Silva RBD recommendations for deflection control of fiber reinforced polymer reinforced concrete beams

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185ACI Structural Journal/May 2020
ACI STRUCTURAL JOURNAL TECHNICAL PAPER
Fiber-reinforced polymers (FRPs), as noncorrosive materials, offer 
a great potential for use as reinforcement in concrete construc-
tion. Nevertheless, the characteristics of these materials have led 
to new challenges in the design of FRP-reinforced concrete (RC) 
components. Design of steel-RC beams usually results in under- 
reinforced beams, with failure governed by the yielding of steel, 
while in the FRP-RC counterparts, concrete crushing is the most 
desirable failure mode. Compared to steel bars, FRP displays higher 
strength and lower Young’s modulus, thus indicating that the design 
of FRP-RC elements will be largely influenced by the serviceability 
limit state of excessive deflections. A significant body of knowl-
edge has been accrued towards the safety of FRP-RC components 
with respect to ultimate limit states; on the contrary, the probabi-
listic assessment of the serviceability of FRP-RC beams is almost 
nonexistent. This study presents a contribution to the development 
of reliability-based design recommendations for deflection control 
of FRP-RC beams. A framework for the probabilistic assessment 
of the deflections of FRP-RC beams designed according to ACI 
440 is described. Monte Carlo simulation is used in the proba-
bilistic description of beam deflections and in the computation of 
the probabilities of excessive deflections (and attendant reliability 
indexes) of 81 representative beams. The results indicate a wide 
range of values for the reliability indexes (from positive up to nega-
tive ones); additionally, all parameters (load ratio, FRP strength, 
concrete compressive strength, and failure mode) have a consider-
able impact on the resulting reliability levels. The use of a smaller 
strength-reduction factor led to a significant improvement in the 
resulting reliability levels for FRP-RC beams.
Keywords: beams; deflections; design codes; fiber-reinforced polymer 
(FRP); Monte Carlo simulation; reinforced concrete (RC); reliability; 
serviceability limit state.
INTRODUCTION
Durability of reinforced concrete (RC) structures and, 
consequently, the service life of these structures is often 
dictated by the corrosion of reinforcing steel usually caused 
by chloride ingress (Lorensini and Diniz 2010) or carbon-
ation (Couto and Diniz 2018). Accordingly, a major chal-
lenge in increasing sustainability of RC structures is to 
reduce/eliminate corrosion of the reinforcing bars. While 
this may be achieved by a number of techniques as presented 
in ACI 222.3R (ACI Committee 222 2011), fiber-reinforced 
polymers (FRPs), as noncorrosive materials, offer a great 
potential for use as reinforcement in concrete construction.
Although the use of FRP bars as structural reinforcement 
shows great promise in terms of durability, the character-
istics of these materials have led to new challenges in the 
design of FRP-RC components. Design of steel-RC beams 
usually results in under-reinforced beams, with the failure 
governed by yielding of steel, while in the FRP-RC coun-
terparts, concrete crushing is the most desirable failure 
mode (Nanni 1993). Furthermore, compared to steel bars, 
FRP displays higher strength and lower Young’s modulus, 
thus indicating that the design of FRP-RC elements will be 
largely influenced by the serviceability limit state of exces-
sive deflections (Mota et al. 2006; La Tegola 1998).
Deflections of RC beams have traditionally been computed 
using an elastic deflection equation that includes an effec-
tive moment of inertia, Ie, originally introduced by Branson 
(1965) for steel-reinforced concrete (Bischoff and Gross 
2011). However, it has been recognized that Branson’s empir-
ical equation gives a too-stiff response for FRP-RC beams 
and underestimates deflection of such members (Nawy and 
Neuwerth 1977; Yost et al. 2003). At the same longitudinal 
reinforcement ratio, the replacement of steel bars with FRP 
would typically result in larger deflections (Gao et al. 1998; 
Tighiouart et al. 1999). Numerous equations have been 
proposed for calculating the effective moment of inertia of 
FRP-RC members (Peña 2010; Silva 2017).
Current design recommendations of steel-reinforced 
members (for example, ACI 318 [ACI Committee 318 2014] 
and Eurocode 2 [CEN 2004]) are based on limit state design 
principles—that is, the member is designed for its required 
strength and then checked for serviceability criteria. The 
limit states format, also known as semi-probabilistic format, 
employs one “characteristic” value of each uncertain param-
eter (usually resistances and load effects) with reduced resis-
tances and majored loads obtained via partial resistance and 
load factors. These factors are currently defined via probabi-
listic methods (Ang and Tang 1984) in a process known as 
code calibration. An example of such a procedure is found 
in Szerszen and Nowak (2003), describing the calibration of 
ACI 318 for different structural components and the perti-
nent ultimate limit states.
While much has been achieved in terms of developing a 
clear rationale for code calibration with respect to ultimate 
limit states of traditional steel RC construction, the same is 
not true for serviceability limit states. Although conceptual 
models have long been available, the implementation of 
reliability-based performance indicators for serviceability 
Title No. 117-S61
Reliability-Based Design Recommendations for Deflection 
Control of Fiber-Reinforced Polymer-Reinforced 
Concrete Beams
by Elayne M. Silva, Sidnea E. C. Ribeiro, and Sofia M. C. Diniz
ACI Structural Journal, V. 117, No. 3, May 2020.
MS No. S-2019-163.R1, doi: 10.14359/51723499, received May 20, 2019, and 
reviewed under Institute publication policies. Copyright © 2020, American Concrete 
Institute. All rights reserved, including the making of copies unless permission is 
obtained from the copyright proprietors. Pertinent discussion including author’s 
closure, if any, will be published ten months from this journal’s date if the discussion 
is received within four months of the paper’s print publication.
186 ACI Structural Journal/May 2020
limit states in current codes has lagged for a number of 
reasons (Ghosn et al. 2016).
RESEARCH SIGNIFICANCE
In spite of the important differences between steel-RC and 
FRP-RC structures, code development regarding the latter 
has followed a similar path as for traditional steel RC, with 
an initial emphasis on the design of safe FRP-RC structures 
and definition of the requisite partial safety factors. To this 
end, a significant body of knowledge has been accrued 
towards the safety of FRP-RC components with respect to 
ultimate limit states (ACI Committee 440 2015; Ribeiro and 
Diniz 2013; Shield et al. 2011). On the contrary, probabi-
listic assessment of the serviceability of FRP-RC beams and 
derivation of reliability-based design recommendations for 
deflection control of such elements is almost nonexistent. In 
this context, this study presents a contribution to the 
development of reliability-based design recommendations 
for deflection control of FRP-RC beams. A framework for 
the probabilistic assessment of deflections of FRP-RC beams 
designed according to ACI 440.1R (ACI Committee 440 
2015) is described. Monte Carlo simulation is used in the 
probabilistic description of the beam deflections, and in the 
computation of the probability of excessive deflections (and 
attendant reliability indexes) of representative beams.
RELIABILITY BASICS FOR DEFLECTION 
CONTROL OF FRP-RC BEAMS
Supply versus demand problem
The basic problem of structural reliability may be cast as 
a supply versus demand problem—that is, to ascertain that 
the supply (for example, strength), X, will be larger than the 
demand (for example, load effects), Y, throughout a refer-
ence period (for example, the service life of the structure), 
represented by (X > Y). Considering the uncertainties asso-ciated to these variables, this assurance is possible only in 
terms of the probability P(X > Y), thus representing a real-
istic measure of the reliability of the structural component 
(Ang and Tang 1984).
Margin of safety
The supply-demand problem may be formulated in terms 
of the margin of safety, M, where M = X – Y. Because X and Y 
are random variables, M is also a random variable with corre-
sponding probability density function, fM(m). Failure corre-
sponds to the condition (M 0 represents “satisfactory performance” and g(X) ρfb) or FRP rupture (ρf ρfb (over-reinforced 
beams), the nominal bending resistance, Mn, is given by
 M A f d a
n f f= −


2
 (8)
where a and the stress level at the FRP reinforcement, ff, are, 
respectively
 a
A f
f b
f f
c
=
′0 85.
 (9)
 f
E f
E E ff
f cu c
f
f cu f cu fu= +
′
−








≤
( ) .
.
ε β
ρ
ε ε
2
1
4
0 85
0 5 
(10)
For ρf


ε
ε ε
 (12)
According to ACI 440.1R (ACI Committee 440 2015), 
to account for long-term exposure to the environment, the 
design tensile strength of FRP bars is
 ffu = CE f*fu (13)
where CE is the environmental reduction factor; and f*fu 
is guaranteed tensile strength of FRP bar, defined as mean 
tensile strength of sample of test specimens minus three 
times the standard deviation.
MONTE CARLO SIMULATION OF DEFLECTIONS 
OF GFRP-RC BEAMS
In this section, the framework of the Monte Carlo simu-
lation of deflections of GFRP-RC beams is detailed. The 
selected beams are presented, a procedure for the calculation 
of the total deflection of GFRP-RC beams is established, the 
statistical description of the basic variables are reviewed, 
and the corresponding statistics of the total deflections of the 
selected beams are obtained.
Selected beams
Eighty-one GFRP-RC beams, designed according to ACI 
440.1R (ACI Committee 440 2015) recommendations for 
flexure, were selected for analysis. Due to the smaller costs 
compared to other types of fibers used in civil engineering 
construction, GFRP has been considered for the reinforcement 
of the selected beams (Ribeiro and Diniz 2013). All beams 
are simply supported, 3 m span, 20 x 30 cm2 rectangular 
cross sections, and subjected to uniformly distributed loads. 
The analysis was planned to verify the influence of concrete 
compressive strength, amount of longitudinal reinforcement, 
GFRP tensile strength, and load ratio on the implicit reliability 
levels with respect to the limit state of excessive deflections.
Three specified concrete compressive strengths—30, 50, 
and 70 MPa—were chosen. Higher concrete compressive 
strengths (50 and 70 MPa) have been included because it has 
been reported that the high tensile strength of FRP is most 
efficiently used when paired with high-strength concrete 
(Nanni 1993). Three GFRP tensile strengths, f*fu—485, 
850, and 1275 MPa, consistent with those suggested in 
ACI 440.1R (ACI Committee 440 2015) for GFRP—were 
selected; the environmental reduction factor, CE, is assumed 
as 0.8. FRP reinforcement ratios in the range of 0.82 to 2.10 
ρfb have been used, representing under-reinforced, in the 
transition zone, and over-reinforced beams. The selected 
GFRP bars have diameters ranging from 6.3 to 22.5 mm. 
Three mean dead load to mean live load ratios (μDL/μLL = 0.5, 
1.0, and 2.0) have been selected.
Each beam is identified by four groups of numbers and/or 
letters. The first group—C30, C50, and C70—represents the 
specified concrete compressive strength in MPa. The second 
group is related to the tensile strength of GFRP bars (P1: 
485 MPa, P2: 850 MPa, and P3: 1275 MPa). The third group 
is related to the load ratio (R5, R1, and R2 correspond to 
μDL/μLL = 0.5, 1.0, and 2.0, respectively). The fourth group is 
associated to the GFRP longitudinal ratio (UR is under- 
reinforced beams, TR is transition zone, and OR is over- 
reinforced beams). For example, Beam C50-P2-R2-UR has 
a specified concrete cylinder strength of 50 MPa, GFRP 
tensile strength of 850 MPa, load ratio equal to 2.0, and is 
under-reinforced.
Deflection equation for GFRP-RC beams
Total deflection, ∆total, is calculated by the sum of the 
immediate deflection, ∆i, and the long-term deflection due to 
creep and shrinkage, ∆(cp + sh).
Immediate deflection, ∆i, of FRP-RC beams is calculated 
by an elastic equation that takes into account the load acting 
on the structure at service pserv, Young’s modulus of concrete, 
Ec, and effective moment of inertia Ie. For simply supported 
beams subjected to uniformly distributed loads, the determin-
istic equation to calculate the immediate deflection is
 ∆ i
serv
c e
p
E I
=
5
384
4
 (14)
Equation (14) requires the effective moment of inertia, Ie, 
which, for steel-reinforced beams, is usually calculated by 
Branson’s equation. In the case of FRP-RC beams, it has been 
reported that Branson’s equation overestimates the effective 
moment of inertia (ACI Committee 440 2015). Therefore, 
different equations have been suggested for the calculation 
of the effective moment of inertia of FRP-RC beams.
In addition to uncertainties inherent in the variables relevant 
to the problem at hand, reliability analysis must include uncer-
tainty in model predictions. In this light, the main problem 
related to Eq. (14) is the selection of the equation for the effec-
tive moment of inertia, Ie, and the incorporation of the atten-
dant uncertainty in the deflection prediction, into the analysis.
Mota et al. (2006) present the statistics of the random vari-
able “model error”—that is, the ratio “experimental deflec-
tion/calculated deflection”, η = ∆exp/∆calc, for FRP-RC beams. 
The calculated deflection, ∆calc, was obtained using different 
equations for the effective moment of inertia in the prediction 
of deflections at service. For GFRP-RC beams, those results 
indicate the ranges 0.49 to 1.45 for the mean model error, μη, 
and 0.063 to 0.082 for the coefficient of variation, COV(η). 
The statistics of the model error, η, may be used in the selec-
tion of the equation to best represent the effective moment of 
inertia. To this end, a mean model error, μη, close to 1.0 and a 
small COV(η) are desirable features of the model.
Considering that the model error associated to the equa-
tion suggested by Yost et al. (2003) has mean μ = 0.95 and 
COV = 0.0698, this is the equation used in this study for the 
prediction of the effective moment of inertia of GFRP-RC 
beams. As a comparison, the corresponding statistics found 
by Mota et al. (2006) for the equation proposed by Gao 
et al. (1998) are μ = 1.24 and COV = 0.074. Accordingly, the 
effective moment of inertia, Ie, is given by (Yost et al. 2003)
189ACI Structural Journal/May 2020
 I M
M
I M
M
I Ie
cr
a
d g
cr
a
cr g= 



+ − 



















≤
3 3
1β (15)
where Mcr is cracking moment; Ma is maximum moment in the 
beam at the stage deflection is computed; Ig is gross moment 
of inertia; Icr is moment of inertia of transformed cracked 
section, βd is reduction coefficient, βd = α([Ef/Es] + 1); α is 
bond dependent coefficient, α = 0.064 (ρf/ρfb) + 0.13; and Es is 
steel modulus of elasticity. The model error, η, is then used as 
a multiplier of the immediate deflection given by Eq. (14), thus 
resulting in the adjusted immediate deflection, ∆i,a
 ∆ i a
serv
c e
p
E I, =
5
384
4

η (16)
The long-term deflection due to creep and shrinkage, 
∆(cp + sh), for FRP-RC beams can be calculated according to 
the following equations (ACI Committee 440 2015)
 ∆(cp + sh) = 0.6λ(∆i)sus (17)
 λ
ξ
ρ
=
+ ′( )1 50
 (18)
where (∆i)sus is the immediate deflection due to sustained 
loads; and ρ′ is the ratio of compression reinforcement. The 
parameter λ in Eq. (17) reduces to ξ because compression 
reinforcement is not considered for FRP-RC members (ρ′ = 
0). Values of the time-dependent factor for sustained loads, 
ξ, are reported in ACI 318 (ACI Committee 318 2014).
In this study, ξ is taken as 2.0 (sustained load duration of 
5 years or more). This value is consistent with the 8-year 
reference period for live loads adopted in this study for 
the serviceability analysis (refer to the section “Loads”). 
Furthermore, it is assumed that 20% of the live load is 
sustained loading. In this way, the long-term deflection of 
GFRP-RC beams is calculated by
 ∆(cp + sh) = 1.2 [(∆i,a)D + 0.2 (∆i,a)L] (19)
where the subscripts D and L in the adjusted immediate 
deflections, ∆i,a, correspond to deflections caused by dead 
and live loads, respectively. Thus, the total deflection is 
given by
 ∆total = [0.8 (∆i,a)L] + ∆(cp + sh) (20)
This deterministic procedure for the computation of the 
beam total deflection is illustrated by the flowchart presented 
in Fig. 2.
Fig. 2—Flowchart of deterministic procedure for computation of total deflection.
190 ACI Structural Journal/May 2020
Statistical descriptionof basic variables
The statistics (mean, coefficient of variation, and type of 
distribution) of the basic variables considered in this study 
(cross section geometry, concrete compressive strength, 
GFRP tensile strength, GFRP modulus of elasticity, dead 
loads, live loads at service, and model error) are summa-
rized in Table 1. The rationale for the assumed statistics 
is presented in the following. Beam span (ℓ = 3.0 m) and 
FRP reinforcement area (nominal area of the FRP bar) are 
assumed as deterministic.
Compressive strength, modulus of elasticity of 
concrete, and cracking moment
Evolution of quality control worldwide has resulted in 
coefficients of variation close to 0.10 for a wide range of 
compressive strengths (Azevedo and Diniz 2008; Nowak 
and Szerszen 2003). Nowak and Szerszen (2003) recom-
mend a Normal distribution to describe the variability of 
the concrete compressive strength; nevertheless, other 
researchers suggest a Lognormal distribution (Azevedo 
and Diniz 2008; Diniz and Frangopol 1997). In this study, a 
Lognormal distribution is assumed for the random variable 
concrete compressive strength (cylinder strength).
For a given specified strength, fc′, obtained from cylinder 
tests, if the coefficient of variation of the concrete compressive 
strength, V, is known, then the mean, μ, may be calculated 
by ACI 318 (ACI Committee 318 2014)
 fc′ = μ(1 – 1.34V) (21)
The random variable in-place concrete compressive 
strength, fc, is obtained by multiplying the random variable 
cylinder strength by a reduction factor, αc, equal to 0.85 for 
fc′ ≤ 55 MPa; or αc = 0.85 – 0.004 (fc′ – 55) ≥ 0.75, for fc′ > 
55 MPa (Diniz and Frangopol 1997).
The modulus of elasticity of concrete, Ec, is taken as a 
random variable derived from the in-place compressive 
strength of concrete, fc
 E fc c= 4700 (22)
with fc given in MPa. Cracking moment, Mcr, is also a 
random variable derived from the modulus of rupture of 
concrete, fr, and the gross moment of inertia, Ig, Mcr = frIg/yt. 
The distance from centroidal axis of gross section to tension 
face, yt, is assumed as deterministic. Modulus of rupture of 
concrete, fr, is taken as a random variable derived from the 
in-place compressive strength, fc, calculated by
 f fr c= 0 62. � (23)
Table 1—Statistics of basic variables
Basic variable μ σ COV Probability distribution
Dimensions*
∆h, ∆b 1.524 mm 6.35 mm 0.0417 Normal
Cover, ∆c 6.35 + 0.004h, mm 4.22 mm — Normal
Concrete compressive strength†
fc′ = 30 MPa 34.64 MPa 3.46 MPa 0.10 Lognormal
fc′ = 50 MPa 57.74 MPa 5.77 MPa 0.10 Lognormal
fc′ = 70 MPa 80.83 MPa 8.08 MPa 0.10 Lognormal
GFRP tensile strength (f*fu)
f*fu = 485 MPa 570.59 MPa 28.53 MPa 0.05 Normal
f*fu = 850 MPa 1000 MPa 50 MPa 0.05 Normal
f*fu = 1275 MPa 1500 MPa 75 MPa 0.05 Normal
GFRP modulus of elasticity (Ef)
Ef = 35 GPa 35 GPa 1750 MPa 0.05 Normal
Ef = 42.5 GPa 42.5 GPa 2125 MPa 0.05 Normal
Ef = 50 GPa 50 GPa 2500 MPa 0.05 Normal
Model error
η 0.95 0.066 0.0698 Normal
Loads
Type μU/Un
‡ COV Probability distribution
Dead load, D 1.05 0.10 Normal
Live load (ULS), LULS 1.00 0.25 Type I
Live load (SLS), LSLS 0.65 0.32 Type I
*Deviations from nominal values.
†Cylinder strength.
‡Ratio mean to unfactored nominal load.
191ACI Structural Journal/May 2020
Gross moment of inertia, Ig, is assumed as a random vari-
able (refer to the section “Cross section geometry” for the 
variability of beam depth, h, and beam width, b). For rectan-
gular cross sections, Ig is given by
 I bh
g =
3
12
 (24)
Mechanical properties of FRP bars
In this sudy, three nominal tensile strengths, f*fu, of the 
GFRP bar are adopted: 485, 850, and 1275 MPa; a Normal 
distribution with coefficient of variation of 0.05 is assumed 
to describe the variability of the tensile strength, ffu (Pilak-
outas et al. 2002). The corresponding mean tensile strength, 
ffu, ave, of the FRP bar is given by ACI Committee 440 (2015)
 ffu, ave = f*fu + 3s (25)
The design modulus of elasticity of FRP, Ef, is the same as 
the value reported by the manufacturer—that is, it is equal to 
the mean elastic modulus (nominal value), Ef,ave, of a sample 
of test specimens (ACI Committee 440 2015)
 Ef = Ef, ave (26)
It is assumed that the modulus of elasticity of FRP, Ef, is 
a random variable with mean 35, 42.5, and 50 GPa for the 
strengths of 485, 850, and 1275 MPa, respectively; it follows 
a Normal distribution and has coefficient of variation of 0.05 
(Pilakoutas et al. 2002).
Cross section geometry
Following Mirza and Macgregor (1979), it is considered 
that the variability of the deviations in the nominal depth, 
∆h, and the nominal width, ∆b, are described by a Normal 
distribution with mean of 1.524 mm and standard deviation 
of 6.35 mm. Concrete cover is also a random variable, and 
the deviation in the nominal value, ∆c, has mean
 μ∆c = 6.35 + 0.004h (27)
and standard deviation equal to 4.22 mm.
Model error
The “model error”, η, is assumed as a Normal random 
variable, with parameters μ = 0.95 and COV = 0.0698 (refer 
to the section “Deflection equation for GFRP-RC beams”).
Loads
Table 1 presents information (ratio of mean to unfactored 
nominal load, coefficient of variation, and type of distribu-
tion) on the random variables: 1) dead load, DL; 2) live load 
at the ultimate limit state (Galambos et al. 1982); LL(ULS), 
and 3) live load at service, LL(SLS), for an 8-year reference 
period (Galambos and Ellingwood 1986). It is assumed that 
only dead and live loads act on the FRP-RC beam. In a reli-
ability analysis for the limit state of excessive deflections, 
the corresponding mean values of dead loads and live loads 
at service must be defined for each beam.
The design flexural capacity of the beam, Md = ϕMn, is 
assumed as identical to the FRP-RC beam flexural demand, 
Mu. The design moment is computed by
 ϕMn = γD MDn + γLMLn (28)
where γD is the dead load factor, equal to 1.2; MDn is the 
nominal moment due to dead load; γL is the live load factor, 
equal to 1.6; and MLn is the nominal moment due to live load. 
For simply supported beams subjected to uniformly distrib-
uted nominal dead and live loads (Dn and Ln, respectively), 
then
 φ γ γM
D L
n D
n
L
n= +
 
2 2
8 8
 (29)
According to the load statistics in Table 1 for the ratio of 
mean to unfactored nominal load, μU/Un at the ultimate limit 
states, it is obtained
 Dn
DL=
µ
1 05.
 (30)
 Ln
LL ULS=
µ ( )
.1 00
 (31)
where μDL and μLL(ULS) are the means of the random vari-
ables dead load, DL, and live load, at the ultimate limit state, 
LL(ULS), respectively. It follows that
 φ γ
µ
γ µMn D
DL
L LL ULS= +




2
8 1 05.
( )
 (32)
In terms of the load ratio, r = μDL/μLL(ULS), Eq. (32) can be 
rewritten as
 φ µ γ γM r
n LL ULS D L= +


( )

2
8 1 05.
 (33)
Thus, resulting for the mean live load at the ultimate limit 
state
 µ
γ γ
LL ULS
d
D L
M
r( )
.
=
+



8
1 05
2

 (34)
The mean live load at service, μLL(SLS), is calculated 
from the data in Table 1 for the serviceability limit state, 
μLL(SLS) = 0.65μLL(ULS).
This procedure is used in the calculation of the mean dead 
load and mean live load at service for each of the 81 repre-
sentative beams. These results are summarized in Table 2, 
which presents design moment Md, mean dead load μDL, 
mean live load at ultimate limit state μLL(ULS), and mean live 
load at service μLL(SLS), for R5, R1, and R2 beams (r = 0.5, 
1.0, and 2.0, respectively).
192 ACI Structural Journal/May 2020
Statistical description of deflections of GFRP-RC 
beams
Monte Carlo simulation was used to obtain the statis-
tics of total deflections, ∆total, for each of the 81 GFRP-RC 
beams designed according to ACI 440 provisions. To this 
end: 1) the deterministic procedure presented in Fig. 2; and 
2) generation of random numbers consistent with the statis-
tics summarized in Tables 1 and 2 are required. For each 
GFRP-RC beam, a sample of 100,000 total deflections, ∆total, 
wasgenerated. This computational procedure was imple-
mented using the Matlab software, version 7.0.1 (and Statis-
tics toolbox). The main steps in the Monte Carlo simulation 
are shown in the flowchart displayed in Fig. 3.
The statistics of the total deflections (minimum, mean, 
and maximum) corresponding to the 81 GFRP-RC beams 
are presented in Tables 3, 4, and 5 for load ratios 0.5, 1.0, 
and 2.0, respectively. These tables also display the nominal 
value of the total deflection computed according to ACI 
440.1R (ACI Committee 440 2006), ∆total,ACI, and the ratio 
μMSC/∆total,ACI (μMSC is the mean total deflection obtained via 
Monte Carlo simulation). From these results, it is seen that, as 
expected, total deflections (immediate plus long-term deflec-
tions) increase as the load ratio increases—that is, as more 
sustained loads act on the beam. For instance, the mean total 
deflections for Beams C30-P3-R5-OR, C30-P3-R1-OR, and 
C30-P3-R2-OR, are 0.0081, 0.0116, and 0.0164 m, respec-
tively. Most importantly, it is seen that most over-reinforced 
beams—which correspond to the desirable failure mode for 
FRP-RC beams—present larger deflections as compared to 
under-reinforced and transition zone beams. This results 
from the fact that comparatively more loads are allowed in 
over-reinforced beams due to larger strength-reduction 
factors for such beams; refer to Eq. (6a) through (6c).
From Tables 3 through 5, it is seen that the ratios 
μMSC/∆total,ACI are in the range of 0.99 to 1.34, 1.01 to 1.28, 
and 1.03 to 1.26 for r = 0.5, 1.0, and 2.0, respectively. 
Considering that in most cases these ratios are larger than 
the unit, it can be concluded that there is a trend for the mean 
Table 2—Design moment Md, mean dead load μDL, mean live load at ultimate state μLL(ULS), and mean live 
load at service μLL(SLS) of GFRP-RC beams
Beam Md, kN·m
R5 R1 R2
μDL, kN μLL(ULS), kN μLL(SLS), kN μDL, kN μLL(ULS), kN μLL(SLS), kN μDL, kN μLL(ULS) , kN μLL(SLS), kN
C30-P1-R*-UR 17.52 3.59 7.17 4.66 5.68 5.68 3.69 8.02 4.01 2.61
C30-P1-R*-TR 28.84 5.90 11.81 7.67 9.35 9.35 6.08 13.19 6.60 4.29
C30-P1-R*-OR 42.26 8.65 17.30 11.24 13.70 13.70 8.90 19.33 9.67 6.28
C30-P2-R*-UR 18.37 3.76 7.52 4.89 5.95 5.95 3.87 8.40 4.20 2.73
C30-P2-R*-TR 24.48 5.01 10.02 6.51 7.93 7.93 5.16 11.20 5.60 3.64
C30-P2-R*-OR 29.94 6.13 12.25 7.97 9.70 9.70 6.31 13.70 6.85 4.45
C30-P3-R*-UR 16.51 3.38 6.76 4.39 5.35 5.35 3.48 7.55 3.78 2.46
C30-P3-R*-TR 20.61 4.22 8.44 5.48 6.68 6.68 4.34 9.43 4.71 3.06
C30-P3-R*-OR 25.72 5.27 10.53 6.84 8.34 8.34 5.42 11.77 5.88 3.82
C50-P1-R*-UR 24.84 4.88 9.76 6.34 7.73 7.73 5.02 10.91 5.45 3.54
C50-P1-R*-TR 40.97 8.38 16.77 10.90 13.28 13.28 8.63 18.74 9.37 6.09
C50-P1-R*-OR 58.88 12.05 24.10 15.67 19.08 19.08 12.40 26.94 13.47 8.76
C50-P2-R*-UR 24.85 5.09 10.17 6.61 8.05 8.05 5.24 11.37 5.69 3.70
C50-P2-R*-TR 32.16 6.58 13.17 8.56 10.42 10.42 6.77 14.71 7.36 4.78
C50-P2-R*-OR 41.27 8.45 16.89 10.98 13.37 13.37 8.69 18.88 9.44 6.14
C50-P3-R*-UR 20.89 4.28 8.55 5.56 6.77 6.77 4.40 9.56 4.78 3.11
C50-P3-R*-TR 28.24 5.78 11.56 7.51 9.15 9.15 5.95 12.92 6.46 4.20
C50-P3-R*-OR 35.36 7.24 14.47 9.41 11.46 11.46 7.45 16.18 8.09 5.26
C70-P1-R*-UR 38.99 7.98 15.96 10.37 12.64 12.64 8.21 17.84 8.92 5.80
C70-P1-R*-TR 54.82 11.22 22.44 14.59 17.76 17.76 11.55 25.08 12.54 8.15
C70-P1-R*-OR 75.75 15.50 31.01 20.16 24.55 24.55 15.96 34.66 17.33 11.26
C70-P2-R*-UR 32.20 6.59 13.18 8.57 10.44 10.44 6.78 14.73 7.37 4.79
C70-P2-R*-TR 42.99 8.80 17.60 11.44 13.93 13.93 9.06 19.67 9.84 6.39
C70-P2-R*-OR 57.76 11.82 23.64 15.37 18.72 18.72 12.17 26.43 13.21 8.59
C70-P3-R*-UR 28.40 5.81 11.63 7.56 9.20 9.20 5.98 13.00 6.50 4.22
C70-P3-R*-TR 37.74 7.72 15.45 10.04 12.23 12.23 7.95 17.27 8.63 5.61
C70-P3-R*-OR 46.12 9.44 18.88 12.27 14.95 14.95 9.71 21.10 10.55 6.86
193ACI Structural Journal/May 2020
total deflection to be larger than those predicted by ACI 
440.1R (ACI Committee 440 2006) procedures. The level 
of conservatism in the computation of the total deflection 
∆total,ACI is approximately the same for all values of the load 
ratio r considered in this study.
Figures 4 through 6 show histograms of deflections for 
some analyzed beams. As a reference, the attendant allowable 
deflection, δa, is indicated in each histogram; δa is assumed as 
deterministic and equal to ℓ/240. Considering that the main 
goal in this study is the assessment of reliability levels corre-
sponding to the limit state of excessive deflections, distribu-
tion fitting of the generated samples was not pursued.
RELIABILITY ANALYSIS OF SERVICEABILITY OF 
GFRP-RC BEAMS
In this study, the performance function represented by 
the margin of safety, g(X) = δa – ∆total, Eq. (4), is used in a 
Monte Carlo simulation procedure for the reliability assess-
ment of 81 GFRP-RC beams with respect to the limit state 
of excessive deflections. With the samples of the random 
variable “total deflection”, ∆total, generated according to the 
procedure previously described and an assumed determin-
istic “allowable deflection”, the probabilities of underperfor-
mance (probabilities of failure) can be easily obtained.
Figures 7 through 9 show the histograms of the margin of 
safety, g(X), for selected beams. As a reference, the line corre-
sponding to the limit state condition—that is, g(X) = 0, is drawn 
in each histogram; the larger the area of the histogram in the 
g(X)0.0006 0.0071 0.0457 0.0066 1.0773
C30-P3-R*-OR 0.0013 0.0116 0.0519 0.0112 1.0368
C50-P1-R*-UR 0.0006 0.0060 0.0296 0.0056 1.0708
C50-P1-R*-TR 0.0038 0.0149 0.0385 0.0146 1.0211
C50-P1-R*-OR 0.0065 0.0152 0.0338 0.0147 1.0315
C50-P2-R*-UR 0.0006 0.0065 0.0382 0.0059 1.0953
C50-P2-R*-TR 0.0015 0.0119 0.0469 0.0113 1.0466
C50-P2-R*-OR 0.0033 0.0171 0.0502 0.0168 1.0238
C50-P3-R*-UR 0.0003 0.0036 0.0335 0.0031 1.1493
C50-P3-R*-TR 0.0009 0.0087 0.0496 0.0080 1.0881
C50-P3-R*-OR 0.0018 0.0139 0.0555 0.0133 1.0451
C70-P1-R*-UR 0.0023 0.0125 0.0364 0.0114 1.0914
C70-P1-R*-TR 0.0056 0.0171 0.0400 0.0162 1.0533
C70-P1-R*-OR 0.0077 0.0167 0.0362 0.0159 1.0494
C70-P2-R*-UR 0.0010 0.0094 0.0435 0.0078 1.2005
C70-P2-R*-TR 0.0027 0.0162 0.0507 0.0146 1.1051
C70-P2-R*-OR 0.0057 0.0202 0.0499 0.0191 1.0580
C70-P3-R*-UR 0.0006 0.0066 0.0448 0.0051 1.2837
C70-P3-R*-TR 0.0016 0.0133 0.0568 0.0113 1.1740
C70-P3-R*-OR 0.0029 0.0184 0.0602 0.0166 1.1098
Table 5—Statistics of total deflection (r = 2.0)
Beam
R2
∆total, m ∆total, ACI, 
m
μMCS/ 
∆total,ACIMinimum Mean Maximum
C30-P1-R*-UR 0.0010 0.0073 0.0281 0.0070 1.0556
C30-P1-R*-TR 0.0048 0.0166 0.0374 0.0162 1.0273
C30-P1-R*-OR 0.0075 0.0168 0.0326 0.0162 1.0384
C30-P2-R*-UR 0.0010 0.0081 0.0360 0.0075 1.0732
C30-P2-R*-TR 0.0027 0.0152 0.0460 0.0147 1.0378
C30-P2-R*-OR 0.0047 0.0197 0.0484 0.0191 1.0287
C30-P3-R*-UR 0.0006 0.0056 0.0333 0.0051 1.1030
C30-P3-R*-TR 0.0013 0.0106 0.0461 0.0099 1.0700
C30-P3-R*-OR 0.0027 0.0164 0.0526 0.0157 1.0409
C50-P1-R*-UR 0.0014 0.0086 0.0300 0.0081 1.0687
C50-P1-R*-TR 0.0066 0.0185 0.0391 0.0178 1.0364
C50-P1-R*-OR 0.0082 0.0179 0.0347 0.0172 1.0431
C50-P2-R*-UR 0.0014 0.0095 0.0387 0.0088 1.0878
C50-P2-R*-TR 0.0032 0.0163 0.0476 0.0155 1.0518
C50-P2-R*-OR 0.0063 0.0220 0.0510 0.0212 1.0375
C50-P3-R*-UR 0.0006 0.0056 0.0326 0.0049 1.1330
C50-P3-R*-TR 0.0019 0.0127 0.0502 0.0118 1.0820
C50-P3-R*-OR 0.0038 0.0192 0.0563 0.0183 1.0504
C70-P1-R*-UR 0.0044 0.0160 0.0370 0.0147 1.0917
C70-P1-R*-TR 0.0087 0.0205 0.0407 0.0193 1.0613
C70-P1-R*-OR 0.0092 0.0195 0.0371 0.0184 1.0580
C70-P2-R*-UR 0.0021 0.0133 0.0441 0.0113 1.1827
C70-P2-R*-TR 0.0052 0.0211 0.0515 0.0192 1.1026
C70-P2-R*-OR 0.0094 0.0247 0.0507 0.0232 1.0651
C70-P3-R*-UR 0.0012 0.0098 0.0442 0.0078 1.2587
C70-P3-R*-TR 0.0033 0.0186 0.0576 0.0161 1.1597
C70-P3-R*-OR 0.0057 0.0243 0.0611 0.0220 1.1064
Fig. 4—Histogram of deflections: C30-P3-R5-UR beam. Fig. 5—Histogram of deflections: C50-P3-R1-TR beam.
195ACI Structural Journal/May 2020
adjusting a probability distribution to the data represented by 
the histogram of the margin of safety can be bypassed.
Table 6 exhibits the probabilities of failure, Pf, obtained 
in this study and the corresponding reliability indexes, β, 
for each of the 81 GFRP-RC beams. It can be seen that all 
parameters (load ratio, GFRP strength, concrete compres-
sive strength, and failure mode) have a considerable impact 
on the resulting reliability levels. These results indicate a 
wide range of values for the reliability index, β, from posi-
tive up to negative ones: 2.54 up to –0.40 (with r = 0.5); 
2.44 up to –1.91 (r =1.0); and 2.08 up to –3.14 (r = 2.0). 
It is reminded that a negative value of β corresponds to Pf 
in excess of 0.5. All other parameters remaining constant, 
beams with a larger load ratio—that is, more sustained 
loading—present a smaller reliability index (and larger Pf). 
For instance, comparing Beams C30-P1-R5-UR, C30-P1-
R1-UR, and C30-P1-R2-UR, it is observed that β is equal to 
2.37, 2.18, and 1.67, respectively.
Regarding the effect of the GFRP strength, all other param-
eters remaining the same, β is the largest for beams with 
the highest GFRP strength considered in this research—that 
is, Beams P3. For instance, for Beams C30-P1-R5-OR and 
C30-P3-R5-OR, the reliability indexes are 0.40 and 1.03, 
respectively. Furthermore, it is observed that β decreases as 
the concrete compressive strength increases. For example, 
for Beams C30-P3-R5-OR, C50-P3-R5-OR, and C70-P3-
R5-OR, the reliability indexes are 1.03, 0.70, and 0.04, 
respectively. With respect to the failure mode, as represented 
by under-reinforced, transition zone, and over-reinforced 
beams, β is the largest for under-reinforced beams and the 
smallest for over-reinforced ones. For example, for Beams 
C30-P1-R5-UR, C30-P1-R5-TR, and C30-P1-R5-OR, the 
reliability indexes are 2.38, 0.74, and 0.40, respectively.
DISCUSSION OF RESULTS
For further analyses of the reliability levels obtained 
in this study for the limit state of excessive deflections of 
GFRP-RC beams, a target reliability index, βtarget, equal to 
1.5, as suggested by Galambos and Ellingwood (1986) for 
the serviceability limit state, is considered. This target reli-
ability corresponds to floor beams under occupancy load 
for an 8-year reference period and is consistent with the 
analysis performed in this research. For this target value, 
it is observed that out of the 81 representative beams, the 
target is met in only 19 beams, none of them being over- 
reinforced (refer to Table 6). Additionally, only two of the 
C70 grade GFRP-RC beams (out of 27), display a reliability 
index above the chosen target value.
Larger strength-reduction factors, ϕ, for over-reinforced 
beams, as recommended by ACI 440 (2015), result in 
comparatively more loads acting on such beams. Submitting 
that the larger ϕ factors for over-reinforced beams are one of 
Fig. 6—Histogram of deflections: C70-P2-R2-OR beam.
Fig. 7—Histogram of margin of safety: C30-P1-R2-OR beam.
Fig. 8—Histogram of margin of safety: C50-P3-R5-UR beam.
Fig. 9—Histogram of margin of safety: C70-P1-R1-TR beam.
196 ACI Structural Journal/May 2020
the reasons for larger deflections in such beams, the use of 
a constant ϕ factor equal to 0.55 in the design process was 
investigated. The corresponding results are shown in Table 7. It 
is observed that, as expected, there is an improvement in the 
reliability levels. For example, for Beam C30-P3-R5-OR, 
β = 1.03 when ϕ = 0.65 (refer to Table 6) while β = 1.80 
for ϕ = 0.55 (refer to Table 7). However, only 22 of the 81 
GFRP-RC beams present reliability indexes that meet the 
target value and two beams are over-reinforced.
In an attempt to further improve the reliability levels for 
the limit state of excessive deflections, the use of a smaller 
factor is also investigated. From the results in Table 8 for 
ϕ = 0.50, it is observed that there is a significant improve-
ment in the reliability levels and βtarget is met in 37 out of 
the 81 GFRP-RC beams, six of them being over-reinforced. 
For a variable ϕ factor (Eq. (6)), ϕ = 0.55, and ϕ = 0.50, 
C50-P3-R1-OR beam have reliability indexes –0.11, 1.00, 
and 1.57, respectively. For ϕ = 0.50, five of the high-strength 
concrete GFRP-RC beams (C70 grade) display a reliability 
index above βtarget.
Based on these results, it is observed that the smaller the 
strength reduction factor, the greater the improvement in 
the reliability index and the implicit reliability levels for the 
limit state of excessive deflections. From the results shown 
in Table 8, it is seen that in general terms, βtarget can be satis-
fied on the condition of using GFRP’s of higher strengths 
(and consequently higher Young’s modulus) and avoiding 
higher concrete compressive strengths. Furthermore, it must 
be emphasized that the adequacy, or not, of the implicit reli-
ability levels are highly dependent on the selected target 
value, a condition for which no consensus exists to date.
SUMMARY AND CONCLUSIONS
In this research, the reliability of GFRP-RC beams 
designed according to ACI 440 (2015), with respect to the 
limit state of excessive deflections, was assessed. To this end, 
Table 6—Probabilities of failure (and reliability 
indexes), limit state of excessive deflections
Beam
R5 R1 R2
Pf β Pf β Pf β
C30-P1-R*-UR 0.0088 2.3756 0.0145 2.1832 0.0471 1.6741
C30-P1-R*-TR 0.2307 0.7366 0.5267 –0.06690.8844 –1.1972
C30-P1-R*-OR 0.3445 0.4002 0.7027 –0.5321 0.9527 –1.6717
C30-P2-R*-UR 0.0159 2.1464 0.0289 1.8968 0.0974 1.2968
C30-P2-R*-TR 0.1280 1.1360 0.3082 0.5011 0.7001 –0.5247
C30-P2-R*-OR 0.3232 0.4587 0.6693 –0.4379 0.9474 –1.6202
C30-P3-R*-UR 0.0057 2.5296 0.0080 2.4103 0.0211 2.0311
C30-P3-R*-TR 0.0390 1.7629 0.0845 1.3752 0.2764 0.5937
C30-P3-R*-OR 0.1516 1.0297 0.3590 0.3612 0.7566 –0.6955
C50-P1-R*-UR 0.0159 2.1462 0.0299 1.8824 0.1041 1.2586
C50-P1-R*-TR 0.3645 0.3463 0.7278 –0.6061 0.9648 –1.8093
C50-P1-R*-OR 0.4736 0.0662 0.8310 –0.9580 0.9832 –2.1246
C50-P2-R*-UR 0.0265 1.9348 0.0542 1.6058 0.1853 0.8952
C50-P2-R*-TR 0.1597 0.9956 0.3799 0.3059 0.7810 –0.7754
C50-P2-R*-OR 0.4638 0.0908 0.8253 –0.9359 0.9854 –2.1794
C50-P3-R*-UR 0.0054 2.5484 0.0072 2.4471 0.0188 2.0788
C50-P3-R*-TR 0.0681 1.4900 0.1577 1.0041 0.4629 0.0932
C50-P3-R*-OR 0.2435 0.6951 0.5427 –0.1073 0.8974 –1.2666
C70-P1-R*-UR 0.1911 0.8737 0.4538 0.1161 0.8407 –0.9972
C70-P1-R*-TR 0.5964 –0.2441 0.9155 –1.3757 0.9951 –2.5814
C70-P1-R*-OR 0.6565 –0.4029 0.9368 –1.5282 0.9962 –2.6711
C70-P2-R*-UR 0.0803 1.4029 0.1912 0.8736 0.5319 –0.0801
C70-P2-R*-TR 0.3846 0.2933 0.7485 –0.6698 0.9706 –1.8901
C70-P2-R*-OR 0.7461 –0.6624 0.9719 –1.9093 0.9992 –3.1382
C70-P3-R*-UR 0.0306 1.8715 0.0632 1.5285 0.2156 0.7871
C70-P3-R*-TR 0.2138 0.7933 0.4894 0.0265 0.8648 –1.1020
C70-P3-R*-OR 0.4832 0.0421 0.8414 –1.0004 0.9876 –2.2445
Table 7—Probabilities of failure (and reliability 
indexes), limit state of excessive deflections (φφ = 0.55) 
ϕ= 0.55
Beam
R5 R1 R2
Pf β Pf β Pf β
C30-P1-R*-UR 0.0088 2.3756 0.0145 2.1832 0.0471 1.6741
C30-P1-R*-TR 0.2008 0.8386 0.4719 0.0705 0.8499 –1.0360
C30-P1-R*-OR 0.1126 1.2129 0.2908 0.5512 0.6778 –0.4615
C30-P2-R*-UR 0.0159 2.1464 0.0289 1.8968 0.0974 1.2968
C30-P2-R*-TR 0.0757 1.4346 0.1804 0.9139 0.5063 –0.0157
C30-P2-R*-OR 0.1012 1.2747 0.2451 0.6901 0.6208 –0.3075
C30-P3-R*-UR 0.0057 2.5296 0.0080 2.4103 0.0211 2.0311
C30-P3-R*-TR 0.0198 2.0575 0.0378 1.7766 0.1289 1.1317
C30-P3-R*-OR 0.0367 1.7908 0.0791 1.4114 0.2644 0.6297
C50-P1-R*-UR 0.0159 2.1462 0.0299 1.8824 0.1041 1.2586
C50-P1-R*-TR 0.2990 0.5274 0.6424 –0.3650 0.9392 –1.5480
C50-P1-R*-OR 0.1823 0.9066 0.4452 0.1379 0.8246 –0.9332
C50-P2-R*-UR 0.0265 1.9348 0.0542 1.6058 0.1853 0.8952
C50-P2-R*-TR 0.1183 1.1837 0.2845 0.5694 0.6772 –0.4597
C50-P2-R*-OR 0.1709 0.9505 0.4072 0.2347 0.8075 –0.8687
C50-P3-R*-UR 0.0054 2.5485 0.0072 2.4471 0.0188 2.0788
C50-P3-R*-TR 0.0366 1.7917 0.0789 1.4125 0.2610 0.6402
C50-P3-R*-OR 0.0677 1.4934 0.1582 1.0018 0.4673 0.0820
C70-P1-R*-UR 0.1911 0.8737 0.4538 0.1161 0.8407 –0.9972
C70-P1-R*-TR 0.5143 –0.0359 0.8660 –1.1076 0.9900 –2.3248
C70-P1-R*-OR 0.3155 0.4802 0.6665 –0.4304 0.9384 –1.5411
C70-P2-R*-UR 0.0803 1.4029 0.1912 0.8736 0.5319 –0.0801
C70-P2-R*-TR 0.3023 0.5178 0.6420 –0.3637 0.9400 –1.5544
C70-P2-R*-OR 0.3909 0.2770 0.7588 –0.7025 0.9729 –1.9251
C70-P3-R*-UR 0.0306 1.8715 0.0632 1.5285 0.2156 0.7871
C70-P3-R*-TR 0.1267 1.1421 0.3039 0.5133 0.7014 –0.5284
C70-P3-R*-OR 0.1818 0.9084 0.4273 0.1834 0.8238 –0.9301
197ACI Structural Journal/May 2020
samples of deflections of the selected beams were generated 
by Monte Carlo simulation, and the corresponding probabil-
ities of excessive deflections were obtained.
Based on the results of the research presented herein, the 
following conclusions can be drawn:
• A comparison of the nominal value of the total deflec-
tion computed according to ACI 440 (2006) recommen-
dations, ∆total,ACI, and the mean total deflection obtained 
via Monte Carlo simulation, μMSC, indicates that there 
is a trend for the mean total deflection to be larger than 
those predicted by ACI 440 procedures;
• The reliability levels associated to the limit state of 
excessive deflections indicate a wide range of values for 
the reliability indexes, β, from positive up to negative 
ones;
• The load ratio has a great influence on the implicit 
reliability levels; all other parameters remaining 
constant, the higher the load ratio μDL/μLL (that is, more 
sustained loading), the smaller the reliability index, and, 
consequently, the larger the probability of excessive 
deflections;
• Over-reinforced beams present larger deflections—and, 
consequently, smaller reliability indexes—as compared 
to under-reinforced and transition zone beams;
• The highest probabilities of failure (exceeding 0.5), are 
found for high-strength concrete GFRP-RC beams (C70 
grade);
• For a target reliability index, βtarget, equal to 1.5 (as 
suggested by Galambos and Ellingwood 1986), only 19 
GFRP-RC beams present reliability indexes that meet 
this target value, none of them being over-reinforced;
• Only two of the C70 grade GFRP-RC beams, out of the 
27, displays a reliability index above the selected target 
value. Therefore, caution should be exercized in the use 
of higher concrete strengths in GFRP-RC beams;
• The use of a constant and smaller strength-reduction 
factor (ϕ = 0.50) lead to a significant improvement in 
the resulting reliability levels for GFRP-RC beams, and 
particularly for C50 beams paired with higher GFRP 
strengths (P2 and P3). This could be a simple alterna-
tive for designing GFRP-RC beams for both safety and 
serviceability.
The reliability results with respect to the limit state of 
excessive deflections reported in this research have been 
obtained for FRP-RC beams designed according to ACI 
440 recommendations. The procedures presented herein 
can be easily extended to the assessment of the implicit reli-
ability levels in design recommendations for FRP-RC beams 
other than ACI 440. Moreover, it must be emphasized that 
the results obtained—and their interpretation—were made 
under a number of assumptions such as allowable deflection 
limit, load ratio, reference period, and target reliability level. 
Particularly, no consensus exist on the target reliability (and 
attendant issues such as reference periods) for the service-
ability limit state of excessive deflections and more research 
is needed on this subject.
AUTHOR BIOS
Elayne M. Silva is a PhD Candidate in the Civil Engineering Department 
of the Federal Center of Technological Education of Minas Gerais, Belo 
Horizonte, Brazil.
Sidnea E. C. Ribeiro is an Associate Professor in the Department of Mate-
rials and Construction at Federal University of Minas Gerais.
Sofia M. C. Diniz, FACI, is a Professor of structural engineering in the 
Department of Structural Engineering at the Federal University of Minas 
Gerais. She is past Chair and current member of ACI Committee 348, 
Structural Reliability and Safety.
ACKNOWLEDGMENTS
The financial support provided by the Brazilian agencies CAPES (Coor-
denação de Aperfeiçoamento de Pessoal de Nível Superior) and CNPq 
(Conselho Nacional de Desenvolvimento Científico e Tecnológico) is grate-
fully acknowledged.
REFERENCES
ACI Committee 222, 2011, “Guide to Design and Construction Prac-
tices to Mitigate Corrosion of Reinforcement in Concrete Structures (ACI 
222.3R-11),” American Concrete Institute, Farmington Hills, MI, 32 pp.
Table 8—Probabilities of failure (and reliability 
indexes), limit state of excessive deflections (φφ = 0.50) 
ϕ= 0.50
Beam
R5 R1 R2
Pf β Pf β Pf β
C30-P1-R*-UR 0.0028 2.7750 0.0035 2.6968 0.0086 2.3833
C30-P1-R*-TR 0.0976 1.2955 0.2421 0.6995 0.6172 –0.2982
C30-P1-R*-OR 0.0506 1.6388 0.1290 1.1312 0.4048 0.2409
C30-P2-R*-UR 0.0057 2.5302 0.0079 2.4121 0.0216 2.0212
C30-P2-R*-TR 0.0318 1.8553 0.0678 1.4925 0.2313 0.7347
C30-P2-R*-OR 0.0441 1.7051 0.1002 1.2805 0.3288 0.4433
C30-P3-R*-UR 0.0016 2.9557 0.0017 2.9218 0.0033 2.7144
C30-P3-R*-TR 0.0073 2.4422 0.0107 2.3001 0.0315 1.8593
C30-P3-R*-OR 0.0143 2.1901 0.0255 1.9510 0.0849 1.3727
C50-P1-R*-UR 0.0056 2.5370 0.0083 2.3972 0.0236 1.9849
C50-P1-R*-TR 0.1593 0.9975 0.3888 0.2824 0.7900 –0.8063
C50-P1-R*-OR 0.0872 1.3581 0.2284 0.7440 0.5930 –0.2353
C50-P2-R*-UR 0.0099 2.3313 0.0163 2.1367 0.0516 1.6299C50-P2-R*-TR 0.0527 1.6194 0.1204 1.1730 0.3811 0.3027
C50-P2-R*-OR 0.0803 1.4033 0.1944 0.8619 0.5407 –0.1021
C50-P3-R*-UR 0.0014 2.9933 0.0016 2.9517 0.0029 2.7646
C50-P3-R*-TR 0.0143 2.1884 0.0249 1.9612 0.0826 1.3881
C50-P3-R*-OR 0.0279 1.9123 0.0579 1.5731 0.1986 0.8467
C70-P1-R*-UR 0.0918 1.3298 0.2276 0.7468 0.5979 –0.2480
C70-P1-R*-TR 0.3176 0.4744 0.6701 –0.4401 0.9475 –1.6206
C70-P1-R*-OR 0.1732 0.9416 0.4282 0.1809 0.8088 –0.8735
C70-P2-R*-UR 0.0340 1.8249 0.0726 1.4567 0.2464 0.6858
C70-P2-R*-TR 0.1604 0.9929 0.3844 0.2939 0.7869 –0.7957
C70-P2-R*-OR 0.2223 0.7645 0.5147 –0.0369 0.8811 –1.1805
C70-P3-R*-UR 0.0117 2.2662 0.0196 2.0629 0.0622 1.5366
C70-P3-R*-TR 0.0569 1.5818 0.1310 1.1217 0.4059 0.2381
C70-P3-R*-OR 0.0865 1.3626 0.2071 0.8164 0.5638 –0.1606
198 ACI Structural Journal/May 2020
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