DEFLECTION OF FRP REINFORCED CONCRETE BEAMS Raed Al ...

DEFLECTION OF FRP REINFORCED CONCRETE BEAMS 

Raed Al-Sunna 1,2 , Kypros Pilakoutas 2 , Peter Waldron 2 and Tareq Al-Hadeed 1 

1 Building Research Centre, Royal Scientific Society, Amman, Jordan. 

2 Centre for Cement and Concrete, Department of Civil and Structural Engineering, 

University of Sheffield, United Kingdom. 

ABSTRACT 

The current provisions in ACI Committee 440 for calculating deflections of 

reinforced concrete (RC) flexural elements with internal fibre reinforced polymer 

(FRP) reinforcement require the use of a bond factor in the determination of the 

effective moment of inertia (I e ). However, it is shown here that this bond factor is 

variable and depends on the reinforcement ratio, even for the same type of rebar. 

This paper presents the deflection results of a testing program of FRP RC beams 

and examines the ACI provisions for deflection in further detail. The form of the 

equation for I e is not fundamentally sound and cannot be used to predict deflections 

of FRP RC in all cases. A more appropriate form is proposed involving the 

reinforcement ratio and elastic properties of the rebar. 

Keywords: beams, concrete, deflection, fibre reinforced polymer, reinforced 

concrete. 

1. INTRODUCTION 

FRP reinforcement has been developed to replace steel in special applications, 

particularly in corrosion-prone RC structures. Compared to steel, FRP 

reinforcement generally possesses a low modulus of elasticity, which leads to 

higher reinforcement strains, wider cracks and larger deflections. Therefore, the 

serviceability limit state may often govern the design of FRP RC. Furthermore, 

due to numerous combinations of matrix, fibre and surface treatments, the bond and 

tension stiffening characteristics of FRP rebars are not readily identified. 

The current provisions in ACI Committee 440 (2003) for calculating short-term 

deflections of FRP RC are based on the effective moment of inertia (I e ), similar to 

ACI Committee 318 (2002) for steel RC, but with modifications to allow for the 

different modulus of elasticity and bond characteristics of the FRP reinforcement. 

The equation for I e , commonly referred to as Branson’s equation, was modified as 

proposed by Gao et al (1998) (equations 1 and 2).

⎡M 

⎤ 

cr 

I 

e 

= I 

cr 

+ ( β 

d 

I 

g 

− I 

cr 

) ⎢ ⎥ ≤ I 

g 

(1) 

⎢ M a 

⎣ ⎥⎦ 

⎡ E 

f ⎤ 

β 

d 

= α 

b ⎢ + 1⎥ 

(2) 

⎣ Es 

⎦ 

in which I g and I cr are the gross and linear cracked moment of inertia, M cr and M a 

are the cracking and applied moment, E f and E s are the FRP and steel modulus of 

elasticity and α b is a bond dependent coefficient, which equals 0.5 for steel rebars. 

A value of 0.5 has also been recommended for all FRP rebar types, until more 

research data becomes available. 

The work presented in this paper is part of an extensive research program on the 

flexural behaviour of FRP RC elements, which was carried out at the Royal 

Scientific Society (Jordan) in collaboration with the University of Sheffield (UK). 

The experimental deflection results of twelve FRP RC beams are used to examine 

the accuracy of those predicted by the approach of ACI committee 440. In 

particular, the factors β d (or α b ) in the equation for I e are evaluated. The main 

variables considered in the tests are the type of FRP rebar and reinforcement ratio. 

3 

2. EXPERIMENTAL PROGRAM 

2.1 Materials 

The type of FRP rebar is one of the variables in this investigation. Two rebar types 

were used: glass (GFRP) and carbon (CFRP) (Fig. 1). The GFRP rebars were 

Aslan (100). Their surface treatment may be classified as helically wrapped, sand 

coated. During the experimental program, the CFRP rebars were still under 

development and are currently commercially available as Aslan (200). Their 

surface treatment may be classified as helically wrapped, indented. Table 1 shows 

the main tensile properties of the FRP rebars, as provided by the manufacturer. 

The tensile properties of the steel rebars were obtained by testing representative 

samples, and are shown in Table 2. 

The concrete was produced locally with 25 mm maximum aggregate size, 0.48 free 

water to cement ratio and 380 kg/m 3 cement content. The fresh concrete slump was 

about 75 mm and the 28-day cube compressive strength was around 35 MPa. 

CFRP rebar 

GFRP rebar 

Fig. 1: CFRP and GFRP rebars

Table 1: Tensile properties of FRP rebars 

Rebar 

type 

Nominal diameter, 

(mm) 

Modulus of elasticity, 

(MPa) 

Guaranteed tensile 

strength, (MPa) 

9.53 40800 760 

GFRP 12.70 40800 690 

19.05 40800 620 

6.35 119750 1250 

CFRP 9.53 122750 1000 

12.70 111750 900 

Table 2: Tensile properties of steel rebars 

Rebar 

type 

Nominal 

diameter, 

(mm) 

Modulus of 

elasticity, 

(MPa) 

Yield 

strength, 

(MPa) 

Steel 12 200000 590 675 

Ultimate 

Strength, 

(MPa) 

2.2 Test Specimens 

The test specimens consisted of three series of GFRP and three series of CFRP RC 

beams. One series of steel RC beams was tested for comparison purposes. To 

ensure repeatability, each series comprised two identical beams. The beam series 

were designated as BG#, BC# or BS#. B stands for beam; # is the series number; 

while G, C and S refer to GFRP, CFRP and steel rebars, respectively. The two 

identical beams, within each series, were identified by adding a or b to the series 

name. The beams were 150 mm wide, 250 mm high, 2550 mm long, with the 

distance between the end-supports being 2300 mm (Fig. 1). All beams were tested 

under four-point loading. The shear span was 767 mm (one third of the beam 

span). The shear span was reinforced with steel stirrups to avoid shear failure, 

while the midspan was free of stirrups. Nominal 6mm GFRP, CFRP or steel rebars 

were used at the top within the shear span to hold the stirrups. The clear concrete 

cover to the rebars was 25 mm in all cases. 

The two beams in each series, along with eight control cubes and eight control 

cylinders were constructed from the same batch of in-situ concrete. The beams and 

control specimens were cured under similar conditions. The cubes were used to 

determine the compressive strength, while the cylinders were used to determine the 

split cylinder tensile strength of the concrete on the day of testing the beam. 

In addition to the type of rebar, the other variable in this study is the reinforcement 

ratio. The GFRP and CFRP beams were designed to have equal flexural capacity, 

equal area of rebars or equal stiffness of rebars, to the steel control beams. 

Thereby, for each type of rebar, a wide range of reinforcement ratios was required 

and several rebar diameters were used. The resulting sections were underreinforced, 

close to balanced or over-reinforced, with failure occurring by rupture 

of bars or crushing of concrete. The geometric and reinforcement details of the test 

beams are shown in Table 3 and Fig. 2.

Table 3: Geometric and reinforcement details of the test beams 

Rebar 

type 

Series 

designation 

Beam 

designation 

Rebar 

details 

Reinforcement 

ratio 

Relation to control 

steel beam 

BG1 

BG1a 

Equal flexural 

2∅9.53 0.00432 

BG1b 

capacity 

GFRP BG2 

BG2a 

Equal area of 

2∅12.7 0.00772 

BG2b 

rebars 

BG3 

BG3a 

Equal stiffness of 

4∅19.05 0.0391 

BG3b 

rebars 

BC1 

BC1a 

Equal flexural 

3∅6.35 0.00286 

BC1b 

capacity 

CFRP BC2 

BC2a 

Equal area of 

3∅9.53 0.00648 

BC2b 

rebars 

BC3 

BC3a 

Equal stiffness of 

3∅12.7 0.0116 

BC3b 

rebars 

Steel BS 

BSa 

BSb 

2∅12 0.00688 - 

2Ø6mm 

stirrups, 

Ø8mm/75mm 

250 

125 

767 

766 767 

125 

2300 

150 

2Ø6mm 

250 

stirrups, 

Ø8mm/75mm 

25mm 

cover to bars 

Fig. 2: Geometric and reinforcement details of the test beams 

2.3 Test Procedure 

All beams were tested under four-point loading (Fig. 3). The load was applied 

centrally by a 600 kN hydraulic jack, and a distribution beam was used to distribute 

the load to the two third-span points. Two dial gauges were used to measure 

settlements at the end supports. Five linear variable displacement transducers 

(LVDT) were used to measure deflections at the third- points of one shear span, at 

the two loading points and at midspan. One LVDT was used to measure the 

average strain and the crack width at the level of the reinforcement. In addition, 

one strain gauge was used to measure the top fibre concrete strain at midspan. 

Though not discussed in this paper, one of the aims of this study was to investigate 

the effect of tension stiffening on deflection and cracking. Therefore, a crack 

inducer, (20 x 0.6) mm, was used to ensure that a crack would occur at midspan. 

Then, a total of ten closely-spaced strain gauges were used to measure rebar strains

on either side. Thus, it was possible to obtain strain profiles between one forced 

crack at midspan and two contiguous natural cracks. Four additional strain gauges 

were used to measure rebar strain at one loading point, at the third-points of one 

shear span, and at one support. 

The testing was carried out using load control. The load was increased at a rate of 

1 kN/min and was paused at about 5 kN intervals to mark and measure the cracks 

and to take notes. Two load cycles were performed. In the first cycle, the load was 

increased to a service load level, which corresponded to a stress of about 45% the 

concrete compressive strength in the top concrete fibre at midspan. In the second 

cycle, the load was increased until failure occurred, either by rupture of bars or by 

crushing of concrete. All data (force, strains and deflections) were collected by a 

data acquisition system, and downloaded to a PC every one second. 

Hydraulic Jack 

LVDT 

Distribution Beam 

LVDT Locations 

Strain Gauge on Rebar 

Strain Gauge on concrete 

Midspan forced crack 

Natural crack 

Natural crack 

90 90 

Strain Gauge Locations 

Fig. 3: Typical beam test setup. 

3. TEST RESULTS AND DISCUSSION 

Fig. 4 and 5 show the experimental load-deflection response at midspan for all 

beams. From these figures, it is clear that the results of the two replicate beams 

within each series are rather identical. A small difference in the initial deflections 

is noticed in the BS series, but this is because beam BSb was accidentally preloaded 

and pre-cracked before testing. Therefore, it can be confirmed that the 

materials used, the production of the beams and the test procedure were all well 

controlled. It is also clear that the test results reasonably satisfy the relation of the 

FRP to the control steel reinforced beams, as initially designed, except for series

BG1 where the GFRP rebars ruptured prematurely. This indicates, that the 

manufacturer tensile material properties of the FRP rebars are also reasonable, 

particularly the supplied modulus of elasticity. Nevertheless, for a more refined 

analysis of the results, tensile tests of representative FRP rebar samples are 

currently underway. 

140 

Beams BG, BS 

120 

100 

Force [kN] 

80 

60 

40 

20 

BG1a 

BG1b 

BG2a 

BG2b 

BG3a 

BG3b 

BSa 

BSb 

0 

0 5 10 15 20 25 30 35 40 45 50 

Midspan Deflection [mm] 

Fig. 4: Experimental force vs. midspan deflection – series BG and BS 

140 

Beams BC, BS 

120 

100 

Force [kN] 

80 

60 

40 

20 

BC1a 

BC1b 

BC2a 

BC2b 

BC3a 

BC3b 

BSa 

BSb 

0 

0 5 10 15 20 25 30 35 40 


Fig. 5: Experimental force vs. midspan deflection – series BC and BS 

The experimental midspan deflections for every beam series are compared to 

predicted deflections in Figs. 6 to 8.

50 

45 

40 

35 

Beam BG1a 

β d =0.09 

(α b =0.07) 

Force [kN] 

30 

25 

20 

15 

10 

5 

0 

BG1a 

ACI 318 (2002) 

ACI 440 (2003) 

linear cracked section 

proposed equations 4 

0 5 10 15 20 25 30 35 40 45 


90 

80 

70 

60 

Beam BG2a 

β d =0.15, ( α b = 0.12) 

Force [kN] 

50 

40 

30 

20 

10 

0 

BG2a 

ACI 318 (2002) 

ACI 440 (2003) 


proposed equation 4 

0 5 10 15 20 25 30 35 40 45 50 


120 

100 

Beam BG3a 

β d =0.5, 

(α b =0.41) 

80 

Force [kN] 

60 

40 

20 

BG3a 

ACI 318 (2002) 

ACI 440 (2003) 



0 

0 5 10 15 20 25 


Fig. 6: Experimental and predicted deflections – GFRP Beams.

80 

70 

60 

Beam BC1a 

β d = 0.19 (α b =0.12) 

Force [kN] 

50 

40 

30 

20 

10 

BC1a 

ACI 318 (2002) 

ACI 440 (2003) 

Linear cracked section 


0 

0 5 10 15 20 25 30 35 40 45 


120 

100 

Beam BC2a 

β d = 0.25 (α b =0.15) 

80 

Force [kN] 

60 

40 

20 


BC2a 

ACI 318 (2002) 

ACI 440 (2003) 

Linear cracked section 


0 

0 5 10 15 20 25 30 35 


140 

120 

Beam BC3a 

β d = 0.5 (α b =0.32) 

100 

Force [kN] 

80 

60 

40 

20 


BC3a 

ACI 318 (2002) 

ACI 440 (2003)] 



0 

0 5 10 15 20 25 


Fig. 7: Experimental and predicted deflections – CFRP Beams.

80 

70 

Beam BSa 

60 

Force [kN] 

50 

40 

30 

BSa 

ACI 318 (2002) and 

proposed equations (4, 5) 


20 

10 

0 

0 5 10 15 20 25 30 


Fig. 8: Experimental and predicted deflections – Steel Beams. 

As may be expected, the original Branson’s equation for I e , results in rather 

accurate deflection predictions for the control steel beams. On the contrary, it 

underestimates deflections of all FRP beams. However, as the reinforcement ratio 

increases, the predicted deflections are better. In fact, for beam series BG3 and 

BC3, which have the highest reinforcement ratios, the deflections are acceptable, 

though still underestimated. This confirms the conclusion of Toutanji and Saafi 

(1999) who proposed using Branson’s equation when [ ρ ( E / E ≥ 0.3] 

, where ρ f 

is the percentage reinforcement ratio. Similarly, Wei et al (1997) recommended no 

modification to Branson’s equation, but without imposing any limits on the 

reinforcement ratio. It is clear that there is a need to modify Branson’s equation, 

particularly for low reinforcement ratios, and it seems reasonable that the 

reinforcement ratio should appear in the equation for I e . 

The coefficient β d in the ACI Committee 440 equation for I e has been determined 

for all beam series, so that the predicted deflections compare best to the 

experiments. The resulting β d (or α b ) vary widely, and increase with reinforcement 

ratio, as shown also in Table 4. Had the proposed ACI approach been adequate, a 

unique value would have resulted for the coefficient β d (or α b ) for each FRP rebar 

type, irrespective of the reinforcement ratio or the rebar diameter. Besides, for the 

CFRP beams, the deflections predicted are not as good as for the GFRP beams. 

Moreover, the currently proposed 0.5 value for α b underestimates deflections, 

particularly for low reinforcement ratios. Therefore, the current ACI Committee 

440 approach for evaluating I e does not seem to be applicable in all cases. This 

implies that the bond characteristics may need to be accounted for in a different 

way. For that purpose, analytical work is underway to investigate the relation 

between deflection behaviour and the rebar strain profiles between cracks due to 

tension stiffening. Yet again, the inclusion of the reinforcement ratio in the 

equation for I e may be necessary. 

f 

f 

s

Table 4: Calibrated coefficients β d or α b proposed in ACI Committee 440 

Rebar Type GFRP CFRP 

Beam Series BG1 BG2 BG3 BC1 BC2 BC3 

Coefficient β d 0.09 0.15 0.5 0.19 0.25 0.5 

Coefficient α b 0.07 0.12 0.41 0.12 0.15 0.32 

ACI Committee 440 equation for I e actually provides a smooth transition between 

the gross and linear cracked moments of inertia, I g and, eventually converging with 

I cr . However, the figures show that the experimental deflections exceed the I cr limit 

on deflections. This indicates that the composite action between the concrete and 

FRP rebars may not be as perfect as assumed. This also means that the form of the 

equation for I e should provide a transition between I g and a certain fraction of I cr . 

Such an equation was proposed by Benmokrane et al (1996), but with very specific 

coefficients for their particular tests (equation 3). 

I 

g ⎡M 

cr 

⎤ 

I 

e 

= α I 

c 

+ ( −αI 

c 

) 

r 

r ⎢ ⎥ (3) 

β ⎣ M 

a ⎦ 

in which α and β were evaluated as equal to 0.84 and 7, respectively. 

3 

Equation 3 has better fundamentals than the current ACI equation. The factor α 

reflects the reduced composite action between the concrete and FRP rebars, as 

discussed above. However, the factor β has no physical significance since there is 

no justification for reducing I g . Nevertheless, its introduction in the equation is 

necessary because it enables a faster transition from I g to I cr , since the degradation 

in stiffness due to the 3 rd power component is very slow. However, a more general 

equation is needed that takes into account the type and amount of reinforcement. 

Considering the experimental results of all beam series, the factor β has been 

modified and expressed as an exponential function of the reinforcement ratio (ρ) 

and the FRP modulus of elasticity (E f ). Therefore, it is proposed to use a more 

general form of equation 3 as follows. 

I 

e 

3 

⎡M 

cr 

⎤ 

= α I 

c 

+ ( βI 

g 

−αI 

c 

) 

r 

r ⎢ ⎥ (4) 

⎣ M 

a ⎦ 

1.2 

330( ρE / ) 

0.1 

f E 

β = e 

s 

(5) 

Fig. 6, 7 and 8 show that the deflections predicted by equation 4 are generally in 

very good agreement with the test results, even in the case of steel reinforcement. 

α−values of 0.9, 0.85 and 1.0 have been used for GFRP, CFRP and steel RC 

beams, respectively. The factor α depends on the rebar bond characteristics and 

needs further investigation. 

Furthermore, the generality of equation 5 has been tested by applying it to the two 

FRP beam series of Benmokrane et al (1996), which was a totally different testing 

program from the one used to derive the equation, with different FRP rebars, beam 

dimensions and material properties. The resulting β values are 0.11 and 0.13, 

which agree well with the 1/7 value proposed in equation 3.

4. CONCLUSIONS 

From the experimental and analytical work presented in this paper, the following 

may be concluded. 

1. The original Branson’s equation for I e works well for steel RC, but 

overestimates the actual I e and underestimates deflections of FRP RC. 

2. The form of the equation for I e , proposed by ACI Committee 440, is not 

fundamentally sound and cannot be used to predict deflections of FRP RC in all 

cases. 

3. A more general form of the equation for I e is proposed by modifying the 

equation adopted by Benmokrane et al (1996). The reinforcement ratio is 

introduced as a new variable and I cr is reduced by a factor α, which is a bond 

dependent coefficient that reflects the degree of concrete-reinforcement 

composite action. 

4. More research work is required to investigate the accuracy of the proposed 

equation. 

ACKNOWLEDGEMENTS 

The authors wish to acknowledge the financial support provided by The Higher 

Council for Science and Technology (Jordan), the Royal Scientific Society 

(Jordan), the Centre for Cement and Concrete,University of Sheffield (UK) and the 

Karim Rida Said Foundation (UK). 

REFERENCES 

ACI Committee 318 (2002): “Building Code Requirements for Structural 

Concrete (ACI 318-02) and Commentary (ACI 318R-02),” American Concrete 

Institute, Farmington Hills, Mich, 367 pp. 

ACI Committee 440 (2003): “Guide for the Design and Construction of Concrete 

Reinforced with FRP Bars (ACI 440.1R-01),” American Concrete Institute, 

Farmington Hills, Mich, 42 pp. 

Benmokrane, B.; Chaallal, O.; Masmoudi, R. (1996): “Flexural Response of 

Concrete Beams Reinforced with FRP Reinforcing Bars,” ACI Structural Journal, 

Vol. 93, No. 1, Jan.-Feb., pp. 46-55. 

Gao, D.; Benmokrane, B.; Masmoudi, R. (1998): “A Calculating Method of 

Flexural Properties of FRP-Reinforced Concrete Beam: Part 1: Crack Width and 

Deflection,” Technical Report, Department of Civil Engineering, University of 

Sherbrooke, Sherbrooke, Quebec, Canada, 24 pp.

Toutanji, H. A.; Saafi M. (1999): “Deflection and Crack Width Predictions of 

Concrete Beams Reinforced with Fiber Reinforced Polymer Bars,” Proceedings, 

Fourth International Symposium on Fiber Reinforced Polymer Reinforcement for 

Reinforced Concrete Structures, Baltimore-USA, SP 188, Charles W. Dolan, Sami 

H. Rizkalla, and Antonio Nanni, editors, American Concrete Institute, Farmington 

Hills, Mich., 1999, pp.1023-1034. 

Wei Z., Pilakoutas K., and Waldron P. (1997): “FRP Reinforced Concrete: 

Calculations for Deflections,” Proceedings of the Third International Symposium 

on Non-Metallic (FRP) Reinforcement for Concrete Structures, Sapporo-Japan, 

Oct. 1997, Vol. 2, pp. 511-518.

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