the eight bars at support A bent down in one layer, appear to have failed in bond after yielding of the reinforcement. They correspond to a bond failure of the type discussed in Section 16, para- graph b. Beams which had four or more bars bent down in two or three layers failed in tension with- out any tendency for stripping of the concrete at the straight bars. However, crushing of the concrete inside the bends was frequently the cause of final failure. Furthermore, diagonal cracks were ob- served to intersect the reinforcement at the bends. These two phenomena were often responsible for a sudden shear-type final collapse of the beams. This occurred, however, well after the yielding of reinforcement. The ratios P/Pf in Table 40 were computed by Eq. 29, using f1= f, and kl2/kkc3 = 0.5 I Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Beam f,' psi 380.1 3060 2 3665 400.1 3158 2 3165 382.1 3315 2 2748 386.1 2870 2 3525 391.1 2892 2 3495 392.1 2818 2 2795 383.1 3082 2 2950 385.1 2985 2 3362 387.1 3398 2 2965 388.1 3260 2 2970 389.1 3210 2 3102 390.1 2905 2 2735 393.1 3155 2 2325 394.1 3145 2 3355 395.1 3120 2 3015 384.1 3080 2 3442 399.1 3352 2 2810 396.1 3410 2 3362 397.1 3235 2 2682 398.1 2682 2 2990 381.1 3070 2 3385 Table 40 Tests by Richart and Larsen, Series 1917. Restrained Beams With Bent-Up Bars Reference: (25) Dimensions: b= 8; d= 15 Spans: f-32 in.; g - 48 in.; h =-48 in.; L = 216 in. Loading: P i P/4; Psi P/4; MA SP in.-k; MB=4P in.-k Long. Reinforcement: Eight %-in. round plain bars at support, from 4 to 8 bars at midspan (See Fig. 21), f,=about 37,600 psi; p=about 1.95% Web Reinforcement: %-in. round plain vertical stirrups; f~.=45,000 psi Age at Test: 60 days Bent-up Bara No. of Bars a Total Layers deg 4 i 22" 5 2 22 6 3 32.5 6 3 32.5 6 3 32.5 6 3 32.5 5 2 32.5 4 2 32.5 4 1 32.5 4 1 32.5 8 4 45 6 3 45 6 3 45 6 3 45 6 3 45 5 2 45 6 2 45 6 2 45 6 2 45 6 2 45 Stirrups No. as in. 2 4 5 12 5 16 kips 102.8 104.0 151.0 149.7 175.5 183.7 188.2 188.0 187.8 172.0 146.4 176.4 183.2 181.5 176.3 190.0 182.3 168.4 173.8 143.2 174.0 169.0 181.2 186.0 164.0 170.0 172.3 185.4 180.6 167.0 176.7 178.6 176.1 184.9 182.6 165.0 192.5 179.6 168.0 173.1 165.0 124.1 Mode of Fail. DT DT DT DT T,DT T,Cr,DT T,Cr T,Cr T T,DT DT T,DT T T,Cr,DT T,Cr T T,Cr,DT T,DT T,B,Cr B,DT T,DT T T,Cr T,Cr T,Cr T,DT T,Cr T,Cr,DT T,Cr,DT T,Cr,DT T,Cr,DT T,Cr,DT T,Cr,DT T,Cr T,Cr T,Cr T,Cr T,Cr DT T,DT Cr,DT DT P. Eq. 18 kips 127.3 139.6 129.5 129.8 133.2 120.0 123.1 136.9 123.7 136.4 121.9 121.2 128.0 125.3 125.4 133.5 133.9 124.8 131.2 125.0 131.4 128.8 124.9 121.0 129.4 108.5 129.6 134.1 129.2 126.6 128.7 136.3 134.0 122.1 135.3 134.1 132.1 119.2 117.9 125.6 127.7 134.3 in the calculation of the flexural capacity P,. Since these ratios are greater than one, the reinforcement was presumably stressed in the work-hardening region at final failure. The main variables intended to be investigated were the angle of inclination and the number and spacing of bends. Even the largest spacing of bent- up bars gave a value of r which was sufficient to prevent shear failures. A few shear failures were obtained, however, when the first bar was located so far from support A that a diagonal crack could form without intersecting any inclined bars. Such failure was observed in Beam 381.2, where the first bend was 24 in. from the support. As seen in Table 40, this beam failed before yielding and at a ratio P/Ps equal to 0.92. Thus, the load at failure was governed by Eq. 18. The companion specimen failed in tension at a higher load, however. Beams 398,397, and 396 were similar to Beams 381 except that they were provided with vertical stirrups as additional web reinforcement. All of these beams failed in tension, although the final failure of Beam 398.1 was a sudden break, called diagonal tension by Richart and Larson. In conclusion, it can be said only that the be- havior and strength of the restrained beams with bent-up bars in these tests is not inconsistent with the behavior of simple-span beams as predicted by Eqs. 18 and 28. Bond failures are outside the scope of these equations; beams must be designed so that the possibility of destroying the bond is eliminated. Care must be taken in the design of bends to avoid crushing of the concrete inside the bends in the reinforcing bars. b. Tests Reported by Moody. Moody reports tests on 96 restrained beams in five series.(12) The dimensions of the beams and the arrangement of reinforcement and loads are shown in Fig. 22. All beams were provided with equal amounts of top and bottom longitudinal reinforcement, four bars placed in two layers. In all but three beams the four top bars and the two lower bottom bars were continuous throughout the total length of the beam; the other two bottom bars were cut off 4 in. Ratio Pt-et P. 0.81 0.75 1.17 1.15 1.32 1.53 1.53 1.37 1.52 1.26 1.20 1.45 1.43 1.45 1.41 1.42 1.36 1.35 1.32 1.15 1.32 1.31 1.45 1.53 1.27 1.56 1.33 1.38 1.40 1.32 1.37 1.31 1.31 1.51 1.35 1.23 1.46 1.50 1.42 1.38 1.29 0.92 Ratio Pts..t P1 0.71 0.70 1.04 1.03 1.19 1.27 1.31 1.27 1.31 1.17 1.02 1.23 1.26 1.26 1.23 1.31 1.26 1.12 1.20 1.04 1.19 1.16 1.24 1.26 1.06 1.23 1.14 1.26 1.24 1.15 1.20 1.28 1.22 1.29 1.24 1.12 1.20 1.27 1.19 1.19 1.14 0.88 ILLINOIS ENGINEERING EXPERIMENT STATION B 7 T i LLQ 0frf~vI f Series II// I7 LLI? 0 Fig. 22. Restrained Beams of Moody from the supports. In the remaining three beams the longitudinal reinforcement was cut off at the supports and the inner load points in accordance with the present ACI Code. The test variables in- cluded the percentage of longitudinal reinforcement, the concrete strength, the dimensions of the beams, and the magnitude of moments and shear as re- flected by different arrangements of loads. Sixty- one beams were tested without web reinforcement, 29 with vertical stirrups, and 6 with 45-deg stirrups. Beams with no web reinforcement are analyzed in Tables 41, 42, and 43. The beams of Series I, II, and IV had continuous longitudinal reinforcement and equal moments at sections A and B. From crack patterns and strain measurements recorded for Beam I-2c, it was observed that bond was de- stroyed in span g so that both the top and bottom reinforcing bars were stressed in tension. The beams failed after developing, in general, two main diag- onal cracks between sections A and B. Conse- quently, Eq. 44 should apply at both these sections. In order to apply Eq. 44, the weakest critical sec- tion must be determined first. Everything else re- maining the same, the shear capacity of a section is determined by the square of its effective depth. This distance was always 0.25 in. larger at section B than that at section A, indicating that A was the critical section. However, it is recalled that there were some differences in the arrangement of the longitudinal reinforcement at these sections. This might have a larger effect on Eq. 44 than the small difference in the values of d. From the con- dition of equal moments and entirely continuous top reinforcement it can be concluded that TA = TB at section B. This assumes that the total tensile force is carried through from section A to section B so that Eq. 44 can be used with TA/TB 1 at section B. At section A, however, only half of the bottom reinforcement is continuous. After bond is destroyed, it is likely that the stress in the continu- ous bars is increased relative to its magnitude be- fore bond failure. In order to transmit the total force TA to section B, the cut-off bars must be com- 4P/14 _T 6.86P _L " 5.33P \ \P/3 SERIES V T_ -15 -Kl- 0 Q- - >- ri c Q '- kQ Series ki J *»f -#i 4-) Bul.428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 41 Tests by Moody, Series I, 1953 Restrained Beams Without Web Reinforcement Reference: (12) Critical Section: Inner Loadpoint Calculation of Ultimate Moment: Eq. 44; TA/TB=1; p.=2p Loading: See Fig. 22; MB 5.33P Dimensions: bo= ; a= 12; t= 0.729; g Beam f/' p Ptest k kips 1950 Series 77 0.623 87 0.615 75 0.673 95 0.664 94 0.641 89 0.715 101 0.696 100 0.587 89 0.591 120 0.631 110 0.631 115 0.672 120 0.685 115 0.578 100 0.580 145 0.629 110 0.630 130 0.664 130 0.670 1952 Series 90 0.405 89 0.486 99 0.550 105 0.469 109 0.541 107 0.467 105 0.534 118 0.466 128 0.527 133 0.572 130 0.628 140 0.659 Table 42 Tests by Moody, Series II and IV, 1953 Restrained Beams Without Web Reinforcement Reference: (12) Critical Section: Inner load point Calculation of Ultimate Moment: Eq. 44; TA/TB= 1; p.= 2p Loading: See Fig. 22 =I 2.67 Beam /' p Patt k A M.t.t A M.t Ratio b/fkAA bdf/'kA Mt^t psi % kips M. Series II; b=7; d. 21; t 0.845; MAf=5.33P; g/d =1.52 17a 2650 2.15 188 0.572 0.518 0.414 0.578 0.451 0.98 b 3000 170 0.561 0.519 0.337 0.579 0.462 1.04 18a 2170 2.72 220 0.626 0.510 0.523 0.571 0.436 0.94 b 2700 " 180 0.609 0.513 0.364 0.572 0.484 1.08 19a 3030 3.46 241 0.642 0.508 0.422 0.575 0.356 0.89 b 3240 " 219 0.637 0.508 0.361 0.565 0.509 1.09 20a 2890 4.25 235 0.679 0.502 0.412 0.568 0.455 1.04 b 2960 249 0.678 0.502 0.427 0.582 0.359 0.95 IIa 3820 0.54 130 0.322 0.547 0.324 0.581 0.340 0.88 b 3720 0.84 145 0.395 0.540 0.316 0.577 0.429 1.11 c 4040 1.20 168 0.446 0.534 0.302 0.577 0.396 1.02 d 3440 1.63 210 0.506 0.527 0.396 0.571 0.349 0.96 Series IV; b=7; d= 12; t-0.729; MB- 6.86P; g/d= 4.0 0.583 0.377 1.06 IVg 3390 0.95 63 9.419 0.601 0.502 0.583 0.313 0.91 h 3750 1.47 70 0.483 0.595 0.442 0.577 0.441 1.25 i 3490 2.10 68 0.548 0.587 0.412 0.577 0. 41 0.95 3600 2.86 83 0.598 0.581 0.451 0.577 0.3 1.05 k 3630 3.76 88 0.644 0.576 0.444 0.572 0.350 1.05 1 3920 4.76 81 0.679 0.571 0.363 0.572 0.386 1.07 0.603 0.440 1.18 0.591 0.462 1.12 0.587 0.488 1.16 0.596 0.409 1.16 0.588 0.469 1.18 0.596 0.399 1.17 0.589 0.402 1.08 0.596 0.429 1.28 0.590 0.431 1.26 0.584 0.353 1.17 0.578 0.388 1.11 0.574 0. 330 1.09 pletely inactive and the continuous bars must resist twice their former stress. However, stress measure- ments in Beam I-2c showed that although the stress increased in the continuous bars, it never reached more than about 120 percent of its former value. This indicates, using the proper subscripts, that the ratio TA/TB is less than one at section A. The smaller is this ratio, the larger is the factor A in Eq. 44 and, consequently, the shear strength of the beam. Thus, section B must be considered as the critical section for Eq. 44, using MA/MB TA/TB = 1. Beams of Series I, II, and IV are analyzed in Tables 41 and 42, and the quantity M/bd2Jc'kA is plotted against fc' in Fig. 23. It is seen that, in general, test results give satisfactory agreement with Eq. 44. Thus, it appears that the assumptions made in deriving this equation are essentially cor- rect and that this equation can be used to deter- mine the shear strength of restrained beams with continuous reinforcement whenever the shear fail- ure takes place subsequent to destruction of bond. This type of failure is still a primary shear failure since the destruction of bond in the high bond stress region does not in itself constitute failure of the beam; it only causes a redistribution of the internal Ratio M. 0.92 0.78 1.11 0.77 0.97 0.85 0.94 0.98 0.81 0.78 0.78 0.95 1.20 1.10 1.00 1.10 1.09 0.92 Table 43 Tests by Moody, Series VI and V, 1953 Restrained Beams Without Web Reinforcement Reference: (12) Critical Section: Support Calculation of Ultimate Moment: Eq. 18 Loading: See Fig. 22 Beam Pc' p Pt.t k k+np' Mt,. Ratio bdf.'(kc+np') M.. psi % kips M. Series VI; b=7; d=11.75 t=0.817; p'-0.5p; MA=6.4P; g/d-2.73 Via 4090 0.95 77 0.300 0.335 0.372 0.96 b 4160 1.47 129 0.351 0.406 0.505 1.31 c 3580 2.10 110 0.401 0.483 0.421 1.03 d 3900 2.86 118 0.435 0.543 0.369 0.93 e 4120 3.76 128 0.467 0.606 0.339 0.88 f 5570 2.10 140 0.383 0.455 0.365 1.14 g 5530 2.86 130 0.422 0.519 0.300 0.94 h 5300 3.76 155 0.457 0.586 0.330 0.99 i 6020 4.76 146 0.483 0.641 0.251 0.84 Series V; b = 7; d= 11.75; p'=0; MA =5.33P; g/d=2.73 Vb 3770 1.47 64.0 0.379 ..... 0.247 0.62 d 3600 2.86 76.5 0.484 ..... 0.242 0.59 f 3380 4.76 74.5 0.574 ..... 0.212 0.51 forces, so that the new combination of the tensile forces at a certain section requires a larger com- pressive force than before. For Moody's beams the new compressive force is about twice as large as that before the destruction of bond. The greatly in- creased compressive force leads to a lower shear strength since the capacity of the compressive zone of the beam is but little greater than that for simple-span beams. Thus the factor A of Eq. 44 can be considered as a reduction factor for re- strained beams which fail after local bond failure. In deriving Eq. 44 it was assumed that the entire tensile force at once section is transmitted to the adjacent section. This assumption is, in general, a conservative estimate since some of the tensile force is probably resisted by partial bond. Tables 41 and 42 show that the ratio Mtest/M, increases as the g/d-ratio increases or as the size of the rein- ILLINOIS ENGINEERING EXPERIMENT STATION Fig. 23. Beams of Moody, Series I, II, and IV. Restrained Beams Without Web Reinforcement forcing bars, as indicated by the percentage of reinforcement, decreases. In both cases the relative importance of partial bond is more pronounced and, consequently, not all of the tensile force is trans- mitted from one section to the adjacent section. The true ratio TA/TB is thus smaller than that ob- tained from the bending moments and the shear strength of the beams is thereby increased. How- ever, as seen in Tables 41 and 42, the increase in the shear capacity was rather small even for the largest value of g/d and the smallest size of rein- forcing bar used in the tests. Furthermore, beams of Series II which had the smallest value of g/d fall even somewhat low in Fig. 23. The limits of applicability of Eq. 44 cannot be determined from Moody's tests. For the beams shown in Fig. 23, the g/d-ratio varied from 1.52 to 4.0. All of these beams failed after redistribution of the internal forces. Thus, it appears that local bond failures are possible with g/d-ratios larger than four. Beams of Series V are analyzed in Table 43. These beams had their longitudinal reinforcement cut off at the supports and the inner load points. Failure took place by a sudden stripping out of the longitudinal reinforcement as discussed in Section 16, paragraph b. The beams were analyzed by Eq. 18 with the support as the critical section, and the load at final bond failure was found to be about one-half the theoretical shear capacity. Beams of Series VI had unequal bending mo- ments, MA being twice Ms. The beams failed, in general, after developing only one main diagonal crack at section A. Thus it is likely that local bond failure had taken place only in the top reinforce- ment, so that both the top and bottom bars were in tension at section B, whereas the bottom bars were still in compression at section A. This possi- bility was illustrated in Fig. 15. Consequently, the shear strength of these beams should be governed by Eq. 18 at section A, and the beams are analyzed in Table 43 accordingly. The quantity M/bd2fc (k + np') is plotted against f,' in Fig. 24, and it is seen that there is good agreement between the measured and calculated moments. Beams with web reinforcement are analyzed in Tables 44 and 45. For beams of Series I and IV the quantity M/Ms is plotted against rfw, in Fig. Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS 'c (ps// Fig. 24. Beams of Moody, Series VI. Restrained Beams Without Web Reinforcement 25. Beams in both these series had equal moments simple-span beams. Beams with vertical stirrups at sections A and B, and the shear moment M, was were tested in two different groups, 6 beams in 1950 calculated by Eq. 44. Figure 25 shows that the and 10 beams in 1952. Beams of 1950 had the beams of Series I which had 45-deg stirrups give closed ends of the stirrups placed toward the lower very good agreement with Eq. 28, derived for face of the beam, whereas the beams of 1952 had Beam // p psi % lOa 3070 4.76 b 2810 lla 3560 b 3180 12a 4000 b 3220 Is 3470 4.76 t 3700 u 3740 2.86 v 3580 w 4210 x 3830 y 4790 4.76 z 4850 a 5070 P 5130 13a 3460 4.76 b 2860 14a 3510 b 3600 15b 3710 16a 3610 4.76 b 3240 Table 44 Tests by Moody, Series I, 1953. Restrained Beams With Web Reinforcement Reference: (12) Critical Section: Inner load point Calculation of Ultimate Moment: Eqs. 44, 28; TA/TB = 1; po=2p Loading: See Fig. 22; MBs= 5.33P Web Reinforcement: Stirrups of inter, grade deformed bars Dimensions: b= 7; d= 12; t=0.729; g/d=2.67 WEB REINFORCEMENT Ptet k A Size s r f/, rf., No. in. % ksi psi kips Vertical Stirrups; 1950 Series 3 6 0.52 47.3 246 163 0.694 0.568 138 0.700 0.568 4 0.95 44.0 418 190 0.685 0.570 174 0.692 0.569 5 1.47 41.2 606 190 0.678 0.571 159 0.691 0.569 Vertical Stirrups; 1952 Series 5 5 1.72 47.6 819 220 0.686 0.570 4 2.14 47.6 1019 240 0.682 0.570 3 6 0.52 53.8 280 160 0.596 0.582 3 4.5 0.70 53.8 377 170 0.599 0.581 4 6 0.95 45.8 435 180 0.589 0.582 5 6 1.47 47.6 700 217 0.594 0.582 3 6 0.52 53.8 280 220 0.669 0.578 3 4.5 0.70 53.8 377 222 0.668 0.578 4 6 0.95 45.8 435 260 0.666 0.573 5 6 1.47 47.6 700 279 0.665 0.573 45-deg Inclined Stirrups 3 6 0.74 47.3 350 185 0.687 0.570 170 0.699 0.568 4 1.35 44.0 594 250 0.686 0.570 240 0.684 0.570 5 2.09 41.2 861 304 0.683 0.570 T-Beams; Vertical Stirrups; b = 23, b'=7, d - 11.75, e=4; t=0.839 Critical Section: Support; Eqs. 44, 28; TA/TB=0.5; p.=0.0714 5 6 1.47 41.2 606 271 0.636 0.675 " 282 0.643 0.675 M. Eq. 44 k-in. 527 500 574 539 609 542 566 586 525 514 554 531 661 664 667 668 566 505 570 577 587 Ratio M. 1.65 1.47 1.76 1.72 1.66 1.57 2.07 2.18 1.62 1.76 1.73 2.18 1.77 1.78 2.08 2.23 1.74 1.79 2.33 2.22 2.76 611 2.36 1.07 576 2.61 1.18 ILLINOIS ENGINEERING EXPERIMENT STATION Table 45 Tests by Moody, Series IV and II, 1953. Restrained Beams With Web Reinforcement Beam ft psi IVm 2860 n 3710 o 3420 21a 3560 b 3640 22a 3000 b 2710 23a 3230 b 3160 IHe 3420 f 3330 the closed ends always in the compression zone. Beams of 1950 give good agreement with Eq. 28 except for the two beams with the largest amount of web reinforcement. These beams were observed to split longitudinally along the reinforcement and the stirrups did not reach yielding at failure. It is possible that longitudinal splitting destroyed the anchorage of stirrups so that they were unable to develop their full effectiveness. In the beams of 1952, longitudinal splitting took place on a more 3.5 3.0 2.5 pn 0.5 oC 0 200 400 bUU rfyw(psi) 800 /OUU /200 Fig. 25. Beams of Moody, Series I and IV. Restrained Beams With Web Reinforcement A M. Ratio Eq. 44 Mtet k-in. M. .568 504 1.99 1.28 .570 586 2.32 1.24 .570 562 2.66 1.11 516 1373 1.20 0.80 516 1387 1.09 0.73 514 1247 1.28 0.70 513 1169 1.32 0.72 515 1301 1.23 0.56 514 1284 1.45 0.66 516 1344 1.55 0.62 515 1323 1.37 0.48 restricted scale, and then only in the region where the stirrups were open-ended. This suggests that longitudinal splitting will not occur if the rein- forcing bars are tied together in the transverse direction. Beams of 1952 give good agreement with Eq. 28 except for two beams with very high values of rfy,. It is noticed, however, that at the value of rfy, at which the beams of 1950 fell below the predicted ultimate moment, the beams of 1952 still agree with Eq. 28. Thus, the anchorage of stirrups was more effective in Series 1952 than in Series 1950; only with very high values of rfy, did the vertical stirrups not develop their full strength at failure. Among the beams of Series I two beams were provided with a 23- by 4-in. flange. Since the flange area increases the compression area of the concrete at section B, the beams are analyzed for section A as the critical section in Table 44. In line with the previous discussion about the effect of cutting off one-half of the bottom bars at the sup- ports, the ratio TA/TB must be less than one at that section. The beams were analyzed with TA/TB equal to 0.5, or only one-half of the tensile force at the inner load point transmitted to the section at the support. The use of this ratio gave satis- factory agreement with Eq. 28, the loads at failure being 7 and 18 percent more than the predicted loads. If the ratio TA/TB had been taken larger than 0.5, the calculated ultimate moment would have been still smaller. Three beams of Series IV were provided with vertical stirrups. These beams were included in Table 45 and Fig. 25. The test moments were found to be from 11 to 28 percent larger than the calcu- lated moments. Beams of Series IV had the largest g/d-ratio used in these tests, g/d = 4. Since the beams without web reinforcement in this series had Reference: (12) Critical Section: Inner Load Point Calculation of Ultimate Moment: Eqs. 44, 28; TA/TB= 1; p.=2p Loading: See Fig. 22 Web Reinforcement: Vertical stirrups of inter, grade deformed bars WEB REINFORCEMENT Ptet k Size a r f/ rf,,, No. in. % ksi pali kips Series IV; b 6= 7; d = 12; t = 0.729; MB = 6.86P; g/d=4.0 3 6 0.52 53.8 280 146 0.698 0 4 " 0.95 45.8 435 198 0.682 0. 5 " 1.47 47.6 700 218 0.687 0 Series II; b =7; d = 21; t =0.845; MsB=5.8P; g/d= 1.52 3 6 0.52 47.3 246 310 0. 590 0. 283 0.589 0. 4 " 0.95 44.0 418 300 0.602 0. 290 0.608 0. 5 1.47 41.2 606 300 0.596 0. 350 0.598 0. 5 1.72 43.5 748 390 0. 593 0. 4 2.14 " 931 340 0.595 0. * A5% -/5% 0 / Series I * Vertical Stirrups, 1950 o Vertical Stirrups, 1952 A 45-degree Stirrups * T-Beams, Vertical Stirrups Series IV a Vertical Stirrups /a V y * ^^ / 0 0 > / ^ Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS only slightly larger shear capacities than given by Eq. 44, the addition of web reinforcement appears to have restricted the development of diagonal cracks so that the relative importance of partial bond was increased. The shear capacity of the beams was thereby increased also. Beams of Series II had the smallest g/d-ratio of all beams, g/d = 1.52. It is seen in Table 45 that these beams failed at a considerably lower load than that given by Eq. 44. Since the beams failed at about 30 percent greater loads than similar beams without web reinforcement, an increase in the amount of web reinforcement apparently did not produce a corresponding increase in shear strength. These beams appear to have failed in shear-proper, and they are analyzed accordingly in Section 19. VI. BEAMS UNDER OTHER TYPES OF LOADING 1 8. Limitations of Shear-Compression Failures Equation 18, the basic equation for shear strength, was derived for simple-span rectangular beams without web reinforcement and under one or two symmetrical concentrated loads. This equation considers shear failures as compression failures. Shearing stresses together with flexural tension stresses are combined in the principal tension stresses and produce a diagonal crack which ex- tends higher than the flexural tension cracks. After this crack has formed, final failure takes place by crushing of the concrete in the reduced compression area. In deriving Eq. 18, the unknown function F(fc') was determined empirically. All available test data, a total of 106 beams, were used in the analysis. The ratio a/d which corresponds to the compressive force-shear ratio in simple-span beams, C/V = a/jd, varied from 1.17 to 4.80 for the beams con- sidered. This variation did not appear to have any effect on the agreement between test results and the values predicted by Eq. 18. Within these limits, consequently, the shear strength of a beam was de- termined entirely by the physical properties of the beam and was not a function of either the magni- tude of the shear or the moment-shear ratio at the section of failure. The beams failed at a limiting moment Ms. This limiting moment was reached by different combinations of V and a and the contri- bution of the shearing stresses was always large enough to produce sufficient diagonal cracking which is a prerequisite for this type of failure. The addition of web reinforcement increased the limit- ing moment to Mw,. Otherwise, the mechanism of failure remained the same as before. As the ratio a/d increases, however, the ratio of moment to shear at the section of maximum mo- ment increases, since M/V = a. Consequently, the contribution of the shearing stress to the principal tension stress decreases relative to the contribution of the flexural stress at a given magnitude of mo- ment. Before cracking, the magnitude of the prin- cipal tension stress is determined by the magnitude of the flexural tension, at the extreme tension fibers of the beam, and by the magnitude of the shear at the neutral axis. Between these two locations, the magnitude of the principal tension stress is de- termined by the relative magnitudes of moment and shear. As the magnitude of shear decreases at a given value of moment, the trajectories of the principal tension stresses become more and more horizontal in the region of maximum moment. They must still intersect the neutral axis at 45 deg, but since the shear force is relatively small, the magni- tude of the principal tension stresses at that loca- tion is also relatively small. Since diagonal cracking is the result of diagonal tension stress, the cracks must start at the location of maximum stress and progress first in an almost vertical direction. Cracking, of course, alters the distribution of prin- cipal tension stress. Furthermore, their subsequent distribution is beyond a theoretical analysis at the present time. However, it is still likely that because of small shear stresses, the cracks might remain vertical or become but slightly inclined. Thus, for large values of a/d, full diagonal cracking might never develop and the beam might fail finally in flexure rather than in shear-compression. This behavior of beams with large ratios of a/d can be observed from tests made by Johnson.30) He tested a number of simple-span beams under two concentrated loads to investigate the effect of compression reinforcement. All beams were heavily reinforced in tension, p = 0.046, and the ratio a/d was equal to about 11. All beams but one failed in flexural compression despite the fact that the mo- ment at failure was considerably larger than the shear-compression moment given by Eq. 18. Fur- thermore, no diagonal cracking developed before failure. It appears that at a/d = M/V equal to 11 the contribution of shear to the principal tension stresses was too small to produce diagonal crack- ing. Consequently, the beams could not fail in shear at the limiting shear-compression moment. The beams continued to take load until their flexural capacity was reached and they failed in flexure. Figure 26 summarizes the above discussion and presents a hypothesis for the limits of shear-com- Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS IUJLAIP C/d Fig. 26. Shear Force V Versus a/d. Possible Modes of Shear Failure for Simple-Span Beams pression failures. Figure 26a shows one-half of a simple-span beam, loaded with two symetrically placed concentrated loads; the loads Po, P1, P2, and P3 represent several possible locations of the load on the beam. Figure 26b shows the corresponding moment diagrams. The ultimate flexural moment, M, and the shear-compression moment Msw, are also plotted on that figure. Both these moments are determined only by the physical properties of the beam cross-section. The corresponding values of shear are plotted in Fig. 26c. These are obtained from Eq. 28 for shear-compression failures and from Eq. 29 for flexural failures. For (a/d)1 < a/d < (a/d)c, it is assumed that the beam fails in shear-compression. Load P1 repre- sents one possible load position in this range. As the beam is loaded, the moment at the section through load P1 increases. Finally, the limiting moment Mw is reached and the beam fails in shear-compression. The corresponding shear force at failure is shown in Fig. 26c; it lies on a curve - - determined from Eq. 28. Any other location of load within the above limits would lead to the same re- sult, the limiting moment remaining constant while the magnitude of shear decreases with increasing ratio a/d as shown in Fig. 26c. With respect to the limit of shear-compression failures, the following hypothesis is advanced: a certain value of a/d, (a/d)cr in Fig. 26, is assumed to be the largest ratio of the internal compression force to shear force at which the contribution of the shearing stress is large enough to develop full diagonal cracking. Thus, at the load location Po, the limiting moment Msw and a critical magnitude of shear, Vo, are reached simultaneously and the beam fails in shear-compression. As the ratio a/d increased beyond (a/d) c, there is a transition region between shear-compression and flexural failures. Considering load location P2 it is seen that as the beam is loaded, the limiting moment Mw is reached at a load at which the cor- responding shear force V2 is below its critical mag- nitude Vo. As a consequence, full diagonal cracking has not taken place and the beam cannot fail in shear at that load. A further increase in load will increase both the bending moment and shear at the section P2. At a certain magnitude of load the shear force reaches its critical value V. while the moment has increased to a value of M2. Considering now moment and shear conditions at a section closer to the support, the section (a/d)c,, it is observed that under the attained load both the shear-compression moment and the critical magnitude of shear are reached simultaneously at that section. This per- mits the formation of full diagonal cracking and makes a shear failure possible. However, the cri- terion of failure is the attainment of the critical magnitude of shear. A sudden formation of di- agonal cracks should occur as soon as this value of shear is reached. In this sense, the resulting sud- den failure is not a shear-compression failure and should be classified as a diagonal tension failure. Furthermore, since the bending moment exceeds the shear-compression moment anywhere between sec- tions P2 and (a/d) , while the shear force remains at the constant value Vo, it is likely that the diag- onal tension failure can take place at any location between these two sections. This type of diagonal tension failure occurs whenever the ratio a/d is between the values (a/d)c, and (a/d),. The critical magnitude of the shear force Vo is determined from the shear-com- ILLINOIS ENGINEERING EXPERIMENT STATION pression moment M,. with the aid of the critical ratio (a/d)c,, if the latter value can be uniquely established. The higher limit of the transition re- gion, (a/d),, is not a constant ratio but a value which depends on the relative magnitudes of M,, and Mf. This is seen in Fig. 26c where the point of intersection between the critical shear force Vo and the shear curve corresponding to flexural failures determines the value of (a/d) 2. Furthermore, a constant shear force gives a linear relation be- tween moment and a/d in the range between (a/d) cr and (a/d)2, as seen in Fig. 26b. The value of (a/d)2 is then determined by the point of inter- section between the flexural moment M, and the straight line through the origin in Fig. 26b which passes through M,, at (a/d)c,. For a/d > (a/d)2, the magnitude of the shear force is never large enough to permit diagonal cracking and the beam must fail in flexure. Con- sidering load location P3 in Fig. 26, it is seen that as the load increases, the bending moment reaches the flexural capacity of the beam before the shear force reaches its critical magnitude Vo. As a con- sequence, diagonal cracking cannot develop, shear failure is not possible, and the beam fails finally in flexure. In conclusion, ample test data is available to justify the concept of shear failures as shear-com- pression failures for values of a/d below a certain limit. Likewise, test data are available to show that beams with high values of a/d fail in flexure although their shear-compression moment has been exceeded at the section of the maximum moment at failure. Very little information is available, how- ever, for beams in the transition region between the two types of failures. The above discussion is presented as a possible explanation for the behavior of beams in this region. It is realized that this hypothesis is not supported by experimental evi- dence. If, however, the validity of this hypothesis can be established by experiments and if the value of the critical ratio (a/d)c, = (M/Vd)cr can be uniquely determined, the behavior of a beam with any value of a/d is fully described by a diagram similar to Fig. 26. This diagram is determined by Ms. and Mf, both of which depend on the physical properties of the beam, and a critical ratio (a/d),r. A few tests on T-beams which fall in the transition region between shear-compression and flexural fail- ures are discussed further in Section 20. For very low values of a/d it is not expected that a beam fails through beam-action. The mode of failure seems to change from shear-compression to what can be called shear-proper; that is, actual shearing off of the concrete. This type of failure is discussed in the following section. 19. Shear-Proper In the range of shear-compression failures, a beam fails, after the formation of diagonal cracks, in compression. However, as the ratio a/d decreases, the mode of failure seems to change. With a con- centrated load close to a support, the cracks open up near the load block in the tension zone of the concrete and progress toward the other load block in the compression zone. Since the load blocks are but a short distance apart, the cracks are almost vertical. The ultimate failure seems to take place by the actual shearing off of the remaining concrete in compression. It is rather difficult to determine the true cri- terion of failure. Cracking of concrete is produced by the principal tension stresses. As load on a beam is increased, more cracks form and the existing cracks both widen and extend higher. Consequently, less and less concrete remains effective to resist the complicated state of stress. Since the shear span is short, the magnitude of the principal tension stresses is also affected by the presence of compres- sive stresses. in the vicinity of the end reaction and the concentrated load. These compressive stresses will reduce the magnitude of the principal tension stresses and will make them less inclined with the axis of the beam. The closer is a load to a support, the larger is the relative importance of the local compressive stresses. Consequently, the tensile stresses are smaller and it is expected that the cracks will form and that the beam will fail at a higher load than if the load were farther from the support. Some quantitative information on this type of shear failure can be obtained from tests reported by Graf in Heft 80.(26) A total of 26 beams were tested, 21 small rectangular beams with the outside dimensions and loading arrangement shown in Fig. 27a and 5 large T-beams as shown in Fig. 27b. The variables included the size of the bearing block for the concentrated load, the amount of longi- tudinal reinforcement, the amount and angle of inclination of bent-up bars, and to a minor extent the compressive strength of concrete. In all tests the distance x between the bearing blocks, Fig. 27a, was either zero or a very small fraction of the depth of the beams. Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS An analysis of the test results shows that, everything else remaining equal, the size of the bearing block had no effect on the ultimate load. This is true despite the fact that an increase in y produced a larger moment under the load bearing block, the load at failure being the same. It was concluded, therefore, that the ultimate load depends on the magnitude of the shear force V and the clear the use of vertical stirrups, however, did not in- crease the ultimate load. Thus, there seems to be a maximum value of a which limits the usefulness of the bent-up reinforcement. The ultimate load increased as the concrete strength increased. However, the range of f,' varied generally only from 1500 to 2000 psi with but one beam of about 3000 psi concrete strength. The above observation suggested that the ulti- mate load could be expressed in terms of a nominal shearing stress in the following form: V Vo =bD = C1 + C2f.' + C3Pt where = A (1 + sin a) bD C1, C2, C3 = numerical coefficients "MIII 1 Fig. 27. Beams of Graf, Heft 80. Shear-Proper Type of Failures shear span x rather than on the a/d-ratio. It also appears that the size of the bearing area was suf- ficiently large in all cases to produce shear-type failures; it is conceivable that local crushing of the concrete can take place under the bearing block when the bearing area is too small. Some of the small beams were without any re- inforcement. The addition of longitudinal steel in- creased the ultimate load. Furthermore, it appears that the use of longitudinal steel was equally ef- fective at any depth in the beam: in the bottom half, at mid-depth, or in the top half of the beam. The use of bent-up bars was more effective than the addition of longitudinal steel, and the effective- ness increased as the angle of inclination increased. Judging from the load at failure, it seems that the effectiveness of the inclined reinforcement increases in proportion to the quantity (1 + sin a), at least to the largest angle of inclination used in these tests, a = 62.7 deg. Since the cracks were almost vertical, and the quantity A,(1 + sin a) refers to the total steel area crossing section A-A, Fig. 27a. When both horizontal and inclined reinforcement is used, the reinforcement ratio Pt must be evaluated for each part separately and the total value used in the calculations. This type of equation was checked against test results. Reasonable agreement was found with the following equation: v, = 200 + 0.188f,' + 21,300 pt where both v and f,' are expressed in pounds per square inch. Since plain beams were included in the analysis, the nominal unit shearing stress was de- termined for the gross section of the beams. For the T-beams of Heft 80 the value of v was calculated by neglecting the flange area outside the web, since the load was applied at a section in the end of the beam where the flange was being tapered off to the width of the web. The physical properties of the beams and the ratios v/v, are shown in Table 46, and in Fig. 28 the quantity v/vc is plotted against x/D, the ratio of the clear distance between the load blocks to the total depth of the beams. It is seen that Eq. 48 gives satisfactory agreement with the test results; only two plain concrete beams with the largest bearing area fall more than 15 percent below the predicted load and three beams are slightly more than 15 percent above. The five large T-beams agree quite well with Eq. 48. &792 ILLINOIS ENGINEERING EXPERIMENT STATION Table 46 Tests by Graf, Heft 80, 1935. Shear-Proper Type of Failures (A) SMALL RECTANGULAR BEAMS Reference: (26) Dimensions: b=7.9 in.; D= 11.8 in.; See Fig. 27 Loading: See Fig. 27 Reinforcement: 0.39-in. plain round bars; f,=49,000 psi; some bars bent as indicated below Concrete Strength: Tests on 7.9-in. cubes; f,'=0.75 fe,' assumed Age at Test: 14 days Horiz. 6 14 7 10 6 8 Arrangement of Reinforcement No. of Bars at A-A A,(1+sin a) 7 16 4 45 6 16 4 60 Physical Properties of Beams and Test Results pe Eq. 47 % 0.79 1.83 2.09 2.20 60.79 1.83 2.09 2.20 60.79 1.83 2.09 2.20 1.79 2.02 1.79 2.02 Group a b d e f g Beam la b c d e 2a b c d e 3a b c d e 4a f g 5a f g Beam 1246 1247 1270 1271 1272 Eq. 48 psi 499 667 889 944 968 482 650 872 927 951 501 669 891 946 970 591 972 1021 563 944 993 (B) LARGE T-BEAMS Reference: (26) Dimensions: b=49.2; b'=9.8; D=22.8; d=21.4; e=3.1; L=137.8; L'=161.4; x=2.0 Loading: One load 11.8 in. from end support; see Fig. 27 Tension Reinforcement: 0.63- and 0.71-in. plain round bars, hooked; f,= 62,000 and 53,400 psi, resp. Web Reinforcement: Bent-up bars and 0.24-in. round vert. stirrups at 7.9 in. Reinforcement in Flange: Four 0.28-in. round horiz. bars; 0.28-in. round transverse bars at 4.9 in. Concrete Strength: Tests on 7.9-in. cubes, f/'=0.75 fI' assumed Age at Test: From 12 to 23 days Reinf. Bars a A,(1-+sin a) pt Ptgt Vtmes Vtest at A-A Eq.47 Horiz. Bent deg in.' % kips kips psi 3-0.71 ...... .... 1.80 0.80 172 157 699 2-0.63 3-0.71 1-0.71 45 3.01 1.34 185 169 753 2-0.63 1-0.63 2-0.71 5-0.71 45 4.99 2.22 247 226 1004 1-0.63 1-0.63 3-0.71 3-0.71 62.7 4.59 2.04 231 212 942 2-0.63 3-0.71 3-0.71 62.7 4.59 2.04 296 271 1206 2-0.63 If psi 1590 1500 1600 2080 1930 /f0' psi 1560 1550 1780 1640 3040 Among previously analyzed test data there were a few beams which failed at a lower load than predicted by the shear-compression equations. Those were the beams tested by Clark(5) which had the shortest shear-span, and two simple-span beams and eight restrained beams of Series II by Moody;(12) all these beams had a very small a/d- ratio and were reinforced with vertical stirrups. The beams for which strain readings were reported failed in general before yielding of the web rein- forcement. These beams are reanalyzed in terms of shear-proper in Table 47. The nominal shearing units stress vc as given by Eq. 48 was computed for each beam, and the ratio v/Vc is plotted against the parameter x/D in Fig. 28. Some of Clark's beams failed in tension and are not included in this comparison. Figure 28 shows that the ratio v/Vc decreases as x/D in- creases. Because Eq. 48 was entirely empirical by nature and the number of tests was rather limited, e.g., there were no beams in the range of x/D from 0.1 to 0.8, no attempt was made to write an expres- sion for the relationship between v/Vc and x/D One possibility is shown by the dashed line in Fig. 28. Beams of Heft 80 with the load very close to the supports showed no evidence that vertical stir- rups increased their shear strength. This is under- Tot. No. of Reinf. Bars 14 14 14 14 12 12 kips 88.2 110.2 154.3 154.3 154.3 88.2 125.7 167.6 165.3 185.2 77.2 132.3 158.7 165.3 176.4 88.2 209.4 231.5 77.2 183.0 207.2 Eq. 48 psi 664 777 1008 943 1206 pe Eq. 47 0679 1.83 2.09 2.20 1.79 2.02 Ratio Va 0.95 0.89 0.93 0.88 0.86 0.98 1.04 1.03 0.96 1.05 0.83 1.06 0.96 0.94 0.98 0.80 1.16 1.22 0.74 1.04 1.12 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS 1.4 1.2 0.8o 0.6 0.6 0.4 ' 0 0 0.2 0.4 0.6 08 1.0 12 x/D Fig. 28. Nominal Shearing Stress Ratio Versus x/D for Failures in Shear-Proper standable since the location of the load forced the formation of almost vertical cracks. However, as x/D increases in the region of shear-proper, cracks follow the edges of the bearing blocks and vertical stirrups crossing the cracks produce a slight in- crease in the ultimate load. This is seen in Table 47 where for any value of x/D the ratio v/vc increases somewhat as the ratio of web reinforcement in- creases. The beams fail, however, before the verti- cal web reinforcement yields. When the load is removed sufficiently far from a support, a regular shear-compression failure takes place. The transition between shear-compression and shear-proper, point (a/d)1 in Fig. 26, seems to de- * Graf, Heft 80, rectangular beams A Graf. Heft 80, T- beams o Clark, a =18" * Moody, simp/e-span beams, Series I//II o Moody, restrained beams, Series II Invest. Beam fI' b D p9 Eq. 47 psi in. in. % Clark Dl-1 3800 8 18 1.39 3 3560 D2-1 3480 1.39 2 3755 D3-1 4090 2.08 D4-1 3350 1.39 Moody 30 3680 7 24 5.57 Series 31 3250 III Moody Series II Table 47 Other Shear-Proper Type of Failures a/d x x/D PLt f, at Fail. % of in. kips fA. 1.17 14.5 0.81 135.4 115.4 130.4 140.4 177.6 140.4 1.52 24 1.00 215 83 228 67 Ratio P., 0.76 0.67 0.69 0.72 0.64 0.55 0.80 0.73 0.82 0.74 0.71 0.73 0.57 0.66 0.62 0.48 Ratio Vte.t vNe V. Ratio Ptit Eq. 48 v"*t kips psi psi 0.91 67.7 470 1210 0.39 0.78 57.7 401 1165 0.34 0.89 65.2 453 1150 0.39 0.94 70.2 487 1202 0.41 0.83 88.8 617 1412 0.44 0.96 70.2 487 1126 0.43 0.67 107.5 640 2081 0.31 0.72 114.0 679 2000 0.34 .... 103.3 615 1883 0.33 .... 94.3 561 1898 0.30 .... 100.0 595 1778 0.33 .... 96.7 576 1723 0.33 .... 100.0 595 1825 0.33 .... 116.7 695 1808 0.38 .... 130.0 774 1857 0.42 .... 113.3 674 1840 0.37 .4 8' * 5-~~ - - - 3 a pend both on the ratio x/D and the amount of web reinforcement used. All beams of Table 47 had cor- responding test specimens without web reinforce- ment and these beams failed in shear-compression in agreement with Eq. 18. Furthermore, Clark's beams with 24-in. shear span having x/D equal to 1.14 and reinforced with vertical stirrups failed in shear-compression. Thus the transition region between the two types of failures seems to lie ap- proximately between x/D equal to 0.8 and 1.1, increasing as the amount of vertical web reinforce- ment increases. The use of inclined web reinforce- ment, however, increases the ultimate load in shear-proper according to Eq. 48. Consequently, whenever the clear shear span x approaches the total depth of the beam, inclined web reinforce- ment should be used instead of vertical stirrups. For restrained beams the distance x was con- sidered in the same way as for simple-span beams - the clear distance between two load blocks. For Series II of Moody's restrained beams this pro- cedure gave good results. It is seen in Fig. 28 that both simple-span and restrained beams with the same x/D-ratio failed at about the same nominal shearing stress. However, if the ratio x/D is con- sidered as a measure of principal tension stresses and the extent of cracking, the use of x as defined above is not strictly correct, since the magnitude of flexural bending stresses for simple-span beams is generally different from that for restrained beams. 20. Transition Region and Flexural Failures There is very little experimental data available for beams with large ratios of a/d. The only tests reported in the literature are those by Johnson,(30) previously discussed in Section 18, and a few T- ILLINOIS ENGINEERING EXPERIMENT STATION Table 48 Tests by Graf, Heft 67, Series II, 1931. Simple-Span T-Beams Under One Unsymmetrical Concentrated Load Reference: (23) Dimensions: b=49.2; b'=9.8; D=22.8; d=20.7; e=3.15; L=212.6; L'=240.2; a/d=2.09 for short segment, 8.18 for long segment Loading: One concentrated load 43.3 in. from support Tension Reinforcement: Ten 0.87-in. round plain bars at load, hooked; f,=about 46,000 psi Reinforcement in Flange: Four 0.28-in. long. bars; 0.28-in. transverse bars at 4.9 in.; f,= about 48,000 psi Web Reinforcement: Bent-up bars and 0.28-in. vertical stirrups Concrete Strength: Tests on 7.9-in. cubes; f/= 0.75 f,.'= 1370 psi Age at Test: 27 to 35 days Ft P. Eq. 35 kips 0.64 51.6 Ratio Pte.t P. 3.20 3.20 2.56 2.35 SHORT SEGMENT rf/. Pts. psi p,; 650 1.39 1.39 1.11 Group Beam PteAt kips 1 1203 165.3 1205 165.3 2 1204 132.3 1206 121.3 beams tested by Graf23) under one unsymmetrical load or several concentrated loads. Beams tested by Graf under one unsymmetri- cally placed concentrated load are reported as Series II in Heft 67.(23) Four such beams were tested; the two beams of Group 1 were reinforced with bent-up bars along the entire length of the beams; the two beams of Group 2 had bent-up bars only in the short segment, whereas the long segment was reinforced with a small amount of vertical stirrups. The beams are analyzed in Table 48. The ratio a/d was 2.09 for the short segment and 8.18 for the long segment. The last value is much larger than the range of a/d for which Eqs. 18, 35, and 26 were derived. It is likely that this ratio corresponds either to the transition region between flexural and shear failures or to the region of flexural failures, Fig. 26. This observation is veri- fied by the test results. The two beams of Group 1 failed in tension at a load 2.18 times larger than the shear strength of the long segment as given by Eq. 26. The two beams of Group 2 failed in shear and the load at failure was up to 2.44 times larger than that given by Eq. 26 for the long segment. It is interesting to note, however, that the beams did not fail under the concentrated load at the section of maximum moment but between the load point and the end reaction in the long segment. The final break took place about 68 in. from the sup- port for Beam 1026 and about 116 in. for Beam 1024. The magnitude of the moment at the actual section of failure was 1.03 and 1.50, respectively, times the shear-compression moment for the beams. In both cases, it was reported that the failure was sudden. Thus it appears that the ultimate load was governed primarily by shear. Because of the long shear span, the shearing stresses were relatively small at the load which corresponded to the shear- compression moment, Ms, from Eqs. 35 and 28, at the section of maximum moment in the more lightly reinforced long segment. This load was less than half the ultimate load. Photographs of the beams show that at that load all cracks were prac- tically vertical. As the load increased, the magni- tude of the shearing stresses increased also and the cracks started to incline. At a certain magnitude of shear force, cracks were sufficiently inclined to lead to a sudden shear failure. Since at that load the moment was larger than the computed ultimate shear moment over most of the beam, any random occurrence of a diagonal crack could produce a shear failure. This might be the reason that the two beams failed at different sections. Beams of Series I in Heft 67 were tested under three equal and symmetrical concentrated loads. Six T-beams were tested in three groups: beams of Group 1 had bent-up bars along the entire length of the beam; Groups 2 and 3 only between the end supports and the first load. All tension reinforce- ment was carried through the two middle segments of Group 2, whereas in Group 3 some of the bars were cut off beyond the moment requirement and hooked in the tension zone of the concrete. The beams are analyzed in Table 49. The quantity a/d has been used as a convenient expression for the compressive force-shear ratio C/V of simple-span beams. An equivalent expres- sion is given by a/d = M/Vd for other types of loading. This ratio is 8.52 for the beams of Series I, thus only a little greater than that for Series II. As a consequence, these beams failed in a manner similar to those of Series II. Beams of Group 1 failed in tension, beams of Group 2 in tension with a shear-type final collapse, and those of Group 3 in shear at the section of the center load before yielding of the tension reinforcement. The failure of the last group of beams appears to have been hastened by diagonal cracks which were initiated at the hooks on the cut-off tension bars. The ratios of the ultimate loads to the loads given by Eq. 26 LONG SEGMENT rfy. Psest psi pK. 230 2.18 " 2.18 20 2.44 . Mode of Fail. T T S 8 1. .k 2. - Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 49 Tests by Graf, Heft 67, Series I, 1931. Simple-Span T-Beams Under Three Concentrated Loads Reference: (23) Dimensions: b=49.2; b' =9.8; D=23.2; d=20.8; e -3.54; L=212.6; L'= 240.2; M/Vd - 8.52 at midspan Loading: Three equal and symmetrical concentrated loads, at midspan and at 35.4 in. from supports Tension Reinforcement: Eleven 0.87-in. round plain bars at midspan, hooked; f,= about 46,000 psi Reinforcement in Flange: Four 0.28-in. long. bars; 0.28-in. transverse bars at 4.9 in; f, =about 48,000 psi Web Reinforcement: Bent-up bars and 0.24-in. vertical stirrups Concrete Strength: Tests on 7.9-in. cubes; f//=0.75 f'= 1490 psi Age at Test: 26 to 41 days .... m.Be p..-. . F M AT FARST LOAD AT MIDSPAN Eq. 35 rf/, Mttt AfMt rf, Mtt M. kips in.-kips psi M, M.. psi M. M,, 1 1197 209 0.64 2118 560 1.75 0.83 230 2.92 2.00 1200 218 " " " 1.83 0.86 " 3.04 2.08 2 1198 198 560 1.66 0.78 20 2.77 2.65 1201 209 1.75 0.83 " 2.92 2.79 3 1199 172 560 1.43 0.67 20 2.39 2.28 1202 187 " 1.56 0.74 2.60 2.49 are comparable to those of Series II since the M/Vd-ratios are nearly the same in both cases. From the results of these tests it is evident that for high values of a/d = M/Vd a beam may fail either in shear at a greater load than that given by the shear-compression moment of Eq. 28 or in flexure before developing any marked diagonal cracking. In accordance with the hypothesis pre- sented in Section 18, it appears that the failure criterion is a critical shear force Vo, determined from the shear-compression moment with the aid of a critical ratio (a/d)cr = (M/Vd)c,. If the actual ratio a/d of a beam at the section of maxi- mum moment is larger than (a/d)c,, the beam fails suddenly in what can be called diagonal tension as soon as the shear force reaches its critical quantity Vo. The corresponding moment at failure is larger than the shear-compression moment M,,,. If, how- ever, the flexural capacity of the beam is reached before the critical shear force V. is attained, the beam fails in flexure. The results of the above tests give the critical value of a/d = M/Vd equal to 3.4-3.7. These val- ues are somewhat lower than the values of a/d at which the rectangular simple-span beams under one or two concentrated loads were still observed to fail in shear-compression. The highest ratio a/d was equal to about 4.8 in that case. However, the present tests are too limited both in number and in scope to provide a check on the validity of the above hypothesis or to permit the setting of a numerical value for the critical ratio (a/d)c,. Furthermore, only T-beams made of rather low concrete strength, about 1000 psi, were tested. This combination leads to very high ratios of Pf/P, = MfM,, up to 2.79 as noted in Tables 48 and 49. For rectangular beams which are without web reinforcement and which are made of more normal concrete strength the flexural capacity rarely exceeds that in shear by more than 50-60 percent. This difference between the two types of beams could also influence the mode of failure which renders it impossible to draw any definite conclusions as to the value of (a/d)cr from these few test results. 21. Beams Under Uniform Load It has been shown that within certain limits of a/d = M/Vd the shear strength of a beam under concentrated loads could be determined by Eqs. 18 and 28 for rectangular beams and by Eqs. 35 and 28 for T-beams. Under this type of loading, the beams tested failed at the section of maximum moment and maximum shear, and the load at fail- ure was determined by the magnitude of moment. As the value of M/Vd increased beyond these limits, however, the actual shear strength was found to be larger than that given by the above equations. Furthermore, the location of failure was not necessarily the section of maximum moment. The upper limit of M/Vd for the applicability of shear-compression equations and the shear strength of a beam in the transition region between shear and flexural failures could not be determined quan- titatively because of insufficient experimental data for beams with high values of M/Vd. For simple-span beams under uniform load the value of M/Vd ranges from zero at the section of no moment to infinity at the section of maximum moment. The beam cannot fail in shear at the section of maximum moment because there are no diagonal cracks at that section. Consequently, if a shear failure is to take place, it must occur at a section where the value of M/Vd is such as to permit diagonal cracking and where the moment itself is sufficient to produce a shear-compression failure. In the following paragraphs, the available test data is analyzed in an attempt to find more Mode of Fail. Tat Midspan T-S at Midspan S at Midspan ILLINOIS ENGINEERING EXPERIMENT STATION quantitative information about the shear strength of beams under uniform loading. No tests could be found of beams under actual uniform load. However, there are reports on tests where uniform loading was simulated by a large number of equal and equally spaced concentrated loads. These beams were tested by Bach and Graf in two series, one series under 16 equal loads as reported in Heft 48,(27' and the other series under 8 equal loads as reported in Heft 20.(28) Beams of Heft 48 were five simple-span T- beams loaded with sixteen equal concentrated loads. The arrangement of loads and reinforcement is shown in Figs. 29 through 32, and Table 50 gives the physical properties and test results for these beams. Beam 1024 had no web reinforcement. It failed at a very low load, the maximum moment at midspan being only 68 percent of the shear-compression moment as given by Eq. 35. A diagonal crack formed at about the third-point of the span shortly before failure. Numerous longitudinal cracks ran f P/16 P/16 P/16 P/16 P/16 P/16 P/16 P/16 from that crack toward the end support. It appears that this beam failed in bond. Beams 1026, 1025, and 1031 had almost identical arrangements of bent-up bars except that the size of the bars was different, the area varying as 1.00:0.53:0.36. Beams 1026 and 1025 failed in tension and Beam 1031 in shear. However Beam 1025 was rather close to its shear strength at failure as indicated by marked diagonal cracking all along the beam. Beam 1032 had only two bent-up bars in the ends and failed in shear. Figures 29 through 32 show the arrangement of loads and reinforcement and the main cracks at failure. Furthermore, the actual ratio M/M, at failure, where M, was computed by Eq. 35, and the corresponding predicted ratio, 1 + 2rfK,/103 from Eq. 28, are plotted along the beam for each indi- vidual beam. The shear-compression moment was calculated for the section at midspan; the reduc- tion of the longitudinal steel area through bending up bars at other sections was not taken into con- f P/16 P/16 P//6 P/16 P//6 P/16 P/16 P//6 Fig. 31. Beam 1031 of Bach and Graf, Heft 48 ( P/16 P/16 P/16 P/16 P/16 P/16 P/16 P/16 Fig. 30. Beam 1025 of Bach and Graf, Heft 48 Fig. 29. Beam 1026 of Bach and Graf, Heft 48 [ P/16 P//16 P/16 P//16 P/16 P/16 P//6 P/16 Fig. 32. Beam 1032 of Bach and Graf, Heft, 48 Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 50 Tests by Bach and Graf, Heft 48, 1921. Simple-Span T-Beams Under Sixteen Equal Concentrated Loads Reference: (27) Dimensions: b=47.2; b'=9.8; D= 27.6; d= 25.2; e=3.94; L=212.6; L'- 244.1 Loading: 16 equal and symmetrical concentrated loads. See Figs. 29-32 Tension Reinforcement: Round plain bars, hooked Reinforcement in Flange: Two 0.28-in. long. bars, 0.28-in. transverse bars at 3.9 in. Web Reinforcement: Bent-up bars and 0.28-in. round vertical stirrups Concrete Strength: Tests on 7.9-in. cubes; f/ = 0.75 f.' Age at Test: 42 to 48 days Beam /,' A. , Size of Pt.t Mt..e F, M. Ratio MM Bent-up Eq. 35 Mt-1 Bars M, psi in.2 ksi in. kips in.-kips in.-k in.-k 1024 3230 1026 3250 1025 3050 1031 2750 1032 2750 5.79 50.5 5.73 50.5 5.78 51.2 5.63 49.8 5.87 50.5 None 105.8 0.98 262.3 0.71 264.6 0.59 211.6 0.98 202.8 sideration. The variation in 1 + 2rf,,w/103 was cal- culated using values of r at mid-height of the beams. If the relationship between the actual and the predicted moment ratios is observed in these figures, it is seen that the beams failed in shear only when the ratio M/Ms approached the quan- tity 1 + 2rfv,/103 at about the fifth load point from the end of the beam. It is recalled that a beam under concentrated loads and in the shear-com- pression region of M/Vd would have failed in shear as soon as the value of M/M, had exceeded that of 1 + 2rfyw/103 at the section of maximum moment. Eq. 28. This difference between the two types o0 --- Beam 1031 Shear - Beam /026 Tension- /675 8.18 / 16.35 77S5 -Beam 1031 Tension Capacity Beam 1025 Tension \ -Beam 1032 Shear -. \ ".. 5O10 3.43 2.32 1.49 0.83 4.57 2.90 /.79 097 03C Q 8 7 6 5 4 J 2 I Load Points Fig. 33. Ratio of Measured to Computed Failure Moment as Function of M/Vd. T-Beams of Heft 48 under Sixteen Concentrated Loads 2859 6973 7031 5625 5390 Ratio Mode Mt-e i of M! Fail. 0.57 4210 0.68 7031 0.41 B? 4210 1.65 6987 1.00 T 4140 1.69 7204 0.98 T 3750 1.50 6857 0.82 S 3793 1.42 7119 0.76 S beams suggests that it might be possible to de- termine empirically the value of M/Vd which limits the region of critical diagonal cracking capable of producing shear-compression failures. Figure 33 shows the ratio between the actual moment at failure and the ultimate shear-compres- sion moment of Eq. 28 plotted along the beams. The values of M/Vd at each side of the load points are also marked in the figure. This figure shows the effect of the M/Vd-ratio more clearly. Beam 1026, which failed in tension, has the ratio Mtest/ Ms. less than one at the fifth load point. Beam 1025, which failed in tension while being very close to a shear failure, has the ratio just above one. Beam 1031, which failed in shear, seems to have failed just as the ratio exceeded one. The ultimate flexural capacity of this beam is shown in the figure also. It is seen that this load, if reached, would have increased the ratio to considerably higher than one. Finally, Beam 1032, which failed in shear, has the ratio somewhat more than one, 1.16. However, Fig. 32 shows that in the case of this beam there is some doubt as to what to con- sider as the value of 1 + 2rf,//103 at the fifth load. The bent-up bars do not cover that particular section; their presence in the vicinity undoubtedly offers some resistance to the formation of diagonal cracks. This, in a sense, would mean an increase in the value of rf,, which would bring the ratio closer to one in Fig. 33. Thus, it appears that the shear-compression equations are applicable for the beams under con- sideration. However, the section at which the shear moment is calculated is not that for maximum moment but that at which the value of M/Vd is equal to about 4.5, corresponding to the fifth load point of the beams of Heft 48. Heft 20 reports tests on 51 simple-span T- beams, tested in groups of three companion speci- L.8 1.6 1.4 1.2 Zo e1t Sw 08 0.6 0.4 0.2 'Vd ,q M. -9 ,q 7 6 5 4 ,5 Z I A// ILLINOIS ENGINEERING EXPERIMENT STATION mens. Sixteen groups of beams were loaded with eight equal concentrated loads as shown in Figs. 34 and 35; one additional group had four loads omitted on one half of the span. The physical properties of the beams and the test results are given in Table 51. The first four groups of beams were reinforced with two 1.57-in. plain round bars. The test vari- ables included the effect of anchoring of the longi- tudinal bars, either straight or hooked, and the effect of web reinforcement which was provided by vertical stirrups placed in accordance with the shear diagram along the entire length of beam. All these beams failed in bond as indicated by the excessive end slip of the longitudinal bars which was measured in most beams. Bond failure led to longitudinal cracking along the reinforcing bars and to the final opening of a diagonal crack, gen- erally between the first and the second load points. Groups 55 and 56 were reinforced with four 1.10-in. plain round bars, two of which were bent P/B P/8 Fig. 34. Beams 60 of Bach and Graf, Heft 20 Fig. 35. Beams 62 of Bach and Graf, Heft 20 up at 13 deg. The ends of the bars were anchored with small 90-deg hooks. Beams of Group 55 had no additional web reinforcement and failed in bond by excessive slipping of the bars. Beams of Group 56 had additional vertical stirrups placed accord- ing to the shear diagram and failed in tension. The remaining beams were reinforced with 6 or 7 round bars of different sizes. Two bars were carried straight to the supports; the rest of the bars were bent up at different locations. The middle portion of the beams, not covered with bent-up bars, was reinforced with vertical stirrups. The beams were tested in companion groups; in one the two straight bars were left unhooked, in the other they were hooked. All bent-up bars were sufficiently hooked in all beams. All beams with the straight bars not hooked failed in bond by excessive end slip. This led to the opening of a diagonal crack at different locations in different beams. All beams with hooked straight bars failed in tension with a secondary crushing of the concrete at midspan. Thus, no beams failed actually in shear. Some indication of the shear strength of the beams can be obtained, however, by analyzing the beams which had the smallest number of bent-up bars. Figures 34 and 35 show beams of Groups 60 and 62 in this category. The arrangement of web reinforce- ment is shown together with the quantity 1 + 2rfw/103 and the ratio M/Ms along the beams. The compressive force-shear conditions are repre- sented by the ratio M/Vd, given at both sides of each load point. It is seen that the M/M.-curve intersects the web reinforcement curve near the third load point, at about M/Vd equal to 5. Since the beams failed in tension, the amount of web reinforcement was sufficient to prevent a failure in shear. Consequently, the critical value of M/Vd for shear failures must be less than 5, which agrees with the previous finding of about 4.5 for beams of Heft 48. From the results of the above two series of tests it appears that the shear strength of beams under uniform load can be represented by the shear- compression equations 18 and 28 for rectangular beams and by Eqs. 35 and 28 for T-beams. Since there are no diagonal cracks in the region of maxi- mum moment and the inclination of cracks is very small for high values of M/Vd, the beam cannot fail in shear unless the bending moment is higher than the shear strength as given by Eq. 28 at a critical value of M/Vd. From the above results, the critical value of M/Vd is set tentatively at about Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS Table 51 Tests by Bach and Graf, Heft 20, 1912. Simple-Span T-Beams Under Eight Equal Concentrated Loads Group No. of Reinf. Bars 51 2 52* 53 54* 55 4 56* 57 6 58 59 60 61 7 62 63 64* 65 66 Reference: (28) Number of Beams: Three in each group Dimensions: b=23.6; b'=7.9; D = 15.7; d= 13.6; e=3.94; L = 157.5; L'= 173.2 Loading: 8 equal and symmetrical concentrated loads. See Figs. 34, 35 Tension Reinforcement: Round plain bars, numerous sizes from 0.39 to 1.57 in. in diam; average /,=46,000 psi Reinforcement in Flange: None Web Reinforcement: Bent-up bars and 0.28-in. plain round vertical stirrups; f/, = 58,300 psi Concrete Strength: Tests on 11.8-in. cubes; f,'= 0.75 f.'= 2490 psi ± 7.7 percent Age at Test: Around 45 days Computed Quantities: Ft=0.77; M.- 1259 in.-k; P.=57.5 kips, average Mff=2135 in.-k, Pf=97.5 kips A. Anch. No. of a Pt.k Ratio Ratio Long. B-Up Pwt Ptmt in.2 Bars Bars deg kips P. Pf 3.90 None .... 47.0 0.82 0.48 3.90 .. 67.6 1.17 0.69 3.90 Hooks .... 51.5 0.90 0.53 3.96 " 94.0 1.63 0.96 3.81 2 13 73.5 1.28 0.75 3.80 100.5 1.75 1.03 3.86 None 4 45 90.5 1.57 0.93 3.86 Hooks 95.5 1.66 0.98 3.91 None 86.1 1.50 0.88 3.91 Hooks 95.5 1.66 0.98 3.92 None 5 85.2 1.48 0.87 3.94 Hooks 99.6 1.73 1.02 3.90 None 90.3 1.57 0.93 3.90 Hooks 106.4 1.85 1.09 3.94 30 100.0 1.74 1.02 3.91 102.0 1.77 1.05 * Vertical stirrups along the entire span. 4.5. It is recalled, however, that the tests were far from being conclusive and that only simple-span T-beams were tested. The validity of the above concept of shear failures of beams under uniform load and a more reliable value of the critical M/Vd must be established by a more comprehensive test program. It appears, however, that the conventional method of reinforcing simple-span beams under uniform load against shear-type failures is incor- rect. Web reinforcement is placed to conform with the shear diagram. This means that the amount of web reinforcement in the region of the critical value of M/Vd is smaller than that for lower values of M/Vd. The above findings suggest, however, that the web reinforcement should be placed at a uni- form spacing between the end reaction and the region of critical M/Vd, say 4.5. Only beyond that region should it be tapered off and reduced to zero at midspan. If it is desired to prevent shear failures altogether, the ultimate flexural and shear moments must be calculated from the properties of the beam by Eqs. 29 and 18 or 35. Then the ratio between the ultimate flexural moment at the section of the critical value of M/Vd and the shear moment of Eq. 18 or 35 must be substituted into Eq. 28 in order to find the necessary amount of web rein- forcement which would force the beam to fail in tension at the section of the maximum moment rather than in shear at the section of the critical M/Vd. Mode of Fail. B B B B B T B T B T B T B T T T VII. SUMMARY AND CONCLUSIONS 22. General Summary and Discussion A general expression for the shear strength of reinforced concrete beams has been derived by con- sidering simple-span beams without web reinforce- ment. It was first assumed that the total shear force is resisted solely by the compression area of the concrete and that the criterion of failure is an ultimate shearing unit stress, related to the com- pressive strength of the concrete. These assump- tions yielded an expression in a form which suggested that the real criterion for shear failures was a limiting moment rather than an ultimate shearing stress. This observation was supported by certain test results reported in the literature.(5,6) It was concluded that shear failures were actually a compression phenomenon. Shear-compression fail- ures differ from flexural compression failures only because the compressive area of the concrete is re- duced by diagonal cracks which extend higher than the flexural tension cracks at failure. a. Simple-Beams Without Web Reinforcement. Treating shear failures as compression failures and assuming that the depth of the compression zone was related to k as determined by the elastic "straight line" theory, Eq. 18* was derived em- pirically to represent with good accuracy the shear strength of rectangular simple-span beams without web reinforcement and under one or two symmetri- cal concentrated loads Equation 18 was based on the test results from 15 different investigations involving 106 beams which failed in shear. These beams were tested over a period of 43 years and had a wide variation in their physical properties as summarized in Table 1. The average ratio of measured to com- puted moments was 0.986 and the standard devia- tion 0.119. The agreement between the measured and computed moments is shown graphically in Fig. 2. Equation 18 was also interpreted theoretically in the light of the conventional theory of compres- sion failures of reinforced concrete beams. From *The equations referred to in this section are summarized for con- venient reference in Section 23. previous test results at the University of Illi- nois"1, 14) the value of kfk3 was approximated by Eq. 20, and for this value of kks3 it was possible to establish a relationship between k, and k, where k, refers to the depth of the compression zone at shear failures, Eq. 22. Since k remains usually within the values of 0.2 and 0.5, Eq. 22 suggests that k, is practically a constant fraction of k. This finding explains why the previous attempt to use the value of k as a measure of k, gave satisfactory agreement with test results. b. Web Reinforcement. The effect of web rein- forcement was investigated next. It was found that the use of web reinforcement increased the shear strength of a beam more than would be accounted for by the internal forces in the stirrups. The total contribution of web reinforcement was expressed empirically by Eqs. 26 and 28. These equations were based on the test results for 80 beams. The average ratio between the meas- ured and calculated moments was 1.017 and the standard deviation 0.089. The range of the physical properties of the beams is summarized in Table 17 and the ratios of P/P, are shown graphically in Fig. 5. The equations were further checked by the help of beams which had failed in flexure. It is seen in Fig. 6 that although the flexural capacity of these beams was reached at different ratios of P/P,, they always failed at a load lower than their strength in shear, given by Eq. 26. Equations 26 and 28 were found to be applicable for all angles of inclination and for different values of yield strength of web reinforcement. It was found also that there was no noticeable difference between the effectiveness of bent-up bars and stir- rups serving as web reinforcement. Equations 26 and 28 show that a given amount of web reinforcement will increase the shear strength of a beam in proportion to its strength without web reinforcement rather than by an amount determined solely by the physical proper- ties of the web reinforcement. It appears that by resisting the extension and widening of diagonal cracks, the presence of web reinforcement increases Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS the available compressive area of the concrete and, conceivably, restricts the concentration of the com- pressive strain of concrete in the region of the main diagonal crack. The relationship between shear-compression and flexural failures was discussed in Section 13. It was found that the amount of web reinforcement neces- sary to prevent shear failures could be determined for any beam by Eqs. 29, 18, and 28. Simple-span rectangular beams reinforced in tension only and designed according to the present ACI Code bal- anced design requirements were found to require about 0.35 percent web reinforcement to ensure ten- sion failures. This assumed that the yield strength of the tension reinforcement was 50,000 psi and that of the web reinforcement 40,000 psi; it also as- sumed that the beams were loaded under one or two symmetrical concentrated loads. c. T-Beams. Since the moment-rotation rela- tionship of a T-beam differs from that of a rec- tangular beam, Eq. 18 had to be modified to apply for T-beams. This was done by the use of a semi- rational shape-factor in the form of Eq. 34. Sub- stituting the compressive area Ac of a T-section as determined by the "straight-line" theory for bkd and using the shape factor of Eq. 34, Eq. 18 was rewritten as Eq. 35, applicable to T-beams. As seen from Fig. 11, Eq. 35 was found to give satisfactory agreement with test results when beams with abnormally large values of d/e and b/b' were excluded. These beams had a lower shear strength because the effective width of their flanges was reduced. However, no attempt was made to de- termine an expression for the effective flange width. Furthermore, it was found that the use of trans- verse reinforcement in the flange effectively counter- acted the reduction in the effective width and thereby increased the scope of Eq. 35. The shear strength of simple-span T-beams with web reinforcement could be determined from Eq. 28 which was derived for rectangular beams, but with the value of r given by Eq. 27a. As seen in Fig. 12, the agreement between the measured and calculated quantities is satisfactory. d. Restrained Beams. Simple-span beams under one or two symmetrical concentrated loads develop just one main diagonal crack under an applied load and fail at that section. In restrained beams, shear and moment conditions are such as to per- mit the formation of more than one main diagonal crack. The beam may fail at any of these cracks, depending on the magnitudes of moment and shear and the arrangement of both longitudinal and web reinforcement. It was found that whenever the pos- sibility of bond failures was precluded, the shear strength of a restrained beam was determined by the same equations as that of a simple-span beam, Eqs. 18 and 28. The critical section was the section of maximum moment. When the longitudinal reinforcement was cut off at some section, a sudden and complete bond failure was possible by stripping out of the cut-off reinforcement. This type of failure was outside the scope of this investigation and was not examined in more detail. Evidently, this is a question of bond characteristics of the reinforcing bars and the length of embedment from a diagonal crack to the end of the bar. Restrained beams with continuous top and bot- tom reinforcement may have another mode of fail- ure. Under certain conditions, e.g., when the dis- tance between a support and a load is short relative to the effective depth of the beam, a local bond failure may take place in the high bond-stress region between the sections of positive and nega- tive moments. As a result of local destruction of bond, both the top and bottom longitudinal rein- forcement is in tension at a certain section. This redistribution of the internal forces results in a reduced shear strength of the beam. Assuming that the entire tension force was transmitted from one section to the adjacent section and that k, was given by Eq. 22, Eq. 44 was derived to represent the shear strength for this type of failure. The validity of Eq. 44 was checked against test results and satisfactory agreement was obtained. Figure 23 shows the measured and calculated mo- ments graphically for all beams which failed after a local bond failure. Most of the test specimens show good agreement with Eq. 44; for some beams a small increase in the shear strength was noticed because of the effect of partial bond. This was dis- cussed in more detail in Section 17, paragraph b. All beams shown in Fig. 23 had equal positive and negative moments and developed, in general, two main diagonal cracks before failure. This re- sulted in a full redistribution of the internal forces and the shear strength of the beams was governed by Eq. 44. For unequal positive and negative mo- ments, however, either one or two cracks may be present at failure. Two cracks will produce, in general, a full redistribution of the internal forces ILLINOIS ENGINEERING EXPERIMENT STATION and the shear strength of a beam will be given by Eq. 44 at the section of maximum moment. One crack will lead to a partial redistribution of the internal forces, so that the shear strength will be governed by Eq. 18 at the section of maximum moment. Beams of Series VI by Moody had un- equal moments at sections A and B and failed at section A after developing only one crack in span g, Fig. 13. The beams were analyzed according to Eq. 18 at that section and Fig. 24 shows that good agreement was obtained between the measured and the calculated moments. From the available test data, it was not possible to determine the limits of Eq. 44. The largest g/d- ratio for which test results were available was 4.0. Since this ratio permitted a redistribution of the internal forces, the limiting g/d-ratio must be larger than four. Furthermore, it is apparent that bond characteristics of the reinforcing bars have an effect on the limiting value of g/d. The above re- sults were reported for beams reinforced with mod- ern deformed bars; plain bars undoubtedly are more susceptible to local bond failures. Likewise, it was not possible to determine the conditions under which two cracks and, consequently, a full redis- tribution of the internal forces will occur for un- equal positive and negative moments. Until such criteria can be established, the more conservative condition of full redistribution should be assumed in determining the shear strength of a restrained beam. It was found that the contribution of web re- inforcement could be determined in restrained beams, as in simple-span rectangular and T-beams, by Eq. 28. Beams reinforced with 45-deg stirrups gave very good agreement with Eq. 28; beams pro- vided with vertical stirrups also agreed with this equation except for two beams with the largest values of rfw. It appears that in beams with rela- tively short shear span inclined stirrups are, in general, more reliable than vertical stirrups. It is conceivable that inclined stirrups have better an- chorage conditions whenever diagonal cracks are forced to form in a restricted space and thereby can develop their full effectiveness. Conversely, the an- chorage of vertical stirrups might be destroyed be- fore their full effectiveness is reached. e. Shear-Flexure Transition. All the shear- compression equations were derived and checked for beams for which the a/d-ratio varied be- tween 1.17 and 4.80. The a/d-ratio represents the compressive force-shear ratio for simple-span beams under one or two concentrated loads; for any other type of loading this ratio can be repre- sented by the equivalent ratio M/Vd. Within these limits of M/Vd, the shear strength of a beam was found to be determined solely by the physical prop- erties of the beam. It was not a function of either the magnitude of the shear or the moment-shear ratio at failure. However, as the M/Vd-ratio increases, the rela- tive importance of shear in connection with the di- agonal tension stresses decreases. Consequently, the extent of diagonal cracking is less pronounced, and it was found that the shear strength of such beams was larger than that given by Eq. 28. It was also noticed that the location of shear failure was not necessarily the section of maximum moment. For sufficiently large values of M/Vd the beams failed in flexure rather than in shear. A hypothesis to explain the behavior of beams with high values of M/Vd was presented in Section 18. As seen in Fig. 26, a critical value a/d = M/Vd was assumed to represent the largest ratio of moment to shear which would permit sufficient diagonal cracking for shear-compression failures. Beams with higher than the critical value of M/Vd did not crack sufficiently and could not fail at the shear-compression mo- ment, Eq. 28. The criterion of failure for those beams was a critical magnitude of shear force, Vo. The magnitude of Vo was determined by the shear- compression moment, Msw, and the critical ratio (M/Vd) cr. The resulting failure was classified as a diagonal tension failure since the beams failed sud- denly as soon as the critical shear force was at- tained. The relative magnitudes of the flexural ultimate and the shear-compression moments and the value of (M/Vd) cr determined the upper limit of the transition region, point (a/d), = (M/Vd)2 in Fig. 26. Beams with ratios M/Vd higher than (M/Vd)2 failed in flexure since their flexural ca- pacity was reached prior to the attainment of the critical shear force, Vo. The validity of this hy- pothesis and the critical value of M/Vd could not be determined because of insufficient experimental data. f. Shear-Proper. Conversely, for very small values of a/d the beams did not fail in shear-com- pression. The mode of failure appeared to be an actual shearing off of the compression zone of the concrete. This type of failure was tentatively called shear-proper. It was also found that the shear Bul. 428. STRENGTH IN SHEAR OF REINFORCED CONCRETE BEAMS strength of such beams depended on the x/D-ratio rather than a/d-ratio, where x denotes the clear distance between the load-bearing blocks and D the total depth of beam. For x/D equal to zero, the shear strength of a beam could be related to a nominal shearing stress. The entirely empirical ex- pression represented by Eq. 48 was found to give satisfactory agreement with test results. As the ratio x/D increased, the ratio between the test and calculated shearing stresses decreased as shown in Fig. 28. Since the number of tests was limited, no expression could be determined for the relationship between v/vc and x/D. For small values of x/D the location of the load-bearing blocks forced the formation of almost vertical cracks and, consequently, vertical stirrups were not found to contribute to the shear strength of the beam. However, as x/D increased in the region of shear-proper, cracks followed the edges of the bearing blocks and vertical stirrups crossing the crack produced a slight increase in the shear strength. The transition region between shear- proper and shear-compression was estimated to lie approximately between x/D equal to 0.8 and 1.1, increasing as the amount of vertical stirrups in- creased. Since the contribution of vertical stirrups is very small, inclined stirrups should be used whenever the x/D-ratio approaches unity. g. Uniform Loading. For simple-span beams under uniform loading, the value of M/Vd ranges from zero at the section of no moment to infinity at the section of maximum moment. It is believed that with certain modifications the shear-compres- sion equations, Eqs. 18, 35, and 28, could be used to determine the shear strength of such beams. Since there are no diagonal cracks in the region of maximum moment and the inclination of cracks is very small for high values of M/Vd, the beam can- not fail in shear unless the bending moment is higher than the shear strength given by Eq. 28 at a critical value of M/Vd. From test results studied, this critical value of M/Vd was set tentatively at about 4.5. However, since only a few T-beams have been tested under conditions which simulated uni- form loading, the validity of the above concept of shear failures and a more reliable value of the critical M/Vd must be established by a more com- prehensive test program. It is conceivable that the same procedure can be used for any type of beam under either uniform or distributed loading to determine its strength in shear. It involves only the determination of critical sections for shear failures. Provided that the value of M/Vd is in the region of shear-compression, Eqs. 18 and 28 can be used directly at sections where maximum shear and maximum moment co- incide. Where these maxima do not coincide, the critical section at which the shear-compression equations should be used is given by the critical value of M/Vd. Since shear-type failures result in a sudden and complete destruction of a structure, they should be avoided in actual construction. In order to deter- mine the amount of web reinforcement necessary to ensure flexural failures, the flexural capacity of the beam should be determined first and the cor- responding loading considered as applied loading. Then, both the applied moment and shear moment of Eqs. 18 and 35 should be determined for critical sections of shear failure. The ratio between the two substituted for M,w/M, in Eq. 28 will determine the amount of web reinforcement required. h. Indeterminate Structures. One additional problem is confronted in statically indeterminate structures whenever redistribution of moments near the ultimate load is considered. In order to utilize the full load-carrying capacity of the structure, its members must be so designed as to permit sufficient rotation at the plastic hinges. Consequently, not only primary but also secondary shear failures after yielding of the reinforcement must be prevented. This is a phase of the phenomenon of shear in re- inforced concrete which has received very little at- tention in the past. 23. Summary of Equations. a. Shear-Proper. For x/D = 0 the shear strength of a beam is determined by the following expression: V VC = bD = 200 + 0.188f,' + 21,300 pt (48) where A, (1 + sin a) Pt = bD (47) as x/D increases, the ratio v/vc decreases. The re- lationship between x/D and v/vc could not be de- termined, although some information is available from Fig. 28. The transition region between shear- proper and shear-compression was estimated to lie between x/D equal to 0.8 and 1.1, depending on the amount of vertical stirrups. Otherwise, the ILLINOIS ENGINEERING EXPERIMENT STATION effect of vertical web reinforcement is neglected in Eq. 47. b. Shear-Compression. In the shear-compres- sion range the shear strength of a beam without web reinforcement and under concentrated loads is given by the following equations for the maximum shear moment, Ms: For rectangular beams: M. = 0.57 - 4.5f' (18) bdf,' (k + np') 105 where k is given for beams reinforced in tension only by k = V (pn)2+ 2pn - pn (14) and for beams reinforced in both tension and compression by k = V [n (p + p')]2 + 2n (p + p' - p't) - n (p + p') where A MB+ 1) (1 - kk,) A-MA k = V (pon)2 + 2pn - pan MA) (45) (42) (43) k. = 1.11 - V/1.23 - 0.926 k (23) k2 = 0.45 The contribution of web reinforcement is de- termined in all cases by the following expression for the ratio of the maximum moment capacity Msw of the beam with web reinforcement to the moment capacity Ms of the same beam without web rein- forcement: Mw + 2rfyw M-. 10, where and where n is given by n=5± 10,000 n = 5For T-beams:+ For T-beams: A r = sw for rectangular beams bs sin a (16) and r = 'A for T-beams b's sin a (27a) M, 4.5fo' A ' 0.57 - 45f (35) Acdf,'Ft 105 where SIT± - Icr Ft = IR + Icr (34) For restrained beams: the shear strength is given by Eq. 18, whenever bond failures are pre- vented, and by the following equation whenever redistribution of internal forces has taken place as a result of local bond failure in the high bond- stress region: M = 0.57 4.5f' (44) bd2f' kA 101 The upper limit of a/d = M/Vd for shear-com- pression failures could not be determined; the high- est value used in tests was 4.8. For high values of M/Vd the shear strength is larger than that given by the above equations. c. Distributed Loading. At a section where maximum moment and maximum shear coincide, the shear strength of a beam under distributed loading can be determined directly by the above shear-compression equations, provided that the value of M/Vd is in the range of applicability of these equations. However, in regions of maximum moment and no shear, the above equations should be used at a section given by M/Vd equal to about 4.5. J