钢筋混凝土梁剪切抗剪强度设计方法的评估

Technical NoteEvaluation of Shear Strength Design Methodologiesfor Slender Shear-Critical RC BeamsZuanfeng Pan1and Bing Li2Abstract:This paper seeks to examine the concrete contribution to shear strength and determine the inclination of the compressive strut within the variable truss model for slender RC shear-critical beams with ing the modified compressionfield theory in place of the conven-tional statistical regression of experimental data,the expression for the concrete contribution to shear strength was derived,and the inclination of compressive struts was determined.A simplified explicit expression for shear strength was then provided,with which shear strength can be calculated without extensive iterative computations.This method was then verified using the available experimental data of209RC rectangular beams with stirrups and compared with the current methods from the American Concrete Institute and the Canadian Standards Association.The theoretical results are shown to be consistent with the experimentally observed behavior of shear-critical RC beams.DOI:10.1061/(ASCE) ST.1943-541X.0000634.©2013American Society of Civil Engineers.CE Database subject headings:Shear strength;Struts;Compression;Concrete beams;Design.Author keywords:Shear strength;Concrete contribution to shear;Inclination of strut;Modified compressionfield theory;Evaluation.IntroductionAlthough theflexural behavior of RC beams is generally well un-derstood,the explanation of shear mechanisms is relatively in-adequate.Over the last century,many researchers have managed to develop semiempirical theories based on extensive experimental data[ASCE-American Concrete Institute(ACI)Committee4261973; ASCE-ACI Committee4451998].Representative models include the limit equilibrium theory,the truss model,the strut and tie model, the plastic theory,and the shear friction theory.However,given the complexity of shear failure mechanisms,none of these theories can offer a complete explanation,and as such,there has been no unani-mously accepted theory.Recent years have seen renewed efforts to develop a theoretical model that is verified by experimental data.Many truss analogy models such as the traditional45°truss model,constant or variable angle truss model,and modified com-pressionfield theory(MCFT)(Vecchio and Collins1986)are widely used as the basis of most shear design methodologies for RC beams. The general methods in LRFD-04(AASHTO2004)and Canadian Standard-04[Canadian Standards Association(CSA)2004]are both based on ing the method in AASHTO LRFD-04,for beams with stirrups,the two factors,b and u,need to be looked up in the data charts.On the other hand,in CSA-04,it is necessary to determine the longitudinal strain at the middepth of the member using extensive iterative computations and a rough gauge of its initial value.The proposed approach in this paper is based on MCFT, rendering it unnecessary for iterative calculations or reference to data tables.The results of the proposed approach are verified using the experimental data of209RC beams with stirrups and compared with the results obtained through the methods mentioned in ACI 318R-08(ACI2008)and CSA-04(CSA2004).Shear Strength for Slender Shear-Critical RC BeamsIt is worthwhile to note that for beams with a small l or deep beams, the hypothesis that plane sections remain plane is not satisfied,and parts of the shear are directly transmitted to the supports by arch action.If the sectional shear design method is used,the results may be conservative without consideration of arch action.For RC beams with stirrups,when l$2.5,the arch action could be considered small(ASCE-ACI Committee4451998).In this paper,the present approach for shear strength based on MCFT is aimed mainly at the slender beams,which means l of the beam is$2.5,because the most practical RC beams are slender,with l ranging from approximately 2.5to6(Kassian1990;Li and Tran2008,2012).Formulas for shear strength in many codes for RC beams take into account the contribution of concrete V c and the contribution of stirrups V s.The MCFT has made an attempt to simplify the transmitting mechanism of concrete using average stresses,average strains,and local variations(Collins and Mithell1991).In the theory, the cracked concrete beam must be capable of resisting the effects of the shear,or the beam will fail before the breakdown of the ag-gregate interlock mechanism,to develop the capacity of a rough and interlocked crack interface for shear transfer.Derived by Collins and Mithell(1991),the contribution of concrete to shear isV c¼min2666640:18ffiffiffiffif0cpbd v0:31þ24ɛ1ðaþ16Þsin usþcos us x,0:33a1a2bd vffiffiffiffif0cpcot u1þffiffiffiffiffiffiffiffiffiffiffiffi500ɛ1p377775ð1ÞFrom Eq.(1),it can be seen that there are two unknowns needed to calculate shear strength:crack angle u and principal tensile strainɛ1.1Lecturer,School of Civil Engineering,Tongji Univ.,Shanghai200092, China.2Associate Professor and Director of Natural Hazards Research Centre (NHRC),Nanyang Technological Univ.,Singapore639798(corresponding author).E-mail:cbli@.sgNote.This manuscript was submitted on July25,2011;approved on July 20,2012;published online on August10,2012.Discussion period open until September1,2013;separate discussions must be submitted for individual papers.This technical note is part of the Journal of Structural Engineering, Vol.139,No.4,April1,2013.©ASCE,ISSN0733-9445/2013/4-619–622/ $25.00.Determination of Crack Angle uThere are three types of shear failure modes for RC beams with stir-rups:crushing of the concrete strut(caused by the dominant arch action),shear compression,and diagonal tension.This paper is based on the premise that the stirrups yield when shear failure occurs in the slender RC beams,and the advantages of this assumption are three-fold.First,when a slender beam fails in the mode of diagonal tension, the shear force at stirrups yielding is approximately equal to the actual shear strength.Second,when a slender beam fails in the mode of shear compression,after stirrups yielding,failure occurs by concrete crushing above the crack.The shear force at stirrups yielding,which is a little lower than the actual shear strength,is taken as the calculated shear strength,which is a little conservative for design but does not sacrifice accuracy.Third,there are calculation methods for theflex-ural crack width in the codes to control theflexural crack width. However,there are no methods in the codes to control the shear crack width.If the peak compressive strain is taken as the identification of shear failure,the yielding of stirrups will induce a larger crack width, whereas the shear crack width can be controlled if the yielding of stirrups is considered as the identification of shear failure for a slender shear-critical beam.From Mohr’s circle of strain of the web,the following equation can be obtained(Collins and Mithell1991):tan2u¼ɛx2ɛ2z2ð2ÞWhen stirrups yield,f15n c tan u(Collins and Mithell1991),and it can be assumed thatɛ25f2=E c,f sx5nE cɛx,and f sz5nE cɛz for simplification.By substituting the preceding equations into the Mohr’s circle for average concrete stresses combined with Eq.(2), the following is obtained:tan2u¼n cot u2n c tan un r sþnðtan uþcot uÞ2n c tan un tan u2n c tan un r vþnðtan uþcot uÞ2n c tan uð3ÞTo determine u,therefore,iterative computations are needed to solve for n c in Eq.(3).For beams with a proper amount of stirrups, the value of V c=V ranges from20to60%.Here,it is assumed that n c50.4n,and this hypothesis has little effect on thefinal value of u.Substituting n c50.4n into Eq.(3)0:6þ0:6n r vtan4u¼1þ1n r s20:4þ0:4n r stan2uð4ÞIt is known that u usually varies from25to45°,the value of u in theright side of Eq.(4)is assumed to be equal to35°,and this hypothesishas little influence on thefinal value of uu¼arctanB@43Â1þ1n r s1þ1n r v1C A0:2526643775ð5ÞRationality of Simplified Equation of Calculating uEq.(5)is derived based on the assumption of n c50.4n,and theinfluence of the n c=n value on u is discussed here.Substitutingn c50.25n into Eq.(3),uðn c50.25nÞ=uðn c50:4nÞ51.07;therefore,there is little difference.Similarly,if n c50.6n,thefinal value ofu has little change.In addition,Eq.(5)is derived based on theassumption of the value of u in the right side of Eq.(4)being35°.Fig.1shows the differences between the calculated values ofEqs.(4)and(5),in which,n5E s=E c56.0,and the calculated valueof Eq.(5)is very close to that of Eq.(4).Hence,Eq.(5)can be usedto calculate u.Solution Algorithm for Shear StrengthAt shear failure when stirrups yield,the Mohr’s circle of concreteaverage strain can be used to calculate the tensile strainɛ1ɛ1¼2Àɛyþɛ2Áj cos2u jþ12ɛ2ð6ÞThe principal compressive strain in concreteɛ2is related to both theprincipal compressive stress f2and the principal tensile strainɛ1(Vecchio and Collins1986).The step-by-step solution process issummarized in theflowchart shown in Fig.2,and the formulas forcalculating V s,f1,f2,v,v ci,max,and f1,max can be found in the MCFT(Collins and Mithell1991).Comparison with Experimental ResultsThe validation of the proposed truss approach is demonstrated bycomparison with published experimental results from previousinvestigations with respect to the shear strength at this state.Thereare209rectangular beams with stirrups with l$2.4(Kim2004;Fig.1.Difference between Eqs.(4)and(5)El-Metwally2004;Lu2007).These beams encompass a wide range of sizes and material properties,and all the selected beams are shear-criticalflexural members.The experimental database based on the 209beams with stirrups was used to evaluate the proposed method. The calculated shear strengths by the proposed method and ex-perimental results are compared in Fig.3.The mean ratio of the experimental to predicted strength and its coefficient of variation are 1.204and0.207for the proposed iterative method,showing a good correlation between the proposed method and the experimental data. Most importantly,the analytical results based on the proposed method are on the safer side as illustrated in Fig.3,because the dowel action and shear carried by the compression zone in the concrete contribution were not taken into account.Determination of Principal Tensile Strainɛ1The calculation method described previously for shear strength requires a program to resolve the iterative computations,which is complex for practical use.Eq.(1)can be used to calculate V c explicitly;however,there is still an unknown:ɛ1.The method to resolve the inclined crack width provides us with the information that there is a relation betweenɛ1andɛz.There are many methods to calculate the inclined crack width,and all are related to the strains of stirrups.De Silva et al.(2008)compared the predicted inclined crack width with the experimental data and concluded thatv¼k w s m uɛzð7Þwhere k w5coefficient for the effect of the web reinforcement angle and is equal to1.2for vertical paring Eq.(7)with v5ɛ1s m u(Collins and Mithell1991),ɛ151.2ɛz.Eq.(1)indicates that it will be unsafe for design ifɛ1is taken to be small.For this reason, using the program shown in Fig.2,based on the database of209 rectangular beams with stirrups,the calculated mean value ofɛ1=ɛy is equal to1.34,and its coefficient of variation is0.08.The relation ɛ1=ɛy51.35is used in this paper to simplify calculation.Therefore, Eq.(1)is changed toV c¼min26666640:18ffiffiffiffif0cpbd v0:31þ32:4f yÀa gþ16ÁE s1sin usþcos us x,0:33a1a2bd vffiffiffiffif0cpcot u1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi675f y=E sp3777775ð8ÞEvaluations of Shear Methodologies Based onShear DatabaseBased on the database of209rectangular beams with stirrups,the mean ratio of the experimental to predicted strength and its coefficient of variation are1.204and0.207,1.213and0.214,1.405and0.256, and1.394and0.228for the proposed iterative method and simplified method and the sectional design methods in ACI318R-08(ACI 2008)and CSA-04(CSA2004),parison of the available models with experimental data indicates that the proposed approach produces better mean ratios of the experimental to pre-dicted strength than others.The methods in ACI318R-08and CSA-04lead to very conservative results compared with experimental tests of shear-critical RC beams.Also,a good correlation betweenthe experimental and predicted strengths across the range of f0c,d v, l,r s,and r v f yv is found,which indicates that the proposed approach represents the effects of these key parameters very well.ConclusionsIn this study,a theoretical method to compute the inclination of struts and predict the shear strength of RC beams is proposedbased Fig.3.Correlation of experimental and predicted shear strength based on proposed methodon MCFT.First,the expression of u is rationally derived as shown by Eq.(5),accounting for the contribution of the tensile stresses in the concrete between the cracks based on MCFT.A program is then developed to calculate the shear strength.However,the iterative computations are complicated and time-consuming.Associating the calculation of shear crack width with the relationship between the principal tensile strain ɛ1and the strain of stirrups ɛz ,a simpli fied explicit expression for V c [shown by Eq.(8)]is given,which does not require reference to a table or iterative computations.A shear database is compiled for slender shear-critical beams with l $2.4,which is used to evaluate the present approach and the methods in ACI 318R-08(ACI 2008)and CSA-04(CSA 2004).There is a good correlation between the shear strengths obtained by the proposed simpli fied method and the published experimental data,with the average ratio of experimental to predicted shear strength of the 209RC rectangular beams and the coef ficient of variation being 1.213and 0.214,respectively.The proposed method therefore provides a potential alternative to the existing techniques.NotationThe following symbols are used in this paper:a g 5maximum aggregate size;b 5web width;d v 5effective shear depth taken as flexural leverarm,which needs not be taken less than 0.9d ;E c 5modulus of elasticity of concrete;f 0c 5cylinder strength of concrete;f ci 5compressive stress on crack surface (assumedas zero in this model);f sx 5tensile stress in the longitudinal steel;f sz 5tensile stress in the transverse steel;f vy 5yielding stress of transverse reinforcing steel;f y 5yielding stress of longitudinal reinforcingsteel;f 15principal tensile stress in cracked concrete;f 25principal compressive stress in crackedconcrete;n 5ratio of modulus of elasticity of reinforcingsteel to modulus of elasticity of concrete;s 5spacing of stirrups;s x 5vertical spacing of longitudinal barsdistributed in the web;V c 5contribution of concrete to shear;V s 5contribution of stirrups to shear;a 15factor accounting for bond characteristics ofreinforcement:a 151.0for deformed bars,a 150.7for plain bars,wires,or bonded strands,a 150for unbonded reinforcement;a 25factor accounting for sustained or repeatedloading,a 251.0for short-term monotonic loading,a 250.7for sustained and/or repeated loads;ɛx 5tensile strain in the longitudinal steel;ɛz 5tensile strain in the transverse steel;ɛ15principal tensile strain in cracked concrete;ɛ25principal compressive strain in crackedconcrete;u 5angle of the inclined strut in cracked concretewith respect to longitudinal axis of member in the variable truss model;l 5shear span to effective depth of section;n 5applied shear stress;n c 5contribution of concrete to shear stress equalto V c =ðbd v Þ;n ci ,max 5maximum shear stress on a crack given widthresisted by aggregate interlock;r s 5ratio of area of longitudinal reinforcement toeffective sectional area of beam;r v 5ratio of volume of shear reinforcement tovolume of the concrete core measured to the outside of stirrups;and v 5crack width.ReferencesAASHTO.(2004).LRFD bridge design speci fications ,3rd Ed.,AASHTO,Washington,DC.American Concrete Institute (ACI).(2008).“Building code requirements for structural concrete and commentary.”ACI 318R-08,Farmington Hills,MI.ASCE-American Concrete Institute (ACI)Committee 426.(1973).“Shear strength of reinforced concrete members.”J.Struct.Eng.,99(6),1091–1187.ASCE-American Concrete Institute (ACI)Committee 445.(1998).“Recent approaches on shear design of structural concrete.”J.Struct.Eng.,124(12),1375–1417.Canadian Standards Association (CSA).(2004).“Design of concrete structures.”CAN CSA A23.3-04,Rexdale,ON,Canada.Collins,M.P.,and Mithell,D.(1991).Prestressed concrete structures ,Prentice Hall,Englewood Cliffs,NJ.De Silva,S.,Mutsuyoshi,H.,and Witchukreangkrai,E.(2008).“Evaluation of shear crack with in I-shaped prestressed reinforced concrete beams.”J.Adv.Concrete Technol.,6(3),443–458.El-Metwally,A.S.(2004).“Shear strength of reinforced and PC beams using shear friction.”Ph.D.thesis,Calgary Univ.,Calgary,AB,Canada.Kassian,W.A.(1990).“Concrete contribution to shear strength in slender beams.”Ph.D.thesis,Univ.of Alberta,Edmonton,AB,Canada.Kim,K.S.(2004).“Shear behavior of reinforced concrete beams and prestressed concrete beams.”Ph.D.thesis,Univ.of Illinois at Urbana-Champaign,Urbana,IL.Li,B.,and Tran,C.T.N.(2008).“Reinforced concrete beam analysis supplementing concrete contribution in truss models.”Eng.Struct.,30(11),3285–3294.Li,B.,and Tran,C.T.N.(2012).“Determination of inclination of strut and shear strength using variable angle truss model for shear-critical RC beams.”Struct.Eng.Mech.,41(4),459–477.Lu,Y.(2007).“Theoretical and experimental research on shear capacity of high strength concrete beams with high strength stirrups.”Ph.D.thesis,Hunan Univ.,Changsha,China.Vecchio,F.J.,and Collins,M.P.(1986).“The modi fied compression field theory for reinforced concrete elements subjected to shear.”ACI J.Proc.,83(2),219–231.。