Rational approach to shear design the 1984 canadian code provisions

ACI JOURNAL TECHNICAL PAPER Title No 83-80 A Rational Approach to Shear Design The 1984 Canadian Code Provisions I by Michael P Collins and Denis Mitchell The 1984 Canadian Concrete Code contains new shear design provisions that are believed to be more rational and more general than the shear regulations of the 1983 ACI Building Code This paper summarizes the new shear design procedures and illustrates their use by means of design examples Keywords: beams (supports); building codes; deep beams; detailing; reinforced concrete; shear properties; shear strength; structural design It is now over ten years since the ACI-ASCE Shear Committee concluded the introduction to its state-of the-art report with the words: "During the next decade it is hoped that the design regulations for shear strength can be integrated, simplified and given a physical significance so that designers can approach unusual design problems in a rational manner." Advances that have been made during the past 15 years enable us to present shear design regulations that we believe meet the above criteria The main objective in developing these new shear design procedures was to move away from the current ACF shear design equations that have been described by MacGregor as "empirical mumbo-jumbo" and move toward procedures comparable in rationality and generality to the ACI plane sections approach for flexure and axial load Chapter 11 - Shear and Torsion of the new Canadian Concrete Code permits the use of two alternative design methods for shear: the simplified method and the general method The simplified method is a much shortened version of the traditional ACI Vc + Vs procedure In this method shear design is viewed as being primarily concerned with the design of transverse reinforcement (shear reinforcement) The influence of shear on the longitudinal reinforcement is assumed to be accounted for by the traditional detailing rules for the development of flexural reinforcement (e.g., "reinforcement shall extend beyond the point at which it is no longer required to resist flexure for a distance equal to the effective depth of the member") For members such as ACI JOURNAL I November-December 1986 deep beams or corbels, the influence of shear on the longitudinal reinforcement cannot be accounted for by these detailing rules Hence, the simplified method is not permitted for such members The new shear design approach which is called the general method is based primarily on the compression field theory5-8 though it also uses concepts from plasticity and truss models developed in Europe over the last decade 9- 12 This method, which considers shear as influencing the design of both the transverse and the longitudinal reinforcement, will be summarized in this paper DETAILED ANALYSIS BY THE GENERAL METHOD In a manner similar to that used in ACI 318, Chapter 10, "Flexure and Axial Loads," the general method commences with a few paragraphs of text stating the general principles Thus, for example, it is stated that: "The resistance of members in shear or in shear combined with torsion shall be determined by satisfying applicable conditions of equilibrium and compatibility of strains and by using appropriate stress-strain relationships for reinforcement and for diagonally cracked concrete Cross-sectional dimensions shall be chosen to insure that the diagonally cracked concrete is capable of resisting the inclined compressive stresses Longitudinal and transverse reinforcement capable of equilibrating this field of diagonal compression shall be provided." This clause permits the resistance and behavior of members in shear to be investigated in detail by performing a sectional analysis that can be considered as a generalization of the plane sections theory Fig illustrates the application of the compression field theory in Received Sept 19, 1985, and reviewed under Institute publication policies Copyright © 1986, American Concrete Institute All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors Pertinent discussion will be published in the September-October 1987 ACI JOURNAL if received by June I, 1987 925 Michael P Collins, FAC1, is a professor of civil engineering a/the University of Toronto He is chairman of AC1-ASCE Committee 445, Shear and Torsion, chairman of the Canadian Standards Association Committee on Concrete Offshore Structures, a Canadian delegate to Comite Euro-1nternational du Beton, and a member_of the Canadian Concrete Code Committee, He is also a member of AC1 Commillees 318, Standard Building Code, 358, Guideways, E 901, Scholarships, and subcommiltee I 8£, Shear and Torsion, Denis Mitchell, FAC1, is a professor in the Department of Civil Engineering and Applied Mechanics at McGill University He is a member of AC1-ASCE Commillee 445, Shear and Torsion, AC1 Committees 408, Reinforcement Bond and Development, and E 90/, Scholarships, and is a member of the Canadian Concrete Code Commillee t:r., ~~& crosssection shear longitudinal stresses strains its most general form The cross section is first divided into a series of horizontal layers Within each layer, the longitudinal strain, the shear stress, and the inclination of principal compressive stress are all assumed to remain constant For each layer, the biaxial stresses and strains are determined by considering the equilibrium, compatibility, and stress-strain requirements It is assumed that the principal stress direction coincides with the principal strain direction The principal compressive stress in the concrete fz is related to both the principal compressive strain t: and the principal tensile strain E in the following manner (see Fig 2) fz = fz max l lJ 2E 0.002 - ( E2 0.002 ) (1) where principal compressive stress trajectories (2) ~MJ: Fig 1-Detailed sectional analysis using the compression field theory E'c (a) Softening of concrete due to transverse tensile strain where >- is a factor accounting for lightweight concrete and ¢c is the material resistance factor for concrete in the Canadian Code (¢c = 0.60) * Note that as the principal tensile strain increases the maximum compressive stress that the concrete can resist decreases (see Fig 2) Applying the compression field theory in this most general form would show (see Fig 1) that the shear stress distribution is not uniform over the depth of the beam, that the direction of principal compressive stresses changes over the depth of the beam, and that the tensile stresses in the concrete between the cracks contribute to the shear resistance of the member See References and 13 for more details The layer-by-layer sectional analysis described previously can be programmed in a few hundred lines of computer code for a microcomputer With the aid of such a program, the detailed response of a given cross section subjected to combined shear, moment, and axial load can be determined An example of the capabilities of such a program is given in Fig This figure 150 compression field theory prediction 0.8 500 \ -;n 100 400 c f 2max ~0- "t'c 600 \ ~ c > MPa ksi z 300.:; 0.4 0.2 layers shear stress OL -~ -L -J ~ -L ~ ~ E1 x10 10 12 (b) Reduction in compressive strength with increasing values of E Fig 2-Compressive stress-strain relationship of diagonally cracked concrete 926 0,001 0,003 0,002 shear strain 0.004 Fig 3-Response of circular reinforced concrete member subjected to shear and axial load Note that the Canadian Code uses load factors of 1.25 for dead load and 1.50 for live load Material resistance factors , = 0,60 for concrete and , = 0.85 for reinforcing bars are used in lieu of the member strength reduction factors of the ACI Building Code ACI JOURNAL I November-December 1986 compares the predicted and observed response of a circular reinforced concrete member subjected to shear and axialload 14 DESIGN BY THE GENERAL METHOD While the analysis procedures described previously can be used to make a detailed prediction of the response of a given section, simpler procedures can be used in design The Canadian Code permits a more direct procedure which concentrates on the conditions at middepth of the beam In this procedure (Fig 4) the shear stresses are assumed to be uniformly distributed over an area bv wide and d, deep, the direction of the principal compressive stresses (defined by angle 0) is assumed to remain constant over the depth d,, and tensile stresses in the cracked concrete are ignored Further details of this more direct general method will be given in the sections following d r~- [-1:"Mc1f/:75l7c ~ ~- VLlUJ_id/2 crosssection shear longitudinal stresses strains T sectional forces Fig 4-More direct design procedure for beam in shear f2 crosssection stresses in web Mohr's circle of stress Fig 5-Concrete stresses in web of beam Stresses and strains at middepth of beam If the principal tensile stresses in the concrete are zero, then the principal compressive stress in the concrete fz will be related to the shear stress on the concrete v = V/(bvdJ by the following equilibrium equation, which can be derived from the Mohr's circle in Fig normal strain t E2 fz ) tan() bvdv = (tan() + - (_!-£_) (3) The cross-sectional dimensions of the member must be sufficiently large to insure that the diagonally cracked concrete is capable of resisting the imposed inclined compressive stresses (i.e., fz < fzmax> If the concrete is extensively cracked resulting in a large principal tensile strain E 1, its ability to resist compressive stresses will be substantially reduced (see Fig 2) The diagonal crushing strength of the concrete fz max is related to the principal tensile strain E by Eq (2) The principal tensile strain E 1, the principal compressive strain E2, the longitudinal strain at middepth En the transverse strain E" and the principal compressive strain direction 0, are interrelated by the requirements of compatibility (see Fig 6) Assuming that at failure the principal compressive strain in the concrete equals 0.002, the principal tensile strain is given by crosssection strains E1 Mohr's circle of strain Fig 6-Concrete strains at middepth of beam diagonal compress1ve stresses, f (4) Design of transverse and longitudinal reinforcement Fig 7-Design of transverse reinforcement for shear Transverse reinforcement must be provided to equilibrate the outward thrust of the diagonal compressive stresses in the concrete (Fig 7) In the Canadian Code this equilibrium requirement is expressed as V = AA>Jy~ r s tan() ACI JOURNAL I November-December 1986 (5) The shear force on the section is resisted by diagonal compressive stresses in the concrete Thus, in Fig it is the vertical component of force D which is carrying the shear The horizontal components of force D is equivalent to an axial compression on the concrete of magnitude l-f/tanO This compression needs to be equilibrated by tensile forces in the longitudinal reinforce- 927 =o= 1::/>:> "lvN," '"~' V I -'- - " _/ \~ crosssection _/ lli-0.5 Nv _ -0.5 Nv sectional forces Fig 8-Longitudinal forces due to shear 0.40 1I I I ! J -t 1/; I I I I I I_ I I I I I transverse remforcement ~ f- yields if below these lines ::: ~+' c E,:; fyiEsl II15: f- 0.30 I I l I~ ~ ~ ~ 0.20 U) !~ Ill , ].,1!1 v (/) ttl ~ I 02~ ~: Oi"' ' ~ :t- ~~ ~- diagonal crushing of h- concrete avoided if below these lines Cf2 < f2maxl ,- ~ I I IL""" II 10° 15° 20° fl.'s I~ t ">1111 0.10 ,1'1j- + t""J '"'o ::- ijL ~~~ f-')r f Q) ~ J 'j ~ t"':._ Cia I I l I 30° 40° 50° (minimum allowed) angle of principal compressive stress, e Fig 9-Choice of{) dvi 'i ,' , -"' L.:'/~-~~~1>1 1 51 factored shear, v, total stirrup capacity requir~d in stirrup band total stirrup capacity required in stirrup band f -'~-="""'i - stirrup band Fig 10- The staggering concept for shear design ment (see Fig 8) Thus, shear causes compressive stresses in the concrete and tensile stresses in the longitudinal reinforcement In terms of the tension in the longitudinal reinforcement the shear is equivalent to an axial tensile load of Nv where Nv 928 = J-jltanO (6) Rather than designing the longitudinal reinforcement at a section to resist moment M and equivalent axial tension Nv, it may be more convenient to design only for a larger moment of M + 0.5 Nvdv Designing for this increased moment will give essentially the same additional longitudinal reinforcement Choice of and ex Two parameters which strongly influence design for shear are the inclination of the principal compressive stresses and the longitudinal strain at middepth Ex· The designer is free to choose the angle which then determines the relative amounts of transverse and longitudinal reinforcement (see Fig and 8) It is usually economical to minimize the amount of transverse reinforcement The transverse reinforcement will be minimized if the lowest possible value of is used Fig gives the values of for which the diagonal compressive stress f2 reaches the diagonal crushing strength f2 max and, hence, gives the lowest possible values of that may be used in design For design, the Canadian Code permits Ex to be taken as 0.002 Lower values of Ex may be appropriate for sections subjected to axial compression, prestress, or low values of moment or for sections containing large amounts of longitudinal reinforcement These stiffer sections will have small web deformations (low Ex and thus low E1) and hence will be able to tolerate higher shear stresses For these sections, the longitudinal strain at middepth Ex can be found by performing a plane sections analysis for the section subjected to the applied moment M 1, the applied axial force N 1, and the equivalent axial tension Nv If the cross-sectional dimensions are adequate it will be possible to choose values of and Ex which insure that the concrete does not crush prematurely and that the transverse reinforcement yields before sectional failure See Fig Additional considerations for member design The discussion so far has concentrated on the design of a section subjected to a given shear and a given moment While a member could be designed on a sectionby-section basis, a more realistic design will result if the behavior of the total member is considered Most members are subjected to shear forces that vary along the length of the beam The question then is ''What shear should the transverse reinforcement at a given section be designed to resist?" In the free body diagram in Fig 10, the stirrups between A and B must resist a force equal to the difference between the upward support reaction and the downward uniform load This difference in force equals the shear at Section B Hence, the stirrups between Sections A and B are designed to resist the lowest shear within this length This has become known as the "staggering concept" for shear design Fig 11 illustrates the influence of support conditions on the required amounts of longitudinal reinforcement The radiating compressive stresses in regions near diACI JOURNAL I November-December 1986 (a) Beam supported by cross-girders Fig 12-Principal compressive stress trajectories in regions near discontinuities nodal zone If I develop tension ~tie force over effective anchorage area tension tie this length (a) Flow of forces (b) End view (b) Beam on direct supports truss node Fig 11-/nfluence of support conditions on the equivalent moment for design of longitudinal reinforcement rect supports, Fig 11(b), will result in less longitudinal reinforcement being required than when the support reactions feed in over the member depth, Fig 11(a) For the beam on direct supports the reinforcement adjacent to the top face need not exceed the amount required to resist the maximum moment Design of disturbed regions near discontinuities The design procedures discussed so far are based on the assumptions that the shear stresses are uniformly distributed over the depth d, and that the principal compressive stress trajectories can be approximated by a series of parallel lines representing the compression field (see Fig 4) In regions near discontinuities such as those shown in Fig 12 these assumptions not apply and, hence, it is necessary to use procedures that more closely approximate the actual flow of the forces The internal flow of the forces in these disturbed regions can be approximated by strut and tie truss models such as that shown in Fig 13 The regions of high unidirectional compressive stress in the concrete are modelled as compressive struts while tension ties are used to model the principal reinforcement The regions of concrete subjected to multidirectional stresses, where the struts and ties meet (the nodes of the truss), are represented by nodal zones ACI JOURNAL I November-December 1986 \_-~ \ :, ', strut compression tension tie force (c) Truss model Fig 13-Strut and tie truss model for a deep beam The Canadian Code requires that the concrete compressive stresses in the nodal zones not exceed 0.85¢J; in nodal zones bounded by compressive struts and bearing areas, 75¢J; in nodal zones anchoring only one tension tie, and 0.60¢J; in nodal zones anchoring tension ties in more than one direction (see Fig 13) In designing a disturbed region, such as the deep beam shown in Fig 13, the first step is to sketch the flow of forces in the region and locate the nodal zones The nodal zones must be chosen large enough to insure that the nodal zone stresses are less than the nodal zone stress limits The geometry of the truss is determined by locating the nodes of the truss at the points of intersection of the forces meeting at the nodal zones (Fig 13) 929 1.0r -., t- - - - - - - - - - - - - - - , £ =~.002.·~ = 0.006 · 0.8 , 2max / E1 ~