Strength Reduction Factor for Flexural RC Members Strengthened with Near-Surface-Mounted Bars

Strength Reduction Factor for Flexural RC MembersStrengthened with Near-Surface-Mounted BarsHany Jawaheri Zadeh, Ph.D.1; Felipe Mejia, Ph.D.2; and Antonio Nanni, F.ASCE3

Abstract: Current American Concrete Institute (ACI) guidelines for the design of flexural RC members strengthened with externally bondedfiber-reinforced polymer (FRP) systems assign an additional partial strength reduction factor to the contribution of FRP, marking a deviationfrom ACI’s approach in building codes. This conservative method finds its rationale in the novelty and the higher variability of FRP, becauseof its nature as a material (compared to steel) and the conditions of its installation (externally bonded). Using the case of near-surface-mounted(NSM) FRP bars, this paper demonstrates that a single strength reduction factor can be formulated, while maintaining the same reliability andsafety required in conventional RC members. Using a comprehensive test matrix of flexural members processed with a computerized MonteCarlo simulation technique, the probabilistic implications of strengthening RC beams and slabs with NSM FRP bars are investigated. Thegenerated statistical data are employed to recommend revised strength reduction factors for flexural RC members strengthened with NSMFRP bars that eliminate the partial factor, and yet, provide a safety level equal to ordinary steel RC members. DOI: 10.1061/(ASCE)CC.1943-5614.0000366. © 2013 American Society of Civil Engineers.

CE Database subject headings: Flexural strength; Fiber reinforced polymer; Reinforced concrete; Safety; Structural members.

Author keywords: Flexure; Fiber-reinforced polymer; Reinforced concrete; Reliability; Resistance; Safety; Strengthening.

Introduction

Assigning a strength reduction factor to RC members externallystrengthened with fiber-reinforced polymer (FRP) bars presentscode writers with a dilemma. On the one hand, any reduction factorhas to address the legitimate concerns arising from adding a newmaterial to RC (e.g., FRP), whose behavior exhibits more unpre-dictability than reinforcing steel. On the other hand, such a reduc-tion factor has to observe continuity. Continuity necessitates theequality of the ultimate strength of an unstrengthened RC memberwith the ultimate strength of the same member with an infinitesimalamount of strengthening. This requirement is violated should asmaller reduction factor be imposed on strengthened members. TheAmerican Concrete Institute (ACI) standard 440.2R-08 (2008) hasopted for an alternative solution: maintaining the strength reductionfactors as stipulated by ACI 318-11 (2011) and dictating a partialreduction factor to the contribution of FRP (e.g., ψf ¼ 0.85). His-torically, the primary reason for this approach was to allow for theuse of an emerging material system whose behavior was not fully

proven. This method, although logical and effective, is currentlyunsatisfactory because it constitutes a departure from ACI’s ap-proach and because it resonates more as an ignorance factor thana safety factor.

The seeming inescapability of a smaller reduction factor forFRP strengthened members stems from the belief that becauseof the higher randomness of the mechanical behavior of FRP thansteel, and because of the uncertainties associated with the installa-tion of FRP, such a strengthened member is necessarily less reliablethan an ordinary RC member; in other words, adding FRP shouldlead to a build-up of uncertainties. This, however, may not alwaysbe true. Certainly, increasing the variance of an existing parameterin a system of random variables increases the randomness of thewhole system. One example is changing a parameter previouslytaken as deterministic to a random variable, which results in highervariation of the system. Nevertheless, according to basic statistics,if a system does not contain a certain parameter, whether as deter-ministic or random, adding this new parameter into the system mayreduce the deviation of the whole system, even if the newly addedvariable has a greater randomness than the former system. The sim-plest of such cases may be explained by two independent randomvariables with equal means and coefficients of variation. The sumof the two forms another random variable with a variation of ap-proximately 70% of the variation of each of its components and ismore, not less, deterministic.

Owing to the provisions of ACI 440.1R-08 (2008), FRP barscommonly reserve a higher portion of their capacity betweenthe point of their nominal design strength and the point of theiractual failure. In statistical terms, FRP bars have a higher bias factorthan steel, which can act as a preapplied or hidden safety factor forFRP bars.

Such arguments prompted the authors of this study to investi-gate the safety of flexural members strengthened with near-surfacemounted (NSM) FRP bars by taking advantage of reliability analy-sis and computerized simulation techniques. Gulbrandsen (2005)employed this technique to calibrate the strength reduction factors

1Dept. of Civil, Architectural, and Environmental Engineering, Univ. ofMiami (UM-CAE), 103 McArthur Engineering Bldg., Coral Gables, FL33146-0630 (corresponding author). E-mail: [email protected]

2Dept. of Civil, Architectural, and Environmental Engineering, Univ. ofMiami (UM-CAE), 103 McArthur Engineering Bldg., Coral Gables, FL33146-0630. E-mail: [email protected]

3Lester and Gwen Fisher Endowed Scholar, Professor, Dept. of Civil,Architectural, and Environmental Engineering, Univ. of Miami (UM-CAE), 103 McArthur Engineering Bldg., Coral Gables, FL 33146-0630;and Professor, Dept. of Structural Engineering, Univ. of Naples, FedericoII, Via Claudio 21, Naples, P.O. Box I-80125, Italy. E-mail: [email protected]

Note. This manuscript was submitted on August 27, 2012; approved onJanuary 22, 2013; published online on January 24, 2013. Discussion periodopen until March 1, 2014; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Composites for Con-struction, Vol. 17, No. 5, October 1, 2013. © ASCE, ISSN 1090-0268/(12)/$25.00.

614 / JOURNAL OF COMPOSITES FOR CONSTRUCTION © ASCE / SEPTEMBER/OCTOBER 2013

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http://dx.doi.org/10.1061/(ASCE)CC.1943-5614.0000366




for FRP-reinforced beams in flexure, which was later adopted byACI 440.1R-08 (2006). This was further investigated by JawaheriZadeh and Nanni (2012), who suggested “comparative reliability,”a reformulation of calibration equations with a different approach tothe target reliability. This study benefits from both.

In this paper, the original task of assessing the uncertainty ofthe behavior of a strengthened RC member paper is broken downinto a series of small steps. This allows the evaluation of theuncertainties associated with each parameter contributing to the re-sistance of the member (e.g., materials and dimensions). None ofthese steps is absolutely judgment-free; however, any time an en-gineering judgment is made, it is quantifiable and so are its effectson the outcome, making the methodology easily repeatable whenricher experimental databases become available. Once the statisti-cal parameters are obtained, the comparative reliability method isutilized to compute the strength reduction factors of the members inquestion.

Structural Model

The nominal flexural capacity of RC members externally strength-ened with FRP, of which NSM system is a subset, may be calcu-lated as the sum of the strengths provided by each reinforcementcomponent, the contribution of steel, Mns, and the contribution ofFRP, Mnf:

Mn ¼ Mns þMnf ð1Þwhere each term on the right of Eq. (1) is a function, namely f and gof resistance variables calculated according to Chapter 10 of ACI440.2R-08 (2008):

Mns ¼ fðAs; fy; f 0c; b; ds;α1; β1; εcu; εbiÞ ð2Þ

Mns ¼ gðAf; ffu;κm;Ef; f 0c; b; df;α1;β1; εcu; εbiÞ ð3Þ

whereMns = contribution of steel reinforcement to nominal flexuralstrength; f = ratio of stress in FRP to its debonding stress; As = areaof steel reinforcement; fy = yield strength of steel reinforcement;f 0c = specified compressive strength of concrete; b = width of

the beam or slab strip; ds = effective depth of steel reinforcement;

α1 = multiplier on for an equivalent rectangular stress distributionfor concrete; β1 = ratio of depth of equivalent rectangular stressblock to depth of the neutral axis; εcu = ultimate axial strain ofconcrete; εbi = strain level in concrete substrate at time of FRPinstallation; Af = area of FRP reinforcement; ffu = ultimatelongitudinal tensile strength of FRP bars; κm = dimensionlessbond-dependent coefficient for flexure; Ef = modulus of elasticityof FRP bars; and df = effective depth of FRP reinforcement. Thetwo components of resistance in Eq. (1) are not totally decoupled,because, as a result of the repositioning of the neutral axis, Mnsdeclines ever so slightly as Mnf increases.

Modes of Failure

According to ACI 440.2R-08 (2008), two distinct modes of failuregovern the behavior of the members externally strengthened withFRP, and thus, affect the calculation of Eqs. (1)–(3). The first modeis initiated by the crushing of the concrete (εcu ¼ 0.003). The de-bonding of the FRP bars initiates the second mode, which is iden-tified by the tensile FRP strain corresponding to the maximum bondstrength. This maximum is referred to as the debonding strain orεfd, discussed later in this study. Also, preexisting reinforcing steelmay or may not have yielded at the point of failure, thus dividingeach of the two aforesaid modes into two submodes for a total offour failure modes.

In this study, by using the customary assumptions of the linear-ity of plane sections in bending and compatibility of strains, theconditions that distinguish between these modes of failure are de-rived based on the member design parameters. The reinforcementindices for steel and FRP, ωs and ωf , are defined as

ωs ¼Asfyf 0cbds

; ωf ¼ Afffdf 0cbds

ð4Þ

The denominator is the same for both parameters; ffd = stress inFRP corresponding to the debonding strain, εfd. The two failuremodes can be predicted by comparison to the balanced condition:�

if ωf ≥ ωfb∶ Concrete crushingif ωf < ωfb∶ FRP debonding ð5Þ

ωfb depends on whether steel yields or not:

ωfb ¼

8>>><>>>:

0:85β1

dfds

εcuεcu þ εfd þ εbi

− ωs∶ if steel yields

0:85β1

dfds


− ωs

�dsdf

�ðεfd þ εbiÞ −

�1 − ds

df

�εcu

εsy∶ if steel does not yield

ð6Þ

ωsb separates the yielding submodes when concrete crushing isthe case:

�if ωs ≤ ωsb∶ steel yieldsif ωs > ωsb∶ steel does not yield

ð7Þ

where

ωsb ¼ 0:85β1

εcuεcu þ εsy

− ωf

�dfds− 1

�εcu þ

�dfds

�εsy − εbi

εfdð8Þ

In Eqs. (6) and (8), εbi = strain level in the concrete substrate atthe time of FRP installation. In practical cases, εbi varies, based on

the loading, in the approximate range of 0.0005 ∼ 0.001, whereasfor laboratory specimens, normally, εbi ¼ 0. For simplicity anduniformity, the latter value is assumed in this study for every in-vestigated case. If the member is deflected at the time of installa-tion, this assumption slightly overpredicts the resistance. However,it has less effect on the randomness of the behavior, which is thefocus of this study.

Eqs. (4)–(8) confirm that the increase in steel reinforcing, ωs,and FRP strengthening, ωf, increase the likelihood of the first modeof failure (i.e., concrete crushing). Generally speaking, debondingis the dominant failure mode and concrete crushing is normallyconfined to the infrequent case of heavily reinforced, heavilystrengthened beams.

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Statistical Model

In lieu of test results upon which the statistical parameters of theflexural capacity (i.e.,Mns,Mnf, andMn) can be established, theseparameters are calculated from the combination of the factors thatgenerate uncertainty in resistance. Each source of uncertainty isdefined by its bias factor (λ, the ratio of the mean to the nominalvalue of a random variable), coefficient of variation (δ or COV) andprobabilistic distribution type. The following recounts how thesedata are assumed or collected from the literature.

Sources of Uncertainty

Although defined by its unique nominal or design value, the actualstrength of a material is a random variable whose mean and SD maybe related to its nominal value from experimental data. Table 1 liststhe probabilistic identifiers of materials and each item is brieflydiscussed in the following.• The compressive strength of concrete, f 0

c, is not typically thecontrolling factor in the flexural capacity of a member; thus,only the most common of concrete types is considered withf 0c ¼ 28 MPa (4.0 ksi), the statistical parameters of which

are taken from the study by Nowak and Szerszen (2003a, b).• For fy, grade 400 MPa (60 ksi) steel (fy ¼ 400 MPa or 60 ksi)

is the common type of reinforcing bar used in RC construction.Nowak and Szerszen (2003a, b) provide the statistical data.

• Regarding ffu, ACI 440 modifies the guaranteed tensilestrength of FRP reinforcement, f�fu, by a reduction factor of0.70 ≤ CE ≤ 1.0 based on the fiber type and conditions ofenvironmental exposure:

ffu ¼ CEf�fu ð9Þ• The tensile strength of FRP bars or laminate, ffu, is a decisive

factor in determining the flexural strength. In this study, it isassumed that CE, regardless of its value, is preapplied; therefore,only ffu takes part in the computations. Ignoring a potentialsource of uncertainty in CE is deemed to be mitigated by anACI 440.2R-08 (2008) statement that its recommended valuesare “conservative estimates.” In other words, FRP materialmight be slightly more volatile, and yet, in contrast, slightlystronger than that presumed in this study.

• Unlike steel, FRP manufacturing protocols are less controllable.This leads to a variation in the mechanical characteristics of FRPproducts, which in turn makes acquiring an objective assess-ment of the probabilistic parameters of FRP materials a challen-ging task, because the test samples are usually biased toward theproducts and manufacturers they represent. In this study, it isassumed that FRP products exhibit a higher level of consistencyof characteristics than that of concrete, but lower than steel. Thisassumption confines the COVof FRP between those of concreteand steel (i.e., 0.05 ≤ δ ≤ 0.10). For FRP bars, circular orrectangular, δ ¼ 0.08 is selected as a reasonable estimate. A biasfactor of λ ¼ 1.20 (Gulbrandsen 2005) that corresponds to sucha deviation completes the couple. Two generic types of FRPare assumed, with Type 1 having significantly higher tensilestrength and modulus of elasticity than Type 2.

• The statistical data for Ef are determined according toGulbrandsen (2005).

• For κm, the debonding strain and stress of the NSM reinforce-ment, εfd and ffd, are defined as

εfd ¼ κmεfu; ffd ¼ κmffu ð10Þ

• For NSM reinforcement, ACI 440.2R-08 (2008) states that0.60 ≤ κm ≤ 0.90, “depending on many factors such as memberdimensions, steel and FRP reinforcement ratios, and surfaceroughness of the FRP bar,” and recommends the use ofκm ¼ 0.70. Assuming a simple uniform random distributionover that range, the statistical parameters are estimated as μκm ¼ð0.90þ 0.60Þ=2 ¼ 0.75, σκm¼ð0.90−0.60Þ=ð12Þ1=2 ¼ 0.0866,and λκm ¼ μκm=κm ¼ 1.071, δκm ¼ σκm=μκm ¼ 0.115.

• The COV of 0.115 makes the debonding factor, κm, the majorsource of material uncertainty.Variations in dimensions and geometry fall under the category

of the fabrication factor. These parameters, as shown in Table 1, arebased on the study by Nowak and Szerszen (2003a, b), except forthose discussed in the following.• For df, the statistical parameters of the effective depth of steel

rebars (2003a, b) are assumed to be applicable to the effectivedepth of FRP bars.

• The concrete cover-to-steel reinforcement center, c, is assumedto be statistically independent of the effective depth of FRP bars,

Table 1. Statistical Parameters

Factor Property or item Nominal value or member Bias (λ) COV (δ) Distribution

Material Concretea f 0c MPa ðksiÞ 28 (4.0) 1.24 0.10 Lognormal

Steel barsa fy MPa ðksiÞ 400 (60) 1.145 0.05 LognormalFRP barsb 1 ffu MPa ðksiÞ 1,750 (250) 1.20b 0.08b Lognormal

Ef GPa ðksiÞ 140 (20,000) 1.04b 0.08b Lognormal2 ffu MPa ðksiÞ 600 (90) 1.20b 0.08b Lognormal

Ef GPa ðksiÞ 40 (6,000) 1.04b 0.08b Lognormalκm 0.70 1.071c 0.115c Uniform

Fabrication dfa Beam 0.99 0.04 Lognormal

Slab 0.92 0.12c Beam 0.99 0.04 Lognormal

Slab 0.92 0.12b Beama 1.01 0.04 Lognormal

Slab 1.00 0.00 DeterministicAs

a Beam, slab 1.00 0.015 LognormalAf

b Beam, slab 1.00 0.03 Lognormal

Professional Psa 1.00 1.02 0.06 Lognormal

Pf 1.00 1.00 0.06 LognormalaNowak and Szerszen (2003a, b).bGulbrandsen (2005).c0.60 ≤ κm ≤ 0.90.


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but with the same probabilistic parameters of bias and variation.Higher variation and smaller bias factor for the effective depthand concrete cover differentiate slabs from beams.

• The effective depth of steel, ds, with negligible approximation,is estimated as

ds ¼ df − c ð11Þ

• Therefore, ds is a random variable whose nominal and meanvalue and its SD depend upon c and df and their relative mag-nitude. These three assumptions about df, c, and ds create aninverse correlation between the effective depth of the bars, ds,and the concrete cover, c, for a given overall depth of h ≈ df ,which simulates the reality.

• The strip unit width used in analysis and design of slabs, b, isassumed to be a deterministic parameter.

• The statistical data for the FRP bars Af, is determined accordingto Gulbrandsen (2005).The professional factor is the ratio of actual to theoretical behav-

ior, excluding the material and fabrication effects. In other words,even if the exact values of material strength and dimensions of amember are known, a difference between the theoretical and testresults might be observed. This difference, known as professionalfactor, is ascribed to the “model error” (Ellingwood et al. 1980) andis caused by common assumptions in structural mechanics(e.g., perfect linearity). Here, the professional factor is independ-ently applied to steel and FRP contributions. In mathematicalterms, material, fabrication, and professional factors can be repre-sented by this statistical model:

R ¼ Rs þ Rf ð12Þ

where R, the total flexural strength of the member, is a randomvariable comprised of steel and FRP random contributions, Rsand Rf:

Rs ¼ PsMs; Rf ¼ PfMf ð13Þ

where Ms and Mf = random variables whose nominal values areMns andMnf [Eqs. (2) and (3)], and contain the collective effect ofthe material and fabrication factors; Ps and Pf = random variablesrepresenting the professional factor and account for the overall un-certainties associated with parameters (other than strength and size)that are presumed to be deterministic when resistance is calculated.There is little reason to believe that the models describing thebehavior of FRP RC beams are any less exact than steel RC beams,once all the mechanical properties (strength, modulus of elasticity,and bonding strength) are assumed to be deterministic. Therefore,the probabilistic parameters of Ps (i.e., λ and δ) according to

Nowak and Szerszen (2003a, b) are deemed to be applicable toPf. However, to remain reasonably conservative, the bias factor(λ) for FRP is lowered to 1.00 (Table 1). Table 1 also displaysthe assumed probabilistic distributions of each parameter. As men-tioned earlier, κm is assigned a uniform distribution (0.60 ≤ κm≤ 0.90). Other parameters are assumed to be lognormally distrib-uted because their physical aspect eliminates the possibility ofnegative values (Haldar and Mahadevan 2000).

Simulation Matrix

Table 2 shows a summary of the simulation matrix. Thirty-two setsof slabs and 30 sets of beams, divided equally between the twotypes of FRP specified in Table 1, are simulated. Each set containsfive members, one unstrengthened and four with different levels ofNSM FRP, resulting in a total of 310 members. All unstrengthenedmembers, representing typical designs, are tension controlled inflexure, according to ACI 318-11 (2011), as shown by the limitsof steel ratios, ρs. Because this study focuses on the flexure, shearresistance is assumed to be sufficient.

A very limited number of simulations pertain to the strengthen-ing range of 100–150%; however, the majority is restricted to ap-proximately 100%, which means that the nominal flexural capacitymay be nearly doubled by strengthening. This is consistent with therecommendations of ACI 440.2R-08, Chapter 9 (2008).

Monte Carlo Simulation

The Monte Carlo method is a technique to generate samples ofnumerical data (e.g., resistance) from which the statistical param-eters may be calculated, such as mean and SD, or equally, biasfactor and COV. Appendix I and Table 6 provides an exampleof how this technique is employed in this study. Tables 3 and 4contain examples of the results of such simulations. For each de-sign, in addition to the nominal values of resistance, the statisticalparameters are also calculated by the Monte Carlo simulation. Ap-pendix II and Table 7 compares the results of the simulation withexperimental results from the technical literature, based on the stan-dard statistical tests, and draws some conclusions on the confidenceof the simulation.

Reliability Analysis and Strength Reduction Factors

The strength reduction factors are calibrated based on the compar-ative reliability equation (Jawaheri Zadeh and Nanni 2012):

Table 2. Summary of Simulated Members

Member df , mm (in.) ds, mm (in.) b, mm (in.) ρs (%) Strengthening level (%) Number of members

Slabs 150 (6.0) 115 (4.5) 1000 (40.0) 0.2, 0.3, 0.4, 0.5a 20–110b 160200 (8.0) 165 (6.5)250 (10.0) 215 (8.5)300 (12.0) 265 (10.5)

Beams 300 (12.0) 235 (9.5) 200 (8.0) 0.5, 1.0, 1.5c 20–150b 150450 (18.0) 385 (15.5) 300 (12.0)600 (24.0) 535 (21.5) 450 (18.0)750 (30.0) 685 (27.5) 450 (18.0)900 (36.0) 835 (33.5) 600 (24.0)

aρs ¼ As=bdf .bApproximate range, divided into five levels.cρs ¼ As=bds.


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Tab

le3.

Examples

ofResults

forNSM

Strengthened

SlabsperUnitWidth

of1,000mm

Slab

Propertiesa

Steelcontributio

nFR

Pcontributio

nTo

talstrength

(NSM

)

Strengthening

level(%

)Failu

remoded

ØNSM

=ØRCe

Set

d f,

mm

(in.)

d s,

mm

(in.)

As,

sq.in.(m

m2)

Afb

(mm

2)

Mns

c

(kN·m

)λ

δM

nfc

(kN·m

)λ

δM

nc

(kN·m

)λ

δ

1(ρ

s¼

0.2%)

150(6.0)

115(4.5)

300(0.48)

013.5

1.090

0.189

0.0

——

13.5

1.090

0.189

0—

1.000

2013.2

1.085

0.190

3.6

1.205

0.200

16.8

1.111

0.176

24II

1.043

4013.2

1.086

0.191

7.1

1.206

0.200

20.3

1.128

0.172

50II

1.066

6013.1

1.086

0.191

10.6

1.205

0.198

23.7

1.139

0.171

76II

1.079

8013.1

1.085

0.190

14.1

1.207

0.200

27.2

1.149

0.172

101

II1.086

2(ρ

s¼

0.3%)

200(8.0)

165(6.5)

600(0.96)

038.3

1.087

0.176

0.0

——

38.3

1.073

0.176

0—

1.000

4037.7

1.087

0.176

9.4

1.207

0.199

47.1

1.111

0.165

23II

1.056

8037.4

1.085

0.177

18.7

1.206

0.202

56.2

1.125

0.165

47II

1.069

120

37.3

1.085

0.177

28.0

1.205

0.200

65.3

1.137

0.164

70II

1.082

160

37.1

1.082

0.177

37.2

1.201

0.198

74.3

1.142

0.165

94II

1.085

3(ρ

s¼

0.4%)

250(10.0)

215(8.5)

1000

(1.60)

082.4

1.087

0.168

0.0

——

82.4

1.077

0.168

0—

1.000

8081.1

1.086

0.169

23.3

1.207

0.201

104.4

1.113

0.160

27II

1.048

160

80.5

1.083

0.170

46.3

1.201

0.198

126.8

1.126

0.160

54II

1.060

240

79.9

1.079

0.172

69.1

1.187

0.200

148.9

1.129

0.165

81II

1.054

320

79.3

1.080

0.175

91.5

1.161

0.204

170.7

1.123

0.174

107

II1.032

4(ρ

s¼

0.5%)

300(12.0)

265(10.5)

1500

(2.40)

0151.0

1.086

0.163

0.0

——

151.0

1.080

0.163

0—

1.000

120

148.8

1.086

0.168

41.6

1.203

0.203

190.4

1.112

0.160

26II

1.035

240

147.4

1.081

0.170

82.6

1.180

0.205

230.0

1.117

0.165

52II

1.031

360

146.0

1.078

0.169

122.9

1.136

0.209

268.9

1.105

0.172

78II

1.007

480

144.5

1.075

0.172

162.3

1.077

0.224

306.8

1.076

0.186

103

II0.957

Note:

1,000mm

¼40in.

a Other

propertiesaccordingto

Table1.

b FRPTy

pe1from

Table1.

c 1.0

kN·m

=m¼

2.7

kip-in:=ft.

d I:concrete

crushing;II:debonding.

e According

toEq.

(14)

with

βT¼

2.5.


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Tab

le4.

Examples

ofResults

forNSM

Strengthened

Beams

Beam

propertiesa

Steelcontributio

nFR

Pcontributio

nTo

talstrength

(NSM

)

Strengthening

level(%

)Failu

remoded

ØNSM

=ØRCe

Set

b,mm

(in.)

d f,

mm

(in.)

d s,

mm

(in.)

As,

mm

2(sq.

in.)

Afb

(mm

2)

Mns

c

(kN·m

)λ

δM

nfc

(kN·m

)λ

δM

nc

(kN·m

)λ

δ

1(ρ

s¼

0.5%)

200(8.0)

300(12.0)

235(9.5)

235(0.38)

021.1

1.165

0.095

0.0

——

21.1

1.159

0.095

0—

1.000

5020.8

1.164

0.095

6.0

1.292

0.161

26.9

1.189

0.087

26II

1.049

100

20.7

1.163

0.096

12.0

1.288

0.158

32.7

1.204

0.089

53II

1.054

150

20.6

1.162

0.097

17.9

1.272

0.150

38.4

1.208

0.092

79II

1.052

200

20.4

1.160

0.098

23.7

1.246

0.143

44.1

1.202

0.094

105

II1.040

2(ρ

s¼

1.0%)

300(12.0)

450(18.0)

385(15.5)

1155

(1.86)

0162.9

1.161

0.094

0.0

——

162.9

1.170

0.094

0—

1.000

375

157.9

1.173

0.095

62.5

1.080

0.169

220.5

1.148

0.091

35I

0.986

750

154.7

1.176

0.095

101.0

1.098

0.167

255.8

1.147

0.097

57I

0.971

1,125

152.3

1.178

0.096

129.8

1.102

0.161

282.1

1.144

0.099

73I

0.962

1,500

150.2

1.178

0.095

153.2

1.104

0.160

303.4

1.142

0.102

86I

0.953

3(ρ

s¼

1.5%)

450(18.0)

600(24.0)

535(21.5)

3611

(5.805)

0675.2

1.171

0.094

0.0

——

675.2

1.171

0.094

0—

1.000

1,500

650.7

1.177

0.095

186.8

1.099

0.20

2837.6

1.160

0.096

24I

0.984

3,000

634.8

1.183

0.096

302.2

1.107

0.19

2937.0

1.159

0.101

39I

0.971

4,500

622.5

1.182

0.097

387.4

1.115

0.18

71009.9

1.157

0.106

50I

0.959

6,000

612.4

1.184

0.099

455.0

1.124

0.18

11067.5

1.159

0.110

58I

0.950

a Other

propertiesaccordingto

Table1.

b FRPTy

pe2from

Table1.

c 1.0

kN·m

¼8.9

kip-in.

d I:concrete

crushing;II:debonding.

e According

toEq.

(14)

with

βT¼

3.5.


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lnðϕNSMϕRC

λRCλNSM

Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2NSM þ δ2RC

p ¼ δRC − δNSMδNSM þ δRC

βT ð14Þ

Eq. (14) relates the strength reduction and statistical parameters(Ø, λ, and δ) of an NSM FRP strengthened member (subindexNSM) to those of an ordinary RC member (subindex RC) actingas the benchmark, if the same level of safety or target reliability ofβT is expected from both member types. In this study, the bench-mark RC members have the same statistical characteristics of re-sistance as the members prior to strengthening.

Tables 3 and 4 display sample simulated sets of five members.The first row of each set (i.e., unstrengthened RC member) consti-tutes the benchmark member. The strength reduction factor for eachbenchmark member is calculated according to ACI 318-11 (2011)and the safety levels of βT ¼ 2.5 for slabs or βT ¼ 3.5 for beamsare selected according to Nowak and Szerszen (2003a, b). Thevalue of βT is considered to be independent of the failure mode,although FRP rupture is more brittle and sudden than concretecrushing. This is in agreement with ACI 440.1R-06 (2006), whichmaintains a minimum reliability index of 3.5 for beams with inter-nal FRP reinforcement, regardless of their failure mode. This maybe justified by the fact that an FRP-governed failure requireslarge deflections, which can provide warning before collapseand act as a substitute for the relative ductility of the other mode(Gulbrandsen 2005).

The failure mode predicted by the design (nominal) values isdenoted by “Mode I” for concrete crushing and “Mode II” for de-bonding. The last column is the ratio of the Ø factor of the strength-ened member, ØNSM, to that of an RC member, ØRC, calculatedaccording to Eq. (14).

Discussion

Investigation of the Results

A general observation can be made from the results of the simu-lations, represented partially by Tables 3 and 4: the FRP contribu-tion has a higher deviation and higher bias factor than the steelcontribution. This is a foregone conclusion as a result of the higherbias and deviation associated with FRP as a material. Further ob-servations include:• Regarding slabs, the strength variation of the NSM FRP

member is normally lower than the variations of each of itscomponents. This is primarily attributable to the previouslymentioned variation reduction; partially, for the increased reduc-tion factors compared to the unstrengthened members (caseswith ØNSM=ØRC > 1.0).

• Regarding beams, the fabrication factors of beams, as opposedto those of slabs, have a markedly lower randomness, whichresults in the more deterministic behavior reflected in theirlower COVs. In the case of beams, the addition of FRP generallyincreases the variation when the failure is governed by concretecrushing. For debonding failures, the trend is primarily similarto that of slabs, which undergo the same failure mode.Aside from these general statements, a pattern is detected in the

calculated strength reduction factors so that they can be formulatedfor practical use. When steel yields, Eqs. (5) and (6) can be re-worked to predict the failure mode based on the ratio of the totalreinforcement index, ωs þ ωf , to the parameter, ωb, defined in thisstudy as

ωb ¼ 0:85β1

dfds


ð15Þ

where ωb marks the combination of the steel and FRP indices,which results in simultaneous FRP debonding and concrete crush-ing, if steel has already yielded. If ðωs þ ωfÞ=ωb ≤ 1.0, the debond-ing mode governs the failure; otherwise, failure is initiated by theconcrete crushing. This ratio can also be interpreted as an indicatorof the deflection before failure. The growth of ðωs þ ωfÞ=ωb leadsto smaller deflections before the flexural failure, and vice versa.Therefore, to formulate the Ø factor, members were regroupedbased on this ratio.

The first group is the one with ðωs þ ωfÞ=ωb ≤ 1.0, or debond-ing failure, which includes all of the simulated slabs and lightlyreinforced beams. Samples of this group are shown in Tables 3and 4 (Mode II). For this group, the calculated strength reductionfactor is almost always larger than ØRC. One exception is presentedin the last row of Table 3, which is, admittedly, a design far frompractical and may be discounted.

For two of other such groups of members (i.e., 1.0 ≤ ðωs þ ωfÞ=ωb ≤ 2.0 and 2.0 ≤ ðωs þ ωfÞ=ωb ≤ 4.0), the calculated strengthreduction factors are plotted against the strengthening levels inFigs. 1 and 2. These conditions, which require high steel or FRPratios, can only be satisfied by beams, so Figs. 1 and 2 only containbeam results. As shown graphically in Figs. 1 and 2 and employinga conservative approach to determine the lower boundary, areduction factor ratio can be suggested:

ØNSM

ØRC¼

8>><>>:

1.0∶ ifωs þ ωf

ωb≤ 1:0

1 − α9≥ 8

9∶ if

ωs þ ωf

ωb≥ 2:0

ð16Þ

0.880

0.890

0.900

0.910

0.920

0.930

0.940

0.950

0.960

0.970

0.980

0.990

1.000

0 20 40 60 80 100 120 140

ØNSM /ØRC

Strengthening level (%)=100α

8/9≤ØNSM/ØRC=1-α/9≤1.0

Fig. 1. Calculated strength reduction factors for beams with 1.0 ≤ðωs þ ωfÞ=ωb ≤ 2.0 (only points with ØNSM=ØRC ≤ 1.0 are shown)

0.880

0.890

0.900

0.910

0.920

0.930

0.940

0.950

0.960

0.970

0.980

0.990

1.000

0 20 40 60 80 100 120 140

ØNSM /ØRC

Strengthening level (%)=100α

8/9≤ØNSM/ØRC=1-α/9≤1.0

Fig. 2. Calculated strength reduction factors for beams with 2.0 ≤ðωs þ ωfÞ=ωb ≤ 4.0 (only points with ØNSM=ØRC ≤ 1.0 are shown)


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where α = strengthening level:

α ¼ Mnjstrengthened −MnjunstrengthenedMnjunstrengthened

ð17Þ

For values of 1.0 ≤ ðωs þ ωfÞ=ωb ≤ 2.0, a linear interpolation isconsidered satisfactory:

ØNSM

ØRC¼ 1 − α

9

�ωs þ ωf

ωb− 1

�≥ 8

9∶ if 1:0 <

ωs þ ωf

ωb< 2:0

ð18Þ

Example

The procedure of the calculation of the ultimate flexural strength,Mu = ØMn, for one of the beams in Table 4 (Set 2, fourth row) isdetailed here.

For the flexural capacity, Mn, of the strengthened member,from beam dimensions and material properties (Tables 1 and 4),the reinforcement indices are calculated as ωs ¼ 0.143, ωf ¼0.146 [Eq. (4)].

Assuming that εbi ¼ 0.00, Eqs. (7) and (8) are used to checkwhether the steel yields: ωsb ¼ 0.395 > ωs; therefore, steel yields.

Debonding can be checked as ωb ¼ 0.188 [Eq. (15)];ðωs þ ωfÞ=ωb ¼ 1.54 > 1.0, so no debonding is expected andconcrete crushing is the failure mode.

In this study, the design equations governing this mode of failure(concrete crushing and steel yielding) are derived as shown in thefollowing.

For the stress ratio in FRP:

f¼ ffffd

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ωsωf− εcuþεbi

εfd

�2þ 3:4β1

ωf

εcuεfd

dfds

r−�ωsωfþ εcuþεbi

εfd

�

2≤ 1 ð19Þ

where ff = stress in FRP. The steel contribution is calculated as

Mns ¼ f 0cωs

�1 − ωsþfωf

1:7

�bd2s ð20Þ

The contribution of FRP is

Mnf ¼ f 0cfωf

�dfds

− ωsþfωf

1:7

�bd2s ð21Þ

Eqs. (19)–(21) can be evaluated by substituting the correspond-ing value as f ¼ 0.697, Mns ¼ 152.3 kN · m (1,347 kip-in.),Mnf ¼ 129.8 kN · m (1,150 kip-in.).

Finally, according to Eq. (1), Mn ¼ 282.1 kN · m(2,497 kip-in.).

The strength reduction factor according to ACI 440.2R-08(2008) is calculated as

Mu ¼ ØRCðMns þ ψfMnfÞ ð22Þin which

0.65 ≤ ØRC ¼ 0.65þ 0.25εs − εsy

0.005 − εsy≤ 0.90; ψf ¼ 0.85

ð23Þwhere εs = strain in steel reinforcement, which can be calculatedfrom the compatibility of strains:

εs ¼dsdf

ðfεfd þ εbiÞ −�1 − ds

df

�εcu ≥ εsy ð24Þ

where εs ¼ 0.0058; therefore, ØRC ¼ 0.90 and Mu ¼ 236.4 kN ·m (2,092 kip-in.).

The strength reduction factor according to this study is1.0 < ðωs þ ωfÞ=ωb ¼ 1.54 < 2.0.

Therefore, the strength reduction factor is calculated fromEq. (18). The strengthening level, α, must be computed. From con-ventional reinforced concrete design:

Mnjunstrengthened ¼ f 0cωs

�1− ωs

1:7

�bd2s ¼ 162:9 kN ·mð1,442 kip-in:Þ

ð25Þ

According to Eq. (17), the strengthening level is α ¼ 0.73,and therefore ØNSM=ØRC ¼ 1 − ð0.73=9Þð1.54 − 1.00Þ ¼ 0.956[Eq. (18)].

Compared to the value of 0.962 obtained by the reliability analy-sis, i.e., Eq. (14) and Table 4, ØRC ¼ 0.90∶ØNSM ¼ 0.860.

Eventually, Mu ¼ ØNSM Mn ¼ 242.6 kNm (2,147 kip-in.).This is 2.6% higher than ACI 440.2R-08 recommendation.

Table 5 summarizes similar calculations for the beams ofTable 4. As indicated in the last column, the proposed reductionfactor commonly allows for a higher usable moment.

Comparison with ACI 440.2R-08

A direct comparison between the strength reduction factors ob-tained in this study [Eqs. (16)–(18)] with that stipulated by ACI440.2R-08 [Eq. (23)] may not seem very streamlined. Nevertheless,if the question is approached from a practical point of view, theissue can be greatly simplified. The lower values of the current re-duction factors from ACI 440.2R-08 (2008) are of relatively lowpractical importance because of a series of reasons, including:• They require low levels of strain on the tension side of the flex-

ural member, which can be translated into low levels of stress inthe strengthening system and its decreased efficiency. Such asituation may defeat the idea of strengthening in the first place.

• The overwhelming majority of the members in need of strength-ening are lightly reinforced, either because of their nature, likeslabs, or because of the reason that originates the requirement ofrepair, like underdesigned beams. Such designs lead to large de-formations on the tension side.

• The additional strength gained by adding FRP hardly ever goesbeyond 50%, with the normal values revolving around 25%.Again, this means less restraint for the member and compara-tively large deflections on the tension side.Tables 4 and 5 present a numerical explanation of these obser-

vations. Set 1 is composed of a lightly reinforced RC beam[ρs ¼ 0.5% slightly higher than the minimum permissible byACI 318 (2011), ρmin ¼ 0.35%] with different degrees of strength-ening, α. Set 3, on the other hand, is heavily reinforced (ρs ¼ 1.5%)and, unlike Set 1, for the strengthening levels in Table 5 requiresØRC factors of smaller than 0.90, according to ACI 440.2R-08(2008). A comparison between the relative ratios of FRP to steel(Af=As in Table 4, or more comprehensively ωf=ωs in Table 5) con-firms the much higher efficiency of the strengthening in Set 1. Theabsence of any slabs, and the rarity and relative impracticality of thebeams that require a factor lower than 0.90 by ACI 440.2R-08(2008) in the simulated designs, confirm that ØRC ¼ 0.90 maybe regarded as the most applicable strength reduction factor recom-mended by the guideline, which, certainly has to be applied in com-bination with the partial reduction factor of ψf ¼ 0.85 imposed onthe FRP contribution.


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Conclusions

Based on the findings of this study, the partial factor of 0.85proposed by ACI 440.2R-08 (2008) can be eliminated and newstrength reduction factors for NSM FRP members can be deter-mined as follows:• Ø ¼ 0.90 for lightly reinforced and strengthened flexural mem-

bers whose failure is governed by debonding; i.e., ðωs þ ωfÞ=ωb ≤ 1.0.

• For other cases, this study recommends a variable but unifiedreduction factor [Eq. (16)] whose effect is comparable to thedouble factor of ACI 440.2R-08 (2008) [Eq. (22)].The obvious advantages can be restated as increased cost-

effectiveness, primarily for slabs, without risking the safety, andrestored ACI tradition in applying the Ø factor to the overallstrength of a member and not to the individual materials.

This study is limited to the NSM systems because only the stat-istical data about FRP bars and their bond to concrete are availableor can be computed. Externally bonded pultruded laminates andin situ layup laminates may be investigated in the same manner,provided that the data concerning their uncertainties are reliablyprovided or estimated.

Appendix I. Example of Simulation Technique

The general procedure for generating random samples, xi, of alognormal variable X with a COV of δX ≤ 0.20, may be describedas (Nowak and Collins 2000):

xi ¼ μXeziδX ð26Þ

zi¼ Φ−1ðuiÞ ð27Þwhere ui = sample of a uniformly distributed variable between 0and 1; and Φ−1 = inverse of the standard normal cumulative dis-tribution; κm is assumed to be uniformly distributed, so its random

sample is generated by a simple interpolation over its range of dis-tribution (0.60 ≤ κm ≤ 0.90):

κmi ¼ 0.60þ 0.30ui ð28Þ

Table 6 shows an example of a simulation for an NSMstrengthened slab (Table 3, Set 2, second row). Excluding thelast the two rows, the nominal values of flexural contribution canbe calculated from the data in the second column, using ACI440.2R-08 (2008): Mns ¼ 37.7 kN · m=m (106.8 kip-in:=ft);Mnf ¼ 9.4 kN · m=m (25.8 kip-in:=ft); Mn ¼ Mns þMnf ¼47.1 kN · m=m (132.6 kip-in:=ft) [Eq. (1)].

These contributions for the ith random sample, denoted bysubindex i, may be calculated from the same data in the last col-umn: Msi ¼ 45.4 kN · m (128.6 kip-in:=ft); Mfi ¼ 13.4 kN · m(36.8 kip-in:=ft); Mi ¼MsiþMfi ¼ 58.8 kN ·m (165.4 kip-in:=ft)[Eq. (1)].

Eventually, applying the professional factors from the last tworows, the random samples of resistance are obtained: Rsi ¼ Psi ·Msi ¼ 45.1 kN · m=m (127.8 kip-in:=ft) [Eq. (13)]; Rfi ¼ Pfi ·Mfi ¼ 13.8 kN · m=m (37.9 kip-in:=ft) [Eq. (13)]; Ri ¼Rsi þ Rfi ¼ 58.9 kN · m=m (165.7 kip-in:=ft) [Eq. (12)].

Here, Ri is one random sample of the flexural strength of amember defined by the nominal or deign values given in the table.The statistical parameters of R can be calculated from the samples,once the number of simulations is large enough to represent thepopulation. This number can be determined by repeating the sim-ulation and measuring the consistency of the outcomes. Fig. 3 por-trays how the sample size is determined in this study. For the slab inthe example, the simulation is repeated 10 times, each time with nsamples (n ¼ 1; 10; : : : ; 100), resulting in 10 different values ofmean strength, μR, for each sample size, n. Let μS and σS representthe mean and SD of the 10 values of μR for any given n. Fig. 3shows how, as expected, the variation of the results declines asthe sample size grows, so that by n ¼ 10,000, the outcome is vir-tually deterministic. To eliminate any reservation, in this study,each simulation contains 20,000 samples.

Table 5. Ultimate Strength of the Beams in Table 4

Set ωb ωs ωf ðωs þ ωfÞ=ωb

Mns(kN · m)

Mnf(kN · m)

Mna

(kN · m) α (%) ØRC ØNSM=ØRCb ØNSM

Muc

(kN · m)Mu

d

(kN · m)Changee

(%)

1 0.205 0.071 0.000 0.35 21.1 0.0 21.1 0 0.900 1.000 0.900 19.0 19.0 0.00.016 0.43 20.8 6.0 26.9 26 0.900 1.000 0.900 24.2 23.3 3.90.032 0.51 20.7 12.0 32.7 53 0.900 1.000 0.900 29.4 27.8 5.80.048 0.59 20.6 17.9 38.4 79 0.900 1.000 0.900 34.6 32.2 7.20.064 0.67 20.4 23.7 44.1 105 0.900 1.000 0.900 39.7 36.5 8.8

2 0.188 0.143 0.000 0.77 162.9 0.0 162.9 0 0.900 1.000 0.900 146.6 146.6 0.00.049 1.03 157.9 62.5 220.5 35 0.900 0.999 0.899 198.2 189.9 4.40.097 1.29 154.7 101.0 255.8 57 0.900 0.982 0.884 226.1 216.5 4.40.146 1.54 152.3 129.8 282.1 73 0.900 0.956 0.860 242.6 236.4 2.60.195 1.82 150.2 153.2 303.4 86 0.900 0.922 0.830 251.8 252.4 −0.2

3 0.180 0.214 0.000 1.20 675.2 0.0 675.2 0 0.900 1.000 0.900 607.7 607.7 0.00.093 1.72 650.7 186.8 837.6 24 0.900 0.981 0.883 739.6 728.5 1.50.187 2.24 634.8 302.2 937.0 39 0.827 0.957 0.791 741.2 737.4 0.50.280 2.76 622.5 387.4 1,009.9 50 0.778 0.944 0.734 741.3 740.5 0.10.374 3.29 612.4 455.0 1,067.5 58 0.744 0.936 0.696 743.0 743.4 −0.1

Note: 1.0 kN · m ¼ 8.9 kip-in.aMn ¼ Mns þMnf .bAccording to Eqs. (16)–(18).cMu ¼ ØNSMMn: according to this study.dMu ¼ ØRCðMns þ 0.85MnfÞ: according to ACI 440.2R-08 (2008).eRelative difference between Mu ¼ ØNSMMn and Mu ¼ ØRCðMns þ 0.85MnfÞ.


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Appendix II. Simulation Model versus ExperimentalResults

The simulation model may be tested against experimental results onNSM FRP strengthened specimens. Table 7 presents the experi-mental ultimate failure loads of 16 NSM specimens. It also pro-vides the corresponding nominal design loads [according to ACI440.1R-08 (2008)], and the means and SDs (according to theMonte Carlo model) for the tested members. The last column ofTable 7 presents the normalized results, upon which the modelcan be tested. The ordered pair of normalized mean and standarddeviation of the test results is calculated according to Table 7as ðμ;σÞexp ¼ ð−0.18; 1.18Þ.

However, the expected statistical parameters of the normalizedpopulation of the simulated samples are ðμ; σÞsim ¼ ð0; 1Þ.

Standard statistical tests may be employed to investigate thesignificance of the differences between the experimental andsimulated results (Hicks and Turner 1999). A significance levelof 10% is selected for both tests.

Test Concerning the Mean

The null hypothesis is H0: μexp < μsim. The test statistic, z, may becalculated as

z ¼ μexp − μsim

σsim=ffiffiffiffiN

p ¼ −0:72 ð29Þ

The sample size is N ¼ 16. For the significance level of 10%,z0.10 ¼ Φ−1ð0.1Þ ¼ −1.28, whereΦ is the CDF of the standard nor-mal distribution, and z0.10 ¼ −1.28 < z ¼ −0.72.

The null hypothesis is rejected at the 10% level of significanceand the difference between the simulation and experiment is notsignificant at this level.

Test Concerning the Standard Deviation

The null hypothesis is H0: σexp > σsim. The test statistic, χ2, may becalculated as

χ2 ¼ ðN − 1Þσ2exp

σ2sim

¼ 20:89 ð30Þ

From the cumulative chi-square distribution with 15 degrees offreedom ðN − 1Þ∶χ2

0.10 ¼ 22.31 > χ2 ¼ 20.89.

Table 7. Experimental versus Simulated Samples

ReferenceNominala

(kN)Meanb

(kN)SDb

(kN)Exp.c

(kN)(Experimental—Mean)/

SD

El-Hacha andRizkalla (2004)

70.7 86.3 8.5 93.8 0.8869.4 84.8 8.0 99.3 1.8177.0 95.1 9.3 110.2 1.62

100.5 122.5 12.0 102.7 −1.65De Lorenziset al. (2000)

176.1 213.4 19.1 203.7 −0.51222.8 273.7 27.1 226.0 −1.76173.5 206.5 16.9 197.0 −0.56

Castro et al.(2007)

223.3 268.6 23.1 245.5 −1.00225.1 270.3 23.1 250.0 −0.88211.3 252.8 21.8 253.5 0.03210.8 252.5 21.4 249.5 −0.14226.8 273.4 23.6 250.0 −0.99225.5 270.9 23.1 226.8 −1.91

Ceroni (2010) 41.4 46.8 4.4 45.8 −0.2241.4 46.8 4.4 50.7 0.8846.7 52.9 4.9 60.5 1.56

μexp ¼ −0.18σexp ¼ 1.18

Note: 1.0 kN ¼ 0.225 kip.aAccording to ACI 440.2R-08 (2008).bAccording to simulation.cTest results.

48

50

52

54

56

58

60

62

1 10 100 1,000 10,000 100,000

R (

kN.m

/m)

µS

µS+σS

µS-σS

n

Fig. 3. Variation of calculated mean resistance versus number of sam-ples for the slab in the example (1.0 kN · m=m ¼ 2.7 kip-in:=ft)

Table 6. Simulation for NSM Strengthened Slab

Item Nominal value (N) Bias (λ) COV (δ) Mean (μ ¼ λN) ui zi xi

f 0c, MPa (ksi) 28 (4.0) 1.24 0.10 34.7 (4.96) 0.766 0.726 36.8 (5.33)

fy, MPa (ksi) 400 (60) 1.145 0.05 458 (68.7) 0.539 0.098 476 (69.0)ffu, MPa (ksi) 1,750 (250) 1.20 0.08 2,100 (300) 0.630 0.332 2126 (308.1)Ef , GPa (ksi) 140 (20,000) 1.04 0.08 145.6 (20,800) 0.544 0.111 1,46.9 (20,986)κm 0.70 1.071 0.115 0.75 0.663 — 0.799df , mm (in.) 200 (8.0) 0.92 0.12 184 (7.36) 0.757 0.697 200 (7.89)c, mm (in.) 35 (1.5) 0.92 0.12 32 (1.38) 0.513 0.033 32 (1.39)ds, mm (in.) 165 (6.5) — — 152 (5.98) — — 168 (6.50)b, mm (in.) 1,000 (40.0) 1.00 0.00 1,000 (40.0) — — 1,000 (40.0)As, mm2 (sq. in.) 600 (0.96) 1.00 0.015 600 (0.96) 0.227 −0.749 593 (0.95)Af , mm2 (sq. in.) 40 (0.064) 1.00 0.03 40 (0.064) 0.703 0.533 41 (0.065)Ps 1.00 1.02 0.06 1.02 0.330 −0.440 0.993Pf 1.00 1.00 0.06 1.00 0.682 0.473 1.029


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Again, the null hypothesis is rejected and the difference betweenthe simulation and experiment is not significant at the significancelevel of 10%.

Plainly speaking, the statistical tests show that these experimen-tal results do not reject the simulation results at the significancelevel of 10%.

Acknowledgments

The authors gratefully acknowledge the financial support for thisresearch provided by the NSF under grant IIP-0933537 as well asthe contribution of the industry members to the NSF Industry/University Cooperative Research Center based at the Universityof Miami.

Notation

The following symbols are used in this paper:Af = area of FRP reinforcement (mm2; sq. in.);As = area of steel reinforcement (mm2; sq. in.);b = width of the beam or slab strip (mm; in.);c = concrete cover to the centroid of steel

reinforcement (mm; in.);CE = environmental reduction factor;df = effective depth of FRP reinforcement (mm; in.);ds = effective depth of steel reinforcement (mm; in.);Ef = modulus of elasticity of FRP bars (GPa; ksi);Es = modulus of elasticity of steel bars (GPa; ksi);f = ratio of stress in FRP to its debonding stress;ff = stress level in FRP reinforcement (MPa; ksi);ffd = design stress of externally bonded FRP reinforcement

(MPa; ksi);ffu = ultimate longitudinal tensile strength of FRP bars

(MPa; ksi);fy = yield strength of steel reinforcement (MPa; ksi);f 0c = specified compressive strength of concrete (MPa; ksi);

f�fu = ultimate tensile strength of the FRP material, as reportedby the manufacturer (MPa; ksi);

Mf = contribution of FRP to flexural resistance disregardingprofessional factor (kN · m; kip-in.);

Mn = nominal flexural strength (kN · m; kip-in.);Mnf = contribution of FRP reinforcement to nominal flexural

strength (kN · m; kip-in.);Mns = contribution of steel reinforcement to nominal flexural

strength (kN · m; kip-in.);Ms = contribution of steel to flexural resistance disregarding

professional factor;Mu = factored moment (kN · m; kip-in.);N = sample size;n = number of simulations;

Pf = professional factor for FRP contribution;Ps = professional factor for steel contribution;R = total flexural resistance (random variable);Rf = contribution of FRP to total flexural resistance (random

variable);Rs = contribution of steel to total flexural resistance (random

variable);ui = ith sample of a uniformly distributed variable between 0

and 1;z = test statistic concerning mean;α = strengthening level;α1 = multiplier on f 0

c for an equivalent rectangular stressdistribution for concrete;

β1 = ratio of depth of equivalent rectangular stress block todepth of the neutral axis;

βT = target reliability;δX = coefficient of variation of a random variable X;εbi = strain level in concrete substrate at time of FRP

installation (mm=mm; in:=in:);εcu = ultimate axial strain of concrete,

0.003mm=mm × ð in:=in:Þ. (mm=mm);εsy = strain corresponding to yield strength of steel

reinforcement (mm=mm; in:=in:);κm = 0.70, dimensionless bond-dependent coefficient for

flexure;λX = bias factor of a random variable X;μX = mean value of a random variable X;ρs = steel reinforcement ratio;σX = standard deviation of a random variable X;Φ = cumulative distribution function (CDF) for standard

normal distribution;χ2 = test statistic concerning variance;ψf = partial reduction factor for FRP, 0.85;ωb = overall reinforcement index producing balanced strain

conditions;ωf = reinforcement index for FRP bars;ωfb = FRP reinforcement index producing balanced strain

conditions;ωs = reinforcement index for steel bars; andωsb = steel reinforcement index producing balanced strain

conditions; andØ = strength reduction factor.

References

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Strength Reduction Factor for Flexural RC Members Strengthened with Near-Surface-Mounted Bars

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