shear behavior of headed anchors with large diameters and deep embedments - discussion title 107-s14...

248 ACI Structural Journal/March-April 2011

DISCUSSIONDisc. 107-S14/From the March-April 2010 ACI Structural Journal, p. 146

Shear Behavior of Headed Anchors with Large Diameters and Deep Embedments. Paper by Nam Ho Lee,Kwang Ryeon Park, and Yong Pyo Suh

Discussion by Donald F. Meinheit and Neil S. AndersonAffiliated Consultant, Wiss, Janney, Elstner Associates, Chicago, IL; Vice President of Engineering, Concrete Reinforcing Steel Institute, Schaumburg, IL

Compliments to the authors for publishing some veryunique and needed data on the shear breakout strength oflarge-diameter and deeply embedded anchors. The discussersare particularly pleased with the conclusion that embedmentdepth hef and anchor diameter da did not influence theconcrete shear breakout strength. This research echoes whatthe Precast/Prestressed Concrete Institute (PCI) has advocatedsince the 4th Edition of the PCI Design Handbook in 1992 9:the concrete shear breakout strength for uncracked concretewas only a function of the concrete compressive strength fc′and edge distance ca1, that is

Vb,PCI 1992 (design) = 12.5 (ca1)1.5 (6)

The form of this equation was further confirmed in an exten-sive PCI-sponsored research program10 on welded headedstuds used as the basis for both the 6th and 7th Editions of thePCI Design Handbook.11,12 The current 5% fractile shearbreakout design equation in the PCI Design Handbook andthe average equation derived from test data on headed studsembedded in uncracked concrete are, respectively, Eq. (7)and (8)

Vb,PCI 2010 (design) = 16.5 (ca1)1.33 (7)

Vb,PCI 2010 (mean) = 22 (ca1)1.33 (8)

Further confirmation of the shear breakout equationformat being a function of fc′ and ca1 was recently presentedby Ronald Cook, University of Florida-Gainesville, duringan ACI 318 Code meeting.13 A design (5% fractile) equationfor cracked concrete and average strength equation foruncracked concrete applicable for many anchors in shear canbe represented by Eq. (9) and (10), respectively

Vb,ACI 2010 (proposed design—cracked) = 9.0 (ca1)1.5 (9)

Vb,ACI 2010 (mean—uncracked) = 17.0 (ca1)1.5 (10)

The average strength equation (Eq. (10)) was derived fromanalysis of both cast-in-place and post-installed anchors innormalweight concrete contained in an international data-base. Equation (10) is a much better concrete strengthpredictor than the current ACI 318-083 Appendix D Eq. (D-25),of which the mean value is presented below as Eq. (11).

Vb,ACI 2008 (mean) = (11)

fc′

fc′

fc′

fc′

fc′

13le

da

-----⎝ ⎠⎛ ⎞ 0.2

da⎝ ⎠⎛ ⎞ fc′ ca1( )1.5

These new test results confirm that a very simple equation,almost identical to that of the tension breakout equation, isaccurate for cast-in embedded bolts and headed studs overthe full range of anchor diameters.

Korean test dataBecause of the importance of the tests reported in the paper,

can the authors please report results for each test in each series,along with the corresponding concrete compressive strength?The paper presents only a test series average for the particularvariable under consideration. The reference list does notidentify another source publication where these test resultscan be found; therefore, it is important that all tests beadequately documented in the literature for future use.

The actual steel strength of the anchor bolts and reinforcingbars used in the specimens are not reported; the authors onlyreported minimum strengths corresponding to the specificstandard for which the material conformed. Likewise, it isunclear if the anchor bolt threads are included or excluded inthe shear plane for these tests. Even though anchor bolt steelstrength did not control, actual strengths are necessaryparameters for future data analysis.

Discussion of test specimensAlthough the researchers took strides to avoid anchor load

frame interference, it appears that some tests might beslightly biased due to the provided clear distance betweenreaction supports. The CCD “mathematical model” purportsa 1.5ca1 spread on each side of the anchor, but unfortunatelythe concrete “does not know that” and reality dictates thisdistance should be upward of double that value, 3ca1, oneach side for testing. Support reactions from the test framecan confine the failure surface outer points, which can resultin higher failure loads.

Tests in Series S7, the series with the largest spacingbetween reaction supports, do not appear to be affected bythe reaction location. As the load-deflection plots (Fig. 7)indicate, cracking occurred at approximately 22 mm (0.87 in.)where there is a change in slope, and then failure. In theother unreinforced tests with smaller edge distances, theload-deflection plots seem to indicate first cracking at 2 mm(0.08 in.), followed by a line slope change; other deflectionreadings around 3.5 to 4 mm (0.14 to 0.16 in.) show anotherline slope change prior to failure. This two-stage slope couldbe indicative of the restraining effect of the support reactions.Can the authors please comment on this effect, and whetherthey believe some restraint may have existed due to the shearreaction location?

Our study of Fig. 5 indicates that the failure surfaces inTests S1-A, S1-C, and S4-E may have been influenced by acorner failure. The shear crack propagation was driven to the

ACI Structural Journal/March-April 2011 249

specimen edge due to the reaction confining effect. Was thischaracteristic observed on any other tests, as only sixrepresentative tests are illustrated in Fig. 5?

Unreinforced testsThe discussers made comparisons of the various mean

prediction equations and the tests results for the unreinforcedand uncracked concrete tests. Our comparisons shown inFig. 12 are in terms of test-to-predicted strength ratios.Included in this comparison is the current ACI 318-083

Appendix D equation. Clearly Fig. 12 shows the currentACI 318 Appendix D mean equation is unconservative bya significant margin. The equation with the best, closest, fitto these data is the proposed ACI 318, 2010 equation,whereas the PCI equations are conservative predictors ofmean strength. The authors should perhaps modify theirconclusion to indicate that the design equation for shearbreakout in the ACI Code, Appendix D, should be changedto a simpler and better prediction equation.

Anchor reinforcement testsThe provided anchor reinforcement strength exceeded the

ACI 318-08 Appendix D mean equation (Eq. (11)). There-fore, there was ample anchor reinforcement in all reinforcedtests to carry test loads larger than the concrete crackingcapacity load. The reinforced tests can be compared to theresults of Series S6 (unreinforced) tests, where da, hef , and c1were all the same.

An analysis of Fig. 7(e), representing four tests fromSeries S6, shows the specimens reached the concreteshear strength when cracks formed at a displacement ofapproximately 5 mm (0.20 in.). Concrete shear crackingevidence appears in the load-deformation plots for theSeries S10 tests, and for some of the tests in Series S8 andS9; cracking is not as evident for Series S8 and S9,perhaps due to the scale of the plots in Fig. 8. It doesappear, however, that there is a slope change of the load-deformation plot at approximately 5 mm (0.2 in.) in eachcase for all tests in Series S8, S9, and S10.

According to Fig. 2, tests in Series S10 had the least amountof anchor reinforcement and reportedly performed the poorest.The Series S10 load-deformation plots clearly show thecracking and transfer of the load to the anchor reinforcing bars.Anchor reinforcement was directly adjacent to and in contactwith the embedded anchor loaded in shear; that is, there wasno intentional concrete between the anchor and the anchorreinforcement, as in Series S8 and some of the tests in SeriesS9. In Series S10, the reinforcement bearing plate waslocated 6 in. (152 mm) below the concrete surface. Althoughnot shown, this location was likely below the shear crackpath initiating at the anchor-concrete surface juncture. Thetests in Series S8 had the highest capacity, likely attributedto the embedded anchor being confined within a reinforcingbar cage, well-distributed across the failure surface.

Series S9 test behaviors are most interesting. All Series S9tests had about the same amount of anchor reinforcement,but the reinforcing bar placement is very informative.Specimen S9-2 had hairpin reinforcement not contacting theembedded anchor, whereas all of the anchor reinforcementwas in contact with the embedded anchor in S9-A through D.Test S9-2 performed better by having a higher ultimate loadand greater stiffness throughout the test (refer to Fig. 8(b)).A companion test, Test S9-1, had two hairpin bars in contactwith the embedded anchor, supplemented with two hookedbars closest to the surface and not in contact with theembedded anchor. Specimen S9-1 shows the greatest stiffnessand almost the same ultimate strength as Specimen S9-2.

From these anchor reinforcing tests, the discussersconclude the anchor reinforcement may not have to be indirect contact with the embedded anchor, and anchorreinforcing bars near the concrete surface are more effectivethan deeper embedded bars.

Another conclusion drawn from these tests is the anchorreinforcements may not be 100% efficient, because theywere not able to fully develop the anchor reinforcementstrength. In each anchor reinforcing test, however, thereinforcement carried the shear load once the concretecracked. In each case, the reinforced specimens were able tocarry at least twice the failure load of unreinforced specimens

Fig. 12—Test-to-calculated ratios for large-diameter anchors in shear.


in Series S6. For Specimen S10, the capacity may have beengreater had the bearing plate assembly been located higher,within the shear failure wedge of concrete. These aresignificant conclusions drawn by the discussers from thisresearch, which have not been previously discussed.

REFERENCES9. Precast/Prestressed Concrete Institute (PCI), PCI Design Handbook,

4th Edition, PCI MNL 120-92, Chicago, IL, 1992.10. Anderson, N. S., and Meinheit, D. F., “A Review of the Chapter 6

Headed Stud Design Criteria in the 6th Edition PCI Handbook,” PCIJournal, V. 52, No. 1, Jan.-Feb. 2007, pp. 84-112.

11. Precast/Prestressed Concrete Institute (PCI), PCI Design Handbook,6th Edition, PCI MNL 120-04, Chicago, IL, 2004.

12. Precast/Prestressed Concrete Institute (PCI), PCI Design Handbook,7th Edition, PCI MNL 120-10, Chicago, IL, 2010.

13. ACI Subcommittee 318-B, Committee correspondence from RonaldA. Cook, University of Florida-Gainesville, proposed code changes forshear breakout of anchors, 2010.

AUTHORS’ CLOSUREThe authors would like to thank the discussers for their

support of the authors’ conclusions and their insightful andconstructive discussion to provide the authors an opportunityto further clarify the findings of the experimental study. Asthe title of the paper implies, the focus was on the concreteshear breakout strength of anchors with large diameters,deep embedments, and relatively short edge distance toanchor diameter. This focus was stated in the researchsignificance section, detailed in the main body of the paper,and reiterated in the summary and conclusions section. Theobjective of the paper was neither to newly develop norvalidate the existing formulas in the current code provisionsfor anchors with relatively smaller diameter and shallowerembedments than those tested in this research.

The discussers indicated that the concrete shear breakoutstrength for uncracked concrete was only a function of theconcrete compressive strength fc′ and edge distance ca1, asthe Precast/Prestressed Concrete Institute (PCI) has advocated.The authors have presented that the embedment depth hefand anchor diameter da influence the concrete shear breakoutstrength, but limited to hef /da = 7 for the concrete shearbreakout strength of the headed anchors based on the testresults with diameters da exceeding 2.5 in (63.5 mm),embedment depths hef exceeding 25 in. (635.0 mm) and hef /da= 7.14 to 10.0. It is beyond the scope of this paper to discussif this research echoes what the Precast/Prestressed ConcreteInstitute (PCI) has advocated. More tests with various ratiosof hef/da less than 7.0 proposed in this paper are thereforerequired to fully support the discussers’ conclusion thatembedment depth hef and anchor diameter da did not influencethe concrete shear breakout strength irrespective of anchordiameter, embedment depth, and edge distance.

Korean test dataThe concrete breakout capacity for each test in each series

is shown in Fig. 7—Measured load-displacement relationship(no supplementary reinforcement), and Fig. 8—Measuredload-displacement relationship (supplementary reinforce-ment). The concrete breakout capacity for each test in eachseries and the corresponding concrete compressive strengthwere eliminated from this paper due to the limitation in lengthbecause they are judged not to be necessary for this paper.The test results were presented in a report in the Koreanlanguage as proprietary material of the sponsors.

This research is intended to evaluate the shear concretebreakout capacity for anchors with large diameters and deepembedments. Thus the test specimens are intentionallydesigned not to induce steel anchor bolt failure but to induceconcrete shear breakout failure. Therefore, the actualstrengths of individual steel anchor bolts, which are generallygreater than nominal strength, were not necessary to bemeasured for these tests. All of the test specimens in thisresearch, however, used the certified anchor bolt material ofASTM A540, equivalent to ASME SA 540 Grade B23 Class 2supplied for Korean nuclear power plants.

Discussion of test specimensThe load increment of about 5.5 kips (24.5 kN) after

deflection readings around 3.5 to 4 mm (0.14 to 0.16 in.) isapproximately 5% of peak load. The displacements atthe last load increment were measured at every 1 minutefor the first 5 minutes and at every 5 minutes for the next20 or 25 minutes. Test results show the displacementsduring the last load increment are increased almost linearlyfrom the previous load increment for around 20 minutesand increased steeply after 20 minutes; that is, no abruptslope change occurred after deflection readings of around 3.5 to4 mm (0.14 to 0.16 in.) at peak load for around 20 minutes. Thedisplacements at peak load shown in Fig. 7 are those observed at25 or 30 minutes, that is, the displacements increased steeplywithout load increment. From these observations, the authorsjudged that there were no restraining effects of the support reac-tions on the peak loads considered as failure loads in this research;however, the displacements obtained from some of the specimensat peak loads were driven by shear crack propagation to thesupport location. This is an indication that there could be a littlerestraint effect due to support reactions.

Unreinforced testsThe current ACI 318 Appendix D equation is for small-

sized anchors with diameters not exceeding 2 in. (50.8 mm)and embedments depth not exceeding 25 in. (635.0 mm).Therefore, it is beyond the scope of this paper to discuss thecurrent ACI 318 Appendix D equation for small-sizedanchors with the findings based on the anchors with largediameters and deep embedments used in this research.

Anchor reinforcement testsThe authors had presented the comparisons of the test

results for reinforced test specimens and the results ofSeries S6 (unreinforced) tests as illustrated in Table 4 andFig. 11. Based on these test results, the discusserssuggested that the anchor reinforcement may not have tobe in direct contact with embedded anchor. However, theauthors would be careful in making such conclusionsbecause the number of test specimens of S9-2 is verylimited. Nevertheless, the authors think that thediscussers’ feedback on the test results of the anchorsreinforced with various types of supplementary reinforce-ments provide more objective and insightful views on theauthors’ reinforced test results and that their conclusionswill be very useful for practical application as well as forthe authors’ evaluation and conclusions.


Disc. 107-S27/From the May-June 2010 ACI Structural Journal, p. 282

Crack Width Estimation for Concrete Plates. Paper by H. Marzouk, M. Hossin, and A. Hussein

Discussion by Andor WindischACI member, PhD, Karlsfeld, Germany

Assessing different national and international codeformulas for prediction of the crack width estimations forthick concrete slabs, the authors recommend a modifiednumerical model based on the tension chord method.

Regarding Eq. (5) and (6) of the Norwegian Code andconsidering the possible scatters of the different terms inthese both formulas reveals that the ratio 1.7 between themaximum characteristic versus average crack width, wk /wm,must rely mostly on the ratio of the crack spacings. It shouldbe also mentioned that the crack width is not the sum of theslips between reinforcing steel and concrete within one crackspacing, but is the result of the slips of same algebraic signon both sides of this crack (within ls,max ; refer to the CEB-FIP (1990) Code), even if the crack distance is easilymeasured and the other lengths are not. The maximum crackdistance and the maximum slip length are neither interrelatednor similar to each other. Moreover, it can be shown that theaverage values of crack distance and crack width are quiteinsensitive to all influencing factors, such as concretestrength, bond properties, and mode of failures. To estimatethe influence of these factors on crack width, the maximumvalues must be considered.

Similarly, the formula (Eq. (7)) for the average crackspacing according to CSA-S474-04 must be questioned aswell: the maximum crack spacing (or maximum slip length)cannot come to 1.7 × 2.0(C + 0.1S) + 1.7k1k2dbehef b/As , asnone of the incorporated parameters have such a high scatterthat could result in the factor 1.7.

Considering the test specimens, it should be mentionedthat the 25 mm (1 in. or No. 8) bar diameter are definitely toothick relative to the slab thickness (200 mm [8 in.]), in particularin the case of 60 mm (2.4 in.) concrete cover. It is reasonableto limit the bar diameter to hef /10. As mentioned previously,experimental results of the average crack width do not allowfor any conclusion.

The authors report that CEB-90 underestimated themaximum crack width by 75%. It could be possible that theircalculation was erroneous. In terms of Eq. (10), the authorsdefine εsr2 as the steel strain at a crack. Specifically, it is thesteel strain in a crack at cracking load.

Equation (15), related to the modified tension chordmodel, yields the average crack width. Nevertheless, inTable 3, characteristic values are given: how were thesevalues calculated? Figures 14 and 15 refer to maximumcrack widths, whereas Fig. 16 and 17 refer to crack widthonly, although the experimental values shown are identicalwith those in Fig. 14 and 15. Which steel stress fs was chosenas the “end of serviceability limit?”

AUTHORS’ CLOSUREThe authors thank the discusser for his interest in the paper

and his request for clarification on some aspects of theexperimental investigation and the analytical modelreported. The discusser is arguing that the ratio 1.7 betweenthe maximum characteristic versus average crack width,

wk/wm, (Eq. (5) and (6) of the Norwegian Code) must relymostly on the ratio of the crack spacing.

This argument is not true in the case of two-way slabs withbar spacing less than 300 mm (12 in.). First of all, it shouldbe noted that the characteristic crack width, wk, is a designvalue given by codes. Test results of more than 120 two-wayslabs (Hossin and Marzouk [2008]; Nawy [1968, 2001];Nawy and Blair [1971]; Rizk and Marzouk [2010]) showedthat the flexural cracks are generated in two orthogonaldirections as an image of the reinforcement when the spacingof intersections of the bars or wires, termed as the grid nodalpoints, is such that the nodal points’ spacing does not exceedapproximately 300 mm (12 in.). This means that the ratio of1.7 is not a reflection of the scatter of the ratio of the crackspacing on the characteristic crack width. There are threemajor factors that contribute to the crack width in a reinforcedconcrete flexural member: strain release in the concrete in thevicinity of the crack; relative slip between concrete and steel;and distance from the neutral axis.

The discusser is arguing that the maximum crack distancels,max (CEB-FIP 1990 Code) and the maximum slip lengthare neither interrelated nor similar to each other. The CEB-FIP technique considers the mechanism of stress transferbetween the concrete and reinforcement to estimate crackwidth and spacing. Prior to cracking, applied tensile loadcauses equal strains in the concrete and steel. The strainsincrease with increasing load until the strain capacity of theconcrete is reached, at which point cracks develop in theconcrete. At the crack locations, the applied tensile load isresisted entirely by the steel. Adjacent to the cracks, there isslip between the concrete and steel, and this slip is thefundamental factor controlling the crack width. The slipcauses transfer of some of the force in the steel to theconcrete by means of interracial stress (called bond stress)acting on the perimeter of the bar. Therefore, the concretebetween the cracks participates in carrying the tensile force.

In the CEB-FIP approach, the crack width is related to thedistance over which slip occurs and to the differencebetween the steel and concrete strains in the slip zones oneither side of the crack. It is recognized that cracking is aprobabilistic process; therefore, the estimated width is acharacteristic crack width, having a low probability of beingexceeded. According to this bond-slip model, intermediatecracks can occur only when the spacing between cracksexceeds ls,max. Thus, crack spacing will range from ls,maxto 0.5ls,max. The average crack spacing is taken to beapproximately 2/3 of ls,max.

Once again, the discusser is arguing that Eq. (7) for theaverage crack spacing according to CSA-S474-04 must bequestioned as well, and the maximum crack spacing (ormaximum slip length) cannot come to 1.7 × 2.0(C + 0.1S) +1.7k1k2dbehefb/As, as none of the incorporated parametershave such a high scatter that could result in the factor 1.7.The argument is not correct for the current experimental testresults, as all test specimens presented in the experimental


work have bar spacing less than 250 mm (10 in.) and theevidence of the experimental work showed that the flexuralcracks developed in two orthogonal directions as a reflectionof the reinforcement grid. Therefore, to control cracking intwo-way slabs, the major parameter to be considered is thereinforcement spacing in two perpendicular directions.Research work by Frosch (1999) that was implemented laterin the ACI 318-05 Code, abandoned the concept of crackwidth calculations to control cracking and replaced it withbar spacing limitation. Hence, the experimental results of theaverage crack width and, consequently, research conclusions,are valid.

The discusser is arguing that the 25 mm (1 in. or No. 8) bardiameter is definitely too thick relative to the slab thickness(200 mm [8 in.]), in particular in the case of 60 mm (2.4 in.)concrete cover. It is reasonable to limit the bar diameter to h/10.

It should be noted that this slab was designed consideringthe maximum capacity of the test frame limitation. The argu-ment could be considered valid for building slabs and it isreasonable to limit the bar diameter to the empirical value(h/10). The argument may not be valid, however, foroffshore concrete structures applications, which is the mainobjective of the presented research work, where large bardiameters and thick concrete covers are common practice forsuch structures. For example, in the case of the Atlantic Oilplatform (Hibernia Offshore NFLd), the ice wall thicknesswas 400 to1000 mm (16 to 40 in.) and it was reinforced with55 mm (2.16 in.) diameter bar in several layers.

Regarding the statement of the crack width values thatwere estimated using CEB-FIP 1990 Code formula is under-estimated by 75%, this is a typographical error and the referenceshould be related to the Eurocode (EC 2) 2004 only—not tothe CEB-FIP 1990 Code. For the list of references, the fourthreference on p. 290 should read “CEN Eurocode 2 (EC 2)

2004-1-1, 2004, “Design of concrete.....“ and not as “BS EN1992-1-1:2004 Design of concrete..” The correctedreference is given in this closure’s References section.

The crack width values given in Table 3 were calculatedusing Eq. (15) related to the modified tension chord modeland were multiplied by 1.7 to yield the characteristic crackwidth values. The word “maximum” should be added to thetitles of Fig. 16 and 17.

The steel stress fs was chosen according to the ACI 318-08Code limit. ACI 318-08, Section 10.6.4, allows the value offs at service loads to be taken as 0.67fy. This value representsthe ratio between service loads and the average factoredloads on a structure. The assumption of fs = 0.67fy is basedon Gegely and Lutz (1968) formula; fs is the calculated stressin reinforcement at service load, which equals the unfactoredmoment divided by the product of steel area and internalmoment arm.

REFERENCESCEN Eurocode 2 (EC 2) 2004-1-1, 2004, “Design of Concrete Struc-

tures—Part 1-1: General Rules and Rules for Buildings,” Comite Européende Normalisation (CEN), Brussels, 230 pp.

Hossin, M., and Marzouk, H., 2008, “Crack Spacing for Offshore Struc-tures,” Canadian Journal of Civil Engineering, V. 35, No. 12, pp. 1446-1454.

Nawy, E., 1968, “Crack Control in Reinforced Concrete Structures,”ACI JOURNAL, Proceedings V. 65, No. 10, Oct., pp. 825-836.

Nawy, E., 2001, “Design for Crack Control in Reinforced andPrestressed Concrete Beams, Two-Way Slabs and Circular Tanks,” Designand Construction Practices to Mitigate Cracking, SP-204, F. G. Barth and R.Frosch, eds., American Concrete Institute, Farmington Hills, MI, pp. 1-42.

Nawy, E., and Blair, K., 1971, “Further Studies on Flexural CrackControl in Structural Slab Systems,” Symposium on Cracking, Deflectionand Ultimate Load of Concrete Slab Systems, SP-30, R. E. Philleo, ed.,American Concrete Institute, Farmington Hills, MI, pp. 1-32.

Rizk, E., and Marzouk, H., 2010, “A New Formula to Calculate CrackSpacing for Concrete Plates,” ACI Structural Journal, V. 107, No. 1, Jan.-Feb., pp. 43-52.

Disc. 107-S32/From the May-June 2010 ACI Structural Journal, p. 330

Shear Strengths of Prestressed Concrete Beams Part 1: Experiments and Shear Design Equations. Paperby Arghadeep Laskar, Thomas T. C. Hsu, and Y. L. Mo

Discussion by Andor WindischACI member, PhD, Karlsfeld, Germany

The authors report on five full-scale prestressed I-beamsand present a new equation for the shear strength ofprestressed concrete beams. This new equation is a functionof the shear span-depth ratio (a/d), the square root of strengthof concrete ( ), the web area (bwd), and the transversesteel ratio (ρt).

The discusser has the following question: How does theprestressing influence the shear strength? The authors referto the results of Lyngberg (1976) where the prestressingforce does not have significant effect on the shear capacity.This statement appears erroneous. The test results ofLyngberg show a very clear and pronounced influence of theeffective prestressing force (N

∞) on the concrete contribution

Vc. The concrete contribution Vc increased linearly by0.12N

∞. This ratio corresponds to the results of other

researchers; thus, the design expressions proposed by theauthors should be corrected accordingly.

Beams B1 to B3 were in the shear valley of Kani (1964),which means that the ultimate shear load increases quitedramatically as the shear span-depth ratio decreases, beginningat a/d ≈ 2.5, which was the case here. Nevertheless, in thecase of Beams 4 and 5 (a/d > ~ 3.5), the span-depth ratio maynot influence the contribution of concrete in shear, Vc, anymore (refer to Fig. 9). Another point may be made that, in thecase of small shear span-depth ratios, the longitudinal size ofthe support, and loading plates, respectively, influence theinclination of the kinematically possible failure surface;hence, a/d is not the prominently effective geometricalinfluencing factor.

The authors refer to Laskar et al. (2006) for the argumentthat the shear strength would not be significantly affected by

fc′


the angle of failure plane. This statement is correct in thecase of shear span-depth ratios greater than ~3.5 only. Infact, the a/d ratio introduced by the authors refers to theangle of failure plane.

The concept of Loov (2002), taking into account therational number of stirrups crossing the failure crack insteadof calculating with smeared stirrups is, in principle, correctand progressive. Nevertheless, in the case of beams withsmall shear span-depth ratios (< 2.5), Eq. (3) and Fig. 8should be corrected. In those cases, the inclination of thefailure plane is less than 45 degrees.

AUTHORS’ CLOSURETable 2, taken from Lyngberg (1976), shows how the level

of prestress influences the ultimate shear strength. It can beseen that concrete strength has more effect than the prestressforce on the ultimate shear strength. Lyngberg stated in hisConclusion 2, “The effect of prestress does not seem tocontribute significantly to the ultimate shear resistance.”This conclusion appears to be correct when the effectiveprestress force varies from 60% to 40% of the tensile strengthof the flexural reinforcement. Sixty percent is typically “fullprestress,” and 40% is about 1/3 replacement of prestressedstrands by mild steel. However, when the effective prestressforce is less than 40% of the tensile strength of the flexuralreinforcement and is approaching zero, the ultimate shearforces could decrease significantly.

The physical meaning is clear: When the web is uncrackedunder large effective prestress, the entire web area bwd is effectivein resisting the ultimate shear. When the web is cracked undersmall effective prestress, however, only the uncrackedcompressive web area is effective in resisting ultimate shear.A recent study shows that in the case of reinforced concretebeams with zero prestress, c is the depth of the compression

zone and is expressed by c/d = . ρw

is the longitudinal steel ratio, and n is the modulus ratio Es/Ec.

2ρwn ρwn( )2+ ρwn–( )

Figure 9 shows that the a/d ratio still affect Vc when it is inthe range greater than 3.5. Specimens by Hernandez (1958),Mattock and Kaar (1961), and Elzanaty et al. (1986) show adecrease in the contribution of concrete in shear Vc when thea/d ratio is greater than 3.5.

Specimens with a/d ratios less than 3.5 were also observedto fail along a 45-degree inclined plane, as shown in Fig. 13.Specimens with a/d ratios greater than 3.5 were also observedto fail along a 45-degree inclined plane, as shown in Fig. 7 ofthe paper. It can be concluded that the shear strength is notsignificantly affected by the angle of failure plane.

Table 2—Shear failures in Lyngberg’s prestressed specimens

SpecimenConcrete

strength, MPaStirrup

strength, kN Prestress, kN Vu,test , kN

2A-3 32.6 62.2 631 506

2B-3 33.9 64.9 629 515

3A-2 31.1 67.0 421 489

3B-2 27.5 63.1 421 433

4A-1 31.5 64.5 217 469

4B-1 30.4 66.5 209 454

Fig. 13—Forty-five-degree crack patterns of Beams B1 andB2 showing web-shear failure.

shear behavior of headed anchors with large diameters and deep embedments - discussion title 107-s14...

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