CHAPTER 4 PREDICTION OF SHEAR STRENGTH OF RC BEAMS

Transcription

1 66 CHAPTER 4 PREDICTION OF SHEAR STRENGTH OF RC BEAMS 4.1 General As mentioned in Chapter 3, the diagonal cracking and ultimate shear strengths of reinforced concrete (RC) beams considering the influence of beam size along with wide ranges of influencing parameters are rarely available. Parametric study using large experimental selective data segregated on the shear strength of RC beams has been carried out to develop predictive design equations through nonlinear regression analysis. These predicted equations are applicable for all practical ranges of influencing parameters. 4.2 Influence of Various Parameters on Shear Strength Primarily, the factors influencing the shear strength of RC beams have been identified. The influencing trend of each one of these factors on the shear strength of RC beams has been understood. A total of 269 for diagonal cracking strength (Appendix A) and 612 data points for ultimate strength (Appendix B) have been segregated and analysed. The comparison of the efficiency of the developed equations has been made with other analytical equations as well as various codes of practice provisions currently in use. Finally, the expressions for

2 67 predicting the ultimate and the diagonal cracking strength have been validated with the experimental observations Influence of Compressive Strength of Concrete on Shear Strength Most of the mechanical and elastic properties of concrete including modulus of elasticity, tensile strength, shear strength, bond strength and stress-strain response are generally expressed as a function of the compressive strength. For all practical purposes, the compressive strength of concrete is recognised as the most important factor for assessing the quality of concrete. In the absence of test data, the modulus of rupture and split tensile strengths are estimated as t r ' c f or f = k f (4.1) where ft = split tensile strength, fr = modulus of rupture, fc = cylinder compressive strength of concrete and k = constant to be obtained from the test results. Further, most of the codes recognise square root of compressive strength, ' f c for estimating the ultimate or the diagonal cracking strength of RC beams. The simplified expression by ACI 318 (2008) for predicting the diagonal cracking strength of a beam without web reinforcement is given by ( 0.17 ) ' cr fc v = which is based on the tests on RC beams made of normal strength concrete (NSC). However, the shear strength of RC beams made of high strength concrete (HSC)

3 68 tends to vary differently from the square root of compressive strength of concrete. The experimental studies by Fenwick and Paulay (1968), Taylor (1970) and Gergely (1969) revealed that the contribution of concrete in compression varies between 20% to 40% (ASCE-ACI 426, 1973). The effect of compressive strength of concrete on shear strength of RC beams as observed by Mphonde and Frantz (1984), Pendyala and Mendis (2000), Elzanty et al. (1986) and Van den Berg (1962) is shown in Fig It can be observed that the rate of increase in the shear strength is not proportional to the rate of increase in compressive strength of concrete. From the observations, a power law ( b y = ax ) seems to fit the experimental data. The shear strength is better correlated with the cubic root of compressive strength ( 3 ' f c ) rather than square root of compressive strength ( ' f c ) when majority of the data consisting of HSC beams. Similar consensus has been reported in ACI 318 (2008) provisions and commentary. Table 4.1 shows the parameters such as compressive strength, a/d ratio, percentage of tensile reinforcement and the beam depth adopted by various authors, which are adopted for evaluating the trend of variation of shear strength with compressive strength of concrete.

4 69 Fig. 4.1 Shear Strength vs. Compressive Strength of Concrete Table 4.1 Range of compressive strength with other parameters by various authors on shear strength Author ' f c, MPa ρ% a/d Ratio d (mm) Mphonde and Frantz (1984) Pendyala and Mendis (2000) Elzanty et al.(1986) Vanden Berg (1962)

5 Influence of Shear Span-to-Depth Ratio on Shear Strength The experimental observations by Taub and Neville (1960), Kani (1966, 1967), Clark (1951), Shin et al. (1999) and Elzanty et al. (1986) showed that there is significant influence of shear-span-to-depth ratio on the shear strength of RC beams without web reinforcement (ACI-ASCE 426, 1973). However, there exists a large deviation in the behaviour of short and normal beams with a/d ratio and also the failure modes are completely different. In short beams, there exists marginal enhancement of resistance by the beam beyond first diagonal crack due to arch action with redistribution of stresses. On the other hand, in normal beam, failure occurs after the first diagonal crack, revealing that there is possible meagre reserve strength beyond diagonal cracking. The effect of a/d ratio on the shear strength of RC beam as observed by Clark (1951), Shin et al. (1999), Kani (1967) and Ahmad et al. (1984) is shown in Fig As the a/d ratio decreases the rate of increase in shear strength increases significantly. The increase in ultimate shear strength of short beams beyond diagonal cracking can vary between 1.1 and 5.0 or even more depending on the shear-spanto-depth ratio. Lower the ratio higher is the increase in the ultimate shear strength. The ranges of a/d ratio adopted by various authors for determining the shear strength, with other parameters constant, are shown in Table 4.2.

6 71 Table 4.2 Range of a/d ratio along with other parameters adopted by various authors on Shear Strength Author Clark (1951) Shin et al. (1999) Kani (1967) a/d Ratio ' f c, N/mm 2 ρl% Ahmad et al. (1984) d (mm) Fig. 4.2 Shear Strength vs. Shear Span-to-Depth Ratio. The trends of variation of shear strength normalised with cubic root of compressive strength of concrete with a/d ratio shows that as the a/d ratio increases the shear strength of RC beams decreases as shown in Fig The shear strength of the beams can effectively be determined if appropriate a/d ratio effect is incorporated in the design equations.

7 Influence of Percentage Longitudinal Reinforcement on Shear Strength Investigations by Krefeld and Thurston (1966), Fenwick and Paulay (1968), Taylor (1969), Gergely (1969) and others showed that the contribution of longitudinal reinforcement through dowel found to vary between 15% and 25% of the total shear action was resistance offered by RC beams. In slabs, the contribution through dowel action is about 30% of the total shear resistance (ASCE-ACI 426, 1973). The influence of the percentage of longitudinal reinforcement on the shear strength of RC beams, from the experimental data reported by Mathey and Watstein (1963), Krefeld and Thurston (1966), Moody et al. (1954) and Ram Mohan Rao et al. (2004) is shown in Fig Fig. 4.3 Ultimate Shear Strength vs. Percentage of Longitudinal Reinforcement

8 73 The increase in the contribution of the percentage longitudinal reinforcement through dowel action has been observed to be not significant beyond a certain percentage of the longitudinal reinforcement as shown in Fig From the variation, it is justifiable to assume that the ultimate shear strength tends to vary as the cube root of percentage longitudinal reinforcement, 3 ρ in beams made of normal strength concrete (NSC), not so for HSC beams as it demands high percentage of the longitudinal reinforcement to enhance the dowel force. The ranges of percentage of longitudinal reinforcement with other parameters adopted by various authors on the ultimate shear strength of RC beams are shown in Table 4.3. Table 4.3 Range of percentage longitudinal reinforcement with other parameters adopted by various authors on Shear Strength Author Mathey and Watstein (1963) Krefeld and Thurston (1966) ' f c, N/mm 2 a/d Ratio Ρ, % d (mm) Ram Mohan Rao (2004) Krefeld and Thurston (1966) Influence of beam depth on Shear Strength The influence of depth of beam on the shear strength is not incorporated in IS 456 (2000) and ACI 318 (2005), while BS 8110 (1997) incorporated the influence of beam depth on the shear strength

9 74 calculations. The size effect is considered to be predominant on the ultimate shear strength rather than on the diagonal cracking strength as reported by (Bazant and Kim, 1984, Walraven and Lehwalter, 1994). The Weibull weakest link-statistical theory indicates that the shear strength is proportional to d -1/4. The term d -1/4 was also recognised by Kani also later by Niwa (1967) from his experimental observations and et al. (1987). The same d -1/4 trend was also suggested by Gustafsson and Hillerborg (1985) using the Fictitious Crack Model, and later by Jenq and Shah (1985). Fig shows the influence of beam depth on the shear strength of RC beams, from the studies of Kani (1967) and Collins et al. (1999). Fig Shear Strength vs. Depth of Beam. The ranges of beam depth along with other parameters adopted by various researchers on the shear strength of RC beams are shown in

10 75 Table 4.4. From the variation of shear strength with beam depth, it appears that the shear strength of RC beams varies as inverse of the fourth root of the beam depth. However, after the analysis the prediction looks better when the shear strength varies with the power of on beam depth. In this analysis, other parameters such as the compressive strength of concrete, a/d ratio and the percentage of the longitudinal reinforcement are maintained constant. For better prediction, the effect of the beam depth is accounted for correctly with power on beam depth. Table 4.4 Range of beam depth with other Parameters adopted by various authors on Shear Strength. Author ' f c, N/mm 2 ρl% a/d d, mm Kani (1967) Kani (1967) Collins & Kuchma (1999) Kim and Park (1994) Development of Predictive Equations Reinforced concrete beams are encountered in tall buildings, foundations, offshore structures, silos and bunkers, and shear walls, which are subjected to two dimensional state of stress under the external loads resulting in nonlinear distribution of strain along the depth. Importantly, the shear deformation in deep beams is

11 76 significant. It is important to estimate the ultimate strength for economic design of RC beams in shear. In such cases, the equations developed from the large experimental data serves satisfactorily for all practical purposes. The research findings indicate that the behaviour and failure mechanism of RC beams in shear change as the shear span-to-depth ratio (a/d) is varied. At small a/d ratios, the flexural stress distribution is different from the Bernoulli-Navier hypothesis. To develop a general expression to measure the shear strength of short and normal beams taking the resisting mechanisms into account becomes very complex and difficult to conclude. Under such circumstances empirical equations based on nonlinear regression analysis is useful for predicting the shear strength incorporating all ranges of influencing factors. The four factors influencing the shear strength and behaviour of RC beams are i) compressive strength of concrete, ' f c, ii) shear spanto-depth a/d ratio, iii) longitudinal reinforcement ratio, ρ, and iv) depth of beam, d. The influence of size of aggregate is neglected based on the earlier reports (Walraven and Lehwalter, 1994) in the range of 8-32mm. In this, different formulations have been attempted and the initial exponents c 1, c 2,....c n, corresponding to the influencing parameters such as compressive strength of concrete, shear-span-to-depth ratio, percentage of longitudinal reinforcement, and depth of beam, are

12 77 assigned certain values based on the evaluation of influence of various parameters, and the range obtained from NLRA. 4.4 Ultimate and Diagonal Cracking Strengths Methodology for Development of Expressions Various equations were developed for estimating the shear strength of RC beams. The equations proposed by Niwa et al. (1987) and Bazant and Sun (1987) are taken for comparison with the equations proposed in this study. Amongst all the forms of equations the one proposed by Niwa et al. (1987) is considered for proposing the diagonal cracking strength and the ultimate strength of concrete beams in shear with due modifications for the indices and coefficients. Fig. 4.5 shows the trends of ultimate and diagonal cracking strength of RC beams vs. shear span-to-depth ratio. Both the strengths level off at a/d ratio beyond 6.0 but the additional load carrying capacity of the short and deep beams can be visualized and also effectively utilized to achieve economy in the design. D ia g o n a l C ra c k in g S t re n g t h U lt im a t e S t re n g t h Shear Strength a / d

13 78 Fig. 4.5 Ultimate and diagonal cracking strengths VS. a/d Ratio The ranges of beam sizes used for developing the unified equations are shown in Fig. 4.6 through histogram. Major fraction of the beam depths are less than 400mm, between 75 to 80% of the data base analysed for the study, for both diagonal cracking and ultimate strength. The data used for the development of model for diagonal cracking strength is shown Appendix A and ultimate strength evaluated by different equations is reported in Appendix B. data points in % ULS - Ultimate Strength (Total 612) DCS - Diagonal Cracking Strength (Total 269) ULS DCS d (mm) Fig. 4.6 Ranges of beam depths used for analysis (NLRA) Niwa et al. (1987) (Eq. 3.7) adopted the shear strength proportional 1 4 to the inverse of fourth root of beam depth, d. The most important factor influencing the shear strength of RC beams is the shear spanto-depth ratio as the failure mechanisms and behaviour of beams change from beam to arch action with decreasing the shear span-todepth ratio as described in the previous sections. In case of beams

14 79 subjected to concentrated loads it is customary to use a/d ratio. However, for beams with uniformly distributed load it is replaced by M/Vd, where M and V are bending moment and shear force respectively at the section, taken at a distance equal to the effective depth from the face of beam support. The proposed equation for estimating the ultimate shear and the diagonal cracking strength of RC beams incorporating the beam size are as follows B ' c3 c c = ( d) [ f ρ d 5 ] (4.2) ( a/d) c1 4 vcr A a + c2 c l v uc D ' c7 c8 c9 = C + [ fc ρ d ] (4.3) c6 ( a/d ) The coefficients A, B, C, D and exponents c1, c2, c3, c4, c5, c6, c7, c8, c9 have been adjusted through nonlinear regression analysis (NLRA). After refining the coefficients in the regression analysis, the diagonal cracking strength equation is as follows a vcr Est 0.25, = d f + cm, MPa 10 ρ d (4.4) a 9 d The average of the ratio of predicted diagonal cracking strength to the experimental diagonal cracking strength is 1.79 with the coefficient of

15 80 variation (CoV) = The total points considered for the analysis are 269 with the regression coefficient (R 2 = 0.745) R= Fig. 4.7 shows the variation of experimental diagonal cracking strength vs. Predicted diagonal cracking strength. The initial point of the line is set to coincide with zero. The trend line falls well within the scattered data points. Fig Variation of experimental vs. Predicted diagonal cracking strength. Fig. 4.8 shows the variation of diagonal strength ratio with compressive strength of concrete. The deviation of the trend line with reference to the strength ratio equal to 1.0 is shown. The effect of compressive strength of concrete on the diagonal cracking strength has been depicted. The cubic root of compressive strength on the diagonal cracking strength shows a mild declining trend from the horizontal. This is due to the fact that majority of the data base

16 81 contains normal strength concrete. If the number of points containing beam tests using high strength concrete increases, the trend line may tend to become more horizontal. However, cubic root of compressive strength looks reasonable to predict the diagonal cracking strength of RC beams. Fig Diagonal cracking strength ratio vs. compressive strength of concrete.

17 82 Fig Diagonal cracking strength ratio vs. shear span-to-depth ratio. Fig. 4.9 shows the variation of diagonal strength ratio with shear span-to-depth ratio. The a/d ratio on the diagonal cracking strength has been observed to be predicting well with = 9 3 ( ) = power. Using this power with a/d ratio, the trend line has been found to be almost horizontal. The prediction of the influence of a/d ratio with power looks reasonably good. Fig Diagonal cracking strength ratio vs. depth of beam. The strength ratio has been observed, as shown in Fig. 4.10, to be influenced by the depth of the beam. The depth has been found to influence as a power of When the diagonal strength ratio is

18 83 plotted with depth, the trend line tends to be inclined slightly down with the horizontal. In Eq. 4.4, the diagonal cracking strength varies as eight root of the percentage of the longitudinal reinforcement is shown in Fig The variation of the diagonal strength ratio has been observed to be horizontal predicting very well with this power on the percentage of longitudinal reinforcement. Fig Diagonal cracking strength ratio vs. tension reinforcement (%). The ultimate shear strength equation is as follows v uc, Est = a d a d f 15 8 d ρ cm, MPa (4.5)

19 84 The average ratio of predicted ultimate shear strength to the experimental ultimate shear strength is 1.32 with coefficient of variation (CoV) = Total points considered for the analysis are 612 and the regression coefficient is (R 2 = 0.80) (R) is Fig shows the variation of experimental ultimate shear strength vs. Predicted ultimate shear strength. The starting point of the line is set to zero. The trend line falls well within the data points y = 1.139x R² = Vuc, Expt Vuc, Est Fig Experimental ultimate shear strength vs. predicted strength. Fig shows the variation of ultimate strength ratio with compressive strength of concrete. The deviation of the trend line from the strength ratio equal to 1.0 is shown in the figure. The effect of compressive strength of concrete on the ultimate shear strength has been depicted. The cubic root of compressive strength on the ultimate

20 85 shear strength shows a slight descending trend from the horizontal. This is due to the fact that majority of the data base contains normal strength concrete. If the number of points from the high strength concrete increases, the trend line will tend to become more horizontal. However, cubic root of compressive strength looks reasonable to predict the ultimate shear strength of RC beams as more and more data points contain high strength concrete. Fig Ultimate shear strength ratio vs. Compressive strength of concrete. Fig shows the variation of ultimate shear strength ratio with shear span-to-depth ratio. The a/d ratio on the ultimate shear strength has been observed to predict well with = ( ) = power. Using power with a/d ratio, the trend line has been found to be very close to the horizontal. The

21 86 prediction of the influence of a/d ratio with power has been observed to be reasonably agreeable. Fig Ultimate shear strength ratio vs. Shear span-to-depth ratio v uc, Expt /v uc, Est Depth, d Fig Ultimate shear strength ratio vs. Depth of beam.

22 87 The strength ratio has been observed, as demonstrated in Fig. 4.15, to be influenced by the depth of the beam. The influence of depth has been found to varying as a power of to the beam depth. When the strength ratio is plotted with depth, the trend line tends to be coinciding with the horizontal. This indicates that the proposed power to the beam depth looks very good. In Eq. 4.5, the influence on the ultimate shear strength as a function of the square root of the percentage of the longitudinal reinforcement is shown in Fig The variation of the ultimate shear strength ratio has been observed to slightly descendd down wrt horizontal using square root of the percentage of longitudinal reinforcement. Fig Ultimate shear strength ratio vs. Tension reinforcement (%).

23 88 The comparison of ultimate shear strength normalized with cubic root of compressive strength vs. depth of beam calculated from the proposed model (Eq. 4.5) for various a/d ratios has been shown in Figs Significantly, the effect of beam depth has been observed on the ultimate shear strength of RC beams. At smaller a/d ratios, the shear strength has been observed to be marginally high. There has been clear gap between the curves drawn for various a/d ratios at any depth of beam. The shear strength of RC beams is predicting well with the power of beam depth varying between to Hence, the fourth root of beam depth in the predicted equation looks reasonably well and satisfying the experimental test results. Fig v u / 3 (f c) vs. Depth [Proposed Model]

24 Comparison of Proposed Models The comparison of ultimate shear strength normalized with concrete compressive strength vs. depth of beam between Eqs. 3.7 and 3.9, the proposed model (Eq. 4.5) and various code provisions is shown in Figs The proposed equations can be used to predict the ultimate shear and diagonal cracking strength of beams without web reinforcement for all types of beams (for wide range of parameters) i.e., i) deep, short and normal RC beams (0.5 a/d 6.0), ii) NSC and HSC members (15 MPa fc 100 MPa), iii) all practical ranges of flexural reinforcement (0.4% ρ l 4.0 %) and iv) over a large depth range of beams (40mm d 1600mm). The strengths predicted by Eqs. 4.4 and 4.5 almost level off for a/d ratios greater than 6.0. The difference in the diagonal cracking strength and the ultimate shear strength for beams with shear span-to-depth ratios in the range of 2.5 to 6.0 can be demonstrated, which were estimated from Eqs. 4.4 and 4.5. This marks the distinct difference in the expressions proposed in the present study and other models reported in the literature. The normalised diagonal/ultimate shear strength versus depth of beam estimated from the proposed equations (Eqs. 4.4 & 4.5), by Bazant and Sun (1987) in Eq. 3.9, Niwa et al. (1987) in Eq 3.7 and various code equations shown in Figs at various a/d ratios have been compared with the experimental shear strength from Kani s observations. It can be observed that Eq. 3.7 (Niwa et al. 1987)

25 90 underestimates and Eq. 3.9 (Bazant and Sun, 1987) overestimates the ultimate shear strength of RC beams without web reinforcement for shear span-to-depth ratio of less than 2.0. However, Eq. 4.5 predicts with fair degree of beams. accuracy the shear strengths for all classes of IS code (IS 456, 2000) and the ACI code (2008) do not account for the size effect in the design provisions. The shear strength predicted by IS and ACI codes for the range of shear span-to-depth ratio, 1 < a/d < 2.5, are very conservative. IS and ACI provisions underestimate the shear strength for small size beams with shear span-to-depth ratio between 1.0 and 2.5 as shown in Figs. 4.18(a) to 4.18 (c) and overestimate the shear strength for large size beams as shown in Figs. 4.18(d) to 4.18 (g). Fig (a). vu / 3 (fc) vs. Depth (Kani, 1967) [a/d = 1.0]

26 91 Fig. 4.18(b).. v u / 3 (f c) vs. Depth (Kani, 1967) [a/d = 2.0] Fig. 4.18(c). vu / 3 (fc) vs. Depth (Kani, 1967) [a/d = 2.5]

27 92 BS 8110 (1997) recognises the effect of beam size up to a depth of 400mm beyond which the size effect has not been accounted for. The strengths predicted by BS code are conservative for beams with shear span-to-depth less than 2.5, as demonstrated in Figs (a) to 4.18 (c), as compared with the experimental studies by Kani (1967). However, prediction of shear strength is better for a/d ratio greater than 2.5. The multiplying factor incorporated by IS 456 and BS 8110 to get the shear strength of short beams is 2 ( a d) and the same has been suggested earlier by Zsutty (1968). As, the multiplying factor 2 ( a d ) is based on the studies on NSC beams of depth less than 400mm, the shear strength suggested by the codes are conservative. Figs (d) to 4.18 (g), show the comparison with the experimental studies by Kani (1967). Fig. 4.18(d). v u / 3 (f c) vs. Depth (Kani, 1967) [a/d = 3.0]

28 93 Fig. 4.18(e). vu / 3 (fc) vs. Depth (Kani, 1967) [a/d = 4.0] Fig. 4.18(f). v u / 3 (f c) vs. Depth (Kani, 1967) [a/d = 5.0]

29 94 Fig. 4.18(g). v u/ 3 (f c) vs. Depth (Kani, 1967) [a/d = 6.0] Efficiency of Proposed Equations From the comparison of % RMSE for beams with a/d ratio 1.0, tested by Kani (1967) it is * (* indicating overestimation of the results) calculated from Bazant and Sun (1987) equation (Eq. 3.9) vs from the present model (Eq. 4.5). Similarly, it can be observed that for shear span-to-depth ratios less than or equal to 1.0, Eq. 3.9 overestimates the ultimate shear strength to a very large extent compared to the proposed model, Eq However, the equation proposed by Niwa et al. (1987) (Eq. 3.7) predicts satisfactorily the diagonal cracking strength of normal size beams and underestimates to a very large extent the strength when applied to short and deep beams. The lower % RMSE from Eq. 4.5 indicates that the correlations

30 95 are better with the experimental data presented in literature, Table 4.5 and Figs (a) to 4.19(f). As shown in Figs (a) through 4.19 (f) the shear strengths are overestimated by the codes and more so on large size beams with low percentage of longitudinal reinforcement. The limiting sizes of beams beyond which the ultimate shear strengths are overestimated by the codes, IS 456 (Eq. 3.22), ACI 318 (Eq. 3.19) and BS 8110 (Eq. 3.21) are presented in Table 4.6. These limiting depths are obtained by fitting a simple power curve (y = ax b ) to the experimental data and then subsequently equating the shear strengths (y-value) predicted by the codes to the measured shear strengths. Table 4.5 Comparison of Percentage Root Mean Square Error for Ultimate Strength. Author Kani (1967) a/d Proposed Eq. (4.5) Bazant Sun (1987) Ultimate Strength Niwa et al. (1987) ACI 318 (2005) BS 8110 (1997) IS 456 (2000) * 12.21* * 05.56* SD Strength overestimated

31 96 Table 4.6 Depth beyond which Ultimate Shear Strength is overestimated by Codes Author a/d ρl% ' f c (MPa) IS-456 Depth (mm) ACI BS Kani (1967) Collins and Kuchma (1999) Kim and Park (1994) Kotosova (2001) Data from Bazant and Sun (1987) The size effect on diagonal cracking strength is clearly manifested in the earlier work of Walraven and Lehwalter (1994). A decrease in strength (42.9%) has been observed on the diagonal cracking strength against 49% on the ultimate strength. The decrease in diagonal cracking strengths is 39.8% and 54.6% for NSC and HSC respectively. Further, the variation in (vcr / 3 ' c f ) test varies as d -0.3 and (vuc / 3 ' f ) c test varies as d thus indicating that the size effect on diagonal cracking strength is not insignificant to be neglected upon. However, contrary observations have been reported by Walraven and Lehwalter (1994) and Tan and Lu (1999). Figs. 4.19(a) to 4.19(f) show the variation of normalised diagonal cracking strength vs. depth of beam from experiments by various researchers. The variation of diagonal cracking strength with depth obtained from Niwa et al. (1987) (Eq. 3.7) shows a better correlation

32 97 with those of experimental results for normal beams. When compared with a/d < 2.5 the shear strength is underestimated to a large extent. Eq. 3.7 is in good agreement with the experimental results for normal beams (a/d > 2.5). The measured diagonal cracking strength vs. the calculated strength from the proposed model in Eq. 4.4 and Eq. 3.7 (Niwa et al. (1987)) are shown. The correlation coefficient for the proposed model, r = 0.85, higher than that of Niwa et al. (1987) (Eq. 3.7) ), r = 0.82, showing a better performance of the proposed model. Fig. 4.19(a). vu/ 3 (fc) vs. Depth (Tan-Lu, 1999) [a/d = 0.84]

33 98 Fig. 4.19(b). vu/ 3 (fc) vs. Depth (Walraven, 1994) [a/d = 1.0] Fig. 4.19(c). vu / 3 (fc) vs. Depth (Walraven, 1994) [a/d = 3.0]

34 99 Fig. 4.19(d). vu / 3 (fc) vs. Depth (Collins, 1999) [a/d = 3.0] Fig. 4.19(e). vu / 3 (fc) vs. Depth (Kim - Park, 1994) [a/d = 3.0]

35 100 Fig. 4.19(f). v u/ 3 (f c) vs. Depth (Kotosova, 2001) [a/d = 3.0] Accuracy of prediction of influence of various parameters on the ultimate shear strength of RC beams is compared with the selected experimental data available. Figs. 20(a) to 20(d) show the comparison of prediction of shear strength with the experimental results by Pendyala (2000), Collins (1999), Mphonde (1984), Pendyala (2000) respectively by varying the compressive strength of concrete. The compressive strength of concrete increases the ultimate shear strength when all other parameters are constant. It has been observed that the prediction of ultimate shear strength of RC beams has been observed to be very good for all ranges of parameters. The diagonal cracking strength is significantly lagging behind the ultimate shear strength. Almost the ultimate shear strength is double the diagonal cracking strength of RC beams as the compressive strength is varying.

36 101 Fig. 4.20(a). vu vs. Depth (Pendyala, 2000) [a/d = 2.0] Fig. 4.20( (b). vu vs. f c (Collins,1999) [a/d = 2.92]

37 Depth = 298 mm ρ = 3.36% Exp. (vu)model (vcr)mod 2.00 v u f c ' (N/mm 2 ) Fig. 4.20(c). v u vs. Depth (Mphonde, 1984) [a/d = 3.6] Depth = 140 mm ρ = 2.0% Experimental (vu)model (vcr)model v u f c (N/mm 2 ) Fig. 4.20(d). v u vs. Depth (Pendyala, 2000) [a/d = 5.0]

38 103 The influence of shear span-to-effective depth ratio on the ultimate and diagonal cracking of the RC beams in the proposed expressions is compared in Figs. 4.21(a) to 4.21(b). The shear strength decreases as the shear span-to-depth ratio increases. The prediction of ultimate shear strength almost coinciding with the experimental test results obtained by Clark (1951) in Fig. 4.21(a) and Shin et al. (1999) in Fig. 4.21(b). It can be seen from Figs. 4.21(a) and 4.21(b) the ultimate shear strength and diagonal cracking strengths are significantly deviating from each other at small a/d ratios. The ultimate shear strength and the diagonal cracking strengths are coinciding at the larger a/d ratios (f c ' ) Avg = MPa ρ = 0.98% Experimental (vu)model (vcr)model 1.2 v u / 3 (f c ) a/d Ratio Fig. 4.21(a). v u / 3 (f c) vs. a/d ratio (Clark, 1951)

39 (f c ' ) Avg = 52.4 MPa ρ = 3.77% Experimental (vu)model (vcr)model v u / 3 (f c ) y = 2.010x a/d Ratio Fig. 4.21(b). vu / 3 (fc) v/s a/d Ratio (Shin et al. 1999) The ultimate shear and the diagonal cracking strengths of RC beams have been predicted and compared with the experimental observations by Ahmad-Kahloo (1986) with varying tensile reinforcement in the beams. It has been observed that the shear strength of RC beams increases as the percentage of the tensile reinforcement increases. At smaller percentages of tensile reinforcement, the difference between the diagonal cracking strength and the ultimate shear strengths are found to be small, whereas the difference widens at higher percentages of tensile reinforcement.

40 (f c ' ) Avg = 64.0 MPa a/d = 1.0 Experimental (vu)model (vcr)model 3.0 v u / 3 (f c ) p t % Fig. 4.22(a). vu/ 3 (fc) vs. pt (Ahmad-Kahloo, 1986) (f c ' ) Avg = MPa a/d = 2.0 Experimental (vu)model (vcr)model 1.5 v u / 3 (f c ) p t % Fig. 4.22(b). vu/ 3 (fc) vs. pt (Ahmad-Kahloo, 1986)

41 106 As demonstrated in Figs. 4.20, 4.21 and 4.22, the prediction of ultimate shear strength of RC beams by the proposed equations in this study has been observed to be reasonably well as the prediction coincides with the ultimate shear strength of beams tested by various researchers by varying different parameters. However, the predicted diagonal cracking strength is significantly deviating from the experimental shear strength of RC beams tested by various researchers. Hence, it is worth mentioning that the estimation of shear strength by various codes of practice is very conservative. This is due to the fact that all the code equations are established from the test results on small size beams of low strength concretes based on diagonal cracking strength. There has been significant reserve strength of RC beams beyond the diagonal cracking strength depending up on the a/d ratio, percentage tensile reinforcement, and depth of the beam. The prediction can be made more reasonable and uniform if the design is based on ultimate shear strength instead of diagonal cracking strength by incorporating appropriate factors on size effect and a/d ratio. 4.5 Summary The parametric study on the variation of shear strength of RC beams without web reinforcement has been presented. Comparison has been made between the results in this study with the code provisions and other expressions. Further, from a large test data, expressions for predicting the ultimate and diagonal cracking strengths have been

42 107 developed to apply for all types of beams. The prediction from the expression on the ultimate strength has been compared and showed better correlation, with experimental results. An expression for diagonal cracking strength has also been developed with nonnegligible size effect. But the size effect on ultimate strength is more dominant than that on diagonal cracking strength. Finally the provisions of codes of practice are without the size effect on large size beams with HSC at low percentage of longitudinal reinforcement exceeds the strength. Hence, the size effect in design expressions for obtaining uniform safety margins and to achieve economy in the design has been emphasised. The expressions proposed have been proved to be useful in predicting the shear strength of RC beams without web reinforcement.