SIZE EFFECT ON SHEAR STRENGTH OF RC BEAMS USING HSC WITHOUT SHEAR REINFORCEMENT

SIZE EFFECT ON SHEAR STRENGTH OF RC BEAMS USING HSC WITHOUT SHEAR REINFORCEMENT (Translation from Proeedings of JSCE, Vol.711/V-56, August 00) Manabu FUJITA Ryoihi SATO Kaori MATSUMOTO Yasuhiro TAKAKI The shear strength of reinfored onrete (RC) beams without shear reinforement is experimentally investigated, with ompressive strength and effetive depth taken as the major experimental fators. The experiment demonstrates that the size effet is more prominent in high-strength onrete (HSC) than in normal-strength onrete beams. It is also shown that the nominal shear stress normalized by onrete tensile strength at diagonal raking is proportional to the minus 1/ power of the ratio of effetive depth to harateristi length. Based on this relationship, a new empirial equation is proposed for alulating the shear strength of RC beams without shear reinforement when made with HSC of ompressive strength of 80-15 N/mm. Keywords : high-strength onrete, size effet, shear strength, frature mehanis Manabu Fujita is a hief researh engineer of Institute of Tehnology & Development at Sumitomo Constrution Co., Ltd., Japan. He reeived his Dr. Eng. degree from Hiroshima University in 00. His researh interests are related to prestressed onrete strutures. He is a member of JSCE and JCI. Ryoihi Sato is a professor of Department of Soial and Environmental Engineering, Graduate Shool of Engineering, Hiroshima University. He reeived his Dr. Eng. degree from Tokyo Institute of Tehnology in 198. His researh interests are related to mehanial properties of onrete strutures. He is a member of JSCE and JCI. Kaori Matsumoto is researh engineer of Institute of Tehnology & Development at Sumitomo Constrution Co., Ltd., Japan. Her researh interests are related to t of prestressed onrete strutures. She is a member of JSCE and JCI. Yasuhiro Takaki is a researh engineer of Institute of Tehnology & Development at Sumitomo Constrution Co., Ltd., Japan. His researh interests are related to prestressed onrete strutures. He is a member of JSCE and JCI. -113-

1. INTRODUCTION In reent years, onrete of ever greater strength has beome available. The Standard Speifiation for Design and Constrution of Conrete Strutures, published by the Japan Soiety of Civil Engineers (JSCE), already inludes onrete with a design standard strength of f' k = 80 N/mm. Among fatory produed onrete items, there are already many examples of high-strength onrete being used, suh as in preast members for large-sale onrete bridges and the like. High-strength onrete has exellent durability and enables member ross setions to be redued, and it also offers the potential for greater to help strutural funtionality, redued osts, and other advantages in future design systems that inorporate performane based design. Aordingly, the range of appliations of high-strength onrete is expeted to inrease. Nevertheless, in the interests of designing members that fully utilize the speial harateristis of high-strength onrete, it is neessary to gain a thorough understanding of failure modes and load-arrying properties in addition to determining material and dynami harateristis. Failure mehanisms and size effets in the high-strength domain are largely unknown, partiularly for shear strength, and there is a pressing need to eluidate these matters and establish alulation formulas. Regarding shear strength alulations for reinfored onrete beam members without shear reinforement, researh thus far both in both Japan and abroad has foused on empirial laws or frature energy properties based on experimental results. In Japan, the aforementioned JSCE Standard Speifiation for Design and Constrution of Conrete Strutures [1] ontains a alulation (the JSCE formula ). This alulation is based on a formula proposed by Niwa et al. [] and derived through analysis and regression of existing experimental results. It takes into aount the effet of ompressive strength, member size, tensile reinforement ratio and axial strength, as well as a size effet orresponding to the effetive depth to the minus 1/4 power. Nevertheless, the data used to obtain this formula, was limited to a omparatively narrow range of ompressive strength, so the JSCE formula is limited to appliations where f' k 80 N/mm. Within this range, the results of FEM analysis arried out using frature mehanis tehniques have been used by, Hillerborg et al. to propose d/l h to the minus 1/4 power, using effetive depth d and harateristi length l h for the size effet of shear strength of ordinary onrete beams [3]. On the other hand, although researh into the shear frature of high-strength onrete (HSC) members has been progressing in reent years, the quantity of data aumulated is not very great. In order to inrease the appliation of HSC to atual strutures, the size effet on the shear strength for HSC must be identified and design equations based on this size effet will be needed. Previously, the authors have onduted shear tests on reinfored onrete simple beams without shear reinforement with the aim of determining the effet of onrete strength and the effet of member size on the shear frature harateristis of RC beam members. In this work, f effetive depth d, and the ratio of shear span a to the effetive depth d ("the shear span ratio, a/d") were adopted as parameters [4][5][6]. The results of these tests onfirmed that shear frature in HSC was haraterized by onspiuous loalization of raking as ompared to ordinary-strength onrete, and that the propagation of these raks was rapid, resulting in more brittle fraturing. Sine the loalization of raking results from tension softening of the onrete, a study using frature mehanis was onduted to determine the size effet on nominal shear stress intensity (the "shear strength") when diagonal raking ours, and a new empirial equation for alulating the shear strength of HSC RC beams without shear reinforement.. EXISTING STUDIES AND DESIGNE CODE FORMULAS The formulas used to alulate the shear strength of RC beams without shear reinforement have been standardized in Japan by the Japan Soiety of Civil Engineers (JSCE). Overseas, standards have been set by the Amerian Conrete Institute (ACI), the Comité Euro-International du Béton and the Fédération Internationale de la Préontrainte (CEB-FIP) and other organizations (Table 1). Hereafter, these formulas will be referred to as the JSCE formula, the ACI formula and the CEB-FIP formula. Where not otherwise speified, the symbols represent the following: : Shear strength when diagonal raking ours (N/mm ) f' : Compressive strength of onrete (N/mm ) f t : Tensile strength of onrete (N/mm ) -114-

Table 1 Formulas about shear strength for RC beams without shear reinforement Existing formulas Range of appliation JSCE (1996) [1] 0 3 3 100 4 =. f ' pw 1000 d (1) 0. 3 f ' 0.7 N/mm ' f 80 N/mm ACI (1999) [7] = 1. 9 f ' + 500p V d M () d : (in), f : (psi) V u : fatored shear fored at setion onsidered M u : fatored moment ourring simultaneously CEB-FIP. f ' p ( d ) (1993) [8] = 0 1 3 3 100 w 1 + 00 w u u ( 3.5 ) : f ' (psi) f ' 100 psi ' (3) f 50 N/mm Niwa et al. [] = 0. 3 f ' 3 100 p w 4 1000 d [ 0.75 + 1. 4 ( a d )] 8 3 5 w ( w ( ) ) p f ' + 3000 p a d = ( 1 + d / 5d ) Bazant And Kim [9] (4) (5) : (psi), f : (psi), d a : maximum aggregate size (mm) a E : Young s modulus of onrete (kn/mm ) d: Effetive depth (mm) b: Girder width (mm) a: Shear span (mm) p w : Tension reinforement ratio (= A s / (b x d)) A s : Setional area of tension side tendons (mm ) In the JSCE and CEB-FIP formulas, shear strength dereases as effetive depth inreases; in other words, the size effet is taken into aount. In ontrast, there is no onsideration of size effet in the ACI formula. All of these formulas establish an upper limit for ompressive strength and limit the appliable range. In addition to the standard formulas desribed above, many other have been proposed for estimating shear. For example, Niwa et al. [] proposed Eq.(4), the foundation of the JSCE formula, based on researh by Okamura and Higai [10] and Iguro et al.. [11]. Investigation of the size effet of shear strength have also been arried out from a frature mehanis approah. Bazant et al. have proposed a theoretial formula, Eq.(5), based on non-linear frature mehanis [9]. Aording to formula [5], the size effet disappears if the maximum aggregate size is proportional to the effetive depth. However, it has been reported that this is ontradited by experiments in whih aggregate diameter was varied [1]. Hillerborg et al. implemented FEM analysis for RC beams using frature mehanis tehniques and disussed the size effet on shear strength based on the relationship between (a) the ratio /f t of shear strength to tensile strength and (b) the ratio d/l h of the effetive depth to the harateristi length l h (as disussed later); they reported that shear strength is proportional to the minus 1/4 power of effetive depth and reported on the effetiveness of frature mehanis in explaining the size effet on shear strength [3]. The evaluated size effet on shear strength differs among these previous studies. For example, with regard to effetive depth, shear strength was noted as being proportional to the minus 1/4 power in Eq.(1) and Eq.(4), and to the minus 1/ power in Eq.(5). However, almost all of these formulas are based on the results of regression analysis using test data for onrete with a ompressive strength f 50N/mm. In this study, in order to hek the predition auray of the existing shear strength estimation formulas with respet to HSC, the upper limit of the appliable ompressive strength range has been ignored. -115-

3. OUTLINE OF TEST Table shows the properties of the test speimens and Fig.1 is a diagram showing their struture. The various test ases (L, M, and U) represent variations in the strength of the mix. Case L is ordinary-strength onrete (f = 36 N/mm ), U represents HSC (f = 100 N/mm ), and M is a value mid way between ordinary- and high-strength onrete (f = 60 N/mm ). In this test, for eah strength level (L, M, and U), two series were established: one designed to examine the size effet, in whih the shear span ratio was onstant (a/d = 3) and the effetive depth was varied, and one to examine the effet of shear span ratio, in whih the effetive depth was onstant (d = 500 mm) and the shear span ratio was varied (a/d = 5). Table 3 shows the onrete mix proportions for the three strengths. The maximum aggregate size G max was made onstant at 0 mm. SD345 was used for the tension reinforement; the reinforement was speified as yield strength of 388 N/mm, a tensile strength of 555 N/mm, and a modulus of elastiity of 196 N/mm. The tension reinforement ratio and the ratio of the sum of the tensile diameters D to the beam width (= D/b) were set to almost exatly the same values for eah test speimen. The tension reinforing bars were anhored with right-angle hooks attahed to the ends. No shear reinforement was provided; the ends of the beams were simply provided with stirrups to prevent pull out of the tension reinforement at failure. Two-point onentrated stati loading was arried out. The load was inreased in monotone fashion until frature. The equimomental setion length was made equal to the effetive depth. One test speimen was tested for the ase with a/d = 5 and three for eah of the other ases, for a total of 48. P tensile reinforing bar d H a d L a 450 950 1130 diameter of tensile reinforing bar d=50mm,500m : D19 d=1,000mm : D3 100 50 80 50 300 100 50 600 3@50=150 3@50=150 60 60 @115=30 350 (unit:mm) d=50mm d=500mm d=1000mm Fig.1 Speimens geometry and load appliation Case Require strength (N/mm ) W/B (%) s/a (%) Table 3 Water W Mix proportion of onrete Unit ontent (kg/m 3 ) Binder B Aggregate Cement C Silia fume SF Sand S Gravel G Chemial admixture C (%) L 36 56.5 47.9 16 87 0 864 966 1.1 M 60 35.0 46.1 165 471 0 789 948 1. U 100 1.0 4. 160 686 76 618 869 1.8-116-

Size effet Effet of the shear span ratio Table Properties of speimens and test results Case L b a d p a/d w mm mm mm mm % f N/mm f t N/mm E kn/mm N/mm Form 37.8 3.05 7.9 1.15 DTF L-5-3 750 150 750 50 3 1.53 36.4.90 31.6 1.48 DTF 36.4.90 31.6 1.50 DTF 37.4.95 7.8 1.55 DTF L-50-3 4500 150 1500 500 3 1.53 36.4.90 31.6 1.05 DTF 36.4.90 31.6 1.16 DTF 35.7.75 7.1 0.91 DTF L-100-3 9000 350 3000 1000 3 1.36 34.7.71 8.7 0.85 DTF 33.7 3.16 8.7 0.95 DTF 68.8 3.90 37.0 1.37 DTF M-5-3 750 150 750 50 3 1.53 51.9 3.46 9.8 1.74 DTF 51.9 3.46 9.8 1.43 DTF 69.4 3.9 36.3 1.16 DTF M-50-3 4500 150 1500 500 3 1.53 51.9 3.46 9.8 1.44 DTF 51.9 3.46 9.8 1.51 DTF 59. 4.05 31.6 0.90 DTF M-100-3 9000 350 3000 1000 3 1.36 53.7 3.43 33. 0.97 DTF 53.0 3.8 33. 0.97 DTF 101 4.47 37.8 1.07 DTF U-5-3 750 150 750 50 3 1.53 9.9 4.91 40.8 1.50 DTF 9.9 4.91 40.8 1.5 DTF 10 4.59 37.5 1.15 DTF U-50-3 4500 150 1500 500 3 1.53 9.9 4.91 40.8 1.10 DTF 9.9 4.91 40.8 1.30 DTF 103 4.75 39.7 0.80 DTF U-100-3 9000 350 3000 1000 3 1.36 89.8 3.9 36.7 0.75 DTF 9.1 4.8 36.7 0.80 DTF 37..9 7.7 1.33 SCF L-50-3500 150 1000 500 1.53 36.4.90 31.6 1.30 SCF 36.4.90 31.6 1.39 SCF 37.7 3.03 7.9 0.94 DTF L-50-4 5500 150 000 500 4 1.53 36.4.90 31.6 1.13 DTF 36.4.90 31.6 1.18 DTF L-50-5 6500 150 500 500 5 1.53 37.8 3.04 7.9 - BF 69.3 3.91 36.5 1.31 DTF M-50-3500 150 1000 500 1.53 51.9 3.46 9.8 1.63 DTF 51.9 3.46 9.8 1.53 SCF 69.8 3.94 35.8 1.09 DTF M-50-4 5500 150 000 500 4 1.53 51.9 3.46 9.8 - BF 51.9 3.46 9.8 1.1 DTF M-50-5 6500 150 500 500 5 1.53 70.4 3.76 35.1 - BF 10 4.67 37.3 1.41 DTF U-50-3500 150 1000 500 1.53 9.9 4.91 40.8 1.6 DTF 9.9 4.91 40.8 1.38 SCF 101 4.43 37.9 0.97 DTF U-50-4 5500 150 000 500 4 1.53 9.9 4.91 40.8 1.05 DTF 9.9 4.91 40.8 1.1 DTF U-50-5 6500 150 500 500 5 1.53 100 4.38 37.9 - BF Note : Results of material tests show data at eah experiment. take into onsideration of weights of speimen and loading devie, and ases of d = 1000 mm are orreted reinforement ratio as p w = 153 %. In this table form suggests the form of failure, and DTF shows the diagonal tension failure, SCF shows the shear ompression failure and BF shows the bending failure. -117-

(N/mm ) Shear strength 1.5 1 JSCE 50 mm 500 mm 1000 mm 0.5 0 60 100 140 Compressive strength f' (N/mm ) CEB-FIP 50 mm 500 mm 1000 mm 50 mm 500 mm 0 60 100 140 0 60 100 140 Compressive strength f' (N/mm ) Compressive strength f' (N/mm ) ACI 1000 mm Fig. Comparison of design ode formulas with test results 4. STUDIES ABOUT FACTORS Table also shows the test results. In this ase of diagonal tension failure and shear ompression failure test speimens, the tension reinforement had not reahed the yield strain when diagonal raking ourred. Moreover, under idential onditions, there was onsiderable variation in some of the test results, but the reason for this is not lear. In these tests, the p w value differed slightly for d = 1,000 mm. Aording to the researh done by Suzuki et al., shear strength at the ourrene of diagonal raking is proportional to p w 1/3 for HSC [13]. Based on these findings, the shear strength for d = 1,000 mm was multiplied by the offset oeffiient (p w0 /p w ) 1/3 for the tension reinforement ratio and onverted to a tension reinforement ratio of p w0 = 1.53 % for d = 50 mm and 500 mm. 4.1 Comparison of the design ode formulas on test results Figure ompares the test results for a/d=3 in Table and the shear strength derived using eah of the design ode formulas given in Setion. It is lear that the JSCE formula and CEB-FIP formula, both of whih take size effet into onsideration, offer evaluations that fall in the safe zone for ompressive strength value within the range of the design odes, even for ases with an effetive depth of 1,000 mm. However, the ACI formula, whih did not take size effet into onsideration, provided evaluations in the danger zone for the ases with effetive depth of 1,000 mm regardless of ompressive strength. With high-strength onrete having a ompressive strength of 90 100 N/mm, whih falls outside the appliable range of the design odes, the JSCE and CEB-FIP formulas offer evaluations that fall slightly inside the danger zone for ases where the effetive depth is 1,000 mm. Other ases, however, yield results that remain with in the safe zone. The ACI formula yields evaluations in the danger zone for all effetive depths, and it is onfirmed that the disrepany between design ode formulas and test results is greater for larger effetive depths. For eah of the design ode formulas, the larger the member size, the greater the disparity with the test results and the greater the trend towards evaluations within the danger zone. This latter trend was partiularly onspiuous with the ACI formula, whih dose not onsiders the size effet. These results demonstrate that appropriate onsideration of the size effet is neessary in order to perform a rational evaluation of shear strength. In partiular, it is lear that the shear strength of HSC annot be properly evaluated using forms of existing design odes. 4. Effet of ompressive strength on shear strength Figure 3 shows a ompares shear strengths derived from Eq.(4) and the test results for ases in Table where the shear span ratio is 3. In order to illustrate the effet of shear strength on onrete strength, the ' 3 vertial axis is the shear strength divided by f 1, in aordane with Eq.(4). The straight lines in the figure represent the values for effetive depth alulated with Eq.(4); they are shown as onstant regardless 1 3 of ompressive strength. However, in the test results, the value of / f ' is different for eah effetive -118-

1/3 ) /(f' 0.5 0.4 0.3 0. 50 mm 500 mm 1000 mm 50 mm 500 mm 1000 mm 0.1 0 60 100 140 Compressive strength f' (N/mm ) Fig.3 Effet of ompressive strength on shear strength depth depending on ompressive strength. For ompressive strengths exeeding 80 N/mm, there is a lear tendeny for shear strength to derease as ompressive strength inreases. Aordingly, it is inferred that the relationship between shear strength and ompressive strength in high-strength onrete is different from that in ordinary-strength onrete. 4.3 Effet of effetive depth on shear strength (N/mm ) Shear strength 1.6 1.4 1. 1 0.8 0.6 0 400 800 100 Fig.4 Comparison of effetive depth to shear strength Figure 4 shows the relationship between effetive depth and the test results for ases in Table where the shear span ratio is 3. For test speimens of all strength, a greater effetive depth means lower shear strength. In Eq.(4), the size effet is speified as the effetive depth to the minus 1/4 power. Based on this, Fig. 5 shows the relationship between effetive depth and the test results, with the shear span ratio given in Table divided by the effetive depth to the minus 1/4 power, /(1000/d) 1/4. Sine shear strength is affeted by ompressive strength, test results for those ases with a shear span ratio of 3, and whih exhibited the same degree of ompressive strength in eah series are shown in the figure. In the U series that orresponds to high-strength onrete, the /(1000/d) 1/4 value for an effetive depth of 1,000 mm is signifiantly lower than the values for the L and M series. This indiates the possibility that the size effet in high-strength onrete may be more marked than speified by effetive depth to the minus 1/4 power. These results onfirm that the size effet of shear strength is dependent on ompressive strength, and that the size effet is greater in high-strength onrete than in ordinary-strength onrete. L M U Effetive depth d(mm) /(1000/d)(1/4) 1.4 1. 1 0.8 L M U 0.6 0 400 800 100 Effetive depth d(mm) Fig.5 Effet of effetive depth on shear strength 4.4 Relation between shear strength and shear span ratio L M U In Eq.(4), the effet of shear span ratio on shear strength is taken into aount by introduing the (0.75 + 1.4 / (a/d)) fator. Hear, to study the effet of shear span ratio on shear strength, the test results in Table are divided by this fator, (0.75 + 1.4 / (a/d)). The study is arried out using data that demonstrates a diagonal tensile failure mode. Figure 6 shows the relationship between shear span ratio and /(0.75 + 1.4 / (a/d)). Sine the / (0.75 + 1.4 / (a/d)) values are almost onstant, the effet of shear span ratio on shear strength an be evaluated using the shear span ratio fator in Eq.(4) regardless of ompressive strength. The above results onfirm that, in extending existing design ode formulas, evaluations of the shear strength of high-strength onrete fall within the danger zone in some ases for lager 0.5 1 3 4 5 Shear span ratio a/d Fig.6 Effet of shear span ratio on shear strength effetive depths, and that the size effet is greater for high-strength onrete than for ordinary-strength onrete. Further, it is also learned that the relationship between shear strength and shear span ratio is almost onstant regardless of ompressive strength. /(0.75+1.4/(a/d)) 1.5 1-119-

L-5-3 L-50-3 L-100-3 M-5-3 M-50-3 M-100-3 U-5-3 U-50-3 U-100-3 Fig.7 Comparison of rak development at the same nominal shear stress intensity ( = 0.80 N/mm ) 5. STUDY OF SHEAR STRENGTH OF RC BEAMS USING FRACTURE MECHANICS In setion 4, it was shown that the size effet of shear strength is greater for high-strength onrete than for ordinary-strength onrete. In this setion, a frature mehanis approah to studying the size effet of shear strength is desribed. 5.1 Charateristis and rak loalization Figure 7 shows the raking state for eah ase when the same nominal shear stress intensity is applied, using the shear strength at shear fraturing in U-100-3 as referene. In order to ompare the effet of ompressive strength and member size on rak propagation, the values are shown on the same sale. Crak loalization ours at a point toward the ompression side from the enter of the ross-setion in eah test speimen. The amount of displaement from the enter inreases with greater effetive depths, and is still more signifiant with greater ompressive strengths. This differene in rak loalization is thought to be related to shear strength and the size effet on shear strength. One fator affeting the relationship between effetive depth and rak loalization is the relative position of the tension reinforement with respet to the effetive depth. It is onjetured that, when the effetive depth is greater, the rak distributing effet of the tension reinforements ats over a relatively smaller range, ausing rak loalization to beome onspiuous at points removed from the reinforement. The results of analytial researh on reinfored onrete beams made of ordinary-onrete without shear reinforement has shown that beams an be divided into two areas around the tensile reinforement: one where suffiient tensile stress transmission an be antiipated even after raking, and a seond unreinfored area where sudden tension softening ours [14] [15]. For the latter, a onstitutive law was applied that took softening into aount by means of frature L-100-3 U-100-3 mehanis, and some beams were shown to 800 demonstrate size effet with regard to shear strength. Diagonal rak Conversely, in HSC, the frature proess zone (FPZ) overs a smaller area in ordinary-strength onrete [16]. It is surmised that, in domains other than the FPZ, behavior is almost ompletely elasti. Consequently, almost all of the elasti energy stored in the test speimen is onsumed near the FPZ, whih is loalized in the domain in front of the rak ends [17], with the result that rak loalization beomes even more onspiuous. Figure 8 ompares the load - defletion urves at the enter of the span for the L-100-3 and U-100-3 test speimens. The differenes in load redution after maximum load are thought to result from differenes in the loalization Load P (kn) 600 400 00 Flexural rak 0 0 5 10 15 0 Midspan defletion (mm) Fig.8 Load-defletion urve with different ompressive strengths δ -10-

Table 4 Results of frature energy tests Case f (N/mm ) f t (N/mm ) E (kn/mm ) γ (g/m 3 ) G f (N/mm) l h (mm) L 35.1.87 9.0.8 0.188 661 M 50. 4.19 3.1.34 0.00 366 U 85.6 5.47 37..40 0.18 71 γ : Conrete density of raking. Further, the ratio of initial rigidity in the two test speimens is almost ompletely in agreement with the Young s modulus ratio obtained from the onrete material tests, so differenes in initial rigidity are thought to result from differenes in Young s modulus. As the FPZ develops, it is loalized by the tension softening harateristis of the onrete [17]. Additionally, rak loalization results from the loalization of the FPZ due to tension softening of the onrete. This rak loalization beomes more onspiuous with member size, and also with higher ompressive. For this reason, it is thought that onsideration of onrete tension softening (in other words, the appliation of frature mehanis) will lead to more rational evaluations of the size effet on shear strength for HSC. Several tests and evaluation methods have been proposed for the in-plane shear-type mode II frature energy. However, at present, frature energy has not even been learly defined [16]. On the other hand, tests and evaluations for tensile-type mode I frature energy have already been drafted by a ommittee of the Japan Conrete Institute (the Test Method for Frature Property of Conrete ommittee) in a doument entitled Test Method for Frature Energy of Plain Conrete (Draft) (referred to hereafter as the proposed test method) [16]. From detailed measurements of the deformation of raking surfaes that lead to frature, the authors have onfirmed that, rather than displaement of the diagonal raked surfae, opening in the vertial diretion is dominant [5]. Furthermore, it has been reported elsewhere that analysis in whih the frature harateristis of diagonal raking are made equivalent to mode I raking harateristis offers omparatively good results with regard to the size effet of shear strength [18]. As a result, in this test, mode I frature energy is applied to study the size effet on shear strength. 5. Evaluation of onrete frature energy [3] Frature energy is evaluated hear in aordane with the proposed test method disussed above. Testing is set up to orrespond with the shear test desribed in the earlier part of this paper, with the same three types of onrete materials and mixes (L, M, and U) as shown in Table 3. The results of the frature energy test are shown in Table 4. These values are the averages for 5-8 test speimens of eah type. The harateristi length in Table 4 is the value proposed by Gustafsson and Hillerborg, and is derived aording to the following formula using the frature energy G f resulting from the test [3]: l h = 1000 E = G f G f f t ( 1 ft ft 1000E ) = G f Ge (6) Where G e is the elasti energy per unit of volume stored in the member until rak propagation. In general, the smaller the harateristi length, the more brittle the frature beomes. From the test results, as ompressive strength inreases, the harateristi length derease, the ompressive strength rises, and the more likely it beomes that brittle frature will our. If the damage is loalized, the total amount of energy released as damage progress (in other words, the energy required to ause frature) is redued. This orresponds well with the fat that the harateristi length is redued as the strength of the onrete inreases. Sine this means that harateristi length an lead to quantitative evaluation of damage loalization (in other words, the tendeny for raking to be loalized), it is a useful indiator for evaluating the size effet of shear strength. -11-

Table 5 Existing test data for investigation Referene Level d (mm) p w (%) f (N/mm ) f t (N/mm ) E (kn/mm ) (N/mm ) L 300 1.3 1.6 4.8.9 0 1.13 1.00 No.19 L 500 1.35 7.3.7 1.8 0.98 1.0 L 950 1. 0.7 0.6 1.87.00 17.6 17.4 0.70 0.77 M 350 1.19 55.1 70.4 3.99 4.0 30.1 34.3 1.6 1.61 U 350 1.19 8.5 4.0 36.7 1.75 No.0 U 550 0.76 84.3 4.71 35.9 1.31 U 750 0.55 87. 4.00 36.8 0.98 M 950 1.10 76.5 3.83 37.4 1.01 L 150.65 3.4.75 8.7 1.5 1.47 L 150.65 38.3 3.14 30.5 1.63 1.5 M 150.65 48.6 3.47 3.7 1.64 1.79 No.1 M 150.65 70.9 5.14 36.9 1.85 1.88 U 150.65 83.4 5.5 37.5.1.36 U 150.65 18 7.84 4.7 1.91 1.89 U 5.55 15 7.8 4.3 1.63 1.54 U 300.58 18 7.84 4.7 1.44 1.7 U 150 1.7 90.6 4.91 43..01 No. U 350 1.3 0.54 0.85 95.5 5.36 43. 1.44 1.49 1.45 U 550 1.06 107 6.08 43. 1. U 650 0.89 108 6.73 43. 1.01 Notie : E of referene No.0 show inferene values by Eq.(7). gives the value onverted to a tension reinforement ratio of p w0 = 1.53 %. 5.3 Derivation and verifiation of shear strength alulation formulas Based on the results given in the preeding paragraph, a study of the size effet of shear strength is arried out on the basis of the harateristi length as obtained from the frature energy tests, and this leads to the derivation of shear strength alulation formulas. a) Approah to size effet using frature mehanis Studies of the size effet were implemented for ases where the shear span ratio was 3. To improve the auray of the study results, the 7 data points are ombined with 34 previously reported data points for shear span ratio (Table 5). All of these existing data points ware for ase with no shear reinforement and two-point or one-point entralized loading. Regarding the shear strength in Table 5, sine the tensile reinforement ratio 0 0 60 100 Compressive strength f' (N/mm ) Fig.9 Investigation of Young s modulus Table 6 Applied frature energy to investigation of the size effet f (N/mm ) G f (N/mm) l h (mm) L ~ 45 0.188 540 ~ 945 M 45 ~ 80 0.00 80 ~ 617 U 80 ~ 0.18 151 ~ 51 was different for eah data point, in aordane with Eq.(4), the shear strength obtained in the test was multiplied by an offset oeffiient for the tensile reinforement (p w0 /p w ) 1/3 to normalize the values to the same p w0 = 1.53 % tensile reinforement ratio as in the ases effetive depth 50 mm and 500 mm in this study. Regarding Young s modulus, whih was not learly indiated in the existing data, estimated values are shown. For f' 80 N/mm, these estimates were derived through linear interpolation of the values [1] in the Standard Speifiation for Design and Constrution of Conrete Strutures. For f' > 80 N/mm, the estimates were derived from a formula by Tomozawa et al. [3] given as Eq.(7). Young's Modulus E (kn/mm ) 50 40 30 0 10 L M U No.19 No.0 Eq.(7) JSCE -1-

1 L M U No.19(L) No.0(U) No.1(L) No.1(M) No.1(U) No.(U) 1 L & M U 1 4 Eq.(8): ( ) / f t d / l k A /f t B /f t 0.1 0.1 1 10 d/l h Fig.10 Relation of d/l h to /f t 1 Eq.(9): ( ) / f t d / l k 0.1 0.1 1 10 d/l h Fig.11 Investigation of size effets 0.3 ( 100 ' 9.8) ( 1000) 9.8 100 E = 9. f γ (7) Conrete densityγ was set at,346 kg/m 3 [0]. Regarding the relationship between ompressive strength and Young s modulus, Fig. 9 ompares the relationship between estimates in this test and the existing data. From this relationship, the measurements and estimates were judged to be generally in agreement, and the Young s modulus values were evaluated as valid. As the frature energy was unknown in the ase of the existing data (Table 5), the measurements in Table 4 were divided into three ategories aording to ompressive strength and applied (Table 6) to determine the harateristi length values. As already noted, Gustafsson and Hillerborg have reported that, with regard to the size effet on the shear strength of ordinary-strength onrete, / ft is proportional to the minus 1/4 power of d/l h [3]. With regard to the tendeny toward raking loalization, whih is onjetured to be a ause of the size effet of shear strength, attention in this study fouses on harateristi length, whih an be evaluated quantitatively, and this approah is applied to the study of the size effet of shear strength arried out using the ombined results of the present test (Table ) and the existing data (Table 5). The results are shown in Fig. 10. Figure 10 onfirms that there is a orrelation between d/l h and / ft, as in the researh by Gustafsson and Hillerborg. The data points for onrete of low- and medium- strength (L and M) are distributed between both grouping A and B marked in Fig. 10. In ontrast, data points for high-strength onrete (U) fall almost entirely in grouping B. This also indiates that the size effet in high-strength onrete tends to be different from that in low- and medium- strength onrete. Figure 11 shows the same orrelation as in Fig. 10, but with the data points plotted as either low- and medium-strength onrete (L and M) or as high-strength onrete (U). Regression analysis for the d/l h and / f t orrelations shows that the size effet is almost exatly proportional to d/l h to the minus 1/4 power and d/l h to the minus 1/ power, respetively, for the two ases. Aordingly, these values are adopted, and relational expressions are determined for eah using the least squares method. In Fig. 11, the red line indiates the urve of Eq.(8), while the blak line indiates the urve of Eq.(9). f t 1 4 ( d l ) h (8) f t 1 ( d l ) h (9) -13-

The test results for low- and medium- strength onrete are expressed by Eq.(8), and this is shown to be in agreement with the existing size effet law (proportional to the effetive depth to the minus 1/4 power) as inorporated into the JSCE formula. The test results for high-strength onrete learly show a different tendeny from that represented by Eq.(8), with ft proportional to d/l h to the minus 1/ power. In other words, that shear strength is proportional to the effetive depth to the minus 1/ power. The results of this frature mehanis based study demonstrated that harateristi length ould be used to evaluate the size effet of shear strength in high-strength onrete using the expression giving by Eq.(9). In the next setion, based on these results, a formula for alulating shear strength in high-strength onrete will be derived and verified. b) Derivation and verifiation of formula to alulate shear strength in high-strength onrete In order to evaluate shear strength in a rational manner, appropriate values must be established for the size effet and various fators affeting shear strength. Equation (4) by Niwa et al., whih was the basis for the JSCE formula, is the produt of fators relating to ompressive strength, tension reinforement ratio, effetive depth, and shear span ratio. For ordinary-strength onrete, this formula yields results that agree losely with the test results []. Aordingly, it was deided to derive a formula for alulating shear strength by onsidering the various fators also as the produt of individual fators. We onsider first the effet of tension reinforement ratio and shear span ratio. It has been reported as noted above that the tension reinforement ratio fator for the shear strength of high-strength onrete is no different from that of ordinary strength onrete evaluated using Eq.(4) [13]. As regards the effet of shear span ratio, it was onfirmed in Setion 4 (4) in that the relationship between shear strength and shear span ratio is not dependent on ompressive strength and an be evaluated from the fators relating to shear span ratio in Eq.(4). Aordingly, the relationship between shear strength and the offset value for shear strength *, from whih the effets of tension reinforement ratio and shear span ratio have been eliminated, an be expressed by the relation given below. 1 3 {( 100 ) ( 0. 75 + 1. ( a d ))} * = p w 4 Next, let us turn to the effet of ompressive strength and effetive depth. The data used in the study of size effet in the previous setion were onstant in terms of shear span ratio (whih was 3), so the data points show no effet of shear span ratio. Moreover, as orretions were made for differenes in tension reinforement ratio, there is thought be almost no effet of tension reinforement ratio. Aordingly, the data points in the study are thought to be almost ompletely ontrolled by material properties (of whih ompressive strength is a typial example) and effetive depth. The effet of effetive depth is none other than the size effet, so the results of the previous setion (Eq.(9)) an be applied. In the CEB-FIP Model Code 90 [8], the frature energy is given by the formula below as the relationship between onrete ompressive strength and maximum aggregate size. (10) G = G f f 0 ( ) 0. f ' f ' 7 0 (11) Where, G f0 : Basi value of frature energy dependent on maximum aggregate size, f' 0 : 10 N/mm, and f' 80 N/mm. This study, whih is onerned with high-strength onrete where f' > 80 N/mm, falls outside the sope of this formula. However, the frature energy in high-strength onrete is also thought to depend on ompressive strength, and here the frature energy in high-strength onrete is postulated in terms of the formula below. G m1 f f ' (1) -14-

where m 1 is a onstant. From the relationship between Eq.(7) and Eq.(1), the following relationship results : E G f 0. 3 m ( ) ( ) 1 m f ' f ' f ' (13) Where m is a onstant. If, as a result of this formula, a proportional onstant n is introdued into Eq.(9) from Eq.(6) and Eq.(13), the result is as follows: * = n f = n t f ' ( d f E G ) m d t -1 f 1 This formula enables * to be expressed as a relationship between ompressive strength and effetive depth (in whih m is a onstant). As a result, it is possible to evaluate shear strength using effetive depth and, as a material property, ompressive strength. Next, from among the results of this test (Table ) and the existing data (Table 5), let us take the data points for high-strength onrete (f' > 80 N/mm ) to determine the relationship between onstants n and m (in other words, */d -1/ and f' ). Figure 1 shows the relationship between */d -1/ and f'. The results of regression analysis (the line in the figure) are shown below. (14) */d-1/ 30 0 Regression analysis Eq.(15) 10 80 100 10 140 Compressive strength f' (N/mm ) Fig.1 Relation of onstants n and m to ompressive strength * 1 d = 190 f ' -0. 51 (Correlation 0.66) Furthermore, the above formula an be used to simplify Eq.(15) in a pratial manner, as indiated by the broken line in Fig. 1. * 1 d = 180 f ' -1 (Correlation 0.66) (15) It an be seen from Fig. 1 that the results given by Eq.(15) are in almost perfet agreement with the regression analysis, and that there is omparatively good orrelation between ompressive strength and */d -1/. Aordingly, Eq.(14) an be expressed as follows: * -1 f ' = 180 d -1 (16) As is lear from Eq.(16), the material property fators in the shear strength of high-strength onrete are evaluated using f' -1/, as opposed to those for ordinary strength onrete whih are evaluated using f' -1/3. From the above, the following formula is proposed as a formula for alulating the shear strength of high-strength onrete (ompressive strength 80 < f' 15 N/mm ). = * = 180 f ( 100 pw ) 1 3 ( 0. 75 + 1. 4 ( a d )) 1-1 3 ' d ( 100 p ) 1 ( 0. 75 + 1. 4 ( a d )) To verify the auray of Eq.(17) for a shear span ratio of 3 and a tensile reinforement ratio of 1.53% a omparison of the results of this test (Table ) and existing researh data (Table 5) orresponding to ompressive strengths of 80, 90, 100, and 15 N/mm, is shown in Fig.13, with alulations of shear strength using eah of the ompressive strength values. Figure 13 shows the test results for a tension reinforement ratio of 1.53% and for offset values with the tension reinforement ratio onverted to 1.53%. With the effetive depth for eah ompressive strength value as a parameter, this figure shows results obtained with w (17) -15-

(N/mm ) Shear strength.5 1.5 f' = 80 N/mm f' = 90 N/mm f' = 100 N/mm f' = 15 N/mm Eq.(4) 1 Eq.(1) a/d = 3, p = 1.53% w Eq.(17) 0.5 0 500 1000 1500 000 Effetive depth d (mm) Fig.13 Comparison of the proposed formula with test results (N/mm ) Shear strength 1.5 1 0.5 d=50 mm d=500 mm d=1000 mm d=000 mm p w =1.53% a/d=3 0 0 40 60 80 100 10 140 Compressive strength f' (N/mm ) Fig.14 The relation of Eq.(1) to Eq.(17) at f = 80 N/mm Eq.(17) as solid lines and those obtained with Eq.(1) (the JSCE formula) and Eq.(4) as the dashed line and the dotted lines, respetively. From Fig. 13, it an be seen that the results obtained with Eq.(1) fall generally in the safe zone for ompressive strengths 80 100 N/mm and effetive depths less than 1,000 mm. In partiular, when the effetive depth is small, the shear strength tends to be evaluated on the low side; onversely, as the effetive depth beomes larger, the safety fator with respet to the test results tends fall. It has been thought that Eq.(1) an also be applied to high-strength onrete as long as a suitable safety oeffiient is employed. However, these results demonstrate that it is not rational to set a uniform safety oeffiient, and thorough onsideration is required partiularly in appliations to strutures with large member sizes. It an also be seen that Eq.(17) aurately mathes the test results, and that the formula maintains the same auray throughout the ompressive strength ranges of 80-15 N/mm and the effetive depths range of 150-1,000 mm. In other words, Eq.(17) provides a valid evaluation of the size effet of high-strength onrete and enables a rational assessment of shear strength based on the speial harateristis of high-strength onrete. Moreover, omparisons of the test results for high-strength onrete with a shear span ratio of 3 (6 data points) in Table and Table 5 with the results of alulation using Eq.(1) and Eq.(17) reveal that the average value of the ratio for the former is 1.7 with a oeffiient of variation of 17.5%, while the average ratio for the latter is 1.01 with a oeffiient of variation of 11.3%. Similarly, a omparison of the ases in whih diagonal tensile failure ourred regardless of shear span ratio (31 data points) and alulations arried out with Eq.(17) reveals that the average value of the ratio is 1.01 with a oeffiient of variation of 10.7%, demonstrating that Eq.(17) retains the same auray even when the shear span ratio differs. Nevertheless, Eq.(17) is an empirial formula based on test data. At this stage, it would be diffiult to say that there is suffiient test data on frature energy and shear for high-strength onrete in the range of large member. Aordingly, it is though that, in applying when Eq.(17) to atual design, the safety oeffiient should be set on the high side so as to ensure that the alulated results fall within the safe zone. Further studies based on data aumulated in the future will further improve the auray of Eq.(17). Finally, Fig. 14 ompares alulations of the relationship between shear strength and ompressive strength arried out using the existing Standard Speifiation for Design and Constrution of Conrete Strutures (Eq.(1)) and the proposed formula (Eq.(17)). In the ase of the Standard Speifiation alulation, a solid line is used to show results within the appliable range (f' 80 N/mm ), while a dotted line indiates results outside this range. For the proposed formula, a thik line is used to show the appliable range (f' > 80 N/mm ). Here the tension reinforement ratio is 1.53% and the shear span ratio is 3. These results show that, for reinfored onrete beams with no shear reinforement and an effetive depth of 1,000 mm or less -16-

(whih were the subjet of this study) the values given by the proposed formula are greater than those given by the Standard Speifiation at 80 N/mm, the boundary between the appliable ranges. However, this differene between the results falls as the effetive depth is inreased, and situation is reversed at an effetive depth of 1,340 mm; at the point, the values given by the proposed formula are less than those given by the Standard Speifiation. Aordingly, within the range of this verifiation, the existing Standard Speifiation formula provides values on the safe side, but in the ase of large sizes members that are outside the range of verifiation, there is a possibility that the evaluation will be too great. The ontinuity of these assessment formulas should be a topi for future study. 6. CONCLUSION In this study, shear tests were onduted on reinfored onrete beams without shear reinforement and with ompressive strengths in the range 36-100 N/mm, effetive depths of 50 1,000 mm, and shear span ratios of 5. This aim was to study shear strength and the size effet of shear strength. Existing researh data was also used to reate a proposal for the shear strength assessment of high-strength onrete, basing the effort on a study of the size effet using frature mehanis. The onlusions drawn from this study an be summarized as follows: (1) In the shear failure of reinfored onrete beams without shear reinforement, the greater the effetive depth and the greater the ompressive strength, the more signifiant the loalization of raking beame, while at the same time the size effet beame more signifiant. () A study of the size effet of shear strength using harateristi length based on frature mehanis revealed that, for low- and medium-strength onrete, the shear strength was proportional to effetive depth to the minus 1/4 power, as reported in other researh results. For high-strength onrete, however, the shear strength was onfirmed to be proportional to the effetive depth to the minus 1/ power. (3) The material properties fator in the shear strength of high-strength onrete was evaluated using f' -1/, unlike ordinary strength onrete, whih is evaluated using f' 1/3. (4) Based on the test results for ompressive strengths 80 < f' 15 N/mm, Eq.(17) was proposed as a formula for assessing the shear strength of reinfored onrete beams made with high-strength onrete without shear reinforement. The predition auray of this formula was onfirmed to be very high; for 31 test speimens with ompressive strengths in the range 80 < f' 15 N/mm and effetive depths of up to 1,000 mm, the average was ratio to the test results was 1.01, with a oeffiient of variation of 10.7%. 7. A FINAL NOTE This researh foused on a study of the shear strength of high-strength reinfored onrete beams without shear reinforement. For reinfored onrete beams with shear reinforement, it has been reported that the self-ompression of high-strength onrete is greater than that of ordinary-strength onrete, and the resulting internal stress and initial defets redues shear strength [4]. Further, this study was based on the results of shear tests onduted on reinfored onrete beams made with high-strength onrete and with a tension reinforement ratio of approximately p w = 1.5%. As no detailed studies of the effet of tension reinforement ratio have been performed, this reinforement ratio was adopted in aordane with the work arried out by Niwa et al., who studied ordinary-strength onrete. The amount of tension reinforement will affet the internal stress resulting from self-ompression of the high-strength onrete. Thus, in future, further studies must be arried out to determine the shear strength harateristis of reinfored onrete beams made of high-strength onrete, fousing on the effet of internal onstraints suh as shear reinforement, tension reinforement, et. -17-

Referenes [1] JSCE, "Standard Speifiation for Design and Constrution of Conrete Struture, Design", 1996. [] Niwa, J., Yamada, K., Yokozawa, K. and Okamura, H., "Reevaluation of the Equation for Shear Strength of Reinfored Conrete Beams without Web Reinforement", Proeedings of the JSCE, No.37/V-5, pp.167-176, 1986. (in Japanese) [3] Gustafsson, P. J. and Hillerborg, A., "Sensitivity in Shear Strength of Longitudinally Reinfored Conrete Beams to Frature Energy of Conrete", ACI Strutural Journal, May-June, pp.86-94, 1988. [4] Fujita, M., Oodate, T., Yasuda, T. and Sato, R., "Experimental Study on Size Effet of High-strength onrete Beams", Proeedings of JCI, vol.0, No.3, pp.349-354, 1998. (in Japanese) [5] Fujita, M., and Oodate, T., "Effet of Conrete Strength on Shear Frature Yield Strength in Conrete Beam Members", Proeedings of JCI, vol.33, pp.955-960, 000. (in Japanese) [6] Fujita, M., Oodate, T., and Matsumoto K., "Influene of Frature Energy on Size Effet in Shear Strength of RC Beams", Proeedings of JCI, vol.3, No.3, pp.751-756, 001. (in Japanese) [7] Amerian Conrete Institute, "ACI Manual of Conrete Pratie Part 3", 1999. [8] CEB-FIP, "Model Code 1990, Comite Euro-International du Beton", Buelletin D Information No.13/14, Lausanne [9] Bazant, Z. P., and Kim, J-K., "Size Effet in Shear Failure of Longitudinally Reinfored Beam", Journal of ACI, Sept.-Ot., pp.456-468, 1984. [10] Okamura, H., and Higai, T., "Proposed Design Equation for Shear Strength of Reinfored Conrete Beams without Web Reinforement", Pro. of JSCE, No.300, pp.131-141, August 1980. [11] Iguro, M., Shioya T., Nojiri Y., and Akiyama, H., "Experimental Studies on the Shear Strength of Large Reinfored Conrete Beams under Uniformly Distributed Load", Proeedings of the JSCE, No.348/V-1, pp.175-184, 1980. (in Japanese) [1] JCI, "JCI Colloquium on Frature Mehanis of Conrete Strutures Part I (Committee Report), 1990. (in Japanese) [13] Suzuki, M., Ozaka, Y. and Imafuku, K., "Shear Charateristis of Ultra High-strength onrete Beam without Shear Reinforement", JCA Proeedings of Cement & Conrete, No.47, pp.558-563, 1993. (in Japanese) [14] An, X., Maekawa, K. and Okamura, H., "Numerial Simulation of Size Effet in Shear Strength of RC Beams", J. Materials Con. Strut., Pavement; JSCE, No.564/V-35, pp.97-316, 1997. [15] An, X., and Maekawa, K., "Numerial Simulation on Shear Failure of RC Beams", Frature Mehanis of Conrete Strutures, Proeedings FRAMCOS-3, pp.1419-148, 1998. [16] JCI, "Report of JCI Tehnial Committee on Test Method for Frature Property of Conrete", 001. (in Japanese) [17] JCI, "Appliations of Frature Mehanis to Conrete Strutures", 1993. (in Japanese) [18] Niwa, J., Nasra, Z., and Tanabe, T., "Size Effet Analysis for Shear Strength of Conrete Beams Based on Frature Mehanis, Proeedings of the JSCE", No.508/V-6, pp.45-53, 1995. (in Japanese) [19] Publi Works Researh Institute, "Report on shear loading test data of large sale RC beams", Tehnial Memorandum of PWRI, No.346, 1996. (in Japanese) [0] Publi Works Researh Institute, "Joint Researh on Design Method of High-strength Conrete Strutures- Study on shear strength of PC beams with High-strength onrete", Cooperative Researh Report of PWRI, No.1, 1995. (in Japanese) [1] Matsui, Y., Uhida, Y., Rokugo, K., and Koyanagi, W., "Shear strength of Reinfored High-strength onrete Beams without Stirrups", Proeedings of JCI, vol.17, No., pp.655-660, 1995. (in Japanese) [] Abe, Y. Ito, K., Matsubara, K., and Suzuki, M., "An Experimental Study on Shear strength of Reinfored Conrete Beams without Shear Reinforement Using Ultra High Strength Materials", Proeedings of JCI, vol.1, No.3, pp.181-186, 1999. (in Japanese) [3] Tomozawa, F., Noguhi, T., and Onoyama, K., "Investigation of Fundamental Mehanial Properties of High Strength and Super High-strength onrete", Summaries of Tehnial Papers of Annual Meeting Arhitetural Institute of JAPAN, pp.497-498, 1990. (in Japanese) [4] Hayakawa, T., Fujita, M., Mise, A., and Sato, R., "Effets of Shrinkage of High-strength onrete on the Strain Behavior in Shear Reinforement", Proeedings of JCI, vol., No.3, pp.589-594, 000. (in Japanese) -18-