FORCE TRANSFER MECHANISMS AND SHEAR STRENGTH OF REINFORCED CONCRETE BEAM-COLUMN ELEMENTS

Transcription

1 4th International Conference on Earthquake Engineering Taipei, Taiwan October 12-13, 2006 Paper No. 117 FORCE TRANSFER MECHANISMS AND SHEAR STRENGTH OF REINFORCED CONCRETE BEAM-COLUMN ELEMENTS Wu-Wei Kuo 1, Shyh-Jiann Hwang 2, and Ying-Zhi Chen 3 ABSTRACT Thi paper tudie the force tranferring mechanim and hear trength of reinforced concrete (RC) beam. The tranferring mechanim of hear force in RC beam can be characterized a the load path in the diturbed region and the beam region. The domain of different region are propoed to be determined by the beam geometry, the longitudinal and tranvere reinforcement. The inclination angle in each region are defined according to the propoed load path of the force tranferring. The diturbed region are failed due to hear compreion failure and hear tenion failure. The failure mode of the beam region i hear tenion failure. The hear tenion failure i the yielding of hear reinforcement and the hear compreion failure i the diagonal cruhing of concrete. The hear trength due to the hear compreion failure can be predicted by the oftened trut-and-tie model, and the hear tenion failure can be calculated according to the ACI code approach. Accuracy of the propoed model i gauged by comparing with the available tet data. Finally, the propoed model of beam i applied to column. Keyword: Dicontinuitie, Force tranfer, Reinforced concrete, Shear, Strength INTRODUCTION Variou behavioral model have been propoed for predicting the hear trength of RC beam. The tru model, originally introduced by Ritter (1899) and Mörch (1909) at the turn of the 19 th century, ha been provided an excellent bai for deign of hear teel. The tructural action can be repreented by a tru, with the main teel providing the tenion chord, the concrete top flange acting a the compreion chord, the tirrup providing the vertical tenion web member, and the concrete between inclined crack acting a compreion diagonal. Furthermore, the tru analogy method ha been greatly extended by the recent work of Schlaich et al. (1987) and alo wa contained in the text by Collin and Mitchell (1991) and MacGregor (1997), and referred a trut-and-tie model. In a RC tructure, the member i claified a either B- (Beam or Bernoulli) Region or D- (Diturbed or Dicontinuity) Region. B-Region are part of a tructure in which Bernoulli' hypothei of traightline train profile applie. D-Region, on the other hand, are part of a tructure with a complex variation in train. D-Region include portion near abrupt change in geometry (geometrical dicontinuitie) or concentrated force (tatic dicontinuitie). St. Venant' principle ugget that a local diturbance uch a a concentrated load will diipate within about one beam depth from the applied point, a hown in Fig. 1. Alo thi concept i included in ACI Appendix A (2005). However, St. Venant principle i only applied in elatic material without cracking. In fact, the 1 PhD Candidate, National Taiwan Univerity of Science and Technology, Taipei, Taiwan, D @mail.ntut.edu.tw 2 Profeor, National Taiwan Univerity, Taipei, Taiwan; Diviion Head, National Center for Reearch on Earthquake Engineering, Taipei, Taiwan, jhwang@mail.ntut.edu.tw 3 Mater, National Taiwan Univerity of Science and Technology, Taipei, Taiwan

2 reinforcement become active and reditribution of internal tree occur after concrete crack. Under thi circumtance, the diviion between B- and D-region hould not follow the imple rule of one ection depth from the dicontinuity. It hould be a function of the concrete trength, main reinforcement, hear reinforcement,... etc. If the diviion between B- and D-region can be defined clearly, more accuracy can be obtained in predicting the behavior for RC beam and column. Figure 1. D-region and dicontinuitie (ACI Committee 318, 2005) In thi paper the force tranferring mechanim and hear trength of reinforced concrete beam are propoed baed on the work of MacGregor (1997). Alo, an analytical model for hear trength prediction i developed, followed by experimental verification. Finally the propoed model of beam i applied to column in order to demontrate the capability and reliability of propoed method. THE SHEAR TRANSFERRING MECHANISMS AND FAILURE MODES OF BEAMS Review of ACI Code Proviion According to Commentary of ACI Appendix A (2005), D-region include the portion of a member within a ditance, h, from a force dicontinuity or a geometry dicontinuity, a hown in Fig. 1. If two D-region overlap or meet a hown in Fig. 2, they can be conidered a a ingle D-region for deign purpoe. The maximum length-to-depth ratio of uch a D-region would be approximately two. Thu, the mallet angle between the trut and the tie in a D-region i arctan 1/2= 26.5 degree, rounded to 25 degree. The trut-and-tie method i emerging a a code-worthy methodology for the deign of D-Region in tructural concrete. If there i a B-region between the D-region in a hear pan, a hown in Fig. 2, the trength of the hear pan i governed by the trength of the B-region if the B- and D-region have imilar geometry and reinforcement (ACI Committee 318, 2005). Thi i becaue the hear trength of a B-region i le than the hear trength of a comparable D-region. Shear pan containing B-region are deigned for hear uing the traditional hear deign procedure ignoring D-region. Propoed Method The tru analogy wa propoing independently by Ritter (1899) and Mörch (1909) in the early 1900 for hear deign of RC beam. In 1997, MacGregor uggeted that the tru model i one of the bet analytical model for hear deign. The contruction of a platic tru model can be illutrated by example in Fig. 3. The vertical applied load, V, mut be tranmitted by diagonal compreion trut (hown by dahed inclined line) to enough tirrup (hown by olid vertical line) to equilibrate thi force. The tirrup tranmit the vertical force to the top joint of the tru, where it i reiting by the vertical component of the force in diagonal trut, and o on. The compreion diagonal originating at the load are referred to a a compreion fan. It can be found in Fig. 3 that each of the compreion fan occur in region of concentrated load a well a in both end upport. Between the compreion fan i a compreion field coniting of the parallel diagonal

3 trut. The compreion fan occur in a D-region, wherea the compreion field i a B-region. For the D-region where the flow of tree i non-uniform, the trut-and-tie model (MacGregor, 1997; Collin and Mitchell, 1991) may be ued to determine internal force effect. And for the B-region the meared concrete model may be ued becaue of tate of uniform tree (Hu and Mo, 1985). The train can meet the requirement of Mohr compatibility in both D- and B-region. The oftening phenomenon of concrete i alo taken into account by the law of contitutive. Strut D-region h a v =2h a v =2h Figure 2. Decription of deep and lender beam (ACI Committee 318, 2005) Figure 3. Contruct of trut and tie (MacGregor, 1997) However, not all the flow of force in a cracked concrete beam go through D-, B-, then D-region a mentioning above. It i believed that a force flow of a ingle D-region i poible in a deep beam with a ratio of hear pan to effective depth ( a / jd ) of 2.0 or le (Fig. 4). The length jd i the lever arm from tenile reinforcement to the center of compreion tre. For beam with ratio of a / jd more than 2.0, the path tranferring load to the upport pae through both B- and D-region, a hown in Fig. 4. To account for the influence of the teel reinforcement location, the ratio of hear pan to effective depth ( a / jd ) i ued intead of a / d in thi paper. a a jd jd l h l h Figure 4. Load path of deep beam and ordinary beam For a deep beam with ratio of a / jd of 2.0 or le, the trut-and-tie model i ued to evaluate the hear trength and failure mode becaue of the non-uniform tree tate all over the beam. In thi paper, the oftened trut-and-tie model developed by Hwang et al. (2000, 2002) i adopted to account for the force tranferring mechanim in D-region. The hear failure of a ingle D-region i et to a cruhing of

4 concrete in a diagonal compreion trut, and it i referred to a hear compreion failure ( V DC ) in thi paper. For an ordinary beam with ratio of a / jd more than 2.0, the flow of internal force tranmitted through D-region, via B-region, and then to D-region, intead of D-region only. When force flow from D- region to B-region, the tree tate would change from non-uniform to uniform. The failure mode of the B-region i diagonal tenion failure( V BT )due to yielding of tirrup. The firt cracking angle, α, i 45 degree under uniform tree without axial load. The hear trength can be predicted b y modified tru analogy (ACI Committee 318, 2005). O n either ide of the B-region i the D-region with diturbed tree. The ACI code (2005) define a D-region a a function of a member within a ditance, h, from a force dicontinuity or a geometry dicontinuity. In theory, thi D-region definition work only on linear elatic material baed on St. Venant' principle. For reinforced concrete tructure, a more rational approach to devide B- and D- region will lead to a better prediction. In fact, parameter uch a reinforcement, tirrup, and concrete trength are uppoed to correlate cloely with the extent of the D-region. According to the concept of MacGregor (1997), the vertical applied load, V, i tranmitted by diagonal compreion trut to enough tirrup to equilibrate thi force vertical load (Fig. 3 and 5). The compreion trut and tenion tie are interconnected at node in equilibrium tate (Fig. 5). Therefore, parameter like concrete trength, reinforcement trength and layout...etc will affect the equilibrium at node. That i, the tronger the hear V i applied, the more node are needed to maintain equilibrium tate. The quantitie of node for equilibrium can be determined by Eq. 1. When etimating the value of n in Eq. 1, the value of hear hould be le than V mn and V BT, in which V mn i the hear trength when nominal moment M n reache. The value of V BT i the hear trength with diagonal tenion failure in B-region, alo it i the nominal hear trength V n according to ACI code (2005). The effectivene of hear reinforcement within D-region wa propoed by Hwang and Lee (2002). In Eq. 2, θ tand for angle of inclination of the diagonal compreion with repect to the horizontal axi. A hown in Fig. 5 and 6, the width of D-region can be determined by horizontal ditance between the point of applied load and the center of reultant of all node in D-region. D Region l v jd jd l h Figure 5. Shear tranferring mechanim in D-region Figure 6. D-region θ = lim n = Vv A f (1) vy tan 1 ( jd / l ) (2) In thee equation above, Vlim i the minimal of VBT and Vmn, n i the number of node to maintain equilibrium, A v i area of hear reinforcement, fvy i yi eld trength of hear reinforcement, θ i angle of average principal tre of concrete, l h i the internal lever arm of the hear couple. V vy It can be found in Eq. (1) : if lim get larger or A v f i maller, more tirrup will be needed to reit, then the area of D-region will extend and the θ angle become mild. According to Thürlimann Vlim h

5 (1979),the hear crack in beam are at an angle between 26.5 and 63.5 degree. A a reult, θ, the angle of principal tre of concrete i auming between 26.5 and 63.5 degree, alo thi aumption i imilar to the definition of ACI (2005). The D-region of a deep beam and an ordinary beam are both the area of diturbed tree, but they have different hear tranferring mechanim. In the D-region of a deep beam, a the internal force i tranmitted directly from the point of concentrated load to the upport, the either end of thi concrete trut i tre-concentrated and well-upported. On the other hand, in the B-region of an ordinary beam, the concentrated compreive tree at the point of the concentrated load are tranmitted by ome trut and tie to get equilibrated. If the equilibrium of trut and tie i maintained, and there i enough hear reinforcement to reit internal tree, then the concrete trut may be cruhed before the hear reinforcement yield. The failure mode i hear compreion failure in D-region( V DC ). A the hear load V get higher, the area of D-region will be larger. If more node are needed to equilibrate the hear load V, the angle θ will become maller, and the cracking width will become larger. At thi time, the tiffne of thi region i too week to develop a trut-and-tie force tranferring mechanim. Therefore, the hear reinforcement tend to yield before the concrete trut get cruhed. The failure mode i diagonal tenion failure in D-region( V DT ). The angle of diagonal tenion o cracking i aumed β = 26.5, and the hear trength i etimated by the internal force equilibrated o along the crack. However, the propoed β ( 26.5 ) repreent the main crack angle of diagonal tenion failure in D-region, not the angle of principal tre in D-region. PREDICTION S OF SHEAR STRENGTH Shear Strength for Concrete with Sh ear Compreion Failure in D-region( ) For predicting the hear trength for concrete with hear compreion failure in D-regio n, the oftened trut-and-tie model (Hwang and Lee, 2002) may be ued. In thi model, the complex flow of internal force in the D-Region under conideration i idealized a a tru carrying the impoed loading through the region to it upport. Alo thi model atifie the requirement of compatibility and the oftening phenomenon of concrete. A implified oftened trut-and-tie method, propoed by Hwang and Lee (2002), wa developed for engineering ue. Baed on the implified method, the diagonal concrete compreion C d and hear trength VDC for concrete with hear compreion failure in D- region can be defined: DC Cd inθ = ' c Atr V DC V = Kζf inθ (3) in which K i trut-and-tie index, a factor repreenting the beneficial effect of the tie force on the ' hear trength; ζ i the oftening coefficient; f c i compreive trength of concrete. The effective area of the diagonal trut in D-region i defined a Atr A = a b (4) tr w here a i depth of the diagonal trut; and b i width of the diagonal trut, that i, the width of beam. T he effective depth a are decided by the depth of compreion zone a b and the width of upport plate a p. 2 b 2 p a = a + a (5) in which ab equal to kd, and coefficient k could be referred to Hwang et al. (2000)

6 On the other hand, preent tudy. V DC decreae a the ductility of member increae. Thi doe not include in Shear Strength with Diagonal Tenion Failure in B-region(V ) BT A for the pred iction of V BT,the equation in ACI code (2005) are modified and adopted a follow: V BT Vud bd jd cotα = Vc + V = ( fc + 120ρw ) + Av fvy, f c in MPa (6) M 7 u in which V denote nominal hear trength provided by concrete due to aggregate interlock; and V i c nominal hear trength provided by vertical hear reinforcement; ρ i ratio of tenion reinforcement; V i deign hear force at ection; and M i deign moment at ection; A i area of hear u u reinforcement within a pacing ; f vy i yield trength of hear reinforcement; α i an angle of 45 degree. Shear Strength with Diagonal Tenion Failure in D-region( V DT ) A the force tranferring mechanim mentioned above, tirrup play an important role in nodal equilibrium. Since the angle of inclined crack i not le than 26.5 degree, the diagonal tenion failure could happen if there are not enough tirrup in D-region. The prediction of hear trength i imilar to that of VBT. The only modification i the calculation of V becaue the angle of β i 26.5 degree, rather than 45 degree. w v V DT V DT Vud bd jd cot β = Vc + V = ( fc + 120ρw ) + Av fvy (7) M 7 u EXPERIMENTAL VERIFICATION The etimation of the propoed method are compared with the 206 pecimen teted by ACI-ASCE Committee 326 (1962) coming from Moretto (1945), Clark (1951), Moody et al. (1954, 1955), Rodriguez et al. (1959), Eltn er et al. (1955), a well a other related reference (Mphonde, 1984; Elzanaty et al., 1986; Johnon and Ramirez, 1989; Roller and Ruell, 1990; Saram and Al-Muawi, 1992; Xie et al., 1994; Kong and Rangan, 1998). A lited in Table 1, pecimen of rectangular ection with vertical or inclined hear reinforcement are included. The failure mode of all pecimen mut be either hear compreion failure or diagonal tenion failure. To enure each pecimen being hear failure, the flexure trength i calculated in advanced and compared with hear trength of tet reult. Beide, the trength reduction factor i taken a unity a the dimenion and material propertie of all pecimen are known. Accuracy for the propoed procedure i gauged in term of a trength ratio, which i defined a the ratio of the meaured to the computed trength. The detailed experimental verification are hown in Table 1 and Fig 7. For deep beam with a / jd of 2.0 or le, the oftened trut-and-tie model uggeted by Hwang and Lee (2002) are ued to predict the hear trength (Eq. 3). The average trength ratio for thee member i 1.17, and the coefficient of variation i And for the beam with a / jd more than 2.0, the average trength ratio for thee member i 1.38, and the coefficient of variation i Almot all the longer beam are diagonal tenion failure except two pecimen are hear compreion failure. A hown in Fig. 7, for the cae of a / jd between 2.0 and 3.0, the prediction of the propoed method are more conervative than thoe of a / jd between 3.0 and 5.0.

7 The average trength ratio are 1.52 and 1.23 repectively. According to ACI code (2005), a member with B-region, and the geometrical and reinforcement configuration imilar to D-region, the trength of B-region will govern and become diagonal tenion failure. Thi point of view agree with that from experimental verification. However, the definition of D-region in about one beam depth will caue an incomplete load path within B-region in beam of a / jd between 2.0 and 3.0. It might be conervative to deign uch member uing B-region with concrete diagonal tenion failure according ACI code (2005). A pan of member get longer, a complete B-region can exit in the member, and the prediction and the tet reult tent to agree with each other. On the other hand, defining D-region uing propoed method lead to another reult, a hown in Fig. 8. Table 1 Tet data of 206 pecimen Author No. a / jd f c ρ v V tet /Vcalc. MPa % Avg. Cov. Moretto (1945) [12] Clark (1951) [13] Moody et al. (1954) [14] Rodriguez et al. (1959) [15] E ltner et al. (1955) [16] M phonde (1984) [17] Elzanaty et al. (1986) [18] Jo hnon and Ramirez (1989) [19] R oller and Ruell (1990) [20] Saram and Al-Muawi (1992) [21] Xie et al. (1994) [22] Kong and Rangan (1998) [23] TOTAL tet V DC V / tet V BT V / a / jd Figure 7. Experimental verification a / jd In Fig. 8, the horizontal axi of ( a 2D) / jd i an index to evaluate B-region exiting in a member or not. D denote length of D-region. In cae ( a 2D) / jd mall than 0, there i no B-region in thi member. If ( a 2D) / jd i between 0 and 1, there i an incomplete B-region in the member. If ( a 2D) / jd i larger than 1, there are both B- and D-region in the member. A hown in Fig. 8, a B-region become more complete, the tet reult can get cloed to the prediction of diagonal tenion trength according to ACI code (2005). But for the member without B-region, it i much more conervative to ue the ame prediction of diagonal tenion trength. Baed on the propoed tranferr ing mechanim, hear reinforcement equilibrate internal force at node. Once the hear reinforcement i not enough, it may caue the angle of compreion trut become 26.5 degree.

8 Along with the length of the member i too hort, two D-region will overlap. A a reult, the diagonal tenion failure ( V DT ) will happen. In predicting the hear trength uing diagonal tenion trength in D-region, the coefficient of variation i till high but cloing to tet reult. It i recommended to ue diagonal tenion trength in D-region to predict the hear trength of member without B-region. However, ome unconervative prediction by the propoed model how that the main crack angle of diagonal tenion failure i not neceary 26.5 degree; therefore, the determination of β could be modified. tet V BT V / tet V DT V / ( a 2D) / jd ( a 2D) / jd Figure 8. Experimental verification Figure 9. Different type of column Figure 10. Force tranfer mechanim of column THE TRANSFERRING MECHANISM OF SHEAR FORCES AND FAILURE MODES IN RC COLUMNS The hear tranferring mechanim in column i imilar to that in beam. The tudy ugget that it may ue a ingle D-region in a very hort column with a ratio of hear pan to effective depth ( a / jd ) of 2.0 or le (Fig. 9). For a hort or lender column with ratio of a / jd of more than 2.0, the path tranferring load to the upport pae through both B- and D-region, a hown in Fig. 9. The failure mode of a very hort column i hear compreion failure in D-region, and the hear trength( V DC ) i calculated uing the model propoed by Hwang and Lee (2002). However, in calculating the effective area of the diagonal trut Atr in the very hort column, the influence of axial load hould take into account. On the other hand, for the column with both B- and D-region, the force tranferring mechanim are imilar to thoe of ordinary beam. Tenion tie (tirrup) and concrete trut tranmit the lateral load to center of the column (Fig. 10). Cloe to both end of the column are D-region, and between the D-region i the B-region. The failure mode of D-region i diagonal tenion ( V DT )or

9 diagonal compreion ( V DC )failure. The prediction of hear trength are the ame a beam except for the calculation of A tr, V c (Eq. 8) and V mn due to axial load. The failure mode of the B-region in column i diagonal tenion( V BT ) failure. The firt cracking angle, α, i no longer 45 degree and can be computed by Eq. 9 propoed by Chen (2006). Nu Vc = 0.167(1 + ) fc bd (8) 14Ag 1 1 σ α = 45 + (tan ) (9) f fr + σ r in which N i axial force applied in column, A i gro area of column ection, σ i compreive u 2 tre of column( kgf / cm ), and fr i allowable tenile tre of concrete( kgf / cm ). g CONCLUSIONS The hear tranferring mechanim and prediction of hear trength of reinforced concrete beam are propoed in thi paper. According to the available tet reult in the literature and their comparion with the propoed model, the following concluion can be made: 1. The definition of D-region hould conider the geometrical and reinforcement configuration of reinforced concrete tructure. The formula expree explicitly the contribution of the hear reinforcement, concrete trength with oftening effect, and cro ectional area of trut. According to the ACI code proviion (2005), D-region are related only to the depth of member. The propoed Eq. 1 and 2, related with hear reinforcement, concrete trength, and geometry of element, are recommended for predicting the D-region. 2. The tree in D-region are diturbed. If there i enough hear reinforcement provided, the hear trength in D-region will be the diagonal compreion trength ( V DC ). If not, the diagonal tenion failure (V DT ) along with inclined crack would happen. The angle of inclined crack i propoed o a β = Eq. 7 i recommended for predicting the diagonal tenion trength in D-region. 3. The tree in B-region are uniform. The tate of node equilibrium could be decribed by the meared concrete model. Eq. 6 i recommended for predicting of diagonal tenion trength. The firt cracking angle, α, equal 45 degree for beam, and ue Eq. 9 for column. 4. The method propoed herein wa compared with 206 pecimen available in the literature, and atifactory correlation wa found ACKNOWLEDGMENTS The work preented in thi paper wa upported by fund from the National Science Council under the Project NSC E Th i financial upport from NSC i gratefully acknowledged. REFERENCES ACI Committee 318, (2005). Building Code Requirement for Structural Concrete (ACI ) and Commentary (ACI 318R-05), American Concrete Intitute. ACI-ASCE Committee 326, Shear and Diagonal Tenion, ACI Journal, Proceeding 59(1), 1-30 (1962); 59(2), (1962); 59(3), (1962). Chen, Y. T., Seimic Aement of Compulory School Building in Taiwan, Mater Thei, Department of Contruction Engineering, National Taiwan Univerity of Science and Technology, Taiwan, R.O.C. (in Chinee) 2

10 Clark, A. P. (1951), Diagonal Tenion in Reinforced Concrete Beam, ACI Journal, Proceeding 48(2), Collin, M. P., and Mitchell, D., (1991). Pretreed Concrete Structure, Prentice Hall Inc., Englewood Cliff, NJ, 766pp. Eltn er, R. C.; Moody, K. G.; Viet, I. M.; and Hognetad, E. (1955), Shear Strength of Reinforced Concrete Beam, Part 3-Tet of Retrained Beam with Web Reinforcement, ACI Journal, Proceeding 26(6), Elzanaty, A. H.; Nilon, A. H.; and Slate, F. O. (1986), Shear Capacity of Reinforced Concrete Beam Uing High-Strength Concrete, ACI JOURNAL, Proceeding 83(2), Hu, T. T. C., and Mo, Y. L. (1985), Softening of Concrete in Low-Rie Shearwall, ACI Journal, 82(6), Hwang, S. J., and Lee, H. J. (2002), Strength Prediction for Dicontinuity Region by Softened Strut-and-Tie Model, Journal of Structural Engineering, ASCE, 128(12), Hwang, S. J.; Lu, W. Y.; and Lee, H. J. (2000), Shear Strength Prediction for Deep Beam, ACI Structural Journal, 97(3), Johnon, M. K., and Ramirez, M. P. (1989), Minimum Shear Reinforcement in Beam with Higher Strength Concrete, ACI Structural Journal, 86(4), Kong, P. Y. L., and Rangan, B. V. (1998), Shear Strength of High-Performance Concrete Beam, ACI Structural Journal, 95(6), MacGregor, J. G., (1997). Reinforced Concrete: Mechanic and Deign, 3rd Edition, Prentice Hall Inc., Englewood Cliff, NJ, 939. Moo dy, K. G.; Viet, I. M.; Eltner, R. C.; and Hognetad, E. (1955), Shear Strength of Reinforced Concrete Beam-Part 1 and 2, ACI Journal, Proceeding 51(4), (1954); 51(5), Moretto, O. (1945), An Invetigation of the Strength of Welded Stirrup in Reinforced Concrete Beam, ACI Journal, Proceeding 42(2), Mörch, E., (1909). Concrete-Steel Contruction, Tranlation of the 3rd German Edition by E. P. Goodrich, McGraw-Hill Book Co., New York, 368. Mphonde, A. G., Shear Strength of High-Strength Concrete Beam, PhD Thei, Univerity of Connecticut, 249. Ritter, W. (1899), The Hennebique Deign Method (Die Bauweie Hennebique), Schweizeriche Bauzeitung, 33(7), Rodriguez, J. J.; Bianchini, A.C.; Viet, I. M.; and Keler, C. E. (1959), Shear Strength of Two-Span Continuou Reinforced Concrete Beam, ACI Journal, Proceeding 55(10), Roller, J. J., and Ruell, H. G. (1990), Shear Strength of High-Strength Concrete Beam with Web Reinforcement, ACI Structural Journal, 87(2), Sara m, K. F., and Al-Muawi, J. M. S. (1992), Shear Deign of High- and Normal Strength Concrete Beam with Web Reinforcement, ACI Structural Journal, 89(6), Schlaich, J., Schäfer, K., and Jennewein, M. (1987), Toward a Conitent Deign of Reinforced Concrete Structure, PCI Journal, 32(3), Thürlimann, B., (1979). Torional Strength of Reinforced and Pretreed Concrete Beam CEB Approach, Bulletin 113, ACI Publication SP-59, Detroit, Mich. Xie, Y., Ahmad, S. H., Yu, T., Hino, S., and Chung, W. (1994), Shear Ductility of Reinforced Concrete Beam of Normal and High-Strength Concrete, ACI Structural Journal, 91(2),