Prestressed Concrete

CE201PrestressedConcreteDesignforPEPreparatoryExamPEPreparatoryExam

T. Y. LIN, Professor of Civil Engineering, Emeritus, University of California, BerkeleyT. Y. LIN, Professor of Civil Engineering, Emeritus, University of California, BerkeleyNED H. BURNS, Professor of Civil Engineering, The University of Texas at AustinNED H. BURNS, Professor of Civil Engineering, The University of Texas at AustinNED H. BURNS, Professor of Civil Engineering, The University of Texas at AustinNED H. BURNS, Professor of Civil Engineering, The University of Texas at AustinJOHN WILEY & SONS THIRD EDITION SI VersionJOHN WILEY & SONS THIRD EDITION SI Version

Tan Tan See See CheeChee Email: Email: [email protected][email protected] 1

Concrete in which there have been introduced internalstresses of such magnitude and distribution that thestresses resulting from given external loadings arecounteracted to a desired degree. In reinforcedconcrete members the prestress is commonlyintroduced by tensioning the steel reinforcement.

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MATERIALMATERIALRESISTINGRESISTING

COMPRESSIONCOMPRESSION

MATERIALSMATERIALSRESISTING TENSIONRESISTING TENSION

& COMPRESSION& COMPRESSION

MATERIALMATERIALRESISTINGRESISTINGTENSIONTENSIONCOMPRESSIONCOMPRESSION & COMPRESSION& COMPRESSIONTENSIONTENSION

STONE BAMBOOSSTONEBRICK

TIMBERBAMBOOSROPES

IRONBARSSTEELWIRESCONCRETE

STRUCTURALSTEEL

REINFORCEDCONCRETE

PASSIVECOMBINATION

HIGHSTRENGTHCONCRETE(HPC)

HIGHSTRENGTHSTEEL(CARBONFIBER)

PRESTRESSCONCRETE

ACTIVECOMBINATION

Fig.11.DevelopmentofbuildingmaterialsCONCRETECOMBINATION

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First Concept Prestressing to Transform Concrete into an Elastic Material@ Serviceability Stage@ Serviceability Stage

This concept treats concrete as an elastic material and is probably still themost common ie point among engineers It is credited to E gene Fre ssinetmost common viewpoint among engineers. It is credited to Eugene Freyssinetwho visualized prestressed concrete as essentially concrete which istransformed from a brittle material into an elastic one by the precompressiongiven to it. Concrete which is weak in tension and strong in compression iscompressed (generally by steel under high tension) so that the brittle concretewould be able to withstand tensile stresses. From this concept the criterion ofno tensile stresses was born. It is generally believed that if there are no tensilestresses in the concrete, there can be no cracks, and the concrete is nolonger a brittle material but becomes an elastic material.longer a brittle material but becomes an elastic material.

GeneralPrinciplesorConceptsofGeneralPrinciplesorConceptsofPrestressedConcrete

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Stress distribution is given by:

Mload pt G

b t tb b

M MM yF Fey FfA I I A Z Z

=

Sectional modulus is used because we are more interested in stresses extreme fibers or Interface :

Fi S di ib i i ll Fig.114.Stressdistributionacrossaneccentricallyprestressedconcretesection 5

This is true only for the statically determinate members wherein externalreactions are not affected by the internal prestressing Not so for staticallyreactions are not affected by the internal prestressing. Not so for staticallyindeterminate systems.

(b) Freebody (Cranking Moment)

Fi P ff i l d i i Fig.117.Prestresseffectisnotrelatedtovariationsawayfromsectionforastaticallydeterminatemember.6

Second Concept Prestressing for Combination of High-Strength Steel withConcrete.

This concept is to consider prestressed concrete as a combination of steeland concrete, similar to reinforced concrete, with steel taking tension and

t t ki i th t th t t i l f i ti lconcrete taking compression so that the two materials form a resisting coupleagainst the external moment.

In prestressed concrete, high-tensile steel is used which will have to beelongated a great deal before its strength is fully utilized. If the high-tensilesteel is simply buried in the concrete, as in ordinary concrete reinforcement,p y , y ,the surrounding concrete will have to crack very seriously before the fullstrength of the steel is developed. Hence it is necessary to prestretch the steelwith respect to the concretewith respect to the concrete.


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Portion of Prestressed Beam Portion of Reinforced Beam

Simplified but valid concept for engineers familiar with reinforced concrete wheresteel supplies a tensile force and concrete supplies a compressive force, forming a

Portion of Prestressed Beam Portion of Reinforced Beam

steel supplies a tensile force and concrete supplies a compressive force, forming acouple with a lever arm between them.

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Prestressed concrete is an extension & modification of the applications ofreinforced concrete to include steels of higher strength. Prestressed concretecannot perform miracles beyond the capacity of the strength of its materials.Although much ingenuity can be exercised in the proper and economic design ofprestressed-concrete structures, there is the eventual necessity of carrying anp , y y gexternal moment by an internal couple, supplied by steel in tension and concretein compression, whether it be prestressed or reinforced concrete. This concepthas been well utilized to determine the ultimate strength of prestressed concretehas been well utilized to determine the ultimate strength of prestressed concretebeams and is also applicable to their elastic behavior.

Simply Reinforced cracks & excessive deflections

Prestressed no cracks and only small deflections

Fig.120.Concretebeamusinghightensilesteel9

Third Concept 10 to Achieve Load Balancing

This concept is to visualize prestressing primarily as an attempt to balance theloads on a member. This concept was essentially developed by Prof. T. Y. LIN.

In the overall design of a prestressed concrete structure, the effect ofprestressing is viewed as the balancing of gravity loads so that membersunder bending such as slabs, beams, and girders will not be subjected toflexural stresses under a given loading condition. This enables thetransformation of a flexural member into a member under direct stress andthus greatly simplifies both the design and analysis of otherwise complicatedstructures.


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The application of this concept requires taking the concrete as a freebody, and replacingthe tendons with forces acting on the concrete along the span.T k f l i l b t d ith b li t d ifTake, for example, a simple beam prestressed with a parabolic tendon if:F = prestressing force; L= span length; h = sag of parabolaThe upward uniform load wb is given by: 28 w LFh

28

8b

bw LFhw and Fh

L= =

Th f i d d if l d th t l d th b i b l dThus, for a given downward uniform load w, the transverse load on the beam is balanced,and the beam is subjected only to the axial force F, which produces uniform stresses inconcrete, f = F/A. The change in stresses from this balanced condition can easily be

t d b th di f l i h i f M/Z Th t i thi i thcomputed by the ordinary formulas in mechanics, f =M/Z. The moment in this case is theunbalanced moment due to (w - wb), the unbalanced load.

Fig.122.Prestressedbeamwithparabolictendon 11

Hence the net downward (unbalanced) load on the concrete beam is (45 - 35.3) = 9.7 kN/m, and the moment at mid-span due to that load is:

2 29.7 7.3 64.68 8mid

wLM kNm= = =

We may over-balance the dead load amount which will result in hogging gg gmoment at mid-span and sagging moment at support.

Fig.125.Example14 12

Externally or Internally PrestressedExternally or Internally Prestressed

Linear or Circular Prestressing (Tank)

Pretensioning and Post-tensioningPretensioning and Post-tensioning

End-Anchored or Non-End-Anchored Tendons

Bonded (grouted) or Unbonded Tendons (greased)Bonded (grouted) or Unbonded Tendons (greased)

Precast, Cast-in-Place, Composite Construction

P i l F ll P i (i BS l )Partial or Full Prestressing (in BS classes)

External Prestressing with HDPE duct

Extradose Prestressing => Stay Cable

ClassificationandTypes13

PrePre--tension & Posttension & Post--tension Prestresstension Prestresstension Prestresstension Prestress

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Prestressing SystemPrestressing System

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One of the considerations to prestressed concrete is thel lit f t f l di t hi h b t tplurality of stages of loading to which a member or structure

is often subjected. For a cast-in-place structure, prestressedconcrete has to be designed (checked) for at least twoconcrete has to be designed (checked) for at least twostages; the initial stage during prestressing and the finalstage under external loadings. For precast members, thirdg g p ,or more stages, e.g., of handling, construction method orcomposite action with second casting has to bei ti t d D i h f th t th iinvestigated. During each of these stages, there are againdifferent periods when the member or structure may beunder different loading conditionsunder different loading conditions.

StagesofLoading16

Steel Stresses not more than the following values:

1 Due to tendon jacking force 0 80f or 0 94f whichever is smaller but not greater than1. Due to tendon jacking force, 0.80fpu or 0.94fpy whichever is smaller, but not greater than maximum value recommended by manufacturer of prestressing tendons or anchorages.

2 Pretensioned tendons immediately after transfer of prestress or post-tensioned2. Pretensioned tendons immediately after transfer of prestress or post-tensioned tendons after anchorage, 0.70fpu

Concrete Stresses not more than the following values:Concrete Stresses not more than the following values:

1. Immediately after transfer of prestress (before losses), extreme fiber stress Compression - 0.60fci, Tension - 0.25fci (except at ends of simply supported Co p ess o 0 60 ci, e s o 0 5 ci (e cept at e ds o s p y suppo tedmembers where 0.50fci is permitted).

2. At service load after allowance for all prestress losses, Compression - 0.45fc, *Tensions - 0.5fcc

when analysis based on cracked sections and bi-linear moment deflection relationships show thatimmediate and long time deflections satisfy Code limits maximum tension is 1.00fc

TABLE12PermissibleStressesforFlexuralTABLE1 2PermissibleStressesforFlexuralMembers(ACICode),cylinderstrength

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Type of stress distribution Allowable compressive stresses

Triangular or near triangular 0.5fci but 0.4fcu Uniform or near uniform 0.4fci but 0.3fcuTransfer & construction condition Allowable tensile stresses

Dead & temporary loads during erection 1 N/mm

Pre-tensioned members 0.45fci Post-tensioned members 0.36fci

Type of loading Allowable compressive stresses

In bending 0.4fcu

In direct compression 0.3fcu

Class Allowable tensile stresses

1 No tensile stresses except at transfer1 No tensile stresses except at transfer

2, Pre-tensioned members 0.45fcu 2, Post-tensioned members 0.36fcu

TABLE1 2aPermissibleStressesforBS5400orBS8110

3 Based on limiting crack width

TABLE12aPermissibleStressesforBS5400orBS8110

Cubestrength18

Initial StageThe member or structure is under prestress but is not subjected to anysuperimposed external loads. This stage can be further subdivided into thefollowing periods, some of which may not be important and therefore may beg p , y p yneglected in certain designs.

Before Prestressing. Before the concrete is prestressed, it is quite weak incarrying load; hence the yielding of its supports must be prevented Provisioncarrying load; hence the yielding of its supports must be prevented. Provisionmust be made for the shrinkage of concrete if it might occur. When it is desirableto minimize or eliminate cracks in prestressed concrete, careful curing beforethe transfer of prestress is very important Drying or sudden change inthe transfer of prestress is very important. Drying or sudden change intemperature must be avoided. Cracks may or may not be closed by theapplication of prestress, depending on many factors. Shrinkage cracks willdestroy the capacity of the concrete to carry tensile stresses and may be destroy the capacity of the concrete to carry tensile stresses and may beobjectionable.

StagesofLoading19

Initial StageDuring Prestressing. This is a critical test for the strength of the tendons. Often,the maximum stress to which the tendons will be subject throughout their lifeoccurs at that period (0.80fpu or 0.94fy). It occasionally happens that an individualp ( pu fy) y ppwire may be broken during prestressing, owing to defects in its manufacture.But this break is seldom significant, since there are often many wires in amember. If a bar is broken in a member with only a few bars, it should beproperly replaced. For concrete, the prestressing operations impose a severetest on the bearing strength at the anchorages. Since the concrete is not aged atthis period while the prestress is at its maximum, crushing of the concrete at theanchorages is possible if its quality is inferior or if the concrete ishoneycombed. Again, unsymmetrical and concentrated prestress from thetendons may produce overstresses in the concrete. Therefore the order of

t i th i t d t ft b t di d b f h dprestressing the various tendons must often be studied beforehand.

StagesofLoading20

Initial StageAt Transfer of Prestress. For pretensioned members, the transfer, of prestress isaccomplished in one operation and within a short period. For post-tensionedmembers, the transfer is often gradual, the prestress in the tendons being, g , p gtransferred to the concrete one by one. In both cases there is no external loadon the member except its own weight.

Thus the initial prestress with little loss as yet taking place imposes a seriousThus the initial prestress, with little loss as yet taking place, imposes a seriouscondition on the concrete and often controls the design of the member. Foreconomic reasons the design of a prestressed member often takes into accountthe weight of the member itself in holding down the cambering effect ofthe weight of the member itself in holding down the cambering effect ofprestressing. E.g., the weight of a simply supported girder is expected to exert amaximum positive moment at mid-span which counteracts the negative momentdue to prestressing. If the girder is cast and prestressed on soft ground withoutdue to prestressing. If the girder is cast and prestressed on soft ground withoutsuitable pedestals at the ends, the expected positive moment may be absent andthe prestressing may produce excessive tensile stresses on top fibers of thegirder.g

StagesofLoading21

Initial StageDecentering and Retensioning. If a member is cast and prestressed in place, itgenerally becomes self-supporting during or after prestressing. Thus the false-work can be removed after prestressing, and no new condition of loading isp g, gimposed on the structure. Some concrete structures are retensioned, that is,prestressed in two or more stages. Then the stresses at various stages oftensioning must be studied.

StagesofLoading22

Intermediate StageThis is the stage during transportation and erection. It occurs only for precastmembers when they are transported to the site and erected in position. It ishighly important to ensure that the members are properly supported andg y p p p y pphandled at all times. For example, a simple beam designed to be supported atthe ends will easily break if lifted at mid-span, a correct way is to be specified.

Not only during the erection of the member itself but also when adding theNot only during the erection of the member itself, but also when adding thesuperimposed dead loads, such as roofing or flooring, attention must be paid tothe conditions of support and loading.

StagesofLoading23

Final StageThis is the stage when the actual working loads come on the structure. As forother types of construction, the designer must consider various combinations oflive loads on different portions of the structure with lateral loads such as windpand earthquake forces, and with strain loads such as those produced bysettlement of supports and temperature effects. For prestressed concretestructures, especially those of unconventional types, it is often necessary toinvestigate their cracking and ultimate loads, their behavior under the actualsustained load in addition to the working load.

StagesofLoading24

Final StageSustained Load. The camber or deflection of a prestressed member under itsactual sustained load (which often consists only of the dead load) is often thecontrolling factor in design, since the effect of flexural creep will eventuallyg g , p ymagnify its value. Hence it is often desirable to limit the camber or deflectionunder sustained load.

Working Load To design for the working load is a check on excessive stressesWorking Load. To design for the working load is a check on excessive stressesand strains. It is not necessarily a guarantee of sufficient strength to carryoverloads. However, an engineer familiar with the strength of prestressedconcrete structures may often design conventional types and proportions on theconcrete structures may often design conventional types and proportions on thebasis of working-load computations, then check strength.

StagesofLoading25

Final StageCracking Load. Cracking in a prestressed concrete member signifies a suddenchange in the bond and shearing stresses. It is sometimes a measure of thefatigue strength. For certain structures, such as tanks and pipes, theg g , p p ,commencement of cracks presents a critical situation. For structures subject tocorrosive influences, for unbonded tendons where cracks are moreobjectionable, or for structures where cracking may result in excessivedeflections, an investigation of the cracking load seems important.

StagesofLoading26

Final StageUltimate Load. Structures designed on the basis of working stresses may notalways possess a sufficient margin for overloads. Since it is required that astructure possess a certain minimum factored load capacity, it is necessary top p y, ydetermine its ultimate strength. In general, the ultimate strength of a structure isdefined by the maximum load it can carry before collapsing. However, beforethis load is reached, permanent yielding of some parts of the structure mayalready have developed. Although any strength beyond the point of permanentyielding may serve as additional guarantee against total collapse, someengineers consider such strength as not usable and prefer to design on thebasis of usable strength rather than the ultimate strength. However, ultimatestrength is more easily computed and is commonly accepted as a criterion fordesign in prestressed concrete as with other structural systems.

StagesofLoading27

Final StageIn addition to the above normal loading conditions, some structures may besubject to repeated loads of appreciable magnitude which might result in fatiguefailures. Some structures may be under heavy loads of long duration, resultingy y g , gin excessive deformations due to creep, while others may be under such lightexternal loads that the camber produced by prestressing may become toopronounced as time goes on. Still others may be subject to undesirablevibrations under dynamic loads. Under a sudden impact load or under the actionof earthquakes, the energy absorption capacity of the member as indicated byits ductility may be of prime importance. These are special conditions which theengineer must consider for his individual case.

StagesofLoading28

ASTM Specifications A-322 and A-29, High-Strength Steel Bars, especially deformed barswith ultimate strength of 1100 MPa are available in sizes 25.4 to 34.9 mm. Ultimatestrength of 1600 MPa is available for these bars with 15 9 mm The deformations on the

N i l B ki N i l A Mi i

strength of 1600 MPa is available for these bars with 15.9 mm. The deformations on thebars serve as threads to fit couplers and anchorage hardware.

mm kN mm2 kNGrade 1720 MPa

6 35 40 23 22 34

Nominal Diameter

Breaking Strength

Nominal Area of Strand

Minimum Load at 1%

6.35 40 23.22 347.94 64.5 37.42 54.79.53 89 51.61 75.6

11.11 120.1 69.68 102.312 7 160 1 9190 136 212.7 160.1 9190 136.2

15.24 240.2 139.35 204.2Grade 1860 MPa

9.53 102.3 54.84 8711 11 137 9 74 19 117 211.11 137.9 74.19 117.212.7 183.7 98.71 156.1

15.24 260.7 140 221.5

Table22PropertiesofuncoatedSevenWireTable2 2PropertiesofuncoatedSeven WireStressRelievedStrand(ASTMA416)

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For prestressed members under the action of design live loads, the stress insteel wires is seldom increased by more than 70 MPa from their effectiveprestress of about 1035 MPa. It is safe to say that, so long as the concrete hasnot cracked, there is little possibility of fatigue failure in steel, even though theworking load is exceeded. After the cracking of concrete, high stress

t ti i t i th i t th k Th hi h t ltconcentrations exist in the wires at the cracks. These high stresses may resultin a partial breakage of bond between steel and concrete near the cracks. Underrepeated loading, either the bond may be completely broken or the steel may berupturedruptured.

1035 0 56 0 56eff fffIn this case or f f= = =0.56 0.56

1860 eff pupuIn this case or f f

f

var 70 0.037 0.0371860 eff pu

f or f ff

= = =1860 eff pupuf

Materials FatigueStrength30

A typical failure envelope for prestressing steel is shown in Fig. 2-7(a). Thisenvelope indicates how the tensile stress can be increased from a given lowerp glevel to a higher level to obtain failure at one million load-cycles. Note that allvalues are expressed as a percentage of the static tensile strength. Thus thesteel may resist a stress range amounting to 0.27fpu if the lower stress limit ispzero, but only a stress range of 0.18fpu, if the lower stress limit is increased to0.40fpu. At a lower stress limit of 0.90fpu, or over, it takes only a negligible stressincrease to fail the steel at one million cycles. While this fatigue envelope variesfor different steels, the curve given here may be considered a typical one.

Materials FatigueStrength31

A typical failure envelope for prestressing steel - failure at one million load-cycles. a stress range 0.27fpu if the lower stress limit is zero, stress range ofy g pu , g0.18fpu, if the lower stress limit is 0.40fpu. At a lower stress limit of 0.90fpu, orover, negligible stress range.

Fig.27.Methodforpredictingfatiguestrengthofprestressedconcretebeams32

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The design of a prestressed concrete member will involve consideration ofthe effective force of the tendon at each significant stage of loading,t th ith i t t i l ti f th t ti i th lif hi ttogether with appropriate material properties for that time in the life historyof the structure. The most common stages to be checked for stresses andbehavior are the following:g1. Immediately following transfer of prestress force to the concrete section,

stresses are evaluated. This check involves the highest force in thetendon acting on the concrete which may be well below its 28 daytendon acting on the concrete which may be well below its 28daystrength, fc. The ACI Code designates concrete strength as fci at thisinitial stage &limits the allowable concrete stresses.

2. At service load after all losses of prestress have occurred & a longtermeffective prestress level has been reached, stresses are checked again asa measure of behavior & sometimes of strength The effective steela measure of behavior & sometimes of strength. The effective steelstress, fse, after losses is assumed for the tendon while the membercarries the service live and dead loads. Also, the concrete strength isassumed to have increased to fc.

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LumpSum Estimates for Prestress LossLump Sum Estimates for Prestress LossTable 41AASHTO LumpSum Losses Table 42Approximate Prestress LossValues for Posttensioning

310 0.167 0.1671860 eff pu

Losses or f f= = = 160 0.10 0.101600 eff pu

Losses or f f= = =230 0.124 0.124

1860 eff puLosses or f f= = = 160 0.145 0.145

1100 eff puLosses or f f= = =

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43 Elastic Shortening of Concrete

( )20 0 4 4GM eF F ef +Where:F = 0 9F (pretensionedmembers)

( )0 0 4 4Gcirf A I I = +

F0= 0.9Fi (pretensionedmembers)fcir = stress in the concrete at c.g.s. due to prestress force F0 which iseffective immediately after prestress has been applied to concrete.

The elastic shortening for the steel may be written in more general form asfollows:follows:

h

( )4 5s cirs circi

E fES f nfE

= = =

WhereN =modular ratio at transfer, Es /Ecifcir = concrete stress from (44)fcir concrete stress from (4 4)Es = 200,000 MPa

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The ACIASCE Recommendation for elastic losses accounts for thesequence of stressing effect on elastic losses is as follows.

E f

Where Kes = 1.0 for pretensioned members & K = 0.5 for posttensioned

( )4 6s ciresci

E fES KE

=

members when tendons are in sequential order to the same tension

where F0 is the total prestress just after transfer, that is, after theh i h k l

( )0 0 4 1ss sc c c

E F nFES f EA E A

= = = =

shortening has taken place.

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The ACIASCE Recommendation for elastic losses accounts for thesequence of stressing effect on elastic losses is as follows.

E f

Where Kes = 1.0 for pretensioned members & K = 0.5 for posttensioned

( )4 6s ciresci

E fES KE

=

members when tendons are in sequential order to the same tension

where F0 is the total prestress just after transfer, that is, after theh i h k l

( )0 0 4 1ss sc c c

E F nFES f EA E A

= = = =

shortening has taken place.

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44TimeDependent Losses (General)

Prestress losses due to creep and shrinkage of concrete and steelrelaxation are both time dependent and interdependent. The materialsp phave properties which are time dependent. The effects becomeinterdependent in a prestressed concrete member. After transfer of

t t i d t i i d b th t l d t hi hprestress, a sustained stress is imposed on both steel and concrete whichwill change with time. To account for these changes with time, a stepbystep procedure can be used in order to account for changes which occur inp p gsuccessive time intervals.

The loss due to elastic shortening can be compensated for in theThe loss due to elastic shortening can be compensated for, in thefabrication of pretensioned members, by tensioning the steel a bit higherthan the stress desired at transfer when the strands are cut. This elasticloss may be computed and compensated for in posttensioning also. Butthe time dependent effects cannot be counterbalanced.

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45 Loss Due to Creep of Concrete (CR)

Creep is assumed to occur with the superimposed permanent dead loadadded to the member after it has been prestressed. Part of the initialadded to t e e be a te t as bee p est essed a t o t e t acompressive strain induced in the concrete immediately after transfer isreduced by the tensile strain resulting from the superimposed permanentd d l d L f t d t i t d f b d d bdead load. Loss of prestress due to creep is computed for bondedmembersfrom the following expression (for normal weight concrete):

WhereK = 2 0 for pretensionedmembers & = 1 6 for posttensioned ones

( ) ( )4 7scr cir cdsc

ECR K f fE

=

Kcr = 2.0 for pretensionedmembers & = 1.6 for posttensioned ones.fcds = stress in concrete at c.g.s. of tendons due to all superimposed deadloads that are applied to themember after it is prestressedEs = modulus of elasticity of prestressing tendonsEc = modulus of elasticity of concrete at 28 days,@ fc

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45 Loss Due to Creep of Concrete (CR)

For unbonded tendons the average compressive stress is used to evaluatelosses due to elastic shortening and creep of concrete losses. The losses ing pthe unbonded tendon arc related to the average member strain ratherthan strain at the point of maximummoment.Thus

Where fcpa = average compressive stress in the concrete along the member( )4 8scr cpa

c

ECR K fE

= cpa g g

length at the c.g.s. of the tendons.

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46 Loss Due to Shrinkage of Concrete

Shrinkage of concrete is influenced by many factors. Those which aremost important: volumetosurface ratio, relative humidity & time fromend of moist curing to application of prestress Since shrinkage is timeend of moist curing to application of prestress. Since shrinkage is timedependent, we would not experience 100% of the ultimate loss for severalyears, but 80% will occur in the first year. The modifying factors fory y y gvolumeto surface ratio (V/S) and relative humidity (RH) are given below:

( )( ) ( )

6

6

550 10 1 0.06 1.5 0.015

8 2 10 1 0 06 100 4 9

shV RHSV RH

= ( ) ( )8.2 10 1 0.06 100 4 9RHS=

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The loss of prestress due to shrinkage is the product of the effectiveshrinkage, sh, and the modulus of elasticity of prestressing steel. The onlyother factor in the shrinkage loss eq ation ( 0) is the coefficient Kother factor in the shrinkage loss equation (410) is the coefficient Kshwhich reflects the fact that the posttensioned members benefit from theshrinkage which occurs prior to the posttensioning (Table 44). This valueg p p g 4 4will be 1.0 for pretensioned beams with very early transfer of prestress andbonded tendons, but for posttensioned beams we may have a significantreduction in shrinkagereduction in shrinkage.

V ( ) ( )68.2 10 1 0.06 100 4 10sh s VSH K E RHS =

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Loss of prestress due to shrinkage is the product of the effectiveshrinkage, sh, and the modulus of elasticity of prestressing steel. The onlyother factor in the shrinkage loss eq ation ( 0) is the coefficient Kother factor in the shrinkage loss equation (410) is the coefficient Kshwhich reflects the fact that the posttensioned members benefit from theshrinkage which occurs prior to the posttensioning (Table 44). This valueg p p g 4 4will be 1.0 for pretensioned beams with very early transfer of prestress andbonded tendons, but for posttensioned beams we may have a significantreduction in shrinkagereduction in shrinkage.

( ) ( )68 2 10 1 0 06 100 4 10h VSH K E RH = ( ) ( )8.2 10 1 0.06 100 4 10sh sSH K E RHS

44

47 Loss Due to Steel Relaxation

Tests of prestressing steel with constant elongation maintained over aperiod of time have shown that the prestress force will decrease graduallyas shown in Fig 4 2 The amount of the decrease depends on both timeas shown in Fig. 42. The amount of the decrease depends on both timeduration and the ratio f /4,,. The loss of prestress force is called relaxation.It was shown in the tests that this source of loss is more significant thanghad been assumed prior to 1963.We can express this loss as follows:

( )log1 0.55 4 1110

p pi

pi py

f ftf f

=

Fig. 4-2. Steel relaxation curves forstress-relieved wire and strandstress-relieved wire and strand

45


With a time interval between the moment of stressing t1 in thepretensioning bed (such as the one shown in Fig. 41) and a later time Iwhen we wish to estimate the remaining force, we can write tire followingequation:

h l i h b f & f /f d

( ) ( )1log log1 0.55 4 1210

p pi

pi py

f ft tf f

=

where log t is to the base of 10 & fpi/fpy, exceeds o.55.

The ACI Code limit on initial prestress @ transfer to fpi = 0.7fpu. Relaxationp pis one reason for limiting fpi. The use of lowrelaxation strand reducesrelaxation losses to about 3.5%maximum.

46


Prestressed beams actually have a constantly changing level of steel strainin the tendons as timedependent creep occurs, and we must modify the

l l i f l i l fl hi h icalculation of relaxation loss, RE, to reflect this. The ACIASCE Committeeaccomplishes approximately the same thing with an equation whichfollows:follows:

( ) ( )reRE= K -J SH+CR+ES C 4 13

Table 4 5 Values of K & J Table 4 6 Values of CTable 4-5 Values of Kre & J Table 4-6 Values of C

47

48 Loss Due toAnchorageTakeUp (Wedge Slip)

For most systems of posttensioning, the anchorage fixtures that aresubject to stresses at transfer will tend to deform, thus allowing thetendon to slacken slightly. Friction wedges employed to hold the wires willslip a little distance before the wires can be firmly gripped. The amount ofslippage depends on the type of wedge and the stress in the wires, rangingslippage depends on the type of wedge and the stress in the wires, rangingfrom 2.56 mm. For direct bearing anchorages, the heads and nuts aresubject to a slight deformation at the release of the jack. An average valuef h d f i b l b 8 f l hifor such deformations may be only about 0.8 mm. If long shims arerequired to hold the elongated wires in place, there will be a deformationin the shims at transfer of prestress. As an example, a shim 0.3 m long mayt e s s at t a s e o p est ess s a e a p e, a s 0 3 o g aydeform 0.3mm.Many of the posttensioning systems have jacking systems where a

iti f h th d f d t i th i di id l t dpositive force pushes the wedges forward to grip the individual strandsbefore releasing the tension onto the anchorage.

48

48 Loss Due toAnchorageTakeUp (Wedge Slip)

A general formula for computing the loss of prestress due to anchoragedeformation a is

Since this loss of prestress is caused by a fixed total amount of shortening,

( )asANC= f = 4 1 4sEL

Since this loss of prestress is caused by a fixed total amount of shortening,the percentage of loss is higher for short wires than for long ones. Hence itis quite difficult to tension short wires accurately, especially for systems of

t i h h l l ti l lprestressing whose anchorage losses are relatively large.

49

49 Loss or Gain Due to Bending of Member

When a member bends, further changes in the prestress may occur: theremay be either a loss or a gain in prestress, depending on the direction ofbending and the location of the tendon. If there are several tendons andthey are placed at different levels, the change of prestress in them willdiffer Then it may be convenient to consider only the centroid of all thediffer. Then it may be convenient to consider only the centroid of all thetendons (the c.g.s. line).This change in prestress will depend on the type of prestressing: whether

i d h h b d d b d d f h d ipre or posttensioned, whether bonded or unbonded. Before the tendon isbonded to the concrete, bending of the member will affect the prestress inthe tendon.the tendon.Neglecting frictional effects, and strain in the tendon will be stretched outalong its entire length, and the prestress in the tendon will be uniformly

difi d Aft th t d i b d d t th t f th b dimodified. After the tendon is bonded to the concrete, any further bendingof the beam will only affect the stress in the tendon locally but will notchange its prestress.g p

50

49 Loss or Gain Due to Bending of Member

After the tendon is bonded to the concrete, the steel and concrete formone section, and any change in stress due to bending of the section iseasily computed by the transformed section method. Hence, it isconvenient to say that prestress does not change as the result ofbending of a beam after the bonding of steel to concrete, although thebending of a beam after the bonding of steel to concrete, although thestress in the tendon does change.The same is true of pretensioned members bending under prestress andh i i h f h f f h b b d dtheir own weight. After the transfer of prestress, the beam bends upwardand the wires shorten because of that bending. For the same reason asdiscussed above, that shortening of the wires due to bending is notd scussed abo e, t at s o te g o t e es due to be d g s otconsidered a loss of prestress, just as the eventual lengthening under loadis not considered a gain in prestress. In both cases, the prestress

id d i th t t th d f th b hi h d tconsidered is the prestress at the ends of the members, which does notchange under bending.

51

410 Frictional Loss, Practical Considerations

First of all it is known that there is some friction in the jacking & anchoringsystem so that the stress existing in the tendon is less than that indicatedby the pressure gage. This is especially true for some systems whose wireschange direction at the anchorage. This friction in the jacking andanchoring system is generally small though not insignificant. It can beanchoring system is generally small though not insignificant. It can bedetermined for each case, if desired, and an overtension can be applied tothe jack so that the calculated prestress will exist in the tendon. It must be

b d h h h h f i i i li i dremembered, though, that the amount of overtensioning is limited tostay within the yield point of the wires. The ACI Code limits the jackingforce to 0.80fpu.pu

52


More serious frictional loss occurs between the tendon and its surroundingmaterial, whether concrete or sheathing, and whether lubricated or not.This frictional loss can be conveniently considered in two parts: the lengtheffect and the curvature effect. The length effect is the amount of frictionthat would be encountered if the tendon is a straight one, that is, one thatthat would be encountered if the tendon is a straight one, that is, one thatis not purposely bent or curved. Since in practice the duct for the tendoncannot be perfectly straight, some friction will exist between the tendon

d it di t i l th h th t d i t t band its surrounding material even though the tendon is meant to bestraight.

53


This is sometimes described as the wobbling effect of the duct and isdependent on the length and stress of the tendon, the coefficient off i i b h i l d h k hi d h dfriction between the contact materials, and the workmanship and methodused in aligning and obtaining the duct. Some approximate values forcoefficients used to compute these losses are given in the Commentary ofcoefficients used to compute these losses are given in the Commentary oftheACI Code,Table 47.

Table 4-7 Friction Coefficients for Posttensioning Tendon

54


The loss of prestress due to curvature effect results from the intendedcurvature of the tendons in addition to the unintended wobble of the duct.This loss is again dependent on the coefficient of friction between thecontact materials and the pressure exerted by the tendon on the concrete.The coefficient of friction, in turn, depends on the smoothness and natureThe coefficient of friction, in turn, depends on the smoothness and natureof the surfaces in contact, the amount and nature of lubricants, andsometimes the length of contact. The pressure between the tendon and

i d d h i h d d h l h iconcrete is dependent on the stress in the tendon and the total change inangle. guide for the normal conditions. Values for the friction that can beexpected with particular type tendons and particular type ducts can bep p yp p ypobtained from themanufacturer of the tendons or system.

55


The effect of overtensioning with a subsequentreleaseback is to put the frictional difference inpthe reverse direction. After releasing, thevariation of stress along the tendon takes someshape as in Fig 4 6 When the frictional loss is ashape as in Fig. 46. When the frictional loss is ahigh percentage of the prestress, it cannot betotally overcome by overtensioning (curve b, Fig.46), since the maximum amount of tensioning islimited by the strength or the yield point of thetendon The portion of the loss that has not beentendon. The portion of the loss that has not beenovercomemust then be allowed for in the design.

Fig. 4-6. variation of stress in tendon due to frictional force.

56

411 Frictional Loss,Theoretical Considerations

The friction loss is obtained from the expression below. Loss of steel stressis given as FR = f1 f2, the steel stress at the jacking end is f1, and thel h h i i i h fi dlength to the point is L, Fig. 48(a). Thus, we find

( ) ( )- -KL1 2 1 1 1FR=f -f =f =f 1-e 4 18 KLf e

Fig 4-8 Approx frictional loss along circular curveFig. 4 8. Approx. frictional loss along circular curve.

57


The formula (418) is theoretically correct and take into account thedecrease in tension and hence the decrease in the pressure as the tendonbends around the curve and gradually loses its stress due to friction. If,however, the total difference between the tension in the tendon at thestart and that at the end of the curve is not excessive (say not more thanstart and that at the end of the curve is not excessive (say not more than15 or 20%), an approximate formula using the initial, tension for the entirecurve will be close enough. On this assumption, a simpler formula can bed i d i l f h b i l f If h l iderived in place of the above exponential form. If the normal pressure isassumed to be constant, the total frictional loss around a curve with angle and length L is, Fig. 48.g , g 4

( ) ( )2 1 1 1 1F -F =-KLF - F =-F K L+ 4 21

58


Transposing terms, we have:

( )2 1F -F =-KL- =- K+ 4 22L The loss of prestress for the entire length of a tendon can be consideredfrom section to section, with each section consisting of either a straight

( )1F R

from section to section, with each section consisting of either a straightline or a simple circular curve. The reduced stress at the end of a segmentcan be used to compute the frictional loss for the next segment, etc.

Fig. 4-9. Approximate determination of central angle for a tendon.

59

412Total Amount of Losses

Initial prestress in steel minus the losses is known as the effective or thedesign prestress.

The jacking stress minus the anchorage loss will be the stress at anchorageafter release and is frequently called the initial prestress.after release and is frequently called the initial prestress.The losses to be deducted will include the elastic shortening and creep andshrinkage in concrete plus the relaxation in steel.

If the stress alter the elastic shortening of concrete is taken as the initialprestress, then the shrinkage and creep in concrete and the relaxation inp , g psteel will be the only losses. For points away from the jacking end, theeffect of friction must be considered in addition.

60


Themagnitude of losses can be expressed. in four ways:1. In unit strains. This is most convenient for losses such as creep,

shrinkage, and elastic shortenings of concrete expressed as strains.2. total strains. This is more convenient for the anchorage losses.3. In unit stresses. All losses when expressed in strains can be3. In unit stresses. All losses when expressed in strains can be

transformed into unit stresses in steel. This is the approach used in theACIASCECommittee method.I % f t Thi ft b tt i t f th i ifi4. In % of prestress. This often conveys a better picture of the significanceof the losses.

61


It is difficult to generalize the amount of loss of prestress, because it isdependent on so many factors: the properties of concrete and steel, curing

d i di i i d d i f li i fand moisture conditions, magnitude and time of application of prestress,and the process of prestressing. For average steel and concrete properties,cured under average air conditions, the tabulated percentages may becured under average air conditions, the tabulated percentages may betaken as representative of the average losses.

Pretensioning, % Posttensioning, %Elastic shortening and bending of concrete 4 1Creep of concrete 6 5Shrinkage of concrete 7 6Shrinkage of concrete 7 6Steel relaxation 8 8

Total loss 25 20In term of fpu 56.25% 60.00%

62


The previous table assumes that proper overtensioning has been appliedto reduce creep in steel and to overcome friction and anchorage losses.A f i ti l L t t b id d i dditiAny frictional Loss not overcome must be considered in addition.Allowance for loss of prestress of about 20% for posttensioning and 25%for pretensioning is seen to be not too far from the probable values forp g pprestressed beams and girders. (when the average prestress in a member(Fe/Ac) is high, say about 7 MPa, losses should be increased to about 30%for pretensioning & 25% for posttensioning When F /A is low say aboutfor pretensioning & 25% for posttensioning. When Fe/Ac is low say about1.7 MPa losses should be reduced to 18% for pretensioning & 15% forposttensioning.

275 0.148 0.1481860 eff pu

Losses or f f= = =

Table 4 8 Limiting Maximum Loss (ACI ASCE Committee)

( )18600.75 0.148 0.602

eff pu

e pu puor F f f= =Table 4-8 Limiting Maximum Loss (ACI-ASCE Committee)

63

51 Introduction and Sign Conventions

Diff ti ti b d b t th l i & d i f t dDifferentiation can be made between the analysis & design of prestressedsections for flexure.

Analysis is the determination of stresses in the steel and concrete when theform and size of a section are already given or assumed. Design of sectioninvolves the choice of a suitable section out of many possible shapes andinvolves the choice of a suitable section out of many possible shapes anddimensions.

In actual practice, it is often necessary to first perform the process of designwhen assuming a section, and then to analyze that assumed section.

For the purpose of study, it is easier to learn first the methods of analysis andthen those of design.

64

51 Introduction and Sign Conventions

A rather controversial point in the analysis of prestressedconcretebeams has been the choice of a proper system of sign conventions.M th i l di h d iti i ( ) f iMany authors including me have used positive sign (+) for compressivestresses and negative sign () for tensile stresses, basing theirconvention on the idea that prestressedconcrete beams are normallyp yunder compression and hence the plus sign should be employed todenote that state of stress. T.Y. LIN prefers to maintain the commonsign convention as used for the design of other structures; that issign convention as used for the design of other structures; that is,minus for compressive and plus for tensile stresses. Plus will stand fortension and minus for compression, whether we are talking of stressesin steel or concrete, prestressed or reinforced.

65

52 Stresses In Concrete Due to Prestress

Then the resultant fiber stress due to the eccentric prestress is given by:

( )5 5ptb tb

MF Fey FfA I A Z

=

Fig. 5-2. Eccentric prestress on a section

66

53 Stresses In Concrete Due to Loads &Combined with Prestress

If the eccentricity does not occur along one of the principal axes of thesection, it is necessary to further resolve the moment into two component

l h i i l i h hmoments along the two principal axes, Fig. 57; then the stress at anypoint is given by:

y M MFeFF F ( )5 5yxx y

px pytb t t

xb yb

yx aM MFeFeF Ff

A I I A Z Z =

Fig. 5-7. Eccentricity of prestress in two directionsin two directions.

67

53 Stresses In Concrete Due to Loads &Combined with Prestress

Only the resulting stresses in concrete are desired, instead of theirseparate values.They are given by the following formula.

( )5 7loadtb

ptb t

b

MMyI Z

MF Fey FfA I A Z

=

When prestress eccentricity and external moments exist along twoprincipal axes, the general elastic formula can be used.

yx x

x y x

tb

yx M xI

FeFeFfA I I

=

( )5 8px py lylxt t t txb yb xb yb

MM M MFA Z Z Z Z

y y

68

54 Stresses in Steel Due to Loads

In order to get a clear understanding of the behavior of a prestressedconcrete beam, it will be interesting to first study the variation of steelstress as the load increases For the mid span section of a simple beamstress as the load increases. For the midspan section of a simple beam,the variation of steel stress with load on the beam is shown in Fig. 511.Along the Xaxis is plotted the load on the beam, and along the Yaxisg p gis plotted the stress in the steel.

Fig 5-11 Variation of steelFig. 5-11. Variation of steel stress with load.

69

55 CrackIng Moment

The moment producing first hair cracks in a prestressed concrete beamis computed by the elastic theory, assuming that cracking starts whenthe tensile stress in the e treme fiber of concrete reaches its mod l sthe tensile stress in the extreme fiber of concrete reaches its modulusof rupture. Most available test data seem to indicate that the elastictheory is sufficiently accurate up to the point of cracking. The ACI Codey y p p gvalue for modulus of rupture, fr, is 0.62fc with units for both fr and fcas psi.We have the value of cracking moment given by

( )2 5 11& 12r rf I IrFe or ec c

fFIM FAc c

+ + + + = where frI/c gives the resisting moment due to modulus of

rupture of concrete rupture of concrete,

Fig. 5-14. Cracking moment.

70

56Ultimate Moment BondedTendons

Exact analysis for the ultimate strength of a prestressedconcretesection under flexure is a complicated theoretical problem, becauseb th t l d t ll t d b d th i l tiboth steel and concrete are generally stressed beyond their elasticrange. The following section develops such an analysis technique forbonded beams. However, for the purpose of practical design, where an, p p p g ,accuracy of 510% is considered sufficient, relatively simple procedurescan be developed.

A simple method for determining ultimate flexural strength followingthe ACI Code is presented herewith, is limited to the followingconditions (PTO).

71


1. The failure is primarily flexural, with no shear bond, or anchoragefailure whichmight decrease the strength of the section.

2. The beams are bonded. Unbonded beams possess differentultimate strength and are discussed later.The beams are statically determinate Although the discussions3. The beams are statically determinate. Although the discussionsapply equally well to individual sections of continuous beams, theultimate strength of continuous beams as a whole is explained bythe plastic hinge theory to be discussed later.

4. The load considered is the ultimate load obtained as the result of ah t t ti t t I t f ti l ti l di tshort static test. Impact, fatigue, or longtime loadings are notconsidered.

72


There is no sharp line of demarcation between the percentage ofreinforcement for an overreinforced beam and that for an

d i f d Th t iti f t t th t kunderreinforced one. The transition from one type to another takesplace gradually as the percentage of steel is varied. A sharp definitionof balanced condition cannot be made since the steel used forprestressing does not exhibit a sharp yield point. For the materialspresently used in prestressed work, the reinforcement index,p, whichapproximates the limiting value to assure that the prestressed steelapproximates the limiting value to assure that the prestressed steel(Aps) will be slightly into its yield range, is given by the ACI Code asfollows:

( )' 0.30 5 13psp pc

ff

= ps

p

Awhere

bd =

73


There are situations where prestressing steel (Aps) and ordinaryreinforcing bars (As) are used together in a prestressed beam. In thisease the total of all the tension steel is considered along with theease the total of all the tension steel is considered along with thepossibility of compression steel (As). The limiting reinforcement ratio isgiven as:g

( )' 0.30 5 14p + '

' '; ' ' 'y ys s

c c

f fA Awhere and andf bd f bd

= = = =

74


ACI Code Bonded Beams. For underreinforced bonded beams, the steelis stressed to a stress level which approaches its ultimate strength atpp gthe point of failure for the beam in flexure. For the purpose of practicaldesign, it will be sufficiently accurate to assume that the steel isstressed to the stress level f Provided the effective prestress f isstressed to the stress level, fps. Provided the effective prestress, fse, isnot less than 0.5fpu, the following approximate value for the steel stressat ultimate moment capacity:

Note that as the steel ratio is reduced the member is increasingly'1 0.5

pups pu p

c

ff f

f = Note that as the steel ratio p, is reduced, the member is increasingly

underreinforced; and the steel stress fp, approaches the ultimatestrength of prestressing steel.

c

75


The computation of the ultimate resisting moment is a relativelysimple matter and can be carried out as follows. Referring to Fig. 516,p g g 5 ,the ultimate compressive force in the concrete C equals the ultimatetensile force in the steelT thus:

' 'C T A f' '

' ' ' 's ps

s ps n

C T A f

M T a A f a M

= == = =

Choosing the simplest stress block, a rectangle, for the ultimatecompression in concrete, the depth to the ultimate neutral axis kd iscomputed by:computed by:

'1 ' '

1 1

'' ' ' ps pscc c

A fCC k f k bd k dk f b k f b

= = =

( )'1

' 5 16s psc

A fk

k f bd=

Fig 5-16 Ultimate momentFig. 5-16. Ultimate moment.

76


where k1fc, is the average compressive stress in concrete at rupture.These formulas apply if the compressive flange has a uniform width bat failure. Locating C at the center of the rectangular stress block, wehave the lever arm

'd k

( )' ' 1

2 2'' 1 5 18

d ka d k d

kHence M A f d M

= = = = ( ), 1 5 182ps ps nHence M A f d M= =

Fig 5 16 Ultimate moment

According to Whitneys plastic theory of reinforced-concrete beams, k1 shouldbe 0.85, based on cylinder strength. According to some authors in Europe, k1

Fig. 5-16. Ultimate moment.

should be 0.60 - 0.70 based on the cube strength; since cube strength is 25%higher than cylinder strength, this would give approximately 0.75 to 0.88 for k1based on the cylinder strength.

77


Variation of the value of k1 does not appreciably affect the lever arm a.For a rectangular section for the compression area, we can let p =pAps/bd.Then we have the following formula:

( )0.59' 1 5 21p psfM A f d ( )'' 1 5 21'

p psps ps

c

M A f df

and k d a

= =

( )( 5 222n ps psaM A f d =

Fig. 5-16. Ultimate moment.

The ACI Code introduces, the strength reduction factor , and writes equation 5-21 in terms of wp as follows:

( ) ( )( )

1 0.59 5 23

( 5 24

ps ps pMu A f d

aM A f d

= = ( )( 5 242u ps psM A f d=

78


For flanged sections, we may still use equation 515 to estimate thesteel stress at ultimate, fps. The total area of prestressed steel, isdivided into two parts with A developing the flanges and Adivided into two parts with Apf developing the flanges and Apwdeveloping the web as shown in Fig. 518, The ultimate moment issimply computed from the two parts: the flange part has thep y p p g pcompression resultant force acting at middepth of flange, hf/2, and thearm of the moment couple is (dhf/2); the web part has the compressionresultant force acting at a/2 from the top the beam and the arm of theresultant force acting at a/2 from the top the beam, and the arm of themoment couple is (da/2).

Fig. 5-18. Flanged section

79


The equivalent rectangular stress block is assumed as before and thedepth a is determined by the compression area required based on equaltotal compression and tension forces at ultimate The commentary oftotal compression and tension forces at ultimate. The commentary ofthe ACI Code contains equations for Mu to cover this case which it termsflanged section.

( ) ( )'( 0.85 5 252 2

fu pw ps c w f

haM A f d f b b h d = + ( )

( ) ( )'5 26

0.85 5 27

pw ps pf

fpf c w

where A A A

hand A f b b

=

= ( ) ( )0.85 5 27pf c wps

and A f b bf

( )( )

( )

'

'

0.30 5 28

, &

&

w pw w

w pwwhere tension normal prestressed steel

i t l

+ Fig. 5-18. Flanged section

( )& w compression steel80

Elastic Design Simple relations exist between stress distribution and thelocation of C, according to the elastic theory, Fig. 66.

loadtb

M yF FeyfA I I

= p Gt tb b

A I IM MF

A Z Z

b b

Stresses at extreme fibers or interface.

Fig. 6-4. Stress distribution in concrete by the elastic theory.81

If C coincides with the top or bottom kern point, stress distribution will bei l If C f ll i hi h k h i i ill b dtriangular. If C falls within the kern, the entire section will be under

compression; if outside the kern, some tension will exist. If C coincideswith c.g.c., stress will be uniform over the entire concrete section.g ,

The actual design of prestressedconcrete sections requires a certaint f t i l damount of trial and error.

1. There is the general layout of the structure at the start but which maybemodified as the process of design develops.p g p

2. There is the dead weight of the member which influences the designbut whichmust be assumed for themoment calculations.

3 The approximate shape of the concrete section governed by both3. The approximate shape of the concrete section, governed by bothpractical and theoretical considerations.

Because of these variables, we need some trial and error, guided by knownrelations to enable the final results to be obtained without excessive work.

82

64 Elastic Design, Remarks onAllowingTension

The design of prestressed concrete sections allowing no tensile stressesmay often be an extravagance that cannot be justified other thany g jsegmental construction. When compared to reinforced concrete wherehigh tensile stresses and cracks are always present under working load, itseems only logical that at least some tensile stresses should be permittedseems only logical that at least some tensile stresses should be permittedin prestressed concrete. On the other hand there are several reasons forlimiting the tensile stresses in prestressed concrete:

1. High tensile stress in prestressed concrete may indicate an insufficientfactor of safety against ultimate failurefactor of safety against ultimate failure.

2. The existence of tensile stress may indicate an insufficient factor ofsafety against cracking and may easily result in cracking if the concretehas been previously cracked. Cracking also signifies a change in thenature of bond and shearing stressing stresses.

83

ACI Building Code permit tensile stresses as follows:

1. Stresses at transfer:Tension in members without auxiliary reinforcement

f

'0.25 cifAt ends of precast simple beams Tension in members with properly designed auxiliary reinforcement no limit.

'0.5 cif

no limit.

Fig. 6-11. Bigger arm for steel when allowing tension In concrete.

84


2. Stresses or service loads:Tension in precompressed tensile zone. '0.5 cfTension in excess of above limiting values may be permitted whenshown to be not detrimental to proper structural behavior. '1.0 cf

It is clear from the above that, while empirical limits are often specifiedfor convenience in design and checking, the magnitude of permissible

il h ld i h h di i d il btensile stresses should vary with the conditions and cannot easily befixed at one or two definite values.

85


When tensile stresses are permitted under working loads, the termpartial prestressing is often employed, indicating that the concrete isonly partially compressed by the prestress. It is the opinion of theT.Y.LIN that there is really no basic difference between partial and fullprestress. The only difference is that, in partial prestressing, thereprestress. The only difference is that, in partial prestressing, thereexists a certain amount of tension in concrete under working loads.Since most structures are subject to occasional overloads, tensile

ill ll i i b h i l d f ll istresses will actually exist in both partial and full prestressing, at onetime or another. Hence there is no basic difference between them.

86

It is sometimes argued that allowing of tension in concrete is adangerous procedure since the concrete might have cracked previouslydangerous procedure, since the concrete might have cracked previouslyand could not take any tension. This is true if the tensile force inconcrete is a significant portion of the tensile force in the steel, in which

f fevent it will be necessary to neglect tensile force furnished by theconcrete, Fig. 612. But tensile force in the concrete is only a smallproportion of that in steel, the calculations will not be very differentproportion of that in steel, the calculations will not be very differentwhether it is neglected or included.

Fig. 6-12. Relative significance of tension in concrete.

87

65 Elastic Design, Allowing andConsideringTension

This method should be used with the understanding that the stressescalculated are not exactly correct if the tension stress exceeds thecracking stress for the concrete. It is a convenient method & yieldsresults comparable to those of the method which neglects the tensionin the concrete when the tensile force in concrete considered is only ain the concrete when the tensile force in concrete considered is only asmall portion of the total tension. This method is usually followed indesign usingACI Code allowable stresses.

Fig 6 13 Allowing and considering tension in concreteFig. 6-13. Allowing and considering tension in concrete.

88

66 Elastic Design, Composite Sections

Composite section consists of a precast prestressed portion to becombined with another castinplace portion which usually forms part

f f f for all of the top or bottom flange of the beam. The design of compositesections is slightly more complicated than that of simple ones becausethere are many possible combinations in the makeup of a compositethere are many possible combinations in the make up of a compositesection.

, ' , ', '

t t t tb load bt t

bFey M yFf

A I IM M

=

, ' , 'p Gt t t tb b

M MFA Z Z

Fig. 6-16. Elastic design of composite sections.

89


E

'

slab

beam

EElastic StressDesignE

f

'slab

beam

c

c

fUltimate StrengthDesign

f

Fig. 6-22. Design of composite system with standard precast beams.

90


In the case considered here (Fig 616) the precast portion forms thelower flange and the web while part or the whole of the top flange iscast in place. Tension is usually permitted in the top flange at transferand often also in the bottom flange under working load. For suchcomposite sections compressive stress in the castinplace portion willcomposite sections compressive stress in the cast in place portion willneed to be checked where superimposed or live load is significant(bridges). When the castinplace portion becomes the major part ofh b h f l k l d hthe web, or when falseworks are employed to support the precastportion during casting, the design method has to be modifiedaccordingly based on the load stages.g y g

91

67Ultimate Design Sizing

Ultimate design for simple sections with bonded tendons will bediscussed here. Basically the procedure is also applicable to they p ppultimate design of composite sections.

Preliminary Design:Preliminary Design:The ultimate flexural strength of sections can be expressed by simplesemiempirical formulas. For preliminary design, it can be assumedthat the ultimate resisting moment of bonded prestressed sections isgiven by the ultimate strength of steel acting with a lever arm. This armlever varies with the shape of section and generally ranges betweenlever varies with the shape of section and generally ranges between0.6h and 0.9h, with a common value of 0.8h. Hence the area of steelrequired is approximated by: (PTO)

92

67Ultimate Design Sizing

M ( )6 210.80

Ts

ps

M mAh f

wher m load factor

= =

( )' ' 5 220.80 0.85Tc cwher m load factor

M mAh f=

Assuming that the concrete on the compressive side is stressed to0.85fc.

The main difficulty in ultimate design lies in the proper choice of thefactor of safety or the load factor, which will depend on the Code beingy , p gfollowed in the design. It may be assumed that a load factor of 2.0 willbe sufficient for steel and one of 2.5 for concrete.

93

67Ultimate Design

i l iFinal Design

In final design the following factors must be considered.a des g t e o o g acto s ust be co s de ed1. Proper and accurate load factors must be chosen for steel and concrete,

related to the design load and possible overloads for the particulart tstructure.

2. Compressive stresses at transfer must be investigated for the tensileflange, generally by the elastic theory. The tensile flange should beg , g y y y gcapable of housing the steel.

3. The approximate location of the ultimate neutral axis may not be easilydetermined for certain sectionsdetermined for certain sections.

4. Design of the web will depend on shear and other factors.5. The effective lever arm for the internal resisting couple may have to be

more accurately computed.6. Checks for excessive deflection and overstresses may have to be

performedperformed.

94

Ultimate vs. Elastic Design

Both the elastic and the ultimate designs are used for design ofprestressed concrete, the majority of designers still following thep , j y g gelastic theory. whichever method is used for design, the other onemust often be applied for checking. For example, when the elastictheory is used in design it is the practice to check for the ultimatetheory is used in design, it is the practice to check for the ultimatestrength of the section in order to find out whether it has sufficientreserve strength to carry overloads. When the ultimate design is used,g y gthe elastic theory must be applied to determine whether the section isoverstressed under certain conditions of loading and whether thedeflections are excessive When we delve into new types anddeflections are excessive. When we delve into new types andproportions, it is possible that elastic design alone might not yield asufficiently safe structure under overloads, while the ultimate designby itself might give no guarantee against excessive overstress underworking conditions.

95

610Arrangement of Steel Prestressing in Stages

The arrangement of steel is governed by a basic principle: in order toobtain the maximum lever arm for the internal resisting moment, itg ,must be placed as near the tensile edge as possible. This is the same forprestressed as for reinforced sections.

But, for prestressed concrete, one more condition must be considered:The initial condition at the transfer of prestress. If the c.g.s. is very neargthe tensile edge, and if there is no significant girder moment to bringthe center of pressure near or within the kern, Fig. 625, the tensionflange may be overcompressed at transfer while the compressionflange may be overcompressed at transfer while the compressionflange may be under high tensile stress. Hence this brings up a specialcondition in prestressed concrete: a heavy moment is desirable attransfer so that the steel can be placed as near the edge as possible.

96

610Arrangement of Steel Prestressing in Stages

Another method used in order to permit placement of steel near thetensile edge is to prestress the structure in two or more stages; this isk t t i At th fi t t h th tknown as stage prestressing. At the first stage, when the moment onthe beam is small, only a portion 0f the prestress will be applied; thetotal prestress will be applied only when additional dead load is placedp pp y pon the beam producing heavier moment on. (transfer beam, bridge)

Fig 6-25 Girder moment insufficient to bring C within kernFig. 6 25. Girder moment insufficient to bring C within kern.

97

For certain sections, the tendons are placed in the compression flangell i th t i fl Fi 6 G ll ki itas well as in the tension flange, Fig. 629. Generally speaking, it moves

the c.g.s. nearer to the c.g.c. and thereby decrease the resisting leverarm. At the ultimate range, tendons in the compressive flange mayg , p g yneutralize some of its compressive capacity, as only those in thetension flange are effective in resisting moment. However, undercertain circumstances it may be necessary to put tendons in bothcertain circumstances it may be necessary to put tendons in bothflanges in spite of the resulting disadvantages.These conditions are:1. When the member is to be subject to loads producing both +ve Mj p g

and ve M in the section.2. When the member might be subject to unexpected moments of

opposite sign during its handling processopposite sign, during its handling process.3. When the MG/MT ratio is small and the tendons cannot be suitably

grouped near the kern point. Then the tendons will be placed inboth the tension and the compression flanges with the resultingc.g.s. lying near the kern.

98

ACI Building Code

Fig. 6-29. Prestressing steel In both flanges reduces lever arm for resisting moment.

ACI Building Code:Minimum clear spacing at each end of the member 4 of individual wiresor 3 of strands, in order to properly develop the transfer bond inp p y ppretensioning steel. Alongmidspan, bundling is always permitted.

Ducts may be arranged closely together vertically when provision is madeDucts may be arranged closely together vertically when provision is madeto prevent the tendon from breaking through into an adjacent duct. Whenthe tendons are sharply curved, the radial thrusts exerted on the concretealong the bends may be considerable. Horizontal disposition of ducts shallallow proper placement of concrete > 25mm.

99

h l id i71 Shear, General Considerations

It is recognised that prestressedconcrete beams possess greater reliabilityIt is recognised that prestressed concrete beams possess greater reliabilityin shear resistance than reinforcedconcrete beams, because prestressingwill usually prevent the occurrence of shrinkage cracks which could

i bl d t th h i t f th i f d tconceivably destroy the shear resistance of the reinforcedconcretebeams, especially near the point of contraflexure.

Consider three beams carrying transverse loads as shown in Fig. 71. Beam(a) is prestressed by a straight tendon. Taking an arbitrary section AA, theshear V at that section is carried entirely by the concrete none by theshear V at that section is carried entirely by the concrete, none by thetendon which is stressed in a direction perpendicular to the shear. Beam(b) is prestressed with an inclined tendon. Section BB shows that thetransverse component of the tendon carries part of the shear, leaving onlya portion to be carried by the concrete, thus, nett shear:Vc =V Vp

100

71 Shear, General Considerations

It must be noted that a horizontal tendon, though inclined to the axisof the beam, does not carry any vertical shear, as illustrated by sectionC C in beam (c) Whenever the tendon is not perpendicular to theCC in beam (c). Whenever the tendon is not perpendicular to thedirection of shear, then it does assist in carrying the shear, for example,section DD. It is interesting to note that, in some rare instances, thegtransverse component of the prestress increases the shear in concrete.

Fig. 7-1. Sheer carried by concrete and tendons (stay or suspended cable).

101

Ultimate Resisting Shear (BS5400/8110)The design of shear reinforcement fully complied with BS 5400The design of shear reinforcement fully complied with BS 5400.Section 6.3.4, Shear Resistance of Beam. The computer printoutrelates closely with the above clause, and is selfexplanatory.relates closely with the above clause, and is self explanatory.Uncracked section, all Classes: (BS5400 equation 28)

Cracked section, Class 1 and 2: (BS5400 equation 29)20.67co t cp tV bh f f f= +

Cracked section, Class 3: (BS5400 equation 30, BS8110 equation 55)0.037 crcr cuMV bd f VM

= + ( )0.37cr cu pt IM f f y= +

Maximum combined shear and torsional stress shall be limited to

1 0.55 pecr c opu

f VV v bd Mf M

= + o pt IM f y=

0.75fcu, up to maximum value of 5.8 N/mm.

102

77 Bearing atAnchorage (Bursting Stress)

For tendons with end anchorages, where the prestress is transferredto the concrete by direct bearing, various designs may be used forto the concrete by direct bearing, various designs may be used fortransmitting the prestress: steel plates, steel blocks, or reinforcedconcrete ones.The design of an anchorage consists of two parts: determining thebearing area required for concrete, and designing for the strengthand detail of the anchorage itself Anchorages are designed byand detail of the anchorage itself. Anchorages are designed byexperience, tests, and usage rather than by theory. Anchorages aregenerally supplied by prestressing specialists having their ownge e a y supp ed by p est ess g spec a sts a g t e ostandards for different tendons, the engineer does not have to designfor them. Anchorages that have been successfully adopted areusually considered reliable, and no theoretical check on theft stressesis necessary.

103

Prestressing SystemPrestressing System

104

78TransverseTension@ End Block (Spalling Stress)

The portion of a prestressed member surrounding the anchorages ofthe tendons is often termed the end block. Throughout the length ofg gthe end block, prestress is transferred frommore or less concentratedareas and distributed through the entire beam section. Thetheoretical length of the end block is the distance through which thischange takes place and is sometimes called the lead length. It isknown from theoretical and experimental investigations that thisknown from theoretical and experimental investigations that thislead is not more than the height of the beam and often is muchsmaller except for pretensioned beams with long transfer length.p p g g

105


Fig. 7-23. Stresses at end block. Fig. 7-24. lsobars for transverse tension in end block (In terms of average compression f).

Shaded areas represent compressive zones.p p

106


The allowable tensile stress @ end block can be set at about a tenthof that for compression, that is, about 0.04fc. Wherever the tensionof that for compression, that is, about 0.04fc . Wherever the tensionexceeds that value, steel reinforcement should be designed to takethe entire amount of tension on the basis of the usual allowablestress in steel (0.5fy, 138230)

In the spalling zone the tensile stresses are very high and willIn the spalling zone, the tensile stresses are very high and willgenerally exceed the allowable value. However, these stresses act ononly a small area, and the total tensile force is therefore small. Foro y a s a a ea, a d t e tota te s e o ce s t e e o e s a omost cases, it has been found sufficient to provide steel for a totaltransverse tension of 0.03F. For posttensioning, this steel is placed asclose to the end as possible. Either wire mesh or steel bars may beused.

107


To carry the tension in the bursting zone, either stirrups or spiral steelmay be used. For local reinforcement under the anchorage, 6.4 mmmay be used. For local reinforcement under the anchorage, 6.4 mmspirals at 50 mm pitch or 9.5 mm spirals at 38 mm pitch aresometimes adopted. For overall reinforcement, stirrups can beefficiently employed.

In pretensioned beams vertical stirrups acting@ 140 MPa is to resistIn pretensioned beams, vertical stirrups acting@ 140 MPa, is to resistat least 4% of the total prestressing force. They are placed within d/4from the end of the beam, and be as close to the end of the beam aso t e e d o t e bea , a d be as c ose to t e e d o t e bea aspracticable. For at least the distance d from the end of the beam,nominal reinforcement shall be placed to enclose the prestressingsteel in the bottom flange. For box girders, transverse reinforcementshall be provided and anchored by extending the legs into the webs.

108


EXAMPLE 77The end of a prestressed beam isrectangular in section and is actedon by two prestressing tendonsanchored as shown The initialanchored as shown. The initialprestress is 760 kN per tendon. Fc= 28 MPa. Design the28 MPa. Design thereinforcement for the end block,allowing a maximum of 0.85 MPafor the tension in the concrete.

Fig. 7-25. Transverse Tensile Stress @ End Block

109

79Torsional Strength

Because of the high shear strength of concrete coupled with its lowtensile strength, the failure of concrete beams in torsion seldomresults from shearing stresses as such but rather from principalresults from shearing stresses as such, but rather from principaltensile stress produced by the shearing stress. When the shearingstress v is combined with direct stress fc.stress v is combined with direct stress fc.

Fig. 7.26. Ultimate torque for reinforced and prestressed concrete members.

110

Fig. 8-7. Layouts for pretensioned

Fig. 8-8. Layouts for post-tensioned beams.(Principle of Superposition applies, Design cable

profile may not be actual cable profile forbeams.

profile may not be actual cable profile for construction so long as c.g.s. is maintained)

111

Equivalent Prestress Equivalent Prestress LoadLoad

Equivalent LoadPrinciple of Free Body applies

,0.0; 0.0 ( )x yM V for each segment= = @ 0.0 ( )Reaction Support Body forces=

112

Calculation of Calculation of Tendon GeometryTendon Geometry

Designer only needs to specify g y p yhigh, low and inflexion points. Specialist contractor will provide shop drawing for the whole cable profile for approval.

113

Calculation of Calculation of Tendon GeometryTendon GeometryPoint of InflexionPoint of Inflexion

114

Calculation of Calculation of Tendon Tendon GeometryGeometryyy

115

85 SpanDepth Ratio Limitations

For reasons of economy and aesthetics, higher spandepth ratios arealmost always used for prestressed concrete than for reinforced

i h i ibl b d fl i b hconcrete. Higher ratios are possible because deflection can be muchbetter controlled in prestressed design. But when these ratios get toohigh camber and deflection become quite sensitive to variations inhigh, camber and deflection become quite sensitive to variations inloadings, in properties of materials, in magnitude and location ofprestress. Furthermore, the effects of vibration become morep ,pronounced.

Table 8-3 Approximate Limits for Span-Depth Ratios

116

10 1 Continuity Pros andCons101 Continuity, Pros andConsA simple comparison between the strength of simply supported and acontinuous beam will demonstrate the basic economy inherent incontinuous construction of prestressed concrete.

Fig. 10-1. Load-carrying g y gcapacity of a simple beam.

( )2 2' 8 ' '' ' ' 10 18w L T aT a w

L= =

( )' 2 ' 216 ' '2 ' ' 10 28c cw L T aT a w

L= =

Fig. 10-2. Load-carrying capacity of a continuous

beam.117

101 Continuity, Pros andCons

Comparing Fig. 101(c) with Fig. 102(c), or equation 101 with equation 102, it is readily seen that wc = 2w. This means that twice the load on thesimple span can be carried by the continuous span for the same amount ofsimple span can be carried by the continuous span for the same amount ofconcrete and steel. This represents a very significant basic economyrealized by continuous prestressed concrete structures. This strengthi h t i ti t ti ll t ti finherent in continuous construction means smaller concrete sections forthe same load and span, thus economy in design.

In reinforced concrete, the ve steel laps with the positive steel bars, andboth sets of ban are extended for additional anchorage, thus cancellingsome of the economy of continuity In prestressed concrete the samesome of the economy of continuity. In prestressed concrete, the samecable for the +M is bent over the other side to resist the M, with no loss ofoverlapping. In addition, continuity in prestressed concrete saves endpp g y panchorages otherwise required over the intermediate supports.

118

fig. 10-5. Layouts for partially continuous beamsbeams.

PC continuity over support.

RC continuity over support.

Fig. 10-4. Layouts for fully continuous beams.

119

103Analysis, ElasticTheory

i h li i f h i i bOwing to the application of prestress, the moments in a continuous beamare directly affected by the prestress and indirectly by the supportreactions induced by the bending of the beam. In a simple beam, or anyreactions induced by the bending of the beam. In a simple beam, or anyother statically determinate beam no support reactions can be induced byprestressing.

Consider a prestressed simple beam, Fig. 108(a). Nomatter howmuch thebeam is prestressed, only the internal stresses will be affected byp , y yprestressing. The external reactions, being determined by statics, willdepend on the dead and live load (including the weight of the beam), butare not affected by the prestress Without load on the beam no matterare not affected by the prestress. Without load on the beam, no matterhow we prestress the beam internally, the external reactions will be zero,hence the external moment will be zero.

120


Wi h l h b h i l i iWith no external moment on the beam, the internal resisting momentmust be zero, hence the Cline (which is the line of pressure in theconcrete): must coincide with the Tline in the steel (which is the c.g.s.) ( gline), as in (b). The Cline in the concrete being known, the moment in theconcrete at any section can be determined byM Te CeM =Te =Ce.

Fig. 10-8. Moment in concrete due to prestressing in a simple beam.

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ffThe difference between a simple and acontinuous beam under prestress can berepresented by the existence of secondaryrepresented by the existence of secondarymoments. Once these moments over thesupports are determined they can bei t l t d f i t l th binterpolated for any point along the beamwhich vary linearly. These moments are calledsecondary, because they are by products ofy, y y pprestressing and because they do not exist in astatically determinate beam. The termsecondary may be misleading sincesecondary may be misleading, sincesometimes the moments are not secondary inmagnitude but play a most important part ingthe stresses and strength of the beam.

Fig. 10-9. Moment In concrete due to prestressing in a p gcontinuous beam.

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Now, the moment in the concrete given by the eccentricity of the prestressis designated as the primary moment, such as would exist if the beam weresimple With known primary moment acting on a continuous beam thesimple. With known primary moment acting on a continuous beam, thesecondary moments caused by the induced reactions can be computed. Theresulting moment due to prestress, then, is the algebraic sum of theprimary and secondary moments.The following gives a procedure for computing directly the resultingmoments in the concrete sections over the supports, based on the momentmoments in the concrete sections over the supports, based on the momentdistribution method. Once the resulting moments are obtained, thesecondary moments can be computed from the relation:

Secondary moment (Ms) + Primary moment (Te) = Resulting moment (Mp)

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There is no significant reason for preferring a concordant cable, there iseven less justification for locating a nonconcordant cable for the sake ofnonconcordancy The real choice of a good c g s location depends on thenon concordancy. The real choice of a good c.g.s. location depends on theproduction of a desirable Cline and the satisfaction of other practicalrequirements, but not on the concordancy or nonconcordancy of the

blcable.

Fig. 10-17. Properties of non-concordant cables.

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106Cracking andUltimate Strength

Tests have shown that the elastic theory can be applied to continuousprestressed concrete beams with great accuracy as long as the concretep est essed co c ete bea s t g eat accu acy as o g as t e co c etehas not cracked.The percentage by which the support moment increased or decreased in

ti ith i l ti t di t ib ti i i b th f ll iconnection with inelastic moment redistribution is given by the followingexpression from theACI Code:

' + ( )'

' '

20 10 60.30

; ' ' '

p

y ys s

a

f fA Awhere and andf bd f bd

+ = = = =

The secondary moment acting should be considered in checking ultimatestrength of continuous members. When se

Prestressed Concrete

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