design and behavior of dapped-end beams - pci journal... · 2018-11-01 · the dapped end as if it...

Design and Behavior ofDapped-End Beams

Alan H. MattockProfessor of Civil EngineeringUniversity of WashingtonSeattle, Washington

Timothy C. ChanFormer Graduate StudentUniversity of Washington

Seattle, Washington

The dapped-end beam is a use-ful concept. It enables the con-

struction depth of a precast con-crete floor or roof structure to bereduced, by recessing the sup-porting corbels into the depth ofthe beams supported.

In a "cantilever and suspendedspan" type of structure, the sus-pended span is a dapped-endbeam, and the ends of the sup-porting cantilevers are similar tothe ends of the dapped-endbeams, but inverted.

The use of dapped-end beamsfacilitates the erection of a precastconcrete structure, due to thegreater lateral stability of an iso-lated dapped-end beam than thatof an isolated beam supported atits bottom face.

Despite the fairly extensive usemade of this form of construction,

few studies 1' 2 appear to have beenmade of its behavior. These wereprimarily analytical, utilizing thefinite element method to analyzethe stresses in and around thedapped end under service loadconditions.

From these analyses, Wernerand Dilger2 were able to predictthe shear at which diagonal ten-sion cracking would occur at there-entrant corner. They also pro-posed that the shear strength ofthe dapped end could be calcu-lated using:

Vn=Ve+Vp +V8

where

NOTE: This paper, which is based on astudy supported by a PCI ResearchFellowship, has been reviewed by the PCITechnical Activities Committee and ishereby endorsed for publication in the PCIJOURNAL.

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V, = shear at diagonal tensioncracking

V p = vertical component of pre-stress force for tendon an-chored in the dapped end

V, = shear force in web reinforce-ment near end face of beam

The present study was under-taken to provide an improved un-derstanding of the behavior ofdapped-end beams both at serviceload and at ultimate, with a viewto developing a rational designprocedure.

Preliminary ConsiderationsIn many respects the nib (or re-

duced depth part) of a dapped end re-sembles an inverted corbel. However,in the case of the corbel, the inclinedconcrete compression force in thecorbel is resisted by a compressionforce in the column (Fig. 1a); but inthe case of the dapped end, the in-clined compression force in the nibmust be resisted by a tension force inthe stirrup reinforcement placed closeto the full-depth end face of the beam(Fig. 1b). For equilibrium of the nib,it appears that this stirrup reinforce-

ment must provide a tension forceequal to the shear force acting on thenib.

The strength of the full-depth partof the dapped end will be adverselyaffected by the formation of diagonaltension cracks. The principal diagonaltension cracks will be those originat-ing at the re-entrant corner A and atthe bottom corner B of the full-depthbeam (see Fig. 2). Sufficient rein-forcement must cross these cracks toprevent failure.

In the light of these considerations,the design of the test specimens wasbased on the following initial pro-posals:

1. Design the reduced depth part ofthe dapped end as if it were a corbel,using the design proposals of Mat-tock.3

2. Provide a group of closed stir-rups close to the end face of the fulldepth beam, to resist the verticalcomponent of the inclined compres-sion force in the nib; that is, the yieldstrength of this stirrup group A„hfv , tobe not less than V/4.

3.3. Design the full depth part of thebeam so as to satisfy moment andforce equilibrium requirements acrossinclined cracks AY and BZ shown in

PCI JOURNAL/November-December 1979 29

Balancing compressionforce in column

^^— Inclined compressionforces

Inclinedcompressionforce

— — ^—Anfy

--t

J1LtLL by

_ J^__ A fN __^ I I g y

VvU

stirrup tensionBalancing

force Aynfy

(a) (b)

Fig. 1. Comparison of internal force systems: (a) in corbel on a column and (b) in adapped-end beam.

Interface between nib Potential diagonal

and full-depth beam tension cracks

"Nib' ( „^--1 Y Z

An / /

h(l2")d //

1Asa b A

'c H(24°)

N /4 \/ /

4 // rA y(4.5°)

Hanger /reinforcement Avh (closed

stirrups) BBeam flexural Beam shearreinforcement reinforcement

Fig. 2. Typical dapped-end reinforcement and location of potential diagonal tensioncracks. Note that the dimensions in parentheses are those of the test specimens.

30

X ^—

h d A --}Ahf)

AsfyNUNu

Fr Vu iAvnfy 0

(a) ,.vh'y

Fig. 3. Forces assumed acting on free bodies cut off by diagonal tension cracks infull-depth beam.

Fig. 2 in addition to carrying out theusual design of sections normal to thelongitudinal axis of the beam for flex-ure and shear.

The horizontal stirrup reinforce-ment A h was provided in the form ofclosed ties which extended 1.5 l d [18in. (or 457 mm)] beyond the nib-beaminterface, as recommended in the PCIDesign Handbook.' The main nibreinforcement A S was extended a dis-tance 1.4 l d beyond the point at whichthe potential crack BZ crosses it, inorder to develop its yield strength atthat location.

When checking moment equilib-rium about Point Y of that part of thedapped end to the left of AY in Fig. 2,it was assumed that A 3 , A h and A 3,, alldeveloped their yield strength, as in-dicated in Fig. 3a.

When checking moment equilib-rium about Point Z of that part of thedapped end to the left of BZ in Fig. 2,it was assumed that A 3 , A V ,,, and thebeam stirrup A„ all developed theiryield strength, as indicated in Fig. 3b.In both cases, the depth of the com-pression zone and the location of theresultant concrete compression forceC were calculated using the equiva-lent rectangular stress distribution ofthe ACI Building Code.5

Experimental Study

The test program was conducted atthe structural engineering laboratoriesof the University of Washington.

Test Specimens

Eight dapped ends were tested,four being subjected to vertical loadonly, and four to a combination ofvertical and horizontal loads. Thedapped ends were formed on oppositeends of 5 x 24-in. (127 x 610 mm) crosssection beams, 10 ft (3.05 m) long. Thenibs all had a length of 8 in. (203 mm)and an overall depth of 12 in. (305mm), i.e., one-half the overall depth ofthe beam.

Typical reinforcement details areshown in Fig. 2. The sizes andamounts of reinforcement in eachspecimen are listed in Table 1, to-gether with the concrete strength atthe time of testing. A 3/4-in. (19 mm)cover was provided to the stirrups andthe main dapped-end reinforcement.

The design of successive specimensdiffered. The design criteria werechanged after study of the behavior ofpreceding specimens. The design ofthe specimens will therefore be dis-cussed along with their behavior.


X

Dapped-End under t1st

Testing machinehead extension

Load cell 7Testbeam

Hydraulic ram

-Ef_- G--- _

Roller— Strut

Horizontal force

"Idlerpin

Pin throughbeam

-96"------Elevation

l8s°, IA-363O", 4A & 4B Plan

Fig. 4. Testing arrangements for dapped-end beam.

Materials and FabricationThe concrete was made from Type

III portland cement, sand and 3/a-in.(19 mm) maximum size gravel, in theproportions 1.0:2.77:3.45 by weight.The freshly cast concrete was moistcured for 2 days, then cured in the airof the laboratory until test at about 5days.

The deformed reinforcing bars ofsize #3 and larger conformed toASTM Specification A 615. The #2bars used for horizontal stirrup rein-forcement had deformations similar tothose on the larger bars conforming toASTM Specification A 615.

The dapped-end bearing plateswere welded to the main reinforce-ment A,, so as to be able to transmitthe horizontal force N directly to thatreinforcement.

Testing Arrangementsand Instrumentation

The dapped ends were tested inde-pendently by supporting the 10-ft(3.05 m) long beam through thedapped end at one end of the beam,and under the beam bottom face at theopposite end, using an 8-ft (2.4 m)span between centers of support.

Typical arrangements for test areshown in Fig. 4. A shot of the testsetup in which both horizontal andvertical loads are acting on a dapped-end beam is shown in Fig. 5.

After test of one dapped end, dam-age was mostly confined to the regionof that dapped end. It was thereforepossible to turn the beam end-for-end,and test the other dapped end.

Specimens 1A, 1B, 2A, 2B, 3A, and3B were loaded so that the outer edge

32

Table 1. Specimen reinforcement details and concrete strength.

Sped-men

Main dapped-endreinforcement

Horizontalstirrups

Hangerreinforcement

Concretestrength

No. Bars A f,, Stir- A„ f,. Stir- A,,h(in. 2 ) (ksi) rups (in. 2) (ksi) rups (in.2) (ksi) (psi)

1A 2#3 0.22 69.1 1#2 0.10 67.0 3#3 0.66 65.5 48751 B 2#6 0.88 59.8 2#2 0.20 66.0 3#3 0.66 67.7 44252A 3#3 0.33 69.4 2#2 0.20 67.0 2#3 0.44 67.1 47852B 2#6 0.88 59.8 2#2 0.20 66.8 2#3 0.44 68.2 44753A 3#3 0.33 69.1 2#2 0.20 65.0 2#3 0.44 68.2 5370

+1#2 +0.10 65.03B 2#6 0.88 63.6 2#2 0.20 70.2 2#3 0.44 70.9 4590

+1#2 +0.10 70.24A 3#3 0.33 69.1 2#2 0.20 63.2 2#3 0.44 69.1 4590

+1#2 +0.10 63.24B 2#6 0.88 63.6 2#2 0.20 67.0 2#3 0.44 71.3 4260

+1#2 +0.10 67.0

*Measured on 6 x 12-in. cylinders.Note: 1 in. = 25.4 mm; 1 in.2 = 645.16 mm2, 1 psi = 6.895 kPa; 1 ksi = 6.895 MPa.

of the 4 x 5-in. (102 x 127 mm) loadingplate was at Point Y in Fig. 2. Thisload location was chosen so that, forthese specimens, the behavior shouldbe controlled by the behavior of thenib and/or by the behavior of thereinforcement crossing the diagonaltension crack originating at the re-en-trant corner A in Fig. 2.

In the case of Specimens 4A and 4B,the outer edge of the loading platewas at Point Z in Fig. 2, so that theinfluence on behavior of the diagonaltension crack originating at the bottomcorner (Point B in Fig. 2) could bestudied.

In tests with vertical load only, the4 x 5-in. (102 x 127 mm) dapped-endbearing plate was supported on a freeroller.

In tests with combined vertical andhorizontal loads (Fig. 5), the face ofthe dapped-end bearing plate had a I/2

x 2-in. (13 x 51 mm) transverse groove.This plate was supported by anotherplate with stub axles on each end, anda 1/2 x 2-in. (13 x 51 mm) transverseupstand which engaged with thegroove in the bearing plate.

The second plate was supported ona free roller, and equal horizontalforces were applied to each stub axleby hydraulic rams acting throughtransducer links as shown in Fig. 4. Inthis way it was possible to apply to thedapped end, independently controlledvertical and horizontal forces. The hy-draulic rams reacted against a 2-in. (51mm) diameter steel pin which passed

Fig. 5. Test setup in which both hori-zontal and vertical loads are acting ondapped-end beam.


Table 2. Test results of dapped-end beams.

Maxi- Dis- Mea-mum tance of suredcrack flexural force inwidth at failure hangerservice plane rein-loads from force-

Sped- N„ V„(calc) V„(test) V(test) V„(test) V(test) (in.) Corner ment atV„(calc) V„(calc)men (kips) (kips) (kips) (kips) A (in.) V,(test)

No. (kips)

1A 0 36.12 32.40 23.49 0.90 0.65 0.017 2.42 26.71 B 30 44.13 42.93 29.97 0.97 0.68 0.011 2.13 20.62A 0 36.98 40.10 31.25 1.08 0.85 0.019 3.19 27.92B 25 38.84 38.10 31.00 0.98 0.80 0.018 3.64 30.0*3A 0 37.04 48.52 35.50 1.31 0.96 0.015 2.28 35.03B 28 38.91 39.70 36.50 1.02 0.94 0.016 2.43 38.2*4A 0 36.74 42.43 32.00 1.15 0.87 0.009 2.96 30.24B 28 38.67 39.78 34.30 1.03 0.89 0.006 2.83 34.8

`Yielded.Note: 1 in. = 25.4 mm; 1 kip = 4.448 kN.

through the center of the beam, whichwas locally reinforced.

The strain in the main dapped endreinforcement A., was measured at lo-cations a and c in Fig. 2, and that inthe hanger reinforcement A „n at Pointb, using electrical resistance gages.

The load cell for vertical load, thetransducer links for horizontal load,and the strain gages, were all moni-tored continuously during the testusing a Sanborn strip chart recorder.

Test ProcedureIt was considered desirable to ob-

tain a measure of serviceability aswell as strength, by measuring crackwidths at service load. It was arbitrar-ily decided to consider that the deadload shear V D and the live load shearVL were equal. Then with (h = 0.85,we have V D = V L = 0.275V,, in thelimit, since the ACI Codes requiresthat

corbel be treated as a live load, theservice horizontal force N = N u/1.7,i.e., N = 4N../1.7 = 0.5N,,.

The following loading sequencewas therefore used:

1. N = 0, V increased incrementallyto V D = 0.275V,,.

2. V = V, N increased incremen-tally to N = 0.5N,,.

3. N = 0.5N, V - increased incre-mentally to (V D + V L ) = 0.55V,,.

4. V = (V D + V L ), N increased in-crementally to N,,.

5. N = N n, , V increased incremen-tally until failure occurred.

The maximum crack widths atV = (V D + V L ) and N = 0.5N,, wererecorded, this being regarded as theservice load. In all tests, the crackswere marked at each increment ofIoading.

Specimen Design and Behavior

4V,, ; 1.4V D + 1.7V L. All Specimens-The design of allthe specimens followed the initial

Also, since the ACI Code requires proposals, with modifications as de-that the horizontal force acting on a scribed below. In the design of the

34

(a) (b)Fig. 6. Typical cracking patterns of dapped-end beams tested: (a) at service loadand (b) just before failure.

"nib" as a corbel, the "Modified ShearFriction" method 3 was used for sheardesign rather than the "Shear Fric-tion" method of the ACI Code, so asto be the least conservative.

The initial "target" design strengthsfor the specimens were V,, = 40 kips(178 kN) and N. = 30 kips (133 kN).However, the strengths and sizes ofreinforcing bars available for use inthe tests were not exactly suitable.The design strengths were thereforevaried as necessary in order to matchthe available reinforcing bars.

The design strengths V,, and N. foreach specimen, and the strengths ob-tained in the tests, are summarized inTable 2. In this table, V,,(test) is theshear strength measured in the test(i.e., the applied shear at failure), V,, isthe shear acting when yield of thedapped-end main reinforcement A 3

occurred. The design of the dapped-end main reinforcement was alwayscontrolled by flexure, so in all thespecimens V n (calc) is the calculatedshear corresponding to a flexural fail-ure of the dapped end.

Specimens 1A and 1B—In the de-sign of these specimens, the shearspan "a" used in the design of the"nib" as an inverted corbel, was takento be the distance from the center ofsupport to the interface between thenib and the full-depth beam, i.e.,a = 1,,. This was as indicated in thePCI Design Handbook. 3 The amountof hanger reinforcement A „ h was cho-sen to make A„,,f,, = V. as closely aspossible.

The behavior of Specimens 1A and1B was unsatisfactory, both at serviceload and at ultimate.

The general process of cracking wassimilar for all specimens. The firstcrack initiated at the re-entrant cornerA at about 20 percent of ultimate. Thiscrack propagated at approximately 45deg to the horizontal, and at serviceload extended to about two-thirds theheight of the nib.

At this time additional diagonal ten-sion cracks occurred in the nib and inthe full depth beam, as shown in Fig.6a. The original crack had the greatestwidth at service load and this oc-


curred close to the re-entrant cornerA.

As the load was further increased,additional cracks formed and existingcracks lengthened. The diagonal ten-sion cracks in the nib assumed a flat-ter trajectory on reaching the hangerreinforcement, propagating toward theloading plate. A typical cracking pat-tern just prior to failure is shown inFig. 6b.

In all cases, the dapped-end mainreinforcement yielded beforemaximum load was reached. At thattime, the cracks crossing this rein-forcement widened markedly. Shortlybefore failure, compression spallingand bulging of the top face of thebeam occurred adjacent to the loadingplate. At failure, inclined crushing ofthe concrete occurred in this same re-gion. In some cases this crushingspread downwards and backwardsinto the nib as failure progressed.

In the case of Specimens 1A and 1B,the main dapped-end reinforcementyielded at about 70 percent of V n(test)and about 66 percent of V „(calc). Thisresulted in undesirable permanentlywide cracks occurring at about 20 per-cent above service load.

Since the flexural strength of an un-der-reinforced section can be calcu-lated with good accuracy, it was de-duced that the interface between thenib and the full-depth beam was notthe location of flexural failure, as as-sumed in the design calculations. As-suming that the flexural strength M.could be calculated satisfactorily, thedistance "a" from the load V to thevertical plane in which flexural failurewas apparently occurring, was calcu-lated using:

M„–N(h–d)a=

V5(test)

The values of "a" so calculated forSpecimens 1A and 1B were 6.92 and6.63 in. (176 and 168 mm), respec-tively. This corresponds to the flexural

failure plane being approximately atthe center of gravity of the hangerreinforcement. This is in agreementwith the concept of a truss-like forcesystem developing in the dapped end,as shown in Fig. 1b.

It was therefore decided that, for allsucceeding specimens, flexural designof the dapped end would be based ona shear span "a" measured from thecenter of -support to the center ofgravity of the hanger reinforcement.

Specimens 2A and 2B —Thesespecimens were tested to determinewhether the amount of hanger rein-forcement could be reduced. Thehanger reinforcement was designed tohave a yield strength A„hf,, equal to(V n – V a ), where V, is the shear thatcould be carried by the concrete in abeam having the same cross section asthe nib, according to ACI 318-77. (Inthis case, V,, was taken to be 2 fbd.)

In designing the main dapped-endreinforcement for flexure, it was as-sumed that the center of gravity of thehanger reinforcement would be 2 in.(51 mm) from the interface betweenthe nib and the full-depth beam, i.e.,"a" = 6.5 in. (165 mm).

The behavior of Specimens 2A and2B was unsatisfactory both at serviceload and at ultimate load. At serviceload, the cracks were excessively wideand the main dapped-end reinforce-ment yielded at about 80 percent ofthe ultimate shear. At yield of themain dapped-end reinforcement inSpecimen 2B, the hanger reinforce-ment had yielded, and in Specimen2A was very close to yield. It wastherefore decided to check whether anincrease in the hanger reinforcementwould improve behavior.

Specimens 3A and 3B —In thesespecimens, the hanger reinforcementwas designed to have a yield strengthA,hf,, equal to V. In designing themain dapped-end reinforcement forflexure, it was again assumed that the

36

center of gravity of the hanger rein-forcement would be 2 in. (51 mm)from the interface between the niband the full-depth beam, i.e., "a"= 6.5 in. (165 mm).

Specimens 3A and 3B behaved in amore satisfactory manner, the maindapped-end reinforcement not yield-ing until about 95 percent of V(calc.)Also, the maximum crack widths atservice load were reduced.

Specimens 4A and 4B—The previ-ous specimens were tested with theedge of the loading plate-at.Point Y inFig. 2. This was done to examine be-havior associated with failure modesinvolving the nib itself and the prin-cipal diagonal tension crack originat-ing at the re-entrant corner "A".

Specimens 4A and 4B were de-signed in the same way as Specimens3A and 3B, but were tested with theedge of the loading plate at Point Z inFig. 2. This was done to check the in-fluence on strength and behavior of adiagonal tension crack propagatingfreely from the bottom corner of thebeam (Point B in Fig. 2).

The behavior of Specimens 4A and4B was considered satisfactory. Al-though the main flexural reinforce-ment yielded at a slightly lower frac-tion of V„(calc.) than in the case ofSpecimens 3A and 3B, the maximumcrack widths at service load were lessthan in Specimens 3A and 3B.

It should be noted that, in Speci-mens 4A and 4B, the main dapped-end reinforcement yielded beforefailure at Point "c" in Fig. 2, i.e.,where the bar intercepted the diag-onal tension crack originating at thebottom corner of the beam. This wasalso the widest crack at high loads.

It is clearly necessary to extend themain dapped-end reinforcement suffi-ciently far into the beam so that it candevelop its yield strength at the pointwhere it is intersected by a line run-ning upwards at 45 deg from the bot-tom corner of the beam.

General CommentsThe measured ultimate shear,

V(test), was greater than V„(calc.) forall of Specimens 3A, 3B, 4A, and 4B.Also, the shear at yield of the maindapped-end reinforcement was satis-factorily high for these same speci-mens.

The crack control provisions of ACI318-77 are equivalent to allowingmaximum crack widths of 0.013 and0.016 in. (0.33 and 0.41 mm) for ex-terior and interior exposures, respec-tively. The maximum crack widths forSpecimens 4A and 4B are well belowthese allowable values, and the load-ing used for these specimens is prob-ably more representative of actualpractice than that used for the othertests.

If the crack widths had been mea-sured at a service load related to thenominal yield strength of 60 ksi (414MPa) (as is the case in practical de-sign), rather than to the actual yieldstrengths of up to 70 ksi (483 MPa), itis probable that the maximum crackwidths in Specimens 3A and 3Bwould not have exceeded the allowa-ble width for exterior exposure.

Conclusions from TestsThe following conclusions may be

derived from the test program:1. The reduced depth part of the

dapped end may be designed as if itwere a corbel, using the design pro-posals of Mattock, 3 providing theshear span "a" used in design is takenequal to the distance from the centerof action of the vertical load to thecenter of gravity of the hanger rein-forcement A „ h, . (Note that for the cor-bel design proposals to be used, aidmust be < 1.0.)

2. A group of closed stirrups A h,having a yield strength A,, f,, not lessthan V u/O, should be provided close tothe end face of the full-depth beam toresist the vertical component of the

PCI JOURNAUNovember-December 1979 37

inclined compression force in the nib.This reinforcement must be positivelyanchored at both top and bottom bywrapping around longitudinal rein-forcing bars.

3. The full-depth part of the beamshould be designed so as to satisfymoment and force equilibrium re-quirements across inclined cracks AYand BZ in Fig. 2, in addition to carry-ing out the usual design of sectionsnormal to the longitudinal axis of thebeam for flexure, shear, and axialforce.

4. The main nib reinforcement A$should be provided with a positiveanchorage as close to the end face ofthe beam as possible. These barsshould also extend into the beam adistance (H – d + l d ) beyond the re-entrant corner, so that they can de-velop their yield strength where inter-sected by a diagonal tension crackoriginating at the bottom corner of thebeam. [Note that if these bars A $ are12 in. (305 mm) or more above thebottom of the beam they are "topbars" according to the ACI Code s andl d must be increased accordingly.]

5. The horizontal stirrups A h shouldbe positively anchored near the endface of the beam by wrapping aroundvertical bars in each corner (as shownin Figs. 2 and 6). They should projectbeyond the re-entrant corner a dis-tance 1.7 l d as recommended in thePCI Design Handbook.6

1978 PCI DesignRecommendations

The design provisions for dapped-end beams contained in the PCI De-sign Handbook were revised in the1978 editions With reference to Fig.2, they now require the followingsteps in design:

1. Design of the nib for flexure andaxial force. The critical section forflexure being taken to be the interface

between the nib and the full depthbeam, i.e., a = 1.

2. Design of the interface betweenthe nib and the full-depth beam for di-rect shear. (One-third of the shear-friction reinforcement A „f to be pro-vided as horizontal stirrups A h . Theremaining shear-friction reinforce-ment to be combined with the rein-forcement for the direct force N„ togive A,, if % A „f is greater than thereinforcement required for flexure A f.)

3. Design of the hanger reinforce-ment A„h to carry the total shear V.across the diagonal tension crackoriginating at the re-entrant corner A.

4. Design of the nib for diagonaltension, providing vertical stirrups(total area A „) in the nib in addition tothe horizontal stirrups A,,, so as tosatisfy the following equation:

V u = 4 V n = çb(Avfb+AJ.

+ 2Abd f^)where

A = 1.0 for normal weight concreteA = 0.85 for sanded-lightweight

concreteA = 0.75 for all-lightweight con-

creted = effective depth ofA, in the nib

It is also specified that A„ is to benot less than one-half the total rein-forcement required according to thisequation.

5. Design of bearing for V,, actingon the nib.

The PCI design provisions specifythat both the main dapped-end rein-forcement A S and the horizontal stir-rups A,, should extend a distance 1.7 1dinto the beam beyond the interfacebetween the nib and the full-depthbeam. The main reinforcement is alsoto be positively anchored at the end ofthe beam by welding to a cross-bar,angle or plate.

These revised provisions are a dis-tinct improvement on those contained

38

in the 1971 PCI Design Handbook;i.e., they recognize that the rein-forcement A „h close to the re-entrantcorner A must be designed to carrythe total shear V,. The correspondingreinforcement in the 1971 PCI DesignHandbook was, in effect, only de-signed to carry a force V „/µ 2 in thecase of vertical load only, i.e., 0.5V n

for µ = 1.4.However, in view of the behavior of

Specimens 1-A and 1B which were de-signed for flexure as specified in the1978 PCI provisions, it appears thatflexural design of the nib according tothe 1978 PCI Design Handbook is in-adequate.

The behavior of Specimens 3A, 3B,4A, and 4B indicates that the criticalsection for flexure should be taken atthe center of gravity of the hangerreinforcement; i.e., "a" should be thedistance from the center of support tothe center of gravity of the hangerreinforcement.

The 1978 PCI Design Handbook in-dicates that the main dapped-endreinforcement A 8 should extend adistance 1.7 1d into the beam beyondthe re-entrant corner. Dependingupon the proportions of the beam andthe size of reinforcing bar used for A9,this distance may be too short..

These bars should extend a distancenot less than their developmentlength beyond Point "c" in Fig. 2.This is necessary so that the yieldstrength of these bars can be de-veloped where the diagonal tensioncrack originating at the bottom cornerof the beam crosses them. This is re-quired in order to satisfy momentequilibrium about the top of thiscrack (i.e., Point A in Fig. 2).

It should also be noted that in manycases these bars will be more than 12in. (305 mm) from the bottom of thebeam, and that they are consequently"top bars" when calculating their re-quired development length accordingto ACI 318-77.5

The 1978 PCI design provisions re-quire the provision of both horizontaland vertical stirrups in the nib, andalso limit the shear stress in the nib(V9 l4bd) to 8 f, . The tests reportedhere indicate that, if ald is _- 1.0, hori-zontal stirrups alone are satisfactory,providing their total cross section ismade not less than the greater of A,/2and A 9f/3, as recommended for corbeldesign by Mattock.3

Using this amount of horizontal stir-rups in the nib, the average ultimateshear stress for Specimens 3A, 3B, 4A,and 4B was 11.4 f^. The limitationof 8 Z, appears to be conservative ifthe stirrup reinforcement recom-mended in this paper is used.

If aid is greater than 1.0 then bothvertical and horizontal stirrups shouldbe provided in the nib. It is recom-mended that they be designed usingthe procedures proposed for deepbeam design by ACI Committee426. 8,9 Using the notation of thispaper, these procedures may be ex-pressed as follows for the case of anib:

V n =V,Ic V, +V8 +Vh

where

V 8 = shear carried by vertical stir-rups

V 8 =A 9f9 (1 – 0.5d1a),but A„f,, d/a

A„ = total area of vertical stirrups innib (not less than 50 balfv)with spacing of vertical stir-rups not more than d/4.

V h = shear carried by horizontalstirrups

V h =AJf,(1.5 – aid), but Ahf9V 8 +V h f 8bd f^V = shear carried by concrete

(3.5d/a)bd X f,'V,, +V 3 +V hf, 0.2bdf,

nor 800 bd (lbs)

The expression for V, is an ap-proximation to Eq. (19) of Reference8, for purpose of simplification.


Fig. 5.9.1 of the 1978 PCI DesignHandbook does not indicate that boththe vertical and horizontal stirrupsshould be positively anchored at theirouter ends by wrapping around rein-forcing bars running at right angles tothe stirrups. This positive anchorageis essential if the stirrups are to de-velop their yield strength. If verticalstirrups are used in the nib, they mustwrap around the main nib reinforce-ment A 3 or be welded to the bearingplate if they are to be effective.

No bearing plate is shown in Fig.5.9.1, although one is shown in Fig.5.9.2 illustrating the design example.If a horizontal force N n acts on thedapped end, a bearing plate should beprovided. It should either be weldeddirectly to the main nib reinforce-ment, so as to transfer N,, directly intoA 3, or it may be attached to the nib byheaded stud shear connectors.

However, in this latter case thestuds must lie inside the positive an-chorage of the main nib reinforce-ment. It is also recommended that abearing plate or armouring angle bewelded to the main nib reinforcement,in all cases, if the bearing area extendsbeyond the positive anchorage of themain nib reinforcement.

Proposed DesignRecommendations

On the basis of the test results re-ported here and the foregoing discus-sion, it is proposed that dapped-endbeams having aid ratios -_1.0* be de-signed as follows:

1. Check that shear stress in nibdue to factored shear V, is not morethan 0.2f, i.e., V nl(/db) 0.2f.(This limit corresponds to themaximum ultimate shear stress de-

"See Fig. 2 and Appendix A for definitions ofsymbols.

veloped in these tests. Future testsmay indicate a higher ultimate shearstress is possible).

2. Design the hanger reinforcementA „h to carry the total shear V,, due tofactored loads using:

_ V.Avh Of.

Provide this reinforcement in theform of closed stirrups and place asclose to the re-entrant corner as possi-ble.

3. Calculate the design momentdue to factored loads using:

Mn=Vna+N,,(h–d)

where a is the distance from thecenter of action of V n to the center ofgravity of the hanger reinforcement

Calculate the required flexuralreinforcement area A f so that

n4iMM.4. Calculate reinforcement area A

to resist horizontal force N n due tofactored loads using:

A n = Nulçbfv

5. Calculate reinforcement area A „fto transfer shear across interface be-tween nib and full depth beam. Use"Shear Friction" provisions of ACI318-77; Section 5.6, Shear Friction, ofthe 1978 PCI Design Handbook; orthe following "Modified Shear Fric-tion" 3 equation:

A nt = {Vn/(0.84') – Kbd]If,but not less than 0.2 bd fu

where K= 0.5 for normal weightconcrete

orK = 0.25 for all-lightweightconcrete

orK = 0.31 for sanded light-weight concrete

In the above equation, V n is in kips,b and d are in inches and f, is inkips/in.2.

40

6. Check whether 2/3A 31 is greaterthan or less thanA f.

If 213A „f is greater than A f:

A s = 2/sAvf+An

If %A „f is less than A f:

A3=Af+An

7. Provide a positive anchorage forA 3 at the outer end. These bars mustalso extend into the beam a distance(H – d + i d) beyond the re-entrantcorner. (This is to ensure that thesebars can develop their yield strength,where intersected by a diagonal ten-sion crack originating at the bottomcorner of the beam.)

8. Provide horizontal stirrups in thelower two-thirds of the depth of thenib, having total area:

A h = 0.5(A3–A..)

(This provision automatically providesthe greater ofA„f13 and A 1/2.)

These stirrups should be closed attheir outer end but may be closed oropen at the other end. They shouldproject beyond the interface betweenthe nib and the full-depth beam adistance 1. 7 l d.

9. The hanger reinforcement mustbe positively anchored at both top andbottom. At the bottom it should passaround longitudinal reinforcing barshaving a total area not less than that ofthe hanger reinforcement. These lon-gitudinal bars should be positivelyanchored at their outer ends. Theywould normally be a continuation ofthe beam flexural reinforcement in areinforced concrete beam. In a pre-stressed concrete beam they shouldextend into the beam the greater of 1.7l d (for the bars) and the developmentlength of the strand.

10. Use strength reduction factor(A = 0.85 in all reinforcement designcalculations.

11. Design for bearing as per 1978PCI Design Handbook. The center of

action of the vertical force V,. shouldlie inside the positive anchorage ofthe main nib reinforcement. If themain nib reinforcement takes the formof a hairpin type bar, then in accor-dance with the recommendations ofACI Committee 426 8 for corbels, thebearing area should not project be-yond the straight portion of the bars.

A bearing plate should be providedif a horizontal force Nu acts on the nib.It should be welded directly to themain nib reinforcement or may be at-tached to the nib by headed studshear connector.

In either case, the detail should bedesigned to transfer the force N u intothe nib. (The studs should lie insidethe positive anchorage of the main nibreinforcement.)

A bearing plate or armouring angleshould be welded to the main nibreinforcement in all cases, if thebearing area extends beyond thepositive anchorage of the main nibreinforcement.

Additional CommentsAlthough it is necessary to satisfy

equilibrium for those parts of thedapped end cut off by the two diag-onal tension cracks AY and BZ in Fig.2, it is not necessary to make anexplicit check if the reinforcement isdesigned as recommended above. Inthis case, sufficient reinforcement tosatisfy these equilibrium require-ments is automatically provided, asshown in Appendix B.

It is considered appropriate to use avalue of 0.85 for the strength reduc-tion factor 0 in all calculations be-cause of the uncertainties regarding:

(a) Exact location of center of ac-tion of V.

(b) Exact value of horizontal forceN u, and

(c) Exact location of critical sectionfor flexure within the end of thebeam.


Design Example

Given: A 16RB28 beam with dappedend as shown in Fig. 7. Assume thefollowing data are known:

V. = 100 kipsN= 15 kips

= 5000 psi (normal weight con-crete)

= 60 ksi (all reinforcement)

Required: Design the reinforcementfor the 16RB28 beam with dapped end(Fig. 7).

Solution:1 Shear Sfr, c

Assuming the effective depth of thenib is 15 in., determine the ulti-mate shear stress from:

V. = Vul¢bd= 100,000/(0.85) (16) (15)= 490 psi<0.2f,' = 1000 psi (ok)

2. Hanger Reinforcement

Avh=Vul4f,,= 100/(0.85) (60)= 1.96 in.2

Assuming that the stirrups can bebundled together and providing1'/2 in. cover to end face, the dis-tance from re-entrant corner tocenter of gravity of hanger rein-forcement is about 23/4 in.

3. Reinforcement to Resist Flexurea 4.5 + 2.75 = 7.25 in.aid = 7.25/15 = 0.48 (i.e., < 1.0)M,,= V,,a +N.(h-d)

= 100 (7.25) + 15 (16 - 15)= 740 kip-in.

MuR = 4bd2f,= 740

= 0.0480.85 (16) (15) 2 (5)

p = (f,'/f,,) (0.85 - 0.72 - 1.7R)

= L1 60J

1 0.85 - 0.72 - 1.7 (0.048)]

= 0.0043 < p ra., = 0.0252for f,' = 5000 psi and f,, = 60 ksi.

.A f = 0.0043 (16) (15) = 1.03 in.2

4. Reinforcement to Resist HorizontalForceA„ = N u/4f,,

= 15/(0.85 x 60)= 199 in.2

Use 5 #4 closed ties, A,,,, = 2.00 5. Shear Transfer Reinforcement

in.2These can be arranged as two,four-legged stirrups and one,two-legged stirrup.

Using Modified Shear-Friction:

A„f = I Vu - 0.5bd /f.10.80

but 0.2bdif,,I 100

A °r = [ 0.8(0.85) - 0.5(16)(15) X60

= 0.45 in.2

0.2bdlf,, = 0.2(16)(15)/60= 0.80 in.2

.'.A„f = 0.80 in.'

6. Main Reinforcement of Nib%A„f = 0.53 in. 2 <A,= 1.03 in.2

.•.AB =Af+An= 1.03 + 0.29 = 1.32 in.2

Use 3 #6 bars, A R = 1.32 in.2

42

"l

IJ VFig. 7. Design example for dapped-end beam.

7. AnchorageThe main nib reinforcementwill be anchored at the outer endby welding a #6 bar transverselyacross the bars. These bars must beextended a distance beyond Point"c" in Fig. 2 sufficient to developtheir yield strength at this point.These are "top bars," being morethan 12 in. from the bottom of thebeam; hence, the required 1d forf C' = 5000 psi and f y = 60 ksi is1.4(18) = 25 in.H–d+l d = 28– 15+25= 38 in.Therefore, extend the maindapped-end reinforcement A 3 adistance 38 in. beyond the re-en-trant corner A.

8. Horizontal StirrupsA h = 0.5(A 3 – A.) = 0.5(1.03)

= 0.52 in.2Use 3 #3 U-bars, A h, = 0.66 in.1.7I d = 1.7(12) = 21 in.Therefore, extend these stirrups 21in. beyond the interface betweenthe nib and the full depth beam.

9. Hanger Reinforcement AnchorageThe hanger reinforcement must

pass around at least 2.00 in. 2 of thelongitudinal reinforcement at thebottom of the beam. This rein-forcement must have a positive endanchorage and must extend at least1.7 l d (for the bars) or 1a for thestrand if a prestressed beam orwould be a continuation of beamflexural reinforcement in a rein-forced concrete beam.

10. Strength Reduction FactorNote that a strength reduction fac-tor 0 = 0.85 is used in all the rein-forcement design calculations.

11. BearingDesign for bearing as per the PCIDesign Handbook.

References

1. Sargious, M., and Tadros G., "Stressesin Prestressed Concrete Stepped Can-tilevers Under Concentrated Loads,"Proceedings, Sixth Congress of the FIP(Prague), 1970.

2. Werner, M. P., and Dilger, W. H.,"Shear Design of Prestressed ConcreteStepped Beams," PCI JOURNAL, V.


18, No. 4, July-August 1973, pp. 37-49.3. Mattock, A. H., "Design Proposals for

Reinforced Concrete Corbels," PCIJOURNAL, V. 21, No. 3, May-June1976, pp. 18-42.

4. PCI Design Handbook (First Edition),Prestressed Concrete Institute, Chicago,1971.

5. "Building Code Requirements forReinforced Concrete (ACI 318-77),"American Concrete Institute, Detroit,1977.

6. PCI Design Handbook (Second Edi-tion), Prestressed Concrete Institute,Chicago, 1978.

7. Mattock, A. H., Chen, K. C., andSoongswang, K., "The Behavior ofReinforced Concrete Corbels," PCIJOURNAL, V. 21, No. 2, March-April1976, pp. 52-77.

8. ACI-ASCE Committee 426, "SuggestedRevisions to Shear Provisions forBuilding Codes," American ConcreteInstitute, Detroit, 1979, 82 pp.

9. MacGregor, J. G., and Hawkins, N. M.,"Suggested Revisions to ACI BuildingCode Clauses Dealing with Shear Fric-tion and Shear in Deep Beams and Cor-bels," ACI Journal, V. 74, No. 11,November 1977, pp. 537-545.

APPENDIX A-NOTATIONa = shear span; distance from cen-

ter of action of shear force V,, tocritical section for flexure, in.

A f = area of reinforcement requiredto resist flexure, in.2

A h = total area of horizontal stirrups,in.2

A„ = area of reinforcement requiredto resist horizontal force N u, in.2

A 3 = total area of dapped-end mainreinforcement, in.2

A„ = area of vertical stirrup rein-forcement, in.2

A „f = area of shear transfer reinforce-ment required to resist shearV,,, 1n.2

A„h = area of hanger reinforcement,in.2

b = width of nib, in.d = effective depth of nib, in.

= specified compressive strengthof concrete (measured on 6 x12-in, cylinders), psi

= specified yield strength of rein-forcement, ksi

h = total depth of nib, in.H = total depth of beam, in.K = coefficient for type of con-

crete in modified shear-frictiondesign equation

l d = reinforcing bar developmentlength, in.

l„ = distance from center of action of

shear V,, to interface betweennib and full-depth beam, in.

= nominal moment strength, kip-in.

M u = moment due to factored loads,kip-in.

N n = nominal strength resisting hori-zontal loads, kips

N. = horizontal load due to factoredloads, kips

V,, = shear carried by concrete, kipsV D = shear due to unfactored dead

load, kipsV 5 = shear carried by horizontal stir-

rups, kipsV L = shear due to unfactored live

load, kipsV,, = nominal shear strength, kipsV S = shear carried by vertical stir-

rups, kipsV,, = total shear due to factored

loads, kipsV,, = shear at yield of main dapped-

end reinforcement, kipsA = coefficient for type of concrete

in PCI equation for nib shearstrength

µ = coefficient of friction used inshear-friction calculations

p = A,lbd= strength reduction factor= 0.85 for all calculations in

dapped-end reinforcement de-sign

44

c, A. H., Chen, K. C., andyang, K., “The Behavior of:ed Concrete Corbels,” PCIL, V. 21, No. 2, March-April52-77.

E Committee 426, “Suggestedns to Shear Provisions for

Codes,” American ConcreteDetroit, 1979, 82 pp.

;or, J. C., and Hawkins, N. M.,:ed Revisions to ACI Buildingauses Dealing with Shear Eric-Shear in Deep Beams and CortCI Journal, V. 74, No. 11,er 1977, pp. 537-545.

[iONar V, to interface between

and full-depth beam, in.ninal moment strength, kip

ment due to factored loads,

ninal strength resisting horital loads, kipsizontal load due to factoredds, kipsar carried by concrete, kipsar due to unfactored deadd, kipsar carried by horizontal stir)S, kipsar due to unfactored live•d, kipsaiinal shear strength, kipsar carried by vertical stir)S, kips:al shear due to factoredtds, kipsar at yield of main dappedd reinforcement, kipsefficient for type of concretePCI equation for nib shear

engthefficient of friction used inear-friction calculations‘bdength reduction factor35 for all calculations inpped-end reinforcement de

APPENDIX B—EQUILIBRIUM OFPARTS OF BEAM CUT OFF BYDIAGONAL TENSION CRACKS

Consider moment equilibrium of parts of the beam cut off by diagonaltension cracks of both Sections AY and BZ (see Fig. 2). Letx be the depthof the center of action of the concrete compression force C in Fig. 3.

1. Beam Section Cut Off by Crack AYIn the case of that pai-t of the beam cut off by Crack AY, consider moment

equilibrium about a point at distance x below Point Y.Moment due to factored loads:

= V(h + l) + N(h — x)Resistance moment due to internal forces (not including any forces in

stirrups A h)

= 4 Af5(d — x) + bA ,J5(h + l — a) or,çbM cbAff5(d — x) + AJ5(d — x)+ qSA,f5(h + l, — a)

4Affy(d — x) + 4A,f5(h — x) — çbAJ5(h — d) +A,J5(1i + l,— a)

[SinceA8is(Af+Afl).j

Now, the depth of the concrete compression zone will be the same at Points

Y and Z as at the center of gravity of the hanger reinforcement, since it

corresponds to the reinforcement force Affy only. Hence, x will be the same atall three locations.

We can therefore write that:Aff(d — z) = moment in plane at center of gravity ofA,,Ajfy(d — x) = Va + N4(h — d)

Also, 4.Af5(h — x) = N(h — x)

qiAf5(h — d) = N(h — d) and,

A,J5(h +l,—a)=V(h +l,— a)—a)

V(h + l) + N(h — x)

i.e. MflMU

Hence, the resistance moment is equal to or greater than the moment due tofactored loads.

2. Beam Section Cut Ott by Crack BZ

In the case of that part of the beam cut off by Crack BZ, consider moment

equilibrium about a point distance x below Point Z.Moment due to fictored loads:

= V(H + l) + N(h — x)

Resistance moment due to internal forces:

qSM= 4AJ5(d— x)+ 4AV,J,(H + ii,— a)

By a series of substitutions similar to those made in Case 1 above, it can beshown that the resistance moment is equal to or greater than the moment due tothe factored loads.


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