Shear TransfShear transfer strength of reinforced concrete
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ACI Structural Journal/July-August 2010 419
ACI Structural Journal, V. 107, No. 4, July-August 2010.MS No. S-2009-105.R3 received July 24, 2009, and reviewed under Institute
publication policies. Copyright 2010, American Concrete Institute. All rights reserved,including the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including authors closure, if any, will be published in the May-June2011 ACI Structural Journal if the discussion is received by January 1, 2011.
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
A recently developed model for the calculation of shear strength inreinforced concrete membrane elements subjected to in-planestresses and in beams subjected to shear and torsion is applied tothe shear-transfer problem. The modeling is different from thecommonly used shear-friction concept, and relates the strength ata shear interface to the state of stress in a membrane elementalong this interface. The shear strength is hence related not only tothe concrete strength and clamping steel, but also to the steelparallel to the shear-transfer plane. The calculations of the simplemodel are compared to the experimental results from 114 normal-weight pushoff specimens and 15 composite beams available in theliterature and are found to be in very good agreement. The modelis also used to derive the empirically based coefficients of existingmethods that relate the shear-transfer strength to the square root ofthe clamping stress.
Keywords: composite beams; reinforced concrete; shear-friction; sheartransfer; strength.
INTRODUCTIONNumerous design cases require the calculation of the
amount of reinforcement necessary to resist shear transferacross an interface between two concrete members that canslip relative to each other. The interface can be susceptible toa potential crack or can be cracked due to previous conditionssuch as external tension and shrinkage, and can be a coldjoint. The interface between a precast girder and a cast-in-place deck slab, and the bearing zones in precast girders,corbels, and horizontal construction joints in walls, areexamples of shear-transfer cases. Refer to Fig. 1(a).
The design for shear transfer has been largely based onempirical and semi-empirical methods that were developedusing the experimental results from pushoff specimens andcomposite beam specimens. Figure 1(b) shows a typicalpushoff specimen similar to that used in the early tests byHofbeck et al.1 The applied compressive forces createshearing stresses (v) along the critical plane, which could beeither precracked or uncracked. The shearing stresses atultimate conditions are typically assumed to be constantalong the interface plane, and an average shearing strengthalong the plane is calculated. These shearing stresses act incombination with compressive stresses (Fig. 1(b)).
There has been a considerable amount of experimentaltests on pushoff specimens,1-8 which led to the developmentof numerous models.9-17 The well-known shear-frictionmodel of the ACI Code17 is based on the assumption that acrack exists along the shear plane before the load is applied.The failure occurs by sliding along the shear plane and theopening of the crack around the aggregates. The clampingsteel is stressed to its yield strength and a friction forceproportional to the clamping yield force is activated. TheACI nominal shear strength is given by
vACI = y fy y (1)
but not greater than (0.2fc ) or 5.5 MPa (800 psi), where y isthe ratio of clamping reinforcement (in the y-direction), fy yis the yield strength of clamping reinforcement (in the
Title no. 107-S41
Shear-Transfer Strength of Reinforced Concreteby Khaldoun N. Rahal
Fig. 1(a) Examples of shear transfer in reinforced concretestructures; and (b) typical pushoff specimen and state ofstress along shear transfer plane.
ACI Structural Journal/July-August 2010420
y-direction), fc is the compressive strength of the concrete,and is the coefficient to account for friction.
Walraven et al.6 developed a model based on 88 test specimenswith concrete strength ranging from 21 to 68 MPa (3000 to9900 psi). The model relates the shear strength to theclamping reinforcement as well as fc , but does not place anupper limit on the shear strength
where C1 = 0.88(fc )0.406, C2 = 0.167(fc )0.303, and C3 = 1 inMPa (C1 = 16.76(fc )0.406, C2 = 0.0371(fc )0.303, and C3 =0.007 in psi).
Hsu et al.13 adopted a more rational approach by consideringthe concrete along the shear-transfer plane to be a membraneelement subjected to combined shearing and normal stresses.They used the equations of the softened truss model tocalculate the shear strength and overall behavior. Thisanalysis differs in concept from the more commonly usedshear-friction models; however, the solution procedure iscomputationally demanding and requires the use of acomputer. A simple semi-empirical equation was subsequentlyproposed by Mau and Hsu.14
Loov and Patnaik15 developed a similar equation for thenominal shear strength
where is the factor to account for lightweight aggregates.The factor 0.1 is replaced by 15 in psi, and has a negligibleeffect at relatively large clamping steel. It was included toavoid the discontinuities in the present codes at lowclamping stresses.15
Mattock16 proposed a trilinear model calibrated using theresults from 189 normalweight and lightweight test specimenswith fc ranging from 16 to 99 MPa (2300 to 14,350 psi)
vWFP C1 C3y fy y( )C2
vMH 0.66 fc( ) y fy y 0.3 fc=
vLP 0.6 fc( ) 0.1 y fy y+ 0.25 fc in MPa( )=
vMat 2.25y fy y when y fy y K1 1.45=
but not greater than 0.3fc or 16.6 MPa (2400 psi) for normal-weight concrete and 0.2fc or 8.3 MPa (1200 psi) for sand-lightweight concrete and all lightweight concrete. The factorK1 is taken as 0.1fc but not greater than 5.5 MPa (800 psi).
To account for a normal stress y acting perpendicular tothe shear plane, the superposition of steel can be appliedand, consequently, the term y fy y is replaced by y fy y y,where y is positive if tensile.
Equations (1) to (5) show that existing models are simple,but are empirical or semi-empirical. More rational models,such as those by Hsu et al.,13 have the advantage of beingapplicable to other shear cases, but are iterative and, hence,are not readily suitable for use in a design office. The challenge isto develop a more rational model that shares the simplicityand accuracy of empirical methods.
A recently developed model called the simplified modelfor combined stress resultants (SMCS) is a simple, noniterativemodel for the calculation of the shear strength and the modeof failure of membrane elements subjected to in-planeshearing and normal stresses.18 The model was generalizedto apply to reinforced and prestressed concrete beamssubjected to shear combined with flexure and axial forces,19to pure torsion,20 and to torsion combined to flexure.21 Thispaper extends the applicability of the model to solve theshear-transfer problem.
RESEARCH SIGNIFICANCEMost simple methods available to solve the shear-transfer
problem in shear-friction specimens and across cold joints incomposite beams are semi-empirical, and their application islimited as they cannot be applied to other types of shearproblems such as shear and torsion in beams and shear inmembrane elements. This paper presents a simple, noniterativemodel that is developed based on a rational theory and isapplicable to other types of shear problems. The proposedmodel has a favorable combination of simplicity, generality,and accuracy in comparison with existing models.
SMCS FOR PURE SHEARThe SMCS model developed for pure shear was applied
without modification to the case of shear friction. A briefbackground of the development of the SMCS is presented.Full details of the model can be obtained from Rahal.18
Figure 2 shows a membrane element reinforced withorthogonal steel subjected to in-plane shearing stresses and asummary of the SMCS equations. The equations are developedfor the case of pure shear, and the effects of the normalstresses are accounted for using the concept of superposition.
The model assumes that the main factors that affect thepure shear strength of membrane elements are the amountsand the strength of the orthogonal steel and the concretecompressive strength. Other factors, such the maximum sizeof the coarse aggregate and the spacing and diameter of thereinforcement, have limited effects and are neglected in thesimplified model. The three main factors are efficientlycombined in the following reinforcement indexes
vMat K1 0.8y fy y+ when y fy y K1 1.45>=
xx fy x
y fy yfc
ACI member Khaldoun N. Rahal is a Professor of civil engineering at KuwaitUniversity, Kuwait City, Kuwait. He is Director and Past President of the ACI-KuwaitChapter and is a member of Joint ACI-ASCE Committee 445, Shear and Torsion.
Fig. 2Membrane element subjected to in-plane stressesand summary of SMCS equations.
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where x,y are the reinforcement indexes in x- and y-directions,respectively; x , y is the reinforcement ratios in x- and y-directions, respectively; and fy x , fy y is the yield strengthof reinforcement in x-