ductility of prestressed and partially prestressed concrete beam sections

Robert ParkProfessor and HeadDepartment of Civil

EngineeringUniversity of Canterbury

Christchurch, New Zealand

Kevin J. ThompsonSenior EngineerMinistry of Works and

DevelopmentInvercargill, New Zealand

Ductility of Prestressedand Partially PrestressedConcrete Beam Sections

Dynamic analyses of structures re-sponding elastically to ground

motions recorded during severeearthquakes have shown that thetheoretical response inertia loads aregenerally significantly greater thanthe static design lateral loads recom-mended by codes. Hence, structuresdesigned for the lateral earthquakeloads recommended by codes can onlysurvive severe earthquakes if theyhave sufficient ductility to absorb anddissipate seismic energy by inelasticdeformations.'-"

Prestressed concrete has beenwidely used for structures carryinggravity loads but has not had the sameacceptance for use in structural sys-tems which resist seismic loading.Part of this caution in the use of pre-stressed concrete for earthquake re-sistant structures has been due to the

paucity of experimental and theoreti-cal studies of prestressed concretestructures subjected to seismic typeloading. A survey of the researchwhich has been conducted on theseismic resistance of prestressed con-crete was published in 1970. 9 Parmeioand Hawkins" have published morerecent reviews of the state of the art ofseismic resistance of prestressed andprecast concrete.

There has been a lack of detailedbuilding code provisions in theUnited States for the seismic design ofprestressed and precast concrete. Forexample, the ACI Code,' the SEAOCrecommendations, 2 the UniformBuilding Code, 3 and the tentativeprovisions of the ATC 4 all containspecial provisions for the seismic de-sign of cast-in-place reinforced con-crete structures, but do not have

46

corresponding special provisions forprestressed or precast concrete struc-ture s.

According to those codes 2' 3, ' precastreinforced concrete frames must com-ply with all code provisions pertainingto cast-in-place reinforced concreteframes. Englekirk 12 using an examplebuilding developed by Freeman, 13 hasrecently demonstrated the difficulty ofapplying the UBC provisions, 3 whichwere developed for cast-in-placereinforced concrete structures, toductile precast concrete frames.

American codes discourage the useof prestressed concrete for primaryseismic resisting members. For exam-ple, the SEAOC recommendations2include in its commentary the fol-lowing statement: "The use of pre-stressing to develop ductile momentcapacity will require testing and is asubject for further study." It is appar-ent that a similar view is held by otherAmerican code committees. The lackof seismic design provisions for duc-tile prestressed concrete frames ispreventing designers from making aproper evaluation of the use of pre-stressed concrete as an alternative toreinforced concrete.

More progress in the developmentof seismic provisions for ductile pre-stressed concrete frames has beenmade elsewhere. For example, theCommission on Seismic Structures ofthe FIP7 has developed such recom-mendations. Also, the Standards As-sociation of New Zealand5,6 has re-cently drafted seismic design recom-mendations for prestressed concretebased on the significant additionaltheoretical and experimental informa-tion which has become available fromrecent studies. A summary of thatNew Zealand work is given in Refer-ence 14.

Prior to developing the main topicof this paper, it is necessary to reviewthe concept of ductility and discuss itsdesign implications.

PCI JOURNAL/March-April 1980 47

Design Criteria forDuctility Demand

A ductile structure is one which iscapable of large inelastic deformationsat near maximum load carrying capac-ity without brittle failure. Typicalductile load-displacement relationsare shown in Fig. 1. The elasto-plasticrelation is shown as a dashed line.The relation typical of prestressedconcrete is shown as a full curvedline.

Nonlinear dynamic analyses ofcode-designed structures respondingto typical severe earthquake motionshave given an indication of the orderof post-elastic deformations, andhence the "ductility" factor, required.However, the number of variables isso great that no more than qualitativestatements can be made at present.

Confusion has existed in the mindsof some designers regarding the def-inition of "ductility factor," since itcan be expressed in terms of dis-placements, rotations or curvatures.

The displacement ductility factor,µ = 0 u/0 where:

A„ = maximum lateral (horizontal)deflection of the structure and

Load

First yield displacement

Displacement

Fig. 1. Load-displacement relation andpossible definition of "first yield"displacement. Prestressed concrete isrepresented by full curved line andelasto-plastic behavior by dashed line.

Dy = lateral deflection of the struc-ture at first yield,

is the value commonly determined innonlinear dynamic analyses.

Some dynamic analyses of frameshave used the rotational ductility fac-tor of members, defined at 9 u /B,,,where:

Bu = maximum plastic hinge rota-tion of end of member and

e v = rotation at end of member atfirst yield

The fundamental informationneeded by the designer concerns therequired member section behavior atthe plastic hinge expressed by thecurvature ductility factor 4I0,where:

= maximum curvature at the sec-tion and

O„ = curvature at the section at firstyield

Thus, the required 4/4 value value is afar more meaningful index for ductil-ity demand than the other pos-sibilities. It needs to be recognizedthat there can be a significant differ-ence between the magnitudes of thedisplacement, rotational and curvatureductility factors. This is because onceplastic hinging has commenced in astructure the deformations concentrateat the plastic hinge positions and fur-ther displacement occurs mainly byrotation of the plastic hinges. Thus,the required O„/d, ratio will begreater than the Ol0 ratio.8

When calculating ductility factors,the definition of first yield deforma-tion (displacement, rotation or curva-ture) often causes difficulty when theload or moment-deformation curve isnot elasto-plastic. This will occur forexample due to the stress-strain curveof prestressing steel not having a welldefined yield plateau, or to non-pre-stressed steel bars at different depthsin the section yielding at differentload levels, or to plastic hinges not

Plastichinge

Frame Column sides way Beam sides waymechanism mechanism

Fig. 2. Building frame under seismic loading and possible mechanisms due toformation of plastic hinges.

forming simultaneously in all mem-bers.

In New Zealand5 the "first yield"displacement is taken as the dis-placement calculated for the structureassuming elastic behavior up to thestrength of the structure in the firstload application to yield, as illustratedin Fig. 1. A similar definition can beadopted for first yield rotation andcurvature. Such a definition for firstyield allows comparison of the effectof different loop shapes, with thesame initial stiffness and strength, onthe ductility demand.

It is apparent that the sequence ofplastic hinge development in struc-tures will influence the curvatureductility demand. Nonlinear dynamicanalyses have indicated that ductilitydemand concentrates in the weakparts of structures and that the curva-ture ductility demand there may beseveral times greater than for wellproportioned structures. This can alsobe illustrated by examination of staticcollapse mechanisms. Fig. 2 shows aframe used for seismic resistance.Possible mechanisms which couldform due to development of plastichinges are also shown in the figure. Ifplastic hinging commences in the col-umns of a frame before the beams, acolumn sidesway mechanism canform. In the worst case the plastic

hinges may form in the columns ofonly one story, since the columns ofthe other stories are stronger. Such amechanism can make very large cur-vature ductility demands on the plas-tic hinges of the critical story8 par-ticularly for tall buildings.

On the other hand, if plastic hingingcommences in the beams before in thecolumns a beam sidesway mechanism,as illustrated in Fig. 2, will develop8which makes more moderate demandson the curvature ductility required atthe plastic hinges in the beams and atthe column bases. Therefore, a beamsidesway mechanism is the preferredmode of inelastic deformation, par-ticularly since the straightening andrepair of columns is difficult. Hence,for frames, a strong column-weakbeam approach is advocated to ensurebeam hinging. In the actual dynamicsituation higher modes of vibrationinfluence the moment pattern and ithas been found8 that plastic hinges inthe beams move up the frame inwaves involving a few stories at atime. Thus, the mechanisms shown inFig. 2 are idealized since they involvebehavior under code type static load-ing. Nevertheless, considerations as inFig. 2 give the designer a reasonablefeel for the situation.

Typical values for the displacementductility factor µ required of code de-


T

(al Prestressed Concrete System

(b) Reinforced Concrete System

signed ductile frames in order to sur-vive very severe earthquakes are 3 to 5.The tentative provisions of the ATC4for ductile frames are based on a valuefor of about 7. The 4u/4, values re-quired of the plastic hinge sections ina strong column-weak beam design ofa framed structure will depend on the

geometry of the members. However, itwould seem that 4/4 of at least 3µshould be available in the plastichinge regions. Codes do not generallyrequire designers to calculate the ac-tual ductility required at the plastichinges and then to match that de-mand. Instead, for reinforced concreteframes the seismic design provisionsof codes are intended to ensure thatthe plastic hinge region will be ade-quately ductile.

The suspicions that exist concerningprestressed concrete in seismic designare generally a concern whether theenergy dissipation at the plastic hingesections is adequate, and whethersuch sections can achieve the re-quired ductility. The first issue arisesbecause for prestressed concretemembers the initial elastic tensilestrain in the tendons due to prestresscauses a large deflection recovery,even after large deflections. Hence,the energy dissipation (area within theloop) of prestressed concrete memberswill be lower than that of reinforcedconcrete members of similar strength.

Figs. 3(a) and (b) show comparedidealized moment-curvature hys-teresis loops for cyclically loaded pre-stressed and reinforced concretemembers. Nonlinear dynamicanalyses of single degree of freedomsystems responding to very severeearthquakes have shown that themaximum displacement of a pre-stressed concrete system is on average1.3 times that of a reinforced concretesystem with the same code designstrength, viscous damping ratio andinitial stiffness.'' However, this effectof lower hysteretic damping shouldnot be an obstacle in design since theductility required at the sections canbe provided in suitably detailed pre-stressed members.

Note that the moment-curvatureloops of prestressed concrete mem-bers can be "fattened," and hence thedisplacement response reduced, if

(c) Partially Prestressed Concrete System

Fig. 3. Idealized moment-curvaturehysteresis loops for structural concretesystems.

non-prestressed steel is added to themember to provide the energy dissi-pation. The presence of longitudinalnon-prestressed reinforcing steel alsoimproves the ductility by acting ascompression reinforcement. Fig. 3(c)shows a typical moment-curvatureloop for partially prestressed concrete.

Scope of Paper

A measure of the ductility of a sec-tion can be obtained from the mo-ment-curvature relation plotted tohigh curvatures. Moment-curvaturerelations can be derived analyticallyusing idealized models for the stress-strain behavior of the concrete andsteel.

In this paper an analytical study ispresented which examines the effectof the percentage of longitudinal pre-stressing steel and its distribution, thepercentage of longitudinal non-pre-stressed steel content and its distribu-tion, the transverse steel content, andthe concrete cover thickness, on thecurvature ductility of rectangularbeam sections.

The aim of the study is to make rec-ommendations for seismic design con-cerning the distribution of longitudi-nal and transverse steel within thesections of potential plastic hinge re-gions in beams of prestressed andpartially prestressed concrete frames.

The results summarized in thispaper are reported in more detail inReference 15.

Theoretical Moment-Curvature Analysis

An analysis procedure was de-veloped 15 to determine the moment-curvature relations for prestressed,partially prestressed and reinforcedconcrete rectangular sections withmonotonically applied bending,

which allows an assessment of theavailable ductility. The analysis en-sures compatibility of strains andequilibrium of forces and is based onidealizations for the stress-strain be-havior of concrete and steel.

Stress-Strain Model for ConcreteThe stress-strain relation for con-

fined concrete used is that derived byKent and Parkes which covers the fullstrain range, allowing for the effect ofconcrete strength and confinement byrectangular hoops on the slope of the"falling branch region." The relationis conservative in that it ignores anyenhancement in the maximum con-crete strength due to confinement byrectangular hoops. For confinementby typical quantities of closely spacedstirrup ties in the plastic hinge zonesof beams the concrete strength may be10 percent greater than assumed.However, a 10 percent increase in theconcrete strength will make little dif-ference to the computed flexuralstrength of beams. Tests conducted bythe Portland Cement Association l'concluded that the relation gave asimple and reasonable approximationfor the stress-strain curve for concreteconfined by rectangular hoops.

The stress-strain relation for con-crete is illustrated in Fig. 4(a). Ex-pressions for the three compressiveregions are as follows:Region AB: E, -_ Eo

z

fc — fc L Eco Eco) ]

(1)

in whichfc = concrete stressf,' = concrete cylinder strengthE c = concrete strain, andE,o = 0.002 is the strain in concrete

at maximum stress f^

Region BC: e, 0 < E, = E2oc

fc = Jc [ 1 — Z(E c — E co)] (2)


D

fc

Bfc -- I z- tone

Confined0.5 f^

concrete

i i I

C0.2f^ --'---' - -! - e

nconlined concreteA

Ec0 E50u e50c E20c

ft.

(a)

A • EpC E^ Ep

E50c = E 50u + E50h

– f 3 +0.002f,'1 + 3 „

L f-1000] 4 p ,Vl s

in which (4)

p" = ratio of volume of hoops tovolume of concrete core

b" = width of concrete cores = hoop spacingNote that f'c is in psi (1 psi =

0.00689 MPa).

The derivation of the empirical Eq.(4) is given in Reference 16.

Region CD: E 20, < Ec

fc = 0.2ff (5)

The maximum tensile strength ofthe concrete (modulus of rupture) isassumed to be given by the expres-sion:

fpfpufpc

fpb

(b)

Fig. 4. Assumed stress-strain relationsfor concrete (top), prestressing steel(middle), and non-prestressed steel(bottom).

in which E2(, is the strain at 0.2f f on

.fi = 7.5 f,' (6)

in which ff and fi are in psi.The stress-strain relation in tension

is assumed to follow the slope of theparabola of Region AB of Fig. 4(a) atthe origin.

Stress-Strain Model forPrestressing Steel

The stress-strain relation for theprestressing steel in tension used is ofthe form derived by Blakeley andPark 18 and illustrated in Fig. 4(b). Therelation comprises three regionswhich are defined by the followingequations:Region AB: Ep -- Ep

fs

f5',

fy

the falling branch of the stress-strain fp = E pE p (7)curve, and Z defines the slope of thefalling branch as follows: in which

E p = steel strainZ = 0.5 (3) Epb = steel strain at Point B (the

C 50C — E co

limit of proportionality)f„ = steel stress, and

where E p = modulus of elasticity of steel

52

Region BC:Epb < Ep-_E,

1'p — fWEnc J nbEPb +J Epc -

E nbEncllnb fnc) (8)E,(E nc —Enb)

in whichE, = steel strain at Point C

= steel stress at Point Bfx = steel stress at Point C

Region CD: E ,,, < E p -- E

f^ = f + I EP E_ ] f) (9)E PU — Ep,

in whichE p,, = ultimate steel strainfpu = ultimate steel stress

Numerical values for the stressesand strains at Points B, C, and D needto be obtained from experimentallymeasured stress-strain curves.

Stress-Strain Model forNon-prestressed Steel

The stress-strain relation for non-prestressed reinforcing steel used is ofthe form adopted by Park and Paulay8and illustrated in Fig. 4(c). The rela-tion is assumed to be identical in ten-sion and compression and comprisesthree regions which are defined bythe following equations:Region AB: E, -- E,

.fs ° E se y (10)in which

E $ = steel strainEy = steel strain at first yieldfs = steel stressE, = modulus of elasticity of steel

Region BC: E y < E s -- Esh

fs = f. (11)

in whiche sh = steel strain at commencement

of strain hardening, and= steel stress at yield

Region CD: E sh < E s ' Esu

fs= fv1Q(Es— Esh) + 2 +60(Es – e ,,,)+ 2

(Es Esh) (60 Q )] (12)2 (30q + 1)2

where

-8" (30q + 1)2 – 60q – 1Q .fY (13)15g

2

and

q = E su — E sh (14)

in whichE. = ultimate steel strain, andf. = ultimate steel stress

Numerical values for stresses andstrains at Points B, C, and D need tobe obtained from experimentally mea-sured stress-strain curves.

Basic AssumptionsThe following assumptions are

made for the analysis of the moment-curvature characteristics of pre-stressed, partially prestressed, andreinforced concrete sections withbonded steel:

1. Plane sections before flexure re-main plane after flexure.

2. The bond at the interface of con-crete and steel is such that no slip oc-curs.

3. No time-dependent effects, suchas creep, occur during the course ofloading.

4. The stress-strain relations forprestressing steel and non-prestressedsteel are as given by Eqs. (7) to (14)and as shown in Fig. 4(b) and (c).. 5. The stress-strain relation forconfined concrete, defined as thatconcrete within the outside of theperimeter of the hoops, is given byEqs. (1) to (6) and as shown in Fig.4(a).

PCI JOURNAUMarch-April 1980 53

6. The stress-strain relation for thecover concrete, defined as that con-crete outside the perimeter of thehoops, is given by one of the follow-ing four models:

Model 1—The cover concrete fol-lows the same stress-strain relation asthe confined concrete core at all strainlevels.

Model 2—The cover concrete fol-lows the same stress-strain relation asthe confined concrete core up to astrain of 0.004, at which the coverconcrete is assumed to spall and makeno further contribution at higherstrains.

Model 3 —The cover concrete fol-lows the same stress-strain relation asthe confined concrete up to a strain of0.002, and then follows the stress-strain relation for unconfined con-crete.

Model 4 —The cover concrete fol-lows the same stress-strain relation asthe confined concrete up to a strain of0.002, and then follows the stress-strain relation for confined concretewith a Z value of twice that of theconcrete core.

The accuracy of the various modelsproposed for the cover concrete willbe checked by comparison of analyti-cal results with measured test results.Models 1 and 2 represent extremes ofbehavior and Models 3 and 4 are ex-pected to give more accurate idealiza-tions of the behavior of the cover con-crete.

Methods of AnalysisA computer program was de-

veloped 15 to compute the moment-curvature relations for rectangularconcrete sections containing up to tenbonded prestressing tendon positionsand up to ten non-prestressed steelpositions anywhere within the sec-tion. The moment-curvature relationwas found through two loading stages:first, where the neutral axis lies out-side the section; and, second, where

the neutral axis lies within the sec-tion.

An iterative procedure, which satis-fies strain compatibility and equilib-rium of forces, was used to trace themoment-curvature relation. In the firstloading stage a range of moments andtheir corresponding curvatures werefound from a range of neutral axis po-sitions; in the second loading stage arange of moments and their corre-sponding curvatures were found for a.range of extreme fiber concrete com-pression strains.

At all stages the stress in each pre-stressing tendon is that correspondingto a total steel strain which is the sumof the steel strain due to prestresswhen the adjacent concrete strain iszero plus the concrete strain whichexists at the level of the steel. Theanalytical procedure for moment-curvature analyses of sections is wellknown. Further details may be seen inReferences 8, 15, 16 and 18.

Comparison ofExperimental andAnalytical Results

In order to assess the accuracy ofthe analytical method for the determi-nation of moment-curvature relations,experimental moment-curvature rela-tions measured for a range of pre-stressed and partially prestressed con-crete beams 15' 19 were compared withanalytical moment-curvature relationsderived for the sections.

Experimental ResultsThe experimental moment-curva-

ture relations were measured in theplastic hinge regions of the beams ofthe beam-column assembly shown inFig. 5. The beams and the columnwere cast in place and represented thepart of a continuous frame betweenthe midspan of beams and the mid-height of columns. The columns were

54

loaded axially as shown and by verti-cal shears in opposite directions at theends of the beams.

Horizontal reactive loads were in-duced at the column ends, and theends of the column were held in thesame vertical line. By reversing thedirection of the beam end loads, andhence the direction of the horizontalreactive loads at the column ends, theeffects of earthquake loading weresimulated. The columns were strongerthan the beams and hence plastichinging occurred in the beams nearthe column faces rather than in thecolumns.

The plastic hinge rotation in thebeams was measured by dial gagesattached to steel frameworks which inturn were attached to pins in the con-crete. The curvature plotted was themean obtained from the measured ro-tation over a 10 in. (254 mm) or 12 in.(305 mm) gage length in the left beamnear the column face of the beam-col-umn test unit during the elastic load-ing runs and the initial run into thepost-elastic range. 15,19 The momentcorresponding to this mean measuredcurvature was taken as that at thecenter of the gage length.

The curvatures along the beamswere also measured along smallergage lengths during the tests and theequivalent plastic hinge length, 1,was found to be approximately 9 in.(229 mm), which is one-half of thebeam overall depth. It should be ap-preciated that this is the equivalentplastic hinge length 1 p for when theplastic curvature distribution is as-sumed to be at its maximum valueover 1 p (that is, an assumed rectangu-lar distribution), whereas the actualdistribution of plastic curvature iscloser to triangular in shape and ex-tends over a length greater than 1p.

In the subsequent cyclic loadingruns into the post-elastic range therewas a degradation of stiffness due toloss of cover concrete and cracking in

^-- 16(406mm) 100 tons

12" (996kN)A=A (305mm)

n •_0.•(3.353m

0

9° T

'29mm) 100 tons

18" (996kN)p

U (457m n)BZd

Fig. 5. Dimensions and loading of beam -column test units (References 3 and 6).

the compression zone as a result ofloading in the previous direction.19However, the flexural strengthreached in the beams in the firstloading cycle could, in most beams,be reached again in the subsequentload cycles, although at larger deflec-tions. Degradation of flexural strengthin subsequent load cycles was onlysignificant in those prestressed beamswhere non-prestressed compressionsteel was not present. Idealizations forcyclic load behavior are shown in Fig.3.

The experimental moment-curva-ture relations measured in the initialloading run into the post-elastic rangefor the beams of Units 1, 7, 6, and 3are shown in Figs. 6, 7, 8, and 9. Themain steel and concrete properties ofthe test beams are shown on the fig-ures. Further details of the test beamsare given in References 15 and 19. Allbeams had approximately the sameflexural strength. The beams of Units1, 7, 6, and 3 had a uniform concretecompressive stress due to prestress attransfer of 1160, 675, 371, and 0 psi(8.0, 4.66, 2.56, and 0 MPa), respec-tively. The beams of Units 1, 7, 6, and3 contained non-prestressed lon-gitudinal Grade 40 (f1, = 275 MPa)deformed bars with equal tension and


=Elastic limit £= 0.003E

^fpc=0.01 /ecc= E20cover 1Ep= 0.02

1500 ^^^- f _^ -

Q • LEGENDExperimental Curve

1000 9 3i8dia hoops• '* for 12in gouge length in left.c • • beam adjacent to column faceEiv • at 7" crs.

i Analytical CurvesZ Mcr (theor) ? Model 1 Zcore =77.3 Zcover = 77.3o her (exper. ) 5J 3-12w/0.200" - - - - Model 2 Zcore = 77.3, Zcover = 77.3

dia. tendons - .- Model 3 Z =77.3,2 =362.8

500 18 1 core cover2 I - Model 4 Zcore = 77 3 • Zcover =154.64 - 3'§ dia Material Properties

32 Gr 40 bars ff = 4,628psi (31.9MPa) tin = 25.4mm(1 2" cover to fpu = 235,800psi (1626MP0) 1 psi = 0.00689MPahoops) Cpu = 0.054 1 kip in =113 Nm

0 fy = 4 1,100psi (283.3 MPa) Iran/in =0.0394 rod/mm

0 500 1000 1500 2000 2500Curvature (rod/in x 10 6)

Fig. 6. Experimental and analytical moment-curvature relations for beam of Unit 1.

=0.01 C =0.003 Es=Esh £p=0.03 1£p(top,=0-oc co cc' 20cover --

1500

_

LEGENDEs(tap)6y

p0.02 Experimental Curves

c Ep= Elastic o-o for 12in. gauge length in leftIII limit beam adjacent to column face.

1000III 9,,

2^" 3" dia. hoops•-• for 10in, gauge length in left

beam fin, from column face.<Pi 3' 8 at 32 "crs. Analytical Curves

1 Model I Zcore°32'1•Zcover=32.1p• 9

----Model2Z =32.1 =32.1core Zcovero I 3-4w/0.276" _. -Mode/3Z -32.1,Z -423.6care - coverm Mcr (theor. 18 ..- dia. tendons - -Model 4 Zcore = 32.1• Zcover= 64.4500 II Material Properties•

Mcr (exper) 4-4 dia. ff = 5326psi (36.7MPa) tin = 25.4mmGr. 40 bars. fpu= 232100psi (1b0OMPa) fpsi= 0.00689MPa(1k' cover to Epu= 0.046 /kip in = 113Nm

21" hoops) fy = 4 (500psi(286.1 MPa) 1 rad/in = 0.03940 I I I I rod/mm.

0 500 1000 1500 2000 2500Curvature (rod/in. x 10)

Fig. 7. Experimental and analytical moment -curvature relations for beam of Unit 7.

- 56

£s = £y Ecc= £co Cs = EshE = 0 003

C = Elastic cc ,, .- • -p limit ^ ^_ .. • • ' _

1500 E 0.01 `^- - - - - LEGEND

Ti o ES(tapir Cy ACC= C20cover Experimental Curves

9" dia. hoo s3 •• o-o for 12in.gauge length in leftP column

a q/•• s ^•• beam adjacent to colum face

at 32 crs. ._. for 10in. gauge length in leftbeam fin. from column face.

1000 4 Analytical Curves

o ^^oocore cover 1-12w/0.200" -Model l Z =32.2,Z 32.2=

dia. tendon- - Zoom -32.2

IIa 18" - - core cover

- •-Model 3 Zcore = 32.2, Zcover= 435.6

2-^,.dia. -"-Model4Zcore=31.2, -64.42 Zcover4•• - . ......J: Gr. 40 bars Material Properties

500Mcr 24(theor) (1z••cover to f/ = 5,356psi (36.9 MPo) fin = 25.4mm

hoops) fur 236,050psi (1,627MPa) 1psi= 0.00689MPaMcr (exper.) -1"dia

fpu= 0.046 lkip in = 113Nm

8r.G40bars fy = 42,690psi(294.3MPa) lrad/in=0.0394

rad/mm0

0 500 1000 1500 2000 2500Curvature (rad/inx10 6)

Fig. 8. Experimental and analytical moment -curvature relations for beam of Unit 6.

Cs = Esh Ecc C20cover ^^ ^•

£cc =1 Cco

Cs - Cy \ Ccc=0003000 LEGEND

1500Experimental Curvee_* for loin, gauge length in left

beam fin. from column faceAnalytical CurvesI - -- Model l Zcore 32.2, Zcover 32.2

• •. 9" Model 2 Zc 'e = 32.2, Zcover = 32.2c̀ 1000 2i 3 dia. hoops - . - Model 3 Zcore = 32.2, Zcover = 440.0

8 at 32'" crs. _ ..-Model4Z =32.2,2 = 64.4core cover f - Material Properties

o P'.fc = 5,400psi (37.2MPa)a f = 42,300 si (291 6 MP)Pm 8„ 4 _ 1^ ,.did. y

5 Gr.40, bars tin = 25.4mm.Mu. (theor.) 1psi = 0.00689MPa

2-1"dia. lkip in = 113NmMcr(exper.l. Gr.40bars. lrad/in = 0.039hrad/mm

21s (12 "'cover to hoops)

C0 500 1000 1500 2000 2500

Curvature (rod/in x 10 i

Fig. 9. Experimental and analytical moment-curvature relations for beam of Unit 3.


The analytical moment-curvaturerelations derived for the beams ofUnits 1, 7, 6, and 3 are shown in Figs.6, 7, 8, and 9. The analytical relationswere derived using the actual proper-ties of each beam and for the fourcover concrete models discussed inthe assumptions. The strains whenvarious significant stages are reachedin the analysis are marked on theanalytical curves. The strains shownare:

e,, = strain in concrete in extremecompression fiber

e v = strain in prestressing steelnearest extreme fiber and

e 8 = strain in non-prestressed steelnearest extreme fiber

Comparison of Experimental andAnalytical Results

Comparison of the analytical andexperimental moment-curvature re-sults shown in Figs. 6, 7, 8, and 9 (andfor the other six beams given else-where 15 ) indicates that the analyticalcurves fit the experimental resultswell providing that the appropriatemodel for the behavior of the coverconcrete is selected. When the coverconcrete thickness is a small propor-tion of the section dimensions, the in-fluence of the cover behavior on themoment-curvature relation will besmall. In the beam sections shown inFigs. 6, 7, 8, and 9 the ratio of coverconcrete thickness to beam width was1.5/9 = 0.167, which is an upper limiton the ratio likely in beams in prac-tice.

Comparison of the experimental andanalytical moment-curvature relationsindicates that when a significantamount of non-prestressed compres-sion steel is present (for example, see

Stress Core concreteZ= tan g°32.2

f^=5500psi --- Cover concrete(37. 9 MPa) Z -tan 8 = 64.4

Strain.0.002 0.006 0.010 0.014

Fig. 10. Assumed stress-strain relationfor core concrete and cover concrete(Model 4).

Fig. 9), the influence of the cover con-crete behavior on the moment-curva-ture curve is very small. It has beenobserved previously 18 that the pres-ence of hoops at small spacing resultsin a plane of weakness between thecover concrete and the core concreteand tends to precipitate the spalling ofthe cover concrete at high strains.Conversely, with a large hoop spacingthe cover and core concrete will actmore monolithically and tend to fol-low the same stress-strain curve.

Inspection of the experimental andanalytical moment-curvature relationscompared in the figures shows that forthe beam of Unit 1 with a hoop spac-ing of 0.39 times the beam overalldepth, Model 4 (Z CO9e,= 2Z core ) gavethe best fit and Model 3 (Z C07er =Z ufCO1r;,,ed ) gave the next best fit. Forthe beams of the other units with ahoop spacing of 0.19 times the beamoverall depth, Model 3 (Zco,,erZunco„ri„ea) gave the best fit and Model4 (Zco ,e,. = 2Zcore ) and Model 2 (Zcover= Zcorei but cover concrete ignored atstrains greater than 0.004) also gavereasonable fits. In general, it appearsthat either Model 4 or Model 3 couldbe used in analysis with good results,that Model 2 is generally conserva-tive, and that Model 1 is generally un-conservative.

This analytical approach does notconsider deformations due to shear.The maximum shear forces present inthe beams of the test units were nothigh, the maximum nominal shear

compression steel ratios of p = p' =0.14, 0.57, 1.46, and 1.99 percent, re-spectively.

Analytical Results

58

stress V a/0.8hb present in the beamsbeing only .1.78 f,' psi (0.147 f,'MPa). The diagonal tension crackingwas not significant in the beam plastichinge zones and it was evident that forthese beams shear had little effect onthe moment-curvature characteristics.

specified Z values and ratio of coverconcrete thickness to beam sectiondimension.

Fig. 10 shows the idealized stress-strain curves for the concrete whenZ = 32.2 and Z = 64.4, and illustratesthe degree of ductility available fromconfinement corresponding to those Zvalues.

General AnalyticalMoment-Curvature Study

The effects of longitudinal pre-stressing steel and non-prestressedsteel content and distribution, trans-verse steel content and cover concretethickness, on the moment-curvaturecharacteristics of rectangular concretesections were studied analyticallywith particular emphasis on the be-havior in the post-elastic range at highcurvatures.

For all cases studied the ultimatetensile strain of the prestressing steelwas taken as 0.035, which was theminimum value obtained from the ex-perimental tests on the prestressingsteel associated with the tests re-ported.'5' 19 Also, the strain in the pre-stressing steel due to prestress alonewas taken as 0.0057, which was thestrain which gave an initial steel stressequal to 70 percent of the ultimatetensile stress.

The beams are assumed to be con-fined by rectangular hoops. For thosesections in which Z core = 32.2 isspecified, the confinement can be re-garded as being provided by % in. (9.5mm) diameter hoops at 3½ in. (89 mm)centers confining a 15 in. (381 mm)deep by 6 in. (152 mm) wide concretecore, and the cover concrete can beregarded as being that 1' in. (38 mm)thickness of concrete between thehoops and the edges of the 9 in. (228mm) wide by 18 in. (457 mm) deepsection. However, since the curveshave been plotted non-dimensionally,they apply generally to other geomet-rically similar sections with the

Effect of Content of PrestressingSteel on Ductility

The effect of prestressing steel,content* on the moment-curvaturecharacteristics, and hence the ductil-ity, of sections is of interest. Theanalytical moment-curvature relationsfor rectangular beam sections usingthe dimensionless coordinates (ph andMY fbh 2, and the assumed section pa-rameters are plotted in Figs. 11, 12,and 13. The cover concrete behavioris assumed to be described by coverconcrete Model 4 with Z eore = 32.2and ZC04er = 64.4. The abrupt reduc-tion of moment at high curvatures inFigs. 12 and 13 (and in some sub-sequent figures) occurs when the pre-stressing steel reaches its ultimate(fracture) strain.

Fig. 11 shows curves for a sectioneccentrically prestressed by one ten-don at a depth of 0.8h for a range ofprestressing steel contents specifiedasA 5/bh ratios.

ACI 318-77' and the UBC 3 requirethat for prestressed concrete beams ingravity load design the longitudinalsteel used to calculate the flexuralstrength should satisfy:

A,fj$lbdff -_ 0.3 (15)

where f1, is the prestressing steelstress at maximum moment.

This requirement is intended to en-sure that beams designed to resistgravity loading have some ductility.

*The term "prestressing steel content" ex-presses the ratio A,,/bh.


Aps /bh = 0.012 = 5,500psi (37.9MPa)0.8h fps= 234,500psi (1617MPa)0.24 0.010 h Model 4 Stress Block

A s Zcore = 32.20.20 0.008 Zcover = 64.4

• = Cracking Moment0.007 0.67b I 1psi = 0.00689MPo

0.16 0.006 Fib

M

f^ bh20.12 0.004

0.080.002

0.04

00 0.01 0.02 0.03 0.04 0.05

q5h

Fig. 11. Effect of content of prestressing steel on analytical moment-curvaturerelation with one tendon position.

Eq. (15) gives A„8/bh -- 0.0069 whend = 0.8h, f,' = 5500 psi (37.9 MPa),f = 234,500 psi (1617 MPa) and f 3 isas given by the ACI Code' and theUBC 3 for members with bonded pre-stressing tendons. A moment-curva-ture curve for A,,,/bh = 0.007 is plot-ted in Fig. 11 and it is evident thatthis code limitation on prestressingsteel area for flexural strength calcu-lations results in moderate ductilitywhich is probably satisfactory forgravity loading. However, these codesallow the steel area placed to exceedthat specified by Eq. (15), providedthat the excess steel area is not in-cluded in the flexural strength calcu-lations. It is evident that, if a greatersteel area than is specified by Eq. (15)is placed, a brittle flexural failure canactually occur, although at a highermoment, even though the additionalsteel is ignored in the flexuralstrength calculations.

For seismic loading, it is desirablefor the moment to be maintained near

the maximum value over a greatercurvature range. Hence, a more severelimitation on the maximum steel areashould be used, and this maximumshould not be exceeded even if anyexcess steel area is not included in theflexural strength calculations. Any ex-cess steel area will lower the sectionductility and the enhanced flexuralstrength may result in a shear failure.To ensure reasonable ductility inseismic design, it is suggested that the0.3 on the right hand side of Eq. (15)be replaced by 0.2. This would givean absolute limiting value of A,,/bh = 0.0046 in Fig. 11, which wouldresult in the section having signifi-cantly better ductility. Thus, for seis-mic design, the requirement for plas-tic hinge regions of beams when theprestressing tendons are concentratednear the extreme tension fiber shouldbe:

Aggff$lbdff -_ 0.2 (16)Figs. 12 and 13 show the moment-

curvature relations for sections con-

[ 1s

• = Cracking Moment f = 5,500psi (37.9MPa)Ipsi = 0.00689MPo fpu= 234,500psi (1617MPa)

Model 4 Stress Block

0.24 2core = 32.2A 5/bh = 0.020 Zcover = 64.4

0.20 0.018 0. h

0.016

ps/20.8h0.014 h

0.160.012 L

• 0.010

0.12 0.008 b

0.006 I I I

0.08 0.004 I I I I I I

......°.°°±..........0,04

I

MU'

0 0 0.01 0.02 0.03 0.04 0.05

q5h

Fig. 12. Effect of content of prestressing steel on analytical moment-curvaturerelation with two tendon positions.

f^ = 5500psi (37.9 MPa )•0.2h •

f = 234,500psi (1617MPa)O.Sh + p^

0.8h I • Aps/3 ' Model 4 Stress Block•7----=.?2.2

a24

0.20

0.16M

ff bh2

0.12

0.08

0.04

0 0.01 0.02 0.03 0.04 0.05Oh

Fig. 13. Effect of content of prestressing steel on analytical moment-curvaturerelation with three tendon positions.

PCI JOURNAUMarch-April 1980

61

centrically prestressed with two andthree tendons, respectively, for arange of A P8/bh ratios. These two fig-ures show that with prestressing steelpresent in the concrete compressionregion, an increase in prestressingsteel content does not lead to a reduc-tion in ductility. This is because, pro-vided that the top prestressing steel isrestrained from buckling in compres-sion by the surrounding confined con-crete and confining steel, the top pre-stressing steel can act as compressionreinforcement at high concrete com-pression strains, thus maintaining thecompression force at a position nearthe top of the section.

Successive cycles of reversed flex-ure during seismic loading may causedamage to concrete leading to buckl-ing of tendons. Hence, it does not ap-pear advisable to take advantage ofthe extremely high ductilities thatmay be available at very high concretestrains. It would appear reasonable torequire that all beam sections are ca-pable of reaching a specified curva-ture at a given extreme fiber compres-sive concrete strain.

This criterion means that at the ul-timate moment the neutral axis depthshould not exceed some limitingvalue. For example, Eq. (16) suggeststhat for beams with rectangular sec-tions having prestressing steel con-centrated near the extreme tensionfiber, the requirement should beA PgfjB -_ 0.2ffbd. Since is thetendon tensile force this means thatthe maximum possible compressiveforce in the concrete at the flexuralstrength is 0.2ffbd. The compressiveforce in the concrete is given by0.85f ab at this stage and hence themaximum possible depth of the rec-tangular concrete compressive stressblock is:

0.2f bda = 0.85f 'b

= 0.235d

But since d is approximately 0.8hand a is about 0.75c [when ff = 6000psi (41.4 MPa)], where c is the neutralaxis depth at the flexural strength,then

a -- 0.2h (17a)

or c -_ 0.25h (17b)

For sections with tendons at variouspositions down the depth of themember, it is difficult to set a limitingvalue of A psfpglbdff because tendonsat various levels result in sectionswith different moment-curvaturecharacteristics from the case when allthe tendons are placed near the ex-treme tension fiber. Rather thanstipulating different limiting valuesfor A pgfpglbdff for various tendon po-sitions in the section, it is more con-venient to require alh -_ 0.2 orclh -_ 0.25 for all sections.

This requirement achieves the sameend result for all sections since thecurvature at ultimate moment will al-ways be at least equal to that of thesection with all tendons placed nearthe extreme tension fiber. Eq. (17) hasbeen adopted by the FIP recommen-dations 7 and the draft New ZealandConcrete Design Code 6 for seismicdesign of prestressed and partiallyprestressed concrete sections.

Effect of Distribution ofPrestressing Steel on Ductility

In seismic design the reversals ofmoment in beams near column facesduring earthquake shaking will re-quire the section to have both nega-tive and positive moment strength,and hence tendons will exist nearboth extreme fibers of the section andnear middepth. Thus, the effect of thedistribution of the prestressing steelwithin the section, while maintaininga concentric steel arrangement, is avariable of interest.

Moment-curvature curves were de-rived for sections with from one to

62

0.12

0.10 N = 2N=3N=4

0.08 N=5M Aps/bh = 0.00696

f' bh 2 a1h f' = 5S00psi (37.9MPa) N = 1c0.06 a h 2 qps/N fpo= 234500psi (1617 MPa l

N Model 4 Stress Blockh A,5/N ' Zcore = 32.2

0.04

- A

S

/N Zcover = 64.4

^ N = Number of Tendonst b h/b =2

0.02 Ipsi = 0.00689MPa

0 0 0.01 0.02

Fig. 14. Effect of distribution of prestressinrelation.

five tendons symmetrically dis-tributed down the depth. The totalsteel content is the same for each ofthe five cases being A 58/bh = 0.00696(which corresponds to that of Unit1.15,19) The stress-strain behavior of thecover concrete is described by Model4, withZ,o ,.e = 32.2 and Zco„e,. = 64.6.

The analytical moment-curvaturecurves using the dimensionless coor-dinates 4h and Mlf,bh 2 are comparedin Fig. 14. The moment-curvaturecurves for N = 2 to N = 5, where N isthe number of tendons, are reasonablysimilar, and these sections are able tomaintain near maximum momentcapacity at high curvatures. For thesections with one tendon, N = 1, theinternal lever arm is smaller and thereare no tendons to act as "compressionreinforcement." Hence, when N = 1the section is more sensitive to anyincrease in the neutral axis depth dueto deterioration of the compressedconcrete, and the moment capacityreduces significantly at high curva-tures. Thus, for sections subjected to

PCI JOURNAL/March-April 1980

N 1 2 3 4 5a1 0.19 0.19 0.19 0.19

0.2 0.40 0.350.3 0.5 0.5 0.5a4 0.60 0.a5 0.81 0.81 0.81 0.81

0.03 0.04 0.05

Oh

g steel on analytical moment-curvature

seismic load reversals it is desirable tohave two or more prestressing tendonswithin the section, with at least onetendon near the top and one near thebottom edges of the section.

Effect of TransverseReinforcement on Ductility

A significant variable affecting theductility of prestressed concretemembers is the degree of confinementof the concrete in the compression re-gion provided by transverse rein-forcement in the form of rectangularsteel hoops (also referred to as stirrupties).

Fig. 15 shows analytical moment-curvature curves using the dimen-sionless coordinates (ph and M/f,bh2drawn for sections with three pre-stressing tendons and various trans-verse steel contents. The prestressingarrangement is similar to that of thebeam of Unit 1.15,19 The curves aredrawn for % in. (9.5 mm) diameterhoops at spacings varying betweens = 7 to 1 in. (178 to 25 mm) with 11/2

63

5 -0 —^

0 12 1in (25.4 mm)tin (50.8mm)3in (76.2mm)

0.10

0.08M

ç bhp I Aps/bh =0.00696

0.05 0.2h •ff =5,500psi(37.9MPo)0.5h - ^3 fpu = 234500psi (1617MPa)0.8h ps

h _ .' S=S 4 StrespSp ing

0.04 APs'3 s = Stirrup Spacingb = 9in (229mm)h = 18 in (457mm)

•/ b 1in = 25.4 mm

0.02 1psi= 0.00689MPa? 0 hoops8

0' i i i0 0.01 0.02 0.03 0.04

Oh

Fig. 15. Effect of hoop spacing on analytical moment-curvature relation.

s = 4in (101.6mm)Sin (127mm)6in(152.4mm17in(177. 8mm)

0.05

in. (38.1 mm) cover to the hoops. Asimilar model for the cover concretewas used for all cases, this beingModel 4 (Z co = 2Z co, e ), since it canbe applied with reasonable accuracyto beams with both small and largehoop spacings.

Fig. 15 indicates that some advan-tage is to be gained from extra coreconfinement. For the particular sec-tion being considered, with an overalldepth of h = 18 in. (457 mm), a hoopspacing of d/4 where d = 0.8h wouldmean a spacing of 3.6 in. (91 mm)which would lead to reasonable duc-tility. Unfortunately, the tendency forthe cover concrete to spall increasesas the hoop spacing decreases andthus the moment capacity of sectionswith close hoop spacing at high cur-vatures will not be as great as shownin Fig. 15, since Model 2 would bemore applicable to the cover concrete.

Some advantages of close hoopspacing not evident from this study ofbeams under monotonic loading arethe reduction of the crushing of the

core concrete between the hoopsduring cyclic loading, and the con-finement of the concrete surroundingthe prestressing steel in the compres-sion region thus preventing possiblesteel buckling.8'1N19

Effect of Cover Thicknesson Ductility

After a prestressed concretemember has sustained large curva-tures the cover concrete partly crushesand disintegrates. After several cyclesof reversed loading have been sus-tained in the inelastic range the coverconcrete tends to spall off completelyand does not contribute to the mo-ment-curvature behavior. In order tostudy the effect of the loss of the coverconcrete on the moment capacity andductility of prestressed concretemembers, the analytical moment-curvature relations of a prestressedconcrete section (with the same size,prestressing steel content and ar-rangement as Unit 1119) are plotted inFig. 16 using the, dimensionless coor-

64

=0— 4 in (6.35mmJ— 8 in (9.53mm)

fin (12.70mm)0.10

0.08M

I bh20.06

0.04

0.02

Aps/bh = 0.00696ff = 5,500psi (37.9MPO)fpu = 234500 psi (1617MP0)Modell Stress Blocktc = Cover to Stirrups1psi = 0.00689MPotin = 25.4 mm

b = 9in (229mm )h = 18in(457mm)

tc = 4 in (19.05mm)1in (25.4 mm)

1 i in (31.75mm)— 1-fin (38.10mm)

0.0 0.01 0.02 0.03 0.04Oh

Fig. 16. Effect of cover concrete thickness on analytical moment-curvature relation.

0.05

dinates i$h and M/ffbh 2 for values ofcover concrete thickness tc from 0 to1'/2 in. (38 mm). Model 2 is used todescribe the cover concrete behavior;that is, the cover concrete follows thesame stress-strain relation as the coreconcrete until a concrete compressivestrain of 0.004 is reached, and thenspalls and makes no further contribu-tion.

The curves in Fig. 16 show that themoment capacity of the section ofwidth 9 in. (229 mm) and overalldepth of 18 in. (457 mm) and with 11/2

in. (38 mm or 0.083h) cover thicknessdrops 20 percent when the cover con-crete spalls, while the section with %in. (9.5 mm or 0.021h) cover thicknessshows a reduction of only 3.5 percentin moment capacity subsequent to thespalling of the cover concrete. If thesections with a cover thickness of lessthan 1 1/2 in. (38 mm) are considered tobe scale models, then for larger mem-bers with cover thicknesses conform-ing to ACI 318-77 1 and to the UBC3the effect of the loss of cover concrete

is not as great as for a smaller memberconforming to the same cover re-quirements.

Effect of Prestressing SteelContent in PartiallyPrestressed Beams

Fig. 17 shows the analytical mo-ment-curvature relations derived for apartially prestressed concrete sectionwith equal quantities of non-pre-stressed steel in the top and bottom ofthe section (A $lbh = A8/bh = 0.0062)and for various values of prestressingsteel content expressed by the ratiosof A,,lbh. This figure shows that anincrease in strength is achieved withincreasing A pglbh without a great re-duction in ductility.

It can be concluded that providedsufficient non-prestressed compressionreinforcement is present, the inclu-sion of a central prestressing tendonin the section can benefit the behaviorof the beam by delaying cracking andincreasing the strength without muchreduction in ductility, at least up to


A^/bh =0.0060.16

0.005

0.004

0.12 0.003M

ff bh20.002

0.08 0.001

b0.13h

A5 _T 0.5hA5

-0.87h

AS

As /bh = As /bh = 0.0062f^ = 5S00psi (37.9MPal

0.04 fpu = 234,500psi (1617MPa)fy = 40,000psi (275MPaJModel 4 Stress Block1psi = 0.00589MPa

0'0 0.01

0.02 0.03

0.04 0.050h

Fig. 17. Effect of prestressing steel content on analytical moment-curvature relationof partially prestressed concrete beam.

the limiting alh ratio proposed in Eq.(17). Also, tests' 19 have demonstratedclearly that in seismic design it is de-sirable to have some non-prestressedlongitudinal steel present near theextreme fibers of the section since itleads to an increase in ductility byacting as compression reinforcementand results in an increase in energydissipation by fattening out the mo-ment-curvature hysteresis loops undercyclic flexure.

Conclusions

The following is a summary of themajor conclusions based on the resultsof the investigation of the moment-curvature behavior, and hence theductility, of prestressed and partiallyprestressed concrete beam sections:

1. Comparison of Analytical andExperimental Results forPrestressed and PartiallyPrestressed Concrete Beams

The analytical moment-curvaturerelations fitted experimentally mea-

sured moment-curvature relationswell, providing that the appropriatemodel for the stress-strain behavior ofthe cover concrete is chosen. When asignificant amount of non-prestressedcompression steel is present, or thethickness of cover concrete is small,the influence of concrete cover be-havior on the moment-curvature rela-tion is not significant. However, alarge thickness of concrete cover overthe hoops in prestressed concretebeams will normally have a significantinfluence on the moment-curvaturebehavior.

In the moment-curvature studies itwas found to be unconservative to as-sume that the concrete cover had thesame stress-strain behavior as theconfined concrete core, and overlyconservative to assume that the con-crete cover crushes and is lost atstrains greater than 0.004. The best fitwith the experimental results was ob-tained when the concrete cover wasassumed to have a stress-strain curvewith a falling branch slope of eitherthan of unconfined concrete or that oftwice the confined concrete.

66

2. Prestressed Concrete Beams(a) Effect of Content of Prestressing

Steel on Ductility — The analyticalstudy showed that moderately ductilebehavior can be achieved in pre-stressed concrete sections eccentri-cally prestressed up to the ACI 318-77and UBC limitation of:

A,J,S1bdf, _- 0.3

but that a more satisfactory limit forprestressing steel content, which en-sures more reasonable ductility forseismic design, would be given by:

A,J,,)bdf,' _- 0.2

In beam sections which are 'con-centrically prestressed with a symmet-rical arrangement of tendons, includ-ing tendons in both the top and bot-tom of the section, satisfactory ductil-ity can be achieved almost regardlessof the prestressing steel content pro-viding that buckling of the prestress-ing tendons does not occur. A suita-ble criterion for seismic design whichwould ensure ductility regardless ofthe positions of the tendons in thesection would be to require a -- 0.2hin all cases.

(b) Effect of Distribution of Pre-stressing Steel on Ductility — Whensubject to reversed moments, such asin seismic design, prestressed con-crete beams need to have both posi-tive and negative moment strength.The analytical study showed that pre-stressed concrete beam sectionswhich are concentrically prestressedwith two or more tendons placedsymmetrically in the section, includ-ing at least one tendon in the top andone in the bottom of the section,achieve a greater ductility than a sec-tion with the same prestressing steelcontent concentrated only at the mid-depth of the section. Hence, reversedmoment strength is best obtained byplacing some tendons near both ex-

treme fibers of the section rather thanplacing all the tendons at middepth.

(c) Effect of Transverse Reinforce-ment on Ductility — The analyticalstudy showed that the presence ofhoops had a beneficial effect on theductility of prestressed concretemembers- because of the confinementof the concrete. Such transverse steelis also necessary to prevent bucklingof longitudinal steel and to preventprogressive damage of core concretewhen subject to cyclic loading.

(d) Effect of Cover Thickness onDuctility — The analytical studyshowed that a small concrete coverthickness ensures that a prestressedconcrete beam section when subjectto large curvatures does not undergo asignificant reduction in the momentcapacity once the concrete coverspalls. From the viewpoint of ductil-ity, the concrete cover thicknessshould be made as small as possible.

3. Partially Prestressed ConcreteBeams

The analytical study showed thatthe introduction of a prestressing ten-don at the middepth of a concretesection which is doubly reinforcedequally top and bottom by non-pre-stressed steel, increases the crackingand flexural strengths of the sectionwithout significant reduction in duc-tility, at least up to the proposed limitof alh ratio. The presence of non-pre-stressed steel in the section is desira-ble in seismic design since it acts ascompression reinforcement, reducesdegradation of moment strength dur-ing cyclic loading and increases theenergy dissipation of the member.

Design Recommendations

Based on the findings of this study,plus the authors' judgment and ex-perience, the following design rec-ommendations are suggested for the


detailing of prestressed and partiallyprestressed concrete beams for duc-tility subject to seismic loading:

1. The content of prestressed plusnon-prestressed steel should be suchthat at the flexural strength at poten-tial plastic hinge regions, alh -_ 0.2,where a = depth of rectangular con-crete compressive stress block andh = overall depth of section.

2. In potential plastic hinge regionsof beams, the spacing of hoops shouldnot exceed d/4 or 6 in. (150 mm),where d = effective depth of sectionbutd need not be more than 0.8h. Thedistance between the vertical legs ofhoops across the section should notexceed 8 in. (200 mm) in each set ofhoops.

3. Except as provided in Item 4below, the prestressing tendons inpotential plastic hinge regions ofbeams should be grouted and shouldbe placed so that at least one tendon islocated at not more than 6 in. (150mm) from the beam top, at least onewithin the middle third of the beamdepth, and at least one at not morethan 6 in. (150 mm) from the beambottom.

4. When partially prestressedbeams are designed with non-pre-stressed steel reinforcement in theextreme fibers of potential plastichinge regions providing at least 80percent of the seismic resistance, theprestress may be provided by one ormore tendons located within the mid-dle third of the beam depth. In suchcases post-tensioned tendons may beungrouted, provided anchorages aredetailed to ensure that anchorage fail-ure, or tendon detensioning, cannotoccur under seismic loads.

Acknowledgment

The work described herein formed partof an investigation conducted in the De-partment of Civil Engineering of the Uni-versity of Canterbury, New Zealand, byK. J. Thompson during his PhD studiesunder the supervision of R. Park. The au-thors are particularly grateful for the finan-cial assistance of the University of Canter-bury, University Grants Committee,Ministry of Works and Development,Building Research Association (New Zea-land) and the New Zealand PrestressedConcrete Institute.

Note: A Notation section is given on thepage following the references.

Discussion of this paper is invited.Please send your comments to PCIHeadquarters by Sept. 1, 1980.

68

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15. Thompson, K. J., "Ductility of Con-crete Frames Under Seismic Loading,"Doctor of Philosophy Thesis, Univer-sity of Canterbury, New Zealand,1975, 341 pp. plus appendices.

16. Kent, D. C., and Park, R., "FlexuralMembers with Confined Concrete,"Journal of the Structural Division,ASCE, V. 97, No. ST7, July 1971, pp.1969-1990.

17. Kaar, P. H., Fiorata, A. E., Carpenter,J. E., and Corley, W. G., "EarthquakeResistant Structural Walls—ConcreteConfined by Rectangular Hoops," Re-search and Development ConcreteConstruction Laboratories, PortlandCement Association, Skokie, 1976, 30PP.

18. Blakeley, R. W. G. and Park, R., "Pre-stressed Concrete Sections With Seis-mic Loading," Journal of the Struc-tural Division, ASCE, V. 99, ST8, Au-gust 1973, pp. 1717-1742.

10. Parme, A. L., "American Practice in 19.Seismic Design," PCI JOURNAL, V.17, No. 4, July-August 1972, pp. 31-44.

11. Hawkins, N. M., "State-of-the-Art Re-port on Seismic Resistance of Pre-stressed and Precast Concrete Struc-

Park, R., and Thompson, K. J., "CyclicLoad Tests on Prestressed and Par-tially Prestressed Concrete Beam-Col-umn Joints," PCI JOURNAL, V. 22,No. 5, September-October 1977, pp.84-110.

PCI JOURNAL/March-April 1980- 69

NOTATION

b

d

t c = cover concrete thickness 0uV„ = shear force 4>„

= total prestressing steel area= total area of non-prestressed ten-

sion steel= total area of non-prestressed com-

pression steel= depth of concrete rectangular

compressive stress block at flex-ural strength

= width of concrete section= width of confined core measured

to outside of hoops= neutral axis depth at flexural

strength measured from extremeconcrete compression fiber

= distance from extreme compres-sion fiber to centroid of tensionsteel but in the case of a pre-stressed concrete member notless than 0.8 times the overalldepth of the member

= modulus of elasticity of prestress-ing steel

= modulus of elasticity of non-pre-stressed steel

= concrete stress= concrete compressive cylinder

strength= prestressing steel stress= prestressing steel stress defined

in Fig. 4(b)= prestressing steel stress defined

in Fig. 4(b)= stress in prestressing steel at flex-

ural strength= ultimate strength of prestressing

steel= stress in non-prestressed steel= ultimate strength of non-pre-

stressed steel= tensile strength (modulus of rup-

ture) of concrete= yield strength of non-prestressed

steel= overall depth of concrete section= equivalent plastic hinge length= moment= number of prestressing steel ten-

dons= spacing of transverse reinforce-

ment

Z = slope of falling branch of concretestress-strain curve defined by Eq.(3)

Zco,.e = value of Z for core concreteZC01,r = value of Z for cover concreteA„ = ultimate displacementA,, = displacement at first yieldEg0C = concrete strain at 0.2f f on falling

branch of concrete stress-straincurve

E50c =-concrete strain at 0.5f f on fallingbranch of concrete stress-straincurve [ see Eq. (4)]

E50h = additional strain due to confine-ment of concrete at 0.5f f on fall-ing branch of concrete stress-strain curve

E50u = strain at 0.5fc' on falling branchof unconfined concrete stress-strain curve

Ec = concrete strainEcc = concrete strain at top of sectionE co = 0.002 = concrete strain at maxi-

mum stress fEp = prestressing steel strain€99 = prestressing steel strain defined

in Fig. 4(b)Eyc = prestressing steel strain de-

fined in Fig. 4(b)Epu = ultimate strain of prestressing

steelE8 = strain in non-prestressed steelEsh = strain in non-prestressed steel at

commencement of strain harden-ing

E = ultimate strain of non-prestressed

steelEu = strain in non-prestressed steel

first yield0„ = ultimate rotation9v = rotation at first yieldFi = displacement ductility factor =

AnlOyP = A,lbd for non-prestressed steel

P = A8/bd for non-prestressed steelPS

= ratio of volume of hoops to vol-ume of confined concrete core

0 = curvature, defined as the rotation

per unit length of member =ECC/C

= ultimate curvature= curvature at first yield

AmA,

A;

E,

E.

fcfe

Jp199

fac

f,,8

fpu

Jsfs,ft

fv

hisMN

70

ductility of prestressed and partially prestressed concrete beam sections

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