Application of the StrengthDesign Method for FlexuralMembers at Prestress Transfer

Panya Noppakunwijai, Ph.D.Research Assistant Professor

Department of Civil EngineeringUniversity of Nebraska-Lincoln

Omaha, Nebraska

Maher K. Tadros, Ph.D.,RE., FPCECharles J. Vranek ProfessorDepartment of Civil EngineeringUniversity of Nebraska-LincolnOmaha, Nebraska

Chuanbing SunPh.D. Student

Civil Engineering DepartmentUniversity of Nebraska-Lincoln

Omaha, Nebraska

This paper illustrates the application of the theorypresented in “Strength Design of PretensionedFlexura I Concrete Members at Prestress Transfer,”published in the January-February 2001 PCIJOURNAL. That paper recommended that workingstress design of prestressed members at time oftransfer of prestress to concrete be replaced withstrength design. This paper shows how this theorycould be implemented using a standardspreadsheet program, such as Microsoft Excel. Aparametric study justifying relaxation of the loadand resistance factors proposed in the earlierpaper is summarized. Applications to standardproducts such as I-beams, inverted tee beams anddouble tee products are given.

In the paper by Noppakunwijai et al.,1 a strength designmethod for prestress transfer was proposed for prestressed concrete members as a more rational alternative

to the standard working stress design method. It was proposed that strength design replace the empirical concretecompressive stress limit of 0.6 given in the ACT 318-02Code2 and AASHTO LRFD Bridge Design Specifications.3

The tension face of the cross section is designed in current practice to meet one of two conditions:

Condition (a): The section is not allowed to crack attransfer. For Condition (a), no changes are proposed byNoppakunwijai et al.

Condition (b): The section is allowed to crack as long asbonded reinforcement is provided to control cracking. Eventhough the section is recognized to be cracked, the building

62 PCI JOURNAL

and bridge design codes allow it to betreated as an uncracked section forcalculating the bonded reinforcementarea. They further permit that the tension force, calculated using the concrete stresses due to unfactored initialprestress combined with memberweight moment, be divided by an average steel stress of 30 ksi (207 MPa)to determine the required bonded reinforcement area.

The AASHTO LRFD Specificationsrequire that 120 percent of the tensionforce be used, whereas ACT 318 doesnot impose the 120 percent factor.Both codes fail to recognize that physical constraints may not permit coincidence of the centroid of the reinforcement with the location of the tensionresultant in concrete expected to be resisted by that reinforcement. Thebonded reinforcement may even haveto be placed on the compression sideof the concrete stress diagram, whichmakes calculation of the amount ofbonded reinforcement totally inconsistent with actual member behavior.

For Condition (b), Noppakunwijai etal. proposed that the current code empirical calculation of the requiredamount of bonded reinforcement bereplaced by determining an amountsuch that its stress at strength limitstate be limited to a value of0.0021(29,000) 60 ksi (414 MPa). Itwas found that a limiting strain of0.0021 and the corresponding stressalways resulted in strain at unfactoredload levels well below the strain of0.001 and the corresponding stress of30 ksi (207 MPa), the limits used incurrent practice for this application.

This approach results in eliminationof the need for empirical workingstress calculations of cracked sectionstress using uncracked section properties, as specified in current codes, orthe need for complicated cracked section calculations, which are avoided incurrent codes.

In Reference 1, a load factor of 1.2was proposed to be applied to the initial prestress force just before prestresstransfer, P1. A load factor of 0.8 or 1.2was proposed for self-weight moment,Mg. A strength resistance factor of 0.7was proposed to be applied to thenominal axial load capacity, P, andbending moment capacity, M, to de

termine the design capacity. Note thatthe so-called strength resistance factorused in this paper is the same as thestrength reduction factor. This paperpresents further evaluation of thesefactors and provides justification forrevising them to 1.15, 0.85, and 0.75,respectively.

A Microsoft Excel spreadsheet program is presented here to aid designerswho wish to make the transition tostrength-based design. The spreadsheet can be accurately used for common rectangular, I-shaped, double tee,and inverted tee sections. Appendix Bprovides details of formulas and conditions used in its development.

This information may be used tofurther develop the spreadsheet byprecast concrete producers who wishto supply design tables and charts forthe standard cross sections they produce, or by designers and researcherswho wish to explore the efficiency ofvarious cross section geometries orstrand arrangements.

An example of the program is presented in Appendix C, and verifiedusing approximation and a hand-heldcalculator.

OVERVIEW OFSTRENGTH DESIGN

Analysis is based on the standardstrength design procedure for compression members subjected to axialload combined with flexure. The axialcompression force is the prestressforce, multiplied by the factor y,,, to account for possible variability of thatforce at the time it is applied to theconcrete member.

The bending moment is caused bytwo effects: the eccentricity of the prestressing force and the member self-weight moment introduced at the timeof prestress transfer. ‘When prestress istransferred to the concrete member,the member generally cambers upwardand begins to resist moment due toself-weight.

This self-weight moment offsetssome of the effect of prestressing, andis thus associated with a load factor,)m’ smaller than unity. Some relativelylong precast concrete members arelifted out of the prestressing bed usinglifting inserts placed at a relatively

large distance from the member end.In this situation, the self-weight moment and the prestressing moment canact in the same direction and shouldboth be associated with load factorslarger than unity.

When the “loads” are applied to the“reinforced” concrete section, the concrete goes into compression and the“reinforcement” goes into compression or tension, depending on its location and the degree of bending of thesection. The strands used to impart themajority of prestressing on the sectionare generally concentrated in the bottom of the section and are expected toexperience a compressive stress increment due to the application of the prestressing force.

It can be argued that the net tensilestress in the prestressing strands, traditionally viewed as initial prestress including “elastic loss,” would justifyusing yet a lower applied load than indicated by elastic theory. Thisuniquely self-relieving characteristicof prestressed concrete, as comparedto externally loaded columns, shouldprovide additional confidence in theconservative approach proposed inReference 1.

The axial and flexural equilibriumconditions at the strength limit stateare as follows:

c[o.85fcAcc (1)

y(Af8y) ± YrnMg =

f—i

_2AfY] (2)

where= area of concrete in compres

sion= area of group j of reinforce

ment

f.j = initial steel stress in group jjust before prestress transfer

= compressive strength of concrete at time of prestress transfer

Mg = moment due to self-weightn = number of different steel

groups, having common initial

September-October 2003 63

Fig. 1. Load andstrength reduction

factors compared toworking stressdesign (WSD).

prestress, centroidal distanceand modulus of elasticity. Inthis paper, this number istaken as 2. The first group isthe bottom prestressingstrands. The second group iseither a top group of prestressing strands or a top group ofbonded mild steel reinforcingbars.

= centroidal distance of concretecompression area measuredfrom bottom fiber

= centroidal distance of steelgroup j from bottom fibersteel stress change at group jdue to application of factoredprestress and self-weight (negative for a decrease in tension)

‘I’m = self-weight moment load factorinitial prestress load factor

= strength resistance (reduction)factor

The right-hand sides of Eqs. (1) and(2) represent the axial and bending resistance values for a given depth ofneutral axis. The plot of axial loadversus bending moment for variousvalues of neutral axis depth is calledan interaction diagram. Several commercial computer programs are available for constructing interaction diagrams of compression members.However, these programs were generally developed for conventionally re

inforced, externally loaded columns,where the concrete strength is specified as input and the total steel area issolved for as the output.

in the current application, the steelarea and arrangement are initiallyspecified and the concrete strength issolved for. If the tensile stress in thebonded tension reinforcement at thetop of the section is found to be largerthan 60 ksi (414 MPa), the top reinforcement is increased until the stressdrops to within the specified limit.Commercial column analysis programs can still be used for the designof prestressed members for prestresstransfer, as long as the differences insteel type, initial steel stress, and loadand resistance factors are recognizedand accounted for.

EVALUATION OF LOADAND RESISTANCE FACTORS

The load and resistance factors proposed in Reference 1 had been conservatively selected based on experiencewith similar applications. The 1.20load factor is the same as that appliedto the post-tensioning force for the design of end anchorages in theAASHTO LRFD Specifications. The0.70 resistance factor is the same asthat applied to the capacity of tiedcompression members.

Further evaluation of load factors

since the publication of Reference Ihas indicated that the factors proposedin Reference I may be unnecessarilyconservative. It will be shown beLowthat the load factor of 1.20 and thestrength resistance factor of 0.70should be revised to 1.15 and 0.75, respectively. When the dead load moment counteracts the prestress moment, it is proposed that the loadfactor be revised from 0.80 to 0.85.

A load factor of 1.15 applied to theinitial prestress force, just before release, is more realistic than a 1.20 factor. The strand jacking stress is normally 0.75f for low-relaxationstrands, which are commonly used incurrent practice. Since the stress justbefore prestress transfer is normallyvery slightly less than the jackingstress, a load factor of 1.20 would correspond to a strand stress of nearly

(1.20)(0.75)f = the yieldstrength of low-relaxation strands. Aload factor of 1.15 would correspondto about 1.15 (202.5) = 233 ksi (1607MPa) for Grade 270 strands, which isstill very conservative. It is also consistent with the factor used in the Australian Code.4

A load factor of 1.15 is proposed tobe applied to self-weight moment, Mg,at the sections where the moment dueto self-weight is negative, i.e., in thesame direction as the moment due toprestress. Using a lower factor is justi

f0If’

1.0

0.9

0.8

0.7

__________________

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5(XIL)

Strength Design with 1.15,0.85 and 0.75 factors

‘—

Strength Design with 1.2,0.8 and 0.7 factors

Span 88 ft (26.82 m) with 22-0.5 in. strands

________

1[ee=5.57inf

__________

_________

U e=17.48in. U

___________

Double Tee 10DT32I I

64 PCI JOURNAL

36

Fig. 2. Number ofrequired debondedstrands usingstrength design (SD)versus workingstress design (WSD)fora 130 ft(39.62m) long, 53.1 in.(1349 mm) deepNU 1350 girderwith 58-0.6 in.(152 mm) diameterstrands.

fiable since the member is generallyplant produced, using precise steelformwork and production practices.

A dead load factor of 0.85 is usedwhen the member weight momentcounteracts the prestress moment. Thisis generally applicable to sections nearthe member midspan, and also immediately after transfer at the transferpoint sections near the member ends.For this very temporary loading condition and also for the reasons listedabove, it is reasonable to use the 0.85factor in place of the originally recommended 0.80 factor.

A strength resistance factor of 0.70was originally proposed to be appliedto the nominal resistance, to matchthat used for tied reinforced concretecompression members. However, asexplained earlier, the compressive prestressing force transferred to the concrete is internally induced and has aself-relieving ability. The concretestrength at transfer is precisely known,as cylinders must be tested before theprestress is allowed to be released tothe concrete member.

Unlike conventionally reinforcedtied columns, strands do not buckleunder factored load application. Thecompression increment due to the application of factored prestress and self-weight is much smaller than the initialtension, and the net result is still a tensile stress.

Over the last several decades, therehave been no reported crushing failures at prestress transfer in membersdesigned to meet the working stressdesign limit of The researchwork done by Frost et al. shows thatan allowable compression stress at release equal to 0.68f is recommendedin codes for pretensioned building andbridge products.5

Calibration of strength designagainst equivalent working stress design cases would produce reasonableestimates of the load and resistancefactors. Consider a cross section of amember that is pretensioned with concentric prestress force, e.g., a columnor pile section. Further, assume that theelastic shortening loss of prestress isequal to 10 percent. Thus, P0 = 0.9P1.

Applying strength design Eq. (1):

=[o.85fA. —AsiL1fsi]

Using the load factor = 1.15 and astrength resistance factor = 0.75, andignoring the contribution of steel tosection resistance, Eq. (1) reduces to:

l.15(P0 /0.9) =0.75 {0.85fcAg]

Thus, the required f according toEq. (4) is:

2.00I= 2.00f (5)

Note that the gross section area Ag isused in the equation for as the section is fully compressed. The stressis the equivalent working stress designextreme fiber compressive stress atprestress transfer. If the working stressdesign method was used, instead ofthe strength design method, to determine the required concrete releasestrength, f, thenf would have to belimited to 0.6f.

Thus, in this case of “axial loading,”the strength design method gives a 20percent higher required concretestrength f than that given by theworking stress design method. Thismay indicate over-conservatism in therevised load and resistance factors forthis loading case.

(3) For flexural members, the prestressing eccentricity near member endsoften corresponds to a larger bendingmoment than the self-weight moment.The proposed load and resistance factors would generally result in f, values lower than those resulting fromworking stress design. Comparison ofthe results of working stress design,and those of strength design with twosets of load and resistance factors, isgiven in Fig. 1.

With working stress design, is

U,

C

7.-

(0

4,C0

4,0

0z

32

28

24

20

16

12

8

4

0

0.00 L/L 0.10 0.20 0.30 0.40

thstance (XIL)

0.50

(4)

September-October 2003 65

constant, equal to 0.60. Strength design with 1.20 and 0.80 load factorsand a 0.70 resistance factor, producesresults that are unnecessarily restrictive. Strength design with 1.15 and0.85 load factors and 0.75 resistancefactor are more realistic. Considering

the full range of prestressed members,it is reasonable at this time to usethese factors. Future refinements maybe justifiable as experience is gainedin applying this design method.

Fig. 1 shows a comparison of theproposed load and resistance factors

versus those previously proposed, aswell as the current compression limitfor a I 0DT32 double tee member. Fig.2 gives the comparison for an NU I-girder. Using the revised load and resistance factors gives more realistic resuits near midspan and provides morerelief at member ends than those corresponding to the previous factors.

CRACK CONTROL. USINGSTRENGTH DESIGN

As indicated earlier, it was determined in Reference 1 that limiting thestress in the bonded reinforcement onthe tension face of the cross section to60 ksi (414 MPa) would ensure a steelstress due to unfactored load of lessthan 30 ksi (207 MPa), the acceptedstress limit for crack control. Changing the load and resistance factors asproposed in this paper has necessitatedthat the stress level be reinvestigatedfor the range of cross section shapesencountered in practice.

Table 1 shows a summary of someof the cases considered. They includethe PCI Design Handbook standardrectangular and inverted tee sectionsand standard NU inverted tee sections.Sections designed by the strength design method were re-analyzed ascracked sections subjected to unfactored loads.6 In all cases, the bonded

Fig. 3. Requiredconcrete strength

for lifting at 3 ft(0.914 m) from

member endof IT 700.

Table 1. Bonded tensile steel stress using cracked section analysis for caseswhere strength analysis produces a 60 ksi (414 MPa) stress.

No. of A5 e Neutral axis Steel stress,

Section strands (sq in.) (in.) depth (in.) I f’5 (ksi)

12RB20 8 0.67 7.25 -_____ 10.75

______

19.58

12RB24 10 0.91 9.00 12.77 21.85

12RB28 12 1.17 10.50 15.21 23.11

12RB32ji812.31 J 17.24 22.22

16RB32 18 1.92 12.33

16RB36j 20 12514.10

16RB40 22 2.56 15.82

281T28 13 1.68 9.09

17.27

19.40

21.55

13.15

281T44

281T60

341T24

341T40

341T60

401T40

401T48

401T52

NU-1T500

NU-1T600

NU-1T700

NU-1T800

NU-1T900

20

28

17

30

42

38

40

48

22

22

22

22

22

2.40 1433

3.09 21.02

2.15 8.03

3.70 13.92

4.80 22.07

4.74 14.47

5.23 - - 17.68

5.64 19.32

0.3! 4.01

0.56 5.52

0.80 7.11

1.0! 8.76

1.21 0.46

22.47

23.00

23.30

23.63

25.27

25.27

22.36

24.76

24.84

24.72

24.86

24.83

18.6!

22.06

23.28

23.63

23.64

20.46

28.44 -

11.80

19.7!

29.74

19.97

24.95

26.09

11.30

12.84

14.68

16.67

18.71

Note; tin. = 25.4 mm; 1 sq in. = 645 mm2; I ksi = 6.9 MPa.

f (psi)8,000

10%topslrarxltension

7,000

4,000 \--:.“-‘

-

Tension 1e’e1 vars

1 000 , , .. Nnniber varies* WSDis based on conipsesslon stiess limit 0 6f5,

0

2 4 6 8 10 12 14 16 18 20 22

Number of0.5 in. (12.7 mm diameter bottom strands

66 PCI JOURNAL

STRENGTH DESIGN FOR PRESTRESS TRANSFER

Fig. 4. Sample inputand output of theExcel spreadsheet.

tension steel area was selected suchthat the strength analysis produced astress of 60 ksi (414 MPa).

Moment due to self-weight, Mg, Wasassumed to be zero in developing thesteel stresses in Table 1. This assumption has a negligible impact on theconclusions drawn from the table. Thetable establishes the relationship between steel stress at service and at ultimate for a given “loading” level andsteel content. Whether a dead load isincluded or not, the important factor isto use the same loading to obtain therelative values of steel stress at serviceand at ultimate.

As shown in Table 1, the corresponding working stress design steelstress has been found to be in therange of 19 to 25 ksi (131 to 172MPa). This indicates that the 60 ksi

(414 MPa) limit at strength level isstill a valid value for crack controlwith the new load and resistance factors. This recommendation is limitedonly to sections that are allowed to becracked at prestress transfer. If nocracking is allowed by the designer orthe governing code, an elastic uncracked section analysis would have tobe performed and the concrete stresskept under the tensile strength limit.

EQUIVALENT WORKINGSTRESS DESIGN

COMPRESSION LIMITSStrength design for prestress trans

fer is more tedious and less familiar todesigners than working stress design.Considerable effort was expended bythe authors to find an equivalent corn-

pressive stress limit that allows for retention of the uniform safety factorsfeatured in strength design, yet allowsdesigners to use the working stress design formula of stress = P/A + MIS.However, the many parameters involved have prevented a simple revision to the limit 0.6f.

The limit was found to vary fromabout 0.80f for double tee sections to0.6f for inverted tee sections for prestress-loading combinations existingnear member ends where self-weightmoments are relatively small. Theequivalent compression limit level decreases as the section approachesmidspan, where gravity load momentsoffset a significant portion of the prestress moment.

It was found to be impractical tohave a number of different limits for

Strength Resultants

/lyI

-- x12—

—— XII —.

•1I

External Le

C

Af2A 2

I

_______

r

Af1A51F C

S 0.85f51

Concrete Dimensions

.I

Ssb-ueuuonyffL) I1 5.50 23.62 23.62

?i 8.6875 8.69 6.313 I 27.5575 6.31 6.31

Uttimate Concrete Strain E I

- r

Steel Information Steel iuyer,i J_g(in.Z) J_ y5 (in.) E5 (ksi) u (ksi)

1 1.736 3.75 28500 202.50L 2 j 0.434 - 675__28500101.25

Self-weight Moment M9(p-Th.) -

l.._z1 I

Load&ReststanceFactor z_.J

_____

1.15 1.150.75

OUTPUT

Concrete Strength fd (ksO

L

lnterrnediateResultsJ__P1(kips)Jt__J_a(in.) c(in.)

395.5 0.85 I 19.91 23.43Steel Stress _ShsettayeLj,j.(kSi) Final Sess limit

1 .L :7181. ]2 7.08 108.33 243

Verification of Results E F = 0? M = 0? J0 0

September-October 2003 67

various section shapes, section locations, and tendon profiles. This finding has directed the authors to concentrate their effort towards providingdesign aids for applying the strengthdesign method, and to abandon theworking stress design method as anaccurate alternative.

CRITICAL SECTIONS ANDSAMPLE DESIGN CHARTS

When prestress is released to a flexural concrete member, the memberusually cambers upward and begins to

resist moment due to self-weight. Formembers with straight strands, thecritical section at this stage is at atransfer-length distance from themember end, where the prestress forceis assumed to be fully transferred.The critical section may shift elsewhere if the strands are draped or ifsome of the straight strands aredebonded. If the member is lifted offthe prestressing bed shortly after prestress transfer, and if the lifting pointsare a significant distance away frommember ends, then the lifting pointsection should be checked as it is sub-

jected to negative self-weight moment,acting in the same direction as the prestressing moment.

The computer program presented inthe following section may be used atvarious sections along the span to determine such design factors as the required concrete strength at transfer,area of bonded tension reinforcement,number of debonded strands, length ofdebonding, and number of drapedstrands.

For standard precast concrete sections where draped strands are not desirable or feasible, the precast concreteproducers in a given region can makeavailable design tables or charts to aiddesigners using these products. An example is the Nebraska inverted teesused in short and medium spanbridges.

Fig. 3 shows a chart for the Nebraska IT 700, which was developedfor a section at 3 ft (0.91 m) from themember end. The required compressive strength is given in terms of thenumber of bottom strands. In all cases,two strands are placed at the top of thesection and are used to control concrete tensile stresses and cracking.

Three levels of tension of these topstrands are considered: 10, 50 and 100percent of the full tension of 0.75f.For comparison purposes, the chartalso shows the required concretestrength using the current workingstress design and allowable compression limit of 0.6f.

The results show a slight advantagegained with increased pretension inthe top strands. In all cases, strengthdesign requires lower concretestrength at transfer than working stressdesign.

EXCEL SPREADSHEETPROGRAM

The strength design method for prestress transfer does not lend itself tohand calculation except for simplecross section shapes where the compression block width can be assumedto be uniform (i.e., for rectangularsections and some inverted tee sections). Although a few available commercial programs such as PCACOL7can be used, some calculations for

H—+6.3 1”

—-48”

11299 BIV-48

r—36”r-—-— —72’---——--- -j-----36”---—-

7:

‘U

3” Fillet

2.56”

BT-72 12DT32

1.77”

j

4

NUI 350 16RB40

Fig. 5. Example of standard section shapes that can be analyzed using the Excelspreadsheet.

68 PCI JOURNAL

input data and careful interpretationof the output are still required.

No commercial software has beenspecifically developed for this application. For this reason, a Microsoft Excelspreadsheet program has been developed by the authors. This spreadsheetprogram is available for download fromthe University of Nebraska Web site atwww.structuresprograms.unomaha.edu.This article is available for downloadfrom the PCI Web site at www.pci.org.

The program is used to analyze a “reinforced concrete column” subjected tobending moment combined with axialcompression force. It is based on thestrain compatibility method with the assumption that compression failure willoccur when the concrete strain reachesits ultimate concrete strain, e, whichdefaults to 0.003.

Fig. 4 shows a sample input andoutput of the program. Appendix Bgives details of the equations and design conditions used in the development of the program. Appendix Cgives a numerical example using theprogram, followed by a hand-calculation check of its results.

The user needs to input the sectiongeometry as a combination of threetrapezoid segments. This allows use ofthe program in its current version formost standard sections, such as rectangular, inverted tee, double tee, and I-girder sections. Fig. 5 and Table 2 showexamples of some standard section dimensions that can be input into the program. Area, centroidal distance, modulus of elasticity, and initial stress ofeach steel group are required for input.

In the current version, only two steelgroups are allowed, i.e., the bottomprestressing steel group and the topbonded tensile reinforcement group.The top bonded steel can be either pretensioned strands or mild reinforcingbars. A trial amount of the top reinforcement is initially input. If the output shows its stress increment due toapplied loads, L1j7, to be greater thanthe limit of 60 ksi (414 MPa) or the required concrete strength, f, to be undesirably high, then a larger amount oftop reinforcement can be selected andthe program re-run.

Note: 1 in. = 25.4 mm.

The program was written in U.S.customary units. The user must makethe conversion from other sets of unitsbefore using it.

SUMMARY ANDCONCLUSIONS

Based on information presented inthis paper and in Reference 1, the following conclusions and summary remarks are provided:

1. It is shown that strength-baseddesign for prestress transfer is not onlya more rational approach than currentworking stress design methods, butalso generally results in lower required concrete strength and lower demand for strand debonding and draping at member ends.

2. This paper is a follow-up to Reference 1, in which the theory is fullydiscussed. In this paper it is shownthat lowering the initial prestress loadfactor from 1.20 to 1.15 and raisingthe strength resistance factor from0.70 to 0.75 is justified and results insignificant advantages. When self-weight moment counteracts the prestress moment, the revised load factorof 0.85, rather than the previous 0.80,is shown to conservatively account formember weight variability.

3. An Excel spreadsheet computerprogram is offered for strength designof prestressed concrete members forconditions at prestress transfer. Examples are given for its use. The equations and procedures used for its development are also provided.

4. Precast concrete producers areencouraged to use this spreadsheet ordevelop their own programs for thestandard section shapes prevailing in

their regions, for the purpose of generating design charts and tables for selection of the required concretestrength at release and the amount, ifany, of the bonded tension reinforcement for crack control.

REFERENCES1. Noppakunwijai, P., Tadros, M. K., Ma

Zhongguo (John), and Mast, R. F.,“Strength Design of PretensionedFlexural Concrete Members at Prestress Transfer,” PCI JOURNAL, V.46, No. 1, January-February 2001, pp.34-52.

2. ACI Committee 318, “Building CodeRequirements for Structural Concrete(ACI 318-02) and Commentary (ACI3 l8R-02),” American Concrete Institute, Farmington Hills, MI, 2002.

3. AASHTO, AASHTO LRFD BridgeDesign Spec(fications, Second Edition,American Association of State Highway and Transportation Officials,Washington, DC, 1998.

4. Warner, R. F., Rangan, B. V., Hall, A.S., and Faulkes, K. A., ConcreteStructures, Addison Wesley LongmanAustralia Pty Limited, Sydney, Australia, 1998, pp. 197-202.

5. Frost, R. Jerry, “Rational Development of Release Strength Criteria,”Master’s Thesis, The PennsylvaniaState University, University Park, PA,August 1997, 96 pp.

6. Tadros, M. K., “Expedient ServiceAnalysis of Cracked Prestressed Concrete Sections,” PCI JOURNAL, V.27, No. 6, November-December 1982,

pp. 86-111.7. PCACOL, “Design and Investigation

of Reinforced Concrete Column Sections, Computer Software Version3.0,” Developed by Portland CementAssociation, Skokie, IL, 1999.

Table 2. Input data dimensions (in.) of standard sections given in Fig. 5.

Geometry input for the programSections X X12 J xxT X3t

. Vt Y2 Lx1T700 23.62 23.62 8.69 6.31 63116.31 5.50 6.687527.5575BTV-48 48.0 48.QJ_ 16.0 10.0 10.0 5.5 8.5 42BT-72 J_26.0 26.0 L 26.0 6.0 6.0 : 6.0 6.0 [iqJ 7212DT32J 5 15.5] 15i 27.5 144.0140 27.0 30.0 32NU1350 38.4 38.4 38.4 5.9 5.9 5.9 5.3 10.8 53.116RB40 - 16.0 16.0 16.0 16.0 [16.0 39.998 39.999 40.0

September-October 2003 69

APPENDIX A — NOTATION

= area of concrete in compression= area of group j of reinforcement

A3 = gross cross-sectional areac = neutral axis depth

f = concrete compressive stress

f = compressive strength of concrete at time of pre

stress transfer

f = initial steel stress in group j just before prestress

transferMg = moment due to self-weightn = number of different steel groupsF, = prestress force immediately before prestress re

lease

P0 prestress force immediately after prestress release

= centroidal distance of concrete compression area

measured from bottom fibercentroidal distance of steel group j from bottom

fiber

4f steel stress change at group j due to application of

factored prestress and self-weight= strain change for steel layerj= ultimate concrete strain

Ym self-weight moment load factorinitial prestress load factorstrength resistance (reduction) factor

SPREADSHEET COMPUTER PROGRAM

The computer program described in the paper is now available for down load

ing from the following Web site:

www.structuresprograms.unomaha.edu

The primary purpose of this paper is to show designers how little time and ef

fort are needed with the new method, if they have access to the computer pro

gram. Readers should be aware that the program may be downloaded for free.

Also, readers need to watch for future upgrades. Planned upgrades will include

more than the two steel area clusters in a cross section as now given: the bot

tom (main) prestressing steel for flexural resistance to external loads, and the

top bonded (strand or bar, pretensioned or untensioned) crack control steel.

Also included will be more than three concrete trapezoidal area components to

allow for more accurate modeling of geometrically complex cross sections; for

example, with an I-beam section, the bottom flange and one rectangle and one

trapezium and the web as one rectangle are currently used. These three com

ponents exhaust the program’s current capacity and so far have worked well as

the top flange is basically in tension under factored prestress transfer loading

conditions.

Finally, the authors would like to invite readers (especially the engineering staff

of precast concrete producers) to further develop the spreadsheet on their own

and to share further developments with other PCI JOURNAL readers.

70 PCI JOURNAL

APPENDIX B — SPREADSHEET PROGRAM DOCUMENTATION

In calculating the strength of a cross section subjected toaxial compression and bending, equilibrium and strain compatibility conditions need to be satisfied. The followingcommonly used assumptions are made:

1. Sign convention: compressive stress in concrete andtensile stress in steel are considered positive. Prestress forceis always positive. Moment in the same sense as prestressforce moment is positive.

2. Plane cross sections remain plane after bending.3. Concrete has no tensile strength.4. The standard equivalent rectangular compression stress

block is assumed, with a stress intensity of 0.85f and deptha = fl1c, where c is the depth of neutral axis and is a coefficient that depends on concrete strength.

5. f3 varies linearly from 0.85 forf, = 4 ksi (27.6 MPa) to0.65 forf = 8 ksi (55.2 MPa).

Plane sections remaining plane and strain compatibilityresult in the following relationship:

where

AE

c = neutral axis depth= centroidal distance of steel group j from bottom

fiber= strain change for steel layerj= ultimate concrete strain, generally assumed =

0.003The steel strain change in each layer can be used to calcu

late the stress change:

Llf =Z1EqEj

2 2Af 4 0.85fA —

A34fj=1 jl

Two equilibrium conditions must be satisfied:

± YmMg =

2 1- Afy I

j=1 j

where

A(.C = area of concrete in compression= area of group j of reinforcement

A9 = gross cross-sectional area

f initial steel stress in group j just before prestresstransfer

f = compressive strength of concrete at time of prestress transfer

Mg = moment due to self-weight

Ycc = centroidal distance of concrete compression areameasured from bottom fiber

= centroidal distance of steel group j from bottomfiber

Af = steel stress change at group j due to application offactored prestress and self-weight

= self-weight moment load factorinitial prestress load factor

= strength resistance (reduction) factorIn the current version of the program, is taken as gross

area, Ag. For a more accurate analysis, the net areashould be used or the stress in the steel located in the corn

(Bi) pression zone reduced by 0.85f.Using Eqs. (Bi) to (B4), one can solve for f, 4f,

and L1f2. There are a number of ways to proceed for solutionof these equations. The steps used in the current Excelspreadsheet version are:

1. Combine Eqs. (Bi) and (B2) into Eqs. (B3) and (B4).Initially, 13i is assumed to be 0.85.

2. Calculate the axial load and bending moment and substitute into Eqs. (B3) and (B4). Eqs. (B3) and (B4) are thusreduced to equations with two unknowns, the compressionblock depth, “a”, and compressive strength of concrete attime of prestress transfer, f.

(B2) 3. Substitute f from Eq. (B3) into Eq. (B4). And solvefor “a”.

4. Once “a” is known, calculatef.(B3) 5. Repeat Steps 1 through 4 with a new f3 corresponding

to that calculated in Step 4, until convergence. Only threecycles were found to give adequate accuracy.

6. Calculate stresses, forces and check equilibrium.Note that, even though only a maximum of three trape

zoidal concrete area components are allowed in the currentversion, more complex shapes can still be handled by thisversion. Any number of concrete components can be corn

(B4) bined into one equivalent component. The equivalent component must have the same area, centroidal depth, and totaldepth of components being combined.

September-October 2003 71

APPENDIX C — NUMERICAL EXAMPLE

Given an NUl 100 girder with the following strand profile:At midspan: 58 - 0.6 in. diameter strands in the bottom

flange and four strands in the top flange, as shown in Fig. Cl(a). The bottom strands are fully tensioned to 202.5 ksi(1396 MPa) and the top strands are tensioned to 20.25 ksi(140 MPa). At the lifting point, which is 5 ft (1.52 m) from

the girder end, ten bottom strands are debonded, with theremaining 48 bottom strands and four top remaining effective [see Fig. Cl (b)].

Determine the required concrete strength f’ using theproposed strength design method and compare it with thatfound from the conventional working stress design method.

Fig. Cl. NU 1100I-girder with

strand pattern.

Fig. C2. Excelspreadsheet

program inputand output.

Concrete Strength f (ksi)

8.012

43.3l’

(a) At Midspan (b) At Lifting Insert

STRENGTH DESIGN FOR PRESTRESS TRANSFER

y,

INPUT

Concrete Dimensions I Sub-uedionluyu (in.> X1(in.) r X,2 (in.)

1’__i W038.40ffJ

Ultimate Concrete

Steel Information Tii 7Tks1 10.416 5.08 28500 202.50

2 0.868 41.31 28500 225

Self-weight Moment }iin.)

_______

I1.15 1.15 0.75 j

Load&RceFacto! “ -

OUTPUT

Intermediate Results P(kips) a (in.) c (in.)2126.8 0.65 19.20 29.54

Steel Stress Steel layer, j f(ksi) Final a (ksi) Stress limit

verification of Results Z F 0? M = 0?

0 0

1 -70.79 131.71 243

2 34.06 54.31 243

72 PCI JOURNAL

—29.6 kips

Fig. C3. Load andresistance atstrength limit for theNU 1100 example.

A. Spreadsheet Solution:

The results of the spreadsheet are shown in Fig. C2.

B. Hand Calculation Check of Spreadsheet:

A hand-calculation check, if desired, can be done as follows:

Assume that the depth of compression block, a, andf obtained from the spreadsheet are correct and verify stressesand equilibrium using Eqs. (B3) and (B4).

Check equilibrium Eq. (B3)Left-hand-side (LHS):

= 1. 15{48(0.21 7)202.5 + 4(0.217)20.25]

= 2446 kips

Fig. C3 (a) shows the compressed concrete area:= 38.4(5.3) + 0.5(5.9 + 38.4)5.5 + (19.2—5.3 — 5.5)5.9= 374.9 sq in.

0.85fA = 0.85(8.012)374.9= 2553.1 kips

The steel stress increment in the top and bottom strandscan be obtained using strain compatibility:

ysi \Ej =—(—C!

where

0.85 — 0.05(8.012 — 4)0.05= 0.65

ac=—

f319.2

0.65= 29.54 in.

Thus,

AE1=—E (i_“ c)

=-0.0031-5.083

29.54)=

— 0.0024838

s2 (i_—

U

= -0.0031-41.31

29.54)

= —0.0011953

zif =

=— 0.0024838(28500)

= —70.79 ksi

=&52ES2

= 0.0011953(28500)

= 34.06 ksi

The right-handside (RHS) of Eq. (B3):

= 1. 15[48(0.217)202.5 ÷ 4(0.217)20.25]

= 2446 kips

which is equal to the LHS of Eq. (B3).

Check equilibrium Eq. (B4)

LHS:

Fa=l2

/3”

YmMg125 in-kips

-)=2446 kips

‘4

Y,=5.382”

0.85f1 =6.81 ksi

(a) Compression zone (b) Stress resultants (c) External loads

— 2 109.2(5.083) + 17.58(41.3 1)- 2109.2+17.58= 5.382 in.

September-October 2003 73

± YmMg = ±

= 2446(5.382) — 1.15(109)

= 13,039 in.-kips

RHS:

Yce =

Assuming the initial prestress loss is 10 percent of theinitial prestress:

= 0.9F

= 0.9[48(0.217)202.5 + 4(0.217)20.25]

= 0.9(2109.2 + 17.58)

= 1914.1 kips

38.4(5.3)5.3/2+l/2(16.25)5.5(2)(5.5/3+5.3)+13.9(5.9)(13.9/2+5.3)

= 5.819 in.374.9

= 2553.1(5.819)= 14857 in.-kips

= —737.3(5.083)—3748 in.-kips

4f2A2y2—_ 29.6(41.3 1)= 1223 in.-kips

- ASJALJYJ]

= 0.75(14857 + 3748—1223)

= 13,037 in.-kips

Therefore, LHS = RHS.

C. Working Stress Design Solution:

P Fe Mf = —- + —f---- + 0.6f

Ag S SCI

For NU 1100, Ag = 694.6 sq in., Yb = 19.6 in., S, = 9299 in.3

e=l9.6—y,= 19.6— 5.382= 14.218 in.

— 1914.1 + 1914.1(14.218) — 109

- 694.6 9299 9299

= 2.756 ÷ 2.927 — 0.012

= 5.671 ksi

Thus, the requiredf = 5.67 1/0.6 = 9.452 ksi.This value is 18.0 percent higher than the 8.012 ksi re

quired by strength design. More importantly, the difference

in required strength may cause the prestressing bed turnover to be delayed by a day or two, and result in significantfinancial burden to the precast concrete producer.

74 PCI JOURNAL