STATISTICAL MODEL FOR THE PREDICTION OF SHEAR STRENGTH OF HIGH STRENGTH REINFORCED CONCRETE BEAMS

STATISTICAL MODEL FOR THE PREDICTION OF SHEAR STRENGTH OF HIGH STRENGTH REINFORCED CONCRETE BEAMS Attaullah Shah* Allama Iqbal Open University Islamabad Pakistan Saeed Ahmad Department of Civil Engineering, University of Engineering & Tehnology Pakistan E-mail: dr_sahmad@yahoo.om. INTRODUCTION Shear strength of reinfored onrete (RC) is determined with the help of ertain empirial equations based on experimental results from the normal strength reinfored onrete (NSRC) beams. Aording to Amerian Conrete Institute Building Code 38 [], the shear strength of onrete members without transverse reinforement subjet to shear and flexure is given by the following equation: V d V = f + b d () u (.6 7 ρ ) w M u where f = 28 days ompressive strength of onrete, ρ = longitudinal reinforement in the web, V u = fatored shear fore at the setion, d = effetive depth of beam and b w = web thikness. Vud V < 3.5 b w d and M <. in omputing V by Equation (), where M u is fatored moment ourring simultaneously u with V u at setion onsidered. * Corresponding author: E-mail: pdaiou@yahoo.om Paper Reeived July 8, 27; Paper Rewritten February 9, 29; Paper Aepted Marh 27, 29 Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 399

Based on the experimental researh on high strength onrete beams, Maphonde and Frantz [2], Sarsam and Al- Musawi [3], and Ahmad et al. [4] have shown that Equation () overestimates the effet of the ompressive strength of onrete and underestimates the effet of shear span to depth ratio on the shear strength of HSRC beams. Hene, Equation () is mostly valid for NSRC beams. Due to lak of test data on ompressive strengths of onrete greater than 7 MPa (, psi), the 989 edition of the ACI ode imposed a maximum value of.7 MPa (psi), for use in the alulation of shear strength of onrete beams, joists, and slabs. Exeptions to this limit were permitted in beams and joists when the transverse reinforement satisfied an inreased value for the minimum amount of web reinforement. Vehhio and Collins [5] developed the Compression Field Theory (CFT) to study the effet of tensile stresses on the shear strength of RC beams in the raked region. The nominal shear apaity V n of the reinfored onrete setion is given as: Vn = Vs + V p V = shear strength provided by the raked onrete V s = shear strength provided tensile stress in stirrups V p = vertial omponent of applied pre-stressed tendons w w v v p V n = β f b d + A f Cotθ + V (3) (2) β = onrete tensile stress fator indiating the ability of diagonally raked onrete to resist shear d w.9 d = the minimum web depth Rameriz and Breen [6] suggested the following model for nominal shear strength of onrete beams without web reinforement: V =.5(3 V θ ) b d (4) n r w V r = shear stress resulting in the first diagonal tension raking in the onrete θ = rak angle in radians Gambarova [7] and Dei Poli et al. [8] developed the approah of the truss model, whih is based on the assumption that the fores are transferred aross the rak by the frition whih depends on the rak displaement (slip and rak width). They proposed the following equation for the ontribution of web reinforement in resisting the shear in RC beams: V s Av f ycotβ r = (5) s where β r = rak inlination d v = inner lever arm s = stirrup spaing Karim et al. [9] proposed the following equation for prediation of ultimate shear stress in beams without web reinforement. V u f ( ρd) ν = =.4 + ( 3 A d ) ( SI Units) (6) bd a where A d = a d for. < a d <2.5 and 2.5 for a d > 2.5 Zararis [] has proposed the following models for the shear strength of beams without web reinforement: a V r =.2.2( ) d f t bd d d (7) where.2.2 a d.65 d (d in meters) f t =.3 ( ) 2 f 3 is depth of ompression zone whih is determined by the following quadrati equation: 4 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

2 6( ρ + ρ ) 6 d ρ ρ + + = d f d f d (SI Units) (8) For beams with shear reinforement, the steel ontribution is added, whih is expressed as a V s=.5 +.25 ρ y f vybd d (SI units) (9) The shear strength of HSRC beams is still determined by using the equations of Normal Strength Reinfored Conrete (NSRC) beams by most of the odes. However, various researhers have proposed ertain empirial equations for shear strength of HSRC beams on the basis of test results and mathematial models. Sarkar et al. [] worked on reinfored onrete beams with ompressive strength ranging from 4 MPa to MPa. They proposed the following two equations for the shear stress of reinfored onrete beams without web reinforement. For beams having a/d 2 ν = 4.3 ( f. ρ. d).66 For beams having a/d > 2 ν 3.5 ( f. ρ. d).55 = (SI Units) () Bazant and Kim [2] proposed a very reliable expression for omputing the shear strength of RC beams without transverse reinforement, whih is given as /3 / 2 5/6 f ν = ζ.83ρ + 26.9ρ uv a d 5/2 (SI Units) () d Where ζ = / a + is a funtion taking into aount the size effet of aggregates, and where d a stands for 25d a aggregates sizes. Russo et al. [3] proposed the following expression for shear strength of HSRC onrete beams without transverse reinforement based on Bazant and Kim s equation: ν ζ ρ ρ.46 / 2.9.38.95 uv =.97 f +.2 f f yl a d 2.33 (2) They further proposed the following expression for the shear strength of HSRC beams with transverse reinforement using the above expression: ν ζ ρ ρ ρ d 2.33.46 / 2.9.38.95.97.2 a uv = f + f f yl +.75Ib v f yv (3) The fator I b is given by the equation I b.46 / 2 f.97ρ =.46 / 2.9.38.95 a.97ρ f +.2ρ f f yl d 2.33 (4) To hek whether the shear failure is due to beam ation or arh ation, the author further proposed a ritial value as ρ / =.57 f ( a d ).9 / 2 f yl.5 (SI units) (5) Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 4

Hene I b =.57, whih means that for i). a/d < (a/d) I b <.57, arh ation prevails ii). a/d > (a/d) I b >.57,beam ation prevails Cladera and Mari [4,5] proposed the following equations for the shear strength of beams without web reinforement /2.2 V =.225 ζ( ρ) f bwd (SI Units) (6) For beams with web reinforement, the shear strength is given as V /2.2 /3 =.7 ξ( ρ) f τ b w d (7) V s dvaw =. (8) sf ( dot θ ) yw In the present researh, seventy high-strength onrete-reinfored (HSRC) beams with and without web shear have been tested under monotoni load at mid span. Based on the test results, two regression equations have been proposed for prediting the shear strength of HSRC beams. The results have been ompared with the existing models proposed by different researhers. 2. DETAILS OF MATERIAL AND TEST SAMPLES To study the behavior of high strength onrete beams in shear, with and without shear reinforement, seventy beams in two series of thirty-five beams eah of size 23m x 3 m (9 in x 2 in) were prepared. Seven values of shear span to depth (a/d) ratios were used to study mainly the slender beams (shear span to depth ratio a/d were taken as 3, 3.5, 4, 4.5, 5, 5.5, 6). For eah value of a/d, five types of longitudinal steel ratios were used (ρ =.33,.75,.,.5,.2) to study the effet of longitudinal steel ratios on the shear strength of HSRC beams. For series-i, thirty-five beams were used without transverse reinforement, whereas in series-ii, thirty-five beams having shear reinforement with #2 bars @ 6 / were used, whih orresponds to minimum shear reinforement as per ACI-38 ode provisions. 2.. Material 2... Reinforing steel For main reinforement, deformed steel bars having nominal yield stress of 44 MPa (6, psi) have been used. For shear reinforement, plain steel bars of yield stress 276 MPa (4, psi) were used. 2..2. Conrete The onrete mix design of the beams used in this experimental program has been given in Table. Coarse aggregates of size ¾ in (2mm) and fine aggregates onforming to ASTM standards with modulus of fineness as 2.67 were used in the onrete. High range water reduers onforming to ASTM C-494 type F standards was used at.7 % by weight of ement to ontrol the water ement ratio and enhane the ompressive strength of onrete. The details of beam sizes, main reinforement, shear reinforement, and a/d ratios are shown in Table 2 and a typial setion of the beams is given in Figure. The loading arrangements are shown in Figure 2. Table. Mix Proportioning/ Designing of High Strength Conrete Constituent Proportion Type- I Cement 628 kg/m 3 Fine aggregates 484 kg/m 3 Coarse aggregates 28 kg/m 3 HRWR @ by weight of ement.7 kg/m 3 Water @.25 w/ ratio 57 kg/m 3 Average Design Cylinder Compressive strength 5-54 MPa ( 28 days) f 42 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

3m 23m 23m Figure. Typial setion of beams without and with stirrups Proving Ring Hydrauli Loading Shear span Span Figure 2. Typial loading arrangement for testing of beams Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 43

Beam Table 2. Details of Beams With and Without Shear Reinforement Beams without stirrups Series-I ρ ( %) a/d Span ( m) Beam ρ (%) Beams with stirrups Series-II a/d Span ( m) Stirrups ρ v (%) B-.33 3. 52.4 Bs-.33 3. 52.4. B-2.33 3.5 77.8 Bs-2.33 3.5 77.8. B-3.33 4. 23.2 Bs-3.33 4. 23.2. B-4.33 4.5 228.6 Bs-4.33 4.5 228.6. B-5.33 5. 254. Bs-5.33 5. 254.. B-6.33 5.5 279.4 Bs-6.33 5.5 279.4. B-7.33 6. 34.8 Bs-7.33 6. 34.8. B2-.73 3. 52.4 Bs2-.73 3. 52.4. B2-2.73 3.5 77.8 Bs2-2.73 3.5 77.8. B2-3.73 4. 23.2 Bs2-3.73 4. 23.2. B2-4.73 4.5 228.6 Bs2-4.73 4.5 228.6. B2-5.73 5. 254. Bs2-5.73 5. 254.. B2-6.73 5.5 279.4 Bs2-6.73 5.5 279.4. B2-7.73 6. 34.8 Bs2-7.73 6. 34.8. B3-. 3. 52.4 Bs3-. 3. 52.4. B3-2. 3.5 77.8 Bs3-2. 3.5 77.8. B3-3. 4. 23.2 Bs3-3. 4. 23.2. B3-4. 4.5 228.6 Bs3-4. 4.5 228.6. B3-5. 5. 254. Bs3-5. 5. 254.. B3-6. 5.5 279.4 B3-6. 5.5 279.4. B3-7. 6. 34.8 Bs3-7. 6. 34.8. B4-.5 3. 52.4 Bs4-.5 3. 52.4. B4-2.5 3.5 77.8 B4-2.5 3.5 77.8. B4-3.5 4. 23.2 Bs4-3.5 4. 23.2. B4-4.5 4.5 228.6 Bs4-4.5 4.5 228.6. B4-5.5 5. 254. Bs4-5.5 5. 254.. B4-6.5 5.5 279.4 Bs4-6.5 5.5 279.4. B4-7.5 6. 34.8 Bs4-7.5 6. 34.8. B5-2. 3. 52.4 Bs5-2. 3. 52.4. B5-2 2. 3.5 77.8 Bs5-2 2. 3.5 77.8. B5-3 2. 4. 23.2 Bs5-3 2. 4. 23.2. B5-4 2. 4.5 228.6 Bs5-4 2. 4.5 228.6. B5-5 2. 5. 254. Bs5-5 2. 5. 254.. B5-6 2. 5.5 279.4 Bs5-6 2. 5.5 279.4. B5-7 2. 6. 34.8 Bs5-7 2. 6. 34.8. 44 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

3. TESTING AND OBSERVATIONS The beams were tested under monotoni onentrated load at the mid span. The shemati testing arrangements of the beams is shown in Figure 2. The beams were ured with moist sand in the uring yard as shown in Figure 3. When the load was applied and inreased gradually, flexural raks appeared in the beams near the mid span of the beams, whih were more or less vertial in nature. With further inrease of load, inlined shear raks developed in the beams, whih are sometimes alled primary shear raks as well. Typial raking in the slender beams without transverse reinforement leading to the failure involved two branhes. The first branh was a slightly inlined shear rak and was typially of the height of the other flexural raks developed on the surfae of beams. The seond branh of the rak, also alled seondary shear rak, initiated from the tip of the first rak at a relatively flatter angle, splitting the onrete in the ompression zone. This rak further extended in the ompression zone and finally met the loading point, leading to the ollapse of the beam as shown in Figure 4. The nominal shear strength at the initiation of the seond branh rak was taken as the shear apaity of the beams. In ase of beams without transverse reinforement, the seondary shear rak formed shortly after the development of the primary shear rak and the shear failure was sudden, as shown in Figure 5 In the ase of beams with transverse reinforement, the formation of seondary shear rak was not abrupt and beams arried more loads before failure as ompared to beams without web reinforement. In both ases, the shear strength of the beams was taken at the point when seondary shear raks appeared.. Figure 3. Beams are staked in the testing yard after asting and uring Figure 4. The formation of seondary raks originating from the primary raks leading to failure of beams without web reinforing Figure 5. Typial shear failure of beam without web reinforement. The failure is more brittle and sudden. Brik has been plaed under the beam to avoid its splitting in two parts (ρ =.5 %, a/d = 5.5 span = 32 m). Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 45

4. REGRESSION MODEL FOR SHEAR STRENGTH OF BEAMS WITHOUT WEB REINFORCEMENT The shear strength of HSRC beams was investigated as a funtion of longitudinal steel ρ expressed in perentage (%), shear span to depth ratio ( a/d), ompressive strength of onrete, and f ( Mpa) for beams without shear reinforement. The Ordinary Least Square (OLS) estimate for the linear regression model is given as V =.26f +.57ρ.28( a/ d) bd (9) The atual and predited values of shear stress based on the linear regression model have been ompared in Figure 6 and Figure 7. 5. LINEAR REGRESSION MODEL FOR SHEAR STRENGTH OF BEAMS WITH WEB REINFORCEMENT Aording to ACI-38, the shear ontribution of stirrups in beams with web reinforement is given as Av f y V s = d (2) s Aording to ACI-38, the nominal shear strength of RC beams with web reinforement is the sum of the individual ontributions of onrete and steel. V = V + V = ( ν + ν ) b d (2) n s s v However, researh by Sarsam and Al-Musawi [6] and P. Regan [7] has revealed that the behavior of stirrups in resisting the shear of RC beams is more ompliated and irregular. The atual results of 35 beams with web reinforement have also shown that the inrease in shear strength due to stirrups exhibits a non-linear and nonuniform behavior in resisting the shear. The linear regression model o was worked out for the data of 35 beams tested in the researh and the following equation was obtained: V =.f +.57ρ.28( a/ d) + 4.53ρv f bd y The atual and predited values by the models have been ompared in Figure 8 and Figure 9. ] (22) 6. COMPARISON OF THE PROPOSED MODELS WITH ACI-38 CODE AND OTHER MODELS 6.. Beams without Shear Reinforement The shear strength of beams without shear reinforement was ompared with the ACI equations [] and other models proposed by Bezant and Kim [2] and G. Russo.et.al. [3]. The omparison has been given in Figure. 6.2. Beams with Shear Reinforement The predited results of shear strength for beams with shear reinforement has been ompared with the results of ACI Equation [] and model proposed by G. Russo et al. [3]. The omparison has been given in Figure. 46 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

.9.8.7.6.5.4.3.2. ρ=.33%.4.2 ρ=.75%.8.6.4.2.4.2.8.6.4.2 ρ=% Figure 6. Comparison of atual and predited values of shear stress in beams without web reinforement for ρ<% Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 47

2.8.6.4.2.8.6.4.2 ρ=.5% 2.8.6.4.2.8.6.4.2 ρ=2% Figure 7. Comparison of atual and predited values of shear stress in beams without web reinforement for ρ>% 48 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

.4.2 ρ=.33%.8.6.4.2.8.6 ρ=.75%.4.2.8.6.4.2.8.6.4.2.8.6.4.2 ρ=% Figure 8. Comparison of atual and predited values of shear stress in beams with web reinforement for ρ<% Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 49

2.5 2.5.5 ρ=.5% 2.5 ρ=2% 2.5.5 Figure 9. Comparison of atual and predited values of shear stress in beams with web reinforement for ρ>% 4 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

Shear stress of beam (MPa).4.2.8.6.4.2 ρ= % a 3 3.5 4. 4.5 5. 5.5 6 shear span to depth ratio a/d ACI Bazant Russo Shear stress of beam (MPa) 2.8.6.4.2.8.6.4.2 ρ=.5% 3 3.5 4. 4.5 5. 5.5 6 shear span to depth ratio a/d ACI Bazant Russo Shear stress of beam (Mpa) 2.8.6.4.2.8.6.4.2 3 3.5 4. 4.5 5. 5.5 6 ρ= 2% ACI Bazant Russo shear span to depth ratio a/d Figure. Comparison of atual shear stress of beams having no shear reinforement with the proposed equation and other models for ρ>% Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 4

Shear strength of beam (Mpa).8.6.4.2.8.6.4.2 ρ= % 3 3.5 4. 4.5 5. 5.5 6 shear span to depth ratio a/d ACI Russo Shear strength of beam (MPa) 2.5 2.5.5 3 3.5 4. 4.5 5. 5.5 6 shear span to depth ratio a/d ρ=.5% ACI Russo 2.5 ρ= 2% Shear strength of beam (Mpa) 2.5.5 ACI Russo 3 3.5 4. 4.5 5. 5.5 6 shear span to depth ratio a/d Figure. Comparison of atual shear stress of beams having shear reinforement with the proposed equation and other models for ρ>% 42 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29

7. CONCLUSION AND RECOMMENDATIONS From omparison of the atual and predited values of shear stress of beams, the following results have been observed: i. For beams without shear reinforement, the proposed equation is onservatives for ρ =.75% and % and un-onservative for ρ =.33%. For ρ=2%, the equation gives loser values to the atual observations. ii. For beams with shear reinforement, the proposed regression equation is onservative for ρ =.33%,.75% and un-onservative for ρ =%. iii. For ρ=.5% and 2%, the equation gives loser values to the atual. From omparison of the predited values of shear stress and values proposed by ACI, the Russo et al. equation, and the Bazant et al. equation, the following results have been observed: i. The Bazant et al. equation is un-onservative in estimating the shear stress for the HSRC beams without web reinforement as it overestimates the shear stress for all values of longitudinal steel. ii. The Russo et al. equation is more onservative as it underestimates the shear stress of the HSRC beams without web reinforement. iii. The ACI-38 equation for shear stress of HSRC beams gives some reasonable values when ompared with the atual and predited values. iv. The Russo et al. equation, on the other hand, is un-onservative for shear stress of HSRC beams with web reinforement. v. The proposed regression equation better estimates the shear stress of beams as ompared to the other models of ACI, Russo and Bazant. However, this may be due to the fat that the equations are based on the data of the tests results. Hene, for generalization of the regression equation to other sets of tests data, more experimental researh is required. ACKNOWLEDGMENTS The authors are highly grateful to the staff of the Civil Engineering Department, the Strutural and Conrete Laboratories of UET Taxila, Pakistan, and the Higher Eduation Commission Pakistan for their assistane and funding of the work under this faulty researh projet, jointly finaned by Engineering University, Taxila, Pakistan and the Higher Eduation Commission (HEC). REFERENCES [] ACI Committee 38, Building Code Requirements for Reinfored Conrete (ACI 38-6) and Commentary-ACI38RM-6, Amerian Conrete Institute, Detroit, 26. [2] G. Maphonde and G. C. Frantz, Shear Tests of High and Low Strength Conrete Beams Without Stirrups, ACI Journal Proeedings, 8(4) (984), pp. 35 357. [3] K. F. Sarsam and J. M. S. Al-Musawi, Shear Design of High and Normal Strength Conrete Beams With Web Reinforement ACI Strutural Journal, 89(6)(992), pp 658 664. [4] S. H Ahmad, A. R. Khallo, and A. Poveda, Shear Capaity of Reinfored High Strength Conrete Beams, ACI Strutural Journal, 82(2)(986), pp. 297 35. [5] F. J. Vehio and M. P. Collins, The Modified Compression Field Theory for Reinfored Conrete Elements Subjeted to Shear, ACI Strutural Journal, 83(2)(986), pp 29 23. [6] J. A. Ramirez and J. E. Breen, Evaluation of Modified Truss Model Approah in Shear, ACI Strutural Journal, 88(5)(99), pp 562 57. [7] P. G. Gambaorva, On Aggregate Interloking Mehanism in Reinfored Conrete Plates With Exessive Craking, IASBE olloquium Zurih, 25(2)(988), pp 5 34. [8] S. Dei Poli, M. D. Priso, and P. G. Gambarova, Stress field in Web RCC Thin Web Beams Failing in Shear Journal of Struture Engg ASCE, 6(9)(99), pp 2496 255. [9] S. R. Karim, F. Javier, and M. Frabizzio, New Shear Predition for Conrete Members Using Statistial and Interpolation Funtion Tehniques, 8th ASCE Speialty Conferene on Probabilisti Mehanis and Strutural Reliability, University of Notre Dame, July 2. Otober 29 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B 43

[] P. D. Zararis, Shear Strength and Minimum Shear Reinforement of Reinfored Conrete Slender Beams, ACI Strutural Journal, (2)(23), pp. 23 25. [] S. Sarkar, O. Adwan, and B. Bose, Shear Stress Contribution and Failure Mehanisms of High Strength Conrete Beams, Material and Strutures-RILEM, 32(999), pp. 2 6. [2] Z. P. Bazant and J. K. Kim, The Size Effet in Shear Failure of Longitudinally Reinfored Beams, ACI Struture Journal, 8(5)(984), pp. 456 468. [3] G. Russo, G. Somma and P. Angeli, Design Shear Strength Formula for High Strength Conrete Beams, Journal of Material and Strutures, (37)(24), pp. 59 527. [4] A. Cladera and A. R. Mari, Shear Design Proedure for Reinfored Conrete Beams Using Artifiial Neural Network, Engineering Strutures, 26(7)(24), pp. 927 936. [5] A. Cladera and A. R. Mari, Experimental Study of High Strength Conrete Beams Failing in Shear, Engineering Strutures, 27(25), pp. 59 527. [6] K. F. Sarsam and J. M. S. Al-Musawi, Shear Design of High and Normal Strength Conrete Beams With Web Reinforement, ACI Journal of Conrete, 89(6)(992), pp. 659 664. [7] P. Regan, Researh on Shear: A Benefit to Humanity or a Waste of Time?, The Strutural Engineer, 7(9)(993), pp. 337 347. 44 The Arabian Journal for Siene and Engineering, Volume 34, Number 2B Otober 29