1、ACI 445R-99 became effective November 22, 1999.Copyright 2000, American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by anymeans, including the making of copies by any photo process, or by electronic ormechanical device, printed, written, or oral, or
2、 recording for sound or visual reproduc-tion or for use in any knowledge or retrieval system or device, unless permission inwriting is obtained from the copyright proprietors.445R-1ACI Committee Reports, Guides, Manuals, and Commentariesare intended for guidance in planning, designing, executing,and
3、 inspecting construction. This document is intended for theuse of individuals who are competent to evaluate thesignificance and limitations of its content and recommendationsand who will accept responsibility for the application of thematerial it contains. The American Concrete Institute disclaimsan
4、y and all responsibility for the stated principles. The Instituteshall not be liable for any loss or damage arising therefrom.Reference to this document shall not be made in contractdocuments. If items found in this document are desired by theArchitect/Engineer to be a part of the contract documents
5、, theyshall be restated in mandatory language for incorporation bythe Architect/Engineer.Truss model approaches and related theories for the design of reinforcedconcrete members to resist shear are presented. Realistic models for thedesign of deep beams, corbels, and other nonstandard structural mem
6、bersare illustrated. The background theories and the complementary nature ofa number of different approaches for the shear design of structural con-crete are discussed. These relatively new procedures provide a unified,intelligible, and safe design framework for proportioning structural con-crete un
7、der combined load effects.Keywords: beams (supports); concrete; design; detailing; failure; models;shear strength; structural concrete; strut and tie.CONTENTSChapter 1Introduction, p. 445R-21.1Scope and objectives1.2Historical development of shear design provisions1.3Overview of current ACI design p
8、rocedures1.4SummaryChapter 2Compression field approaches, p. 445R-52.1Introduction2.2Compression field theoryRecent Approaches to Shear Designof Structural ConcreteReported by Joint ACI-ASCE Committee 445ACI 445R-99(Reapproved 2009)2.3Stress-strain relationships for diagonally crackedconcrete2.4Modi
9、fied compression field theory2.5Rotating-angle softened-truss model2.6Design procedure based on modified compressionfield theoryChapter 3Truss approaches with concrete contribution, p. 445R-173.1Introduction3.2Overview of recent European codes3.3Modified sectional-truss model approach3.4Truss models
10、 with crack friction3.5Fixed-angle softened-truss models3.6SummaryChapter 4Members without transverse reinforcement, p. 445R-254.1Introduction4.2Empirical methods4.3Mechanisms of shear transfer4.4Models for members without transverse reinforcement4.5Important parameters influencing shear capacity4.6
11、ConclusionsChapter 5Shear friction, p. 445R-355.1Introduction5.2Shear-friction hypothesis5.3Empirical developmentsP. E. Adebar*N. M. Hawkins D. Mitchell*D. M. Rogowsky*J. F. Bonacci T. T. C. Hsu R. G. Oesterle G. M. SabnisM. P. Collins G. J. Klein M. A. Polak D. H. SandersD. Darwin T. Krauthammer K.
12、 S. Rajagopalan J. K. WightW. H. Dilger J. G. MacGregor K. H. Reineck*P. ZiaA. B. Gogate*Members of Subcommittee 445-1 who prepared this report.J. A. Ramirez*ChairC. W. FrenchSecretary445R-2 MANUAL OF CONCRETE PRACTICE5.4Analytical developments5.5Code developmentsChapter 6Design with strut-and-tie m
13、odels,p. 445R-376.1Introduction6.2Design of B regions6.3Design of D regionsChapter 7Summary, p. 445R-467.1Introduction7.2Truss models7.3Members without transverse reinforcement7.4Additional workAppendix AACI 318M-95 shear design approach for beams, p. 445R-49Appendix BReferences, p. 445R-50CHAPTER 1
14、INTRODUCTION1.1Scope and objectivesDesign procedures proposed for regulatory standardsshould be safe, correct in concept, simple to understand, andshould not necessarily add to either design or constructioncosts. These procedures are most effective if they are basedon relatively simple conceptual mo
15、dels rather than oncomplex empirical equations. This report introduces designengineers to some approaches for the shear design of one-way structural concrete members. Although the approachesexplained in the subsequent chapters of this report are rela-tively new, some of them have reached a sufficien
16、tly maturestate that they have been implemented in codes of practice.This report builds upon the landmark state-of-the-art reportby the ASCE-ACI Committee 426 (1973), The ShearStrength of Reinforced Concrete Members, which reviewedthe large body of experimental work on shear and gave thebackground t
17、o many of the current American Concrete Insti-tute (ACI) shear design provisions. After reviewing themany different empirical equations for shear design,Committee 426 expressed in 1973 the hope that “the designregulations for shear strength can be integrated, simplified,and given a physical signific
18、ance so that designers canapproach unusual design problems in a rational manner.”The purpose of this report is to answer that challenge andreview some of the new design approaches that have evolvedsince 1973 (CEB 1978, 1982; Walraven 1987; IABSE1991a,b; Regan 1993). Truss model approaches and relate
19、dtheories are discussed and the common basis for these newapproaches are highlighted. These new procedures provide aunified, rational, and safe design framework for structuralconcrete under combined actions, including the effects ofaxial load, bending, torsion, and prestressing.Chapter 1 presents a
20、brief historical background of thedevelopment of the shear design provisions and a summaryof the current ACI design equations for beams. Chapter 2discusses a sectional design procedure for structural-concrete one-way members using a compression fieldapproach. Chapter 3 addresses several approaches i
21、ncorpo-rating the “concrete contribution.” It includes brief reviewsof European Code EC2, Part 1 and the Comit Euro-Interna-tional du BtonFdration International de la Prcontrainte(CEB-FIP) Model Code, both based on strut-and-tie models.The behavior of members without or with low amounts ofshear rein
22、forcement is discussed in Chapter 4. An explanationof the concept of shear friction is presented in Chapter 5.Chapter 6 presents a design procedure using strut-and-tiemodels (STM), which can be used to design regions havinga complex flow of stresses and may also be used to designentire members. Chap
23、ter 7 contains a summary of the reportand suggestions for future work.1.2Historical development of shear design provisionsMost codes of practice use sectional methods for design ofconventional beams under bending and shear. ACI BuildingCode 318M-95 assumes that flexure and shear can behandled separa
24、tely for the worst combination of flexure andshear at a given section. The interaction between flexure andshear is addressed indirectly by detailing rules for flexuralreinforcement cutoff points. In addition, specific checks onthe level of concrete stresses in the member are introduced toensure suff
25、iciently ductile behavior and control of diagonalcrack widths at service load levels.In the early 1900s, truss models were used as conceptualtools in the analysis and design of reinforced concrete beams.Ritter (1899) postulated that after a reinforced concrete beamcracks due to diagonal tension stre
26、sses, it can be idealized asa parallel chord truss with compression diagonals inclined at457 with respect to the longitudinal axis of the beam. Mrsch(1920, 1922) later introduced the use of truss models fortorsion. These truss models neglected the contribution of theconcrete in tension. Withey (1907
27、, 1908) introduced Ritterstruss model into the American literature and pointed out thatthis approach gave conservative results when compared withtest evidence. Talbot (1909) confirmed this finding.Historically, shear design in the United States has includeda concrete contribution Vcto supplement the
28、 45 degreesectional truss model to reflect test results in beams and slabswith little or no shear reinforcement and ensure economy inthe practical design of such members. ACI Standard Specifi-cation No. 23 (1920) permitted an allowable shear stress of0.025 fc, but not more than 0.41 MPa, for beams w
29、ithout webreinforcement, and with longitudinal reinforcement that didnot have mechanical anchorage. If the longitudinal reinforce-ment was anchored with 180 degree hooks or with platesrigidly connected to the bars, the allowable shear stress wasincreased to 0.03fc or a maximum of 0.62 MPa (Fig. 1.1)
30、.Web reinforcement was designed by the equationAvFv= Vs sin /jd (1-1)whereAv= area of shear reinforcement within distance s;fv= allowable tensile stress in the shear reinforcement;jd = flexural lever arm;RECENT APPROACHES TO SHEAR DESIGN OF STRUCTURAL CONCRETE 445R-3V = total shear minus 0.02 fcbjd
31、(or 0.03 fcbjd withspecial anchorage);b = width of the web;s = spacing of shear steel measured perpendicular to itsdirection; and = angle of inclination of the web reinforcement withrespect to the horizontal axis of the beam.The limiting value for the allowable shear stresses atservice loads was 0.0
32、6fc or a maximum of 1.24 MPa, or withanchorage of longitudinal steel 0.12fc or a maximum of2.48 MPa. This shear stress was intended to prevent diagonalcrushing failures of the web concrete before yielding of thestirrups. These specifications of the code calculated thenominal shear stress as v = V/bj
33、d.This procedure, which formed the basis for future ACIcodes, lasted from 1921 to 1951 with each edition providingsomewhat less-conservative design procedures. In 1951 thedistinction between members with and without mechanicalanchorage was omitted and replaced by the requirement thatall plain bars m
34、ust be hooked, and deformed bars must meetASTM A 305. Therefore, the maximum allowable shearstress on the concrete for beams without web reinforcement(ACI 318-51) was 0.03fc and the maximum allowable shearstress for beams with web reinforcement was 0.12fc.ACI 318-51, based on allowable stresses, spe
35、cified thatweb reinforcement must be provided for the excess shear ifthe shear stress at service loads exceeded 0.03fc.Calculationof the area of shear reinforcement continued to be based ona 45 degree truss analogy in which the web reinforcementmust be designed to carry the difference between the to
36、talshear and the shear assumed to be carried by the concrete.The August 1955 shear failure of beams in the warehouseat Wilkins Air Force Depot in Shelby, Ohio, brought intoquestion the traditional ACI shear design procedures. Theseshear failures, in conjunction with intensified research,clearly indi
37、cated that shear and diagonal tension was acomplex problem involving many variables and resulted in areturn to forgotten fundamentals.Talbot (1909) pointed out the fallacies of such proceduresas early as 1909 in talking about the failure of beamswithout web reinforcement. Based on 106 beam tests, he
38、concluded thatIt will be found that the value of v shear stress atfailure will vary with the amount of reinforcement,with the relative length of the beam, and with otherfactors which affect the stiffness of the beam. Inbeams without web reinforcement, web resistancedepends upon the quality and stren
39、gth of theconcrete. The stiffer the beam the larger thevertical stresses which may be developed. Short,deep beams give higher results than long slenderones, and beams with high percentage of reinforce-ment give higher results than beams with a smallamount of metal.Unfortunately, Talbots findings abo
40、ut the influence of thepercentage of longitudinal reinforcement and the length-to-depth ratio were not reflected in the design equations untilmuch later. The research triggered by the 1956 Wilkinswarehouse failures brought these important concepts back tothe forefront.More recently, several design p
41、rocedures were developedto economize on the design of the stirrup reinforcement. Oneapproach has been to add a concrete contribution term to theshear reinforcement capacity obtained, assuming a 45 degreetruss (for example, ACI 318-95). Another procedure hasbeen the use of a truss with a variable ang
42、le of inclination ofthe diagonals. The inclination of the truss diagonals is allowedto differ from 45 degree within certain limits suggested onthe basis of the theory of plasticity. This approach is oftenreferred to as the “standard truss model with no concretecontribution” and is explained by the e
43、xistence of aggregateinterlock and dowel forces in the cracks, which allow a lowerinclination of the compression diagonals and the furthermobilization of the stirrup reinforcement. A combination ofthe variable-angle truss and a concrete contribution has alsobeen proposed. This procedure has been ref
44、erred to as themodified truss model approach (CEB 1978; Ramirez andBreen 1991). In this approach, in addition to a variable angleof inclination of the diagonals, the concrete contribution fornonprestressed concrete members diminishes with the levelof shear stress. For prestressed concrete members, t
45、heconcrete contribution is not considered to vary with the levelof shear stress and is taken as a function of the level ofprestress and the stress in the extreme tension fiber.As mentioned previously, the truss model does not directlyaccount for the components of the shear failure mechanism,such as
46、aggregate interlock and friction, dowel action of thelongitudinal steel, and shear carried across uncrackedconcrete. For prestressed beams, the larger the amount ofprestressing, the lower the angle of inclination at first diagonalcracking. Therefore, depending on the level of compressivestress due t
47、o prestress, prestressed concrete beams typicallyhave much lower angles of inclined cracks at failure thannonprestressed beams and require smaller amounts of stirrups.Traditionally in North American practice, the additionalarea of longitudinal tension steel for shear has been providedby extending th
48、e bars a distance equal to d beyond the flexuralcutoff point. Although adequate for a truss model with 45 degreediagonals, this detailing rule is not adequate for trusses withdiagonals inclined at lower angles. The additional longitudinaltension force due to shear can be determined from equilibriumc
49、onditions of the truss model as V cot, with as the angle ofinclination of the truss diagonals. Because the shear stressesare assumed uniformly distributed over the depth of the web,the tension acts at the section middepth.The upper limit of shear strength is established by limitingthe stress in the compression diagonals fdto a fraction of theconcrete cylinder strength. The concrete in the cracked webFig. 1.1American Specification for shear design (1920-1951) based on ACI Standard No. 23, 1920.445R-4 MANUAL OF CONCR
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