SHEAR DESIGN OFPRESTRESSED CONCRETESTEPPED BEAMSM. P. WernerPartnerJ. H. Cook Architects and EngineersCalgary, Canada

W. H. DilgerAssociate ProfessorDepartment of Civil EngineeringThe University of CalgaryCalgary, Canada

It is demonstrated experimentally that the shearforce leading to cracking at the re-entrant corner ofa stepped beam can be considered as the shearresisted by the concrete in the shear design.Recommendations for the detailing of the stepare given.

The prestressed stepped beam connec-tion shown in Fig. 1 provides an eco-nomical and efficient means of connect-ing precast to precast and precast tocast-in-place concrete members. Theconnection is economical because allthe miscellaneous mild steel is elimi-nated and because the span of thedrop-in beam is effectively reduced ifthe connection can be located at the in-flection point of the total span.

The designer has little informationfor the shear design of the steppedbeam end, because this case is not cov-ered by present design codes. Thequestions which arise in the design of astepped beam are: (1) what is theshear force at which shear cracks startto form at the re-entrant corner of thestep and (2) can this cracking shear beconsidered to be the contribution of theconcrete to the shear strength of the

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DROP-INBEAM

Fig. 1. Prestressed stepped beam connection

PCI Journal/July-August 1973ҟ 37

h /4

h/4

h = 13 in. I __"-P2 i h/4

-h/4

Fig. 2. Forces applied tobeam end zone

step? The first question was investi-gated using the finite element tech-nique and the second question wasstudied experimentally.

THEORETICAL STUDY

It is generally accepted' ,2 that thetotal shear resisted by a beam at anycross section is the summation of theshear resisted by the concrete and theshear carried by the shear reinforce-ment. In prestressed beams with drapedtendons the vertical component of theprestressing force is added. The shearresisted by the concrete at a section isassumed to be approximately equal tothe shear force at the section at the for-mation of an inclined crack.

Although the concrete cannot resist

Fig. 3. Fine mesh for finite elementanalysis

any tensile forces after cracking, othermechanisms such as the concrete com-pression zone, the dowel forces of barscrossing the crack, and the aggregateinterlock take over the shear previouslyresisted by the untracked concrete. Noattempt is made here to discuss theshear resistance of these mechanisms.In zones of prestressed concrete beamswhere the shear force is high and themoment small, the shear carried by theconcrete is equal to the web crackingload which is determined by equatingthe principal tensile stress at the cen-troid of the member, to a design tensilestrength of the concrete.1,2

In a stepped beam this approach isnot easily applicable because of thecomplicated stress configuration at there-entrant corner. 3 Also, no evidencehas been available, to date, that theshear force at the formation of a crackat the re-entrant corner can be consid-ered as the shear force resisted by theconcrete. The theoretical part of thisstudy is concerned with the calculationof concrete stresses at the re-entrantcorner.

The finite element technique 4 wasused to determine the elastic stress dis-tribution in the step. (The computerprogram with a brief description of in-put and output data is available fromthe authors on request.)

Four different loading cases (Fig. 2)were investigated for the beam shownin Fig. 5:

38

(I) (2) (3) (4)

P^A A A A

p2 Fi ►

Fig. 4. Principal stresses at re-entrant corner due to unit forces P 1 , P2, V, and H

1. Prestressing force Pl = 1

2. Prestressing force P2 = 1

3. Vertical force V = 1

4. Horizontal force H = 1

These forces were applied to a beamof 1 in. thickness. The finite elementanalysis was carried out in two steps:first, the whole shear zone of the beamwas analyzed using a coarse mesh ofrectangular elements to find the nodalforces at a distance of 10 in. from thestep. These nodal forces were then ap-plied to the fine mesh of Fig. 3. The

stresses at the re-entrant corner weredetermined by averaging the stresses atthe corners of the three elements whichare connected there. The stress compo-nents and the principal stresses at there-entrant corner (Point A) are listed inTable 1 for all four loading cases, andshown in Fig. 4. With the exception ofload P,, all forces cause tensile stressesat Point A. With the stress componentsdue to unit forces known, the crackingloads can be determined if the tensilestrength of the concrete is known. Cal-culation of the cracking load will be

Table 1. Stress Components and Principal Stresses at Re-entrant Corner Due toUnit Forces (Fig. 2).

Stresses P, = 1.0 P2 = 1.0 V = 1.0 H = 1.0

fa —0.248 0.023 0.508 0.801ta, —0.150 0.068 0.575 0.327tzv 0.113 —0.015 —0.315 —0.192t, —0.075 0.073 0.858 0.869f2 —0.322 0.018 0.225 0.259

e 33.3 —16.9 —42.0 —70.5

PCI Journal/July-August 1973ҟ 39

40 1f2 .. 15" 40V2

__RP/2 P/2

----------2_».3 --------- 325

32a 13.0

325

34be" 19 3/4" 34'/' 0'11M-

Fig. 5. Dimensions of test beams and reinforcement of Beam W1.

shown after discussion of the experi-mental investigation.

EXPERIMENTAL STUDY

Five prestressed beams were testedto verify the theoretical studies on crackinitiation, and to determine whether thecracking load is equal to the shear forcewhich can be supported by the con-crete. The principal test variables were:

1. Support condition.2. Type and amount of shear rein-

forcement.

Three beams were supported verti-cally, the remaining two were support-ed such that an inclined reaction devel-oped as the beam was loaded. This in-clined reaction was introduced tosimulate the horizontal tension whichcan develop at the support due tocreep, shrinkage, and temperature ifshortening of the drop-in beams is pre-vented.

TEST SPECIMENS

All the beams had the same cross

X 2,1 2 8. 8' 6, s .. P/2ҟP/2A

t r1 i —— t- - -^ — 2 3̂--r I— I- -^, —Trt t ^i^

2-'3 j 3 STIRRUPS ^I Tal — —

2 "3 ACover

23 h /a 6" 6' P12 P/2

\\ i I "3 STIRRUP S --7

`--- — 4-- ----- ---I--- I

2-*3Fig. 6. Reinforcement of beams with shear reinforcement.

Top: Beams W2 and W4. Bottom: Beams W3 and W5.

40

section, b/h = 6.75 x 13.0 in., an over-all length of 96 in. and a span of 92 in.The steps on each end were 4 in. long.All beams were post-tensioned by two0.6 in. draped deformed bars (Fig. 5).The 1-in, ducts were grouted afterstressing of the bars. Beam W1 had noshear reinforcement, Beams W2 andW4 had vertical stirrups only [Fig. 6(top)], and Beams W3 and W5 hadbars bent up at 45 deg at the beam end[Fig. 6 (bottom)]. Beams Wl, W2,and W3 were supported vertically.

To introduce vertical and tensileforces in Beams W4 and W5 they weresupported on a plane inclined at about30 deg to the beam axis.

STEEL PLATE

Fig. 7. Bell anchor of prestressing bars.Strengthening of anchor zones using

steel plate

150

125ҟPRESTRESSING

STEEL

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AREAҟ(ln2) 0.282 0.11 0.11YIELDҟSTRESS (ksi) 115 57.3 52.5

TENSILEҟSTRENGTH (ksi) 157 64.4 75.7ELONGATIONҟ(%) 28 28

Q2 SMOOTH 3/g BARS

DEFORMED 3/g BARS

25

0.5ҟ1.0ҟ1.5ҟ2.0ҟ2.5

STEEL STRAIN (%)

Fig. 8. Stress-strain diagrams of prestressing and reinforcing bars

PCI Journal/July-August 1973 41

b) Detail A for vertical Reactionҟc)Detail A for vertical andҟd) Detail Bhorizontal Reaction

Fig. 9. Test setup

All the beams were cast in one place-ment using ready-mixed concrete witha 3 in. maximum aggregate size and a2-in. slump. The cylinder strength atprestressing (age 21 days) was 5380psi and at testing (age 42 days) 5800psi. Stressing of the bars was immedi-ately followed by grouting of the ducts.The grout strength was 5100 psi at 28days.

The 0.6 in. bars were Dywidag barswith the deformations of the bars serv-ing as a thread for the anchor nuts.Each anchor nut bore against a bellanchor (Fig. 7). The stress-strain rela-tion for the prestressing steel is shownin Fig. 8.

The steel used for the stirrups and

the nonprestressed longitudinal rein-forcement consisted of plain #3 bars,except the short hooked bars at mid-depth of the beam ends were #3 de-formed bars. Their properties, togetherwith their stress-strain curves, are alsogiven in Fig. 8.

LOADING AND MEASUREMENTS

Two symmetrical loads were appliedincrementally to failure. For the com-bined horizontal and vertical reaction,a triangular bearing plate was used(Fig. 9c) . Measurements of strains anddeflections were taken at the locationsindicated in Fig. 10, and the extent of

42

h

crack propagation was marked at eachload increment.

BEHAVIOR OF TEST BEAMS

Up to the formation of the first crack,each of the beams behaved elastically.The first crack in all the beams formedat the re-entrant corner of the step.These cracks propagated nearly hori-zontally towards the load points. Thebeams supported vertically, cracked atapproximately 17.5 kips (Table 2) . Theones with horizontal and vertical reac-tions cracked under a load of about13.5 kips. The formation of these hori-zontal cracks resulted in a sharp in-crease in the steel stress in the shear re-inforcement. The crack patterns of allbeams after failure are shown in Fig.11. After formation of the first crack at18.0 kips, Beam W1 resisted another12 kips before failure occurred at 30kips. Prior to failure the crack hadopened about 1/s in. at its root.

Beam W2 with vertical stirrupsreached the highest failure load (P. =43.5 kips) . This beam failed by crush-ing of the concrete in the constant mo-ment zone, after yielding of the pre-stressed bars indicating flexural failure.The stirrups did not reach the yieldpoint and the shear crack was narrowup till failure.

Beam W3 failed under a load of P. =31.6 kips far below the flexural ultimateload. Failure occurred due to burstingof the anchorage zone. The bent-upbars with hooks were obviously inade-quate to confine the anchorage zone.The hooks of the plain bars split theconcrete anchorage zone so that no re-sistance to shear was provided by thebent-up bars. As a secondary effect, theshear compression zone near the load-ing point crushed.

Beam W4 (vertical stirrups, horizon-tal and vertical forces) showed somecracks in the anchorage zones under P

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PCI Journal/July-August 1973 43

ALL BEAMS

Fig. 10. Position of strain gages and dial gages

= 8 kips. To avoid distress of the an-chorage zone, a steel plate was provid-ed at the beam ends (Fig. 7). Failureof Beam W4 occurred due to crushingof the shear compression zone near theload point. The stirrups had notreached yield under the failure loadand the width of the shear cracks re-mained small up till failure. The loadreached the theoretical flexural failureload.

Beam W5 (bent-up bars, horizontaland vertical reactions) was alsostrengthened by steel plates at the ends,after cracks were showing under lowloads. This beam collapsed suddenlyafter the bent-up bars had reached theyield stress.

The ultimate load of P,,, = 40.0 kipsis slightly above the theoretical ultimateflexural load, yet the sudden type offailure (Fig. lie) clearly indicated ashear failure.

STEEL STRESSES IN WEBREINFORCEMENT

The steel stresses were measured atthe locations indicated in Fig. 10. Thestrain gages were mounted where theshear crack crossed the stirrups so thatthe measured values can be considered

to be the maximum strains occurring inthese bars. In Fig. 12 (top), represent-ing the results of Beams W2 and W3with vertical reactions, the stress in-crease after formation of the shearcrack at P = 18 kips is quite pro-nounced. In Beam W2 the stirrup No. 2closest to the beam end picks up thehighest stress. From the strains record-ed in Beams W2 and W3 it is obviousthat the shear reinforcement was stillfar from reaching the yield point of57.3 ksi at failure. The stirrups inBeams W4 and W5 with vertical andhorizontal reaction'. also had a markedincrease in stress after crack formation.The stresses in the vertical stirrups ofBeam W4 stayed well below the yieldpoint at failure while yield was reachedin the two inclined bars of Beam W5.

EVALUATION OF TEST RESULTS

CrackingThe stresses resulting from unit loads

(Table 1) may be used to predict thecracking load. Equating the principaltensile stress fl resulting from the vari-ous forces, to the tensile strength of theconcrete f' t, we get:

fl=ft=½(Yf.+Ify)+V ¼ ( f - f) 2 + (1)

44

Rearranging leads to:

fI t2 — f t (:.fx+ 4fy) +

(f) (f) - (fxv)2 = 0 (2)

With the prestressing forces P1 =P2 = 3.87 kips per in. of beam widthand putting the tensile strengths of theconcrete ft = 6 V/f' 0, we can solve Eq.

(2) for the shear force leading to crack-ing.

For the beams supported vertically(H = 0) this procedure results in V=1.22 kips per in., which is equivalent toa cracking load Pc,. = 18.11 kips.

For the beams with vertical and hori-zontal reactions (H = 0.59V) solutionof Eq. (2) results in a cracking load

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P/2 P/2

j2227 32 I 1 I I I 1

W4ҟIҟIҟ1ҟIҟJҟ1ҟIҟ1(d)ҟPU =38kҟIIҟIҟIҟIҟIҟ2 I I I II

P/2 P/2

4\!— 22 271 1 3 1ҟI I --

W5(e) PU = 40k--- I IiҟI I I—_-

Fig. 11. Crack patterns at failure

PCI Journal/July-August 1973ҟ 45

rn

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m

0

STEEL STRESS - KSI

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CRACKING LOADPu = 13 kipҟ̂^

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BEAM W4_P =38.Oki

O BEAM W5: P 40.0 kip ,^ҟOII

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STEEL STRESS - KSI

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CRACKING_LOADP^ = 18 Kip W II

BEAM W3 _hҟPu = 31.6 kip

BEAM W 2_— -- Pu= 43.5Kip

= 13.50 kips. These results comparewell with the experimental crackingloads (see Table 2). The good agree-ment indicates that the finite elementmethod with the mesh selected was ap-propriate for accurately predicting thecracking load at the re-entrant cornerof the step.

Ultimate loadIf we assume that the shear force ex-

isting when cracking occurs at the re- aentrant corner, is equal to the shearforce which continues to be resisted by E''the concrete, then the ultimate load canbe calculated by adding to this forcethe vertical component of the tendonand the shear carried by the web rein-forcement. This is done in Table 3 andthe results are compared with the ex-perimental failure loads. The shearforce due to prestressing is the vertical

°

component of the effective prestressingforce. With Pl = 26.1 kips at an angle 'of 10.7 deg, the vertical component isVp = Pt sin (10.7°) = 4.87 kips. Theshear resisted by the shear reinforce-ment is calculated from the stressesshown in Fig. 12. E

Comparison of the actual failure load Uwith the shear force resulting from theshear components discussed above is ^?shown in Column 7 of Table 3. Theagreement is excellent.

ti

CONCLUSIONS

The shear force which causes shear 0Ocracks to form at the re-entrant cornerof a stepped beam can be accuratelypredicted by a finite element analysis Uusing a tensile strength for the concreteequal to 6 -,/f This cracking load canbe taken as the shear force which is re- vsisted by the concrete. The shear force En

which is in excess of the cracking load cois resisted by the inclined tendon andthe shear reinforcement crossing thecrack.

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PCI Journal/July-August 1973 47

H

SECTION A - A

Fig. 13. Recommended design details

Vertical and inclined shear reinforce-ment seem to be equally efficient in re-sisting shear.

As a result of the cracking of the an-chorage zone under low load it must beconcluded that special attention mustbe devoted to the choice of anchorageused for the tendons, particularly if thestep is short.

RECOMMENDATIONS FOR THESHEAR DESIGN OF THE END

ZONE OF STEPPED BEAMS

For practical design the tensile stressresulting in cracking at the re-entrantcorner should be assumed in accor-dance with present design codes to be

48

equal to 4 f 1.

Accurate analytical methods must beused to predict the cracking shear. Thevertical component of the prestressingforce should be included. The shearforce at ultimate which is not resistedby the concrete and the inclined pre-stressing steel must be supported byshear reinforcement, preferably verticalstirrups placed as close as possible tothe beam end.

Careful detailing of the end zones isrequired to avoid spalling around thetendon anchors. If the step is short, ten-dons with steel anchor plates should beused to prevent bursting of the anchorzone. Horizontal bars at middepth mustextend at least a distance equal to thebeam depth beyond the re-entrant cor-ner and should be provided with down-ward hooks to ensure anchorage belowthe plane of a potential shear crack(Fig. 13) .

ACKNOWLEDGMENT

The authors gratefully acknowledge

the help of Dr. G. S. Tadros in prepar-ing the computer program.

The research for this paper was fi-nancially supported by the National Re-search Council of Canada.

REFERENCES

1. ACI Committee 318, "BuildingCode Requirements for ReinforcedConcrete (ACI 318-71) ," AmericanConcrete Institute, Detroit, 1971,144 pp.

2. Canadian Standards Association,Prestressed Concrete, CSA StandardA 135, 1962.

3. Werner, M. P., "Prestressed SteppedBeams," ME Thesis, Department ofCivil Engineering, The University ofCalgary, 1971.

4. Przmieniecki, J. S., Theory of MatrixStructural Analysis, McGraw-Hill,New York, 1964.

5. Cooper, M. B., "Shear Transfer inPrecast Concrete Structural Ele-ments," MS Thesis, Department ofCivil Engineering, The University ofCalgary, 1968.

Discussion of this paper is invited.Please forward your discussion to PCI Headquartersby December 1, 1973, to permit publication in theJanuary-February 1974 PCI JOURNAL.

PCI Journal/July-August 1973 49