Analytical Modeling of Concrete BeamsReinforced with Carbon FRP Bars

MUHAMMAD MASOOD RAFI,* ALI NADJAI AND FARIS ALI

FireSERT, University of Ulster at Jordanstown, Shore Road

Newtownabbey BT37 0QB, UK

ABSTRACT: A non-linear finite element model is presented for the analysisof simply supported reinforced concrete beams. Cracked concrete is considered as anorthotropic material and the crack formation is simulated as smeared cracks.A rotating crack method is employed based on the total strain crack model. Theanalysis was carried out with the help of 2D isoparametric plane stress elements.Compression and tension softening and tension stiffening effects of the crackedconcrete were considered. The tension reinforcement consisted of either steel or fiberreinforced polymer (FRP) bars. Excellent convergence and numerical stability of theformulation was found. The model showed good agreement with the recorded dataof the tested beams.

KEY WORDS: finite element, smeared crack, isoparametric, secant modulus,tension softening, DIANA, curvature, strain.

INTRODUCTION

THE ADVENT OF modern computing technology over the last 20 years has permittedprofessionals to attempt to solve very complex engineering problems using

sophisticated techniques. The use of finite element modeling (FEM) in the analysisand design of reinforced concrete (RC) structures is one of these successful techniques.The availability of numerous material models and its ability to capture the non-linearbehavior of RC has made FEM a very powerful tool in understanding the behavior ofa structure and to quantify its responses in areas such as strain development, deflection,load carrying capacity, stress distribution, etc. New non-metallic fiber reinforced polymer(FRP) materials have been introduced recently in the construction industry to enhance thedurability of conventional RC structures. A significant amount of research work has beencarried out to investigate the behavior of FRP bar reinforced structures. As a result,a wealth of very useful information is available in the form of literature. The comparisonof experimental results with non-linear FE analysis (FEA) could become helpful in orderto quantify the solution obtained.

*Author to whom correspondence should be addressed. E-mail: rafi-m@ulster.ac.uk

Journal of COMPOSITE MATERIALS, Vol. 41, No. 22/2007 2675

0021-9983/07/22 2675–16 $10.00/0 DOI: 10.1177/0021998307078728� SAGE Publications 2007

Los Angeles, London, New Delhi and Singapore

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This article presents the details of a FE model for RC beams and comparison ofthe analytical and experimental results. The beams were reinforced with steel or carbonFRP (CFRP) bars and were tested at the University of Ulster. The details of the tests andtheir results are being published separately. The notation used to identify the beamsare as follows: the first letter (B) stands for beam; the second letter indicates the testingtemperature as R for room temperature; and the third letter represents the type of tensionreinforcing bar material such as S for steel and C for CFRP bars. The compressive strength( fc) of the concrete on the day of the test is given in Table 1.

The proposed model is implemented in the commercial FE program DIANA [1] with thehelp of available modeling features. The analytical procedure was first tried for the beamstested by the authors to evaluate its success. To study the reproducibility and effectiveness,the investigation is further extended to validate this numerical model against experimentalresults of other investigators.

PROBLEM STATEMENT

The properties of materials play a significant role in the success of an analytical model.The description of material properties in case of an isotropic material, whether elastic orelastic–plastic, is fairly straight forward. Concrete is considered isotropic only beforecracking, as cracked concrete behaves as an orthotropic material. The location andorientation of cracks affect the overall response of a structure to the applied set of loads.The properties of cracked concrete both at material and structure level creates difficulty inmodeling its behavior. As a result, the prediction of response of even simpler structuresand elements becomes difficult. Concrete is characterized by its distinctly inelastic andnon-linear behavior, which is apparent in the early stages of its stress–strain relation. Thenon-linearity becomes even more pronounced close to its ultimate strength. A generalizedstress–strain relationship for unconfined concrete is shown in Figure 1. A number ofconstitutive laws are available to model the non-linear behavior of concrete and one has torely on these, most of the time. The accuracy of the analytical results depends on theeffectiveness of appropriately combining the non-linearities, arising from the constituentmaterials, to improve the accuracy of the solution algorithm. Numerical errors can resultin instabilities to an extent that could lead to the underestimation of the structural loadcapacity.

A smeared cracking approach has been used here to deal with the concrete cracking andis combined with the non-linear softening behavior of concrete. The FE formulation isbased on 2-D plane stress element. Shear reinforcement was not included in the analysisand the response of cracked RC was simulated with the total strain crack model.The model is based on the modified compression field theory (MCFT), which was

Table 1. Mechanical properties of concrete.

BeamCompressivestrength (MPa)

Modulus ofelasticity (MPa)

Tensile strength(MPa)

BRS1 46.52 33,502 3.88BRS2 44.64 33,222 3.78BRC1 42.55 32,686 3.66BRC2 41.71 33,580 3.61

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proposed by Vecchio and Collins [2]. The theory takes the tensile strength of the concretebetween the cracks into account, as opposed to conventional compression field theory.The material stiffness matrix is based on secant moduli in the direction of principalcompressive and tensile stress axes. The formation and propagation of cracks wassimulated with the rotating crack model (RCM). In the RCM a primary crack formsperpendicular to the direction of the major principal stress when the tensile strength of theconcrete is exceeded. As the principal stress rotates, after the formation of a primary crack,its direction will then not remain perpendicular to the crack plane and the tensile strengthof the concrete may be exceeded in another principal direction. Thus the crack rotates atall stages of loading with the change in the strain direction and shear on the crack planedoes not appear. The model described in the following sections can be used for thesuccessful theoretical prediction of the behavior of simply supported beams reinforcedwith any type of tension bars under monotonic loading.

MATERIAL MODEL

The models, which have been employed in order to represent the characteristic behaviorof the reinforcing bars and concrete (in compression and tension), are now discussed.

Concrete Compressive Behavior

The behavior of concrete in compression is of prime importance as the concrete ismostly used to resist compressive stresses. Therefore, the compressive strength is thecriterion of the quality of concrete. Although the concrete is subjected to multi-directional

s

fct

fc

Strain Softening

� �o

Figure 1. Uniaxial stress–strain relation of unconfined concrete.

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stresses simultaneously in a structural element, a uniaxial state of stress is assumed forsimplicity in the design of concrete structures [3]. Compressive strength of concrete isdetermined from the compression test on cylinders. The concrete does not fail immediatelyat its ultimate strength and its load carrying capacity, under the compressive state of stressbeyond this point, reduces progressively with increasing deformation. This type ofresponse to the load is an indication of a certain degree of ductility in the concrete at thematerial level. Thus the stress–strain graph of concrete in compression is comprised of anascending and a gradual descending branch, which is termed the softening branch, asshown in Figure 1. A constitutive model for the concrete in compression suggested byPopovics [4] and latter modified by Thorenfeldt et al. [5] has been used to describe thecompressive behavior of concrete in the direction of the principal compressive strain. Theuniaxial stress–strain relation is expressed by Equation (1)

�

fc¼

"

"0�

n

½n� 1þ ð"="0Þnk�

ð1Þ

where n is the curve fitting factor and k is the post-peak decay term and is taken as 1 for"/"o<1.

Collins and Mitchell [6] suggested expressions for n and k, which are given inEquation (2). It is to be noted that fc is taken in the metric system of units in Equation (2).

n ¼ 0:8þfc17

k ¼ 0:67þfc62

: ð2Þ

A typical stress–strain relation according to Equation (1) is shown in Figure 2.

Strain

Str

ess

fc

s

��o

Figure 2. Thorenfeldt et al. [5] compression curve.

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Concrete Tensile Behavior

Concrete is a weak material in tension and its tensile strength is of very little significancein any direct application. However, it plays a key role in the fracture mechanism ofconcrete, which can influence its behavior at the structure level as discussed in thefollowing.

FRACTURE MECHANISM OF CONCRETEConcrete is considered a brittle material and the stress–strain relation for uncracked

concrete in the direction of maximum principle tensile strain is assumed linear up to thetensile strength (fct) as shown in Figure 3. The formation of cracks in the concrete takesplace when the major principal stress exceeds fct. The directions of the principal strainsthen define the planes of orthotropy. The cracking criterion is based on fracture energy(Gf) and a crack band model is used to model fracture mechanism of the concrete. Theconstitutive model is formulated on smeared crack concept and damage approach.Only Mode I cracks were considered as these are the cracks which do not have any shearstress at their front.

Similar to the behavior in compression, concrete can resist strains higher than "ctand its post-peak behavior can comprise of a tension softening branch as shownin Figure 3. The term tension softening is a relationship between tensile stress andcrack bandwidth (hc) in the zone of fracture in concrete. This falling branch reflects thetension stiffening effects of the concrete between the cracks and the load-displacementresponse of an element is affected by the chosen post-ultimate characteristics of theconcrete in tension [7]. An exponential crack opening law (Figure 3) was adopted to modelthe tension softening and stiffening effects in the concrete. The curve is based on mode Ifracture energy through crack bandwidth of an element. The crack bandwidth was takenas (2A1)

1/2, where A1 is the total area of the element.

fct

ct

Ten

sile

str

ess

Tensile strain

Exponential tension softening

�

Figure 3. Behavior of concrete under tension.

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Reinforcement Behavior

A typical stress–strain curve for steel consists of a linear elastic portion, which isfollowed by a yield plateau. The length of yield plateau generally depends on thestrength of steel. A strain hardening zone follows the yielding of the bar until its failure.The elastic–plastic constitutive behavior of the steel bars was modeled by Von Mises fieldcriterion with associated flow and isotropic hardening. The tensile test results on the steelbars were used to describe the plastic behavior in the post-yielding stage, as shown inFigure 4.

FRP bars exhibit only linear elastic behavior up to failure. Von Mises yieldcriterion with associated flow and isotropic hardening was used to model the CFRPbars (Figure 4).

FINITE ELEMENT MODEL

The displacement at an integration point for any type of individual FE can be related tothe displacement of nodes using the relation given in Equation (3).

fug ¼ ½L�fdg ð3Þ

where {u} is the vector of displacement components at an integration point, {d} is thevector of displacements at nodes, and [L] is the matrix of shape functions.

A 2D quadrilateral plane stress element with eight nodes, as shown in Figure 5,was selected for the FE discretization. The element is based on linear displacementfunctions and quadratic interpolation. It has two degrees of freedom at each node,

Str

ess

Strain

SteelCFRP

Figure 4. Uniaxial stress–strain response of steel and CFRP bars.

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i.e., translation in orthogonal directions. Two systems of orthogonal axes are used for eachelement. The co-ordinates and displacements of each node are defined in relation to theglobal co-ordinate system (X,Y) whereas the shape functions are expressed with respectto the natural co-ordinates (�, �). Once an appropriate material stiffness matrix isdetermined, the element stiffness matrix {k} can be computed by the standard procedures.These procedures are available in the text books and are summarized as [8]:

fkg ¼

Z½B�T½D�½B�dV ð4Þ

where [D] is the material stiffness matrix, V is the volume of the element and [B] is thedisplacement–strain transformation matrix of nodes and is made up of the differentials ofthe shape functions, which are included in the matrix [L]

½B� ¼ ½@�½L� ð5Þ

where ½@� is the differential operator matrix.The model adopts isoparametric formulation of elements. A 3� 3 integration scheme

was performed on the elements and the stiffness matrix was formulated using Gaussintegration.

The non-linear analysis was carried out by incorporating the material models andelement formulation described above into FE program DIANA [1]. Only one half ofthe beam was modeled to take advantage of the symmetry in geometry and loadingconditions. The mesh, which was selected after a few mesh convergence trials, is shownin Figure 6. It consisted of a 4� 37 element grid. The same mesh was used for all models.Only tension reinforcing bars were included in the analysis. These were modeled as

1

2 3

4

56

7

8

X

Y

x

h

Figure 5. A typical eight-nodes quadrilateral element.

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bars embedded in quadrilateral FE. These uniaxial bar elements do not have any degreeof freedom of their own and have strength and stiffness characteristics only in thebar direction (�). A two-point Gauss integration scheme was used for these elements.The longitudinal strain of the bar is compatible with the strain of mother FE, whichensures a perfect bond between the rebar and the surrounding concrete.

ANALYSIS PROCEDURE

The analysis procedure adopts an incremental-iterative algorithm. The regular Newton–Raphson method was used for the solution of the non-linear equations. The method isbased on setting up a tangential stiffness matrix, before each iterative step. The analysiswas carried out by using an incrementally applied load in steps. The successive iterationsare performed with each increment in load, with the stiffness matrix being updated witheach iteration. The load sizes were specified explicitly and were kept small to obtainconvergence even in the highly non-linear state, such as when the concrete cracks. A linesearch algorithm was also used in order to increase the rate of convergence and to stabilizethe convergence behavior. A tolerance of 5� 10�1 on the displacement norm was definedas the convergence criterion for all load steps.

BASIC MATERIAL PROPERTIES

The modulus of elasticity of concrete in compression (Ec) was calculated by Equation (6)included in CP110 [9]. The equation provided consistent strength and stiffness results forall beams compared to the other suggested relations. Ec values of the beams are shownin Table 1.

Ec ¼ 9100ð fcuÞ1=3

ð6Þ

fct of the concrete was calculated using Equation (7) suggested by Eurocode 2 part 1–1 [10]and values for each beam are given in Table 1:

fct ¼ 0:30�ffiffiffiffifc

3p� �2

: ð7Þ

P1 P2 P3

CL

Figure 6. Finite element mesh.

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The fracture energy value of concrete Gf¼ 0.168N/mm was used, which was calculatedusing Equation (8) proposed by Wittmann [11]

Gf ¼ a:�nmax ð8Þ

where a¼ 80.6, n¼ 0.32, and �max is the maximum size of aggregate.The Poisson ratio of concrete before cracking was considered to be 0.2 whereas the

Poisson ratio is taken as zero for cracked concrete.

DISCUSSION ON ANALYTICAL RESULTS

The comparison of analytical predictions with the experimental results of only one beamof each type, which were tested by the authors, has been included in the forthcomingdiscussion whenever a similarity in the results existed. The result up to the last convergedstep has been shown throughout this section.

Strain Distribution

The experimental and analytical strain distribution of the rebar at mid-span for thebeam BRS1 and BRC2 is shown in Figure 7. These were obtained on the bar locationcorresponding to a strain gauge. As mentioned before, a perfect bond between the barsand the surrounding concrete was assumed in the model for both steel and CFRP bars.It can be seen from Figure 7 that although the influence of cracking on the recorded strainis quite evident, the predicted results are fairly close to the experimental plot and a goodcorrelation exists between the two results. The maximum stress on the CFRP barspredicted by the model was 1449MPa, which is fairly close to the experimental stress valueof 1345MPa. This shows that the CFRP bars developed a good bond with the concrete.

A comparison of the theoretical and recorded strain of the CFRP bar in the shear spanof BRC beams is illustrated in Figure 8. The strain gauge was bonded to the bar ata distance of 275mm from the support of the beam. A stiff theoretical response of thebeam, in the post-cracking stage, can be seen in Figure 8 compared to the observed

0

10

20

30

40

50

0 0.002 0.004 0.006 0.008Strain (m/m)

Lo

ad (

kN)

BRS1 (Exp)

BRS1 (Theo)

(a)

0

20

40

60

80

100

0 0.005 0.01 0.015

Lo

ad (

kN)

BRC2 (Exp)

BRC2 (Theo)

(b)

Strain (m/m)

Figure 7. Strain distribution of bar at midspan: (a) beam BRS1 (b) beam BRC2.

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response, especially for the BRC2 beam. However, the overall co-relation is reasonablygood for both BRC beams taking the sudden increase in the recorded strain into account,owing to the crack formation close to the gauge. The results correlate closely withthe observation of an adequate CFRP bar–concrete bond in the BRC beams.

Cracking and Ultimate Load

Table 2 provides a comparison of the recorded and analytical cracking (Pcr) andultimate (Pu) load of the beams. It is evident that the analytical Pcr was not onlyoverestimated but also seems affected by the type of reinforcement. It can be seen inTable 2 that Pu, and thus the failure modes, are fairly accurately predicted by the models.

Moment-Curvature

The comparison of the rebar strain profiles in Figures 7 and 8 validates the assumptionof a perfect bond of the CFRP bars with the concrete and, as a result, concrete strain canbe considered linearly proportional to the distance from the neutral axis (NA), as shown inFigure 9.

From Figure 9

"1 ¼ ’� c1 ð9aÞ

"2 ¼ �’� c2: ð9bÞ

0

20

40

60

80

100

0 0.002 0.004Strain (m/m) Strain (m/m)

Lo

ad (

kN)

BRC1 (Exp)

BRC1 (Theo)

(a)

0

20

40

60

80

100

Lo

ad (

kN)

BRC2 (Exp)

BRC2 (Theo)

(b)

0.006 0 0.002 0.004 0.006

Figure 8. Strain distribution of CFRP bar in the shear span: (a) beam BRC1 (b) beam BRC2.

Table 2. Cracking load, ultimate load and deflection.

Pcr (kN) Pu (kN) � at failure (mm)

Beam Exp. FEA Exp. FEA Exp. FEA

BRS1 7.8 11.25 41.90 45.00 47.37 45.15BRS2 7.5 11.25 40.10 43.75 39.29 37.12BRC1 7.1 10.00 88.90 90.00 37.39 36.17BRC2 7.1 10.00 86.50 87.50 43.43 39.16

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The negative sign shows a compressive strain. Adding absolute strains from Equation (9a)and (9b), we get

"1j j þ "2j j ¼ ðc1 þ c2Þ’: ð10Þ

The curvature (’) of the section can be evaluated with the help of Equation (11).

’ ¼"1j j þ "2j j

d: ð11Þ

Figure 10 traces the theoretical ’ of the BRC and BRS beams. The average values of "1and "2 has been used in Equation (11) to obtain a mean curvature. The strain values at themidspan and at a point 100mm away from the center of the beam were used to calculatethe average. The data, therefore, includes the influence of the cracks, and thus the tensionstiffening, on the theoretical strain over a gauge length. The points of measurement havebeen shown in Figure 6 as P1 and P2. It can be seen in Figure 10 that the initial responseof the beams is quite stiff in the uncracked state as well as immediately after cracking forboth BRS and BRC beams. However a marked difference in the curvature of the two typesof beams is evident in Figure 10, at a later stage. The curvature of the BRC beams

ε1

ϕ

NA

ε2

c2

c1

hd

Figure 9. Strain distribution across a section.

0

4

8

12

16

Mo

men

t (k

N-m

)

BRS1 (Theo)

BRS2 (Theo)

(a)

0

10

20

30

40

Mo

men

t (k

N-m

)

BRC1 (Theo)

BRC2 (Theo)

(b)

0 15 30 45 60

j × 106 (m−1) j × 106 (m−1)

0 20 40 60 80

Figure 10. Theoretical curvature of beams: (a) BRS beams (b) BRC beams.

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increased rapidly in the post cracking state up to around a moment of 13.5 kN-m.Thereafter the rate of increase in the curvature reduces with the applied moment andbecomes nearly linear. The post cracking ’ is stiffer for the BRS beams compared to theBRC beams. The rate of reduction in the curvature for the cracked BRS beams is also lesscompared to the BRC beams beams. Although the curvature of the BRS increased after amoment of 10 kN-m the beam failed shortly after this moment level. It was noted duringthe testing of the beams that the number of cracks stabilized after a load of 30 kN for bothtypes of beams. The load corresponds to a moment of 10.13 kN-m. The rebar straincontributed largely to the curvature beyond this stage compared to the concrete strain. Thebar strain was significantly higher for the cracked BRC beams as compared to the crackedBRS beams (Figure 7). This partly explains the reason of the above noted changes in thecurvature of the two types of beams. It can be seen in Figure 10 that the curvature of bothtypes of beams is similar to its type and is nearly unaffected by the variations in theconcrete strength of the beams.

The average height of the NA in the constant moment region of the beam BRS2 andBRC1 are plotted in Figure 11. The depth of the NA at any load level was calculated withthe help of Equation 9(b). The curvature and the concrete strain in Equation 9(b) weretaken as an average at points P1, P2 and P3 in Figure 6. It is evident in Figure 11 thatthe NA remains at nearly the same height for both types of beams at low load level(up to 20 kN). Beyond this load the depth of the NA in the BRC beams reduces rapidlyin comparison to the BRS beams. The NA is at a higher elevation in the BRC beamscompared to the BRS beams at their failure suggesting a higher maximum concrete strainat the top for the BRC beams.

Deflection Behavior

The analytical and experimental deflection (�) behaviors of the BRS and BRC beamsare traced in Figure 12. It can be seen that the predictions of ultimate capacity and stiffnessof the beams at all stages of loading i.e., pre and post cracking and post yielding arefairly good. The co-relation for the initial stiffness and for the overall non-linear behavioris very exact. It is evident in Figure 12 that the FE model slightly overestimates thepost-cracking stiffness. This may be due to the use of a too stiff tension stiffening model.Nevertheless, this overestimation is typical of this type of analysis [12] as the effects

0

20

40

60

80

100

0 20 40 60 80 100 120

Lo

ad (

kN)

BRC1 (Theo)

BRS2 (Theo)

NA (mm)

Figure 11. Average height of NA in the constant moment region.

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of other factors like local-bond slip and shrinkage stresses are unaccounted for in thetheoretical analysis.

In the BRC beams the drop in the applied load was gradual after the crushingof concrete during the test. However this type of response may not be possible to obtainanalytically in load controlled mode of simulation. A comparison of the experimentaland theoretical deflection at the failure of the beams is shown in Table 2.

VALIDATION OF MODEL

In order to further verify the consistency and reproducibility of the above-describedprocedure, a comparison was made of load–deflection curves reported by various otherresearchers in the published literature. The results of comparison with three reported studiesare shown here. These include beam GB5 [12], beam ISO 1 [13] and beam F–1–GF [14].All these beams were reinforced with the GFRP bars. The geometrical and materialproperties used for these beams are given in Table 3. Figure 13 traces both the experimentaland analytical behaviors of these beams. It is important to note that in the absence of specificinformation, an assumed value of fracture energy of 0.168N/mm has been used in all thesecases. The results of the analysis up to the last converged step are included in Figure 13.

It can be seen in Figure 13 that the simulation results are fairly accurate for allthese beams. It is the experience of the authors that the model is capable ofsuccessfully predicting the theoretical behavior of the beams reinforced with any typeof reinforcing bars.

0

10

20

30

40

50

0 10 20 30 40 50

Lo

ad (

kN)

BRS1 (Exp)

BRS1 (Theo)

(a)

0

10

20

30

40

50

Lo

ad (

kN)

BRS2 (Exp)

BRS2 (Theo)

(b)

(d)

0

20

40

60

80

100

0 10 20 30 40

Deflection (mm)

Lo

ad (

kN)

BRC1 (Exp)

BRC1 (Theo)

(c)

0

20

40

60

80

100

Lo

ad (

kN)

BRC2 (Exp)

BRC2 (Theo)

Deflection (mm)

0 10 20 30 40 50

Deflection (mm)

0 10 20 30 40 50

Deflection (mm)

Figure 12. Experimental and analytical load–deflection curves.

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SUMMARY AND CONCLUSIONS

The analytical flexural behavior of simply supported RC beams was studied usinga non-linear finite element modeling technique. Uncracked concrete was considered to bean isotropic material. The concrete was treated as an orthotropic non-linear elasticmaterial in the post-cracking stage with its stiffness based on secant moduli, which weretaken perpendicular and parallel to the direction of the crack. The model based on totalstrain crack was used to idealize the response of the cracked concrete. The cracks were

0

20

40

60

80

100

120

0 10 20 30 40 50

Deflection (mm)

Lo

ad (

kN)

GB5 (Exp)

FB5 (Theo)

(a)

0

20

40

60

80

0 15 30 45 60

Deflection (mm)

Lo

ad (

kN)

ISO1 (Exp)

ISO1 (Theo)

(b)

0

25

50

75

100

125

150

0 15 30 45 60

Deflection (mm)

Lo

ad (

kN)

F-1-GF(Exp)

F-1-GF(Theo)

(c)

Figure 13. Comparison of load–deflection curves of other researchers: (a) beam GB5, (b) beam ISO 1, (c)beam F-1-GF.

Table 3. Material properties data.

Beams

Geometrical and material properties GB5 [12] ISO 1 [13] F-1-GF [14]

Concretex-section – b� h (mm) 150�250 200�300 154�254Compressive strength (MPa) 26.52 43.00 36.50Tensile strength (MPa) 2.67 3.68 3.30Modulus of elasticity (MPa) 24,980 33,3000 30,208Poisson ratio 0.20 0.20 0.20

GFRP barsArea (mm2) 429.42 573.04 530.14Tensile strength (MPa) 1000 690 586Modulus of elasticity (MPa) 45,000 45,000 34,000

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considered as smeared cracks and a rotating crack approach was employed to simulate theformation and propagation of cracks. The behavior of concrete in compression, effects oftension softening stiffening and the behavior of tension reinforcement were considered inthe model. The FE discretization was carried out using a two-dimensional eight-nodedquadrilateral plane stress element.

An incremental-iterative non-linear solution procedure was used for the analysis.The algorithm considered non-linear behavior of the concrete and allowed smeared crackformation. The iteration scheme consisted of forming tangential stiffness matrix beforeeach iteration to meet the specified convergence criteria. It confirmed excellent stabilityand convergence characteristics up to the ultimate capacity of the beams.

The analytical model simulated the flexural behavior in terms of rebar straindistribution, failure modes stiffness characteristics at all stages of loading reasonablywell and the agreement between the experimental and numerical results was foundsatisfactory. The beams tested by other researchers were also analyzed. The theoreticalbehavior demonstrated good agreement with the recorded data.

NOMENCLATURE

b¼width of beamh¼height of beam"¼ concrete compressive strain"0¼ strain at maximum stress fc"ct¼ultimate tensile strain at fct� ¼ compressive stress at a level of "fcu¼ cube compressive strength

REFERENCES

1. TNO Building and Construction Research (2005). DIANA Finite Element Analysis, User’sManual Release 9. Delft.

2. Vecchio, F.J. and Collins, M.P. (1986). The Modified Compression-field Theory for ReinforcedConcrete Elements Subjected to Shear, ACI Journal, 83(22): 219–231.

3. Park, R. and Pauly, T. (1975). Reinforced Concrete Structures: Basic Assumption of Theoryfor Flexural Strength, John Wiley & Sons, Inc., USA.

4. Popovics, S. (1973). A Numerical Approach to the Complete Stress–Strain Curve of Concrete,Cement and Concrete Research, 3(5): 583–599.

5. Thorenfeldt, E. Tomaszewicz, A. and Jensen, J.J. (1987). Mechanical Properties of High-strengthConcrete and Application in Design, Proceedings of the Symposium Utilization of High-StrengthConcrete, Tapir, Trondheim, pp. 149–159.

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