Seismic Resistance ofFrames Incorporating Precast

Prestressed ConcreteBeam Shells

D. K. BullDesign Engineer

Smith Leuchars Ltd.Consulting Engineers

Wellington. New Zealand

• Robert ParkProfessor and Head of

Civil EngineeringUniversity of CanterburyChristchurch, New Zealand

The use of precast concrete in build-ing frames has a number of attractive

features such as better quality control ofthe product and savings in formworkand construction time. The basic prob-lem in the design ofearthquake resistantbuilding structures incorporating pre-cast concrete elements is in finding aneconomical and practical method forconnecting the precast elements to-gether. The connection between theelements should ensure satisfactorystrength and stiffness against seismicloads and enable the structure toachieve the necessary ductility duringcyclic loading in the inelastic range.

Composite systems of concretebuildings, combining precast and cast-in-place (cast in situ) reinforced con-crete, have a number of advantages in

construction. The incorporation of pre-cast concrete elements has the advan-tage of high quality control and speed ofconstruction, and the cast-in-placereinforced concrete provides the struc-tural continuity and the ductility neces-sary for adequate seismic performance.

A building system which has becomepopular in New Zealand involves theuse of precast concrete beam shells aspermanent formwork for beams. Theprecast shells are typically pretensionedprestressed concrete U-beams and areleft permanently in position after thecast-in-place reinforced concrete corehas been cast. The precast U-beamssupport the self weight and constructionloads and act compositely with thereinforced concrete core when sub-jected to other loading in the finished

54

structure. The precast U-beams are notconnected by steel to the cast-in-placeconcrete of the beam or column,

The typical structural organization ofa building floor and frame system incor-porating the precast pretensioned U-beam units is shown in Fig. 1. Currentconstruction practice is to support theU-beam units on the cover concrete ofthe previously cast reinforced concretecolumn below, with a seating of 40 to 50mm (1.6 to 2.0 in.) and to place a pro-prietary precast concrete floor systembetween the U-beams of adjacentframes.

Some propping may be providedunder the ends of the U-beam units as aback-up measure in case the U-beamseating on the column should prove in-adequate to carry the construction load.Once the precast floor system is inplace, the reinforcement may be placed,and the in-place concrete cast, insidethe beam units, the topping slab and thecolumns of the next story. The section ofthe composite beam in the finishedstructure is shown in Fig. 2. Precastconcrete columns have sometimes beenused rather than cast-in-place concretecolumns.

The precast prestressed concreteU-beam illustrated in Fig. 2 has webstapered from the bottom to the top, toensure ease of removing internal form-work when precast. The inside surfaceis intentionally roughened, by the use ofa chemical retarder and the removal ofthe surface cement paste, to facilitatethe development of interface bond be-tween the precast U-beam concrete andthe cast-in-place concrete core.

The U-beams are pretensioned withseven-wire strands and are designed tocarry at least all of the self weight andimposed loads during construction.Note that the strands terminate in theend of the U-beam and hence are notanchored in the beam-column joint re-gions of the frames.

Initially in New Zealand, precast con-crete U-beams were principally used in

SynopsisThe performance of cast -in-place

reinforced concrete moment resistingframes incorporating precast pre-stressed concrete U-beam shells,subject to seismic loading, is investi-gated. The precast beams act aspermanent formwork and are not con-nected by steel to the cast-in-placeconcrete of the beam or column.

Three full scale beam-exterior col-urnn subassemblies were tested todetermine their seismic performancecharacteristics when plastic hinge re-gions occur in the beams adjacent tothe columns.

Provisions for the seismic design ofsuch composite structures are dis-cussed and additional design recom-mendations based on the test resultsare proposed where necessary. Anumerical design example is includedto illustrate the design approach.

the construction of low rise buildings inwhich the horizontal seismic loads areresisted primarily by other elementssuch as totally cast-in-place reinforcedconcrete structural walls or frames. Anearly example of this type of construc-tion is the Karioi Pulp Mill' (see Fig. 3).

Recent trends have seen this form ofcomposite beam construction used inmultistory moment resisting reinforcedconcrete framed structures. In this ap-plication, the composite beams are re-quired to be adequately ductile to act asthe primary energy dissipating membersduring seismic loading. Doubts havebeen expressed by some designers andbuilding officials concerning the abilityof this form of composite construction tobe able to fulfill that demand.

This paper reviews seismic designconsiderations for frames with suchcomposite beams. The results of tests

PCI JOURNAL/July-August 198655

PropandTapp

Cos-in-BeamCotReinforci

in

Fig. 1. Construction details of a composite structural system (not al! reinforcementis shown).

conducted on three full scale compositebeam-exterior column subassemblies,subjected to simulated seismic loading,are summarized. Design provisionsbased on the test results are proposedand a numerical design example is in-cluded to illustrate the design approach.The results of the tests may be seen re-ported in more detail in Ref. 2.

SEISMIC DESIGNCONSIDERATIONS

In the design of structures for earth-quake resistance, a prime consideration.is to ensure that the structure is capableof deforming in a ductile manner whensubjected to several cycles of horizontalloading well into the inelastic range.This is because it is generally uneco-

nomical to design a structure to respondin the elastic range to the large hori-zontal inertia loads induced by thegreatest likely earthquake.

The recommended level of seismicdesign loads in codes is generally sig-nificantly lower than the elastic re-sponse inertia Ioads during severeearthquakes and the structure may berequired to undergo horizontal dis-placements which are four to six timesthe horizontal displacement at thecommencement of yielding of the frame.The ratio of the maximum displacementto the displacement at first yield iscommonly referred to as the displace-ment ductility factor.

Ideally, the design concept for mo-ment resisting frames should aim at dis-sipating seismic energy by ductileflexural yielding at chosen plastic hinge

56

Cost-in-place Concrete Cast-ir-placeBeam Core Concrete Topping

Precast 'ConcreteFloor Slab Precast PrestressedF

Concrete U-Beam Unit

Fig. 2. Section of composite beam in finished structure(reinforcement is not shown).

Fig. 3. Precast concrete U -beams used as permanent formwork for cast-in -place

reinforced concrete frames (Karioi Pulp Mill Building, New Zealand).

regions when the structure is subjectedto the seismic design loads. The rest ofthe frame should be made sufficientlystrong to ensure that it remains in theelastic range when flexural yielding oc-curs at the chosen plastic hinge loca-tions. This means ensuring that shearfailures and bond failures do not occurand that the preferred energy dissipat-ing mechanism forms.

Mechanisms involving flexuralyielding at plastic hinges are shown inFig. 4. If yielding commences in thecolumn before it occurs in the beams, a

column sidesway mechanism can formas illustrated in Fig. 4b. Such a "softstory" mechanism can make very largecurvature ductility demands on theplastic hinges of the critical story, par-ticularly in the case oftall buildings.

On the other hand, if yielding occursin the beams before it begins in the col-umns a beam sidesway mechanism, il-lustrated in Fig. 4c, can develop whichmakes more moderate demands on thecurvature ductility required at the plas-tic hinges in the beams and at the col-umn bases, even for tall frames. There-

PCI JOURNAL/July-August f986 57

• Plastichinge

Bending®®^ moment

(a) Moment resisting (b) Column sideswayframe mechanism

(c) Beam sideswaymechanism

Fig. 4. Moment resisting frame with horizontal seismic loading and possible mechanisms.

fore, for tall frames, a beam sideswaymechanism is the preferred mode of in-elastic deformation and a strongcolumn-weak beam concept is advo-cated to ensure beam hinging.

For frames with less than about threestories, and for the top story of tallframes, the curvature ductility requiredat the plastic hinges if a column side-sway mechanism develops is not par-ticularly high. Hence, for one- andtwo-story frames, and in the top story oftaller frames, a weak column-strongbeam concept can be permitted 4.S,6 Thisapproach would protect the compositebeams from damage during seismic mo-tions.

However, for tall frames where astrong column-weak beam concept isnecessary, the composite beams willneed to be designed for adequate duc-tility. Seismic design considerations formoment resisting frames when plastichinges form in the composite beams arediscussed in the following sections.

Flexural Strength of BeamsThe critical section for fkcxure in

beams in moment resisting frames sub-jected to gravity and seismic loading isat or near the beam ends. In frameswhere gravity loading effects dominate,the critical sections for positive momentdue to gravity plus seismic loading mayoccur in the beams away from the col-

umn faces. The critical negative mo-ment sections will always occur at thebeam ends. A distinctive feature of thebehavior of the composite beam-columnconnection shown in Fig. 1 is that theprestressing strands of the precast con-crete U-beam terminate at the end of theU-beam and hence are not anchored inthe beam-column joint core.

The negative moment flexuralstrength at the end of the compositebeam will be aided by the presence ofthe U-beam since the bottom flange ofthe U-beam will bear in compressionagainst the cast-in-place column con-crete (see Fig. 5b). Hence, the upperlimit of the negative moment flexuralstrength at the ends of the beam will bethat of the composite section. However,should the beam end bearing on thecolumn concrete and/or the interfacebond between the cast-in-place and pre-cast beam concrete break down duringseismic loading, the available negativemoment flexural strength will reduce toless than the composite section value.The lower limit of negative momentflexural strength at the beam ends is thatprovided by the cast-in-place reinforcedconcrete core alone. The negative mo-ment flexural strength away from theends will be that due to the compositesection.

The positive moment flexural strengthat the end of the beam will be providedonly by the longitudinal reinforcement

58

and the cast-in-place concrete in thebeam core and slab topping (see Fig.5a). Away from the beam ends there willbe some contribution from the precastprestressed U-beam to the positive mo-ment flexural strength, but a full contri-bution from the prestressing strands(and hence full composite action of thesection) can only occur at a distancegreater than approximately 150 stranddiameters from the beam end, which isthe order of length required to developthe tensile strength of the strand.

Hence, the dependable negative andpositive flexural strengths of the com-posite beam at the beam ends should betaken as that provided only by thecast-in-place reinforced concrete beamcore.

Plastic Hinge Behavior ofBeams

The length of the plastic hinge regionin the beams is of interest in seismic de-sign since the plastic hinge length has asignificant effect on the level of dis-placement ductility factor which can beachieved by frames. Longer plastichinge lengths lead to greater availabledisplacement ductility factors for agiven ultimate section curvature .3 In aconventional reinforced concrete framethe length of the beam region overwhich the tensile reinforcement yieldsis typically about equal to the beamdepth and several flexural cracks willform in that region.

In the composite system consideredhere, in which there is no connection bysteel between the end of the precastprestressed U-beam and the column, thelength of the region of reinforcementyielding at the end of the compositebeam when the bending moment is pos-itive will be less than for a beam in aconventional reinforced concrete frame.This is because when positive momentis applied, the first crack to form will beat the contact surface between the endof the precast U-beam and the face of the

column. It is possible that positive mo-ment plastic rotations will concentrate atthis one cracked section, since signifi-cant cracking may nut occur in the flex-urally stronger adjacent composite sec-tions during subsequent loading.

if the flexural cracking in the beamduring positive bending moment doesconcentrate at the column face, the con-sequence would be higher beamcurvatures in the plastic hinge regionthan for conventional reinforced con-crete members. Hence, the concretethere would be subjected to highlocalized compressive strains and thelongitudinal reinforcement in the beamthere would suffer high localized plastictensile strains which would perhapslead to bar fracture when significantplastic hinge rotation occurs. Further,the extensive widening of that crack atlarge plastic hinge rotations may meanthat the shear resistance mechanism dueto aggregate interlock along the (verti-cal) crack will break clown, leading tosliding shear displacements along thatweakened vertical plane.

These opinions concerning the plastichinge behavior during positive momenthave resulted in reservations being ex-pressed by some designers about theperformance of this type of compositebeam when required to act as primaryenergy dissipating members duringseismic loading.

The possible shortening of the lengthof the region of reinforcement yieldingonly applies when the beam moment ispositive. When the beam moment isnegative, the behavior should be similarto a conventional reinforced concretebeam, since the top of the cast-in-placeconcrete core does not have the precastU-beam surrounding it and the plastichinge region should be able to spreadalong the beam.

One possible approach, aimed at im-proving the plastic hinge behavior dur-ing positive moment, would be to con-struct a composite beam in such a man-ner that in the potential plastic hinge re-

PCI JOURNAUJuly-August 1986 59

ngeu- bear,

of

Diagonalcompressionstrut

Tt Cc CsICSIv1'kcc1

b +-yen,

CS w V

(a) Positive Bending Moment Applied to Beam

DiagonalCs Cc T' compressra

strut

h

r

ycof

c>(_Vbfr \_j2

T"' C ' S Flange ofShallow U-beamcompressionstrut

lb) Negative Bending Moment Applied to BeamNote: Not all reinforcement is shown.

Fig. 5. Internal forces acting on a composite beam-exteriorcolumn joint core during positive and negative beam moments.

gions at the ends of the beam the bond atthe interface between the precast pre-stressed U-beam and the cast-in-placeconcrete core is intentionally elimi-nated. The effect of such a detail wouldhe to allow the plastic hinge region tospread along the cast-in-place concretebeam core without hindrance from theU-beam, and so avoid the possible con-centration of the beam plastic hinge ro-tation in the region close to the end ofthe beam.

In the plastic hinge regions of thebeams the reinforced concrete cast-in-place core should have longitudinal andtransverse reinforcing steel which isdetailed according to the seismic designprovisions for reinforced concrete duc-tile frames. This typically means a limi-tation on the maximum area of tensionsteel, the presence of compression steelwith an area of at least one-half of thearea of tension steel, and stirrup tieswith a close spacing so as to confine the

60

compressed concrete and to preventbuckling of longitudinal bars and shearfailure.

In the New Zealand concrete designcodes the spacing of stirrup ties in thepotential plastic hinge regions of beamsis required not to exceed the smaller ofone-quarter of the effective depth of thebeam or six longitudinal bar diametersor 150 mm (6 in.). The potential plastichinge region is taken to extend over alength equal to twice the overall beamdepth.

Shear Strength of BeamsIn the plastic hinge regions at the

ends of composite beams the cast-in-place reinforced concrete core will needto resist all the applied shear forcealone, if the bond at the interface be-tween the precast U-beam and thecast-in-place concrete breaks down or ifthe bond is intentionally eliminated.Therefore, the beam core should be de-signed to have sufficient transversereinforcement to resist the design shearforce, using the seismic design provi-sions for reinforced concrete ductileframes. Away from the ends of the beam,the whole composite section may beconsidered to provide shear resistance.

In order to avoid a shear failure, andhence to ensure that ductile plastichinging of the beams can occur duringsevere seismic loading, the design shearforce for the beams should be that as-sociated with the likely beam over-strength in flexure. To calculate thelikely upper limit of flexural over-strength of the beam, composite actionshould be assumed in plastic hinge re-gions where negative moment isapplied, since the flange of the U-beamcan act as the compression zone of thecomposite member, as previously dis-cussed.

However, for positive bending mo-ment in plastic hinge regions at the endsof the member, only the cast-in-placereinforced concrete beam core need be

considered. If positive moment plastichinges form away from the beam ends,the composite section flexural strengthshould he used if the interface shear andstrand development length require-ments are satisfied.

The stirrup ties provided in the po-tential plastic hinge regions of thebeams should be capable of resisting theentire design shear force by truss actionalone, since the shear carried by theconcrete, Vr , diminishes during severecyclic loading. That is, V. tends to zerodue to a breakdown in the shear trans-ferred by dowel action of the longitudi-naI bars, by aggregate interlock, andacross the compression zone.

Interface Shear TransferBetween Precast U-Beam andCast-in-Place Concrete Core

Composite action of the beam canonly occur if shear can be transferredacross the interface between the ad-joining precast and cast-in-place con-crete surfaces with practically no slip.Shear stress is transferred across theinterface of concrete surfaces by con-crete adhesion, interlock of matedroughened contact surfaces, and friction.

Friction is reliant on a clamping forceorthogonal to the contact plane. In thecomposite beam detail, reinforcementdoes not cross the contact surface andtherefore does not provide a clampingforce. Some small clamping force maybe generated on the side faces of thecast-in-place concrete core by the U-beam webs resisting dilation caused byrelative shear movement along theroughened contact surfaces. Neverthe-less, it would seem appropriate to ignorefriction and to rely only on shear transferby adhesion and interlock of the matedroughened contact surfaces.

The imposed shear stresses at theinterface of the contact surfaces of theU-beam and the cast-in-place concretecore are the summation of stresses froma number of sources. The imposed hori-

PCI JOURNAUJuIy-August 1986 61

zontal shear stresses at the interface ofcontact surfaces between the U-beamand cast-in-place concrete core duringpositive bending moment arise from thetransfer of the prestressing steel tensionforce from the U-beam to the core, andduring negative bending moment arisefrom the transfer of the reinforcing steelforce from the core to the U-beamflange.

The horizontal interface shear stresscould be found from V. Ibc d, where

VV = vertical shear force at factored(ultimate) load

bn = total width of interface (two sidesplus bottom surface) and

d = effective depth of compositesection

This is a simplistic approach to themore complex real behavior of the U-shaped interface.

The imposed vertical shear stresses atthe interface during service loadingarise from the superimposed live loadsbeing supported by the floor system.The service live loads need to be trans-ferred from the U-beam unit, on whichthe floor system is seated, to the cast-in-place concrete core by vertical shearstresses across the interface (see Fig. 2).The self weight of the U-beam unit, theprecast concrete floor system, and thecast-in-place concrete core and floortopping during service loading are car-ried by the U-beam alone and thereforewill not cause vertical shear stresses atthe interface.

However, the transfer of vertical shearstresses across the interface will bemore critical at the factored (ultimate)load if the end support of the U-beam inthe column cover concrete is lost duringseismic loading. In that case the verticalshear stresses will arise from the selfweight of the U-beam, the precast floorsystem and the cast-in-place concretecore and floor topping, as well as fromthe live loads. This more critical case atultimate should he used to determinethe design vertical shear stress at theinterface.

In the New Zealand concrete designcodes an interface shear of 0.55 MPa (80psi) is permitted at the factored (ulti-mate) load for interfaces that have nocross ties, but have the contact surfacescleaned and intentionally roughened toa full amplitude of 5 mm (0.2 in.). A de-sign approach to check interface sheartransfer when the inside face of the pre-cast U-beams have been so roughenedwould he to find the vector sum of theimposed design horizontal and verticalshear stresses at the interface at the fac-tored (ultimate) load and to ensure thatit is less than 0.55 MPa (80 psi).

ColumnsSeismic design provisions for rein-

forced concrete ductile frames shouldbe used to determine for the columnsthe longitudinal reinforcement requiredfor flexure and axial load, and the trans-verse reinforcement required for shear,concrete confinement and restraintagainst buckling of longitudinal bars.

For tall frames a strong column-weakbeam concept is adopted, in order toprevent as far as possible a columnsidesway mechanism (soft story) fromoccurring during a major earthquake.Hence, the column bending momentsfound from elastic frame analysis for thecode factored (ultimate) load combina-tions need to be amplified to give ahigher column design moment, to takeinto account the likely beam over-strength in flexure, the higher mode ef-fects of dynamic loading which cancause much higher column momentsthan calculated from code static loading,and the possible effect of seismic load-ing acting along both principal axes ofthe building simultaneously,3.4.5.6

Similarly, the column shear forcesfound from elastic frame analysis for thecode factored (ultimate) load combina-tions need to be amplified to give ahigher design shear force so as to avoidthe possibility of brittle shear failure ofthe columns, Transverse reinforcement

62

in the column ends is also necessary toprovide flexural ductility there, sincethe amplified column design momentsmay not be sufficiently high to eliminatethe possibility of some column hinging.In particular, a transverse bar spacing ofnot more than six times the longitudinalbar diameter to prevent prematurebuckling of compression steel is an im-portant requirement.

Beam -Column JointsThe design shear forces for the

beam-column joint cores can be basedan the overstrength internal forces frombeams.

During negative bending moment,the greatest beam flexural strengtharises from composite action when theprecast U-beam flange transfers most ofthe compressive force in the beam to thejoint core by direct bearing against thecolumn. Then both the upper and lowerlayers of longitudinal reinforcement inthe beam may be in tension. A joint corediagonal compression mechanism in-volving two struts which transfer part ofthe joint core forces is shown in Fig, 5b.

One strut forms between the bends inthe upper tension steel and the lowerconcrete compression zone. The otherstrut forms at a shallow angle to the hor-izontal between the bends in the lowertension steel and the lower compressionzone. Should the flange of the precastU-beam cease to transfer compression tothe column during seismic loading, thenegative beam moment will be due tothe cast-in-place concrete core aloneand the joint core behavior will be thatof a conventional reinforced concreteframe.

During positive bending moment, thecast-in-place reinforced concrete alonetransfers the beam forces to the beam-column joint core. Hence, for positivemoment in the beam the joint core be-havior is that of a conventional rein-forced concrete frame. A diagonal com-pression strut mechanism which trans-

fers part of the joint core forces is shownin Fig. 5a.

It is apparent that the code approachfor the design of cast-in-place reinforcedconcrete beam-column joints could beused ignoring forces from possible com-posite beam action. That is, the designhorizontal joint core shear forces couldbe found for the cast-in-place concretebeam acting alone. This assumption isobvious for positive beam moments butis an approximation for negative mo-ments, However, for negative beammoments the upper layers of bars intro-duce the horizontal joint core shear forceover the greatest part of the core depth.The horizontal shear force introducedby the lower layers of bars may be as-sumed to he equilibrated by the veryshallow diagonal compression strutshown in Fig. 5b if those bars are in ten-sion.

The joint core mechanism resistingthe applied forces is made up partly bythe diagonal compression strut mecha-nisms described above and partly by atruss mechanism involving transversehoop reinforcement and intermediatecolumn bars, During cyclic loading inthe inelastic range the joint core sheartransferred by the diagonal compressionstrut mechanism decreases, mainly dueto the presence of full depth cracking inthe beam at the column face, and theshear transferred by the truss mecha-nism increases.s'fi

TEST PROGRAM

Three full-scale composite beam-exterior column units have been tested2to assess the seismic performance char-acteristics of the composite frame sys-tem described. The overall dimensionsof the units are shown in Figs. 6, 9 and10. For ease of construction of the units,the T-beam flanges typically resultingfrom the presence of the cast-in-placeconcrete floor topping were not mod-elled.

PCI JOURNAUJuIy-August 1986 63

EN rn ,- 11

T 0r1('1

L L2645

450

;. O

CC('TlnM e-A

^25p ^ 250

— I Cast in placedc) Beam Vic'

Pr¢caSt

a U Beamv

:;

Opp 1 400

cc TIn1u Pct GF^rTIrmJ ('-C'

Fig. 6. Elevation and sections of test units (dimensions in mm).Note: 1 mm = 0.0394 in.

Fig. 7. View of beam and column reinforcement in place during construction of Unit 1.

M

Debonding moteriol3.5mm thick foam

rubber

Length ofdebonded region=depth of cast-in-placecore +U-beam seating

U-beam seating40 -50mm

Lower column

Fig. 8. Method of debonding potential plastic hinge region of Unit 3.

All units were designed using theNew Zealand concrete design codeswith the addition of the suggested sup-plementary seismic design recommen-dations where necessary as discussed inthe previous section. The strength re-duction factors were taken as di = 1 in allcalculations and the overstrength factorfor the longitudinal beam reinforce-ment, used for the calculation of the de-sign shear forces, was taken as 1.25.

Details of Test UnitsUnit 1 was detailed using code provi-

sions for seismic loading, with a poten-tial plastic hinge region in the beam ad-jacent to the column face. Unit 2 was notdetailed for seismic loading, being de-signed using code provisions for gravityloading only. Unit 3 was detailed usingcode provisions for seismic loading andwas identical to Unit 1 in all respectsexcept that the interface between theprecast U-beam and cast-in-place con-

crete core in the potential plastic hingeregion was deliberately dehonded in anattempt to improve the plastic hinge be-havior. The details of the reinforcementin all units is shown in Figs. 9 and 10.

The interior surfaces of all precastU-beams had been roughened with anamplitude typically of 3 mm (0.12 in.).This surface roughness was achieved bychemical retarding of the interior sur-face after initial set and then removal ofthe surface cement paste from aroundthe aggregate by washing with waterand wire brushing.

The in-place concrete of the units wascast in the same orientation as for a pro-totype structure and according to antici-pated site practice. There were twopours of in-place concrete for each unit.First, the lower column was cast up tothe height where the precast U-beamwould be seated on it. The precast U-beam was placed on the edge of the topsurface of this column pour when con-crete strength was gained (see Fig. 7). In

PCI JOURNAL/July-August 1966 65

tn

^7f V00

O

C)0)

R10 hoop

ruL2tie

Stirrups5-R16 3-R16 0

in

2 00GD

o= ^

-u^1 9oa°-

dl

^! N^ LQOsa)

6 -R10 Stirrup ties 13 -RIO Stirrups ties200 crs 120 a 100 crs

r I I r

—U- Beam— L ^2645

RIO 450250 N I

hoops 1^ 7,67,

4-D2

o

iI4PR10O stirrup

^o Q 4-D24

u Beorn2-H 20 3-HD16 400 Q

bars bars each side

SECTION 3 1 400

83 206 83

2 -7.9mmstrands

R6 stirrups oam) 225 crs 0_ a

3-12-5mmstrands 400

SECTION OF U$EAM UNIT SECTION 2 SECTION 1

Fig. 9. Details of reinforcement of Units 1 and 3 (1 mm = 0.0394 in.).

C)_C0C

z

CC

mrn

LLV

10

murt+ CLN aC

^ ca i. v

' 4UU '

SECTION 1

M Fig. 10. Details of reinforcement of Unit 2 (1 mm - 0.0394 in.).

Table 1. Measured Properties of Steel and Concrete.

Reinforcing Steel

Bar designation RIO R12 D20 D24 HD16 HD20

f M Pa 336MPa 467

311463

285437

308469

402789

444704

Note: R = plain round intermediate strength steel, D = deformed intermediatestrength steel, HD = deformed high strength steel, No. – bar diameter in mm.

Seven-Wire Prestressing Strand

Diameter 7.9 mm 12.5 min

0.2 percent proof stress MPa 1808 1678, MPa 1926 1793

Concrete Cylinders: f f at Day of Testing Unit, MPa

Unit Lower column Upper column and beam U-beam

1 30.1 33.6 50.22 24.2 29.2 54.63 28.3 26.1 53.0

Note: 1 mm = 0.6394 in.; 1 MPa = 145 psi.

the next pour the beam core, beam-column joint and upper column werecast. Each unit was damp cured for notless than 7 days. The precast U-beamswere supplied by a precasting firth.

The debonded plastic hinge region inthe beam of Unit 3 was achieved by fix-ing a 3.5 mm (0.14 in.) thick sheet offoamrubber to the inside face of the precastU-beam, over a length equal to thedepth of the cast-in-place core, beforecasting the in-place concrete core. Fig. 8shows the details of the debonded re-gion.

Intermediate strength steel with aspecified yield strength of 275 MPa(40,000 psi) was used for the beam lon-gitudinal reinforcement and all trans-verse reinforcement. High strength steelwith a specified yield strength of 380MPa (55,000 psi) was used for the col-umn longitudinal reinforcement. Themeasured concrete and steel strengths

are shown in Table 1.The steel ratio provided by the top

longitudinal reinforcement in all thebeams was 1.83 percent based on the band d dimensions of the beam core. Thesteel ratio provided by the bottom lon-gitudinal reinforcement in the beamswas 1.42, 0.47 and 1.42 percent for Units1, 2 and 3, respectively, based on the band d dimensions of the beam core.

When the level of column axial loadwas 0.1 A,, f f where f f is the concretecompressive cylinder strength and A e isthe gross area of the column, the sum ofthe ideal flexural strengths of the col-umn sections above and below the beamwas 1.59 times the beam section idealflexural strength, where the idealflexural strength is that calculated usingthe actual strengths of the steel and con-crete arid assuming a strength reductionfactor 0 of 1.0. The Commentary of theNew Zealand concrete design codes re-

68

quires the design column bending mo-ments to be at least 1.8 times the bend-ing moments found from elastic frameanalysis for the code factored (ultimate)load combinations.

The Test RigFigs. 11 and 12 show the loading ar-

rangements and test rig. By alternatingthe directions of the load at the end ofthe beam, earthquake loading wassimulated. The loading cycles wereapplied statically. The axial load on thecolumn during the tests was held con-stant at 0.1f, A 0. Superimposed deadloads were also applied to the beams torepresent a 200 mm (7.9 in.) thick pre-cast concrete voided slab floor with 65mm (2.6 in.) thick concrete toppingspanning between frames in the proto-type building at 6.6 m (21.6 ft) centers.The positioning of the superimposeddead load was organized so that this loadwas applied only on the top horizontalsurface of the precast U-beam webs,which is how the slab system in a realstructure would be supported.

The New Zealand general design andloadings code' specifies that the perfor-mance of a ductile structural assembly issatisfactory for earthquake resistance ifit retains at least 80 percent of itsstrength after being subjected to a min-imum of four cycles of lateral loading toa displacement ductility factor of four ineach direction. The displacement duc-tility factor p. is defined as the ratio ofthe maximum displacement A to thedisplacement at first yield A.

Note that in these tests, A, was takenas 1.33 times the beam end deflectionmeasured at three-quarters of thetheoretical flexural strength of the unit.This definition of,, although arbitrary,gives a convenient reference first yielddisplacement which assumes elastic be-havior up to ultimate, as is also assumedin many dynamic analyses of the earth-quake response of structures.

In these tests a gradual increase in the

Directions of Pe^loads V arereversed duringloading cycles Vcot

0 ^^

VbeamVcoI

Pe

Fig. 11. External loads applied.

imposed displacement ductility factor p.= A/A„ was chosen, to enable the be-havior of the units to he examined atvarious ductility levels. The displace-ment controlled load cycles consisted oftwo cycles top. = ±1, four cycles toµ =

2, four cycles to p.+. _ ± 4, and twocycles to µ = ± 6.

Test ResultsThe beam end load versus beam end

deflection hysteresis loops measured forthe units are shown in Fig. 13. In thefigure the clashed lines marked +P{ and–Pf represent the loads at the theoreticalideal flexural strengths based on thebeam core alone, and –Pi is the load atthe theoretical ideal flexural strengthbased on the composite section, calcu-lated using the measured materialstrengths and assuming a strength re-duction factor ob of 1.0.

All theoretical ideal flexural strengthcalculations were conducted using thestrain compatibility-equilibrium ap-proach, incorporating the measuredstress-strain curves for the prestressingsteel and the reinforcing steel and as-suming a rectangular concrete compres-sive stress block as in the ACI BuildingCode7 and an extreme fiber concrete

PCI JOURNAUJuIy-August 1986 69

Table 2. Ratio of Maximum MeasuredBeam Moments to the Theoretical IdealFlexural Strengths of the Beams.

PositiveUnit moment J

Negativemoment

Mm., Mma: Mmu.rM, Mr ,yt"

1 1.12 1.33 0.902 1.09 1.43 1.03.3 1.03 1,04 0.75

M, = ideal flexural strength of cast-in-place corealone

Mi' = ideal flexural strength ofcomposite section(Both calculated using actual material strengthsand 4n – 11

compression strain of 0.003. Table 2shows the maximum moments reachedin the loading cycles as a percentage ofthe theoretical ideal flexural strengths.It is evident that in design the dependa-ble flexural strength of the beams at thecolumn face should be based an thecast-in-place concrete core alone.

Views of the units at the end of testingare also shown in Fig. 13. At the finalstages of testing there was spalling ofconcrete at the column face in the regionof the beam seating. This damage couldhave been prevented if a gap had beenformed between the sides and bottom ofthe precast U-beam and the cast-in-place column concrete.

cal resistance strain gauges on the lon-gitudinal beam reinforcement indicatedthat at the completion of the test thelength of beam over which yielding ofthe tension steel occurred was aboutone-half of the depth of the cast-in-placeconcrete core in the case of positivemoment and slightly greater than thedepth of the core for negative moment.Hence, the plastic hinge rotation did notconcentrate in the beam at the columnface and no undesirable concentration ofcurvature resulted. In Unit 1 the precastconcrete U-beam became extensivelycracked during the tests.

In Unit 3 the deliberate debonding ofthe interface concrete resulted in alonger measured plastic hinge length inthe cast-in-place concrete core and theprecast concrete U-beam was not dam-aged during the testing. Although bothUnits 1 and 3 displayed satisfactoryductile behavior during seismic loading,it may be considered that the dehondedconstruction used in Unit 3 is to be pre-ferred if it is considered important to re-duce the damage occurring to the pre-cast concrete U-beam shell during se-vere seismic loading.

No ill effects from beam shear wereobserved in Units 1 and 3. The maxi-mum applied nominal shear stress inthe plastic hinge region of the beams,assuming the beam core carried all theshear, was V,n4, /b,o d = 0.28 yJ f,' MPa(3.4 ^f ' psi) for Unit 1 and 0.25 v1'], MPa(3,0 v' f psi) for Unit 3. The stimip tiesin the beam cores of these two unitswere capable of carrying a nominalshear stress by truss action ofA,,f,/bu, s0.41 v77 MPa (4.9 fG psi) for Unit 1 and0.47 [ f^ MPa (5.6 T ,' psi) for Unit 3.

The transverse steel in the potentialplastic hinge regions of the beams wasgoverned by spacing required by thecode ductility provisions. The peakstress reached by the beam-column jointcore shear reinforcement, measured byelectrical resistance strain gauges, was83 percent of the yield strength for Unit1 and one hoop reached yield in Unit 3.

Units 1 and 3Fig. 13a and c show that Units I and 3,

which were designed for seismic load-ing, exhibited-very satisfactory strengthand ductility characteristics. In addition,the hysteresis loops were not pinchedand indicated satisfactory energy dissi-pation characteristics.

In Unit 1 there was a tendency for theplastic hinging to spread along thecast-in-place reinforced concrete corewithin the precast concrete U-beam,even during positive bending moment.The strain readings recorded by electri-

70

Fig. 12. View of test rig.

Diagonal tension cracks were observedin the joint cores (see Fig. 13a and c).The columns of both units remained inthe elastic range with limited crackingduring the tests.

Unit 2Unit 2 was not designed for seismic

loading. That is, the potential plastichinge region was not detailed with

closely spaced stimip ties for ductility.Extensive sliding shear displacementsoccurred along the vertical cracks in thebeam at the face of the column as thetest progressed, which was the mainreason for the pinched hysteresis loopswith low included area (see Fig. 13b). InUnit 2 the spacing of stirrup ties in thepotential plastic hinge region was 250mm (9.8 in.) compared with the 100 mm(3.9 in.) spacing used in Units 1 and 3.

PCI JOURNAL/July-August 1986 71

UNIT 1

BEAM ENDLOAD j 91113 1119 2325

+ P; =132 kN100 1

i-150 -100

-P,=73kN----T,-

6 `26 22 X10 8 4 -10024 218

-P;=112 kN

Fig. 13a. Beam end load versus beam end deflection and view of units atend of testing (1 kN = 224 Ib, 1 mm = 0.0394 in.).

50 100BEAM END DEFLECTION

(mm)

C)C-0Cz

CC

CCC

(0WO)

UNIT

BEAM ENDLOAD(kN)

1005 13^

27 25f 1 7517 23 + P = 70 kN

w

-100—Pi=61 kN — ----

242?0 -

-R' = 95 kN 1876

Fig. 13b. Beam end load versus beam end deflection and view of units atend of testing (f kN = 224 Ib, 1 mm = 0.0394 in.).

-50

—501Z

74

----to8 — --- —100

VA BEAM END

LOAD(kN) g

150 rrr ^r^19 2325

UNIT 3j 3 5 +P =132 kN

100 r.

BEAM END DEFLECTION (mm)

-P =73kN26 24 22 7412

- R'=112 kN 1816 108 -100

Fig. 13c. Beam end load versus beam end deflection and view of units atend of testing (1 kN = 224 Ib, 1 mm = 0.0394 in.).

Evidently, that spacing in Unit 2 wastoo large to assist dowel action of thelongitudinal beam bars to preventsliding shear from occurring at the col-umn face.

The maximum applied nominal shearstress in the plastic hinge region of thebeam, assuming that the beam core car-ried all the shear, was 0.29 , f,' MPa(3.5 vT psi). The stirrup ties providedwere capable of resisting a nominalshear stress of 0.18 r7 MPa (2.2psi) by the truss mechanism. Hence, theremaining nominal shear stress in theplastic hinge region needed to be car-ried by the concrete shear resistingmechanisms.

Also, in Unit 2 the area of longitudinalreinforcement in the bottom of thecast-in-place concrete core was only0.47 percent of the core bd, where b isthe beam core width and d is the beamcore effective depth. The positive mo-ment flexural strength of the beam coreat the column face was insufficient tocause cracking of the precast U-beamwhen acting compositely with the corein the region adjacent to the columnface. As a result, in Unit 2 the plasticrotation in the beam during positivebending moment was undesirablylocalized at a single crack at the columnface.

It is evident that the steel ratios of thebottom reinforcement in the core shouldbe higher than used in Unit 2 (it was 1.4percent of the core bd in Unit 1) in orderto improve the plastic hinge behavior.This points to the importance of havingsufficient longitudinal reinforcement inthe bottom of the cast-in-place concretecore to cause the precast part of thecomposite beam to crack during positivebending moment if bond exists betweenthe U-beam and the core.

The hoops in the beam-column jointcore remained in the elastic range dur-ing the tests. Diagonal tension crackingof the joint core occurred only duringnegative moment loading, since thepositive moment joint core shears were

too small to cause cracking, The columnremained in the elastic range with lim-ited cracking during the tests.

CONCLUSIONSThe theoretical considerations and

test results from three full-scale beam-exterior column test units subjected tosimulated seismic loading lead to thefallowing conclusions:

1. A convenient building system forseismic resisting frames involves thecombination of precast prestressed con-crete U-beams acting compositely withcast-in-place reinforced concrete beamcores. The incorporation of precast con-crete elements has the advantage of highquality control and speed of construc-tion, and the cast-in-place reinforcedconcrete provides the structural conti-nuity and ductility necessary for ade-quate seismic performance. Special at-tention to detailing in design is neces-sary to ensure that composite frames doperform in a ductile manner during se-vere seismic loading.

2. Code provisions do not cover allaspects of the design for ductile be-havior under seismic loading for thistype of construction. Proposals can bemade for additional design recom-mendations where necessary to take intoaccount the presence and directionalinfluence of the precast prestressedconcrete U-beam during severe seismicloading. A summary of design recom-mendations is given in Appendix A anda design example is provided in Appen-dix B.

3. When the composite beam isformed from a precast prestressed con-crete U-bean► and a cast-in-place rein-forced concrete core, and when thepretensioned strand in the U-beam isterminated at the beam end and is notanchored in the reinforced concrete col-umn, the following conclusions from thetest results with regard to seismic per-formance were reached:

(a) During seismic load reversals

PC! JOURNAL/July-August 1986 75

bond deterioration at the interface of thecast-in-place concrete core and the pre-cast concrete U-beam, along withcracking of the precast concrete U-beam, allow yielding of the longitudinalbeam reinforcement to spread along areasonable length of the cast-in-placebeam core. Hence, an undesirable con-centration of plastic hinge rotation doesnot occur at the beam crack at the col-umn face and satisfactory ductile be-havior is achieved. However, a suffi-cient quantity of longitudinal rein-forcement is required in the bottom ofthe cast-in-place beam core in order toensure that the flexural strength of thebeam core is great enough to cause thecomposite section to crack away fromthe beam end, thus permitting this de-sirable spread of yielding during posi-tive bending moment.

(b) Deliberate dehonding of theinterface of the cast-in-place and precastconcrete in the potential plastic hingeregion results in a larger potential plas-tic hinge length in the cast-in-place coreand no damage to the precast concreteU-beam during seismic loading. How-ever, since satisfactory ductile behaviorcan be obtained without debonding thepotential plastic hinge region, the extracost of the debonding operation can onlyhe justified if damage during severeseismic loading is unacceptable.

(e) The dependable flexural strengthsof the composite beams for both positiveand negative bending moment at thecolumn face were found to be that givenby the cast-in-place reinforced concretebeam core alone. The maximum flexuralstrengths required for beam over-strength considerations, for the de-termination of the design shear force,were found to be given by the compositebeam for negative moment and by thecast-in-place reinforced concrete beamcore alone for positive moment.

(d) Close spacing of stirrup ties in thepotential plastic hinge region of thebeam near the column face was essentialto prevent sliding shear of the beam

along a vertical crack at the column face.A spacing of stirrup ties in the beams of0.44h, was too large since it resulted inserious sliding shear along the verticalcrack at the column face, whereas aspacing of 0.18h, prevented slidingshear, where h, is the overall depth ofthe beam core.

(e) The design of shear in the poten-tial plastic hinge zone of the beamshould be based on shear carried by thebeam core alone, and the stirrup tiesshould be capable of resisting the totalshear force by truss action.

(f) The shear stresses at the interfacebetween the precast concrete U-beamand cast-in-place concrete core shouldbe checked. In the units designed forseismic loading the calculated imposedinterface shear stress, assuming com-posite action at the ultimate load, wasslightly in excess of 0.55 MPa (80 psi).The interior surface of the precast U-beams had been roughened with anamplitude of typically 3 mm (0.12 in.).The application of a limit of 0.55 MPa(80 psi) for the calculated interface shearstress outside the plastic hinge zone forthese units appeared to result in satis-factory behavior.

4. A general conclusion from the testsis that Units 1 and 3 which were de-signed for seismic loading would indeedbe satisfactory for use in ductile seismicresisting frames, but that Unit 2 whichwas designed without the special provi-sions for seismic loading would be suit-able for nonseismic resisting frames instructures where seismic loads are car-ried by walls or other structural systems.

ACKNOWLEDGMENTSThe financial assistance for the ex-

perimental work provided by the NewZealand Ministry of Works andDevelopment and the University ofCanterbury is gratefully acknowledged.The precast U-beam units weresupplied by Precision Precasting Ltd.,Otaki, New Zealand.

76

REFERENCES

I. "Karioi Mill," New Zealand ConcreteConstruction, V. 24, June 1980, pp. 18-21.

2. Park, R., and Bull, D. K., "Behaviour ofCast in Situ Reinforced Concrete FramesIncorporating Precast Concrete BeamShells Subjected to Seismic Loading,"Bulletin, New Zealand National Societyfor Earthquake Engineering, V. 17, No. 4,December 1984, pp. 223-250.

3. Park, R., and Paulay, T., Reinforced Con-crete Structures, John Wiley and Sons,New York, N.Y., 1975, p. 769.

4. Code of Practice for General StructuralDesign and Design Loadings for Build-ings, NZS 4203:1984, Standards Associa-

tion of New Zealand, Wellington, 1984, p.100.

5. Code of Practice for the Design of Con-crete Structures, NZS 3101:I982, Stan-dards Association of New Zealand, Well-ington, 1982, p. 127.

S. Park, R., "Ductile Design Approach forReinforced Concrete Frames," Farth-quake Spectra, Professional Journal of theEarthquake Engineering Research Insti-tute, V. 2, No.2, May 1986.

7. ACI Committee 318, "Building Code Re-quirements for Reinforced Concrete (ACI318-83)," American Concrete Institute,Detroit, Michigan, 1983, p. 111.

NOTE: Discussion of this paper is invited. Please submit yourcomments to PCI Headquarters by March 1, 1987.

PCI JOURNALJJuIy-August 1986 77

APPENDIX A - SEISMIC DESIGNRECOMMENDATIONS

Seismic design provisions for cast-in-place reinforced concrete frames in-corporating precast prestressed concreteU-beam shells, subjected to gravity andseismic loading, are not fully covered bydesign codes in New Zealand$ or theUnited States.' Supplementary seismicdesign recommendations are madebelow where necessary to take into ac-count the presence and directional in-fluence of the precast concrete U-beamduring severe seismic loading.

Al Flexural Strength of BeamsThe dependable flexural strength of

the beams should be at least equal tothat required by the factored (ultimate)gravity and seismic loads:

0 fi -- Ma (Al)

where the strength reduction factor 0 forflexure is 0.9, and the ideal (nominal)flexural strength should be calculated asfollows:

Edge of precastfloor system

I

bw

- r - 1 d

I

Outline ofU-beam unit

(a) Positive Moment (b) Negative MomentCAST-/N-PLACE CONCRETE BEAM CORE

h

(c) Positive Moment (dl Negative Moment

COMPOSITE BEAM

Fig. Al. Beam section of parameters.

78

Cast-in-placeconcrete core

'as tconcrete U-beam

(and

Region of degradationof bond between core

Column U-henrn

Fig. A2. Possible failure of precast concrete U-beamin debonded plastic hinge region of compositebeam after loss of seating at cover.

(a) At or near the beam ends, M i isbased on the cast-in-place concretebeam core alone.

(b) Away from the beam ends, M, isbased on the composite section wherethe interface shear between the precastand cast-in-place concrete can be trans-ferred satisfactorily and developmentlength requirements of the prestressingstrand are satisfied. The normal de-velopment length of the prestressingstrand (approximately 150 strand diam-eters) should be increased by 200 mm(7.9 in.) to allow for some degradation ofbond during reversals of severe seismicloading.

In complying with the code require-ments for the longitudinal steel ratios,and other design parameters, the beamsection dimensions are required. Thewidth of the compression face b, the ef-fective depth d, and the overall depth h,all depend on the moment direction andthe location of the section in the beam,and are defined in Fig. Al.

The precast U-beam should containsome top longitudinal steel so that it hasadequate negative moment strength atthe ends of the beam. This is to avoidpossible failure of the type shown inFig. A2 if the bond between the precastU-beam and the cast-in-place beamcore, and the seating of the precast beamat the column face are lost.

A2 Length of Potential PlasticRegion in Beams

In the design of beams that may formplastic hinges during seismic Ioading,the special detailing requirements forductility specified by the code shouldextend over a potential plastic hinge re-gion of length equal to twice the beamdepth. This length represents the likelyregion of yielding of the longitudinalreinforcement. It is suggested conser-vatively that the beam depth here bedefined as the full depth of the compos-ite beam section.

A3 Shear Strength of BeamsThe dependable shear strength of the

beams should be at least equal to thatrequired by the factored (ultimate)gravity loads:

0 u; b.d _- V,, (A2)

The strength reduction factor çb forshear is 0.85. The shear stress v { is thetotal ideal shear stress resisted by theconcrete mechanisms (vt ) and the trussmechanism of the shear reinforcement

A further seismic design requirementis that the ideal shear strength should heat least equal to that associated with theoverstrength beam moments and thefactored gravity loads:5

PCI JOURNAUJuly-August 1986 79

v, b,nd - V,; (A3)

In the potential plastic hinge regionsat the ends of the beams only the cast-in-place reinforced concrete beam coreshould be relied upon to provide shearresistance, and all design parameters arerelated to that core section. The dimen-sion b used in the shear equationscould be taken as the mid-depth widthof the in-place core and the dimension dis as defined in Fig. Ala and b.

In the potential plastic hinge regionthe shear reinforcement in the cast-in-place concrete beam core should be de-signed to resist all the shear force. Thatis, v, = 0 should be assumed in this re-gion, due to degradation of the concreteshear resisting mechanisms during re-versals of severe seismic loading.

Away from the potential plastic hingeregions shear will be resisted by thecomposite section and the shear forcecan be allocated to both the concreteand to the shear reinforcement. Thecomposite section values for d definedin Fig. Alc and d could be used, and b„can be taken as the width of the upperhalf of the cast-in-place beam core.

The determination of the shear forceV is illustrated in Fig. A3 for the casewhere the plastic hinges form at theends of the beam. The overstrengthbeam moments should be taken as:

MUO = a m, (A4)

where a = 1.25 includes the effect ofstrain hardening of reinforcement andthe possibility that the actual yieldstrength is higher than specified, andMI= ideal (nominal) flexural strength isbased on either the cast-in-place con-crete core for positive moment or thefull composite section for negative mo-ment. The use of these overstrengthmoments will ensure that the probabil-ity of shear failure is sufficiently low.

The transverse reinforcement in thepotential plastic hinge regions shouldalso be adequate to confine the com-

pressed concrete and to prevent pre-mature buckling of the compressionreinforcement. In potential plastic hingeregions where moment reversal cancause the nonprestressed longitudinalbars to yield both in tension and com-pression at the top and bottom of thebeam core, the maximum permittedcenter to center spacing of stirrup ties isthe smaller of 150 mm (5.9 in.) or sixlongitudinal bar diameters ord/4, whereconservatively d is the effective depth ofbeam core.'

Also, the yield force of the leg of thestirrup tie should be at least equal toone-sixteenth of the yield force of thecompressed longitudinal bar or bars it isto laterally restrain multiplied by(s/100), where s is the spacing of stirrupties in mm (1 mm = 0.039 in.).

A4 Interface Shear Stresses ofBeams

The composite action of the section isignored in the potential plastic hingeregion due to degradation of bond be-tween the precast and in-place concreteinterfaces during reversals of severeseismic loading. Nevertheless, theinterface shear stresses should bechecked to ensure that they are not ex-cessive. The interface shear stress hascomponents from horizontal and verticalshear stresses.

In general design the interface hori-zontal shear stress 0,d„ can be found fromthe factored (ultimate) shear force at thesection using:

Grth = V. (A5)4h14

where d is the effective depth of thecomposite section and b, is the total"width" of the interface.

As an approximation by = 2h{ + b;,where h{ is the depth of near vertical in-side face of the U-section and b, is thewidth of horizontal inside face of theU-section. Note that this is a simplistic

80

OM(A

M°MuB

(a) BendingMomentDiagram

VA (b) ShearVue Force

Diagram

w= D + 1.3LA per unit length g (c! Actions0

MuB onN'1uA ^„ - eleor span Member

Vo M .4 + Mu8 w`n vo MUA +M,°g_ w?nuA - + u(3

1n 2 In 2

Fig. A3. Calculation of shear force with gravity and seismic loading.

approach to the more complex real be-havior of the U-shaped interface. Thestrength reduction factor ¢ is 0.85.

In seismic design the interface hori-zontal shear stress can be found from:

n

°dk = b a (AG)L

where V is the vertical shear force as-sociated with the overstrength beammoments and the factored gravity loads.

The vertical shear stress at the inter-face, cdn , originating from the U-beamand floor system dead weight and theimposed live loading supported by thefloor .system, should also be considered.If the seating of the U-beam ends at thecolumn face is lost, through spalling ofthe seating concrete, these gravity loadsare transferred from the U-beam entirelyvia the interface to the in-place concretebeam core (and then out to the columnssupporting the beam core). In many

cases the vertical shear stress due togravity load ingv , is small enough to beignored.

The vector sum of the average hori-zontal shear stress v, and the verticalshear stress v,,, should not exceed thepermitted value. For interfaces whichhave no cross ties but have the contactsurfaces cleaned, are free of laitance,and are intentionally roughened to a fullamplitude of 5 mm (0.2 in.), the permit-ted shear stress could be taken as 0.55MPa (80 psi).

Then the shear stress requirement is:

v dh + v d, _- 0.55 MPa (80 psi) (A7)

A5 Columns and Beam-ColumnJoint Cores

The seismic design procedures usedfor ductile moment resisting reinforcedconcrete frames can be used for columnsand beam-column joint cores.

PCI JOURNAL/July-August 1986 81

APPENDIX B - DESIGN EXAMPLE

This Appendix contains a designexample showing how the design rec-ommendationscan be used to design atypical beam in a cast-in-place momentresisting reinforced concrete frame in-corporating precast prestressed concreteU-beam shells, subjected to gravity andseismic loading.

B1 The Structure

The beam to be designed is shown inFig. BI. It is an interior beam of a one-way moment resisting frame with col-umns at 6.0 m (19.7 ft) centers in thedirection of the frame. The one-wayframes are placed at 7 m (23.0 ft) centersin the building. A typical section of thefloor illustrating the concrete dimen-sions is shown in Figs. B2 and B3.

Concrete Strengths:

Precast U-beam and precast floor sys-tem: ff = 40 MPa (5.80 ksi).Cast-in-place beam core and slab top-ping: ff = 25 MPa (3.63 ksi).

Steel Strengths:

Nonprestressed reinforcing steel-= 275 MPa (40 ksi).

750mm

f255 m m! (10.41n)

Seven-wire prestressing strand of 12.5mm (% in.) diameter: f = 1774 MPa(257 ksi).

Service Dead Load:

Element Resulting load on beam

U-beam 3.1 kN/m (0.21 kips/ft)

Cast-in-placebeam core 4.3 kN/m (0.29 kips/ft)

Precast floorplus cast-in-place topping,4.1 kPa(85.2 lb/ftz ) 27.7 kN/m (1.90 kips/ft)

Services andceiling, 0.75 kPa(15.61blf ) 5.1 kN/m (0.35 kipi!R)

Total 40.2 kNlm (2.76 kips/ft)

Service Live Load

1.9 kPa(39.5 lb/ft) 13.3 kN/m (0.91 kips/ft)

Seismic LoadFrom the results of elastic frame

analysis, for the horizontal design seis-

750 mm(29.5in

Precast floor with cast-in-place topping

600mm precosf U -Beom23.&rn1

750mm(29.51n )Square

5.Om (19.7f1) column

Fig. B1. Elevation of typical interior beam of the moment resisting frame.

82

Cast-in-place concrete beamcore and slab topping 55mm (2.61n)

SGm 200mm(9.81n) (7. gin)

Precas 855mmfloor system

206 600mm 34.in)

Precasmm8'- 100mm (23-Sin)

(.3.9in) U -BeamII II

400mm(15.7in )

Fig. 82. Typical section of composite beam and floor(reinforcement not shown).

54mm

2-28 mm2-24mm dia.

12mm dia

dia

2-24mm dio.Stirrup tie _l SGmm

stirrup —_— 1 54 mmdia. 50mm

2-24mm dia

6mm

2-24mm dia.3-12 mm die. --- 2-28mmdra

seven wire144mm

prestressing I 70mstrands Note: -

All reinforcing steel

1mm = 0.0394inis Grade 275(fy >_ 40ksi )

Fig. B3. Details of reinforcement and prestressing steel inprecast U-beam and cast-in-place concrete beam core.

mic forces applied to the frames, thebending moments in the interior beamat the column centerlines are found tobe ± 477 kN-m (352 kip-ft).

B2 U-Beam Supporting DeadLoads

The precast U-beam during the con-struction of the structure supports itsown dead load plus the dead load of the

precast floor system plus the dead loadof the fresh cast-in-place concrete dur-ing construction, totalling:

D = 3.1 + 4.3 + 27.7= 35.1 kNhn (2.40 kips/ft)

The span will be propped during con-struction (see Fig. B4). The span be-tween props is:

Z=6.00-2x0.625=4.75m(15.6Ii)

The maximum service load moment at

PCI JOURNAUJuIy-August 1986 83

Precast U-BeamHi 750mm

(29.5/n)

Prop Prop Squarecolumn

625 mm ^ ^ 4.75m (15.6 ft) ^ I 625mm(24.6 in ) (24.6 in lL &Om (19.7ft)

Fig. B4. Elevation showing precast U-beam positioned on columns beforeprecast floor and cast-in-place concrete is poured.

midspan of the simply supported spanis:

M1AQ1. = D1218

= 35. 1 x 4.75218= 99.0 kN-m (73.1 kip-ft)

The minimum service load moment iszero near the ends of the span.

The gross section properties of the600 x 400 mm (23.6 x 15.7 in.) precastU-beam (see Figs. 82 and B3) are:Distance of centroidal axis xx from bot-tom fiber = 247 mm (9.72 in.)A = 0.126 m2 (195 in. 2 )

4.102 x 10-3 ma (9855 in.4)Zt = 0.0116 ms (708 in.)Z, = 0.0166 m3 (1013 in 3)e = 177 mm (6.97 in.), using three

12.5 mm ( 1/2 in.) diameter strands.Stress in strand after losses;

0.6f,„, = 0.6 x 1774 = 1064 M Pa (154 ksi)Total force in strands:

F = 3 x 93 x 1064 N = 297 kN (66.7kips)

Check Longitudinal Stresses WithMaximum Service Load:

Top fiber, compression:

AF Fe Mmax

= A - r + Zr- 297,000 - 297,000 x 177 +

126,000 0.0116 x 109

99,000,0000.0116 x 109

= 6.36 MPa (922 psi)

which is less than 0.45ff = 18 MPa(2610 psi).

Therefore, top fiber stress is satisfac-tory.Bottom fiber, tension:

297,000297,000 x 177+

fb - 126,000 0.0166 x U?

99,000,0000.0166x10"

= -0.44 MPa(-64 psi)which is greater than -0.5 T f _ -3.16MPa (-458 psi).

Therefore, bottom fiber stress is satis-factory.

Check Longitudinal Stresses WithMinimum Service Load:

Top fiber, tension:

_ 297,000 _ 297,000 x 177

126,000 0.0116 x 1011

= -2.17 MPa (-315 psi)which is greater than -0.5 V+' f,' = -3.16M Pa (-458 psi). Therefore, top fiber stressis satisfactory.Bottom fiber, compression:

_ 297,000 + 297,000 'c177jb 126,000 0.0166 X 1(P

= 5.53 MPa (802 psi)which is less than 0.45f f = 18 MPa (2610psi).

Therefore, bottom fiber stress is satis-factory.

84

Check Shear Strength

At h12 from the prop supports, shearforce due to 1.4D:

Va =1.4X35.1x `#.75 - 0.62 2

= 102 kN (22.9 kips)Imposed nominal shear stress at ulti-mate:

Vu

b.d

whered = 530 mm (20.9 in.)b,, = 400 - (250 + 206)12

= 172 mm (6.8 in.)= 0.85

Therefore,_ 102,000

V1 0.85 x 172 x 530

= 1.32 MPa (191 psi)Since the V.1M„ ratio is high near the

ends of the beam, the nominal shearstress resisted by concrete is:

u= 0.3(J +fa)= 0.3(440+2.36)= 2.61 MPa (378 psi)

Since v; is less than ye, use minimumshear reinforcement: A = 0.35bmslf,.

Use 6 mm (0.24 in.) diameter Grade275 (f„ = 40 ksi) stirrups.

2 x irx3" x 275s =

0.35 x 172- 258 rum (10.1 in.)

sm,=0.75h = 0.75x600= 450 mm (17.7 in.)

or -_ 600 mm (23.6 in.).Therefore, use 6 mm (0.24 in.) diameter

Grade 275 stirrups at 250 mm (9.8 in.)centers.

Check Dependable Flexural Strength

t P v, I -0.5 'lr J

=1774 1 - 0.5 3 x 93 x 1774

150x530 40]

= 1636 MPa (237 ksi)

Therefore:

c i M i = O.f.Ap [d - 0.59f,b

where = 0.9.¢Mr= 0.9x1636x3x93x

1530 - 0.59 1636 x 3 x 93140x150 J

= 199 kN-mwhich is greater than 1.4M,, = 139 kN-m(102 kip-ft). Therefore, flexural strength issatisfactory,

B3 Completed Beam SupportingGravity Plus Seismic Loads

Loading Cases

Loading cases for the factored (ulti-mate) loads are!

U=1.4D+1.7L (B1)

U = 1.OD + 1.3L + E (B2)

U = 0.9D +E (B3)

where

D = service dead load

L = service live load and

E = earthquake loading

Note that Eq. (B3) will not be as criti-cal for the beams as Eq. (B2), and henceEq. (B3) will not be considered further.

Bending Moments

It will be assumed that the dead loadis carried by the beam as if it is simplysupported at the column centerlines,and that the live load and earthquakeload are carried by the beam as part ofthe continuous frame. The bendingmoments for the loading case U = 1.4D+ 1.7L will be found with live load onall spans.

This assumes that sufficient momentredistribution can occur for the patternloading case (that is, the live load pres-ent on alternate spans only) to be disre-

PCI JOURNAL/July-August 1986 85

Column Column

Beam

6m (19.7ft)

-6Z8(-50.0) (-50.0)

-529 287(-390)1 \ (212)

Note:- For Eacting fromright to left 4 25

(3131

InteriorBeam

(a) Bending Momentsdue to LoadingCaseU = 1.ED+ 1.7L

kN-m(kip- ff l

(5) Bending Momentsdue to LoadingCaseU 1.0Dt 1.3L+E

kN-m(kip-ft)

Fig. B5. Bending moments for interior composite beam due to the factoredloading cases.

garded. This is a reasonable assumptionsince the sections are designed for duc-tility, and in any case the moments dueto seismic loading are much greater thanthe moments due to gravity load-ing.For load case U = 1.4D + 1.7L (see Fig.B5a):Negative moment at supports

_ –(1.7L)12/12= –(1.7x x 13.3) x 62112_ –67.8 kN-m (-50.0 kip-ft)

Positive moment at midspan= (1.4D)1 2/8 + (1.7L)12/24_ +(I.4x40.2)x6218+ (1.7x 13.3)x61124= 287 kN-m (212 kip-ft)

For load case U = 1.01) + I.3L +E (seeFig. B5b):Seismic load moment at the supports are

±477 kN-m (352 kip-ft).Negative and positive moments at sup-ports

= –(1,3L )1 2112 + 477= –(1.3 x 13.3) x 62/12 ± 477_ –529 kN-m (-390 kip-ft)

and 425 kN-m (313 kip-ft)Positive moment at midspan

= (1.OD)1 1/8 + (1.3L)12/24= (1.0 x 40.2) x 6218 + (1.3 x 13.3) x 62/24= 207 kN-m (153 kip-ft)The bending moment envelopes for

the loading cases for the design mo-ments at the column faces are shown inFig. B5c. It is apparent that themaximum positive and negative mo-ments due to 1.OD + 1.3L + E occur atthe column faces. Hence, the potentialplastic hinge regions of the beam are the

86

- 529 At column At column -529(-3901 r face face (-3901 Ic) Bending Moment-408(-3011 -408( 3011 Envelope due toLoading CasesU 1.1D+1.7L andU. 1.0D +1.3L +Ewith E Acting

/ / from Both Directions/^+ kN-r» (kip-ft/

` 287425 (212) 425(3131 At column At column (313J

face face126(37L.) '.26(31'.)

Column 5.25 m Column2.1m 2.1m ; 2-28 mm dio.

1.5m 1.5m I^ -24 mmdio.^2-24mm dia.s2-24 mm dio.

Column I 1I228mm^2-24 mm dio.

dface 1.73mL1.53m1_1.73m- 3 12 5 mm d1a.

Region whereI strandsPotential strand is fully Columnplastic developed facehinge region (dl Arrangement of

Longitudinal1mm . 0.0394•n Reinforcement(Transverserein forcernen tnot shown )

Fig. B5 (cont.). Bending moments for interior composite beam due to the factoredloading cases.

end 2h of beam, that is the end 2 x 668mm = 1.73 m (5.67 ft).

Design for Flexure

The maximum moments at the columnfaces of 426 kN-m (314 kip-ft) and –408kN-m (-301 kip-ft) are to be resisted bythe in-place concrete beam core alone.Use equal areas of top and bottom lon-gitudinal nonprestressed steel, and as-sume a steel couple to provide the mo-ment capacity, then:

A, =A – l(d –d')

whereM,, = 426 kN-m (314 kip-ft)

= 0.9 andfv – 275 MPa (40 ksi)

If the steel arrangement shown in Fig.B3 is used, comprising two 28 mm (1.10in.) diameter and four 24 mm (0.94 in.)diameter bars in the top and in the bot-tom, then with 40 mm (1.57 in.) cover at

PCI JOURNALJJuIy-August 1986 87

top and 30 mm (1. 18 in.) cover at bottom:d' = 101 mm (4.0 in.) andd = 674 mm (26.5 in.)Hence, the area of the longitudinal

reinforcement is:_ _ 426,000,000

AR-A8 0.9x275x(674 - 101)3004 mm2 (4.66 in.')

The available A, = A,' provided by thetwo 28 mm plus four 24 mm diameterbars is:

(2 x 615) + (4 x 452) = 3038 mm 2 (4.71in.2)

Therefore, the steel area furnished issatisfactory.

Note that the contribution of the steelarea in the slab topping is conserva-tively neglected,

Checks of Flexural Steel Content ofBeam:

(a) the maximum diameter of beamlongitudinal bar permitted to passthrough beam-column joint core tosatisfy bond requirements' is:

h 750d°

__ 25 = 30 mm (1.18 in.)25

which is greater than 28 mm (1.10 in.).Thus, the bar diameter is satisfactory.

(b) The tension steel content at thecolumn face is:

A, 3038 = 0.020bd 228x674

and the compression steel content p'0.020, where b here is taken at themidheight of the concrete core of theU-beam. Hence as required" p' > 0.5 p.Therefore, p' is satisfactory,

1+0.17I fl, -3IPmnx -

\ 7 ll! (i + '^100 .

= 0.022> 0.02, therefore, satis-factory.

Pm,= = 7/f„ = 7/275 = 0.025 > 0.02therefore, satisfactory.

(c) The maximum moment at midspanof + 287 kN-m (212 kip-ft) is to be re-sisted by the composite section.

The dependable positive momentflexural strength of the composite see-

tion at midspan due to prestressing steelalone (that is, not including the con-tribution of the bottom nonprestressedreinforcing bars in the core) is:

b = 250 mm (9.8 in.) ignoring slabtopping, and ff = 25 MPa (3.63ksi)

d = 865 - 70 = 795 mm (31.3 in.)

.fvu 1 - 0. p5 pff„_\f^ J

(1 3 x 93 1774)=1774-0.5 x J250x795 25

= 1686 MPa (244 ksi)Therefore:

0ME = 0f1A, (d - 0.59 ff A °J

(d 'b= 0.9 x 1686 x 3 x 93 x

(865-70-0.591686x3x93)25 x 250 J= 321 kN-m (237 kip-ft) >

287 kN-m (212 kip-ft)Hence, the prestressing tendons alone

can carry the positive moment atmidspan.

Curtailment of Longitudinal SteelFor 24 min (0.94 in.) diameter rein-

forcing bars, if t d = development lengthas specified by codes and d = 674 mm(26.5 in.), then:

For top bars I d + d = 1.41 m (4.63 ft)For bottom bars I d + d = 1.24 m (4.07

ft)For 12.5 mm (' in.) diameter seven-

wire strand:la = (1P, - 2f /3)d5/7

=(1636-2x 1064/3) x 12.5/7= 1.66 in (5.45 ft)

Allowing 200 mm (7.9 in.) for bonddegradation:

1 d +0.2m= 1.86m(6.10 ft)The top 24 mm (0.94 in.) diameter

longitudinal bars can be cut off withinthe span as shown in Fig. B5d, accord-ing to the bending envelope and the re-quired development lengths. Aminimum of two bars must extendthrough the span. The two 28 mm (1.10in.) bars extending through the span willneed to be lap spliced at midspan.

88

1.0O+ 1.3LWVWV Actions o

cxM; -( t Beam )aMi Beam due to1.OD+ 1.3L +

V°-Column face Overstrength

Beam Actions

4 11 Note - For E (92.51 acting from

right to leftShear ForceDiagram

109 kN (kips)(24.51

5.25m (17.2ft)

411 311 311 Off(92.5) (70.0) (70.0) (92.5) Shear Force

Envelope forE Acting fromBoth Directions

1.73m I 1-73m(5.67ft) (5.57ft1

Potential'plastic hinge regions

Fig. B6. Shear forces for interior composite beam due to factored gravityloading plus overstrength beam actions.

Curtailment of the bottom bars is notpossible. Over the end potential plastichinge regions of 2h = 1.73 m(5.67 ft) thecast-in-place beam core carries the mo-ment alone and the positive momentdoes not diminish sufficiently to permitcurtailment of bottom bars in those re-gions. The prestressing strand developsits strength fp, at 1.86 in (6.10 ft) fromeach column face and the strand alone iscapable of providing the required beamflexural strength over the central 1.53 m(5.02 ft). However, since id + d for thebottom bars is 1.24 m (4.07 ft), there isinsufficient length in this central region

to consider curtailment. Therefore, allbottom bars are extended through thespan with a lap splice at midspan (seeFig. B5d).

Shear Design for Overstrength BeamMoments

The maximum design shear force forload case involving seismic loading (seeFig. A3) is:

Vu _ a (IM; I +M^) + (1.OD + 1,3L)t11 n2

wherea = 1.25 = overstrength factor for

PCI JOURNAL/July-August 1986 89

flexurel„ = clear span = 6 - 0.75 = 5.25 m

(172 ft)D = service dead load = 40.2 kN/m

(2.76 kip/ft)L = service live load = 13.3 kN/m

(0.91 kip/ft)M = positive moment ideal flexural

strength of cast-in-place con-crete core alone

= A Ff9 (d - d'), since A, = A;= 3038x275 (674 - 101) N-mm= 479 kN-m (353 kip-ft)

M f = negative moment ideal flexuralstrength of composite section atface of column

= Aj,, (d - 0.5a), ignoring com-pression steel

where a = A,f"0.85f f b

3038 x 2750.85 x 40 x 400

= 61 mm (2.4 in.)< 100 mm (3.9 in.)

Therefore, the neutral axis lies in theflange of the U-beam.

Therefore:

M1 = 3038x275

1

1865 101 - 21 J

= 613 kN-m (452 kip-ft) 11

andV . 1.25(479+613) +

5.25(40.2 + 1.3 x 13.3) 5.25

2= 260 + 151 = 411 kN (92.5 kips)

The minimum shear force is given by260 - 151 = 109 kN (24.5 kN). The shearforce diagram is shown in Fig. B6.

Transverse Reinforcement in PotentialPlastic Hinge Regions

In each end 1.73 m (5.67 ft) region ofthe member, the shear is resisted by thebeam core alone and ye = 0 is assumed.The spacing of 12 mm (0.47 in.) diame-ter stirrup ties required for shear is:

s =Acf„d/V°= (2 x it x 61 ) x 27.5 x 6741411,000 =

102 mm (4.0 in.)

Use 12 mm (0.47 in) diameter Grade275 stirrup ties at 100 mm (3.9 in.) cen-ters, Checking spacing required for con-crete confinement and to prevent buck-ling of longitudinal steel:

(a) 150 mm(5.9 in.)>s = 100 mm and,therefore, satisfactory

(b) Six longitudinal bar diameters= 6x24= 144mm(5.7in.)>s =100 mm, and, therefore, satisfac-tory

(c) d/4 = 674/4 = 169 mm (6.6 in.) > s= 100 mm and, therefore, satis-factory

(d) Tie force = it x 62 x 275 N = 31.1kN (7.0 kips)

> 1 x longitudinal bar force x 10016 100

_+x1rx142x275N=10.6kN

(2.38 kips), and, therefore,satisfactory

v:411,000= 2.67 MPa

bmd = (400 - 172) x 674= 0.53 0.537 MPa

(6.36 i7 psi)Therefore, satisfactory since shear stressis less than the maximum value permit-ted.

Transverse Reinforcement Outside thePlastic Hinge Regions

In the central 1.79 m (5.87 ft) region ofthe member, the maximum shear force is311 kN (70.0 kips) and is predominantlyin the positive moment region. It will beassumed, as in Fig. Ale, that the shear isresisted by a composite beam of widthbu, = 250 mm (9.8 in_) and effectivedepth d = 684 mm (26.9 in.) (Le., the d tothe centroid of the bottom nonpre-stressed and prestressed steel) with v, =0.17 ,, f MPa (2 SIT psi), and the stir-rups in the precast U-beam will be ig-nored. The spacing of 12 mm (0.47 in.)diameter stirrup ties in the core re-quired for shear on this basis is:

311,000=vCti f,bd +A,f,,dls=(0.17^+25x250x684)+

(2x7r x62 x275x674/s)

90

from whichs = 257 mm (10.1 in.).The spacing is not to exceed d/2 =

684/2 = 342 mm (13.5 in.). Therefore,use 12 mm (0.94 in.) diameter Grade 275stirrups at250 mm (9.8 in,) centers.

Note: Shear design for loading case U= 1.41) + 1.7L will not be more critical.

Check of Interface Shear Stress

Assume that the contact surface isclean, free of laitance and intentionallyroughened to a full amplitude of 5 mm(0.2 in.) and that the permitted interfaceshear stress when the beam over-strength moments develop is 0.55 MPa(80 psi). The shear force diagram isshown in Fig. B6.

Horizontal interface shear stress isfound from Eq. (A6), where

h„ = 500 + 206 + 500 = 1206 mm(47.5 in.)

d = 674 mm (26.5 in.), i.e., for coresection, conservatively

V, = 411 kN (92.5 kips) as in Fig. B6at the column face.

Therefore:411,000

vdn ^

= 0.506 MPa (73 psi)1206x674

Vertical interface shear stress is foundfrom loading case 1.OD + 1.3L. Thevertical interface shear force is35.9 + 1.3 x 13.3 = 53.2 kN/m (3.65 kiplft).Each side face is 500 mm (19.7 in.) deep.

Therefore:53,200 0.053 MPa

lJdU (500 + 500) x ]000 (8 Psi)

and, v2 + vJ = 0.506 2 + 0.0532

= 0.509 MPa (74 psi)< 0.55 MPa (80 psi)

Therefore, the total imposed shear stressat interface is satisfactory.

Note that with regard to the questionof interface shear in seismic over-strength design where the shell beam inthe plastic hinge areas is ignored, itcould be said that interface shear capac-ity has degraded to zero and cannot berelied upon. Therefore, the maximumshear force for the interface shear stresscheck might be considered as the maxi-mum shear of the composite beam sheardesign that is at the end of the plastichinge region adjacent to the compositemidspan region.

In seismic design where there is asignificant shear gradient along thebeam it would be more economic andjustifiably safe to approach the interfacecheck in this manner. In pure gravityload design the interface shear checkreverts hack to the shear force at dfrom the column face. In the above ex-ample then the horizontal interfaceshear would be 311 kN (70.0 kips)rather than 411 kN (92.5 kips) (Fig.S6), resulting in an interface stress of0.38 MPa (55 psi).

Cast-in-Place Topping Slab

The cast-in-place topping slab needsto act as a diaphragm to transfer to theframe system the seismic forcesoriginating from the mass of the floorand the loads placed on it. The rein-forcement placed in the two directionsin the topping slab should not be lessthan that required by the code forshrinkage and temperature reinforce-ment in slabs.

PCI JOURNAL/July-August 1986 91

APPENDIX C - NOTATIONA

= area of precast U-beam sectionAe = gross area of column sectionAp = area of prestressing steelAn = area of top longitudinal rein-

forcement in cast-in-placebeam core

Ae = area of bottom longitudinal re-inforcement in cast-in-placebeam core

A, = area of shear reinforcement atspacings

h

= width of compression face ofbeam

b, = width of precast U-beam mea-sured at bottom of core

hr, = width of interface of sectionbeing considered for horizontalshear

b. = width of web of beamd = distance from extreme com-

pression fiber of concrete tocentroid of tension reinforce-ment

d' = distance from extreme com-pression fiber of concrete tocentroid of compression rein-forcement

U = service dead load

e = eccentricity of prestressingforce in precast U-beam section

E = earthquake loading

A = longitudinal concrete stress in

bottom fiber= compressive cylinder strength

of concretefu, = longitudinal stress in concrete

due to prestress at centroid ofbeam section

fre = stress in prestressing steel after

lossesI,nn = stress in prestressing steel at

flexural strengthfm. = ultimate tensile strength of pre-

stressing steelfnu = ultimate tensile strength of re-

inforcing steelft = longitudinal concrete stress in

top fiberfin = yield strength of reinforcing

steelF = total force in prestressing

strandh = depth of sectionh,pd = depth of columnh t = depth of core of precast U-beam

measured from the top of U-beam

= moment of inertia of precastU-beam about horizontal cen-troidal axis

1 = span of beam between columncenters

I„ = clear span of beamL = service live loadM = bending momenta'bf; = ideal (nominal and theoretical)

flexural strength of cast-in-place concrete core alone cal-culated assuming 0 = 1

M{ = ideal (nominal and theoretical)flexural strength of compositesection calculated assuming ro=1

R!',„ar = maximum measured momentMu = design bending moment from

factored (ultimate) loading+P i = beam end load to cause positive

momentM, to be reached—P f = beam end load to cause nega-

tive moment M ; to be reached—P; = beam end load to cause nega-

tive moment.,' to be reached$ = spacing of shear reinforcementU = design load combinationsv, = shear stress resisted by con-

crete mechanismsvdv = total imposed vertical shear

stress at interfaceVdn = total imposed horizontal shear

stress at interfaceVi = total ideal shear stress resisted

by concrete mechanisms andshear reinforcement

V = shear forceV, = shear force resisted by concrete

shear resisting mechanismsV.Q.r = maximum applied shear force

during the test

92

V„ = design shear force from fac-tored loading

V = design shear force from over-strength beam moments andfactored gravity loading

Zb = section modulus to bottom ofprecast U-beam section

Z, = section modulus to top of pre-cast U-beam section

a = overstrength factor for beam

flexural strengthA = beam end deflectionA, = beam end deflection at first

yieldp = A,Ibdp' = A,/bdpp = APIbd0 = strength reduction factorNc = A/A,, = displacement ductility

factor

PCI JOURNAL/July-August 1986 93