The shear design expressions provided in the ACI

Transcription

1 A Unifie Approach to Shear Design by Robert J. Frosch, Qiang Yu, Gianluca Cusatis, an Zeněk P. Bažant The shear esign expressions provie in the ACI Builing Coe 1 ere evelope over 50 years ago. They ere base on the test results available at the time 2,3 an ith the goal of proviing simplifie esign expressions that conservatively estimate shear strength. As time has passe an aitional test results have become available, it is clear that there are limitations ith the current approach. These limitations relate primarily to lo percentages of flexural reinforcement, high-strength concrete, an member size. Furthermore, ifferent esign approaches are require for nonprestresse an prestresse members. As research has continue to progress, ne perspectives of shear resistance have emerge, inicating that it is possible to eliminate limitations of past practice, unify esign methos, an provie for improve safety of our structures. Unifie Shear Design A unifie approach is propose for the shear esign of structural concrete members. Consistent ith current practice, the nominal shear strength n is the sum of the concrete contribution c an shear reinforcement contribution s : n = c + s. For both nonprestresse an prestresse concrete members, c is calculate using Eq. (1) here is eith in in.; c is the cracke section neutral axis epth in in.; f c is concrete compressive strength in psi; an λ is the moification factor for lighteight concrete. The evelopment of Eq. (1) is presente in References an 5 for nonprestresse concrete, an the equation is further extene to prestresse concrete in References an 7. The benefit of this approach is that this single esign expression can be use to calculate the shear strength of any structural concrete member, incluing nonprestresse, prestresse, partially prestresse, an axially loae (in tension or compression). Furthermore, this same expression can be use to calculate the shear strength of fiber-reinforce polymer (FRP) reinforce members. In fact, Guie for the Design an Construction of (1) Structural Concrete Reinforce ith Fiber-Reinforce Polymer (FRP) Bars (ACI 0.1R-15) 8 uses this esign expression for the calculation of the concrete contribution to shear strength. s is calculate using Eq. (2) here t is the istance in inches from the extreme compression fiber of the member to the centroi of the reinforcement nearest the tension face. Equation (2) is similar to the current expression for s, ith the slight moification of using t in place of the traitional value of effective epth. This moification helps to avoi confusion if multiple layers of reinforcement are provie. Equation (1) is similar to the current expression for c, but there is a major ifference in that c is use rather than, as illustrate in Fig. 1. In effect, the propose expression consiers the average shear stress over the uncracke epth of the cross section, rather than the average shear stress over the effective epth of the member. It is the use of c that unifies the shear strength expression, as c accounts for a number of parameters that affect shear strength. avg b avg bc Flexural Reinforcement Neutral Axis Cracks Fig. 1: Simplifie illustration of istributions of average shear stress τ avg over an c c (2).concreteinternational.com Ci SEPTEMBER

2 The neutral axis epth c is compute at the location here the shear strength is of interest. For nonprestresse members ithout axial force, c is compute as the elastic cracke section neutral axis epth. For a rectangular section ith reinforcement concentrate near the extreme tension fiber, a cracke section analysis results in c presente by Eq. (3). For other members, such as prestresse an axially loae members, c is calculate using strain-compatibility. c = k (3) here, ith the reinforcing ratio given by an the moular ratio given by n = E s / E c. The elastic cracke section neutral axis is appropriate for nonprestresse members ithout axial force for several reasons. First, for shear failures to be consiere vali, the flexural reinforcement must remain belo yiel. Therefore, sections across the shear span remain elastic. Hoever, even hen consiering beams that have resulte in shear failures post-yiel of the flexural reinforcement, shear failure occurs outsie of the yiele region, near locations here the moment is just above the cracking moment an here the elastic neutral axis epth remains applicable. For prestresse an axially loae members, hoever, the failure location oes not necessarily remain in the elastic region, an straincompatibility is use to compute the neutral axis epth. For these members, c is a function of the flexural moment an the axial loa. In general, as the moment is increase, c ecreases. Furthermore, axial compression increases the neutral axis epth hile axial tension ecreases the epth. For prestresse members, thin ebs ith flanges are often use, an eb-shear failures are possible. While the current Coe approach for c can be use, it is possible to provie a esign equation in the same format as that propose for c but incluing the effects of axial stress f pc an the vertical component of the effective prestressing force at the section p (Eq. ()). This expression can be conservatively simplifie by neglecting the axial stress contribution (Eq. (5)), making the equation similar to Eq. (1), but ith the full member epth h replacing c because the section is uncracke. The relationships for a cracke section (Eq. (1)) an an uncracke section (Eq. (5)) thus correspon to the current Coe s relationships for flexure-shear strength ci an eb-shear strength c, respectively. shon in Fig. 2 (test ata consistent ith those provie in Reference ), there is a significant tren toar increasing conservatism ith increasing reinforcement ratio. This plot inclues ata from beams ith fiber-reinforce polymer (FRP) reinforcing bars; therefore, the ata is plotte using the effective reinforcement ratio (ρ eff ), hich is ρ multiplie by the moular ratio (E r /E s ). While a ie range of ρ eff is inclue, the practical range is from 0 to 2%. In focusing on this range, especially belo 1%, it is observe that significantly unconservative results evelop. It is for this reason that ACI Committee 0, Fiber-Reinforce Polymer Reinforcement, aopte a ne expression for the esign of such members. 8 The lack of conservatism for these very practical reinforcement ratios is also of concern for the continue use of this esign expression for members ith steel reinforcement. The influence of the reinforcement ratio is essentially eliminate using the propose esign expression (Fig. 3). As evient across the range of reinforcement ratios, the calculate test calc f b calc c 0.5 Steel FRP Effective Reinforcement Ratio, eff (%) Fig. 2: Influence of reinforcement percentage ACI 318 () (5) Reinforcement Percentage One of the primary limitations of the current ACI calculation proceure for shear strength is relate to its accuracy an conservatism as the reinforcement percentage ρ is varie. As Fig. 3: Influence of reinforcement percentage propose expression 8 SEPTEMBER 2017 Ci.concreteinternational.com

3 shear strengths are conservative, ith the loest results being consistently aroun 1.0. For very lightly reinforce members (particularly FRP members), the calculate shear strengths increase slightly in conservatism. Consiering the smaller number of tests, especially for steel reinforcement in this range, this level of conservatism is reasonable. Concrete Compressive Strength The current esign approach for shear strength limits the contribution of high-strength concrete. While concrete strengths greater than 10,000 psi can be use in a member, ACI 318 limits for calculation to 100 psi unless minimum eb reinforcement is provie. For the propose approach, a limit on the contribution of high-strength concrete is not require. As the concrete strength increases, c ecreases hile increases. The former results in a slightly loer shear strength, hile the latter results in a higher shear strength. The net increase in the value of is greater than the ecrease in c, resulting in slightly greater shear strengths for high-strength concrete. As shon in Fig., the use of the propose esign expression provies consistent results across the ie range of concrete compressive strengths teste. Size Effect The current esign approach for shear strength provies a constant average shear strength ( ) on a shear area of, regarless of the size of the member. Experimental results, hoever, inicate that shear strength oes not increase in proportion to. As shon in Fig. 5, if the reinforcement area is hel constant hile is ouble (Fig. 5(a)), the current expression inicates a oubling of the shear strength hile the propose esign expression inicates only a 50% increase in shear strength. Alternatively, if the reinforcement percentage is hel constant hile is ouble, both equations result in a oubling of the shear strength. The propose expression properly accounts for the reinforcement percentage through a reuction in the shear area ( c), an it provies a result that is not proportional ith beam epth. This example also illustrates the importance of holing appropriate parameters (such as ρ) constant hen evaluating the influence of iniviual variables for a given esign expression. Another limitation of the current esign approach is the use of a constant shear strength ( ). Even hen the shear area is consiere as c (hich accounts for the reinforcement percentage), a ecrease in the average shear strength coefficient (a) Beam A Beam B 2 2 f b c c c (Beam B) = 2 c (Beam A) 5 f bc c c is observe ith increasing (Fig. ). While a coefficient of 5 appears to be satisfactory over the range of epths shon, the average levels of safety ecrease as the effective epth is increase. Therefore, extrapolation of test results beyon the range of the ata shoul be consiere carefully an base on the size effect theory. Consequently, Eq. (1) is moifie by a size effect factor γ, resulting in Eq. (). The factor is base on a proposal by ACI Committee, Fracture Mechanics, 9 hich is reviee in etail by Yu et al. 10 here γ = 1.0 if t < 10 in. or if t < 100 in. an A v A v,min. Otherise here 0 = 10 in. if A v < A v,min or 0 = 100 in. if A v A v,min () (7) A s A s c (Beam B) = 1.5 c (Beam A) (b) 2 f b c c c (Beam C) = 2 c (Beam A) Beam A Beam C A s 2 5 f bc c c c (Beam C) = 2 c (Beam A) 2A s Fig. : Influence of concrete strength propose expression Fig. 5: Influence of oubling beam epth: (a) constant reinforcement area A s; an (b) constant reinforcement percentage ρ.concreteinternational.com Ci SEPTEMBER

4 For beams ithout shear reinforcement, the size effect oes not nee to be consiere if the epth of the extreme tension reinforcement is less than 10 in. Similarly, if minimum shear reinforcement is provie, the size effect oes not nee to be consiere for epths less than 100 in. Therefore, for most practical members, a size moification is not neee. By simply proviing minimum transverse reinforcement, a beam epth must excee 8 ft before the size effect must be inclue in calculations. For members of greater epths, a reuction in shear strength is accounte for by multiplying the base concrete shear strength (Eq. (1)) by the fractional value etermine from Eq. (7). As an example of the influence of size, if a one-ay slab has a t value of 15 in. an is not reinforce for shear, the shear strength from Eq. (1) oul be multiplie by Similarly, for a beam containing minimum shear reinforcement, a epth t of 120 in. oul result in a multiplier of 0.9. Members must become very large for this factor to provie a significant influence. Hoever, consiering the limite number of large beam tests, reuctions base on soun theoretical consierations are appropriate. As shon in Fig. 7, the propose approach (Eq. ()) provies goo agreement ith test results provie in the Joint ACI-ASCE Committee 5 atabase for nonprestresse members. For comparison purposes, the results using ACI 318 are also provie. As is evient, the propose esign proceure significantly improves structural safety. Although irect comparisons ith ra atabases as presente in Fig. 2 through an an 7 are simple an iely use, one must be aare of to hien isavantages: The mean values of seconary variables vary systematically over each variable s range; an The ata are typically croe in one part of the range hile being sparse in another. In the case of Fig. an 7, the mean values of the steel ratio in successive size intervals ecrease by about an orer of magnitue, an the mean values of a/ vary almost as much. Therefore, the size effect plots shon herein actually represent the effect of a certain combine variation of beam size, reinforcement ratio, an shear span. Obviously, this is misleaing if the size effect tren is to be evaluate. To separate the effects of steel ratio an shear span a, one oul nee to either conuct multivariate regression of the atabase or create filtere subbases from hich these seconary effects are remove. To keep the approach presente herein at a simple uniform level, such refine comparisons are omitte Steel FRP 5 test bc f c 8 test calc (a) Effective Depth, (in.) (a) Effective Depth, (in.) Steel FRP 5 test bc f c 8 7 test calc (b) Effective Depth, (in.) Fig. : Influence of member epth: (a) linear scale; an (b) log scale (Note: All ata plotte ithout any filtering for the variation of ρ, a/, an throughout the range) 50 (b) Effective Depth, (in.) Fig. 7: Preictions mae using esign proceures are compare against ata from the Joint ACI-ASCE Committee 5 atabase for nonprestresse members: (a) using propose proceure (Eq. ()); an (b) using current ACI 318 proceure 50 SEPTEMBER 2017 Ci.concreteinternational.com

5 They can, hoever, be foun in Bažant et al. 9 an Yu et al. 10 Strength Limits The ACI Coe limits the maximum shear that can be resiste by a member (Eq. (8)). The intent of this limit, as outline in Commentary Section R of ACI 318-1, is to minimize the likelihoo of iagonal compression failure in the concrete an limit the extent of cracking. (8) It is propose that a similar limit be maintaine such that the maximum contribution of shear provie by the shear reinforcement be limite to c, consistent ith experience an current practice. Therefore, the maximum shear is propose to be limite accoring to Eq. (9). This simple limit, in hich the maximum shear strength is limite to five times the concrete contribution of the shear strength, provies ease of calculation an clarity. (9) Conclusions The propose esign proceure provies a simple approach consistent ith both the current ACI esign philosophy, hich consiers n = c + s, an ith current ACI esign assumptions such as critical sections. By proviing a ne expression for the concrete contribution of shear strength,, a unifie esign proceure is achieve hich eliminates the numerous esign expressions require in the current Coe. This esign expression can be use for the shear esign of structural concrete members reinforce ith either steel or FRP bars an is applicable to both nonprestresse an prestresse members, members incluing both nonprestesse an prestresse reinforcement, an members ith axial forces. This approach has the further avantage of extening to the calculation of to-ay shear strength, 11 here, an is currently provie as the shear esign metho (one-ay an to-ay) for members ith FRP reinforcement. 8 By proviing a single, unifie metho for the shear esign of structural concrete members, it is possible to both simplify the Coe an enhance structural safety. References 1. ACI Committee 318, Builing Coe Requirements for Structural Concrete (ACI 318-1) an Commentary (ACI 318R-1), American Concrete Institute, Farmington Hills, MI, 201, 519 pp. 2. ACI Committee 318, Builing Coe Requirements for Reinforce Concrete ACI Custom Seminars Bring ACI training an eucation expertise to your oorstep. Scheule a custom seminar for your employees, customers, or members. Customize topics for customers Unlimite attenance ith no travel costs Attenees earn continuing eucation creits Organizational an sustaining members receive a seminar iscount $ isit.concreteseminars.com for more info..concreteinternational.com Ci SEPTEMBER

6 (ACI 318-3), American Concrete Institute, Farmington Hills, MI, 193, 1 pp. 3. ACI Committee 318, Commentary on Builing Coe Requirements for Reinforce Concrete (ACI 318-3), SP-10, American Concrete Institute, Farmington Hills, MI, 195, 91 pp.. Tureyen, A.K., an Frosch, R.J., Concrete Shear Strength: Another Perspective, ACI Structural Journal,. 100, No. 5, Sept.-Oct. 2003, pp Tureyen, A.K.; Wolf, T.S.; an Frosch, R.J., Strength in Shear of Reinforce Concrete T-Beams ithout Transverse Reinforcement, ACI Structural Journal,. 103, No. 5, Sept.-Oct. 200, pp Wolf, T.S., an Frosch, R.J., Shear Design of Prestresse Concrete: A Unifie Approach, Journal of Structural Engineering, ASCE,. 133, No. 11, Nov. 2007, pp Saqan, E.I., an Frosch, R.J., Influence of Flexural Reinforcement on Shear Strength of Prestresse Beams ithout Web Reinforcement, ACI Structural Journal,. 10, No. 1, Jan.-Feb. 2009, pp ACI Committee 0, Guie for the Design an Construction of Structural Concrete Reinforce ith Fiber-Reinforce Polymer (FRP) Bars (ACI 0.1R-15), American Concrete Institute, Farmington Hills, MI, 2015, 83 pp. 9. Bažant, Z.P.; Yu, Q.; Gerstle, W.; Hanson, J.; an Ju, J.W., Justification of ACI Proposal for Upating ACI Coe Provisions for Shear Design of Reinforce Concrete Beams, ACI Structural Journal,. 10, No. 5, Sept.-Oct. 2007, pp Yu, Q.; Le, J.; Hubler, M.H.; Wenner, R.; Cusatis, G.; Bažant, Z.P., Comparison of Main Moels for Size Effect on Shear Strength of Reinforce an Prestresse Concrete Beams, Structural Concrete,. 17, No. 5, Dec. 201, pp Ospina, C.O., Alternative Moel for Concentric Punching Capacity, Concrete International,. 27, No. 9, Sept. 2005, pp Receive an reviee uner Institute publication policies. ReaCi ONLINE COER-TO-COER A flip-book version of the entire current issue of CI is available to ACI members by logging in at.concreteinternational.com. Click "vie the flipbook" on the magazines home page. Robert J. Frosch, FACI, is Associate Dean for Resource Planning an Management an Professor of civil engineering at Purue University, West Lafayette, IN. A past member of the ACI Boar of Direction, he is currently Chair of the Financial Avisory Committee an serves on ACI Committee 318, Structural Concrete Builing Coe. He also chairs ACI Subcommittee 318-D, Members. Frosch is a member of ACI Committees 22, Cracking, an 0, Fiber-Reinforce Polymer Reinforcement, as ell as Joint ACI- ASCE Committees 08, Development an Splicing of Deforme Bars, an 5, Shear an Torsion. He is the recipient of the ACI 318 Distinguishe Service Aar an the Arthur J. Boase Aar for outstaning structural research. He receive his BSE from Tulane University, Ne Orleans, LA, an his MSE an PhD from the University of Texas at Austin, Austin, TX. Qiang Yu is an Assistant Professor at University of Pittsburgh, Pittsburgh, PA. His research interests inclue mechanical properties of concrete structures, risk analysis of avance structural materials, an smart materials an hybri structures. He receive his BS from Tongji University, Shanghai, China; his MS from Tsinghua University, Bejing, China; an his PhD from Northestern University, Evanston, IL. ACI member Gianluca Cusatis is Associate Professor of Civil an Environmental Engineering at Northestern University. He is Chair of ACI Committee 209, Creep an Shrinkage in Concrete; Past-Chair of Joint ACI-ASCE Committee, Fracture Mechanics of Concrete; an a member of Joint ACI-ASCE Committee 7, Finite Element Analysis of Reinforce Concrete Structures. Cusatis is also a member of American Society of Civil Engineers (ASCE). He receive his PhD in structural engineering at Politecnico i Milano, Milan, Italy. ACI Honorary Member Zeněk P. Bažant is a McCormick Institute Professor an Walter P. Murphy Professor of Civil an Environmental Engineering at Northestern University. A member of the National Acaemy of Sciences an the National Acaemy of Engineering, he is the founing Chair an a member of Joint ACI-ASCE Committee, Fracture Mechanics of Concrete. Bažant is also a member of five other ACI committees, incluing 209, Creep an Shrinkage in Concrete, an 38, Structural Reliability an Safety. He is a license structural engineer in Illinois. 52 SEPTEMBER 2017 Ci.concreteinternational.com