flexural design of prestressed concrete beams using frp ... journal/2001/march-… · beam,...

Chad R. Burke, P.E.Design EngineerCH2M-HillRedding, California

Charles W. DolanPh.D., P.E.ProfessorCivil and Architectural EngineeringUniversity of WyomingLaramie, Wyoming

Fiber Reinforced Polymer (FRP) tendons are beingconsidered for design in structures exposed toaggressive environments or where non-metallicproperties are desired. FRP tendons requireconsiderable attention to detail during the designprocess. This paper presents a unified approachfor the flexural design of beams with FRP tendons.Equations for flexural strength are presented,failure modes are defined, calibrations with testdata are presented, and strength reduction factorsare recommended. A test program validates thedesign approach and provides some serviceabilitydata. Conclusions from the test program anddesign recommendations are provided.

Flexural testing of Fiber Reinforced Polymer (FRP),prestressed concrete beams began in Japan in the mid1980s under a nationally coordinated program to de-

velop design guidelines for concrete reinforced or pre-stressed with FRP tendons. Work began in Europe and inthe United States in the late eighties.1,2

Several attempts have been made to develop designguidelines for FRP reinforcement, and these are in variousstates of completion. The lack of uniform testing proce-dures, material definitions, and reporting procedures hascaused great difficulty in developing guidelines. Several ofthe developing design guidelines are evaluated and com-pared by Gilstrap.3

Flexural Design of Prestressed Concrete BeamsUsing FRP Tendons

76 PCI JOURNAL

March-April 2001 77

Methods for evaluating the flexuralstrength of an FRP prestressed beamoriginated in 19912 and were refined in1996.4,5 This paper presents a unifiedapproach to flexural design of FRPprestressed members. Derivation of theflexural strength equations and calibra-tion of these equations against testbeams taken from the published litera-ture is presented. An experimental con-firmation of flexural behavior andother aspects of beam design allows anevaluation of serviceability perfor-mance of FRP prestressed beams. De-sign recommendations and strength re-duction factors are recommended.

THEORETICALDEVELOPMENT OF

FLEXURAL CAPACITYFRP tendons behave linearly elasti-

cally to failure. The industry trend isto define the tendon performance bythe design strength of the tendon. Themethods of defining the designstrength and alternative approaches todefining tendon strength are discussedbelow. These definitions are essentialto understanding the failure modes ofan FRP prestressed structure and thecorresponding strength reduction fac-tors used in design.

The basis for the strength designmethodology is the balanced ratio.The balanced ratio, ρb, is the rein-forcement ratio at which simultaneousconcrete compression and tendon rup-ture failure occurs. Any reinforcementratio above this value leads to a pri-mary concrete compression failure,while any reinforcement ratio belowthe balanced ratio indicates a tendonrupture failure. The stress and strainconditions for the balanced ratio areshown in Fig. 1.

The balanced ratio for FRP pre-stressed beams is similar to the bal-anced ratio used in reinforced concretedesign to ensure ductile behavior. Thebehavior is different than that for steeltendons because FRP tendons are abrittle material. This leads to the FRPbalanced ratio being an indicator ofthe failure mode rather than an assur-ance of ductility. By the traditionaldefinition of ductility, beams rein-forced or prestressed with FRP materi-als are not ductile. Fig. 1. Balanced ratio showing stress and strain conditions.

At the same time, the strain to fail-ure of an FRP tendon is greater thanthe strain for a steel tendon to reach ayield state. Consequently, beams pre-stressed with FRP tendons may exhibitlarge deformations prior to failure.The deformations are on the sameorder of magnitude as beams pre-stressed with steel tendons and thebeams provide corresponding warningof incipient failure. In this paper, andfor general use with a FRP prestressedbeam, “under-reinforced” implies atendon failure and “over-reinforced”implies a primary concrete compres-sion failure.

Upon completion of the balancedratio formulation, flexural design rela-tionships for both over-reinforced andunder-reinforced members are devel-oped for bonded tendons. BecauseFRP tendons are linearly elastic tofailure, draping or harping of tendonsresults in a loss of tendon strength inthe vicinity of the harping or drapingpoints. Strength reduction due to cur-vature of the tendons must be includedin the design calculations based on thetendon radius of curvature at the de-flection points.

Definition of Tendon Strength

A key design issue for FRP tendonsis the definition of tendon strength.Due to the lack of a yield plateau orstrain-hardening region, the designermust ensure that the design strength ofthe tendon is not exceeded in service.

Due to the variability in the tendonbreak strength, three different methodsfor reporting break strength have beenused. First, the mean tensile strength isreported. Second, a 95 percent inclu-sion strength is reported. The 95 per-

cent inclusion is 1.65 standard devia-tions less than the mean strength.Third, the “design strength” is re-ported. The design strength is themean break strength reduced by threestandard deviations.

Note that Japanese tendon manufac-turers provide a “guaranteed breakstrength” for their tendons, which cor-responds to the design strength. Often,neither the mean strength nor the stan-dard deviation is reported in the litera-ture. One consequence of this omis-sion is that authors who report designstrength tend to under-predict the ca-pacity of a member.

Industry practice reports the grosscross-sectional area of a prefabricatedtendon and the tendon stress is re-ported based on the gross area. FRPtendon sections consist of 50 to 70percent fibers by volume and the re-maining cross section is a polymerresin. Only the fibers contribute to thestrength. Therefore, the ultimatestresses for FRP tendons based on thegross cross section are lower than theparent fiber strength reported in theliterature.

Balanced ratio

The balanced ratio, ρb, is based onstrain compatibility in the cross sec-tion and is calculated using four basicassumptions: the ultimate concretecompression strain is εcu = 0.003, thenonlinear behavior of the concrete ismodeled using an equivalent rectangu-lar stress block, tendon failure is de-fined as the ultimate tensile strain ofthe tendon, εpu, and the prestressingtendons are placed in a single layer.These conditions are illustrated inFig. 1.

78 PCI JOURNAL

The balanced ratio is valid for bothflanged and rectangular sections pro-vided that the depth of the compres-sion block is within the depth of theflange. The equation for the balancedratio is given in Eq. (1):

where β1 = factor defined as the ratio of

the equivalent rectangularstress block depth to the dis-tance from the extreme com-pression fiber to the neutralaxis depth

f ′c = specified compressive strengthof concrete

ffu = ult imate tensi le s t ress oftendon

εcu = concrete ultimate compressionstrain, 0.003

εpi = initial prestressing strainεfu = ultimate strain capacity of

tendon Parameters β1, f ′c, and εcu are mate-

rial properties of the concrete, ffu andεfu are tendon material properties,while the initial prestress, εpi, is cho-sen by the designer based upon thelevel of desired prestress and the typeof tendons being used.

One conservative simplification isincluded in the formulation of the bal-anced ratio. The strain needed to de-compress the tendon, εd, is ignored.This is a conservative assumption be-cause εd is a negative value and it hasan order of magnitude smaller than theother strains.

To evaluate Eq. (1), only the materialproperties and the initial prestress levelmust be known. The tendon supplierprovides the ultimate tensile stress andstrain for the tendon. If the tendon de-sign strength is reported, the balancedratio will slightly under-predict thetransition from tendon failure to theconcrete compressive failure mode.

Under-Reinforced Beams

Beams with a reinforcement ratio, ρ,less than ρb, fail by tendon rupture andare divided into two different condi-tions: very under-reinforced beamsand under-reinforced beams. Thesetwo conditions are identified because a

designer may be concerned that a sin-gle formulation for flexural strengthmay not properly represent the flexu-ral behavior if the concrete is notstrained into the nonlinear zone.

Very under-reinforced beams havereinforcement ratios less than half ofρb, and a nominally linear stress-strainbehavior in the concrete up to thepoint of tendon failure. Such beamscould be more precisely defined ashaving a concrete compressive stressat failure of less than 0.45 f ′c.

Using 0.5ρb provides a computa-tionally easier solution and avoidscomplex stress checks of the section.The nominal moment capacity, Mn, ofthe very under-reinforced beam isbased on the tendon strength and theinternal moment arm for an elasticsection. The tendon strength is:

Tn = Apffu = ρbdffu (2)

whereAp = area of FRP tendonffu = ultimate tensile strength of

tendonρ = reinforcement ratio = Ap/bdb = width of compression faced = distance from centroid of out-

ermost reinforcement to ex-treme compression fiber

Summing the moment of the forcesaround the compression centroid givesthe nominal moment capacity:

Mn = ρ b d 2 ffu (1 - k/3) (3)

where

and n is the modular ratio.Under-reinforced beams have rein-

forcement ratios between 0.5ρb and ρb.These beams encounter significantnonlinear behavior in the concreteprior to tendon failure. Use of a rect-angular stress block, the tensile capac-ity of the tendon given in Eq. (2), andsumming moments about the centroidof the rectangular equivalent compres-sion block results in:

where

Mn = nominal moment capacityof section

a = depth of equivalent rectangu-lar compression block deter-mined by equilibrium offorces on cross section asshown in Eq. (6):

Modification of this equation is re-quired when multiple layers of rein-forcement are used. The outermosttendon layer is more highly strainedand will rupture first.

The remaining tendon layers may beunable to carry the load and a progres-sive failure of the beam will occur. Asa result, the computation of the bal-anced ratio and the predicted momentcapacity needs to be adjusted for vari-ation in depth and stress level. Thesemodifications are developed else-where.6

Over-Reinforced Beams

In over-reinforced beams, the mo-ment capacity is found by equilibriumand compatibility similar to under-re-inforced beams. In this case, the strainin the tendon is unknown and the con-crete stress field is represented with anequivalent rectangular stress block.The neutral axis is located by solvinga series of strain compatibility equa-tions to locate the neutral axis.

The depth to the neutral axis is de-fined as c = kud, where c is the depthfrom the extreme compression fiber tothe neutral axis. The coefficient ku isdefined by Eq. (7):

where λ is a material constant definedin Eq. (8) using the tendon elasticmodulus, Ef:

The above relationships result in thedepth to the compression block being

λε

β=

′E

ff cu

c0 85 1. (8)

kupi

cu

pi

fu

= + −

−

−

ρλ ρλ εε

ρλ εε

1 (7)

21

2

2

abdf

ffu

c

=′

ρ0 85.

(6)

M bdf da

n fu= −

ρ (5)

2

k n n n= + −( )ρ ρ ρ2 2 (4)

ρ β εε ε εb

c

fu

cu

cu fu pi

f

f= ′

+ −0 85 1.

(1)

March-April 2001 79

Mexp / φ

Mn

Beam Source Tendon f ′

c P1 Α

p b d ρ Μ

exp Mn Ratio Ratio Ratio Ratio Ratio Ratio

No. No. type Condition MPa kN mm2 mm mm kN-m kN-m φ

=1.0

φ

=0.9

φ

=0.85

φ

=0.80

φ

=0.75

φ

=0.70

Slab1 9 Aramid Over-reinforced 69.2 33 88.8 280 70 0.00453 14.25 20.54 0.69 0.77 0.82 0.87 0.92 0.99

BA-1M 13 Aramid Over-reinforced 57.7 684 900 230 405 0.00966 664.4 540.25 1.23 1.37 1.45 1.54 1.64 1.76

BA-6Y 13 Aramid Over-reinforced 57.2 684 900 230 405 0.00966 654.6 537.18 1.22 1.35 1.43 1.52 1.62 1.74

Fibra1 14 Aramid Over-reinforced 41.2 43.8 126 127 162 0.00612 38.9 34.45 1.13 1.25 1.33 1.41 1.51 1.61



AFRP-40 15 Aramid Over-reinforced 35 80 196 150 235 0.00556 50 73.22 0.68 0.76 0.80 0.85 0.91 0.98

AFRP-80 15 Aramid Under-reinforced 85 90 196 150 235 0.00556 50 56.80 0.88 0.98 1.04 1.10 1.17 1.26

CFCC1 14 Carbon Over-reinforced 36.1 85.4 76 127 179 0.00334 47.2 38.10 1.24 1.38 1.46 1.55 1.65 1.77



R-4-5.H 16 Carbon Over-reinforced 47 296.6 201 200 280 0.00359 155.9 189.83 0.82 0.91 0.97 1.03 1.09 1.17

R-4-5.V 16 Carbon Over-reinforced 47 296.6 201 200 255 0.00394 157.4 162.29 0.97 1.08 1.14 1.21 1.29 1.39

R-4-7.V 16 Carbon Over-reinforced 47 415.2 201 200 255 0.00394 170.8 170.21 1 1.12 1.18 1.25 1.34 1.43

3 4 Carbon Under-reinforced 35.3 68 55.7 100 100 0.00557 10.29 8.07 1.28 1.42 1.50 1.59 1.70 1.82



CTL3 11 Carbon Under-reinforced 47.9 187.1 152 254 210 0.00285 66.6 64.82 1.03 1.14 1.21 1.28 1.37 1.47



CFCC1 12 Carbon Under-reinforced 59.78 84.52 76 254 76.2 0.00393 11.32 11.26 1.01 1.12 1.18 1.26 1.34 1.44

CFCC2 12 Carbon Under-reinforced 58.61 84.52 76 254 76.2 0.00393 10.02 11.24 0.89 0.99 1.05 1.11 1.19 1.27

T-4-5.H 16 Carbon Under-reinforced 47 296.6 201 600 280 0.0012 186.1 158.74 1.17 1.30 1.38 1.47 1.56 1.68

T-4-5.V 16 Carbon Under-reinforced 47 296.6 201 600 255 0.00131 172.2 143.91 1.2 1.33 1.41 1.50 1.60 1.71



R-2-5.V 16 Carbon Under-reinforced 47 148.3 101 200 255 0.00197 100.6 70.12 1.43 1.59 1.69 1.79 1.91 2.05

Leadline1 14 E-Glass Over-reinforced 39 41.1 120 127 157 0.00601 31 31.33 0.99 1.10 1.16 1.24 1.32 1.41

Leadline2 14 E-Glass Over-reinforced 39 41.1 120 127 191 0.00494 43 43.69 0.98 1.09 1.16 1.23 1.31 1.41

Leadline3 14 E-Glass Over-reinforced 39 41.1 120 127 191 0.00494 40.2 43.69 0.92 1.02 1.08 1.15 1.23 1.31

Mean = 1.161 1.290 1.366 1.452 1.548 1.659

Standard deviation = 0.188 0.209 0.221 0.235 0.251 0.269

Mean = 1.040 1.156 1.224 1.300 1.387 1.486


Mean = 1.021 1.135 1.202 1.277 1.362 1.459


For Mean - Bσ = 1.0, B = 0.091 0.513 0.725 0.936 1.147 1.358

Mean = 1.149 1.277 1.352 1.437 1.532 1.642


For Mean - Bσ = 1.0, B = 0.840 1.403 1.684 1.965 2.247 2.528

Mean = 1.097 1.219 1.290 1.371 1.462 1.567


Sources of beam data:

Ref. No.

4. Kakizawa, 1993 10

9. Taerwe, 1995 11

11. Arockiasamy, 1995 12

12. Zhao, 1994 13

13. Niitani, 1997 (Draft) 14

14. Currier, 1995 15

15. Gowripalan, 1996 16

16. Abdelrahman, 1997 5

All Beams

Normally Reinforced Beams

Over-Reinforced Beams

Aramid Tendon Beams

Carbon Tendon Beams

Table 1. Calibration of flexural strength calculations.

80 PCI JOURNAL

a = β1c = β1kud. Summing the moment of forces about the tendonleads to Eq. (9) for the nominal mo-ment capacity for an over-reinforcedbeam:

Calibration of Design EquationsAgainst Available Test Data

A review of published materialidentified 30 test specimens for cali-bration (see Table 1). Each beam inTable 1 was analyzed to determine thebalanced ratio and probable failuremode. The nominal moment capacitywas computed using the appropriateequation. Interpretation of test datafrom the literature is complicated be-cause authors have not consistently re-ported tendon strength or prestresstendon area.

The use of mean break strength orthe design strength makes a signifi-cant difference in the predictions.Each paper was reviewed in detail. Inmost cases, North American and Eu-ropean authors reported meanstrength while Japanese authors re-ported design strength. In some in-stances, it is impossible to discernthe tendon properties.

The ratio of the reported experimen-tal moment capacity to theoretical mo-ment capacity is presented for each ofthe test specimens. An analysis of the

M f b k dk

n c uu= ′ −

0 85 121

2 1. β β

(9)

Fig. 2. Typical curve from parametric study of design equations.

adjusted so that the probability offailure is minimized. For a 95 percentinclusion of all data, the ratio of ex-perimental to predicted strengthshould be raised to 1.0 plus 1.65 stan-dard deviations.

Table 1 presents summaries based onthe failure mode and on the tendon ma-terial. Because the section strength re-duction factor is philosophically tied tomaterial performance rather than failuremode, the selection and recommenda-tion of the value for the strength reduc-tion factor is based on the tendon type.

Based on data in Table 1, a strengthreduction factor of 0.85 is recom-mended for carbon tendons and 0.70for aramid tendons. Table 1 alsoshows that strength reduction factorsof 1.0 and 0.9 under-predict too manysamples.

Analytic Validation of Design Equations

An analytic calibration of the designequations consisted of a parametricstudy that examined the behavior ofthe equations over a large range of re-inforcement ratios and compared theequation predictions to an alternativemethod of section analysis. The alter-native analysis consisted of a com-puter program that simulates a sectionby dividing it into thin lamina with thematerial properties being defined indi-vidually for each lamina.

The program varies the compressionstrain at the top of the member over auser-defined range and uses an itera-tive process to solve for the neutralaxis location and moment capacity ateach compression strain level. Theprogram uses a nonlinear stress-strainrelationship for the concrete behaviordefined by Collins and Mitchell,8 anda linear stress-strain relationship forthe tendons.

Two cross sections and four differ-ent tendon types were examined. Atypical calibration curve is shown inFig. 2. This study accomplished threeobjectives:

First, the analysis confirmed that thevery under-reinforced and under-rein-forced equations resulted in valuesthat did not diverge until the over-reinforced range. Thus, while the twoequations are useful for analysis, onlyEq. (5) is needed for design.

ratio of experimental to predictedstrength was conducted. The meanvalue and the standard deviation in theratio are computed for different failuremodes, tendon types and strength re-duction factors.

The strength reduction factor wasvaried from 1.0 to 0.7 in order to eval-uate the number of specimens that aresafely predicted with each reductionfactor. Values are not calculated forbeams with glass tendons, since glassis not recommended for use in preten-sioned beams due to stress corrosionand creep-rupture limitations.7

For all cases, the mean value of theexperimental strength to the predictedstrength slightly exceeds 1.0. This im-plies that the predictive equations aresuitable for design. The beam data arealso analyzed by failure type and ten-don type. Table 1 indicates that pre-dictions of under-reinforced beams areslightly more conservative than forover-reinforced beams.

Carbon tendons provide more con-servative predictions than aramid ten-dons. This latter conclusion is influ-enced by the fact that the CFCCbeams are based on the tendon designstrength rather than the mean strength.Since the correlation between meanand design strength is not provided,the higher apparent performance of thecarbon tendons is not unexpected.

The use of strength reduction fac-tors, φ, allows the ratio of experimen-tal to nominal moment capacity to be

March-April 2001 81

Second, the lamina model verifiedthe balanced ratio calculation. Thelamina model indicates a transition be-tween the over-reinforced beam curveand the under-reinforced beam curveat the balanced ratio for each beam ex-amined in this study.

Third, high strength concrete did notsignificantly increase the moment ca-pacity in the over-reinforced region,but high strength concrete was veryeffective in raising the overall momentcapacity when accompanied with in-creased tendon capacity.

STRENGTH DESIGN ISSUESThe designer must evaluate two is-

sues prior to developing a design:First, the decision must be made

whether the member is over-rein-forced or under-reinforced. Second,the maximum initial prestress valuesmust be selected.

Under-Reinforced VersusOver-Reinforced Beam Design

Data for analyzing under-reinforcedand over-reinforced beams were takenfrom Refs. 5, 10 to 16.

Both under- and over-reinforcedFRP prestressed beams fail in a brittlemode. Fig. 3 shows the failure of anunder-reinforced beam with a rein-forcement ratio near the balancedratio. Currently, a debate exists amongresearchers regarding which failuremode is most desirable. Some re-searchers propose that FRP-pre-stressed beams be designed as over-reinforced members due to an appar-ent gain in ductility.9

The ductility derives from crushing ofthe concrete and it is marginally greaterthan the energy absorbed from tendonrupture. Both cases have significantlyless ductility than steel prestressedmembers. Internally prestressed mem-bers are typically under-reinforced withthe failure being rupture of the tendon.

Most common prestressed shapesare difficult to over-reinforce with in-ternal tendons. Table 2 indicates thenumber of steel strands needed to ex-ceed the balanced ratio and the maxi-mum number of strands typicallyplaced in the section. Data were takenfrom the PCI Design Handbook17 andRocky Mountain Prestress.18

Fig. 3. Bonded FRP prestressed beam after tendon failure ρ ρ = ρb.

Table 2. Number of steel tendons to over-reinforce typical common prestressed sections.

Number of tendons required Maximum number of steel tendonsSection to over-reinforce section possible in cross section

8DT12* 48 8

8DT20* 87 12

10DT34* 190 22

4HC6* 12 9

4HC12* 27 8

FS4* 7 6

G-54✝ 141 28

G-68✝ 177 28

G-72✝ 177 28

Sources:* PCI Design Handbook, Fifth Edition, 1999.17✝ Rocky Mountain Prestress, Form Data Book.18

Based on this analysis, most FRPprestressed beams will be under-rein-forced. If the maximum number oftendons used is governed by allowablestresses instead of beam geometry, ifthe section is shallow, or if externalpost-tensioning is used, then the num-ber of tendons can be increased and anover-reinforced condition may occur.

Maximum Initial PrestressTwo factors combine to establish a

maximum prestress level in FRP ten-dons: creep-rupture behavior and re-serve strain capacity. Creep-rupture isthe failure of an FRP tendon in a finitetime interval when the tendon isstressed less than its full tensilestrength. FRP tendons are susceptibleto creep-rupture at high stress levels.For example, at 90 percent of its ten-

sile capacity, an aramid tendon has alife expectancy of about two hours.7

To prevent creep-rupture failure, andto have the design life of the tendon ex-ceed 100 years, the maximum prestresslevel is limited to 50 to 60 percent ofultimate capacity for carbon tendonsand 40 to 50 percent of ultimate capac-ity for aramid tendons. Glass tendonsare not recommended due to their poorcreep-rupture characteristics in moistalkaline environments.

The creep-rupture limitation on ini-tial prestress is not as severe as it ap-pears because tendon strain reserve isneeded in the tendon for good flexuralbehavior. Referring to Fig. 1, the flex-ural strain to failure, εfu, must be suffi-ciently large to allow the beam to de-form, crack and provide an indicationof incipient failure. Limiting the initial

82 PCI JOURNAL

prestress to 60 percent of ultimate pro-vides the deformability needed to meetthese criteria.

Creep-rupture behavior also affectsstressing practices. FRP tendons maybe overstressed to account for anchorseating losses, but they should not beoverstressed for relaxation losses. Aportion of relaxation loss comes fromthe relaxation of the polymer in thetendons. Over-stressing to compensatethe polymer relaxation increases theinitial strain in the fiber and decreasesthe creep-rupture life expectancy.

The design approach consists of se-lection of a section consistent with thenominal moment requirements. Thereinforcement ratio is selected usingEq. (5) and an initial estimate for thedepth of the compression block. Thereinforcement area is computed and areinforcement pattern is selected.Then, the material strength reductionfactor is chosen.

Checking the balanced ratio, selectingthe appropriate prediction equation andverifying the nominal moment capacityvalidates the design. The nominal ca-pacity times the strength reduction fac-tor must exceed the ultimate moment. Ifnot, the process must be repeated.

EXPERIMENTALCONFIRMATION OFDESIGN APPROACH

To validate this design approach, aseries of prestressed concrete beamswere designed and constructed usingdifferent types of FRP tendons. Thetendons included commercial materi-als and a generic tendon developed forthis project. The generic “strawman”tendon is produced by Glasforms Inc.,and has a simple cross section thatcould be produced from any numberof fibers and resins. The propertiesand materials used for this tendon arelisted in Table 3.

The T-section design loads were de-termined to be approximately one-tenth of an AASHTO LRFD DesignTandem. This design truck was chosenfor its two-axle layout, which is closeto the four-point loading configurationused in flexural testing. Allowableconcrete service stresses were definedusing the AASHTO LRFD DesignSpecification (see Table 4).

Tensile Tensile Tensile TensileVolume strength capacity modulus elongationpercent ksi (MPa) kips (kN) ksi (GPa) percent

Shell 9405 Resin/9470 curing agent 35 9.3 (64) – 401 (1.9) –Glasforms carbon fibers 65 650 (4482) – 34,000 (165) 1.9Theoretical strawman properties 425 (2935) 34.0 (151.0) 22,240 (108) 1.9Tested strawman average – 270 (1862) 21.6 (96.0) 22,240 (146) 1.2

σ – 8.8 (60.7) 0.70 (3.1) 2 (311) 0.12Average –1.65 (95% inclusion) – 20.4 (90.9) – –

Table 3. Strawman tendon properties.

Allowable stresses a transfer of prestress (prior to losses)

(a) Extreme fiber stress in compression 0.60 f ′ci

(b) Extreme fiber stress in tension except for Item (c) 3

(c) Extreme fiber stress in tension at ends 6

Allowable stresses under service loads (following losses)

(a) Extreme fiber stress in compression due to prestress plus sustained loads 0.45 f ′c

(b) Extreme fiber stress in compression due to prestress plus total loads 0.60 f ′c

(c) Extreme fiber stress in precompressed tensile zone 6 ′fc

′fci

′fci

Table 4. Allowable concrete stresses.

Table 5. Characteristics of test beams.

Area ofNumber tendons Pu Pi Ef f ′

c l

of sq in. kips kips ksi psi fttendons (mm2) (kN) (kN) (GPa) (MPa) (mm)

CFCC 1 1 0.12 26.2 18.1 20,500 7400 15.0(76) (161) (80.5) (137) (49.6) (4570)

Fibra 1 1 0.20 39.6 15.9 10,250 9700 15.0

(127) (176.4) (70.6) (68.6) (64.8) (4570)

Strawman 1 2 0.078 21.6 12.0 21,800 6300 15.0

(50.3) (96.2) (53.4) (146) (31.0) (4877)

Strawman 2 2 – 21.6 12.0 21,800 4600 16.0

(50.3) (96.2) (53.4) (146) (31.0) (4877)

Strawman 3 1 – 21.6 12.0 21,800 4600 10.0

(50.3) (96.2) (53.4) (146) (31.0) (3048)

Notes: CFCC = carbon fiber tendonFibra 1 = aramid tendonStrawman = carbon tendon provided from Glasforms Inc.

All of the beams with the exceptionof Strawman l beam were designedwith straight tendons. The Strawman lbeam was designed with two tendons,one straight and one depressed 3 in.(76 mm). The harped tendon waspushed down with a hydraulic jack anda 2 in. (51 mm) long saddle with a ra-dius of curvature of 36 in. (914 mm).The harped tendon was stressed so thatthe harping action plus the axial pre-stress would result in the same maxi-mum stress level as the straight tendon.

The stress increase, σ, due to theharping of a solid tendon should bedetermined by Eq. (10):

whereEf = modulus of elasticity of tendonR = radius of saddler = radius of tendonFive different prestressed beams

were designed. Carbon/nylon FRP

σ =E r

Rf (10)

March-April 2001 83

stirrups were placed outside the con-stant moment section at a spacing ofd/2. No stirrups were placed in theconstant moment section of the beam.The details of the test beams areshown in Table 5, their predictedstrength is given in Table 6, and thebeam cross sections are illustrated inFig. 4.

Flexural Testing

The beams were loaded with a sin-gle 55 kip (245 kN) MTS jack andload cell. A spreader beam created afour-point loading with a 36 in. (914mm) region of constant moment.Strawman 3 beam had a 12 in. (305mm) constant moment section due tothe shorter length of the beam. Thetest setup used is shown in Fig. 5.

Data Acquisition

The data acquisition for this projectmonitored load, deflection, tendon endslip, and crack width. Load readingsfor the flexural testing were taken di-rectly from the load cell on the MTSjack. Deflections were taken by plac-ing direct current displacement trans-ducers (DCDT) on either side of thebeams at midspan. End slip of the ten-dons was measured by linear poten-tiometers (LP).

After release of prestress, the excesstendon was removed, leaving a shortsection protruding from each end ofthe beam. The LPs were attached tothe ends of the tendons and spring-loaded against the end of the beams.The LPs measure end slip of the ten-don and provide an indication whetherthe development of the tendons gov-erned the failure of the beams. Crackwidth data were taken by loading thebeam beyond the cracking load, com-pletely unloading it, and then placing alinear potentiometer across the largestinitial crack.

RESULTS OF FLEXURALSTRENGTH CALIBRATIONThe design models are compared

with the beam behavior to validate theability to predict the failure mode, theactual strength, and overall tendon be-havior and serviceability conditions.Tables 5 and 6 summarize the beam

Fig. 4. Cross section of test beams.

Fig. 5. Testing apparatus setup.

Table 6. Predicted beam behavior and strength.

Predicted strength

kip-ft

Beam ID Tendon type Condition (kN-m)

CFCC Carbon Under-reinforced 27.5

(37.3)

Fibra 1 Aramid Under-reinforced 18.4

(25.0)

Strawman 1 Carbon Under-reinforced 30.1

(40.9)


(40.9)


(18.6)

and tendon properties, test setup andtest results.

Failure ModesAll of the beams failed by tendon rup-

ture as predicted by the balanced ratiocalculation. The failure of the CFCC1beam was the result of the breakage of a

small portion of the tendon outside of theleft load point. This resulted in an energyrelease that caused a long horizontalcrack to form at the tendon level and thebottom of the beam breaking off, leavingthe remaining tendon exposed.

The beam continued to carry loadup to the point where the remaining

84 PCI JOURNAL

tendon broke, at a load over 50 per-cent higher than the load that initiatedthe initial tendon failure. The initialtendon failure occurred in the centerof the beam in a section with no shearreinforcement. The lack of shear rein-forcement allowed rapid propagationof the crack in the concrete in thevicinity of the initial tendon “wire”failure.

The Fibra l tendon fractured slightlyoutside of the left load point. A flexu-ral crack had formed at that location.

The Strawman l tendon fractured si-multaneously with a diagonal crackpropagating at a low angle through theflange. The failure occurred at a pointwell outside of the right load point.The longer portion of this beam wasretested, resulting in a tendon failureunderneath the right load point. Therewas no apparent tendon damage at theharp point. The Strawman 3 tendonfailed outside of the right load point.

Strawman 2 was not tested duringthis program but was held for long-term durability evaluation. The ten-dency of the failure point to fall under-neath or near one of the load points isdue to very slight variations of theload points and tendon eccentricitiescreating one load point being slightly,more sensitive than the other.

Predicted VersusExperimental Strength

All of the beams failed within 10percent of the predicted strength, withthe exception of the Fibra l beam,which broke at a much lower load thanpredicted. The lower than predictedstrength of the Fibra l beam can be at-tributed to possible damage to the ten-don during stressing or can be at-tributed to the variation in beams

ffu – mean value ffu – mean value-1.65σ ffu – mean value-3σMexp Mexp Mexp Mexp Mexp Mexp Mexp Mexp Mexp

Mexp Mcalc Mcalc Mcalc Mcalc Mcalc Mcalc Mcalc Mcalc Mcalc Mcalc

Beam kN-m kN-m φ = 1.0 φ = 0.9 φ = 0.85 φ = 1.0 φ = 0.9 φ = 0.85 φ = 1.0 φ = 0.9 φ = 0.85

CFCC1 34.9 37.3 0.936 1.040 1.102 0.996 1.107 1.172 1.051 1.168 1.237

Fibra1* 19.6 25.0 0.783 0.870 0.921 0.816 0.906 0.960 0.845 0.939 0.994

Strawman1 39.2 40.9 0.959 1.066 1.128 1.013 1.126 1.192 1.062 1.180 1.249

Strawman1-2 39.3 40.9 0.960 1.067 1.130 1.014 1.127 1.193 1.063 1.182 1.251

Strawman 3 19.1 18.6 1.027 1.141 1.208 1.084 1.205 1.276 1.137 1.263 1.337

Table 7. Predicted and experimental beam results.

* Only 3σ strength was available; therefore, statistical data were used for a similar aramid tendon to adjust the strengths. The aramid was chosen since it is another braided product and is similar to the Fibra construction.

Fig. 6. Mexp versus φ Mn with mean tendon strength.

Fig. 7. Mexp versus φ Mn for mean tendon strength less 1.65σ (95 percent inclusion).

prestressed with aramid tendons asseen in Table 2.

The experimental results are com-pared to the calculated nominal capac-ity based on the mean tendon strength,the mean tendon strength reduced by

1.65σ, and the mean tendon strengthreduced by 3σ, each with φ factors of1.0, 0.9, and 0.85. These results aresummarized in Table 7 and in Figs. 6 ,7, and 8. Figs. 6 through 8 also indi-cate the increasing conservatism with

March-April 2001 85

the more restrictive definition of thetendon design strength and lowerstrength reduction factor. The figuresfurther reflect the Fibra tendon failingat below the expected capacity.

In all the design cases, using thetendon design strength, mean teststrength less three standard deviations,and a φ factor of 0.85, provided a de-sign prediction greater than the experi-mental results. A φ factor less than0.85 is desirable for the Fibra tendon.This result is consistent with the re-sults of the larger calibration studypresented earlier (see Table 1).

End Slip

No tendon end slip was observedduring the testing. The linear poten-tiometers on the tendons at the ends ofthe beam showed no movement of anyof the tendons. These data imply thatadequate bond and development wereachieved in all the tendons. Bond de-velopment was not a consideration inthe nominal moment capacity of anyof the beams. The actual embedmentlength and equivalent strand diametersto the closest failure point are given inTable 8.

This test program was not designedto establish development length but itconcludes that full tendon develop-ment is attainable within reasonabledistances from the beam end. Thesetest results also suggest that the actualtendon design development lengthsshould be no greater than the valuesgiven in Table 8.

Fig. 8. Mexp versus φ Mn for mean tendon strength less 3σ (design strength).

Table 8. Minimum bond length to failure point.

Minimum length from beam end to failure point

Beam Length, in. (mm) Strand diameters

CFCC 1 72.5 (1820) 145

Fibra 1 72.5 (1820) 202

Strawman 1 72.5 (1820) 227

Strawman 3 53.9 (1370) 171

Crack Widths

Monitoring crack width proved in-conclusive. The major crack in eachbeam was instrumented. However, in allthree cases, either new cracks formedadjacent to the monitored crack, or a bi-furcation occurred and the monitoredcrack was not representative of the ac-tual crack width. Additional researchand testing is ongoing to monitor thecrack width behavior.

CONCLUSIONSThe following conclusions for flex-

ural behavior may be drawn from theextended calibration and test valida-tion results:

1. The balanced ratio is effective indefining the transition between theunder- and over-reinforced failuremodes.

2. The design equations provide aneffective method of predicting theflexural strength of bonded FRP pre-stressed members and may be usedfor the design for FRP prestressedbeams.

3. The tendon design strength, equalto the mean test strength minus three

standard deviations, is recommendedas the basis of design.

4. The strength reduction factor forflexural design is a function of the ten-don type. Based on the calibrationanalysis, the following φ factors arerecommended:

φ = 0.85 for carbon tendonsφ = 0.70 for aramid tendons5. There is more variability in over-

reinforced beams than in under-rein-forced beams; however, both types ofbeams are safely predicted.

DESIGNRECOMMENDATIONS

These recommendations address onlyflexural strength. The following obser-vations will assist the designer in detail-ing and fabricating FRP prestressedbeams.

1. Harping through small angles didnot have a significant effect on theStrawman 1 beam. The total anglechange for the harped tendon in Straw-man l was slightly less than 2 degrees.This is only a single test, and moretesting needs to be completed to deter-mine if there is a significant reductionin strength with greater harp angles orsaddle radii configurations. Safe de-sign procedures should include the in-crease in stress in the tendon due tocurvature in the harping design. Thisstress increase is given in Eq. (10).

2. The design equations for multiplelayers of reinforcement must be calibrateddue to the possible rupture of the outer-most tendons. These equations and designmethodology are provided in Ref. 6.

3. Tendons should be stressed indi-vidually to prevent uneven stressingand breakage during the prestressingoperation.

4. Use of stirrups over the entirebeam length appears to provide suffi-cient confinement of the concrete inthe event of individual fiber breakagein the tendons.

86 PCI JOURNAL

FUTURE RESEARCHCompletion of a unified design

approach requires completion ofwork on a comprehensive approachto tendon development, serviceabil-ity behavior, crack width, creep-rupture, prestress losses, deforma-bility and fatigue behavior. Thesetopics are currently under develop-

ment by a number of researchersworldwide.

ACKNOWLEDGMENTS

The work presented in this paperis sponsored by the Federal High-way Administration under ContractDTFH61-96-C-0019. The authors

wish to thank Eric Munley for hissupport.

The opinions expressed in this paperare those of the authors and are not con-strued to represent those of the sponsors.

The authors wish to thank the PCIJOURNAL reviewers of this paperfor their insightful and constructivecomments.

REFERENCES

1. Gerritse, A., and Werner, J., “ARAPREE – a Non-MetallicTendon,” Advanced Composite Materials in Civil EngineeringStructures, ASCE, Materials Engineering Division, New York,NY, 1991, pp. 143-154.

2. Dolan, Charles W., “Kevlar Reinforced Prestressing forBridge Decks,” Transportation Research Record 1290, ThirdBridge Engineering Conference, Denver, CO, March 1991,pp. 68-75.

3. Gilstrap, J. M., Burke, C. R., Dowden, D. M., and Dolan,C. W., “Development of FRP Reinforcement Guidelinesfor Prestressed Concrete Structures,” Journal of Compos-ites for Construction, ASCE, V. 1, No. 4, November 1997,pp. 131-139.

4. Dolan, C. W., and Burke, C. R., “Flexural Strength and Designof FRP Prestressed Beams,” Advanced Composite Materials inBridges – Structures, Second International Symposium, Cana-dian Society of Civil Engineers, Montreal, Quebec, Canada,1996, pp. 383-390.

5. Abdelrahman, A., and Rizkalla, S., “Serviceability of ConcreteBeams Prestressed by Carbon Fiber-Reinforced-Plastic Bars,”ACI Structural Journal, V. 94, No. 4, July-August 1997, pp.447-457.

6. Dolan, C. W., and Swanson, D., “Development of Flexural Capacity of a FRP Prestressed Beam with Vertically Dis-tributed Tendons,” Proceedings of ACUN-2, InternationalComposites Conference, University of New South Wales, Syd-ney, Australia, February 14-18, 2000, pp. 96-101.

7. Leu, B. L., Dolan, C. W., and Hundley, A., “Creep-Rupture ofFiber Reinforced Plastics in a Concrete Environment” FR-PRCS-3, V. 2, Sapporo, Japan, October 1997, p. 187.

8. Collins, M. P., and Mitchell, D., Prestressed Concrete Struc-tures, Prentice Hall, New York, NY, 1992, p. 61.

9. Grace, N.F., and Sayed, G. A., “Ductility of Bridges UsingCFRP Strands,” Concrete International, V. 20, No. 6,American Concrete Institute, Farmington Hills, MI, June1998, p. 25.

10. Kakizawa, T., Ohno, S., and Yonezawa, T., “Flexural Behav-ior and Energy Absorption of Carbon FRP Reinforced Con-crete Beams,” FRP Reinforcement in Concrete Structures – In-ternational Symposium SP-138, American Concrete Institute,Farmington Hills, MI, 1993, pp. 585-598.

11. Taerwe, L., and Matthys, S., “Structural Behavior of ConcreteSlabs Prestressed with AFRP Bars,” Non-Metallic (FRP) Rein-forcement for Concrete Structures, Proceedings of the SecondInternational RILEM Symposium (FRPRCS-2), E - FN SPON,London, United Kingdom, 1995, pp. 421-429.

12. Arockiasamy, M., Zhuang, M., and Sandepudi, K., “DurabilityStudies on Prestressed Concrete Beams with CFRP Tendons,”Non-Metallic (FRP) Reinforcement for Concrete Structures,Proceedings of the Second International RILEM Symposium(FRPRCS-2), E - FN SPON, London, United Kingdom, 1995,pp. 456-462.

13. Zhao, L., “Behavior of Carbon Fiber Composite Tendon Pre-stressed Concrete Planks Under Static and Fatigue Loading,”M.S. Thesis, Department of Civil and Architectural Engineer-ing, Laramie, WY, 1994.

14. Niitani, K., Tezuka, M., and Tamura, T., “Flexural Behavior of Pre-stressed Concrete Beams Using AFRP Pre-Tensioning Tendons,”FRPRCS-3, V. 2, Sapporo, Japan, October 1997, pp. 663-670.

15. Currier, J., “Deformation of Prestressed Concrete Beams withFRP Tendons,” M.S. Thesis, Department of Civil and Archi-tectural Engineering, Laramie, WY, 1995.

16. Gowripalan, L., Zou, X. W., and Gilbert, R. I., “Flexural Be-havior of Prestressed Beams Using AFRP Tendons and HighStrength Concrete,” Advanced Composite Materials in Bridgesand Structures, 2nd International Conference, Canadian Soci-ety for Civil Engineering, Montreal, Quebec, Canada, 1996,pp. 325-333.

17. PCI Design Handbook – Precast and Prestressed Concrete, FifthEdition, Precast/Prestressed Concrete Institute, Chicago, IL, 1999.

18. Form Design Sections, Rocky Mountain Prestress, Denver,CO, June 1996.

March-April 2001 87

APPENDIX – NOTATIONa = depth of equivalent rectangular compression

block (mm or in.) Ap = cross-sectional area of prestressed reinforcement

in tensile zone (mm2 or sq in.)B = number of standard deviations to achieve Mexp/Mn

= 1.0 (see Table 1)b = width of compression face of member (mm or in.)c = distance from extreme compression fiber to neu-

tral axis (mm or in.)d = distance from centroid of outermost reinforcement

to extreme compression fiber (mm or in.)dc = distance from centroid of prestressing force to ex-

treme compression fiber for multiple layers of re-inforcement (mm or in.)

Ec = modulus of elasticity of concrete (GPa or psi)Ef = modulus of elasticity of FRP tendon (GPa or psi)f ′c = specified compressive strength of concrete (MPa

or psi)f ′ci = specified concrete compressive strength at time of

initial prestress (MPa or psi)ffu = ultimate tensile stress of FRP strand (MPa or psi)k = ratio of neutral axis depth to structural depth for

beams under service loadsku = ratio of neutral axis depth to structural depth for

over-reinforced beamsl = length of beam (mm or ft)M = design moment (kN-m or kip-in.)Mexp = moment capacity established by experimental

testing (kN-m or kip-in.)

Mn = nominal moment strength at section (kN-m or kip-in.)

n = modular ratio of elasticity equal to Ef / Ec

Pe = effective prestress force in tendon after losses (kNor kips)

Pi = initial prestress force in tendon (kN or kips)Pu = ultimate tensile force in tendon (kN or kips)R = radius of curvature of harping saddler = radius of solid tendonTu = nominal tensile capacity of FRP tendon (GPa or

ksi)β1 = factor defined as the ratio of the equivalent rect-

angular stress block depth to the distance from theextreme compression fiber to the neutral axisdepth

εcu = ultimate compression strain in concrete (0.003)εd = decompression strain, strain in precompressed

zone required to bring section back to zero strainεFRP = strain in tendon under a specified loadεpi = initial prestressing strain in prestressed tendonεfu = ultimate tensile strain capacity of tendon φ = strength reduction factorγ = load factorλ = material constant utilized to calculate ku

ρ = ratio of prestressed reinforcement equal to Ap/bdρb = balanced reinforcement ratioσ = standard deviation for statistical data reductionσ = stress increase in tendon due to harping

flexural design of prestressed concrete beams using frp ... journal/2001/march-… · beam,...

Documents