MINIMUM SHEAR REINFORCEMENT FOR OPTIMUM DUCTILITY OF REINFORCED CONCRETE BEAMS

Transcription

1 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: MINIMUM SHEAR REINFORCEMENT FOR OPTIMUM DUCTILITY OF REINFORCED CONCRETE BEAMS Appa Rao. G 1, Injaganeri. S. S 2 1 Assoiate Proessor, 2 Researh Sholar, Dept. o Ciil Engg, IIT Madras, Chennai , India garao@iitm.a.in Abstrat Failures in reinored onrete (RC) strutures under transerse shear ores are proed to be atastrophi with the presene o web reinorement. The minimum web reinorement reommended b seeral odes o pratie has been intended to maintain adequate strength and deletion dutilit ater the ormation o diagonal raking and to ontain widening o the diagonal raking. Howeer, the expressions or estimating the minimum shear reinorement in the odes o pratie are based on the experimental data base obsered on testing o small size beams made o normal strength onrete (NSC). Suh ode proisions need to be reinestigated on large size beams made o high strength onrete (HSC). Further, there has been lak o onsensus on the quantit o minimum shear reinorement to be proided b dierent odes o pratie, as the dier signiiantl in respet o HSC members. In this paper, man ators inluening the minimum shear reinorement required in RC beams hae been studied. An expression has been proposed inorporating a wide range o parameters. A omparison o the minimum shear reinorement predited b the proposed expression has been made with the odes o pratie. The inluene o shear reinorement on the dutilit o RC beams o aring sizes has been inestigated. The optimum shear reinorement index has been ound to be somewhere between 0.45 and 0.5. Dutilit o RC beams inreases with inreasing the shear reinorement index. Small size beams exhibited signiiant dutilit or the gien shear reinorement index. Kewords: Minimum shear reinorement, resere strength, dutilit, RC beams, HSC *** INTRODUCTION It is known rom earlier stud that the web reinorement in the orm o stirrups does not inluene the diagonal raking strength signiiantl. Howeer, the presene o shear reinorement alters the behaiour o RC beams and enhanes shear apait b modiing the shear transer mehanisms through improed dowel ation, restraining rak propagation, minimizing bond splitting and inreasing the ontribution o onrete in ompression zone. In the design o RC strutures, use o minimum shear reinorement is mandated when the atored shear ore exeeds one-hal o the design shear strength o onrete. The objetie o speiing minimum shear reinorement b the odes o pratie is to preent sudden ailure at the ormation o irst diagonal raking, to ontrol widening o raks at serie loads and also to ensure adequate dutilit beore ailure. Some o the major odes o pratie suh as ACI, CANADIAN and AASHTO spei the minimum shear reinorement as a untion o onrete strength while it is onl a untion o ield strength o shear reinorement and independent o ompressie strength o onrete b IS and BS odes o pratie. 2. REVIEW OF LITERATURE A brie reiew o literature arried out b authors on minimum shear reinorement is presented. Lin and Lee [1] reported that an inrease o tension reinorement inreases the strength but dereases the dutilit o the beams. Further an inrease in ompression steel reinorement and onrete strength enhanes the dutilit o the beams eetiel, and the beams proided with high strength shear reinorement exhibit the same raking resistane as those with normal strength shear reinorement. In another stud, Lin and Lee [2] onluded that the ators aeting the dutilit o RC beams are ratio, spaing o stirrups and strength o shear reinorement. Also it was onluded that b inreasing the quantit and the strength o shear reinorement does not hae apparent inluene on the diagonal raking strength. Howeer, inreasing the strength o onrete and strength o shear reinorement inreases the ultimate strength, whereas the derease in the shear-span-depth ratio and the spaing o stirrups inreases the ultimate strength o beams. For beams with small ratio, the eetieness o the shear reinorement is muh redued. Xie et al. [3] arried out experimental inestigation on 15 RC beams or understanding the dutilit under shear dominant loading with and without web reinorement. The ariables Volume: 02 Issue: 10 Ot-2013, 25

2 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: onsidered in the stud are; ompressie strength o onrete ranging between 40 and 109 MPa, shear span-to-depth ratio aried between 1.0 and 4.0 and the quantit o shear reinorement aried between 0.0 and 0.784%. The post peak response was haraterised through the shear dutilit o RC beams. Based on the omparison o two large sale beams tested b Johnson and Ramirez [4] and the tests on redued size speimens, Robert Frosh [5] onluded that the beam size did not aet the post raking behaiour or the shear strength proided b the stirrups. Howeer, rom the analsis o the test results rom the preious studies, Johnson and Ramirez [4] onluded that the oerall resere strength ater the diagonal tension raking diminished with the inrease in the ompressie strength o onrete, or beams designed with the urrent proisions o minimum shear reinorement. This situation would be more ritial or beams with larger ratios and smaller quantit o longitudinal reinorement. Yoon et al. [6] onluded that or HSC members the rak spaing is a untion o the spaing o longitudinal and transerse reinorement. The spaing o the shear raks inreases as the member size inreased and hene the Canadian Standards Assoiation [7] (CSA) predits lower shear stress at ailure or large size members. Roller and Russell [8] rom their experimental work onluded that the minimum quantit o shear reinorement speiied in the odes must be inreased with the inrease o ompressie strength o onrete. The equation proposed b Yoon et al. [6] was re-ealuated b Ozebe et al. [9], whih onluded that the quantit o shear reinorement ould be 20% smaller than that o the minimum shear reinorement speiied b the ACI ode [10]. The experimental inestigations onduted b Angelakos [11] showed that the minimum quantit o the shear reinorement b the ACI ode [10] resulted in inadequate saet margins. 3. SHEAR REINFORCEMENT PROVISIONS The design proisions or the minimum reinorement in shear speiied b arious odes o pratie are presented in Table 1. Table-1: Minimum Shear Reinorement b arious Codes o Pratie S. No Code Equation Eq. 1 ACI 2 3 CSA 4 A bs s 16 AASHTO A s [12] bs 12 A bs IS 456 [13], A s 0. 4 BS 8110 [14] bs 0.87 s 0.33 (1) (2) (3) (4) Fig1. Reinorement Index s. Compressie Strength Using these proisions the ariation o shear reinorement index, r* (r = As/bS) with ompressie strength o onrete is shown in Fig. 1. It an be obsered that the quantit o minimum shear reinorement is a untion o the ompressie strength o onrete aording to the proisions b ACI, AASHTO and CSA odes. But it is a untion o the strength o the shear reinorement alone as per BS and IS odes. Also, the ariation is muh higher or HSC beams ompared to NSC beams. The shear reinorement is represented in terms o shear reinorement index, whih is the produt o shear reinorement ratio and the ield strength o the shear reinorement. 4. DETERMINATION OF MINIMUM SHEAR REINFORCEMENT The minimum web reinorement proided b dierent odes o pratie is intended mainl to ensure that the apait o a member ater the diagonal raking exeeds the load at whih the inlined raking ours or in other words or a beam with a gien geometr and properties o materials the minimum quantit o shear reinorement is neessar to inrease the shearing strength o the beam to a partiular shear ore V greater than that orresponds to raking strength, Vr. The studies arried out b Johnson and Ramirez [4] showed that the post raking strength dereases with inrease in the ompressie strength o onrete. It would be more ritial with the inrease in the shear-span-to-depth ratio and derease in the longitudinal reinorement. Hene, it is lear that the minimum shear reinorement must be a untion o the shearspan-to-depth ratio and the longitudinal reinorement along with the ompressie strength o onrete. Using the relationship between the diagonal raking and the ultimate shear strength b authors [15] and also rom the ondition (V + Vs) > Vr, in Eq. 5, an expression or the minimum shear reinorement has been deeloped in Eq. 10. V + V s V r (5) Volume: 02 Issue: 10 Ot-2013, 26

3 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: The relationship between the diagonal raking strength and the ultimate shear strength [15] is as shown in Eq. 6, 3 V r = u 1/6 2 ρ The shear strength o onrete or the reinored onrete members as per the ACI ode [10] is, bd (6) 5. VARIATION OF MINIMUM SHEAR REINFORCEMENT The minimum shear reinorement ealuated using the model deeloped in Eq. 10 in the present stud or ube onrete ompressie strength o 20, 40 and 60 MPa or dierent perentages o longitudinal reinorement are shown in Figs. 2 to 4 respetiel in beams with dierent shear span-to-depth ratios. V 0.17 = γ ' bd Taking the lower bound alue or the tensile strength o onrete as a untion o the ompressie strength in [16] proposed an expression or the ultimate shear strength o onrete whih is gien b Eq. 8, (7) Reinorement Index (r*) =1 =2 =3 =4 =5 =6 0.5 ' u = (8) Perentage o Longitudinal Reinorement Taking, V = A s s d s From Eq. 5 substituting or u, Vs, Vr and V we get, 3 A s ' /6 bs = ρ ' (9) (10) Fig2. Reinorement Index s. % Longitudinal Reinorement or 20 MPa Conrete Reinorement Index (r*) =1 =2 =3 =4 =5 =6 The minimum reinorement is a untion o ompressie strength o onrete, shear span to depth ratio and perentage lexural reinorement. For the beams with the shear span-todepth ratio, > 3, the minimum shear reinorement predited b the ACI, CSA and AASHTO proisions is not onseratie and urther inestigations are to be arried out or the higher shear span-to-depth ratios with ompressie strength o onrete greater than 60 MPa. The model proposed in this stud predits satisatoril the minimum shear reinorement or all tpes o beams with arious ompressie strengths o onrete, shear-span-to-depth ratio and longitudinal reinorement Perentage o Longitudinal Reinorement Fig3. Reinorement Index s. % Longitudinal Reinorement or 40 MPa Conrete As shown in Figs. 2 to 4 using 20, 40 and 60 MPa ompressie strength o onrete, the reinorement index dereases with inrease in the perentage longitudinal reinorement. The longitudinal reinorement proided in the tension region o the beam an improe the shear strength o RC beams due to dowel ation o the lexural tensile reinorement. Due to the dowel ation o tension reinorement, the quantit o shear reinorement required to resist the transerse ores has been obsered to be dereased. Volume: 02 Issue: 10 Ot-2013, 27

4 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: Reinorement Index (r*) Perentage o Longitudinal Reinorement =1 =2 =3 =4 =5 =6 Fig4. Reinorement Index s. % Longitudinal Reinorement or 60 MPa Conrete The ariation o the reinorement index s. shear span-todepth ratio in arious beams with onrete ompressie strength o 20, 40 and 60 MPa are shown in Figs. 5 to 7 respetiel with dierent perentage lexural tensile reinorement. The shear reinorement index inreases with inreasing the shear span-to-depth ratio. This an be explained b the at that the shear strength o RC beams, at er low to low shear span-to-depth ratio behaing like a deep and short beams respetiel, is er high. The shear reinorement required is redued at small ratios. In RC beams with large ratios, the ailure tends to be lexure-shear or lexure mode. In suh beams, the shear apait is relatiel less and the ailures are more like lexural ailures. The shear apait is er small. Hene, at large ratios, the minimum shear reinorement tends to be inreased. perentage o the longitudinal reinorement. Figs. 2 to 7 show the ariation o the shear reinorement with perentage lexural reinorement, ρl and the shear span-to-depth, () ratio or dierent ompressie strengths o onrete. Reinorement Index (r*) ρ l =0.5 ρ l =1.0 ρ l =1.5 ρ l =2.0 ρ l =2.5 ρ l = Fig6. Reinorement Index s. Shear span-to-depth ratio or 40 MPa Conrete Reinorement Index (r*) ρ l =0.5 ρ l =1.0 ρ l =1.5 ρ l =2.0 ρ l =2.5 ρ l =3.0 Reinorement Index (r*) ρ l =0.5 ρ l =1.0 ρ l =1.5 ρ l =2.0 ρ l =2.5 ρ l = Fig5. Reinorement Index s. Shear span-to-depth ratio or 20 MPa Conrete Fig7. Reinorement Index s. Shear span-to-depth ratio or 60 MPa Conrete Fig. 8 shows the ariation o the shear reinorement index with ompressie strength or = 3.0 and ρl = 2.4%. It an be obsered that the shear reinorement dereases with inrease in the longitudinal tension reinorement, ρl and inreases with inrease in the shear span-to-depth ratio, ratio. The objetie o odes o pratie or minimum reinorement in shear must also be to ensure minimum dutilit apart rom ensuring the minimum strength o a member is greater than its diagonal raking strength. It is also lear that the addition o een small quantit o web reinorement inreases the dutilit o a member but one needs to quanti the minimum dutilit that has to be deeloped within the member. The minimum shear reinorement alulated aording to the ACI and IS odes orresponding to the ultimate shear strength o the beam, whih is equal to the onrete shear strength o 0.48 MPa (or = 60 MPa) and 0.4 MPa respetiel. Howeer, the proisions b the odes o pratie are independent o the shear-span-to-depth ratio and also the Volume: 02 Issue: 10 Ot-2013, 28

5 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: Fig8: Shear Reinorement Index s. Compressie Strength ( = 3.0, pl = 2.4%). Beam B mm Table-2: Beam Dimensions D mm d/b le mm 6. EXPERIMENTAL INVESTIGATIONS 6.1. Materials and Beam Dimensions le/d Portland Pozzolona Cement was used or preparation o the reinored onrete beams and other ompanion speimens. The width o the beams was maintained onstant at 150 mm maintaining two dimensional similarit. Twele reinored onrete beams o three dierent sizes were ast and tested with depths o beams 200 mm, 400 mm and 600 mm, with the eetie depths o 161mm, 345mm and 536mm respetiel. H H H H H H H H H H H H The eetie span-to-eetie depth ratio or all the six beams tested was 6.0. The shear span-to-eetie depth ratio was 3.0. The beam dimensions are shown in Table 2. The ompressie strength o onrete was aried between rom 50 MPa to 60 MPa. A stress-reinorement index representing the quantit o shear reinorement, (r*) equal to 0.4 was estimated orresponding to the minimum shear reinorement required as per the IS Two more alue o these indies 0.6 and 0.8 were also adopted in this stud antiipating the minimum shear reinorement to be ahieed or high strength onrete with large size beams, as the dutilit dereases with inrease in the strength o onrete and the size o the beam. The beams o depth 200 mm and 400 mm were ast in steel moulds, while 600 mm depth beams were ast in moulds made o well seasoned wood. The beams were demolded ater 24 hrs and subsequentl ured or 28 das beore testing. To obtain the tensile and ompressie strength o onrete ompanion linders o 150 mm diameter and ubes o 150 mm size were ast and tested along with the beams. Twele geometriall similar reinored onrete beams o three dierent sizes made o medium strength onrete were ast and tested under three-point loading using a losed loop MTS testing sstem o 500 kn apait. The properties o the steel reinorement used in this programme are shown in Table 3. The ompressie strength o onrete in the beams is shown in Table 4. The lexural tensile reinorement in all the beams was 2.5%. the number o lexural reinoring bars orresponding to the aboe perentage reinorement is shown in Table 4. The spaing o the losed stirrups was aried to stud the dutilit and minimum shear reinorement proisions b dierent odes o pratie. The shear reinorement in the beams was aried in terms o shear reinorement index. The shear reinorement index is the produt o the shear reinorement ratio multiplied b the ield strength. Table 4 shows the lexural and shear reinorement details in the beam. Table-3: Properties o Web and Longitudinal Reinorement Diameter Area (mm2) Yield Strength (N/mm2) 3 mm mm mm mm mm mm Ultimate Strength (N/mm2) Volume: 02 Issue: 10 Ot-2013, 29

6 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: Table-4: Flexural and Shear Reinorement Details Beam k, MPa Ast As r* H mm Nil 0.0 H mm / 0.4 H mm / 0.6 H mm / 0.8 H mm Nil 0.0 H mm / 0.4 H mm / 0.6 H mm / 0.8 H mm Nil 0.0 H mm / 0.4 H mm / 0.6 H mm / 0.8 Fig9. Load s. Deletion in beam HSC Load s. Deletion The ariations o load with deletion o beams o depths 200 mm, 400 mm and 600 mm are shown in Figs. 9 to 11, whih demonstrate that with inrease in SRI the dutilit o beams inreases. The ultimate loads or 200 mm depth beams with SRI 0.0 and 0.6 were 112 and 150 kn and the orresponding deletions are 2.4 mm and 5.3 mm respetiel. Similarl, or 400 mm depth beams with SRI 0.0 and 0.8 the ultimate loads were 240 kn and 270 kn with the orresponding deletions 6.4 mm and 7.4 mm respetiel. Also, or the beams o depth 600 mm with SRI o 0.0 and 0.8 the ultimate loads were 340 kn and 460 kn and the deletions at the peak loads were 8.5 mm and 9.8 mm respetiel. The omparison o these trends shows that the inrease in the deletion at the peak load with the orresponding inrease o SRI or 200 mm depth beams seems to be higher. This learl indiates that the stirrups are more eetie in resisting the shear ore in the members with smaller depths than the beams o larger depths. On the ontrar the ailures o the large size beams are more brittle than the small size beams. This phenomenon hanges the partiipation o stirrup or dierent depths, whih needs serious ealuation. Figs. 9 to 11 reeal that the minimum quantit o reinorement speiied b odes ma be suiient or small size beams as adequate dutilit has been ahieed. Howeer, it is inadequate or the large size beams (depth o 600 mm) made o HSC beause the web reinorement snapped ater reahing the ultimate load or r* = 0.4 and 0.6 or beams o depth 600 mm. Fig10. Load s. Deletion in beam HSC-400. Fig11. Load s. Deletion in beam HSC-600. Volume: 02 Issue: 10 Ot-2013, 30

7 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: Modes o Failure O all the twele beams tested, three o them were without web reinorement ailed in shear with a er distint diagonal rak orming on one side o the beam. The rak widths, rak pattern and also how the raks were propagated with the appliation o the load was monitored and also reorded. The raks were ormed smmetriall on either side o the beam with urther loading but the inal ailure was due to one major diagonal rak widening up. The raks in all the beams extended rom the load point to the adjoining support H-0.0 Series The mode o ailure or all the three depths o beams without web reinorement (SRI equal to 0.0) is shown in Fig. 12. The beams deeloped seondar raks along the longitudinal reinorement due to deterioration o the bond and the ailure was due to the shear-tension ailure. Sine suiient end anhorage was designed, there was no anhorage ailure obsered. The beams o depth 400 mm and 600 mm depth were proided with longitudinal reinorement in two laers. The ailure pattern in H0.0 beams is shown in Fig kn 40 kn 40 kn 180 kn 90 kn 90 kn 260 kn Fig13. Crak Proile or H-0.6 Series beams at Ultimate Stress leel H-0.6 Series The beam H60 o H-0.6 series exhibited loal rushing o onrete in the iinit o the load point due to the ombined ation o the shear and ompression. Spalling o onrete took plae in addition to the diagonal raking prior to the ailure. The beams o depth 200 mm and 600 mm deeloped the splitting raks along the longitudinal reinorement. Howeer, the H40 beams ailed in shear-ompression mode without an seondar raking along the longitudinal reinorement. The ailure patterns or this series are shown in Fig H-0.8 Series 150 kn 75kN k 139 kn 139kN 420 kn All the beams in this series ailed in shear-tension mode aompanied b loal rushing and spalling o onrete. 130 kn 130 kn Fig12. Crak Proile or H-0.0 Series beams at 1.7 MPa Stress leel H-0.4 Series The beams H-0.4 had a web reinorement orresponding to SRI 0.4 MPa. The beams with 200 mm depth exhibited a dutile behaiour, whereas the beams o 400 mm depth exhibited a er sudden ailure beore swithing on rom the load ontrol to the displaement ontrol sstem during testing. All the stirrups aross the diagonal rak passing through it was snapped. Though the ailure o all the beams in this series was also due to shear-tension ailure but onl a single seondar rak along the longitudinal reinorement was obsered unlike in the ase o beams without web reinorement where two distint seondar raks were obsered. CONCLUSIONS 1. The minimum shear reinorement speiied b IS and BS ode are independent o the ompressie strength o onrete, and inadequate or HSC beams. Howeer, ACI, CSA and AASHTO proisions represent as a untion o ompressie strength o onrete but are independent o ratio and ρ l. 2. The model proposed in this stud predits reasonabl well the minimum web reinorement or all tpe o beams based on the ompressie strength o onrete, ratio and ρ l. 3. From the load deletion response or the beams tested with SRI = 0.4 MPa and rom the nature o ailure obsered, it has been ound that the proision o SRI o 0.4 MPa is inadequate. 4. For ariation in SRI rom 0.0 to 0.4, there is no signiiant enhanement o shear strength or all depths o beams thus indiating that the minimum web reinorement proposed b the IS and BS odes o pratie need to be re-ealuated. Volume: 02 Issue: 10 Ot-2013, 31

8 IJRET: International Journal o Researh in Engineering and Tehnolog eissn: pissn: Based on the load-deletion response it has been obsered that with inrease in SRI (r ) the shear dutilit inreases onl up to a ertain ritial limit o 0.6. Thereore, or a gien strength o onrete inreasing the web reinorement beond the ritial limit does not improe the dutilit signiiantl. 6. The energ absorption ratio is not enhaned signiiantl b inreasing SRI rom 0.0 to 0.4 or the beams o 400mm and 600mm depths ompared to the beams o 200mm depth. Partiipation o shear reinorement is more eetie in small size beams ompared to the large size beams. REFERENCES [1] Hung Lin and Feng-Sheng Lee, Dutilit o High Perormane Conrete Beams with High Strength Lateral Reinorement, ACI Strutural Journal, V. 98, No.4, Jul-Aug. 2001, pp [2] Lin and Lee, Shear Behaiour o High Workabilit Conrete Beams, ACI Strutural Journal, V. 100, No.5, Sept.-Ot. 2003, pp [3] Yuliang Xie., Shuaib. H. Ahmad., Tiejun, Yu., and Chung. W., Shear Dutilit o RC Beams o Normal and High Strength Conrete ACI Journal, V. 91 No.2, 1994, pp [4] Johnson, M.K., and Ramirez, J. A., Minimum Shear Reinorement in Beams with HSC, ACI Strutural Journal, V. 86 No.4, Jul-August 1989, pp [5] Robert Frosh, Behaiour o large Sale Reinored Conrete Beams with Minimum Sear Reinorement, ACI Strutural Journal, V. 97 No.6, No.-De. 2000, pp [6] Yoon, Y.S., Cook, W D., and Mithell, D., Minimum Shear Reinorement in Normal Medium and High Strength Conrete Beams, ACI Strutural Journal, V. 93 No.5, Sept.-Ot. 1996, pp [7] Canadian Standards Assoiation, Rexdale., Design o Con. Strutures or Buildings 1984 [8] Roller, J.J., and Russell, H. G., Shear Strength o HSC Beams with Web Reinorement, ACI Strutural Journal, V. 87 No.2, Marh-April 1990, pp [9] Ozebe, G., and Erso, U., and Tankut, T., Ealuation o Minimum Shear Reinorement Requirements or HSC, ACI Journal, V. 96 No.3, Ma-June 1999, pp [10] ACI Committee 318, Building Code requirements or Strutural Conrete (ACI ), Amerian Conrete institute, Mihigan. [11] Agelakos, D., Bentz, E.C., and Collins, M.P., Eet o onrete strength and Minimum Stirrups on Shear Strength o Large Members ACI Strutural Journal, V.98, No.3, Ma-June 2001, pp [12] AASHTO LRFD Bridge design Speiiations and Commentar, 2 nd edn., Amerian Assoiation o State Highwa Transportation Oiial, Washington DC., 2000 [13] IS 456: 2000, Indian Standard, Code o Pratie or Plain and Reinored Conrete, Fourth Reision, Bureau o Indian Standards, New Delhi. [14] BS 8110: Part 1:1985, Code o Pratie or Design and Constrution, British Standards [15] Injaganeri, S S., Studies on Size Eet on Design Strength and Dutilit o Reinored Conrete Beams in Shear, Ph. D. Thesis, IIT Madras. [16] Shah, S P., and Ahmad, S H., High Perormane Conrete: Properties and Appliations, MGraw-Hill In., Nework. Notation a = shear span = shear span-to-depth ratio A s = area o tension reinorement A s = Area o shear reinorement b = breadth o beam d = eetie depth in mm = lindrial ompressie strength in MPa = ield strength o longitudinal steel s = ield strength o shear reinorement ρ l = longitudinal reinorement ratio r = A s / bs S = Spaing o stirrups r = diagonal raking strength u = ultimate shear strength V ult = atored shear ore at the ritial setion Volume: 02 Issue: 10 Ot-2013, 32