This paper addresses the justification for removalof two compressive stress limits, called hereafterLimits 1 and 2, which are currently used in boththe ACI 318 Building Code and AASHTO BridgeSpecifications to limit concrete stresses due toeffective prestress combined with applied loads.Removal of these limits provides relief in sizing ofmembers that have small compression zones thatare subjected to large bending moments, such asinverted tee bridge members in the positivemoment zones and I-girder bridges madecontinuous for superimposed loads in the negativemoment zone. Limit 1 is 0.6 f ′c, where f ′c is thespecified compressive strength at service. It isimposed on concrete stresses due to the effectiveprestress combined with full dead plus live loads.Limit 2 is 0.45f ′c. It is imposed on stresses due tothe effective prestress combined with dead loads.It is proposed that strength be used as the primarydesign criterion to determine member capacity incompression at various loading stages, includingprestress transfer, lifting, erection, deck weight,superimposed dead load and live load. Variousserviceability checks can additionally be used asneeded to control camber, deflection, vibration,and cracking. Design steps and numericalexamples are given. Also, proposed changes to theACI 318 Building Code and to the AASHTO-LRFDBridge Design Specifications are given.

Elimination of PrestressedConcrete Compression Limits at Service Load

2 PCI JOURNAL

Panya Noppakunwijai, Ph.D.Research Assistant Professor Department of Civil EngineeringUniversity of Nebraska-LincolnOmaha, Nebraska

Nabil Al-Omaishi, Ph.D., P.E.Research Assistant Professor

Department of Civil EngineeringUniversity of Nebraska-Lincoln

Omaha, Nebraska

Maher K. Tadros, Ph.D., P.E.Cheryl Prewett ProfessorDepartment of Civil EngineeringUniversity of Nebraska-LincolnOmaha, Nebraska

Gary L. Krause Ph.D., P.E.Associate Professor

Department of Civil EngineeringUniversity of Nebraska-Lincoln

Omaha, Nebraska

November-December 2002 3

The restrictions on concrete com-pressive stress at service limitstate in the design of prestressed

concrete flexural members in theAmerican Concrete Institute BuildingCode (ACI 318-02),1 and AmericanAssociation of State Highway andTransportation Officials (AASHTO)Load and Resistance Factor Design(LRFD) Bridge Design Specifications2

appear to be based on historical justifi-cation. They neither directly addressserviceability conditions, such as cam-ber and deflection limits, nor ensureadequate strength under factoredloads.

When an inverted tee bridge super-structure is designed and the workingstress design compressive stress limitsare checked, its capacity is often con-trolled by the limit imposed by the ef-fective prestress plus dead load, as cal-culated by the standard elasticitytheory without regard to stress-strainnonlinearity or creep redistribution ofstresses between the precast beam andthe cast-in-place topping.

The same problem sometimes arisesin bridge I-girders in integral bridgesystems, i.e., when the girders aremade continuous with cast-in-placeconcrete diaphragms and continuitynegative moment reinforcement in thedeck. Bottom fiber stresses can exceedthe allowable limits, even though thestrength of the negative moment sec-tion meets all code requirements. Thissituation forces designers to either ig-nore the code requirements or unnec-essarily select a larger member size.

Removal of the unnecessary com-pression limits could result in creationof a series of more efficient precast,prestressed concrete products. This,coupled with increasing acceptance ofself-compacting concrete, would makeproducers and their engineers develop,for example, efficient inverted doubletee members that would be connectedin the field with a thin topping, creat-ing a very efficient building floor witha shallow depth and a smooth soffit.Many existing products could be opti-mized, reducing their weights and in-creasing their capacities.

As prestressed concrete was beingintroduced in North America, about 50years ago, it was believed to be impor-tant to double-check its adequacy with

both “working stress design,” whichwas the prevalent design method, and“strength design,” which was gettingintroduced at that time. Convention-ally reinforced concrete, on the otherhand, was allowed to be designedstarting with the 1963 edition of theACI 318 Code by either the workingstress design method or the strengthdesign method, but not both methods.The overly conservative compressivestress limits in current working stressdesign practice of prestressed concretecontrol the design in a number of ap-plications and thus limit the utilizationof the full potential of prestressed con-crete.

In an earlier paper,3 a strength-baseddesign method for prestressed con-crete flexural members at transfer ofprestress was presented as a replace-ment of working stress design and thecompressive stress limit of 0.60f ′ci,where f ′ci is the specified compressivestrength of concrete at initial prestresstransfer.

This paper presents a historicalbackground of compressive stress lim-its at service load conditions, and pro-vides a justification for total removalof these limits. The two limits beingimposed on concrete in compressionare Limit 1, 0.60f ′c, on concrete stressdue to effective prestress, after al-lowance for long-term losses, com-bined with full loads, and Limit 2,0.45f ′c, on stress due to effective pre-

stress plus dead load. Limits 1 and 2in the AASHTO-LRFD Specificationsare identical to those in the ACI 318Building Code.

A third limit, unique to bridge de-sign, is 0.40f ′c. It is required to be ap-plied in AASHTO-LRFD to concretestresses due to 50 percent of effectiveprestress plus 50 percent of dead loadplus 100 percent of live load. It is in-tended to control fatigue due to liveload, which is a repeated cyclic loadcaused by moving vehicles during thelife of a bridge. However, the provi-sions for that limit should be movedfrom their current location to the sec-tion that deals with design for fatigueeffects. In that section, the fatiguetruck loading model, rather than thestandard truck loading model used forstrength calculations, should be usedto determine the live load stresses.With these two suggested modifica-tions, this limit should not be re-moved, and thus the topic is not dis-cussed further in this paper.

The AASHTO Standard Specifica-tions for Highway Bridges(AASHTO-STD)4 are being phasedout. They have similar limits to thosein AASHTO-LRFD, except that Limit2 in AASHTO-STD is 0.40f ′c ratherthan 0.45f ′c. Both values were intro-duced in the 1994 editions of the twosets of AASHTO Specifications. Thedifference is believed to have occurredby an arbitrary decision by the com-

Fig. 1. Concrete compression stress limits according to AASHTO Specifications.

Limit 1: fd + fl ≤ 0.6f ′c

Limit 2: fd ≤ 0.45f ′c

Limit 3: fd + 2fl ≤ 0.8f ′c

4 PCI JOURNAL

mittee in charge of maintaining thesespecifications. For clarity, only the0.45f ′c value is used in further discus-sion in this paper.

The three limits of allowable com-pressive stresses in AASHTO-LRFDcan be written as follows:

Limit 1: fd + fl ≤ 0.6f ′c (1)

Limit 2: fd ≤ 0.45f ′c (2)

Limit 3: fd + 2fl ≤ 0.8f ′c (3)

where fd = compressive stresses due to

the sum of effective prestressand permanent loads after al-lowance for time-dependentlosses

fl = compressive stresses due tolive load

Fig. 1 shows the domain of limits onfd and fl to satisfy each of Limits 1, 2and 3. For example, when the live loadstress, fl, is equal to 0.2f ′c or higher,the maximum dead load plus effectiveprestress, fd, is controlled by Limit 3which controls concrete fatigue. Whenfl is less than 0.15, Limit 2 controlsthe maximum allowed fd. Thus, Limit1 has practically no impact on designby the current LRFD provisions, asseen in Fig. 1.

It can be shown, using a similargraph as in Fig. 1, that Limit 1 nevercontrols design according to theAASHTO-STD Specifications. There-fore, removal of Limit 1 saves an un-necessary design check since eitherLimit 2 or Limit 3 supersedes it. Theproposal in this paper, however, is toremove both Limits 1 and 2.

In the following sections, a histori-cal overview is given. It is followed

by a discussion justification for re-moval of Limits 1 and 2. An alternatestrength-based design procedure ispresented and illustrated with numeri-cal examples. Changes to the affectedsections in the ACI 318 Code and inthe AASHTO-LRFD Specificationsare proposed.

HISTORICALBACKGROUND

The Prestressed Concrete Institute(PCI) issued the first specifications5

for bonded pretensioned prestressedconcrete on October 7, 1954. In thosespecifications, the maximum allow-able compressive stresses under finaldead and live loads was 0.40f ′c forbridge members and 0.44f ′c for build-ing members.

In 1954, the Bureau of Public Roads(BPR) published the first edition ofCriteria for Prestressed ConcreteBridges, by E. L. Erickson,6 who atthe time was chief of the Bridge Divi-sion, Bureau of Public Roads (BPR).That bureau was later renamed theFederal Highway Administration(FHWA).

At that time, there was a consider-able difference of opinion among thevarious authorities as to how muchcompression to allow in the extremeconcrete fiber under dead, live, andimpact load. Several authorities (forexample, Hajnal-Konyi, Dobell, Leon-hardt, Muller, Billig) at that time had

proposed different limits ranging from0.33f ′c to 0.5f ′c as reported in Refer-ence 6. According to Article 3.4.11 ofthe 1953 “AASHO Standard Specifi-cations for Highway Bridges,” an al-lowable compressive stress limit of0.4f ′c was given for conventional con-crete subjected to dead, live, or impactloads. The Bureau of Public Roadsconservatively adopted the same stresslimit for prestressed concrete struc-tures.

The allowable compression stresslimit of 0.45f ′c appeared in the firstpublished draft of the ACI-ASCEJoint Committee 323 Report “Recom-mended Practice for Prestressed Con-crete”7 in 1958. The early code writersfelt that this limit was conservativelyestablished to decrease the probabilityof failure of prestressed concretemembers due to repeated loads and topreclude excessive creep deforma-tions. In the 1958 report, “TentativeRecommendations for PrestressedConcrete” and 1959 report “PCI Stan-dard Building Code for PrestressedConcrete (Tentative),” the 0.45f ′c limitwas retained, but there was no clearcommentary as to why this was done.

Although the 0.45f ′c limit was totallyremoved for non-prestressed concreteat the time of the introduction ofstrength design, this limit was kept un-changed until 1995 for prestressedconcrete. The ACI 318-95, the 1994AASHTO-LRFD and the 1996AASHTO-STD Specifications adopted

Fig. 2. Cross section of axially prestressedmember example.

Fig. 3. Comparison between results of linear and nonlinear analyses.

November-December 2002 5

changes based on the work by Huo etal.8 That reference gave justificationfor a 33 percent increase in the limit to0.60f ′c, and for the introduction ofLimit 2, due to the effective prestressplus dead load. It was believed, bysome code committee members, thatthe 0.45f ′c limit should not be ex-ceeded to avoid dealing with the non-linear creep of concrete at higherstress levels.

Since live load is considered tran-sient, it was indicated that a 33 percentadditional stress would not be detri-mental. Huo et al.8 indicated that thechanges were intended to be transi-tional, with the eventual goal of totalelimination of compressive stress lim-its. It was further demonstrated that, ifworking stress had to be performed ona conventionally reinforced rectangu-lar section example that was alreadydesigned with strength design, the ex-treme fiber compressive stress couldbe as high 0.83f ′c.

FLEXURAL BEHAVIOR OFPRESTRESSED MEMBERS

Two sources of nonlinearity exist inthe flexural behavior of prestressedconcrete, namely, nonlinearity due toincreasing load and nonlinearity due totime-dependent volume change effects.The first part of this section deals withthe nonlinearity of the concrete stress-strain relationship and its effect onmember condition at time of prestresstransfer. It will be shown that the linearelastic calculation of compressivestress is not compatible with the 0.6f ′climit which falls outside of the linearrange of the stress-strain diagram, noris it representative of a constant safetyindex against concrete crushing.

The second part of this sectiondemonstrates that continuous stress re-distribution occurs due to creep andshrinkage of concrete in a compositemember, consisting of two concretecomponents. When that effect is takeninto account, the top fiber stresses ofthe precast concrete component of themember undergo considerable relief.Therefore, a linear elastic analysis forcompressive stress, and limiting it to0.6f ′c, does not correspond to the ac-tual member behavior. On the otherhand, methods are available, which

Table 1. Inverted tee section properties. Area Total depth yb I Sb St (beam) St (deck)

Section Ed/Eb (sq in.) (in.) (in.) (in.4) (in.3) (in.3) (in.3)Beam – 252.0 24.00 8.14 12,235 1503 771 –Deck – 144.0 6.00 27.00 432 – – –

Composite 1.00 396.0 30.0 15.00 45,252 3017 5028 3017Age-adjusted

0.66 347.0 30.0 13.31 37,053 2784 3466 2220composite

Note: 1 in. = 25.4 mm; 1 sq in. = 645 mm2; 1 in.3 =16,387 mm3; 1 in.4 = 416,231 mm4.

will be summarized below, that allowdesigners to directly check for mem-ber strength against concrete crushing,or for member serviceability againstexcessive deflections.

Analysis at Transfer due toIncreasing Prestress Level

Huo et al.9 considered an exampleof an axially prestressed concretemember, summarized below. Thecross section is shown in Fig. 2. Con-crete and steel stresses and strains dueto a progressively increased number ofprestressing strands up to the failureload, are shown in Fig. 3. The strandsare assumed to be arranged to producea concentric load as their number in-creases.

Two methods of analysis, namely,standard linear analysis and nonlinearanalysis, are considered. As shown inFig. 3, both standard linear analysisand the more accurate nonlinear anal-ysis produce about the same results, amaximum of 26 strands, when theconcrete stress is at the 0.6f ′ci codelimit. The strains are not as close asthe stresses between the two models.

Linear analysis appears to indicatethat concrete reaches the strength f ′ci

when 44 strands are used. However,nonlinear analysis more correctly indi-cates that concrete continues to resistplacement of more prestressing up to62 strands when the strain reaches theassumed ultimate strain of 0.003.

This figure confirms three facts: 1. Concrete crushes when its strain

reaches ultimate strain, not when thestress reaches its peak elastic analysisvalue.

2. As prestressing increases, so doesconcrete strain and prestress losses.Thus, unlike externally induced com-pression, prestressing has a self-reliev-ing mechanism.

3. Member failure cannot be accu-rately predicted without utilization ofthe entire stress-strain diagramthrough nonlinear analysis andstrength based design.

Analysis of Composite MembersAfter Prestress Transfer

In standard practice in current use,linear elastic behavior is employed tocheck tension and compression limitsin concrete and to calculate camberand deflection. Concrete, as well assteel, is assumed to follow Hooke’sLaw, i.e., stress equals the product of

Fig. 4. Cross-sectionaldimensions ofinverted tee, 82 ftspan bridge girder.

6 PCI JOURNAL

strain and modulus of elasticity. Pre-stress losses due to creep and shrink-age of concrete and relaxation of pre-stressing steel are calculatedseparately and assumed to be exter-nally introduced as “negative pre-stress” in the linear elastic analysis.

When these assumptions are made,true time-dependent stress redistribu-tion cannot be determined and design-ers are left with, at best, nominal stressvalues that may be highly exagger-ated. The following four numerical ex-amples relate to the same bridgebeam: • The first example illustrates the con-

ventional elastic stress analysis forstresses at various stages of loading.

• The second example shows howtime-dependent analysis can be con-ducted and how much the results dif-fer from those of the elastic analysis.

• The third example deals with cam-

ber and deflection calculations. • The fourth example shows how

safety can be directly satisfiedagainst failure by conducting astrength analysis of the member.Due to space limitations, only seg-ments of the full example will begiven. A more detailed treatment ofthe topic is given in Reference 10.

Example 1: Elastic Analysis of anInverted Tee Bridge Beam

A precast concrete bridge super-structure consists of an inverted teebeam, with a span of 82 ft (24.99 m),and a spacing of 2 ft (0.61 m). Thebeam is made composite with a 6 in.(152 mm) thick cast-in-place slab. Fig.4 shows the concrete dimensions andsteel arrangement. Table 1 gives thesection properties of the precast beamand composite section.

Concrete strength of the precast sec-tion at transfer, f ′ci = 6 ksi (41.37 MPa)and at service, f ′c = 8 ksi (55.16 MPa).The deck concrete strength is 6 ksi(41.37 MPa).

Moments at midspan: Mg = 2648kip-in. (299 kN-m), Md = 1513 kip-in.(171 kN-m), Ml = 5295 kip-in. (598kN-m) due to self-weight, deckweight, and live load, respectively.

The stress in the prestressing steelhas been determined through time-de-pendent analysis to be 184.21 ksi(1270 MPa), 173.49 ksi (1196 MPa)and 160.28 ksi (1105 MPa) immedi-ately after transfer, at time of deckplacement and at time infinity, respec-tively. The corresponding prestressforce values are: 620.1, 584.0 and539.5 kips (2758, 2598 and 2400 kN).

The precast concrete top fiber stress,ft, is calculated using the followingelastic stress analysis formula:

Fig. 5. Concretestresses in aninverted tee

bridge girder atvarious loading

stages.

-0.536 ksi

-0.286 ksi

1.755 ksi

November-December 2002 7

where P = prestress forcee = eccentricity of the prestressing

forceM = external load momentA = section areaSt = section modulus

At transfer:

At erection:

At final time:

The stresses at other locations in the

section are calculated in a similar way.The results are shown in Fig. 5 andTable 2.

Both the ACI 318 Code and theAASHTO LRFD Specifications re-quire that the stress due to permanentloads not exceed 0.45f ′c = 3.6 ksi(24.82 MPa). The stress at erection of3.817 ksi (26.32 MPa) exceeds thatlimit. Both codes require that thestress due to full load not exceed0.60f ′c = 4.8 ksi (33.09 MPa). Thestress of 4.990 ksi (34.41 MPa) ex-ceeds that limit. Thus, according toelastic analysis, the beam fails to meetthe code requirements due to bothloading combinations.

Time-Dependent StressRedistribution in Composite Member

The purpose of this section is todemonstrate two points:

1. If there is a need for serviceabil-ity checks, means already exist to cal-culate the design parameters and tocheck against required limits. This ap-plies to deflection, camber, cracking,and other parameters.

2. Composite members consisting ofprecast, prestressed concrete and cast-in-place concrete components undergocontinuous stress redistribution that

results in significant reduction in theelastic compressive stress at the topfiber of the precast component. Suchredistribution cannot be determinedthrough elastic analysis (see, for ex-ample, Kamel11). Thus, the standardpractice of elastic analysis, which isbased on limiting the top fiber com-pressive stress, is misleading.

The theory that Kamel used in-volved an elaborate analysis of time-dependent effects, based on earlierwork by Tadros et al.12 The time his-tory, in that analysis, is divided intomany small steps. As the size of thetime-step decreases, the accuracy ofthe analysis increases. However, thatmethod is suited for commercial soft-ware solutions.

A slightly less accurate, but greatlysimplified, method presented by Dil-ger13 is suitable for hand-calculationand designer-developed spreadsheetsolutions. Dilger’s “age-adjusted ef-fective modulus” method is the oneadopted in this paper for further dis-cussion.

The age-adjusted effective modulusmethod is similar to conventional elas-tic analysis. Instead of using the con-ventional modulus of elasticity to de-termine transformed sectionproperties, the age-adjusted effectivemodulus is used. In addition, deforma-

ft = − +

+ +

=

539 50

252

539 50 5 14

7712648

771

1513

771

5295

50284 990

. . ( . )

.

ksi (34.41 MPa)

ft = −

+ +

=

583 97

252

583 97 5 14

7712648

771

1513

7713 817

. . ( . )

.

ksi (26.32 MPa)

ft = − +

=

620 05

252

620 5 5 14

771

2648

7711 759

. . ( . )

. ksi (12.13 MPa)

fP

A

Pe

S

M

Stt t

= + + (4)

Table 2. Summary of stresses at various loading stages. Linear elastic analysis

Transfer Erection FinalTop beam Bottom beam Top beam Bottom beam Top deck Bottom deck Top beam Bottom beam

Loads (ksi) (ksi) (ksi) (ksi) (ksi) (ksi) (ksi) (ksi)P/A 2.461 2.461 2.317 2.317 – – 2.141 2.141Pe/S -4.133 2.122 -3.892 1.999 – – -3.596 1.847Mg/S 3.431 -1.762 3.431 -1.762 – – 3.431 -1.762Md/S – – 1.961 -1.007 – – 1.961 -1.007Ml/S – – – – 1.755 1.053 1.053 -1.755

Total 1.759 2.821 3.817 1.547 1.755 1.053 4.990 -0.536

Limiting 0.6f ′ci 0.6f ′ci 0.45f ′c 0.45f ′c 0.6f ′c 0.6f ′c 0.6f ′c 6

stresses 3.600 3.600 3.600 3.600 3.600 3.600 4.800 -0.537OK OK High OK OK OK High OK

Time-dependent analysisAt erection 3.817 1.547 – – 3.817 1.547Differentialcreep and – – 0.198 0.209 -0.830 0.251shrinkage

Prestress loss – – – – 0.120 -0.329Live load 1.755 1.053 1.053 -1.755

Total 3.817 1.547 1.953 1.262 4.160 -0.286High OK OK OK OK OK

′fc

Note: 1 ksi = 6.895 MPa.

8 PCI JOURNAL

tion due to shrinkage and creep ofconcrete, and relaxation of prestress-ing steel, are considered during a timeinterval as “initial” strains, much likepseudo-elastic analysis for effects oftemperature change.

To incorporate initial strains into thestress analysis, three steps are fol-lowed:

1. Separate various concrete andsteel components of the cross sectionand allow them to deform freely.

2. Restrain the components of thesection and calculate the restrainingforces to bring the initial strains calcu-lated in Step 1 to zero.

3. Reattach the various componentsto each other and restore equilibriumby applying equal and opposite forcesto those calculated in Step 2.

The sum of stresses in Steps 2 and 3is the net stress, and the deformationin Step 3 is the net deformation. InSteps 2 and 3, the modulus of elastic-ity of concrete is adjusted by dividingEc by (1 + χψ), where χ is the agingcoefficient to allow for gradual intro-duction of time-dependent stress in-crements, and ψ is the creep coeffi-cient. To illustrate this procedure, thefollowing example provides a detailedcalculation of stress, strain and defor-mation due to time-dependent effectsin a composite section.

Example 2: Time-DependentAnalysis of the Beam of Example 1

This example shows the differencein results with the elastic analysiswhen time-dependent stress redistribu-tion is accounted for. The input data isthe same as in Example 1. Addition-ally, the following time-dependentproperties have been estimated usingthe formulas in Reference 14 assum-ing the member is subject to ambientrelative humidity of 70 percent. Othersources of prediction of material prop-erties are AASHTO LRFD Specifica-tions, and the PCI Bridge DesignManual (PCI-BMD).15

The modulus of elasticity of the pre-cast concrete at service, Eb = 5422 ksi(37.39 GPa). An average compressivestrength of the deck of 6 ksi (41.37MPa) is assumed for the period be-tween the time it starts to act compos-itely with the girder and time infinity.

The corresponding modulus of elastic-ity, Ed = 4696 ksi (32.38 GPa).

Creep and shrinkage of precast con-crete between erection and time infin-ity, ψbdf = 0.65, and εbsh = 150 × 10-6.

Creep and shrinkage of deck con-crete ψddf = 1.3, and εdsh = 300 × 10-6.

The aging coefficient is taken con-stant for both materials, χb = χd = 0.7.For clarity of presentation, the age-ad-justed transformed section propertiesare calculated assuming an area ofsteel equal to zero.

Note that a more detailed set of cal-culations of the same example, includ-ing the interaction of the prestressingsteel may be found in Reference 10.Using the actual area of steel in calcu-lation would automatically account fortime-dependent prestress losses. Inthis example, the prestress losses esti-mated for Example 1 will be assumed.

Because of the simplified assump-tions used in this example, the con-crete stress increments due to variousactions are identical to those in Exam-ple 1, with one exception. Once thedeck becomes composite with thegirder, differential creep and shrinkagebetween the two concretes cause con-tinuous redistribution in the concretestresses. The calculations will, thus,focus on the time interval betweenerection and time infinity.

The analysis steps explained earlierwill be followed.

Step 1

Separate the two section compo-nents and allow each to deform freely.

Axial strain in precast beam:

Curvature in precast beam:

= 1.134 × 10-5 in.-1 (4.464 × 10-7 mm-1)

Axial strain in deck = 0.0003

Step 2

Calculate the forces and the corre-sponding stresses in each componentthat cancel the deformations in Step 1.For this analysis, the age-adjusted ef-fective modulus for each material isused.

Age-adjusted effective modulus forprecast beam:

Age-adjusted effective modulus fordeck:

Axial restraining force in precastbeam:

= –(free axial strain)E*bAb

= –4.278(3727)(252)(10-4) = –401.75 kips (–1787.08 kN)

Corresponding stress in precastbeam:

= –401.75/252= –1.594 ksi (–10.99 MPa)

The bending restraint force in thebeam and the axial restraining force inthe deck can be calculated similarly tobe –516.97 kips-in. (–2300 kN) and–106.21 kips (–472.46 kN).

The corresponding stresses in thebeam are:

Top fibers, –2.264 ksi (–15.61MPa);

Bottom fibers, –1.250 ksi (–8.62MPa); and in the deck are: top fibers =bottom fibers = –0.738 ksi (–5.09MPa).

Step 3

Now that the compatibility of thetwo components has been restored inStep 2, the two components can be

EE

dd

b bdf

*

( . )( . )

=+

=+

=

1

4696

1 0 7 1 3

2459 ksi (16.95 GPa)

χ ψ

EE

bb

b bdf

*

( . )( . )

=+

=+

=

1

5422

1 0 7 0 65

3727

ksi (25.70 GPa)

χ ψ

M

EI bdfψ

=+[ ]2648 1513 583 97

5422 12 235

– .

,

(5.14)

( )(0.65)

P

AE bdf bshψ ε

+

= +

= × −

583 97

252 54220 65 0 00015

4 278 10 4

.

( )( . ) .

.

November-December 2002 9

“reattached” and equilibrium restoredby applying equal and opposite forcesto the forces obtained in Step 2. Toundertake the analysis in this step, theage-adjusted composite section prop-erties must be calculated and used.These can be found using a deck-to-beam modular ratio = E *

d/E *b =

2459/3727 = 0.66. The section proper-ties are shown in Table 1.

All the component forces in thisstep are then combined into an axialforce at the centroid of the age-ad-justed composite section.

Axial force: = 401.75 + 106.21 = 507.96 kips (2259.53 kN)

Bending moment:= –102.58 kip-in. (–11.59 kN-m)

The corresponding top fiber stress inthe beam is:

The stresses at the extreme fibers ofthe deck and at the bottom fibers ofthe beam are calculated similarly.

Step 4

The summation of all stresses anddeformations in Steps 1 through 3 pro-duce the net incremental values due tocreep and shrinkage between at timeof erection and time infinity. Thetime-dependent top fiber stress in thebeam, for example, equals –2.264 +1.434 = –0.830 ksi (–5.72 MPa). Othervalues are given in Table 2.

The additional prestress losses insteel between erection and final timewere assumed to be the same as in Ex-ample 1. Thus, a concrete stresschange due to a steel stress loss of:

= 160.28 – 173.49 = –13.21 ksi (–91.08 MPa)

needs to be included. As shown inTable 2, the stress change at the topfibers is 0.12 ksi (0.83 MPa).

The net stresses, including time-de-pendent effects, are equal to thestresses at the erection plus the time-

ft = − −

=

507 96

347

102 58 24 13 31

37 053

. . ( . )

,

1.434 ksi (9.89 MPa)

dependent stress increments. For thebeam top fibers, that net stress is:

3.817 – 0.830 + 0.12 = 3.107 ksi (21.42 MPa)

This stress is below the code limitof 0.45f ′c = 3.6 ksi (24.82 MPa).

Adding live load stresses as shownin Example 1, the beam top fiberstress equals:

3.107 + 1.053 = 4.160 ksi (28.68 MPa).

This again is less than the code limitof 0.6f ′c = 4.8 ksi (33.09 MPa).

A comparison of the results of Ex-amples 1 and 2 leads to the conclusionthat a linear elastic analysis does notaccount for the generally significantstress relief of the compressive stressin the top fibers of the precast compo-nent of a composite section, and leadsto the false impression that thesefibers are overstressed.

It should be emphasized that thevalue of service load analysis is tocheck member serviceability, i.e., tocheck for excessive deformation and

cracking, and not to check againstconcrete crushing which is the func-tion of strength design.

Example 3: Camber and DeflectionCalculations of the Beam ofExamples 1 and 2

This example illustrates the calcula-tion of the camber and deflection atthe midspan of the beam of Examples1 and 2. Numerical integration of thecurvature may be used to obtain thedeflection.

For this example, the deflection-cur-vature relationships δ = φcl2/8 due toprestressing and δ = 5φcl2/48 due togravity load, given in Reference 16, areused. The symbols δ, φc, and l repre-sent midspan deflection, midspan cur-vature, and span length, respectively.

(1) Instantaneous camber at initialprestress transfer

Design for prestress transfer, usingthe strength design method proposedin Reference 3, indicates that no drap-ing or debonding of the strands isneeded if the concrete strength at pre-stress transfer, f ′ci, is set at 5.74 ksi

Fig. 6. Comparison of stress diagrams between working stress design and strength design.

10 PCI JOURNAL

(39.58 MPa) and if two No. 4 Grade60 bonded crack control mild steel re-inforcing bars are placed at the top ofthe member in the end zone. Thus, itcan be assumed for curvature andcamber calculations that the prestressimposes a constant bending momentover the full span length.

(a) Due to prestress force

=–5.550 ×10-5 in.-1 (–2.185 ×10-6 mm-1)

(b) Due to self-weight moment

= 4.608 × 10-5 in.-1 (1.814 × 10-6 mm-1)

Net initial camber at transfer:

= –6.72 + 4.65 = –2.07 in. (–53 mm)

(2) Camber increment betweentransfer and erection due to initialloads

To account for concrete creep andshrinkage, the age-adjusted modulusof elasticity calculated in Example 3,is used in this step to determine the in-cremental curvature.

=–6.124 ×10-6 in.-1 (–2.411 ×10-7 mm-1)

(3) Deflection increment due toprestress loss between release anderection

∆P = (173.49 – 184.21)3.366 = –36.08 kips (–160.51 kN)

= 2.797 × 10-6 in.-1 (1.101 × 10-7 mm-1)

δ φ=

= × ×

=

−

cl2

6 2

8

2 797 10 82 12

80 34

. ( )

. in. (9 mm)

φc

Pe

EI=

=

∆

36 08 5 14

5422 12 235

. ( . )

( , )

δ φ=

= × ×

=

−

cl2

6 2

8

6 124 10 82 12

8

–

– 0.74 in. (–19 mm)

. ( )

φ ψc

bidM

EI=

= − ××

( . . )( . )

,

2648 620 05 5 14 0 65

4696 12 235

δ φ=

= × ×

=

−

5

48

10 82 12

484 65

2

5 2

cl

5(4.608

in. (118 mm)

)( )

.

φcgM

EI=

=×

2648

4696 12 235,

δ φ=

= × ×( )

=

−

cl2

5 2

8

5 550 10 82 12

8

–

– 6.72 in. (–171 mm)

.

φc

Pe

EI=

=×

–620 05 5 14

4696 12 235

. ( . )

,

Net camber before deck placement:

= –2.07 – 0.74 + 0.34 = –2.47 in. (–63 mm)

(4) Instantaneous deflection due todeck weight

= 2.281 × 10-5 in.-1 (8.980 × 10-7 mm-1)

Net camber immediately after deckplacement

= –2.47 + 2.30 = –0.17 in. (–4 mm)

An accurate estimate of camber justbefore deck placement allows for acorrect prediction of concrete deckthickness directly over the girder(haunch thickness), that would accom-modate member camber and produce asmooth roadway profile. An accurateprediction of camber is also an indica-tion of good correlation between thedesign calculations and actual behav-ior. There is no code limit on camberat erection. Some designers prefer tohave a positive, i.e., upward, camberjust after deck placement, for aestheticpurposes.

(5) Time-dependent camber afterdeck placement to final load

The curvature change due to time-dependent effects can be calculatedbased on a bending moment obtainedfrom Step 3 of the Example 2:

= –7.429 × 10-7 in.-1

(–2.925 × 10-8 mm-1)

∆φcb

M

E I=

=×

*

.

,

102 58

3727 37 053

δ φ=

= × ×

=

−

5

48

5 2 281 10 82 12

48

2

5 2

cl

( . )( )

2.30 in. (58 mm)

φcdM

EI=

= 1513

5422 12 235( , )

Fig. 7. Time-dependent deflection.

November-December 2002 11

(6) Deflection due to prestress lossafter deck placement

= 3.447 × 10-6 in.-1 (1.357 × 10-7 mm-1)

Net deflection before live load ap-plication:

= –0.17 – 0.09 + 0.42 = 0.16 in. (4 mm)

(7) Deflection due to live load

= 2.158 × 10-5 in.-1 (8.496 × 10-7 mm-1)

The recommended live load deflec-tion limit for bridge superstructures is:

The calculated live load deflection,using a simple span assumption, indi-cates that the member is too shallowor the span is too long. A simple spanlength of 69 ft (21 m) would corre-spond to an acceptable live load de-flection. However, the importance ofintegral abutment diaphragms, whichare standard practice in Nebraska,

l

800

82 12

8001 23

= ×( )

= (31 mm).

δ φ=

= × ×

=

−

5

48

5 2 158 10 82 12

48

2

5 2

cl

( . )( )

2.18 in. (55 mm)

φclM

EI=

= 5295

5422 45 252( , )

δ φ=

= × ×

=

−

cl2

6 2

8

3 447 10 82 12

8

. ( )

0.42 in. (11 mm)

φc

Pe

EI=

= −

∆

–( . . )( . )( . )

( , )

160 38 173 49 3 366 5 14

5422 12 235

δ φ=

= × ×

=

−

cl2

7 2

8

7 429 10 82 12

80 09

–

– in. (–2 mm)

. ( )

.

should be noted. They create end rota-tional restraints which can reduce de-flection by up to 80 percent.

Example 4: Strength Analysis of theBeam of Examples 1 to 3

In addition to limiting service loadcompressive concrete stresses, theACI 318 Code and AASHTO Specifi-cations require that the strength of pre-stressed concrete members due to ef-fective prestress plus full loads bechecked. For composite members, thestrength of the precast concrete mem-ber due to all loads applied to it beforeit becomes composite with the deck, isnot required to be checked.

In both ACI 318 and AASHTO,non-prestressed reinforced concrete isrequired to be designed for strengthonly. In this example, strength designwill be performed to check the flexu-ral capacity of the precast memberalone and the capacity of the compos-ite member. The load and resistancefactors of the AASHTO LRFD Speci-fications are applied.

(a) Strength of the precast mem-ber. Loads applied are the memberweight and the deck weight. The re-quired moment is:

Mu = 1.25(Mg + Md) = 1.25(2648 + 1513) = 5201 kip-in. (587.59 kN-m)

The steel stress at flexural strengthmay be obtained using the strain com-patibility analysis procedure and the

stress-strain power formula given inthe PCI Bridge Design Manual:

where fps is the stress in seven-wire,270 ksi (1860 MPa) low relaxationstrands, corresponding to a total strainεps, which is the strain in the concreteat the same level as the steel row beingconsidered plus decompression strain(which is approximately taken as theratio of effective prestress to modulusof elasticity).

The results of the strain compatibil-ity analysis produce a neutral axisdepth, c = 21.57 in. (548 mm), com-pression block depth a = β1c =0.65(21.57) = 14.02 in. (356 mm),strain at the centroid of the tension re-inforcement = 0.006, and averagestress in the prestressing steel:

= 170.0 ksi (1172 ksi)

+ ×{ }

27 613

1 112 4 0 006007 7 36 1 7 36

,

( . . ) . / .

= +

0 006007 887.

fps

ps

ps

=

++{ }

εε

88727 613

1 112 4 7 36 1 7 36

,

( . ) . / .

fps

ps

ps

=

++{ }

(5)

εε

88727 613

1 112 4 7 36 1 7 36

,

( . ) . / .

Fig. 8. Beam specimen under loading.

12 PCI JOURNAL

For equilibrium, the tension forcemust equal the compression force (T =C).

fpsAp = 170.0(3.366) = 0.85f ′cba= 572 kips (2544 kN) OK

The stress level in the prestressingsteel is significantly lower than theyield strength of the strand of 0.9(270)= 243 ksi (1675 MPa). Thus, this pre-cast-only section should be consideredover-reinforced for this analysis. Thisis not a surprising outcome as the deckhas not yet hardened at this loadingstage.

Using Mast’s resistance factor inter-polation function given in the PCI

Bridge Design Manual:

0.7 ≤ φ = 0.2 + 0.3(dext/c) ≤ 1.0 (6)

φ = 0.2 + 0.3(22/21.57) = 0.51 ≤ 0.7

Therefore, use φ = 0.7

φMn = φfpsAp(dp – a/2) = 0.7(170.00×3.366)(21 – 14.02/2)= 5603 kip-in. (633 kN-m) > Mu = 5201 kip-in. (587.59 kN-m)

Thus, the section is adequate.

(b) Strength of the compositemember due to full dead plus liveload:

Table 3. NU 900 precast beam and composite section properties.

Total depth Cross-sectional Moment of Distance between Weight ofh area A inertia I centroid and bottom section

Section (in.) (sq in.) (in.4) fibers yb (in.) (lb/ft)Precast Beam

35.4 648.1 110,262 16.1 675.1NU 900

Composite 43.4 1237.9 282,790 27.27 1544

Note: 1 in. = 25.4 mm; 1 sq in. = 645 mm2; 1 in.4 = 416,231 mm4; 1 lb/ft = 0.0146 kN/m.

Mu = 1.25(Mg + Md) + 1.75(Ml) = 1.25(2648 + 1513) + 1.75(5295) = 12,467 kip-in. (1409 kN-m)

Using the strain compatibilitymethod, c = 8.07 in. (205 mm), a =5.25 in. (133 mm), εps = 0.0126, fps =254.50 ksi (1755 MPa), φ = 1.0, andφMn = 20,881 kip-in. (2359 kN-m) >Mu.

The strength analysis of the com-posite section in this example indi-cates that the deck slab provides thecompression resistance of the flexurecouple and that the top of the beamhas no impact on the capacity as thecompression block depth is within theflange. The compressive stress distri-bution near member failure is consis-tent with experimental observationsand is quite different from that ob-served in the working stress analysisof Examples 1 and 2, in which the topfibers of the beam appear to be themost stressed fibers and the slab ap-pears to be lightly stressed (see Fig.5). Thus, working stress analysisshould only be used for serviceabilitychecks under unfactored loads andstrength analysis should be used tocheck strength against overload, usingload magnification factors.

Discussion of Examples 1 to 4

The solution of Examples 1 to 4 hasdemonstrated the following:

1. Working stress design using thecustomary linear elastic analysis as-sumptions may grossly overestimatecompressive stresses at the extremefibers of the precast concrete compo-nent of a composite section.

2. Time-dependent analysis for dif-ferential creep and shrinkage betweenthe precast concrete beam and thecast-in-place composite topping isavailable to designers using existingsoftware and spreadsheets, such asExcel spreadsheet calculations. Theanalysis shows that the top fiber stressin the precast section is well below thecode limits, even though linear elasticanalysis would show an unacceptabledesign.

3. Spreadsheet time-dependent anal-ysis can be effectively used to calcu-late member camber at various stagesof construction and service life of the

Fig. 9. Cross section of bridge.

Fig. 10. Strand pattern at midspan and end of girder.

November-December 2002 13

member. This approach, rather thanartificial concrete stress limits, shouldbe the procedure for deflection con-trol.

4. Working stress analysis is onlyneeded for satisfaction of the bottomfiber tensile stress limit. It may also beneeded in the rare occasions of fatigueanalysis of the compression zone dueto live loads. Even in these cases, a re-alistic time-dependent analysis, ratherthan the traditional linear elastic anal-ysis, should be used.

Concrete capacity to resist the ap-plied loads with adequate safetyagainst crushing is best handledthrough strength analysis of both theprecast component of the member forloads introduced before the compositetopping is hardened, and again for thetotal section due to full loads. The lat-ter check is a current code require-ment.

Fig. 6 shows a comparison betweena typical stress diagram based onworking stress design and strength de-sign along the beam depth of the non-composite and composite sections. Astrength analysis of a composite sec-tion indicates that the depth of theequivalent rectangular compressionstress block is likely within the thick-ness of the topping deck. Thus, the im-pact of high compression in the topfiber of the precast component onmember strength is virtually nonexis-tent. A check of the strength of theprecast-only section for applicableloads should be added as a code re-quirement and the compression limitsshould be removed as proposed in thispaper.

EXPERIMENTAL PROGRAMSeveral full-scale specimens have

been tested in this research program.One objective of the experiments wasto determine whether camber and de-flection can be accurately estimatedwhen the code compression limits areexceeded. A second objective is toverify that no strength or ductility dis-advantages result from waiving thecode required compression limits. Afull description of the experimentalprogram is given in Reference 10.Here, only the results of one invertedtee specimen will be shown.

Table 4. Results of proposed versus conventional design of member inExample 5.

Conventional ProposedNumber of strands and Design Controlling Design Controlling

concrete strength parameter criterion parameter criterionNumber of strands

36Bottom stress

36Bottom stress

at midspan at service at serviceNumber of draped strands

4Compression limit

4Strength at

at ends (0.6f ′ci) transferPrecast concrete strength

7.06Compression limit

6.0Strength at

at transfer (ksi) (0.6f ′ci) transfer Precast concrete strength

7.11Compression limit

6.57Strength at

at service (ksi) (0.45f ′c) deck placementCast-in-place deck

4.0Strength under

4.0Strength under

concrete strength (ksi) full load full load

Note: 1 ksi = 6.895 MPa.

A 50 ft (15.24 m) long, 15.75 in.(400 mm) deep IT-400 was preten-sioned with 18 0.5 in. (12.7 mm) diam-eter bottom strands and two 0.5 in.(12.7 mm) diameter top strands, at theRinker Materials, Inc. plant inLaPlatte, Nebraska. The beam wasshipped to the Structures Laboratory ofthe University of Nebraska, and placedon supports for a period of 52 days be-fore a 6 in. (150 mm) thick by 48 in.(1220 mm) wide composite toppingwas placed. The composite beam wastested 20 days after the slab was cast.

The maximum compressive concretestress at transfer was 4.062 ksi (28MPa) which was 0.74 of the specifiedconcrete strength of 5.5 ksi (38 MPa)and 0.60 of the actual strength of 6.82ksi (47 MPa). In both cases, the allow-able limit of 0.6f ′c was exceeded. Themaximum compressive stress at thetop fiber of the precast member at thetime of deck placement, using linearelastic analysis was 0.66 of the speci-fied concrete strength of 8 ksi (55MPa) and 0.65 of the actual concretestrength of 8.14 ksi (56 MPa).

This was the primary criterion beingtested and the calculated stress wasmuch higher than the code limit of0.45f ′c. Fig. 7 shows the predicted de-flection-time diagram, and the mea-sured values at key events. The agree-ment between experimental andpredicted values is excellent.

The beam was then tested to failure.Two hydraulic jacks with a 10 in. (250mm) stroke were not adequate to ac-commodate the large deflection of thebeam before it failed (see Fig. 8). Theload had to be removed and the speci-

men shimmed several times before itfinally collapsed.

The maximum measured deflectionand load just before failure were 14.7in. (373 mm) and 54 kips (240 kN).Obviously, the large deflection showsa tremendous amount of ductility inthe beam and no signs of compres-sion-controlled brittle failure. Themeasured load was close to that pre-dicted by theory, 51.9 kips (231 kN).

PROPOSED FLEXURALDESIGN CRITERIA

The following design modificationsare proposed for the flexural design ofprestressed concrete members:• Eliminate the requirements for con-

crete compression limits at prestresstransfer, 0.6f ′ci, due to effective pre-stress combined with dead loads,0.45f ′c, and due to effective prestresscombined with full loads, 0.6f ′c.

• Retain the requirement for concretecompression limit due to fatigueloading, 0.4f ′c, for bridge membersand other relevant applications.

• Retain service load checks for con-crete tensile stresses at transfer andat service. If stress exceeds themodulus of rupture of concrete,generally limited to 6 (inpounds per square inch), bonded re-inforcement is to be added as percurrent practice to control flexuralcracking.

• Retain all other serviceability crite-ria such as deflection control, cam-ber control, vibrations, and other pa-rameters. With increasing use ofhigh strength concrete, such criteria

′fc

14 PCI JOURNAL

become increasingly important.• Add the requirement of strength de-

sign for prestress transfer to ensuremember capacity against concretecrushing.

• Retain the strength design of fullsection due to total loads, and usethis criterion as the primary designcriterion.

• Add the requirement that precast-only sections have adequate flexuralstrength due to loads applied beforethe sections become composite withcast-in-place topping. The steps to be used in design may

vary according to designer preference.The following steps may be used forstandard precast concrete bridge andbuilding members:

Step 1: Select section shape andsize based on prior experience.

Step 2: Determine area of prestress-ing steel required for acceptable ca-pacity of the member at final condi-tions due to full loads. There are twocriteria for this step: (a) strength ofcomposite section must be not lessthan the factored total load moment,and (b) concrete tensile stress due tounfactored service loads must besmaller than the code limit. In thisstep, the specified compressivestrength of the deck concrete can bedetermined such that the compressionblock is confined within the deck slab.

Step 3: Determine the concretestrength of the precast component ofthe section, based on strength analysisof the section for the loads applied be-fore it becomes composite with thedeck concrete.

Step 4: Using strength design, de-termine the concrete strength at trans-fer, and debonding and/or draping re-quirements at the end section. Also,determine if top bonded reinforcementat member end is required to controlcracking at transfer.

Step 5: Check remaining service-ability criteria, such as deflection andfatigue.

The following example illustratesthe proposed design steps.

Example 5: Flexural Design of aNU-900 I-Beam

It is required to design an interiorbeam of the center span of a three-

span bridge. The span lengths are 80 -100 - 80 ft (24.4 - 30.5 - 24.4 m). Thesuperstructure consists of five NU-900I-beams spaced at 9 ft (2.74 m) oncenter, as shown in Fig. 9. The beamsare designed to act compositely with a7.5 in. (191 mm) cast-in-place con-crete deck slab.

A 0.5 in. (12.7 mm) wearing surfaceis considered to be an integral part ofthe slab. Thus, the slab is consideredto be 7.5 in. (191 mm) thick for resis-tance calculations. The design loadsand resistance factors of AASHTOLRFD Bridge Design Specificationswill be used.

Midspan moment due to weight ofthe precast member, Mg = 810.5 kip-ft(1099 kN-m); due to deck weight, Md

= 1110.3 kip-ft (1505 kN-m); due tobarrier weight, Mb = 52.0 kip-ft (70.5kN-m); due to future wearing surface,Mws = 95.3 kip-ft (129.2 kN-m); anddue to live load-plus-impact, MLL+I =1184.8 kip-ft (1606.37 kN-m).

Seven-wire, 0.6 in. (15.4 mm) diam-eter, Grade 270, low-relaxationstrands at 2 in. (50.8 mm) spacing areused. The steel stress is assumed to be202.5 ksi (1396 MPa), just beforetransfer, 180 ksi (1241 MPa) just aftertransfer, and 155 ksi (1069 MPa) afterallowance for all time-dependentlosses.

The following concrete strengths areinitially assumed, based on currentstandard practice in Nebraska: precastconcrete strength at prestress transfer= 6.5 ksi (45 MPa), and at service = 8ksi (55 MPa); cast-in-place deck con-crete = 4 ksi (28 MPa). Table 3 showsthe properties of the non-compositeand composite cross sections.

Step 1: Section shape and size

A limited superstructure depth is as-sumed to be a constraint for thisbridge. A relatively shallow, 900 mm(35.4 in.) NU I-girder size is selected.

Step 2: Required area ofreinforcement based on flexuraldesign at midspan

Initially, select 36 - 0.6 in. (15.24mm) diameter strands, arranged asshown in Fig. 10. Check the tensilestresses at midspan of the composite

section for AASHTO LRFD ServiceLevel III loading combination:

= –0.448 ksi (–3.09 MPa)

This stress is within the correspondingtensile stress limit of –6 = –0.537ksi (–3.70 MPa).

Required strength of composite sec-tion:

Mu = 1.25(Mg + Md + Mb) + 1.5(Mws)+ 1.75 (MLL + I)

= [1.25(810.5 + 1110.3 + 52.0) + 1.5(95.3) + 1.75(1186.6)](12)

= 56,226 kip-in. (6353 kN-m)

Strength analysis indicates that thedesign strength, φMn = 77,589 kip-in.(8766 kN-m), which is significantlylarger than the required strength, Mu,indicating that the strength at serviceis not a critical design criterion. Thedepth of the equivalent rectangularstress block, a = 5.62 in. (143 mm)which is less than the 7.5 in. (191 mm)slab thickness, indicating that a 4 ksi(28 MPa) deck concrete is adequate.

Step 3: Required strength ofprecast section at time of deck placement

Mu = 1.25(Mg + Md) = 28,812 kip-in. (3255.3 kN-m)

The analysis for this loading case isdone using the strain compatibilitymethod of the PCI Bridge DesignManual which incorporates Mast’svariable strength reduction formula, asdescribed in Example 4. A minimum

′fc

= + ×

− + ×

− + × −

× ×

1210 86

648 1

1210 86 13 1

6 849

810 5 1110 3 12

6 849

52 0 95 3 12

10 370

0 81184 8 12

10 370

.

.

. .

,

( . . )

,

( . . )

,

..

,

fP

A

P e

S

M M

S

M M

S

M

S

be e

b

g d

b

b ws

bc

LL I

bc

= + −+

− + − × +

( )

( ).

( )0 8

November-December 2002 15

precast concrete strength of 6.57 ksi(45.23 MPa) is needed to satisfy thestrength requirements for this loadingcase. The corresponding nominalstrength, Mn is 41,179 kip-in. (4653kN-m).

The analysis indicates that the sec-tion is compression-controlled andthat the resistance factor should betaken = 0.7. Thus, φMn = 0.7 × 41,179= 28,825 kip-in. (3257 kN-m) >28,812 kip-in. (3255.3 kN-m)

Step 4: Required strength ofprecast section at transfer

Using the strength design method ofReference 3 for design at time of pre-stress transfer, it can be establishedthat the required concrete strength attransfer satisfies the midspan sectionconditions. For this case, the analysisis done for a section subjected to afactored moment = 0.85Mg = 8267kip-in. (934 MPa) combined with afactored pretensioning force = 1.15(36)(0.217)(202.5) = 1819.22 kips(205.54 kN). The corresponding mini-mum concrete strength is 4.73 ksi(31.61 MPa).

The analysis is repeated at the lift-ing point, which is assumed to be 5 ft(1.52 m) away from the member end,to determine if any strand draping isrequired. For that section, the factoredmoment due to self weight =(1.15)(0.675)(5)2(12)/2 = 116 kip-in.(13.11 kN-m). With the full preten-sioning force used, i.e., no strandsdraped, the required concrete strengthis 9.8 ksi (67.57 MPa) which is obvi-ously too high.

Now, if four strands are draped, thecorresponding strength at release is6.0 ksi (41 MPa). Because of the re-quirement of draping, the 0.4l sectionwould need to be checked. For thatsection, the required concrete strengthis 4.8 ksi (33.09 MPa) which is belowthat required for the section at the lift-ing point.

Step 5: Check deflection and fatigue

An analysis for these design criteriaindicates that fatigue stress and deflec-tion are below the recommendedAASHTO LRFD limits.

For comparison purposes, an analy-sis of the same example using the cur-rent conventional criteria was under-taken. The results are given in Table 4.

CONCLUSIONS1. The present restrictions on con-

crete compressive stress at servicelimit state in the design of prestressedconcrete flexural members neither di-rectly address serviceability condi-tions, nor ensure adequate strength un-derfactored loads.

2. In composite members, the cur-rent allowable compressive stress of0.6f ′c, due to effective prestress plusfull loads, gives the false impressionthat the highest compressed fibers, andmost likely to “crush” under factoredload, are the extreme precast concretefibers. This is in conflict with thestrength design method and experi-mental evidence which indicates thatthe highest stressed fibers are those inthe composite topping.

3. Both the ACI 318 Code andAASHTO Specifications already re-quire that the flexural strength of amember be checked. This appliesequally to prestressed and non-pre-stressed conventionally reinforcedmembers. The compressive stress lim-its of 0.6f ′c, due to full loads, and0.45f ′c due to sustained (dead) loads,are imposed on prestressed concretemembers only. Conventionally rein-forced members have not been sub-jected to these limits since the strengthdesign approach was introduced in thecodes about 50 years ago, without anyrelated reported deficiencies.

4. Imposing unnecessary compres-sive stress limits on prestressed con-crete members inhibits developmentand use of efficient sections such usinverted tees and U-shaped precastsections with composite cast-in-placetopping slabs.

5. It is proposed that the compres-sive stress limits of 0.6f ′c and 0.45f ′cimposed on concrete stresses due tofull loads and due to dead loads, re-spectively, be deleted as design re-quirements.

6. It is proposed that compositemembers be checked for strength at allcritical loading stages, particularly

loads introduced just before the mem-ber becomes composite with the top-ping.

7. Mast’s unified strength designmethod which provides an interpola-tion function for transition of the resis-tance factor from tension-controlled tocompression-controlled behavior is themost appropriate for analysis of bridgebeams, especially when consideringthe precast component before compos-ite action takes effect. The method hasbeen adopted as the standard methodin ACI 318-02. It is the recommendedmethod in the PCI Bridge DesignManual. The AASHTO LRFD Speci-fications should be revised to allowfor use of this method in bridge de-sign.

8. Strength design should be usedfor conditions at prestress transfer, andat lifting out of the casting bed, as rec-ommended by Noppakunwijai et al.3

9. Service load limits should remainimposed on such serviceability crite-ria as camber, deflection, crack con-trol and fatigue as appropriate for thestructure being designed. For exam-ple, if bridges and some buildingstructures are not desired to becracked due to effective prestress plusfull loads, the tensile stress due tothese effects should be limited to6 as currently required in bothACI and AASHTO.

10. Limiting the compressive stressdue to half of the effective prestressand half of the dead loads combinedwith the full live load is a fatiguecheck for bridge members. It shouldbe retained. However, it is inconsistentto use a standard truck load. Rather,the “Fatigue Truck,” and other rele-vant provisions to the fatigue limitstate in the AASHTO Specificationsshould be used.

11. Time-dependent analysis, usingthe age-adjusted effective modulusmethod, as described by Dilger, and inthe PCI Bridge Design Manual onlyrequire modest programming with astandard Excel or similar spreadsheet.It should be used for all serviceabilitychecks, including tensile stress andcamber control. Computer software,such as CREEP III17 and CONSPLICEPT,18 can also be used to design forservice limit states.

′fc

16 PCI JOURNAL

This research was funded by the Nebraska Department ofRoads, Project No. SPR-PL-1 (037) P527. Additional supportwas provided by the Center for Infrastructure Research of theUniversity of Nebraska-Lincoln (UNL), Omaha, Nebraska.

The authors especially thank Mr. Lyman Freemon, BridgeEngineer, Nebraska Department of Roads (NDOR) withoutwhose vision, leadership and encouragement much of theprestressed concrete bridge advancement in Nebraska would

not have been possible. Mr. Sam Fallaha, Assistant BridgeEngineer, NDOR, and Dr. Sherif Yehia, Research AssistantProfessor, UNL, and Mr. Chuanbing Sun, Ph.D. student,provided significant assistance throughout this project.

The authors also want to thank several PCI JOURNAL re-viewers for their thoughtful and constructive comments.They include Henry Bollman, Leslie Martin, Robert Mast,George Nasser and Stephen Seguirant.

1. ACI Committee 318, “Building Code Requirements for Struc-tural Concrete (ACI 318-02),” American Concrete Institute,Farmington Hills, MI, 2002.

2. AASHTO, AASHTO LRFD Bridges Design Specifications,Second Edition, American Association of State Highway andTransportation Officials, Washington DC, 1998.

3. Noppakunwijai, P., Tadros, M. K., Zhongguo (John) M., andMast, R.F., “Strength Design of Pretensioned Flexural Con-crete Members at Prestress Transfer,” PCI JOURNAL, V. 46,No. 1, January-February 2001, pp.34-52.

4. AASHTO, AASHTO Standard Specifications for HighwayBridges, 16th Edition, American Association of State Highwayand Transportation Officials, Washington, DC, 1996.

5. PCI’s First Specifications for Pretensioned Bonded Pre-stressed Concrete Products, First Edition, Prestressed Con-crete Institute, Lakeland, FL, 1954.

6. Erickson, E. L., Criteria for Prestressed Concrete Bridges,Bureau of Public Roads, U.S. Department of Commerce, U.S.Government Printing Office, Washington, DC, 1954.

7. ACI-ASCE Joint Committee 323, “Tentative Recommenda-tions for Prestressed Concrete,” Title No. 54-30, ACI Journal,V. 29, No. 7, January 1958, pp. 545-578.

8. Huo, X., Savage, J. M., and Tadros, M. K., “Reexamination ofService Load Limit Compressive Stress in Prestressed Con-crete Members,” ACI Structural Journal, V. 92, No. 2, March-April 1995, pp. 199-210.

9. Huo, X., and Tadros, M. K., “Allowable Compressive Strengthof Concrete at Prestress Release,” Open Forum, PCI JOUR-

NAL, V. 42, No. 1, January-February 1997, pp. 95-99.10. NDOR, “Allowable Compression Limits in Prestressed Con-

crete Members,” NDOR Report, Nebraska Department ofRoads, Lincoln, NE, 2002.

11. Kamel, M., “Innovative Precast Concrete Composite BridgeSystem,” Ph.D. Dissertation, Department of Civil Engineering,University of Nebraska-Lincoln, Lincoln, NE, 1996.

12. Tadros, M. K., Ghali, A., and Dilger, W. H., “Time-DependentAnalysis of Composite Frames,” ASCE Journal of StructuralEngineering, V. 103, No. 4, 1977-B, pp. 871-884.

13. Dilger, W. H., “Creep Analysis of Prestressed Concrete Struc-tures Using Creep Transformed Section Properties,” PCIJOURNAL, V. 27, No. 1, January-February 1982, pp. 89-117.

14. NCHRP, “Prestress Losses in Prestentioned High-StrengthConcrete Bridge Girders,” National Cooperative Highway Re-search Program, Washington, DC, 2002.

15. PCI Bridge Design Manual, First Edition, Precast/PrestressedConcrete Institute, Chicago, IL, 1997.

16. Tadros, M. K., Ghali, A., and Meyer, A. W., “Prestress Lossand Deflection of Precast Concrete Members,” PCI JOUR-NAL, V. 30, No. 1, January-February 1985, pp. 114-141.

17. Karim A., Ahmad M., and Tadros, M. K., “Computer Analysisof Spliced Girder Bridges,” ACI Structural Journal, V. 90, No.1, January-February 1993, pp. 21-31.

18. CONSPLICE PT, Design and Analysis of Spliced Pre-stressed/Precast Bridge Girders and CIP Slabs, “ComputerSoftware Version 1,” Developed by LEAP Software Inc.,Tampa, FL, 2001.

REFERENCES

ACKNOWLEDGMENT

November-December 2002 17

a = depth of equivalent rectangular stress block A = cross-sectional areaAb = area of precast concrete beamAp = area of pretensioning steelb = width of compression face of memberc = distance from extreme compression fiber to neu-

tral axis C = compression forcedext = depth of extreme steel layer from extreme com-

pression fiberdp = distance from extreme compression fiber to cen-

troid of pretensioning tendonse = eccentricity of prestressing strandsE = modulus of elasticity Ec = modulus of elasticity of concrete Eb = modulus of elasticity of beam concrete Ed = modulus of elasticity of deck E*

b = age-adjusted, effective modulus of elasticity ofprecast concrete beam for a gradually appliedload at time of deck placement

E*d = age-adjusted, effective modulus of elasticity of

concrete deck for a gradually applied load at timeof deck placement

f ′c = specified compressive strength of concrete at 28days

f ′ci = compressive strength of concrete at time of initialprestress

fd = compressive stresses due to effective prestressplus dead load

fl = compressive stresses due to live loadfps = stress in a given layer of prestressed reinforce-

ment whose strain is εps

ft = concrete stress at top fiber of precast beamh = overall depth of memberI = moment of inertial = span lengthM = momentMb = unfactored bending moment due to barrier weightMg = unfactored bending moment due to beam self-

weight

Md = unfactored bending moment due to deck weightMl = unfactored bending moment due to live loadMLL+I = unfactored bending moment due to live load and

impactMWS = unfactored bending moment due future wearing

surfaceMn = nominal flexural resistance Mu = factored bending moment at a sectionP = prestress force∆P = change of prestress forcePe = effective prestress force after allowing for all

lossesSb = section modulus for extreme bottom fiber of non-

composite precast beamSbc = composite section modulus for extreme bottom

fiber of precast beam St = section modulus for extreme top fiber T = tension forceyb = distance between section centroid and extreme

bottom fiberβ1 = ratio of depth of equivalent uniformly stressed

compression zone assumed in strength limit stateto depth of actual compression zone

εps = strain in a given layer of reinforcementεbsh = shrinkage strain of precast beam εdsh = shrinkage strain of deck φ = strength reduction factorφ = curvatureφc = curvature at midspan∆φ = change of curvatureχ = aging coefficient χb = aging coefficient for precast beamχd = aging coefficient for concrete deckψ = creep coefficient ψbid = creep coefficient of precast beam at time of deck

placementψbdf = creep coefficient of precast beam at final place-

mentψddf = creep coefficient of cast-in-place slab at final

placement

APPENDIX A — NOTATION

18 PCI JOURNAL

ITEM #1: Add the following paragraph at the begin-ning of Article 5.5.3.1:

Fatigue of concrete in compression shall be checked usingthe fatigue load factor specified in Table 3.4.1-1 in combi-nation with truck loading configuration specified in Article3.6.1.4. Live load stresses shall be computed in accordancewith Table 3.3.1-1 and Article 3.6.1.4. The total concretecompressive stress due to live load plus one-half the sum ofeffective prestress and permanent loads shall not exceed0.4f ′c (ksi).

Fatigue of reinforcement need not be investigated for…….(Remainder of Article remains unchanged)

ITEM#2: Revise the first paragraph of Article 5.5.4.1as follows:

The strength limit state issues to be considered shall bethose of strength and stability. For pretensioned concretemembers, this method shall be considered as the primary de-sign method for satisfaction of member capacity in com-pression at various loading stages, including prestress trans-fer, lifting, erection, deck placement, superimposed deadloads and live loads.

ITEM #3: Add the following paragraph at the end ofArticle 5.9.4.2.1:

Compressive stress limits specified in Table 5.9.4.2.1-1may be waived for flexural member if the strength limitstate provisions according to Article 5.7.3.2. are satisfied atprestress transfer, at time of cast-in-place deck placement,and at final loading. Further, the deflection and camber atvarious loading stages must be checked, using conventionalanalysis methods, for satisfaction of various design and con-struction limitations.

ITEM #4: In Table 5.9.4.2.1-1, revise the 1st, 2nd, 3rdand 4th bullets, and “Stress Limit” as follows:• In other than segmentally constructed due to the sum of

effective prestress and permanent loads• In segmentally constructed bridges dDue to the sum of ef-

fective prestress and permanent loads. This limit may bewaived if strength design is performed.0.45f ′c

• In other than segmentally constructed bridges due to liveload and one-half the sum of effective prestress and per-manent loads.

• Due to the sum of effective prestress, permanent loads,

and transient loads and during shipping and handling.This limit may be waived if strength design is performed. 0.60ϕwf ′c

Other Affected Articles:None

Background:The provisions for fatigue should be moved from their

current location of Article 5.9.4.2.1, Service Limit State, toArticle 5.5.3, Fatigue Limit State. In Article 5.5.3.1, the fa-tigue truck configuration of Article 3.6.1.4 rather than thestandard truck loading used for strength design calculationsshould be used to determine the live load stresses.

Relative to Item # 4, the stress limit for segmentally con-structed bridges and “other than segmentally constructedbridges” is the same and the two bullets should be combinedinto one bullet. The provisions for fatigue limit should bemoved from their current location of Article 5.9.4.2.1, Ser-vice Limit State, to Article 5.5.3, Fatigue Limit State.

Working stress design should be replaced with thestrength design method, consistently with the design of non-prestressed concrete members. Study in 1993, ReferenceX1, showed that non-prestressed concrete members that areheavily reinforced could experience extreme compressivestress significantly greater than 0.6f ′c if the unfactored loadand linear stress-strain theory were employed, with no re-ported negative effects. Additionally, a study in 1997, Ref-erence X2, showed that the stress-strain relationship for con-crete deviates considerably from a linear relationship as thestress in concrete exceeds about 0.5f ′c. Therefore the appar-ent compressive stress using the inaccurate linear analysis isexaggerated. References X3 and X4 shows that strength de-sign is adequate in satisfying the concrete capacity require-ments. It shall be noted that members made composite withmultistage concrete placement shall be checked usingstrength design at each stage of concrete placement and thefactored loads that exist at each of these stages.

Anticipated Effect on Bridges:• Consistent design for fatigue for all structural materials

and systems. • Consistent design for conventionally reinforced and pre-

stressed concrete flexural members. • Improved account for the influence of concrete strength

on design.

APPENDIX B — PROPOSED CHANGES TO AASHTO LRFD SPECIFICATIONS

X1. Huo, X., Savage, J. M., and Tadros, M. K., “Reexamination ofService Load Limit Compressive Stress in Prestressed Con-crete Members,” ACI Structural Journal, V. 92, No. 2, March-April 1995, pp.199-210.

X2. Huo, X., and Tadros, M. K., “Allowable CompressiveStrength of Concrete at Prestress Release,” PCI JOURNAL,Open Forum, V. 42, No. 1, January-February 1997, pp. 95-99.

X3. Noppakunwijai, P., Al-Omaishi, N., Tadros, M. K., andKrause, G. L., “Elimination of Prestressed Concrete Compres-sion Limits at Service Load,” PCI JOURNAL, V. 47, No. 6,November-December 2002, pp. 00-00.

X4. Noppakunwijai, P., “Allowable Compression Limits in Pre-stressed Concrete Members,” Ph.D. Dissertation, University ofNebraska-Lincoln, Lincoln, NE, 2001.

REFERENCES

November-December 2002 19

Change Submittal: CG XYZSubject: Modifications of ACI 318 Code wording that

permit the strength design theory to be the primary methodof design of flexural member.

Current Sections Affected: 18.4.3, 18.4.4, R18.4.3,R18.4.4

Reason: Permit the strength design theory to be the pri-mary method of design of flexural members at various load-ing stages during construction and at service loads after ap-plication of full external loads.

Proposed Code Change:Add a new Section 18.4.3 18.4.3 — Compressive stress limits specified in Section

18.4.2 (a) and (b) may be waived for flexural members if thestrength limit state provisions according to Section 18.7 aresatisfied at time of cast-in-place deck placement, and at finalloading. Further, the deflection and camber at various load-ing stages must be checked, using conventional analysismethods, for satisfaction of various design and constructionlimitations.

Renumber Section 18.4.3 to Section 18.4.418.4.34Proposed Commentary Change:Add Commentary R18.4.3 to read:R18.4.3 –If creep deformation due to dead load is a con-

cern, it could be evaluated and a satisfactory limit on deadload deflection imposed. For precast members made com-posite with cast in place topping, the 0.45f ′c limit is some-times reached at the top fiber of the precast component ofthe section immediately after the topping concrete is placed.However, the following two points provide justification forremoval of this requirement:

1. Redistribution of stress, due to creep between precastand cast-in-place components of the member, cause gradualreduction of compression in the top fiber of the precast com-ponent and a corresponding increase in compression in thecast-in-place component. Therefore, calculated high stress atthe time of topping placement is only temporary and gradu-ally decreases upon hardening of topping concrete.

2. Strength analysis of the composite section often indi-cates that the equivalent rectangular compression stressblock is limited within the thickness of the topping and theimpact of the high compression in the top fibers of the pre-cast component on member strength is virtually nonexistent.

The 0.6f ′c limit due to full loads plus effective prestress isonly imposed on prestressed members. Study in 1993, Ref-erence 18.X1, showed that non-prestressed concrete mem-bers that are heavily reinforced could experience extremecompressive stress significantly greater than 0.6f ′c if the un-factored load and linear stress-strain theory were employed,with no reported negative effects. Additionally, a study in1997, Reference 18.X2, showed that the stress-strain rela-tionship for concrete deviates considerably from a linear re-lationship as the stress in concrete exceeds about 0.5f ′c.Therefore, the apparent compressive stress using the inaccu-rate linear analysis is exaggerated. References 18.X3 and18.X4 show that strength design is adequate to satisfy theconcrete capacity requirements. It shall be noted that mem-bers made composite with multistage concrete placementshall be checked using strength design at each stage of con-crete placement and the factored loads that exist at each ofthese stages.

Renumber Commentary Section 18.4.3 to CommentarySection 18.4.4

R18.4.34

Add new References 18.X1, 18.X2, 18.X3 and 18.X4:

APPENDIX C —PROPOSED CHANGES IN ACI CODE

18.X1. Huo, X., Savage, J. M. and Tadros, M. K., “ Reexamina-tion of Service Load Limit Compressive Stress in Pre-stressed Concrete Members,” ACI Structural Journal, V.92, No. 2, March-April 1995, pp. 199-210.

18.X2. Huo, X., and Tadros, M. K., “Allowable CompressiveStrength of Concrete at Prestress Release,” PCI Journal,Open Forum Section, V. 42, No. 1, January-February 1997,pp. 95-99.

18.X3. Noppakunwijai, P., Al-Omaishi, N., Tadros, M. K., andKrause, G. L., “Elimination of Prestressed Concrete Com-pression Limits at Service Load,” PCI JOURNAL, V. 47,No. 6, November-December 2002, pp. 00-00.

18.X4. Noppakunwijai, P., “Allowable Compression Limits inPrestressed Concrete Members,” Ph.D. Dissertation, Uni-versity of Nebraska-Lincoln, Lincoln, NE, 2001.

REFERENCES