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    SheartransferstrengthofreinforcedconcreteARTICLEinACISTRUCTURALJOURNALJULY2010ImpactFactor:0.96

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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

    (2)

    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

    (3)

    Loov and Patnaik15 developed a similar equation for thenominal shear strength

    (4)

    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)

    (5a)

    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=

    (5b)

    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

    (6)

    vMat K1 0.8y fy y+ when y fy y K1 1.45>=

    xx fy x

    fc---------------;y

    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.

  • ACI Structural Journal/July-August 2010 421

    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- and y-directions, respectively.

    Several advanced models can be used to calculate thestrength and mode of failure of the membrane elementshown in Fig. 2. One such model is the modified compressionfield theory (MCFT), which is capable of calculating the fullresponse of membrane elements subjected to in-planestresses.22 The equations of the MCFT were used to calculatethe relationship between the reinforcement indexes on oneside and the shear strength and mode of failure on the otherside. The spacing of the diagonal cracks was assumed to be300 mm (12 in.) and the maximum coarse aggregate size wastaken as 19 mm (0.75 in.), and the analysis was run on 40 MPa(5800 psi) concrete elements. Figure 3(a) shows the relationshipbetween the normalized strength vn/fc and the index y,when x remained constant (= 0.2). For y values below thatcorresponding to the point marked A, the y-reinforcementyielded before concrete crushing at ultimate conditions. Forlarger values, the strains remained below yield levels at ultimate.The index corresponding to Point A is referred to as thebalanced reinforcement index in the y-direction. For theresults shown in Fig. 3(a), the amount of x-reinforcement (x =0.2) was smaller than the balanced ratio in the x-direction,and yielded before concrete crushing.

    The analysis was repeated for different levels of x and thecurves shown in Fig. 3(b) were generated. Two balancedreinforcement curves were obtained. For x smaller than thebalanced x-indexes, the x-reinforcement yields when ultimateconditions are reached. Similarly, for y smaller than the balancedy-indexes, the y-reinforcement yields when ultimate conditions arereached. This gives rise to four classifications of reinforcementcorresponding to four different modes of failures: 1) fullyunder-reinforced (UR), where both x- and y-reinforcementyield before ultimate conditions are reached; 2) partiallyunder-reinforced (PUR), where only the x-reinforcementyields at ultimate conditions; 3) partially under-reinforced(PUR), where only the y-reinforcement yields at ultimateconditions; and 4) fully over-reinforced (OR), whereconcrete crushes before yielding in any steel.

    For fully under-reinforced membranes, yielding in both thex- and y-reinforcement prevents any restraint on the openingin the diagonal crack and, hence, reduces the concretecontribution provided mainly by aggregate interlock to nearlyzero. For this reason, the curves in Zone 1 (UR) can be calculatedusing the plasticity theory,23 as follows.

    (7)

    In the remaining three zones of Fig. 3(b), the concretecontribution is considerable, and is built-in in the calculatedultimate shear strength vn.

    Figure 3(a) shows that Eq. (7) adequately calculates thestrength of fully under-reinforced elements, but overestimates thestrength of partially under-reinforced and fully over-reinforcedelements. It also shows that the additional strength obtainedby increasing the amount of steel beyond the balancedlevel is limited. To maintain the simplicity of the model, thestrength provided by reinforcement in excess of the balancedvalues is neglected and, hence, the amount of usable reinforce-ment is limited to this value. In Fig. 3(b), the upper limit on

    vn

    fc----- xy=

    the reinforcement in one direction is shown to depend onthe amount of reinforcement in the other direction. Thisis neglected in the simplified version of the model and asingle value is adopted for a given fc.

    Equation (7) implies that the upper limit on the indexesleads to an upper limit on the shear strength. Based onexperimental results from normal- and high-strengthconcrete, the upper limit on the normalized shear strength vn/fcwas found to depend on fc and can be calculated18 using

    (8)

    where is the upper limit on shear strength (and amount ofusable reinforcement index). The coefficient 900 is in MPa,and is replaced by (130,500) if fc is in psi.

    The effects of in-plane normal stresses were accuratelyaccounted for in SMCS18,19,21 using the concept ofsuperposition of reinforcement. Using the concept ofsuperposition and introducing the upper limit on theamount of usable reinforcement gives

    (9)

    13---

    fc900---------= in MPa( )

    xx fy x x

    fc--------------------------- ; y

    y fy y yfc

    --------------------------- ==

    Fig. 3(a) Relationship between normalized shear capacityand reinforcement index; and (b) shear strength and modeof failure curves for membrane elements.

  • 422 ACI Structural Journal/July-August 2010

    where x, y are the normal stresses in the x- and y-directions(taken as positive if tensile). The upper limit on x and yleads to an automatic limit on the normalized shear stress.

    (10)

    Consequently, the equations of SMCS are reduced to Eq. (8)to (10). Refer to Fig. 2. Comparison between the calculationsof the SMCS model and the results from 84 membraneelements subjected to in-plane shearing and normal stressesshowed that its accuracy was very similar to those of theMCFT22 and the plasticity theory,23,24 and provided morefavorable results than the plasticity theory in the identificationof the mode of failure. The model was also generalized toapply 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

    SMCS FOR SHEAR-TRANSFER SPECIMENSFigure 1(b) shows an element along the shear transfer

    plane of a pushoff specimen. This element is subjected to acombination of a shearing stress along with a normalcompressive stress in the x-direction, x. The shear strengthof such elements can be calculated using the SMCS model ifx is available or assumed. The compressive stress is proportionalto the shearing stress and provides an increase over the pureshear strength of the element. An accurate calculation of xis not simple, especially because it varies considerably alongthe shear plane.

    To maintain the simplicity of the model while providingconservative results, the stresses (x) are neglected. Neglectingx does not have a significant effect on the results of the bulkof the shear-friction specimens because the x-reinforcementprovided in most of these specimens was relatively large, andthe elements were over-reinforced in the x-direction.Figure 3(b) shows that in membranes over-reinforced in thex-direction, an increase in x (that can be caused by x) doesnot have a significant influence on the shearing strength. Inthe SMCS equations, this increase is neglected by placingthe limit . By neglecting the compressive stresses, thestrength of the shear-transfer specimens is reduced to thecalculation of the strength of the membranes subjected topure shear or shear combined with y.

    It is to be noted that similar to theoretical models, manyexperimental studies did not consider the importance of thex-steel and, hence, did not report its full details. Forexample, Mattock and Hawkins2 and Walraven andStroband7 did not report the bar size or fy x , and Mattocket al.3 did not report fy x.

    DERIVATION OF COEFFICIENTSIN EMPIRICAL METHODS

    Equation (10) is a general equation that is applicable tovarious values of x and fc . For normal-strength membersover-reinforced in the x-direction, Eq. (1) can be reduced toa format similar to those of existing methods that weredeveloped based on results from such members. Forexample, all 33 specimens tested by Hofbeck et al.1 werereinforced with x = 5.7% and were normal-strengthconcrete (average fc = 25.9 MPa [3750 psi]). This largeamount of steel caused the specimens to be over-reinforced inthe x-direction and, consequently, the usable x islimited to (equal to approximately 0.30 for such

    vSMCS fc xy =

    concrete). Substituting this value of in Eq. (10) androunding the numbers gives

    (11)

    Equation (11) is very similar in format to Eq. (4) by Loov andPatnaik15 and Eq. (3) by Mau and Hsu,14 but gives 8% and16% smaller strength, respectively. The upper limits are alsowithin the same range. For normal-strength concrete wherefc is known, Eq. (11) can be further simplified by substitutingthe value of the fc (the average of the tests) to give

    vn = 2.80 MPa (33.8 psi) (12)

    with an upper limit of 7.9 MPa (1140 psi). Equation (12) isvery similar in format to Eq. (13) proposed by Birkeland (asreported in Reference 12) and Eq. (14) proposed by Raths.11

    vn = 2.78 MPa (33.5 psi) (13)

    vn = 3.11 MPa (37.42 psi) (14)

    Equations (13) and (14) differ from Eq. (12) by only 1% and10%, respectively. Consequently, the strength equationsproposed by Loov and Patnaik,15 Mau and Hsu,14 Birkeland(as reported in Reference 12), and Raths11 can be seen asspecial cases of the more general strength equation of theSMCS method (Eq. (10)), and are most suitable for specimensover-reinforced in the x-direction.

    EXPERIMENTAL EVALUATIONOF PROPOSED MODEL

    The calculations of the SMCS method and other methodsare compared with the experimental results from 114normalweight precracked and uncracked concrete pushoffspecimens and 15 composite beam specimens. Table 1summarizes the results of the comparison between theobserved strength and the calculations of six models. A moredetailed comparison is shown in Table 1A (Appendix A*) forSMCS, the ACI code17 equations, and for the modelproposed by Mattock.16 The Mattock model is selected forthe detailed comparison because it was developed based ona larger database, and because this database includes morehigh-strength concrete specimens.

    Hofbeck et al.1 pushoff specimensHofbeck et al.1 tested five series of normal-strength concrete

    pushoff specimens to study the effects of precracking; concretestrength; and the size, arrangement, and yield strength ofthe clamping y-reinforcement. The 12 uncracked specimens ofSeries 1 had an average fc of 28.4 MPa (4100 psi) and avariable amount of y-steel achieved using 9.5 mm (No. 3)bars at a variable spacing. The average provided x was0.62, which was larger than the upper limit = 0.302. Hence,the usable x was taken as 0.302. The comparison betweenthe observed and calculated nominal shearing strength isshown in Fig. 4(a).

    *The Appendix is available at www.concrete.org in PDF format as an addendum tothe published paper. It is also available in hard copy from ACI headquarters for a feeequal to the cost of reproduction plus handling at the time of the request.

    vn 0.55 fc( ) y fy y 0.3fc=

    y fy y y fy y

    y fy y y fy y

    y fy y y fy y

  • ACI Structural Journal/July-August 2010 423

    The 16 specimens of Series 2, 3, and 4 were cracked, andtheir average fc was 26.6 MPa (3850 psi). The variableamount of y-reinforcement was achieved by changing thenumber of the 9.5 mm (No. 3) bars in Series 2 and bychanging the size of the bars while maintaining their numberconstant in Series 3. Series 4 was similar to Series 2 but theclamping reinforcement had a larger yield strength. Theaverage provided x was 0.66 and, hence, the usable valuewas limited to the upper limit = 0.302. The results of thecomparison are shown in Fig. 4(b). The specimens of Series5 had a relatively lower concrete strength, with an average fcof 17.5 MPa (2540 psi), and the amount of y-reinforcementwas varied. The usable x was limited to = 0.314. Theresults of the comparison are shown in Fig. 4(c).

    The comparisons in Fig. 4 and Table 1 show that the SMCScaptures the trends observed in the tests and that the results were,in general, similar to those of the other methods. On the otherhand, the ACI calculations were more conservative but lessaccurate than the other methods.

    Importance of reinforcement parallelto transfer plane

    Nagle and Kuchma8 tested 18 large-scale high-strengthconcrete specimens with the shear transfer steel inclined atangles of 25 and 35 degrees to the shear transfer plane. Theprecracked specimens modeled the shear transfer along diagonalcracks in large-scale bridge girders. The concrete compressivestrength ranged from 93 to 121 MPa (13,500 to 17,550 psi)and the average was 104 MPa (15,080 psi).

    In all of these tests, the ratio of x-reinforcement x wasapproximately 0.0123, which was only 22% of the x =0.057 used in the Hofbeck et al.1 tests. The x-directionreinforcement index x was, on average, 0.0567 and, hence,was not limited to the upper limit of = 0.218 calculatedusing an average fc of 104 MPa (15,080 psi). These experimentalresults offer a test of the importance of considering the steel

    parallel to the shear transfer interface, and a test of theadequacy of the SMCS and the other methods.

    Substituting the average x and in Eq. (10) reduces it tothe following

    (15a)

    Substituting the value of the average fc reduces it further to

    vn = 2.43 MPa (29.3 psi) (15b)

    with an upper limit of 22.6 MPa (3280 psi). Equation (15)gives only 36% of the strength calculated using Mau andHsu's equation (Eq. (3)), approximately 40% of that by theLoov and Patnaik equation (Eq. (4)), approximately 87% ofthat by Birkelands equation (Eq. (13)), and 78% of that byRaths (Eq. (14)).

    Figure 5 compares the experimental with the calculatedstrength. All methods described in the Introduction sectionare included in the comparison to test their adequacy to thiscase of low x index. The reported values of the clampingstress and the experimental shear strength were adjusted byNagle and Kuchma8 for the inclination of the clamping steel.The results show an inadequacy of the methods proposed byMau and Hsu,14 Walraven et al.,6 and Loov and Patnaik.15The calculations of all 18 specimens were severelyunconservative, with calculated strength in some casesexceeding three times the observed strength. The Walravenet al.6 method was developed based on specimens with aconcrete compressive strength below 70 MPa (10,150 psi).Even if such a conservative value is taken as a limit for theapplicability of the method, the results remained severelyunconservative, as shown in Fig. 5. The same is true for theMau and Hsu14 and for the Loov and Patnaik15 methods. The

    vn 0.24 fc( ) y fy y 0.22fc=

    y fy y y fy y

    Table 1Summary of correlation of calculations of six methods with experimental shear capacities

    Source of testsNo.

    of tests fc , MPay fy y ,

    MPax fy x ,

    MPa y , MPa

    Average of observed-to-calculated shear strength (coefficient of variation, %)Method

    SMCS Mattock16 Mau and Hsu14Walraven

    et al.6Loov and Patnaik15 ACI 318-0517

    Hofbeck et al.1 33 16 to 31 0.34 to 10 17.7 0 1.17 (12.6) 1.18 (20.5) 1.04 (13.4) 1.05 (11.8) 1.18 (15.9) 1.69 (28.0)Mattock and

    Hawkins2 3 28 to 40 2.15 to 6.8* 0 to 5.61 1.08 (8.86) 1.03 (14.2) 0.99 (19.1) 0.96 (10.1) 1.42 (21.1) 1.66 (13.7)

    Mattock et al.3 9 26 to 29 3.65 to 5.6 16.6 2.76 to 0 1.05 (9.24) 1.05 (9.98) 0.88 (9.18) 0.94 (10.1) 0.95 (9.59) 1.40 (18.8)Mattock4 8 41 1.6 to 13.3 * 0 1.09 (8.40) 1.06 (18.0) 0.93 (12.4) 0.91 (8.57) 1.06 (16.5) 1.97 (22.8)

    Mattock et al.5 6 28 1.5 to 7.7 24.3 0 1.00 (8.04) 0.99 (4.28) 0.89 (10.8) 0.91 (9.77) 1.02 (15.6) 1.38 (10.2)Walraven

    et al.6 31 17 to 48 1.1 to 15.2 9.45 0 1.17 (15.3) 1.12 (11.5) 1.03 (11.5) 0.99 (11.8) 1.18 (13.4) 1.70 (18.5)

    Walraven and Stroband7 6 99 3.3 to 14.9

    * 0 0.92 (12.6) 1.02 (10.5) 0.66 (12.6) 0.53 (6.13) 0.72 (12.9) 2.43 (31.4)

    Nagle and Kuchma8 18 93 to 121 0.73 to 6.3 5.84 0 1.18 (19.8) 0.91 (21.4) 0.42 (19.9) 0.43 (15.5) 0.45 (21.0) 1.48 (21.3)

    Loov and Patnaik15 15 19 to 48 0.4 to 7.7

    || 0 1.19 (9.76) 1.56 (35.3) 0.98 (9.85) 1.03 (9.38) 1.04 (8.96) 1.64 (14.1)

    All tests 129 16 to 121 0.34 to 15.2 2.76 to 5.61 1.14 (14.8) 1.14 (26.7) 0.90 (26.5) 0.90 (26.8) 1.02 (29.2) 1.67 (25.7)*Longitudinal reinforcement relatively large, upper limit (=) is assumed.Based on assumed value of fy x = 460 MPa (69 ksi) (results do not change for any fy x > 240 MPa [35 ksi]).Calculated based on assumed value of fy x = 460 MPa [69 ksi].Composite beam specimens.||Near flexural compression zone, upper limit (=) is assumed.Note: 1 MPa = 145 psi.

  • 424 ACI Structural Journal/July-August 2010

    results of the SMCS method were generally adequate,whereas those by the ACI Code were more conservative.Mattocks method does not take into consideration thex-reinforcement, and overestimated the strength of aspecimen by 85%. However, it yielded adequate overallresults for this case because Eq. (5a) underestimates thestrength of specimens with relatively small clampingstresses. This is shown in Specimen 3 (Hofbeck et al.1; referto Table A1) and is also shown in a following section usingthe Loov and Patnaik tests.15 In conclusion, the Nagle andKuchma8 tests clearly show the importance of accounting for

    the steel parallel to the shear plane to avoid unconservativedesigns in empirically based methods.

    External clamping stressesThe effects of an external compressive or tensile stress

    perpendicular to the transfer plane can be accounted for bythe superposition of reinforcement. Mattock et al.3 tested twoseries of specimens subjected to tensile stresses. Series E and Fhad an average y fy y of 3.73 and 5.52 MPa (540 and800 psi), respectively. The yield strength of the x-reinforcementwas not reported, but the results using the SMCS methodwould not change for any fy x larger than 240 MPa (35 ksi)for this series of specimens. A value of 460 MPa (69 ksi) wasused to report the x fy x shown in Tables 1 and A1.

    The effects of a compressive clamping stress were investigatedby Mattock and Hawkins2 in their series of 10 specimens testedusing a modified pushoff test setup. The x-reinforcementwas not reported, and a value of x equal to is assumed.

    Fig. 5Shear transfer in large-scale high-strength concretepushoff specimens with relatively low level of x-reinforcement.8

    Fig. 4Shear strength of normal-strength reinforced concretepushoff specimens.1

    Fig. 6Effect of stress in direction perpendicular to transferplane2,3 on shear strength.

  • ACI Structural Journal/July-August 2010 425

    Figure 6 shows the experimental results from the threeseries of specimens (E, F, and 10) subjected to shear and aclamping stress, and compares them with the strength calculatedusing SMCS. A very good agreement is obtained. Theaverage and coefficients of variation values reported inTable 1 show that good accuracy was obtained by usingMattocks method.16 The tables also show that the ACIequations were conservative, especially in combined shearand compression. The upper limit of 5.5 MPa (800 psi) onthe strength was critical and did not allow benefit from thecompressive clamping stresses.

    Tables 1 and A1 include comparisons with experimentalresults from other pushoff specimens.4-7

    Composite beam specimensLoov and Patnaik15 tested the shear transfer between the

    webs and the flanges of 16 composite beams. The webs werecast first and the top surfaces were left as-cast with the coarseaggregates left protruding without efforts to produce roughsurfaces. Stirrups extended through the cold joints andprovided the clamping forces after the flanges were cast. Thejoint in one beam was smooth and was not included in theevaluation. The beams were tested in a three-point loadingset up and, hence, the web-flange interface was located nearthe flexural compression zone. This proximity provided astate of considerable level of compression in the x-direction(parallel to the shear interface), and, hence, for the SMCScalculations, x is considered to be large enough to belimited by the upper value .

    The ACI Code17 provides special design provisions for thehorizontal interface between precast girders and cast-in-place slabs when the shearing stress is smaller than 3.5 MPa(500 psi).

    (16a)

    (16b)

    For clean and intentionally roughened joints, and for cleanbut not intentionally roughened joints with minimum transversesteel as per ACI 318-05, Section 11.5.6, a 0.55 MPa (80 psi)stress is allowed.

    In 12 of the beams, fc was approximately 35 MPa (5075 psi)and the clamping force was variable. Figure 7 shows theexperimental shear strengths and compares them with thecalculations of the three methods, in addition to the results ofLoov and Patnaiks15 own method, which was calibratedusing these test results. The ACI equations provided veryconservative calculations, whereas Mattocks16 equationsgave very conservative results at low levels of clampingforces, showing the disadvantages of a linear relationshipbetween the strength and the clamping stress. The SMCSprovided an accurate lower-bound of the results and wasmore suitable than the empirically developed methods.

    Overall correlationTable 1 shows the average and the coefficient of variation

    for the ratios of the experimental to the calculated shearstrength for the 129 test results included in this study. Acomparison between the results of the six methods shows thatthe proposed SMCS model provided the most accurate results.

    vACI 1.8 0.6y fy y+( ) 3.5 MPa=

    vACI 260 0.6y fy y+( ) 500 psi=

    CONCLUSIONSThis paper showed that the SMCS method can be applied

    to the shear-transfer problem. The calculations of thismethod and five existing semi-empirical methods werecompared to the experimental results from 15 compositebeams and 114 normalweight concrete pushoff specimens,and the proposed method was shown to provide the mostaccurate results.

    The better accuracy of the proposed method was mainlydue to its ability to account for the effects of the reinforcementin the direction parallel to the shear-transfer plane. The fiveother models that were discussed in this paper do not accountfor this factor, and were calibrated using test specimenswhere this reinforcement was relatively very large. Four ofthe methods were severely unconservative when used tocalculate the shear strength of recently tested pushoff specimenswith moderate (and practical) levels of such reinforcement.Meanwhile, the ACI code results were adequate for these testsbecause of the markedly large margin of conservatism built-inin the ACI shear-friction model.

    It was also shown that the semi-empirical equations thatrelate the shear strength to the square root of the clampingstress can be seen as special cases of the more generalstrength equation of the proposed SMCS method. The semi-empirical equations can be derived from the SMCS methodfor the case where the specimens are over-reinforced in thedirection parallel to the shear-transfer plane.

    The noniterative SMCS model is a generalized modelapplicable also to membrane elements subjected to in-planestresses and to beams subjected to combined shear andflexure and to combined torsion and flexure. Hence, it has afavorable combination of generality, accuracy, and simplicity.

    NOTATIONvACI = ultimate shear strength calculated using ACI equationsvexp = experimentally observed ultimate shear strengthvLP = ultimate shear strength calculated using model by Loov and Patnaik15vMat = ultimate shear strength calculated using model by Mattock16vMH = ultimate shear strength calculated using model by Mau and Hsu14vn = nominal shear strengthvSMCS = ultimate shear strength calculated using proposed SMCS modelvWFP = ultimate shear strength calculated using model by Walraven et al.6 = angle of inclination of clamping reinforcement with respect to

    shear transfer plane

    Fig. 7Shear transfer across cold joint between webs andflanges of composite beams.15

  • ACI Structural Journal/July-August 2010426

    REFERENCES1. Hofbeck, J. A.; Ibrahim, I. O.; and Mattock, A. H., Shear Transfer in

    Reinforced Concrete, ACI JOURNAL, Proceedings V. 66, No. 2, Feb. 1969,pp. 119-128.

    2. Mattock, A. H., and Hawkins, N. M., Shear Transfer in ReinforcedConcreteRecent Research, Journal of the Prestressed ConcreteInstitute, V. 17, No. 2, 1972, pp. 55-75.

    3. Mattock, A. H.; Johal, L.; and Chow, H. C., Shear Transfer in ReinforcedConcrete with Moment or Tension Acting across the Shear Plane, Journalof the Prestressed Concrete Institute, V. 20, No. 4, 1975, pp. 76-93.

    4. Mattock, A. H., Shear Transfer under Monotonic Loading across anInterface between Concretes Cast at Different Times, University of WashingtonReport SM 76-3, Sept. 1976, 66 pp.

    5. Mattock, A. H.; Li, W. K.; and Wang, T. C., Shear Transfer in Light-WeightReinforced Concrete, Journal of the Prestressed Concrete Institute, V. 21,No. 1, 1976, pp. 20-39.

    6. Walraven, J. C.; Frenay, J.; and Pruijssers, A., Influence of ConcreteStrength and Load History on the Shear Friction Capacity of ConcreteMembers, Journal of the Prestressed Concrete Institute, V. 32, No. 1,1987, pp. 66-84.

    7. Walraven, J. C., and Stroband, J., Shear Friction in High-StrengthConcrete, High-Performance Concrete, High-Performance Concrete,SP-149, V. M. Malhotra, ed., American Concrete Institute, FarmingtonHills, MI, 1994, pp. 311-330.

    8. Nagle, T. J., and Kuchma, D. A., Shear Transfer Resistance in High-Strength Concrete Girders, Magazine of Concrete Research, V. 59, No. 8,2007, pp. 611-620.

    9. Walraven, J. C., Fundamental Analysis of Aggregate Interlock,Journal of the Structural Division, ASCE, V. 107, No. 11, Nov. 1981,pp. 2245-2270.

    10. Birkeland, P. W., and Birkeland, H. W., Connections in PrecastConcrete Construction, ACI JOURNAL, Proceedings V. 63, No. 3, Mar.1966, pp. 345-368.

    11. Raths, C. H., discussion of the paper, Design Proposals forReinforced Concrete Corbels, by A. H. Mattock, PCI Journal, V. 22,No. 2, 1977, pp. 93-98.

    12. Shaikh, A. F., Proposed Revisions to Shear-Friction Provisions,Journal of the Prestressed Concrete Institute, V. 23, No. 2, 1978, pp. 12-21.

    13. Hsu, T. T. C.; Mau, S. T.; and Chen, B., Theory of Shear Transfer ofReinforced Concrete, ACI Structural Journal, V. 84, No. 2, Mar.-Apr.1987, pp. 149-160.

    14. Mau, S. T., and Hsu, T. T. C., Readers Comments on Influence ofConcrete Strength and Load History on the Shear Friction Capacity ofConcrete Members, Journal of the Prestressed Concrete Institute, V. 33,No. 1, 1988, pp. 166-168.

    15. Loov, R. E., and Patnaik, A. K., Horizontal Shear Strength ofComposite Beams with a Rough Interface, Journal of the PrestressedConcrete Institute, V. 39, No. 1, 1994, pp. 48-69.

    16. Mattock, A. L., Shear-Friction and High-Strength Concrete, ACIStructural Journal, V. 98, No. 1, Jan.-Feb. 2001, pp. 50-59.

    17. ACI 318-05, Building Code Requirements for Structural Concrete(ACI 318-05) and Commentary (318R-05), American Concrete Institute,Farmington Hills, MI, 2005, 430 pp.

    18. Rahal, K. N., Simplified Design and Capacity Calculation of ShearStrength in Reinforced Concrete Membrane Elements, EngineeringStructures, V. 30, No. 10, 2008, pp. 2782-2791.

    19. Rahal, K. N., Shear Strength of Reinforced Concrete, Part II: BeamsSubjected to Shear, Bending Moment and Axial Loads, ACI StructuralJournal, V. 97, No. 2, Mar.-Apr. 2000, pp. 219-224.

    20. Rahal, K. N., Torsional Strength of Reinforced Concrete Beams,Canadian Journal of Civil Engineering, V. 27, No. 3, June 2000, pp. 445-453.

    21. Rahal, K. N., Combined Torsion and Bending in Reinforced andPrestressed Concrete beams Using SMCS, ACI Structural Journal, V. 104,No. 4, July-Aug. 2007, pp. 402-411.

    22. Vecchio, F. J., and Collins, M. P., Modified Compression FieldTheory for Reinforced Concrete Elements Subjected to Shear, ACIJOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

    23. Brstrup, M. W., Plastic Analysis of Shear in ReinforcedConcrete, Magazine of Concrete Research, V. 26, 1974, pp. 221-228.

    24. Brstrup, M. W., discussion of Shear Strength of ReinforcedConcrete, Part I: Membrane Elements Subjected To Pure Shear, by K. N.Rahal and closure by author, ACI Structural Journal, V. 97, No. 6, Nov.-Dec. 2000, pp. 910-913.

  • Appendix A

    This Appendix provides detailed listing of the analyzed specimens and the results from three of the models

    (SMCS, ACI code and Mattock16

    Method).

    Table A1: Properties of Test Specimens and Results of SMCS, ACI and Mattock16

    Methods

    Type

    of

    test

    ID fc

    (MPa)

    y fy-y

    (MPa)

    x fy-x

    (MPa)

    y

    (MPa)

    vexp

    (MPa)

    vSMCS

    (MPa)

    vMat

    (MPa)

    vACI

    (MPa)

    exp

    SMCS

    v

    v

    exp

    Mat

    v

    v

    exp

    ACI

    v

    v

    Push

    -off

    1

    1.1A 27.0 1.54

    17.7 0

    5.17 3.55 3.46 2.15 1.46 1.49 2.40

    1.1B 29.9 1.46 5.82 3.62 3.28 2.04 1.61 1.78 2.85

    1.2A 26.5 3.08 6.90 4.98 5.11 4.31 1.39 1.35 1.60

    1.2B 28.8 2.91 6.76 5.03 5.21 4.08 1.34 1.30 1.66

    1.3A 26.5 4.62 7.59 6.09 6.34 5.30 1.24 1.20 1.43

    1.3B 27.0 4.37 7.38 5.99 6.20 5.41 1.23 1.19 1.36

    1.4A 31.1 6.15 9.38 7.56 8.03 5.50 1.24 1.17 1.71

    1.4B 26.6 5.83 8.83 6.86 7.32 5.32 1.29 1.21 1.66

    1.5A 31.1 7.69 9.66 8.45 9.26 5.50 1.14 1.04 1.76

    1.5B 28.0 7.28 9.54 7.85 8.41 5.50 1.22 1.13 1.74

    1.6A 29.7 9.23 9.88 8.93 8.92 5.50 1.11 1.11 1.80

    1.6B 27.9 8.74 9.79 8.44 8.38 5.50 1.16 1.17 1.78

    2.1 21.4 1.54 4.07 3.19 3.37 2.15 1.28 1.21 1.89

    2.2 21.4 3.08 4.69 4.51 4.60 4.28 1.04 1.02 1.10

    2.3 26.9 4.62 5.79 6.14 6.38 5.38 0.94 0.91 1.08

    2.4 26.9 6.15 6.90 7.09 7.61 5.38 0.97 0.91 1.28

    2.5 28.8 7.69 8.97 8.17 8.65 5.50 1.10 1.04 1.63

    2.6 28.8 9.23 9.55 8.69 8.65 5.50 1.10 1.10 1.74

    3.1 27.9 0.34 1.66 1.70 0.78 0.48 0.97 2.13 3.43

    3.2 27.7 1.56 3.59 3.62 3.52 2.19 0.99 1.02 1.64

    3.3 21.4 3.08 4.69 4.51 4.60 4.28 1.04 1.02 1.10

    3.4 27.9 5.21 7.09 6.62 6.95 5.50 1.07 1.02 1.29

    3.5 27.9 7.25 7.94 7.82 8.36 5.50 1.02 0.95 1.44

    4.1 28.1 2.01 4.86 4.12 4.41 2.81 1.18 1.10 1.73

    4.2 28.1 4.01 6.76 5.83 6.02 5.50 1.16 1.12 1.23

    4.3 29.9 6.02 8.14 7.35 7.81 5.50 1.11 1.04 1.48

    4.4 29.9 8.02 9.66 8.49 8.98 5.50 1.14 1.08 1.76

    4.5 23.4 10.03 9.10 7.19 7.01 4.68 1.27 1.30 1.95

    5.1 16.9 1.54 3.52 2.86 2.92 2.15 1.23 1.20 1.63

    5.2 18.1 3.08 4.83 4.17 4.27 3.61 1.16 1.13 1.34

    5.3 16.4 4.62 5.59 4.89 4.93 3.29 1.14 1.13 1.70

    5.4 17.8 6.15 5.48 5.58 5.34 3.56 0.98 1.03 1.54

    5.5 18.1 7.69 6.97 5.66 5.42 3.61 1.23 1.28 1.93

    Push

    -

    off

    2

    10.7 27.7 6.63

    2.67 9.97 8.39 8.32 5.50 1.19 1.20 1.81

    10.8 27.7 6.79 0.00 7.69 7.55 8.21 5.50 1.02 0.94 1.40

    10.1 40.0 2.15 5.61 9.72 9.47 10.21 5.50 1.03 0.95 1.77

    Pu

    sh-o

    ff 3

    E1C 26.6 3.74

    16.6#

    0.00 6.08 5.50 5.65 5.24 1.10 1.07 1.16

    E2C 29.1 3.77 -0.69 6.41 5.19 5.37 4.31 1.23 1.19 1.49

    E3C 27.3 3.81 -1.12 4.92 4.71 4.88 3.76 1.05 1.01 1.31

    E4C 26.3 3.65 -1.38 4.64 4.26 4.45 3.18 1.09 1.04 1.46

    E5C 27.7 3.78 -2.07 3.63 3.79 3.85 2.39 0.96 0.94 1.52

    E6C 27.5 3.68 -2.76 2.54 2.76 2.06 1.28 0.92 1.23 1.98

    F1C 29.1 5.43 0.00 6.81 6.90 7.25 5.50 0.99 0.94 1.24

    F4C 26.8 5.56 -1.38 5.79 5.83 6.03 5.37 0.99 0.96 1.08

    F6C 28.6 5.60 -2.76 5.54 4.95 5.14 3.98 1.12 1.08 1.39

  • Push

    -off

    4

    A1 41.5 1.57

    0

    5.24 4.32 3.52 2.19 1.21 1.49 2.39

    A2 41.5 3.13 5.52 6.11 6.66 4.38 0.90 0.83 1.26

    A3 40.1 5.05 7.93 7.65 8.05 5.50 1.04 0.98 1.44

    A4 40.6 6.73 9.79 8.87 9.44 5.50 1.10 1.04 1.78

    A5 42.2 7.78 10.34 9.70 10.45 5.50 1.07 0.99 1.88

    A6 40.7 10.59 12.14 11.14 12.21 5.50 1.09 0.99 2.21

    A6A 41.2 10.59 12.83 11.20 12.35 5.50 1.15 1.04 2.33

    A7 41.2 13.30 13.38 11.84 12.35 5.50 1.13 1.08 2.43

    Push

    -off

    5

    N1 28.8 1.54 24.3

    0

    3.17 3.66 3.48 2.16 0.87 0.91 1.47

    N2 26.9 3.20 24.3 5.38 5.11 5.25 4.48 1.05 1.02 1.20

    N3 27.6 4.76 24.3 6.62 6.30 6.56 5.50 1.05 1.01 1.20

    N4 28.6 6.18 24.3 7.93 7.30 7.81 5.50 1.09 1.02 1.44

    N5 27.1 7.72 24.3 8.10 7.97 8.14 5.43 1.02 1.00 1.49

    N6 28.4 7.72 24.3 8.21 8.57 8.52 5.50 0.96 0.96 1.49

    Push

    -off

    6

    110208t 30.5 2.43

    9.45## 0

    5.08 4.71 5.00 3.40 1.08 1.02 1.49

    110208 26.1 2.43 5.50 4.39 4.55 3.40 1.25 1.21 1.62

    110208g 25.0 2.43 5.08 4.31 4.44 3.40 1.18 1.14 1.49

    110408 26.1 4.86 6.44 6.21 6.50 5.22 1.04 0.99 1.23

    110608 26.1 7.29 7.39 7.61 7.83 5.22 0.97 0.94 1.42

    110808h 25.0 9.72 8.39 7.64 7.50 5.00 1.10 1.12 1.68

    110808hg 25.0 9.72 8.58 7.64 7.50 5.00 1.12 1.14 1.72

    110706 26.9 5.58 7.19 6.75 7.16 5.39 1.06 1.00 1.33

    210204 31.1 1.06 3.22 3.14 2.39 1.48 1.03 1.35 2.17

    210608 31.1 7.29 9.72 8.23 8.94 5.50 1.18 1.09 1.77

    210216 31.1 10.12 9.25 9.29 9.33 5.50 1.00 0.99 1.68

    210316 31.1 15.17 10.11 9.29 9.33 5.50 1.09 1.08 1.84

    210808 21.4 9.72 7.97 6.63 6.43 4.28 1.20 1.24 1.86

    120208 25.1 2.43 5.36 4.31 4.45 3.40 1.24 1.20 1.58

    120408 25.1 4.86 6.53 6.10 6.40 5.02 1.07 1.02 1.30

    120608 25.1 7.29 6.78 7.47 7.52 5.02 0.91 0.90 1.35

    120808 25.1 9.72 7.31 7.66 7.52 5.02 0.95 0.97 1.46

    120706 24.8 5.58 6.92 6.51 6.95 4.96 1.06 1.00 1.39

    120216 24.8 10.12 6.53 7.59 7.45 4.96 0.86 0.88 1.32

    230208 47.7 2.43 6.72 4.79 5.47 3.40 1.40 1.23 1.98

    230408 47.7 4.87 10.83 6.78 8.66 5.50 1.60 1.25 1.97

    230608 47.7 7.29 12.56 8.30 10.60 5.50 1.51 1.18 2.28

    230808 47.7 9.72 14.19 9.58 12.54 5.50 1.48 1.13 2.58

    240208 16.9 2.43 4.65 3.60 3.64 3.38 1.29 1.28 1.37

    240408 16.9 4.86 6.04 5.09 5.07 3.38 1.19 1.19 1.79

    240608 16.9 7.29 6.55 5.32 5.07 3.38 1.23 1.29 1.94

    240808 16.9 9.72 6.29 5.32 5.07 3.38 1.18 1.24 1.86

    250208 32.5 2.43 6.83 4.79 5.19 3.40 1.43 1.32 2.01

    250408 32.5 4.86 8.69 6.78 7.14 5.50 1.28 1.22 1.58

    250608 32.5 7.29 9.65 8.30 9.08 5.50 1.16 1.06 1.75

    250808 32.5 9.72 9.94 9.55 9.74 5.50 1.04 1.02 1.81

    Push

    -off

    7

    10 99.0 3.33

    0

    6.30 8.58 7.49 4.66 0.73 0.84 1.35

    11 99.0 6.67 11.20 12.14 10.84 5.50 0.92 1.03 2.04

    12 99.0 10.00 15.00 14.87 13.50 5.50 1.01 1.11 2.73

    13 99.0 13.27 18.10 17.13 16.12 5.50 1.06 1.12 3.29

    14 99.0 7.50 11.00 12.88 11.50 5.50 0.85 0.96 2.00

    15 99.0 14.94 17.61 18.17 16.55 5.50 0.97 1.06 3.20

    Pu

    sh-o

    ff 8

    glsh_4_13_25 92.7 2.24

    5.84 0

    4.21 3.44 4.57 2.84 1.22 0.92 1.48

    glsl_2_13_25 92.7 1.12 2.01 2.44 2.28 1.42 0.82 0.88 1.41

    glsh_4_13_35 92.7 2.03 3.70 3.12 3.74 2.33 1.19 0.99 1.59

    glsl_2_13_35 92.7 1.01 2.53 2.20 1.87 1.16 1.15 1.35 2.17

    g2sh_4_16_25 97.6 4.28 5.62 4.76 8.60 5.43 1.18 0.65 1.04

  • g2sl_2_16_25 97.6 2.77 3.96 3.83 5.64 3.51 1.03 0.70 1.13

    g2sh_4_16_35 97.6 3.86 3.85 4.30 7.11 4.42 0.90 0.54 0.87

    g2sl_2_16_35 97.6 2.50 3.24 3.46 4.61 2.87 0.94 0.70 1.13

    g3sh_4_13_25 114.4 3.26 5.52 4.15 6.65 4.14 1.33 0.83 1.34

    g3sl_4_13_25 114.4 2.17 4.36 3.39 4.43 2.75 1.29 0.99 1.58

    g3sh_4_13_35 114.4 2.95 4.92 3.76 5.44 3.38 1.31 0.90 1.45

    g3sl_4_13_35 114.4 1.97 3.96 3.07 3.62 2.25 1.29 1.09 1.76

    g4sh_4_16_25 114.4 6.28 8.88 5.77 10.05 5.50 1.54 0.88 1.61

    g4sl_4_16_35 114.4 5.79 7.62 5.26 9.29 5.50 1.45 0.82 1.39

    g5sh_2_10_25 120.6 0.81 1.80 2.07 1.65 1.02 0.87 1.09 1.76

    g5sh_2_10_35 120.6 0.73 1.57 1.87 1.35 0.84 0.84 1.16 1.87

    g6sh_4_16_25 93.0 3.20 6.37 4.11 6.52 4.06 1.55 0.98 1.57

    g6sh_4_16_35 93.0 2.82 4.90 3.67 5.20 3.23 1.33 0.94 1.52

    Com

    po

    site

    Bea

    m 1

    5

    1 37.4 4.36

    0

    7.76 6.90 7.23 4.36 1.13 1.07 1.78

    2 34.9 1.66 4.27 4.13 3.74 2.80 1.03 1.14 1.53

    3 30.5 2.73 6.82 4.99 5.23 3.44 1.37 1.30 1.98

    4 34.7 6.03 8.10 7.85 8.29 5.50 1.03 0.98 1.47

    5 34.8 1.63 5.54 4.09 3.67 2.78 1.36 1.51 1.99

    6 37.1 1.62 5.25 4.19 3.65 2.77 1.25 1.44 1.89

    7 35.8 6.06 9.25 7.98 8.43 5.50 1.16 1.10 1.68

    8 35.6 0.77 3.12 2.84 1.73 2.26 1.10 1.80 1.38

    9 37.1 1.62 4.64 4.19 3.65 2.77 1.11 1.27 1.67

    10 37.6 0.77 3.46 2.91 1.73 2.26 1.19 2.00 1.53

    11 32.7 0.40 2.57 1.97 0.90 2.04 1.30 2.86 1.26

    12 34.6 7.72 9.20 8.88 9.64 5.50 1.04 0.95 1.67

    13 19.2 0.82 2.92 2.22 1.85 2.29 1.32 1.58 1.27

    15 44.0 0.80 3.94 3.16 1.80 2.28 1.25 2.19 1.73

    16 48.3 0.80 4.01 3.29 1.80 2.28 1.22 2.23 1.76 Longitudinal reinforcement relatively large, upper limit (=) is assumed.

    Near flexural compression zone, upper limit (=) is assumed. #based on assumed value of fy-x=460 MPa (69 ksi) (results do not change for any fy-x >240 MPa or 35 ksi)

    ##calculated based on assumed value of fy-x=460 MPa (69 ksi)

    Note: 1 MPa = 145 psi