analytical and numerical study of frp retro tted rc beams...

Transaction A: Civil EngineeringVol. 16, No. 5, pp. 415{428c Sharif University of Technology, October 2009

Analytical and Numerical Studyof FRP Retro�tted RC Beams

Under Low Velocity Impact

M.Z. Kabir1;� and E. Shafei1

Abstract. An analytical along with numerical analysis has been carried out to investigate the behaviorof concrete simply supported beams strengthened with Fiber-Reinforced Polymer (FRP) unidirectionallaminates under impact loading. Concrete beams are reinforced with a minimum ratio of exural andshear steel rebars and �nally retro�tted with epoxy-bonded high strength carbon FRP laminates in their exure surfaces. The impact force was applied with a solid steel cylinder drop weight. Analytical resultsshowed that composite laminates externally bonded to reinforced concrete beam substrate can signi�cantlyenhance the performance of theses structural members to resist impact loadings. Also, based on theobtained results, in retro�tted beams the crack propagation happens in a desirable mode; an increasingof yielded rebar zones and residual beam sti�ness. Retro�tted beams were sti�er than unretro�tted onesin their �rst impact response. The residual sti�ness of impacted concrete beams depends on their initialsti�ness and the impact energy level. The analytical method uses an idealized elastic spring-mass modeland exural wave propagation theory to calculate the dynamic response of assumed beams. Analyticalresponses are adjusted due to an inelastic response, and �nally are used for the simpli�ed designation ofimpact resisting reinforced concrete beams retro�tting laminates.

Keywords: Impact loading; Reinforced concrete beam; Composite laminate; Retro�tting; FEM.

INTRODUCTION

In practical cases, reinforced concrete structures canbe subjected to sudden dynamic impacts, like dropprojectile impacts, blasts and ocean waves. The maincharacteristics of impact load are a high loading rateand a very short period of application that results inhigh material strain rates. Meanwhile, the mechanicalbehavior of structural materials can be changed dueto high strain rates too. So, traditional static analysismethods cannot be used as a solution to this complexcase. This subject has been paid attention to by manyengineers in recent past years.

Projectile induced impacts can be classi�ed as lowvelocity and high velocity according to the projectilemass and velocity. As a general view, a low-mass

1. Department of Civil and Environmental Engineering, Amirk-abir University of Technology, Tehran, P.O. Box 15875-4413,Iran.

*. Corresponding author. E-mail: [email protected]

Received 11 October 2008; received in revised form 12 February2009; accepted 16 March 2009

projectile at high velocity may cause a high velocityimpact, but a high-mass projectile at low velocitycan cause a low velocity impact. This paper investi-gates the retro�tting e�ect of unidirectional compositelaminates in controlling the structural behavior ofreinforced concrete beams subjected to low velocityimpacts.

The advantages of laminated composites like highstrength-to-weight ratio, high tensile rupture strengthand their easy application procedures cause theirfrequent usage in civil and mechanical engineeringprojects. Composite laminates are almost fabricatedwith carbon, glass and aramid �bers bonded withepoxy or other thermo-set resins and have variousply lay-up con�gurations. Carbon-based compositelaminates (CFRP) have a high tensile modulus andstrength and can service well in high temperature �elds.Aramid and glass-based laminates (KFRP and GFRP)have lower strength and tensile strengths but higherfailure strain values. All the above-mentioned laminatetypes can be quite suitable for the impact retro�ttingof concrete structures. Most laminated �ber-reinforced

416 M.Z. Kabir and E. Shafei

composites behave in a linearly elastic regime up totheir ultimate failure rupture limit.

In recent years, much research has been un-dertaken in the FRP-based retro�tting of concretestructures to resist static and earthquake loadings, butrelatively few studies have been done on the retro�ttingof these members to withstand sudden and impactloadings.

RESEARCH SIGNIFICANCE

The lack of strength in traditional reinforced concretemembers against impact loading has been consideredextremely important during last few years. Theabsorbed impact energy, local and global response ofthese structures can severely a�ect their designationprocedure. The main objective of this paper is togive a general view about the behavior of unretro�ttedreinforced concrete beams subjected to low velocityimpact loading and their performance changes whensubjected under retro�tting using composite unidirec-tional laminates. The variations of sti�ness, de ection,absorbed impact energy, crack modes, yielded zonesof steel rebars and probable debonding interfaces aretaken into account as investigated parameters. Basedon the numerical simulation results and mathematicalmanipulations, the total reaction force and maximumde ection can be estimated in an appropriate manner.

EXISTING LITERATURES

Erki and Meier [1] investigated the impact behaviorof four 8-meter reinforced concrete beams retro�ttedwith steel plates and Kevlar sheets (KFRP) in anexperimental case. The impact load was inducedby lifting one end of a simply supported beam anddropping it from given heights. Comparisons are madebetween the dynamic impact behavior of the beamsstrengthened with KFRP laminates and steel plates.The beams externally strengthened with KFRP lami-nates performed well under impact loading, althoughthey could not provide the same energy absorption asthe beams externally strengthened with steel plates.They also concluded that additional anchoring at theends of laminates would improve the impact resistanceof these beams [1].

White, Soudki and Erki [2] also investigatedexperimentally the e�ects of strain rate on the behaviorof reinforced concrete beams strengthened with CFRPlaminates. Nine 3-m reinforced concrete beams, oneun-strengthened and eight strengthened CFRP lami-nates were suddenly loaded. The stroke rates rangedfrom 0.0167 mm/s (slow loading rate) to 36 mm/s (fastloading rate). The rapidly loaded beams showed anincrease of approximately 5% in capacity, sti�ness, and

energy absorption. Ductility and the failure modeswere not directly a�ected by loading rate changes [2].

Tang and Saadatmanesh [3] also tested a seriesof 27 concrete beams to investigate the behavior ofconcrete beams strengthened with FRP laminates un-der impact loading in the laboratory. Two out ofthe 27 beams were not retro�tted and were used ascontrol specimens. They applied the impact force witha steel drop weight [3]. The conclusion was devoted tothe fact that composite laminates bonded to concretebeams could signi�cantly improve the serviceabilityof this type of structure to withstand impacts. Inaddition, bonding laminates increased cracking and exural strength as well as residual sti�ness of thebeams. Furthermore, it reduced the number of cracks,crack widths and the maximum de ection [3].

ANALYTICAL APPROACH

In general, two types of response in a concrete beamunder impact loading can be observed, that is local andglobal deformations. Local deformations are directlycaused by the contact occurred between the projectileand beam surface, but global responses are caused bythe dynamic vibration of the member. Because ofthe high intensity of global response rather than localdeformations, in low velocity impacts, global deforma-tions dictate the majority of principles related to theanalysis and designation of impact resistance reinforcedconcrete beams. So, in analytical formulations, theglobal sti�ness and deformation regime were taken intoaccount. But, it should be mentioned that the globalpoint of view must be correlated according to locale�ects [4].

Impact Force

In order to determine an elastic estimation of theimpact force applied to a simply supported beam, anidealized spring-mass can be used. The analyticalmodel is assumed as a two-freedom-degree system witha global elastic sti�ness as shown in Figure 1. Thedynamic motion equations of the total system can be

Figure 1. Spring-mass idealization of impactphenomenon.

Low Velocity Impact Behavior of FRP Retro�tted RC Beam 417

expressed as the following relations:

m1�y1 + F = 0;

m2�y2 +Kbsy2 +Kmy32 + F = 0: (1)

Here, m1 and m2 are the masses of the projectile andbeam, respectively; y1 and y2 are the displacementsof the projectile and beam; F is the contact forcebetween the projectile and beam; Kbs is the bending-shear sti�ness of the beam; and Km is the membranesti�ness. The initial conditions are expressed at t = 0(before contact):

_y(0) = V; y1(0) = y2(0) = 0:0: (2)

Hence, V is the initial velocity of the impactor justbefore contact occurs. If the axial deformation andthe projectile indentation are assumed insigni�cant,the model may be considerably simpli�ed to a singledegree of freedom system with the following equationof motion:

m1�y +Kbsy = 0: (3)

For further simpli�cation of the equilibrium equation,the e�ective mass of the structure is neglected. As aresult, the structure and projectile will vibrate togetheras soon as the contact takes place.

y(t) =V!

sin(!t); ! =pKbs=m1: (4)

Because the contact force, F , is equal to the force inthe linear elastic spring, Kbs, the contact force historycan be expressed as follows:

F =

(VpKbsm1 sin(!t) !t � �

0:0 !t > �(5)

The above equation is set based on the assumptionthat the beam sti�ness remains constant during theimpact. But, the sti�ness of reinforced concrete beamsdecreases as it subjects under the crack failure. So,the impact force should be corrected with a reductionfactor, like RF , to incorporate the e�ects of reducedsti�ness [4]. The impact force reduction factor willbe estimated using the numerical inelastic dynamicanalysis results.

De ection

The governing equation of motion for a beam underimpact loading can be developed using a exural wavetheory as the following equilibrium formula [5]:

@2

@x2

�EI

@2y@x2

�+ �A

@2y@t2

= q(x; t): (6)

Here, y, �, A and EI are the beam displacement,density, section area and the elastic bending sti�ness,respectively. The initial conditions of the di�erentialequation are as follows:

y(x; 0) = 0; _y(x; 0) = 0:

An explicit solution of the following boundary valuefourth-order PDE problem needs a Laplace transfor-mation to be applied. If the initial dynamic conditionsare merged with the new obtained equation, the trans-formed equation can be expressed as follows [5]:

EI�A

�y(4)(x; s) + s2�y(x; s) =�q(x; s)�A

: (7)

As can be shown, s, �q and �y are the transformed shapeof time, load, and de ection parameters, respectively.Using the orthogonal in�nite series, the expansion-based general solution of the equilibrium equation canbe stated as follows [5]:

�y(x; s) =2�Al

1Xn=1

Yn(x)(a2�4

n + s2)

lZ0

�q(u; s)Yn(u)du;

Yn(x) = sinn�xl; �n =

n�l: (8)

Here, l is the length of the beam and Yn, and �n arethe orthogonal-type de ection and mode shapes of thevibrating beam. Using the inverse Laplace-transformfunction, one can obtain the real response of the beamas a function of its elastic sti�ness, applied load andinitial-boundary conditions as follows:

y(x; t) =2�Al

1Xn=1;3;5

Yn(x)a�2

n24 lZ0

tZ0

q(u; �)Yn(u) sin a�2n(t� �)d�du

35 ;(9a)

!n = a�2n =

�n�l

�2sEI�A

: (9b)

Here, !n is the nth-mode natural vibration frequencyin radian/sec. The presented equation can calculatethe closed-form elastic response of a simply supportedbeam subjected to any general time-dependent loading.Now, consider a case where the external load is animpact type and is applied at the beginning of the timedomain. The impact load has total magnitude of Pand is applied in the x0 location of the beam span.Considering the Dirac delta function, the mentionedload function can be expressed as follows [5]:

q(x; t) = P�(x� x0)�(t): (10)


Here, �(x � x0) and �(t) are the Dirac delta functionsof the impact force in the x0 location of the beam spanand in the t = 0 moment of the response time domain,respectively. By substituting the Laplace-transformedform of the Dirac delta force function into Equations 9and doing some mathematical manipulation and sim-pli�cations, the �nal solution of the de ection equationcan be expressed as below [5]:

y(x; t)=2P�Al

1Xn=1;3;5

(�1)(n�1)=2 sin(n�xl ) sin(!nt)!n

;(11a)

P =Z �

!

0VpKbsm1 sin(!t)dt; ! =

rKbs

m1:(11b)

Because of the inelastic response of the beam andits frequent cracking modes, the system will absorbmore impact force and, then, may de ect less thanits analytical value. So, the subsequent numericalde ection values will be less than the analytical onesand therefore analytical results should be scaled downwith a reduction factor like RD [5]. The maximumde ection reduction factor can be determined using thenumerical inelastic dynamic analysis results and willbe presented after the �nite element analysis of thestructure.

MATERIAL CONSTITUTIVE MODELS

Materials used in the �nite element analysis of rein-forced concrete beams under projectile impact loadinginclude steel reinforcing rebars, concrete, epoxy resinand FRP laminates. Many reliable and applicable con-stitutive material models are available in commercial�nite element analysis source code libraries [6]. So,their input material properties and associated consti-tutive models have to be only introduced brie y. It isassumed that all materials, except unidirectional FRPlaminates, will behave in an elastic, plastic and fracturenonlinear regime. FRP laminates almost behave in alinear elastic mode up to the rupture. The failure modeof stated composite laminates can be determined onthe critical stress ratio of each failure mode for furtherinvestigations.

Concrete

Under multi-axial combinations of loading, the failurestrengths of concrete are di�erent from those observedunder uniaxial conditions. However, the case of amaximum strength envelope under multiaxial stressseems to be independent of the load path [7]. AMohr-Coulomb type compression surface together witha crack detection surface is used to model the yieldand failure surfaces of concrete as shown in Figure 2.

Figure 2. Schematic concrete failure surfaces inmeridian-deviatoric plane.

If the principal stress components of concrete arecompressive, the response of the concrete is modeled byan elastic-plastic theory with a non-associated ow ruleand an isotropic hardening plasticity. In tension, oncecracking is de�ned to occur (the crack detection surfacecriteria is satis�ed), the orientation of the cracks inintegration points are stored (non-rotating smearedcrack theory). Damaged elasticity and fracture energyis used to model the mesh-independent cracking phe-nomenon [7].

Here, p, q and �uc are, respectively, the �rst invari-ant of the volumetric stress sensor, the second invariantof the deviatoric stress tensor, and the maximumuniaxial compressive failure stress. When the crackingof concrete takes place, a smeared model is used torepresent the orthotropic macro-crack behavior. It isknown that the cracked concrete element can still carrysome tensile stress in the direction normal to the crack,which is termed as tension sti�ening [7]. In this study,a simple linear elastic fracture-based damaged modelproposed by Hillerborg & B�azant [8] is used to modelthis tension sti�ening phenomenon. This method canalso result in the mesh independency of the responseand localization modeling. The cracking behavior canbe represented by a stress-displacement relationshipusing fracture energy. It is assumed that the fractureenergy required to form a unit area of the crack surface,Gf , is a material property. This value can be calculatedfrom measuring the tensile stress as a function ofthe crack opening displacement. Typical values ofGf range from 40 N/m for a traditional constructionconcrete (f 0c = 20 MPa) to 120 N/m for a high strengthconcrete (f 0c = 40 MPa) [8]. A schematic view of aconcrete fracture model is shown in Figure 3. Theconcrete material used in the numerical analysis hadcharacteristic constants as stated in Table 1. Thesesvalues were calibrated on the previous experimentaltests and results [3].

Here the dilation angle, c, is the angle of thecompression surface line in a p � q plane with respectto the p axis, and it can be accounted as the lateralcon�nement implementation parameter [7]. The strain


Table 1. Concrete material model input values.

Elastic Modulus Ec 28240 (MPa)

Poisson's Ratio �c 0.19

Density �c 2402.77 (kg/m3)

Compressive Strength f 0c 36.1 (MPa)

Dilation Angle c 36.34 (degree)

Biaxial to Uniaxial Failure Stress Ratio �b0=�c0 1.16

Fracture Energy Gf 104.4 (N/m)

Figure 3. Crack formation based on fracture energy.

rate e�ect is also modeled by magnifying the maximumcompressive strength and elastic modulus, and bydecreasing the failure compressive strain. The semi-analytical modifying relations are presented by B�azant,Mander and Park [9]. The compressive strength ofconcrete becomes more sensitive by increasing thestrain rate, rather than by any change in its elasticmodulus. However, increasing the strain rate wouldresult in a decrease of the compressive strength value.There is no signi�cant change in the ultimate failurestrain value of concrete in the uniaxial behavior. Thestrain rate e�ect in the uniaxial behavior of modeledconcrete is shown in Figure 4.

Steel Reinforcing Rebar

The stress-strain curve of the reinforcing rebars (lon-gitudinal and transverse) is assumed to be elastic, fol-lowed by a multi-linear kinematic hardening plasticitybased on the von-Mises associated ow rule as shownin Figure 5. The steel reinforcements are treated as atwo-node truss elements network embedded discretelyin concrete brick elements, and the full bond e�ectbetween concrete and steel elements are imposed byintegration points and nodal constrains. In orderto properly model the constitutive behavior of thereinforcement, the cross sectional area, spacing andposition of all reinforcements are exactly modeled ina �nite element con�guration of a three-dimensional

Figure 4. Strain rate e�ect on compressive uniaxialbehavior.

Figure 5. Elastic-hardening plastic uniaxial model forsteel reinforcing bar.

reinforced concrete beam model. The steel materialused in the numerical analysis was the ASTM A615Grade 40 type, which had characteristic constantsas stated in Table 2. Values are based on recentexperimental tests and results [3].

Fiber-Reinforced Polymeric Laminate (FRP)

For �ber-reinforced polymers, each lamina can beconsidered as an orthotropic layer in a plane stress


Table 2. Steel material model input values.

Elastic Modulus Es 199947.98 (MPa)

Poisson's Ratio �s 0.29

Density �s 7849.05 (kg/m3)

Yield Strength fy 275.79 (MPa)

Rupture Strength fu 413.68 (MPa)

Hardening Characteristic - kinematic

condition. It is known that unidirectional �brous com-posites behave elastically and linearly up to their failurecriteria. It should also be mentioned that each laminabehaves isotropic in its transverse directions. Thelaminate used in this investigation is a unidirectionalcarbon based FRP laminate (CFRP) with the followingmaterial characteristics. These special type orthotropicmaterials have nine independent elastic constants. Forsimpli�cation of the analysis process, the maximumstress of each failure mode had been used as the failurecriteria of the unidirectional laminate (stress-basedRankin failure criteria). In Table 3, E, G, X, Y and Spresent elastic modulus, shear modulus, �ber-directionstrength, matrix-direction strength and shear strengthof laminate in its local coordinates, respectively. Axis 1is parallel to the �ber direction, axis 2 is the in-planematrix direction and, �nally, axis 3 is the out of planematrix direction (normal).

Epoxy Bonding Resin

Most high-strength bonding resins remain in the elasticlinear zone up to their failure limit. But, in thepost failure region, they soften linearly in a fracturedprocedure and their sti�ness degrades. The sti�nessin any part of the softening process depends on thedissipated fracture energy in the loading path. Thelimit behavior of the bonding resin consists of thefollowing three stages:

1. Damage initiation,2. Damage evolution,3. Sti�ness degradation.

This modeling method is proposed by Camanho andDavila in the National Aeronautic and Space Associ-ation, US (NASA) [10]. The assumed linear damagedelasticity model for bonding resin is quite similar tothe fracture of concrete in tension. Resins used formechanical and civil engineering purposes are mostlyun-reinforced tough-epoxy and have an isotropic ma-terial con�guration. The mechanical characteristics ofthe resin used in this investigation are stated in Table 4.

Here, E, G1 and G2 are the elastic modulusnormal to the bond interface and the shear modulusin two transverse directions in the bonding plane,respectively. It is assumed that the debonding phe-nomenon can be initiated from the mid-plane of thebonding interface thickness. For determination ofthe damage initiation limit, values like bond normalstrength, N0, and transverse bond shear strengths,T0 and S0, should be checked according to interfaceexisting stresses. For the damage evolution process, afracture energy interactive criterion, called Benzeggah-Kenane, has been applied to the debonding prob-lem [11]. In this criterion, the sti�ness degradation ofthe normal and transverse behavior of resin is modi�edby the dissipated energy resulted from interface relativedisplacements. The degraded sti�ness of the bondingresin can be obtained using the interface damage index,D, which is calculated from the recent deformed statusof the interface [11].

NUMERICAL APPROACH

Geometry of Beams and FE Models

For investigation of a concrete beam, considering itsnonlinear behavior under low velocity impact loading,

Table 3. CFRP laminate material model input values.

Plastic Modulus Poisson's Mode-Failure Strengths (MPa) Shear Modulus Shear Strength(MPa) Ratio Tensile Compressive (MPa) (MPa)

E1 144700 �12 0.3 XT 1240 XC 870 G12 5200 S12 77

E2 9650 �13 0.3 YT 52 YC 230 G13 5200 S13 77

E3 9650 �23 0.45 YT 52 YC 230 G23 3400 S23 38

Table 4. Bonding epoxy resin material model input values.

Density (kg/m3) Elastic Modulus (MPa) Failure Strengths (MPa) Fracture Energy (N/m)

E 4300 N0 61 GCn 751260 G1 1600 T0 68 GCt 547

G2 1600 S0 68 GCs 547


smeared-plasticity based FE models were proposedand used for determination of the global and localresponse of beams. A constant regular discrete mesh isimplemented for the numerical modeling. Furthermore,the constitutive relations of materials are formulatedat the element integration points. To increase the e�-ciency of numerical results, the aspect ratio of elementsare kept equal to one for solid concrete and epoxyresin materials. The dynamic responses of models arecalculated using the central di�erence formulation ofstructural motion equations. In the numerical analyses,sixteen simply supported reinforced concrete beamswith various sti�ness and impact energy levels, areconsidered. It should be mentioned that the sheardeformation is more pronounced in short beams ratherthan long ones and, so, the shear sti�ness of all modelsshould be considered in analytical solutions. To studythe in uence of retro�tting laminates, the minimumallowed ACI 318-08 code (speci�ed shear and exuralreinforcement ratios) is considered in the proposedmodels [12]. For the concrete core continuum, aneight-node brick element (C3D8) has been used thatcan consider nonlinear geometrical e�ects, plasticityand damaged deformations. Three orthogonal crackplanes are accounted for in all integration points of thebrick element. The tensile behavior of the concreteis implemented by the damaged-elasticity fracturedmodel.

For retro�tting CFRP laminates, a four-node shellelement (S4) with a composite lay-up section is used forthe �nite element simulation of strengthened models.A plane stress condition is assumed in the constitutiverelations of each lamina sti�ness relation. CFRPlaminates are tied to epoxy resin elements at the topand bottom surfaces of beams and are of unidirectional0.5-mm thickness, CFRP T300/977-2, in type.

The bonding resin interface is also modeled withan eight-node cohesive element (COH3D8) with fourintegration points in its mid-plane interface. Resinelements are merged with laminate shell elements andconcrete brick elements in the exural surfaces ofbeams. A traction-separation constitutive relationcombined with a Benzeggah-Kenane fracture energymodel is used in the simulation of the resin interfacedebonding. The bonding interface has a 2-mm thickM10 epoxy resin material property.

Modeled RC beams with simply supportedboundary conditions are in a rest status (zero velocityinitial condition). Before applying the impact, modelsare statically analyzed to gravitational loading. Then,a 22.73 - kilogram steel cylinder is dropped freely from1 and 1.5 meter heights to the top surface of beamsrepeatedly. The location of the dropping is assumedin the beam midspan for maximum de ection. Fornumerical simpli�cation of models, only one quarterof each bean is modeled as a result of symmetric

boundary conditions. Regarding the higher sti�nessvalue of steel rather than concrete, the projectile ismodeled by discrete rigid 3D elements. The contactbetween the projectile and beam is considered as ahard contact. Therefore, the resultant contact forcewill incrementally be calculated in the time domain bytaking into account projectile penetration and beamdeformation. The typical FE model idealization isshown in Figures 6 and 7.

The rigid body formulation of the projectile con-tact zone is de�ned by the penalty contact method.The projectile is assumed to have no frictional contactand can be separated after completion of the penetra-tion. The proposed models are categorized into two un-retro�tted concrete beams (initiated with a S symbol)and laminate-retro�tted concrete beams (initiated witha R symbol). The geometrical and impact propertiesof beams used in this study are presented in Table 5.All concrete beams are 12 centimeters in width.

Minimum required exural - shear reinforcementratio rebars are calculated according to the ACI 318-08 structural code [12] as steel reinforcing rebars. Theretro�tting capability of CFRP unidirectional lami-nates can be revealed well, if the minimum requiredreinforcement is provided. Models are loaded uniformlywith their gravitational weights �rst and analyzedstatically. The deformed and stressed models are thenstroked by a falling drop projectile in the middle zoneof their span. The nonlinear dynamic analysis period

Figure 6. Finite element mesh con�guration andconnectivity.

Figure 7. Idealization of simply supported concretebeams under impact loading.


Table 5. Properties of concrete beams used and their reinforcement details.

Model Steel RebarReinforced

CFRPRetro�tted

Projectile FropHeight (m)

Beam Height(cm)

Span Length(m)

Rebar Stirrups

S1h1L1p � 1.0 10 1.6 4�6 2�4@40 mm

S1h1L2p � 1.0 10 2.0 4�6 2�4@40 mm

S1h2L1p � 1.0 16 1.6 4�8 2�4@80 mm

S1h2L2p � 1.0 16 2.0 4�8 2�4@80 mm

S2h1L1p � 1.5 10 1.6 4�6 2�4@40 mm

S2h1L2p � 1.5 10 2.0 4�6 2�4@40 mm

S2h2L1p � 1.5 16 1.6 4�8 2�4@80 mm

S2h2L2p � 1.5 16 2.0 4�8 2�4@80 mm

R1h1L1p p

1.0 10 1.6 4�6 2�4@40 mm

R1h1L2p p

1.0 10 2.0 4�6 2�4@40 mm

R1h2L1p p

1.0 16 1.6 4�8 2�4@80 mm

R1h2L2p p

1.0 16 2.0 4�8 2�4@80 mm

R2h1L1p p

1.5 10 1.6 4�6 2�4@40 mm

R2h1L2p p

1.5 10 2.0 4�6 2�4@40 mm

R2h2L1p p

1.5 16 1.6 4�8 2�4@80 mm

R2h2L2p p

1.5 16 2.0 4�8 2�4@80 mm

of each beam was about 0.1 seconds and the minimumvalue of the time increment was about 5.81e-9 seconds.

In impact problems, the wave propagation ve-locity is much higher than the deformation velocityof the structure. Therefore, the time increment inthe analysis should be taken quite small, so that thedynamic explicit solution of the structure approachesto an appropriate convergence. The time history ofmidspan de ection, total reaction force, cracking modeof RC beams, yielded zones of steel reinforcements andthe probable debonding interfaces of resin are taken asoutput data.

RESULTS AND DISCUSSION

De ection of Models Under Impact Loading

The average of the midspan node displacement is takenas the maximum midspan de ection. The general timehistory of the midspan de ection for beams S1h1L2,S2h2L1, R1h1L2 and R2h2L1 are plotted in Figures 8to 11.

The damage rate of a S2h2L1 reinforced concretebeam is seen higher than beam S1h1L2. Also, vibrationof the sti�er RC beam is damped more quickly than thesofter one. The S1h1L2 beam vibrated when strokedfor the third time, but the sti�er beam, S2h2L1, rapidlydamped after some few vibrations. The rapid dampingof vibrations in the S2h2L1 beam resulted in severedamage. It should be mentioned that the natural

Figure 8. De ection of S1h1L2 for �rst, second and thirdimpact.

Figure 9. De ection of S2h2L1 for �rst, second and thirdimpact.


Figure 10. De ection of R1h1L2 for �rst, second andthird impact.

Figure 11. De ection of R2h2L1 for �rst, second andthird impact.

damping of reinforced concrete beams depends on theirplastic deformations and their concrete cracking modes.Beams that are vibrated well in their exural modescan dissipate the imposed impact energy as a crackingand crushing formation. Therefore, beams with highervalues of sti�ness and little plastic deformation capabil-ity may receive more impact load and can then undergomore damage. The analytical and numerical maximumde ection values of all analyzed beams are reported inTable 6.

The retro�tting e�ect of CFRP laminate can beeasily shown by comparing Figures 9 and 11. FRPlaminates have controlled the variation domain of thede ection in its transient and steady state responsezone. CFRP-retro�tted reinforced concrete beams hadmuch lower maximum de ection values and also hada special convergence in residual de ection values inall impact repeats. It shows that FRP laminatescan distribute the local deformations to a wider rangeof the beam span and take the contribution of thewhole beam for controlling the impacted zone de ectionof the structure. Continuity e�ects of compositelaminates were valid for all analyzed models. But,the de ection controlling e�ect was more apparent inlong low-depth ( exure controlled) reinforced concretebeams. By increasing the impact energy level, the

average de ection value of concrete beams is increased,but FRP laminates constantly reserved their de ectioncontrolling e�ect in these cases too. Furthermore, theanalytic mathematical method resulted in higher valuesof de ection with respect to the numerical method asa result of the severe inelastic behavior of beams.

Reaction Force

A reaction force response occurred more rapidly thanthe de ection response. The summation of end supportreaction forces is taken as the total impact- inducedreaction force. It is obvious that the time historyvariation of the reaction force depends on reducedsti�ness, impact energy level, type of beam (long low-depth or short high-depth) and reinforcement ratio(steel rebar and FRP). The time history response ofthe beams, S1h1L2, S2h2L1, R1h1L2 and R2h2L1, areplotted in Figures 12 to 15. It is revealed that themaximum value of the reaction force (absorbed impactforce) is decreased by increasing the impact applyingtimes. This is due to the sti�ness degradation of thestructure as a result of damage evolution.

It can be observed that the beams with lowersti�ness values absorbed less impact force (reaction

Figure 12. Reaction force of S1h1L2 for �rst, second andthird impact.

Figure 13. Reaction force of S2h2L1 for �rst, second andthird impact.


Table 6. Maximum de ection values of analyzed reinforced concrete beams.

Model Sti�ness (N/m) Analytical (N) Numerical (N) Ratio (RD)

S1h1L1 3.51E+06 26.04 18.49 1.408

S1h1L2 1.80E+06 32.55 23.63 1.378

S1h2L1 1.48E+07 9.93 6.78 1.465

S1h2L2 7.68E+06 12.41 8.53 1.455

S2h1L1 3.51E+06 31.89 20.12 1.585

S2h1L2 1.80E+06 39.86 25.47 1.565

S2h2L1 1.48E+07 12.16 7.46 1.630

S2h2L2 7.68E+06 15.20 9.52 1.597

R1h1L1 4.02E+06 24.34 14.02 1.736

R1h1L2 2.06E+06 30.42 19.91 1.528

R1h2L1 1.61E+07 8.54 4.48 1.908

R1h2L2 8.32E+06 11.91 6.25 1.905

R2h1L1 4.02E+06 29.81 15.85 1.880

R2h1L2 2.06E+06 37.26 22.42 1.662

R2h2L1 1.61E+07 10.46 5.18 2.021

R2h2L2 8.32E+06 14.59 7.29 2.001

Figure 14. Reaction force of R1h1L2 for �rst, second andthird impact.

force) than those with higher ones. These beam typesalso had less reaction force variation during their steadystate time domain. Beams with high sti�ness valueslost a major proportion of their load carrying capacityduring the second and third impacts in comparisonwith lower sti�ness beams. This phenomenon impliesthat RC beams with higher sti�ness values may bedamaged more severely under impact loading thanlower sti�ness ones. In other words, the impact-induced damage rate of a RC beam is proportional toits sti�ness value. It also should be mentioned thatan increase in the impact energy level can result in anincreasing of the reaction force and a reduction in thebeam sti�ness. The analytical and numerical results ofthe maximum reaction force are stated in Table 7.

By analyzing Figures 14 and 15, it can be easilyseen that retro�tted RC beams showed less reaction

Figure 15. Reaction force of R2h2L1 for �rst, second andthird impact.

force variations and also preserved their impact loadcarrying capacity in their transient and steady statedynamic responses. The response-continuity e�ectof CFRP laminates has increased the impact loadcarrying capacity of retro�tted beams too. The di�er-ence between analytical and numerical reaction forceresults is directly a�ected by increasing the impactenergy level. But, retro�tting concrete beams reducesthese di�erences and addresses a global behavior ratherthan a local response to retro�tted members. This is apositive e�ect produced by FRP laminates.

Comparing Tables 6 and 7, one can see that theanalytical-numerical reaction force di�erences are muchless than de ection di�erences. In retro�tted beams,this result gap is more apparent. This shows thatCFRP laminates work as a de ection-reducer ratherthan an impact force-absorber. With lower failure


Table 7. Maximum reaction force values of analyzed reinforced concrete beams.

Model Sti�ness (N/m) Analytical (N) Numerical (N) Ratio (RF )S1h1L2 3.51E+06 39490.70 27332.56 1.445S1h1L1 1.80E+06 28308.00 19292.07 1.4674S1h2L2 1.48E+07 81198.25 54984.38 1.477S1h2L1 7.68E+06 58377.68 39363.68 1.483S2h1L2 3.51E+06 48366.04 30192.27 1.602S2h1L1 1.80E+06 34670.07 21215.84 1.634S2h2L2 1.48E+07 99447.14 59203.44 1.680S2h2L1 7.68E+06 71497.76 41621.53 1.718R1h1L2 4.02E+06 42223.32 29460.05 1.433R1h1L1 2.06E+06 30274.60 20927.10 1.447R1h2L2 1.61E+07 84519.59 58064.51 1.456R1h2L1 8.32E+06 60789.81 41577.67 1.462R2h1L2 4.02E+06 51712.79 32626.02 1.585R2h1L1 2.06E+06 37078.66 22852.00 1.623R2h2L2 1.61E+07 103514.93 62475.37 1.657R2h2L1 8.32E+06 74452.01 44101.16 1.688

de ection, the system can withstand more ultimateimpact forces, so, these two retro�tting criteria arenot independent and are somehow related to eachother.

Impact-Induced Cracking and Failure Modes

After applying the third impact, the cracks in theconcrete core and yielded zones of the reinforcementnetwork were appeared. The debonding probableinterface zones, resin damage index distribution andstrength status of the CFRP laminate are also inves-tigated. Two types of concrete cracking mode areobserved in reinforced concrete beams under impact:

1. Flexural cracks,

2. Shear cracks.

In intact beams, exural cracks are localized in theimpacted zone of the beam and resulted in the localyielding of longitudinal rebars. But, for the retro�ttedbeams, exural cracks are distributed in a wider rangeof concrete beams and, therefore, resulted in an in-crease of yielded rebar lengths. CFRP laminate alsocontrolled the depth and width of exural cracks inall models and reduced the local e�ect of the inducedimpact. The cracking modes and yielded zones of steelrebars are plotted in Figures 16 to 19. The crack modesin retro�tted concrete beams are di�erent from intactones. The number and depth of cracks are dependenton the beam type (long or short) and the impact energylevel.

By comparing Figures 16 and 18, it can be con-cluded that CFRP laminates made results in a wider

Figure 16. Concrete crack distribution and steel rebaryield mode of S1h1L2.

Figure 17. Concrete crack distribution and steel rebaryield mode of S2h2L1.

distribution of exural cracks and, �nally, increased theductility of the ultimate failure of members. Globalresponses are more desirable than local deformationsand are taken into account in the designation ofstructural members. FRP laminate assists the globalresponse in becoming more apparent. The CFRPlaminate also made the top longitudinal steel rebarbecome locally yielded. The important thing to noticeis that in long and low-depth RC beams, there are some


Figure 18. Concrete crack distribution and steel rebaryield mode of R1h1L2.

Figure 19. Concrete crack distribution and steel rebaryield mode of R2h2L1.

local longitudinal cracks concentrated at the ends ofthe beam in laminate cuto� zones. These cracks aredistributed in concrete cover zones and can result inthe separation of this part. This failure case is notrevealed well in short high-depth reinforced concretebeams.

The CFRP laminate also was e�cient in thebetter distribution of exural and shear concrete cracksof short high-depth beams (compare Figures 17 and19). Shear cracks are occurred more in these typesof beam. However, it can easily be veri�ed thatCFRP laminate was more controlling on long low-depth beams than the mentioned short high-depthbeams. The composite laminate-based retro�tting ofreinforced concrete beams made the exural cracksbecome slightly declined due to the constrain in uenceof the laminate in controlling the strain values of theretro�tted beam. The impact energy level increasemade the cracks be redistributed wider and deeper.It also made the local stirrups to be yielded in theimpacted zone.

Bonding Interface Status

The bonding interface strength and damage statuswas also investigated in this study. Considering thedebonding probability, retro�tted reinforced concretebeams with lower sti�ness were more critical thanbeams with higher values of sti�ness. It was shownthat longer beams with lower depth are so sensitiveto debonding phenomenon and this is due to more

exural cracked zones of the concrete interface. Themain e�ect of CFRP laminate was established very wellwhen the concrete cracks and the local internal forceis transmitted to composite laminate. Debonding ofthe CFRP laminate will be initiated where the damageindex reaches the critical value one. If the damageindex of the bonding resin is equal to one, it meansthat no more interfacial stresses could carry on and,therefore, the interfacial sti�ness approaches to zero.The NASA proposed constitutive model was quite ex-act with regard to experimental results driven by recentinvestigations [11]. The bonding interface damageindex of the retro�tted concrete beams, R1h1L2 andR2h2L1, were plotted in Figures 20 and 21.

The debonding probable zones of the resin in-terface were located in the midspan and end supportof the concrete beams. Consequently, the safe zoneof the bond interface was located in the �rst quarterand third-quarter of the beam span. Because of local

Figure 20. Bonding resin damage distribution contoursof R1h1L2.

Figure 21. Bonding resin damage distribution contoursof R2h2L1.


e�ects of the impact projectile, the top interface of thebeam was more critical than the bottom interface atthe midspan location. But, in general, on behalf of exural cracks at the bottom surface, this interface ismore likely to debond than the top interface at anylocation of the beam span. The damage index variationin beams with lower sti�ness is much higher than high-sti�ness beams. It is also proposed that the cuto� zonesof the CFRP laminate at the end beam supports shouldbe anchored to the beam in order to prevent laminatedebonding (matrix cracking) or local concrete coverseparation. It can easily be proved that the damageindex values of the interface are getting more criticalby increasing the impact energy level.

VARIATION OF BEAM RESPONSES

Considering the numerical and analytical results ob-tained from the dynamic analysis of the models, thevariations of the maximum reaction force, maximumde ection and the sti�ness variation of beams can beinvestigated. If the residual sti�ness and de ection ofretro�tted reinforced concrete beams were in the al-lowed service range, the remained strength of the mem-ber can be considered acceptable for further appliedimpact loads. The maximum reaction force variationis plotted with respect to the sti�ness variation for twointact and retro�tted beam categories in Figure 22.It is clear that for a constant value of sti�ness, aretro�tted beam will tolerate more impact load thana virgin one. Also, there is a little di�erence in theload capacity ranges of simple and retro�tted reinforcedconcrete beams.

The maximum midspan de ection variation isalso plotted as a function of the beam sti�ness forintact and retro�tted beam categories in Figure 23.Considering the maximum midspan variation with re-spect to sti�ness, one can conclude that for lower valuesof sti�ness, the CFRP laminate was more e�cient in

Figure 22. Maximum reaction force variation withregard to beam sti�ness.

Figure 23. Maximum midspan de ection variation withregard to beam sti�ness.

controlling the maximum de ection. The sti�ness ofreinforced concrete beams has a critical value and forvalues higher than this limit, the de ection reductionfactor of retro�tted beams converges to a constantvalue. It should be stated that in retro�tted beams,the maximum de ection value is more sensitive to theimpact energy level.

Sti�ness variations of intact and retro�tted RCbeams are plotted in Figures 24 and 25. Comparingthese �gures, one can deduce that unretro�tted beamsare more sensitive to the times when the projectile wasdropped onto the beam. Unretro�tted beams with alower sti�ness lost a major part of their sti�ness atthe moment of impact energy increase. Unretro�ttedshort beams resisted reserving their sti�ness at �rst andsecond impacts, but �nally lost a major part of theirsti�ness at the third impact.

It is veri�ed that the laminate based retro�ttingprocedure is more e�ective in beams with higherlengths and lower depths. Retro�tted beams showeda much di�erent performance compares to the intactones. Retro�tted beams with higher sti�ness reserved

Figure 24. Sti�ness degradation of intact concrete beamwith respect to impact times.


Figure 25. Sti�ness degradation of retro�tted concretebeam with respect to impact times.

their sti�ness in a much desired manner, rather thanunretro�tted ones at all impact times.

CONCLUSIONS

The impact e�ects on concrete beams strengthenedwith CFRP unidirectional composite laminates werestudied thoroughly. The following conclusions aredrawn based on the obtained results through analyticaland numerical studies:

1. The retro�tting technique of RC beams usingepoxy-bonded CFRP laminates can ensure the in-terface integrity and global deformation of membersunder impact loading.

2. Composite laminates increase the beam initial andresidual sti�ness. CFRP laminates also decreasedthe maximum de ection values and increased themaximum impact loading capacity.

3. CFRP laminate made the �nal failure mode of theretro�tted beam more ductile by increasing thenumber of exural-shear cracks and extending theyielded lengths of steel rebar reinforcements.

4. For long low-depth reinforced concrete beams, aCFRP cuto� concrete cover separation failure modeis highly probable. Therefore, an FRP anchoringtechnique in this zone is highly recommended.

5. FRP-based retro�tting techniques are more likely tocontrol the maximum de ection of a member thancontrolling the maximum absorbed impact force.

6. Long span low depth beams have more exuralcracks than shear cracks. However, the producedcracks in short span high-depth beams are almost ina shear mode. Thus, in long beams, the debondingstatus of the laminate bond interface is more criticalthan for short ones.

7. Unretro�tted RC beams having higher values of ini-tial sti�ness and induced impact energy were more

heavily damaged than similar beams with lowervalues of impact energy and sti�ness. Retro�ttedbeams with similar cases were damaged less thanintact ones.

REFERENCES

1. Erki, M.A. and Meier, U. \Impact loading of concretebeams externally strengthened with CFRP laminates",Journal of Composites for Construction, pp. 3645-3659(August 1999).

2. White, W.T., Soudki, A.K. and Erki, M.A. \Responseof RC beams strengthened with CFRP laminatesand subjected to a high rate of loading", Journalof Composites for Construction, pp. 153-162 (August2001).

3. Tang, T. and Saadatmanesh, H. \Behavior of concretebeams strengthened with �ber-reinforced polymer lam-inates under impact loading", Journal of Compositesfor Construction, ASCE, 7(3), pp. 65-67 (August2005).

4. Krauthammer, T. \Blast mitigation technologies: De-velopment and numerical considerations for behaviorassessment and design", Structures under Shock andImpact, Computational Mechanics Publications, pp. 1-12 (1998).

5. Brillouin, L., Wave Propagation and Group Velocity,Academic Press, New York (1960).

6. Abaqus Inc., ABAQUS 6.7 User's Manual, SIMULIA,USA (2007).

7. B�azant, Z.P. \Concrete fracture models: Testing andpractice", Engineering Fracture Mechanics, 69, pp.165-205 (2002).

8. Hillerborg, A., Modeer, M. and Peterson, P.E. \Anal-ysis of crack formation and crack growth in concreteby means of fracture mechanics and �nite elements",Cement and Concrete Research, 6, pp. 773-782 (1976).

9. Mander, J.B., Priestley, M.J.N. and Park, R. \The-oretical stress-strain model for con�ned concrete",Journal of Structural Engineering, ASCE, 114(8), pp.1804-1825 (August 1990).

10. Camanho, P.P. and Davila, C.G. \Mixed-mode decohe-sion �nite elements for the simulation of delaminationin composite materials", NASA/TM-2002-211737, pp.1-37 (2002).

11. Benzeggagh, M.L. and Kenane, M. \Measurement ofmixed-mode delamination fracture toughness of uni-directional glass/epoxy composites with mixed-modebending apparatus", Composites Science and Technol-ogy, 56, pp. 439-449 (1996).

12. ACI \Building code requirements for structural con-crete (ACI 318-08) and commentary", American Con-crete Institute, Farmington Hills, MI 48331, USA(2008).

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