- Tehnial Paper - THE EQUATION CONSIDERING CONCRETE STRENGTH AND STIRRUPS FOR DIAGONAL COMPRESSIE CAPACITY OF RC BEAM Patarapol TANTIPIDOK *, Koji MATSUMOTO *, Ken WATANABE *3 and Junihiro NIWA *4 ABSTRACT Objetives of this study are to investigate the effet of ompressive strength, stirrup ratio and stirrup spaing on the diagonal ompressive apaity and propose the prediting equation for the diagonal ompressive apaity of RC beams onsidering the effet of ompressive strength and stirrup spaing. Five I-beams ere tested by three-point bending. As a result, the effet of ompressive strength and stirrup spaing on the diagonal ompressive apaity is interrelated. The proposed equation provides a better estimation in ase of using high-strength onrete than the existing equations. Keyords: diagonal ompressive apaity, high-strength onrete, eb rushing, predition equation. INTRODUCTION Reently, high-strength onrete ith the ompressive strength (f ) more than N/mm and high-strength reinforing bars ith yield strength (f y ) more than 685 N/mm have been developed. Suh advaned materials have been applied into the onstrution pratie sine designers an take their advantages and propose eonomial infrastruture by reduing material volumes, alloing the inrease in the span length of onrete bridges and the redution of ross-setional area of members. Hoever, in thin eb T- or I-shaped reinfored onrete (RC) beams ith exessive shear reinforements, an unommon type of shear failure may our. This type of failure is knon as diagonal ompression failure. It is aused by the rushing of eb onrete prior to the yielding of stirrups. Researh on the mehanism of diagonal ompression failure as insuffiient sine it as usually avoided. The prediting equation for the diagonal ompressive apaity of RC beams in the urrent JSCE Standard Speifiations for Conrete Strutures [] has underestimated the diagonal ompressive apaity and not been appropriate for the appliation to RC beams using high-strength onrete as reported by Kobayashi et al. []. Besides, there are limited studies on diagonal ompression failure in RC beams using high-strength onrete exept for Kobayashi et al. []. They reported that the fators affeting the diagonal ompressive apaity ere f, the shear reinforement ratio (r ), the shear-span to effetive depth ratio (a/d) and the spaing of stirrups (s). Hoever, in regard to some fators, e.g. f, r and s, there is a lak of experimental data espeially in f > N/mm and r < %. Objetives of this researh are to investigate the effet of f, shear reinforement ratio and stirrup spaing on the diagonal ompressive apaity of RC beams and to propose the prediting equation for the diagonal ompressive apaity of RC beams onsidering the effet of f and stirrup spaing. Furthermore, the auray of existing equations for the diagonal ompressive apaity is verified and ompared ith the proposed equation.. REIEW OF EXISTING EQUATIONS FOR DIAGONAL COMPRESSIE CAPACITY OF RC BEAMS. The equation by Plaas and Regan Plaas and Regan proposed an empirial equation for evaluating the diagonal ompressive apaity as the folloing [3]: Plaas (.4.r ) f ' b d () The fators involving the diagonal ompressive apaity in this equation are f / and the ratio of stirrup r (%).. JSCE Standard In JSCE standard speifiations [], only f / is the influening parameter of the diagonal ompressive apaity. This equation is a onservative estimation and only valid for onrete ith f loer than 5 N/mm..5 f b d () JSCE '.3 Euroode 99--:4 In Euroode [4], the variable strut inlination method as used to determine the maximum shear * Graduate student, Graduate shool of Civil Engineering, Tokyo Institute of Tehnology, JCI Member * Post-dotoral fello, Dept. of Civil Engineering, Tokyo Institute of Tehnology, Dr. E., JCI Member *3 Assistant Prof., Dept. of Civil Engineering, Tokyo Institute of Tehnology, Ph. D., JCI Member *4 Prof., Dept. of Civil Engineering, Tokyo Institute of Tehnology, Dr. E., JCI Member
fore hih an be sustained by the member, limited by rushing of the ompression struts. Assuming the angle of onrete strut, θ, to be 45 degree, the folloing expression an be obtained. EC vf ' b. 9d (3) here; v is a oeffiient that takes into aount the inrease of fragility and the redution of shear transfer by aggregate interlok ith the inrease of f. It may be taken to be.6 for f 6 N/mm and the minimum beteen.9-(f /) and.5 for f > 6 N/mm. Sine the existing equations did not onsider some influential fators and are not appliable to high-strength range, further investigation is required. 3. EXPERIMENTAL PROGRAMS 3. Speimen details The experimental program prepared five RC beams ith I-shaped ross setion. Three-point bending tests ere onduted by a kn apaity hydrauli testing mahine. The summary of test variables and details of speimens are provided in Table and Fig.. The main parameters ere ompressive strength of onrete f, stirrup ratio r ith different diameter and spaing s. The experimental ases an be lassified into to series: one is the ases for the effet of r and s in high-strength onrete beam ith various diameters of stirrups (7.,, 3 mm). The other is the ases for the effet of the s ith different f. The other parameters and the speimen s dimension ere the same as in the study by Kobayashi et al. [] in order to ompare and disuss ith their results. The onstant variables ere the folloing: diameter of tensile bars of mm, shear span (a) of 66 mm, the effetive depth (d) of mm, a/d ratio of 3. and the total length of 8 mm. All speimens ere designed to be symmetri and be able to resist against the flexure failure and the diagonal tension failure by using high-strength reinforing bars (f y > 93 N/mm ) as tensile and shear reinforements. In addition, the ombination of thin eb ross setion ith dense reinforement ill ause speimens to exhibit the diagonal ompression failure. In order to avoid the loal failure, the eb idth outside support as inreased to that of the bottom flange. Anhor plates and bolts ere used to ensure the suffiient anhorage of the tensile bars and prevent anhorage failure. 3. Instrumentation and test proedures For all speimens, applied load, mid-span defletions and strains of onrete, longitudinal bars and stirrups ere measured. Strain gauges ere attahed at the mid span to measure the strain of longitudinal bars hereas at the distane of d/ from ompression fiber for all stirrups in the shear spans. Besides, both surfaes of all speimens ere painted by hite olor to ease the draing and observation of rak during the experiments. 4. EXPERIMENTAL RESULTS The experimental results inluding Kobayashi et al. [] are summarized in Table. Data of the strains of Speimen f' [N/mm ] b d Table List of experimental ases a/d b * * p f /b D *3 f y [N/mm ] r *4 d *5 s *6 f y *7 [N/mm ] UH.. 5 UH.5 4.5 953 UH.8 4 3. 6.5 8.8..8 SUHs9 3.5 9 3 93 3 SUHs6. 6 955 * upper flange idth, * longitudinal reinforement ratio, *3 nominal diameter of longitudinal bars, *4 stirrup ratio, *5 diameter of stirrups, *6 spaing of stirrups, *7 yield strength of stirrups. B B 4 5 C L s 66 66 5 6 A 6 A : Strain gauge Load [kn] 3 5 5 5 UH. UH.5 UH.8 SUHs9 SUHs6 5 5 Unit: mm A-A setion B-B setion Fig. Dimensions and steel layout of speimens 4 6 8 Defletion Fig. Load-defletion relationships
Table Summary of experimental results Speimen f' [N/mm ] r s a/d b f /b exp * [kn] exp / Plaas exp / JSCE exp / EC * [N/mm ] /f / UH. 5. 5.7.84.9.49.7.4 UH.5.5 8..86.98.54.3. UH.8 7.8 34.3.99.8.63 5.3.47 SUHs9 3 3.5 9 9.3.69.3.5 4.7.8 SUHs6 8. 6..7.88.43.5. UH []. 9 37...3.68 5.6.54 UH3 [] 98. 3. 6 44.9.95.33.75 6.5.66 UH4 [] 99. 4. 45 67.6.97.53.85 9..9 UHs5 []. 5 68.5.9.46.78 9..83 UHs6 [] 3. 6.5 5. 6 96.6.67.88.43..3 SUH3 [] 33 3. 6 6..96.8.6 8.4.6 N6 [] 3..63 6 59.4.97.95.78 6.7.9 N [] 33.4. 6 58.6.88.9.74 6.7.5 N [] 35.8. 8 6.6.76.94.7 7..7 N3 [] 35..9 55 6.7.68.95.74 7..8 H [] 73.8. 8 4.5.99..74 3..5 H3 [] 6.5.9 55 7.8.89.5.75.3.56 SSUH3 [] 65 3. 6 9..97.36.5.8.7 * the shear apaity from experimental result, * shear stress from experiment (= exp /b d). longitudinal bars and stirrups revealed no yielding and the values ere muh less than their yield strength. It implies that the failure mode is neither flexure failure nor diagonal tension failure. 4. Load-defletion relationship Load-defletion relationship is illustrated in Fig.. Firstly, speimens behaved in elasti manner until the first flexural rak ourred, hih is refleted in the graph as a rate of inlination dereased. After the first flexure rak, the load-defletion urve remains to advane linearly ith the initiation of diagonal rak at the eb onrete. In the pre-peak region, the defletion inreased ith a relatively small inrease in applied load until the eb onrete began to rush. Data from the strain gage attahed in stirrups and tensile bars reveals no yielding strain in all stirrups and tensile bars. After the peak load, applied load rapidly dereased. 4. Effet of r and s () Effet of r and s in speimens ith different diameter of stirrups Kobayashi et al. [] adapted a method to eliminate the effet of f by normalized the obtained / shear apaities by f hih is used in the prediting equations of JSCE [] and. This method is also applied in this study. The relationships beteen r and /f / for the speimens ith f = N/mm are demonstrated in Fig. 3. With the inrease in r, the diagonal ompressive apaity inreases hen using stirrups of d = mm. A linear relationship agrees ell ith the test results. Hoever, in the ase of onstant value of r = % ith different d and s, / /f differs signifiantly. Furthermore, the experimental results of the same set of data ere plotted against s despite of r in Fig 4. Notithstanding the diameter of stirrups and r are different, the diagonal ompressive apaities are distributed along the same linear line. It means that the diagonal ompressive apaity an be represented by not r but s. Therefore, the diagonal ompressive apaity ill be disussed by s in the folloing setion. The differene in the diagonal ompressive apaity ith the hange in s an be explained by to mehanisms []. One is the inrease of onfinement effet on raks by providing loser shear reinforements. It results in smaller diagonal rak idth (); therefore, the ritial average stress in eb onrete (f max ) ould be greater beause affeted the diagonal ompressive apaity as reported by Shäfer et al. [6] and Reinnek [7]. Seond is the loalization of ompressive strut. Fig. 5 explains the model of ompressive strut formation under different stirrup spaing proposed by Kobayashi et al. []. Fig. 5(a) demonstrates that the diagonal stress generates uniformly along the beam ith lose-spaing stirrups. In ontrary, as shon in Fig. 5(b), the diagonal stress as onentrated in a loal portion of the beam ith ide-spaing stirrups. This stress onentration aused early rush in the eb onrete; hene the diagonal ompressive apaity dereased. This mehanism an also be observed in this study. Fig. 6 illustrates rak patterns after loading. The thiker lines and the grey areas represent the ider rak idth and rushing areas, respetively. In speimens ith lose spaing (UH.8 and SUHs9), raks distribute more finely and the rushing area at eb distributes more idely than that of the ide-spaing speimens (UH. and SUHs6). It implies that the failure loalization ourred in UH. and SUHs6. Sine the onfinement of stirrups helps stress to transfer aross a rak after the rak initiation, the stress an distribute along the beam. Therefore, the loalization of ompressive strut seemed to be indued
by a lak of the onfinement effet. Hoever, further study by some analytial approah is required to larify atual stress flo. () Effet of s ith different f / Fig. 7 shos /f as a funtion of s for speimens ith different f. It is onfirmed that the effet of s depends on f. There ere slightly variation in /f / by the hange in s hen using normal strength onrete hile the greater effet of s an be observed in high-strength onrete beams. As for beams ith f > 6 N/mm, the diagonal ompressive apaity redues as s inreases in linear relationship. With higher f, the inlination of trend line inreases ith a limitation hen f approximates N/mm. It an be explained by onsidering that the apaity depends on not only f' but also the idth of ompressive strut. If the idth of ompressive strut beomes smaller ith higher f, there is a possibility that the diagonal ompressive apaity is limited. 4.3 Effet of f Fig. 8 plots the shear strength (= exp /b d) as a funtion of f. Sine there as slight effet of s hen using f = 3 N/mm, the speimen N3 [] is used as a representative of speimen ith f = 3 N/mm and s = 6mm in this figure. It an be observed that inreases ith the inrease in f. As disussed before, the f and s are interrelated. It an be observed again in Fig. 8 as the inremental rate of is different hen using different s and this inrement rate dereased as f inreased. From Fig. 6, there is not muh differene in rak patterns after loading beteen speimens ith different f but approximately the same s. 5. DEELOPMENT OF EQUATION 5. Auray of the existing equations The shear apaity from experimental results of d =mm () d =mm (Kobayashi et al.) [] d =3mm (Kobayashi et al.) [] d =7.mm (Kobayashi et al.) [] r =.% () r =.5% () r =.8% () r =% (Kobayashi et al.) [] r =3% (Kobayashi et al.) [] r =4% (Kobayashi et al.) [] Stress is uniform along the beam / /f'.5.5 d = mm / /f'.5.5 (a) Narroly spaed stirrups Stress is onentrated in the speifi part.5.5 3 4 5 r Fig. 3 Effet of r (f = N/mm ) 5 5 s Fig. 4 Effet of s (f = N/mm ) (b) Widely spaed stirrups Fig. 5 Effet of spaing of stirrups UH. UH.5 f' =N/mm () f' =3N/mm () f' =N/mm (Kobayashi et al.)[] f' =3N/mm (Kobayashi et al.) [] f' =35N/mm (Kobayashi et al.) [] f' =65N/mm (Kobayashi et al.) [] s=6 mm() s=6 mm (Kobayashi at al.) [] s~6mm (Kobayashi et al.) [] UH.8 SUHs9 / /f'.5.5 f = N/mm f = 3 N/mm v [N/mm ] exp 5 s=6mm 5 SUHs6.5 f = 65 N/mm f = 35 N/mm 5 5 s s=6mm 5 5 Fig. 6 Crak patterns after loading Fig. 7 Effet of s ith different f Fig. 8 Effet of f ( - f ) 5 f' [N/mm ]
/ exp Plaas Kobayashi et al. [] / exp JSCE Kobayashi et al. [] exp / EC Kobayashi et al. [].5.5.5 Avg. =.8 C.. = 3.5 %.5 Avg. =.96 C.. = 8.4 % 5 5 f' [N/mm ] (a) Plaas (Eq. ).5.5 Avg. =. C.. = 9. % 5 5 5 5 f' [N/mm ] f' [N/mm ] (b) JSCE (Eq. ) () Euroode (Eq. 3) Fig. 9 Auray of the existing equations the total of 3 beams inluding Kobayashi et al. [], and as used to illustrate the auray of existing equations. The ratio beteen shear apaity and results alulated by the existing equation revieed in hapter are also listed in Table. The average of these ratios (avg.) ith a oeffiient of variation (C..) is presented in Fig. 9. The average of exp / Plaas =.96 ith a C.. of 8.4%. It implies that Plaas equation an evaluate an average value of the diagonal ompressive apaity inluding the ase of f > N/mm. In ontrary, C.. of 8.4% implies that there are other fators affeting the diagonal ompressive apaity exept f and r. In addition, Fig. 9(a) presents that in the ase of f > 5 N/mm, the equation of Plaas et al. overestimates the diagonal ompressive apaity. JSCE standard demonstrated the average of exp / JSCE =.. Fig. 9(b) indiates that JSCE equation underestimates the apaity in almost all speimens. The speifiation may alulate the diagonal ompressive apaity onservatively for safety reason. In the same ay as Plaas equation, the results alulated by the speifiation shoed large variation as C.. = 9.%. The auray of Euroode for prediting the diagonal ompressive apaity as shon in Fig. 9(). The avg. of exp / EC is.8 ith a C.. of 3.5 %. Similar to Eq., Eq. 3 overestimates the apaity in ase of high-strength onrete and shos large sattering. From the reasons mentioned above, the urrent design equations are not aurate enough to evaluate the diagonal ompressive apaity of RC beams, espeially in high-strength onrete. Besides, onsidering only f and r in the existing equation are not appropriate. 5. Equation proposal Assume a free body diagram ut in a vertial diretion as shon in Fig.. The fores ating on utting surfae are ) ompressive fore of onrete fiber, C, ) diagonal ompressive fore of strut, D and 3) tensile fore of tension reinforement, T. Considering the equilibrium in a vertial diretion, the shear fore, must be resisted by the vertial omponent of D. With the assumptions that the diagonal ompressive fore auses the failure of the member ithout yielding of stirrups, the folloing expression an be obtained. max D' max sin f maxb jd os sin (4) here; max : shear fore arried by eb at eb rushing stage (N). θ : angle of inlination of onrete struts to the longitudinal reinforement (degree) f max : ritial average ompressive stress in the eb onrete (N/mm ) From influential fators disussed in hapter 4, f max an be expressed by (f max = ). With the use of trigonometry, Eq. 5 is proposed based on Eq. 4 max b jd sin (5) represents the effet of f and s. As mentioned in setion 4., the relationship beteen the diagonal ompressive apaity and s demonstrated inverse variation. In addition, the effet of s and f are interrelated. Setion 4.3 indiated that the diagonal ompressive apaity inreases as f inreases ith different inrement ratio depending on the value of s. This ratio dereases as f inreases. The 3-D expression of -f -s as used to formulate having minimal variation and orresponding to the relationship observed from the experimental results. The as developed through a surfae-fitting tehnique and as the folloing equation: x ' 3.93(.5 x ) f (6) s here; x.7 735 (f : N/mm, s: mm) (7)
C' Residual of [N/mm ] Kobayashi et al. [] exp / max Kobayashi et al. [] θ T D' jd f [N/mm ] Fitting surfae s.5.5 Avg. =. C.. = 7.8 % 5 5 f' [N/mm ] Fig. Free-body diagram Fig. Residual plot of -f -s Fig. Auray of proposed equation (Eq. 7) The residual plot of -f -s demonstrated in Fig. shos a good auray of the fitting surfae. Kobayashi et al. measured the angle of prinipal ompressive strain of struts of speimens NAD45 and UHAD35 [8]. It is revealed that this angle is onstant and approximately 3 degrees at the peak load. This study assumes the angel of ompressive strut, θ to be same as that of prinipal ompressive strain. By substituting Eqs. 6 and 7 into Eq. 5, jd = (7/8)d as reommended by JSCE [] and θ = 3 degrees, the diagonal ompressive apaity an be alulated. In Fig., the auray of the proposed equation as evaluated ith the same database as in setion 5.. Although the average of. and C.. of 7.8% are not satisfatory, Eq. 6 an predit the diagonal ompressive apaity ell ith slight variation in ase of f > 5 N/mm. It is beause this equation takes into aount only the effet of f and s. Hoever, a/d and b f /b of Plaas et al. s speimens ere designed to be 3.6 and 9.6, onsequently. Kobayashi et al. observed a slight effet of a/d in ase of a/d > 3. []. It implies that the influene of b f /b should be onsidered. Although b f /b of 5.4-6.35 in Rangan s study approximates to the authors speimens, a/d of Rangan s speimens ere.5. It implies that the effet of a/d is signifiant in ase of a/d =.5 to 3.. It is hoped that the development of the equation onsidering all influential fators an be performed. 6. CONCLUSIONS () The effet of ompressive strength and stirrup spaing on the diagonal ompressive apaity of RC beams is interrelated. The greater effet of stirrup ratio as observed in high-strength onrete. Shear apaity inreases as ompressive strength inreases ith different inrement ratio depending on stirrup spaing. The inrement ratio dereased as ompressive strength inreased. () The prediting equation for the diagonal ompressive apaity of RC beams hih takes into aount the effet of ompressive strength and stirrup spaing as developed based on the experimental results. The proposed equation provided a better estimation of the diagonal ompressive apaity in ase of beams using high-strength onrete than the existing equations. ACKNOWLEDGEMENT The authors aknoledge Neturen Co., Ltd. for the support of longitudinal reinforements. REFERENCES [] Japan Soiety of Civil Engineers (JSCE): Standard Speifiation for Conrete Strutures [Strutural Performane erifiation], 7 [] Kobayashi, C., Watanabe, K., Nia, J.: Evaluation for Diagonal Compressive Capaity of RC Beams, Proeedings of the Japan Conrete Institute, ol.3, No., pp.577-58, 9 (In Japanese) [3] Plaas, A. and Regan, P.E.: Shear Failure of Reinfored Conrete Beam, ACI Journal, ol.68, No., pp.763-774, 97 [4] European Committee for Standardization (CEN): Euroode : Design of Conrete Strutures, Part, General Rules and Rules for Buildings, EN 99--:4, pp.84-9, De. 4 [5] Rangan, B..: Web Crushing Strength of Reinfored and Prestressed Conrete Beams, ACI Strutural Journal, ol.88, pp.-6, Jan.-Feb. 99 [6] Shäfer, K., Shelling, G. and Kuhler, T.: Compression and Transverse Tension in Reinfored Conrete Elements, Deutsher Ausshuss für Stahlbeton No.48, Beuth erlag, GmbH, Berlin, pp.-85, 99 [7] Reinnek, K. H.: Theoretial Considerations and Experimental Evidene on Web Compression Failures of High Strength Conrete Beams, CEB Bulletin D Information No.93, pp.59-74, De. 989 [8] Kobayashi, C.: Proposed Method for Evaluating the Diagonal Compressive Capaity of Reinfored Conrete Beams, Master s Thesis, Tokyo Institute of Tehnology, 9