1. INTRODUCTIONFiber-reinforced polymer (FRP) bars have emerged asan alternative reinforcement for concrete structures.On the one hand, this kind of reinforcement exhibitssuch properties as corrosion resistance, electromag-netic neutrality and high cuttability [1]. As a result itcan have many applications, especially in structuresused in marine environments, in chemical plants,when electromagnetic neutrality is needed, or in tem-porary structures.

On the other hand, FRP bars have low modulus ofelasticity and high tensile strength [2]. Due to theirmechanical properties, deflections and cracking ofFRP RC flexural members are larger than of tradi-tional RC members. As a result, the design of FRP RCbeams is often governed by the serviceability limitstates [3].This paper presents the results of a numerical study inwhich three GFRP RC beams were tested in four-point bending. The aim of this simulation was toexamine the failure mechanism and deflections of sim-

NUMERICAL STUDY OF THE FLEXURAL BEHAVIOUR OF GFRP RC BEAMS

Dawid PAWŁOWSKI a*, Maciej SZUMIGAŁA b

a PhD Student; Faculty of Civil and Environmental Engineering, Poznań University of Technology, Piotrowo 5,60-965 PoznańE-mail address: dawid.p.pawlowski@doctorate.put.poznan.pl

b Associate Prof.; Faculty of Civil and Environmental Engineering, Poznan University of Technology, Piotrowo 5,60-965 Poznań, Poland

Received: 15.10.2014; Revised: 15.10.2014; Accepted: 28.01.2015

A b s t r a c tFiber-reinforced polymer (FRP) bars have been commercially available in civil engineering in the last twenty years. Due totheir mechanical and physical properties, the behaviour of FRP reinforced concrete (RC) members is significantly differentto that of traditional steel RC. This paper presents the results and discussion of a numerical study of the flexural behav-iour of simply supported glass fiber-reinforced polymer (GFRP) RC beams under short-term static loads. The numericalanalysis was performed using the Finite Element Method (FEM). All calculations were carried out in the environment ofthe Abaqus. The main objective of this paper was to investigate the flexural behaviour of GFRP RC members depending onthe reinforcement ratio. The results of the numerical analysis were examined and compared with code formulations andwith the results of experiments.

S t r e s z c z e n i eZbrojenie kompozytowe (FRP) jest wykorzystywane w budownictwie od ponad dwudziestu lat. Ze względu na jego mecha-niczne i fizyczne właściwości, zachowanie elementów zbrojonych prętami niemetalicznymi jest odmienne od zachowaniaklasycznych elementów żelbetowych. W pracy przedstawiono wyniki symulacji komputerowej zachowania swobodnie pod-partych belek zbrojonych prętami z włókna szklanego (GFRP) poddanych działaniu krótkotrwałego obciążenia statycznego.Badanie numeryczne przeprowadzono przy zastosowaniu oprogramowania Abaqus wykorzystującego Metodę ElementówSkończonych (MES).Głównym celem pracy było zbadanie wpływu stopnia zbrojenia na nośność i sztywność elementów zbrojonych prętami GFRP.Wyniki symulacji komputerowej porównano z wynikami normowymi oraz z rezultatami doświadczeń laboratoryjnych.

K e y w o r d s : Composite materials; GFRP reinforcement; GFRP RC beams; Flexural behaviour; FEM analysis.

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ply supported GFRP RC beams depending on thereinforcement ratio. The dimensions of the speci-mens and properties of concrete and GFRP barswere assumed on the basis of an experimental study[4]. The results of the numerical simulation werecompared with code formulations [2, 5] and with theresults of experimental tests [4].

2. NUMERICAL SIMULATION2.1. Test specimensThe numerical model of beams was created on thebasis of the beam which is shown in Fig. 1.The numerical study consisted in investigating theflexural behaviour of three beams with varying GFRPreinforcement (Tab. 1). All beams had a cross-sectionof 0.14 × 0.19 m2, a total length of 2.05 m and a spanof 1.80 m. The shear reinforcement consisted of8 mm round steel stirrups placed at intervals of70 mm. In the pure bending zone no stirrups wereprovided. Two 6 mm steel bars were used as top rein-forcement to hold the stirrups.

2.2. Materials propertiesConcreteAll beams had a target concrete compressive strengthof 30 MPa. The properties of concrete were evaluatedfrom cylindrical specimens. They are presented inTab. 2.

GFRPGFRP ribbed bars were used as a flexural reinforce-ment. The experimentally determined mechanicalproperties of reinforcement are shown in Tab 3.

2.3. Model of the beamThe finite element (FE) model of considered beamswas implemented in ABAQUS environment [6]. Theanalysis was performed on 2D model and the follow-ing assumptions were adopted:•Concrete damage plasticity (CDP) model of con-

crete [7] was assumed,•Tension stiffening effect was taken into account,•GFRP reinforcement was assumed as a linear elas-

tic isotropic material,•Steel reinforcement was assumed as a linear elastic-

plastic material with isotropic hardening,•The reinforcement was modelled as 2-node truss

elements embedded in 4-node elements of planestress (Fig. 2).

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Figure 1.Geometry and reinforcement of specimen [mm]

Table 1.Characteristics of specimens

Table 2.Mechanical properties of concrete

Beamdesignation Main bar

Reinforcementratio ρf [%]

…-2#12 2#12 0.99

…-2#16 2#16 1.77

…-3#16 3#16 2.66

Table 3.Mechanical properties of GFRP reinforcement

Modulusof elasticity

Ec [GPa]

Compressivestrengthfc [MPa]

Tensilestrengthfct [MPa]

25.8 32.1 2.8

Diameter [mm]Tensile

strengthffu [MPa]

Modulusof elasticity

Ef [GPa]

12 1321 63.416 1015 64.6

N U M E R I C A L S T U D Y O F T H E F L E X U R A L B E H A V I O U R O F G F R P R C B E A M S

The model of beams consisted of two different typesof finite element:•T2D2 – 2-node 2D truss elements,•CPS4R – 4-node plane stress elements with reduced

integration.The concrete was modelled as concrete damage plas-ticity material, which is based on the brittle-plasticdegradation model [8]. For concrete under uniaxialcompression, the stress-strain curve shown in Fig. 3was adopted. It is composed of the Eurocode 2 [5]parabolic ascending branch and a descending branchextended up to the ultimate strain εcu [9].

The tension stiffening effect was taken into accountby applying a modified Wang&Hsu [10] formula(Eq. 1) to describe the behaviour of concrete undertension (Fig. 4):

where Ec is the modulus of elasticity of concrete, εt isthe tensile strain of concrete, εcr is the tensile strain atconcrete cracking, fctm is the average tensile strengthof concrete and n is the rate of weakening.

3. NUMERICAL SIMULATION RESULTS,DATA ANALYSIS AND DISCUSSIONS3.1. Failure mode and ultimate loadThe results of the numerical simulation and theexperimental studies [4] shown that all the beamsfailed in a brittle mode due to concrete crushing. Theresults of FE analysis are presented in Fig. 5. It wasassumed that the value of the maximum compressiveconcrete strains is about 0.0042.

According to ACI 440.1R-06 [2], the failure mode isgoverned by concrete crushing when the reinforce-ment ratio ρf is greater than the balanced reinforce-ment ratio ρfb:

where Af is the area of GFRP reinforcement, b is thewidth of the section and d is the effective depth. InEq. (3), b1 is the ratio of depth of equivalent rectan-

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Figure 2.Scheme of the 2D FE model

>=

≤=

crt

n

t

crctmt

crttct

f

E

εεεεσ

εεεσ

,

,

(1) bdAf

f =ρ (2)

fucuf

cuf

fu

cfb fE

Eff

+=

εε

βρ 185.0 (3)

Figure 3.Response of concrete to uniaxial loading in compression

Figure 4.Response of concrete to uniaxial loading in tension

Figure 5.Ultimate loads (FEM results)

ec

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gular stress block to depth of the neutral axis, fc is theconcrete compressive strength, ffu is the design tensilestrength of GFRP reinforcement, Ef is the modulusof elasticity of FRP, and εcu is the maximum concretestrain (0.003 for ACI 440.1R-06 [2]). The actual andbalanced reinforcement ratios are compared inTab. 4. All the beams had higher reinforcement ratiosthan ρfb, hence according to code [2], failure by con-crete crushing was expected in all of them. This modeof failure was confirmed by the numerical analysisand the results of experiments [4].

Experimental (EXP), numerical (FEM) and theoret-ical (ACI for ACI 440.1R-06 and EC2 forEurocode 2) ultimate loads are compared in Tab. 5.There is good agreement between the experimentaland numerical results, whereas ultimate loads calcu-lated according to the codes [2, 5] are underestimat-ed. Their values are lower than the values of loadsobtained in the experimental tests [4] by about 26-31% and 10-15% for ACI and EC2, respectively.These differences can be caused by the value of themaximum concrete compressive strain εcu which isassumed in these codes – 0.0030 for ACI and 0.0035for EC2. The results of experiments [11] show thatthe actual ultimate concrete strain εcu is about0.0042-0.0047.On the basis of the results shown in Tab. 5, it can besaid that the reinforcement ratio has an influence onthe flexural strength of the beams. The increase inthe reinforcement ratio results in the increase in theultimate loads of the beams.

3.2. DeflectionsFigs. 6-8 show the numerical, theoretical and experi-mental load-deflection curves for all beams. Theresults of the numerical analysis correspond well withthe results obtained in the experiments.

Comparing theoretical predictions obtained based onACI (Eq. 4) and EC2 (Eq. 5) with the results ofexperimental tests, it can be observed that up to theservice load (deflection d<L/250) there is goodagreement between theoretical and actual values ofdeflections. For higher loads these codes underesti-mate deflections. These differences can be connectedwith the fact that these theoretical approaches use asimplified linear stress-strain constitutive relation-ship for concrete.

Eq. 4 shows the expression for an effective moment ofinertia Ie of the concrete section according to ACI,where Ig is the gross moment of inertia of concrete sec-tion, Icr is the moment of inertia of the cracked section,Mcr is the cracking moment, Ma is the maximummoment in the member and bd is the reduction coeffi-cient related to the reduced tension stiffening effect.

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gcra

crgd

a

cre II

MMI

MMI ≤−+=

33

1β (4)

Table 4.Balanced and actual reinforcement ratio

Beamdesignation

Actual reinforcementratio ρf [%]

Balanced reinforcementratio ρbf [%]

2#12 0.99 0.222#16 1.77 0.363#16 2.66 0.36

Table 5.Ultimate loads

Beam designationReinforcement

ratio ρf [%]Pu EXP

[kN]Pu FEM

[kN]Pu ACI

[kN]Pu EC2

[kN]…2#12 0.99 84 79 64 73…2#16 1.77 100 99 78 89…3#16 2.66 112 112 89 102

Figure 6.Experimental, numerical and theoretical load – midspandeflection curves (2#12 beam)

N U M E R I C A L S T U D Y O F T H E F L E X U R A L B E H A V I O U R O F G F R P R C B E A M S

Eq. 5 shows the formulation for deflections d accord-ing to Eurcode 2, where dI is uncracked-state deflec-tion, dII is fully cracked-state deflection and z is thecoefficient related to the tension stiffening effect.

As can be observed in Figs. 6-8, the reinforcementratio has a significant effect on the stiffness of the RCbeams. As expected, higher deflections are obtainedfor lower reinforcement ratios and vice versa.

4. CONCLUSIONSThis paper presents the results of numerical, theoret-ical and experimental study of the flexural behaviourof GFRP RC beams. Based on these results, the fol-lowing conclusions may be drawn:• The reinforcement ratio has a significant effect on

the flexural behaviour of the GFRP RC beams. Theincrease in the reinforcement ratio results in theincrease in the ultimate loads and in the stiffness ofthe beams.

• The failure mode is governed by concrete crushingwhen the reinforcement ratio ρf is greater than thebalanced reinforcement ratio ρfb (according to ACI440.1R-06). All beams behave almost linearly up tothe moment of failure, which takes place at rela-tively large deflections.

• At the service load level, the deflections calculatedaccording to ACI 440.1R-06 and Eurocode 2 are inclose agreement with the results of the experi-ments. For higher loads these codes underestimatedeflections.

• The ultimate loads calculated according to ACI440.1R-06 and Eurocode 2 are underestimated.This underestimation can be caused by the value ofthe ultimate concrete strain εcu which is assumed inthese codes. It is lower than the value of εcu obtainsin experiments.

• The nonlinear model of concrete, which was adopt-ed in the study, reflects relatively well the behav-iour of the actual concrete.

ACKNOWLEDGMENTSThe paper was presented at the 8th InternationalConference AMCM 2014 – Analytical Models andNew Concepts in Concrete and Masonry Structures(AMCM’2014), Wrocław, June 2014.

REFERENCES[1] Federation International du Beton, FRP reinforce-

ment in RC structures, Fib Task, Group 9.3, 2007,pp.3-30

[2] ACI 440.1R-06, Guide for the design and construc-tion of structural concrete reinforced with FRP bars,ACI Committee 440, 2006

[3] Nanni A.; North American design guidelines for con-crete reinforcement and strengthening using FRP:Principals, applications and unresolved issues,Construction and Building Materials, Vol.17, No.6-7,2003; p.439-446

[4] Barris C., Torres L., Comas J., Mias C.; Cracking anddeflections in GFRP RC beams: an experimentalstudy, Composites: Part B, Vol.55, 2013; p.580-590

[5] EN 1992-1-1:2003, Eurocode 2: Design of concretestructures – Prat 1-1: General rules and rules forbuildings, Technical Committee CEN/TC250, 2004

[6] ABAQUS, Abaqus analysis user’s manual, Version6.10, Dassault Systemes, 2010

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Figure 8.Experimental, numerical and theoretical load – midspandeflection curves (3#16 beam)

III δζζδδ )1( −+= (5)

Figure 7.Experimental, numerical and theoretical load – midspandeflection curves (2#16 beam)

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[7] Lubliner J., Oliver J., Oller S., Onate E.; A plastic-dam-age model for concrete, International Journal ofSolids Mechanics, Vol.25, No.3, 1989; p.299-326

[8] Lee J., Fenves G.L.; Plastic-damage model forcyclic loading of concrete structures, Journal ofEngineering Mechanics, Vol.124, No.8, 1998;p.892-900

[9] Park R., Paulay T.; Reinforced concrete structures,John Willey & Sons, 1975

[10] Wang T., Hsu T.T.C.; Nonlinear Finite ElementAnalysis of concrete structures using new constitu-tive models, Computers and Structures, Vol.79,No.32, 2001, p.2781-2791

[11] Barris C., Torres L., Tauron A., Baena M., Catalan A.;An experimental study of the flexural behaviour ofGFRP RC beams and comparison with prediction-models, Composites Structures, Vol.91, 2009;p.586-295

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