1

A study of the shear behaviour of CFRP strengthened beams

incorporating a shear plane

W. Al-Juboori and Dr. L. Weekes

Abstract

The contribution of CFRP to the shear behaviour of reinforced concrete beams has been the

subject of much recent and current research. This paper presents an experimental and

numerical study of beams strengthened using CFRP across a shear plane which has been

incorporated. The shear plane represents a pre-cracked scenario, and the results demonstrate

clearly the effect of the orientation of the carbon fibre. The numerical finite element studies

(using ANSYS ver. 12) corroborate these results using a truss analogy in the modelling, and

the technique employed is suggested for use in further studies. It has been found that the

results provide insight into the effects of the orientation of the CFRP, and the incorporation of

a predetermined ‘shear crack’ provides a clear understanding of the contribution to shear

strength.

1 Introduction

The increase in service loads on structural elements requires the strength of these elements to

be upgraded to meet the new requirements. In certain situations, it may not be economically

feasible to replace an outdated structure with a new one, and design and construction of a

new structure to replace the existing one is quite often not an attractive economic solution. In

addition, the structural elements may be subjected to aggressive environmental conditions.

Therefore the maintenance of serviceability and strength of a particular structure throughout

its life can be one of a number of problems facing the structural engineer. A potential solution

is the use of new technologies to upgrade and strengthen the deficient structures. In

addressing the need to develop economic and efficient methods to upgrade, repair, or

strengthen existing reinforced concrete structures, Fibre Reinforced Polymer (FRP) materials

have been found to be successful at flexural strengthening, shear strengthening and ductility

enhancement of concrete structures [1-4]. External reinforcement is a promising, and

sometimes the only, alternative to partial demolition and replacement of many of these

structures. Externally bonded FRP plates increase the load carrying capacity with a negligible

increase in construction depth and weight of the existing structures.

2

This paper focuses on shear strengthening using Carbon Fibre Reinforced Polymer (CFRP).

The understanding of concrete structures designed for strengthening in shear is still an area

where uniform design rules do not exist or are treated very briefly. The reason for this is

probably that shear failure is a complicated mechanism [3]. When a strengthening material is

applied to the concrete, the shear mechanism will be even more complicated. To be able to

understand the behaviour of CFRP shear strengthened structures, theoretical as well as

experimental studies are needed. This is also important if the CFRP strengthening method is

to reach full acceptance worldwide. Normal RC design dictates that failure occurs in a

controlled ductile flexural mode rather than through brittle shearing. In addition, shear failure

of reinforced concrete beams can occur suddenly without any advance warning. Furthermore,

many existing RC beams have been found to be deficient in shear strength and need to be

strengthened. Several factors need to be considered in shear deficient structures such as lack

of shear reinforcement or reduction in steel area due to corrosion, construction faults, old

design codes, and increased service load beyond the original design. The amount and weight

of traffic has increased significantly since most bridges were built, and are well in excess of

those envisaged by their designers. Since 1999, the maximum allowable gross vehicle weight

within the UK has increased from 38 tonnes to 40 tonnes and it is likely that a further

increase to 44 tonnes will occur sometime in the near future. As a result there has been much

research in recent years to improve our understanding of the contribution of the CFRP

strengthening and to develop new analysis methods.

In traditional shear design [4-7] the nominal shear strength of an RC section, is the sum of

the nominal shear strengths of concrete, (for a cracked section this depends on the dowel

action of the longitudinal reinforcement, the diagonal tensile strength of the un-cracked part

of the concrete and the aggregate interlocking effect) and steel shear reinforcement, . For

beams strengthened with externally bonded FRP reinforcement, the shear strength may be

computed by the addition of a third term to account of the FRP contribution, . This is

expressed as follows:

To use externally bonded FRP reinforcement in design or retrofit, it is necessary to be able to

predict its contribution to the ultimate shear strength. Therefore, this study is focused on the

3

contribution from the composite, . The general equation to account for the shear

contribution of the FRP shear reinforcement of almost existing design methods is given by:

where is the bonded area of CFRP sheets , is the effective stress of CFRP

sheets at rupture, is the depth of CFRP reinforcement (usually equal to for rectangular

section and for T-sections), the spacing and the effective depth of CFRP (see

Figure 1, and is nominal concrete compressive strength (MPa). Note that for continuous

vertical shear reinforcement, the spacing of the strip, , and the width of the strip, , are

equal.

Figure 1.Dimensions used to define the area of FRP. (a) Vertical oriented FRP strips. (b) Inclined

strips.

2 Proposed Model

The objective of this research is to develop a new approach to examining the shear

performance and contribution of CFRP, using a simplified experimental model which allows

a parametric study of shear strengthening. The arrangement is effectively a simulation of a

beam subject to three points loading. The model is illustrated in Figure 2 and comprises two

separate parts (the double inclined lines CD represent the boundary surface between the two).

The purpose of this was to provide a shear failure plane. This model has some specific

characteristics which can be explained as follows:

Incorporating the shear plane allows the effect of the contribution of the bonded material

to be isolated.

The size of the model allows many configurations to be examined.

4

The position of the shear 'crack', which is provided by the boundary surface between the

two parts, is constant. This facilitates focusing on the failure of CFRP and hence selects

the best arrangement for any instrumentation.

As a result of this model arrangement, parametric studies are more easily addressed,

providing a straightforward means of understanding easy way to understand the behaviour

of CFRP sheets. This in turn will lead to clear theoretical analysis for homogeneous

materials such as aluminium and allow the study to focus on the behaviour of CFRP

material.

Figure 2 Model to examine the shear performance and contribution of CFRP.

3 Experimental Programme

3.1 Material properties

Before any tests could begin, a series of laboratory test were devised to determine material

properties of aluminium, CFRP, and adhesive. These need to be known to simulate the model

using the FE method (laboratory tests were derived to determine the common properties).

The first test was a tensile test (similar to dog bone tests as shown in Figure 3) which was

required to find the modulus of elasticity and Poisson’s ratio for aluminium. In this test, four

resistance strain gauges were used to check the strain. As shown in Photograph 1, the gauges

were positioned both horizontally and vertically on each side of the test specimen.

The results of this test are shown in Figure 4. From these results the modulus of elasticity for

the sample of the aluminium was Ex = Ey = 64.4 GPa and the Poisson’s ratio was n = 0.3. As

the maximum capacity of the machine was 50 kN, the sample did not initially fail, therefore a

retest in a tensile machine of greater capacity was used to achieve the eventual fracture load

of 55.5 kN.

P

A B C

D

5

Photograph 1 Aluminium tensile test (dog bone ). (A) The sample in the machine . (B) The

sample after eventual fracture.

Figure 3 Dimension of the aluminium dog bone (in mm)

6

Figure 4 Stress-strain curves of aluminium sample from dog bone test

The second important step towards developing a three-dimensional finite element

representation for the model in this test was to find the relationship between the slip

and the local bond stress. Figure 5 shows the detail of the bond test. Two pieces of

aluminium were bonded together using two sheets of CFRP on both sides of the

specimen (200 x 43.4 mm2). Material properties for the unidirectional CFRP sheets

were based upon information supplied by [Weber]. The nominal thickness, tf , of the

carbon fibres contained in the FRP sheet was equal to 0.1178 mm, and the tensile

strength and the Young’s modulus of the composite sheet based on the fibre thickness

were equal to 3900 MPa and 240 GPa, respectively (same material property of CFRP

sheet which was used for aluminium beam test). After the two surfaces of aluminium

were cleaned, the epoxy resin used consists of two components supplied by Weber

[8]. The epoxy resin used has an ultimate tensile strength of 19 MPa and an elastic

modulus of 10 GPa.

Figure 5 Schematic of the bond test.

The specimen was mounted in a tensile testing machine and subjected to pure tensile

force until total failure of the bonded system took place as a result of direct shear on

the laminates. It was not possible, however, to avoid a slight moment caused by a very

small eccentricity between the top and bottom grips. The results of this test are

illustrated in Figure 6 (there are four curves in this Figure). Each curve represents the

values of the displacement from each gauge. As shown in Figure 5, there are two

Eddy current gauges and two LVDT gauges. These gauges should in theory yield

similar results, but, due to the eccentricity of the load and the gauges, there are slight

differences in the results. The average of these results was taken to obtain a more

7

accurate load-slip relationship. This relationship was used to simulate the bond

strength between the CFRP sheet and aluminium sections in the finite element

analysis.

Figure 6 The bond test results between CFRP sheet and aluminium section.

3.2 Test set-up

This study comprised physical tests on an aluminium beam consisting of two separate

parts. The total span of the beam model was 13.5 inch (342.9mm), and its cross-section

was rectangular (3 inch (76.2mm) wide and 6 inch (152.4mm) deep). The cross-section

was hollow, as illustrated in Figure 7(a). Before testing the aluminium beams, initial

studies using timber beams were conducted to examine the relative geometric movement

of the two separate parts of the model (which effectively have a shear plane already

incorporated). The two parts were held together initially with duct tape (width 50 mm, cut

into strips) to provide insight into how load might distribute in any externally bonded

material. The detail of support B is shown in Figure 7(b).

(a) The dimension of aluminium model (dimensions are in mm).

152.5

P

63.5

76.2 38.1

38.1

152.5 38.1

38.1

38.1

8

(b) Three dimension of aluminium model

Figure 7 Representation of the aluminium model

The initial experiments were carried out for checking of the instrumentation only. Again,

the aluminium sections were bonded together using duct tape. Photograph 2 illustrates the

checking test before application of the load. Four linear variable differential transducer

(LVDTs) were used in these tests to measure the vertical and horizontal movement of the

aluminium parts as shown in Figure 8. Further experimental tests in this paper utilize the

same aluminium model with CFRP. The aim of these was to simulate the shear

contribution of CFRP, which is used to strengthen the new model, and also to compare

this model with finite element models created using the commercially available finite

element analysis package ANSYS ver12.

Photograph 4 Initial test of the aluminium model for verification.

9

Figure 8 Schematic representation of test set-up for specimen with instrumentation (dimensions

are in mm).

Eight tests were carried out for the aluminium model strengthened by CFRP as follows

(see Figure 9 and Photograph 3):

Four specimens, AC1, AC2, AC3, and AC4 were strengthened with an inclined

sheet of CFRP at 45o

Two specimens, AV1, and AV2 were strengthened with a vertical sheet of CFRP

Two specimens, AH1, and AH2 were strengthened with a horizontal sheet of

CFRP.

(b) Incline direction {AC1, AC2, AC3, and AC4}

P

(c ) Vertical direction {AV1, and AV2}

P

(d) Horizontal direction {AhH1, and AH2}

P

(a) Tape

107mm

107mm

P

57.15 25.4

146

V1 V2

H1

H2

10

Figure 9 Samples of aluminium model with different orientations.

Photograph 3 Aluminium specimens under load application. (A) Specimen AC1. (B) Specimen

AV2.

4 Test results and discussion

The load – deflection curves for all of the tested beams are shown in Figures 10. As

expected, the load capacity of control beam (without any CFRP strengthening) is equal to

zero. From observation of the Figures, it can be seen that the orientation of CFRP with an

angle of 45o provides the optimum strength. For orientation of CFRP with an angle of 45

o,

the test was repeated four times to determine the repeatability of the results. For all

beams, the load decreased abruptly after reaching the peak load because almost all cases

failed due to de-bonding (with the exception of horizontal orientation). For beams AH1

and AH2 (beams with horizontal orientation of CFRP), the load was very low when

compared with other samples. The failure for these two beams was as a result of the

separation between fibres of the sheet. The capacity of the sample with horizontal

orientation was approximately 10 kN, and with vertical orientation approximately 37 kN.

In contrast, the sample with a 45o angle of CFRP sheet produced results between 50 and

60 kN. All these values present the sole contribution of CFRP because the sample without

any strengthening could not carry any load. In general, the results for same orientation

had similar behaviour. Slight differences in the test conditions produced some slight

variations in the test results, as can be seen from the data of AV2 with vertical orientation

of CFRP sheets (initial test) and AC2 (reload). However, in general good repeatability

was demonstrated. Figure 11 shows the general comparison between the three cases of

orientation on the same scale. The modes of the failure for some specimens are illustrated

11

in Photograph 4. If we examine the failure of specimens in Photograph 6, it appears

complex. Sheets of CFRP were distorted in a few cases, but the general failure was

identified as de-bonding. The previous studies have established clearly that such

strengthened beams fail in shear mainly due to one of two modes: FRP rupture or de-

bonding [7]. Therefore, the main failure of samples was de-bonding, and there was

rupture failure in a limited area of CFRP sheets for some specimens, as shown in

Photograph 4 (A) and (B).

Figure 10 Load-Displacement relationship for vertical orientation of CFRP

Figure 11 Load-Displacement relationship for different orientations of CFRP

12

Photograph 4 Failure models of some specimens. (A) Specimen AC2. (B) Specimen AC3. (C)

Specimen AV1. (D) Specimen AV2. (E) Specimen AH1. (F) Specimen AH2.

5 Numerical analysis

The Finite Element method is a numerical method which can approximate and solve complex

structural problems to within acceptable boundaries. Finite element analysis was first

developed by the aircraft industry to predict the behaviour of metals forming for wings. The

ANSYS finite element program has been comprehensively developed to the extent that it has

applications across the whole engineering spectrum [9]. In particular, civil engineers are

frequently interested in modelling materials such as steel and concrete, the latter requiring

complex methodology in its representation. As concrete is an orthotropic material that

exhibits nonlinear behaviour during loading, this behaviour is numerically implemented in

ANSYS [10]. A number of previous researchers have used the finite element method to

provide insight into the behaviour of the FRP-concrete bonded joints, and CFRP-strengthened

13

RC beams. Hemmaty et al. [11] considered a nonlinear adherence-shear law based on the

experimental studies between concrete and reinforcement in the modelling of reinforced

concrete elements. While modelling the adherence-shear relationship, they used a nonlinear

spring/damper element COMBIN39 (element in ANSYS) for their main modelling. Also, Lu

et al. in 2009 [12] used COMBIN39 to model the interface between the FRP elements and the

supports. Lu et al. [12] presented a numerical study of the FRP stress distribution at de-

bonding failure in U-jacketed or side-bonded beams using a rigorous FRP-to-concrete bond–

slip model and assuming several different crack width distributions. This element type

COMBIN39 is used in the present study. Huyse et. al. [13] presented a paper concerning

analysis of reinforced concrete structures using the ANSYS nonlinear concrete model. This

paper considers the practical application of nonlinear models in the analysis of reinforced

concrete structures. The results of some analyses performed using the reinforced concrete

model of ANSYS are presented and discussed. The differences observed in the response of

the same reinforced concrete beam, as some variations are made in a material model that is

always basically the same, are emphasized. The consequences of small changes in modelling

are discussed and it is shown that satisfactory results may be obtained from relatively simple

and limited models.

Santhakumar et. al. in 2004 [14] presented a numerical study to simulate the behaviour of

retrofitted reinforced concrete beams strengthened with CFRP laminates using ANSYS. The

effect of retrofitting on un-cracked and pre-cracked reinforced concrete beams was studied,

and the behaviour of beams obtained from the numerical study showed good agreement with

the experimental data. There was no significant difference in behaviour between the un-

cracked and pre-cracked retrofitted beams. Al-Mahaidi et al. [15] studied the behaviour of

three shear deficient T-beams strengthened using web-bonded CFRP plate. The experimental

results have shown that repairing the beams with CFRP strips enhances their shear capacity.

The increase in strength ranged between 68% and 87%. Nonlinear finite element modelling

and analysis with DIANA was used to investigate the behaviour of these beams, assuming

plane stress conditions and perfect bond between the concrete surface and the web bonded

CFRP strips. It was shown that finite element analysis was capable of predicting the ultimate

strength, stiffness of the beams and strain levels in CFRP plates with reasonable accuracy.

The cracking patterns and crack inclinations produced by the finite element model were also

comparable to the patterns observed from testing. Fanning [16] presented nonlinear models

for reinforced and post-tensioned concrete beams. The finite element software used (ANSYS)

14

included dedicated numerical models for the nonlinear response of concrete under loading.

These models usually included a smeared crack analogy to account for the relatively poor

tensile strength of concrete, a plasticity algorithm to facilitate concrete crushing in

compression regions, and a method of specifying the amount, the distribution and the

orientation of any internal reinforcement. The numerical model adopted by ANSYS is

discussed in this paper. Appropriate numerical modelling strategies are recommended and

comparisons with experimental load-deflection responses are discussed for ordinary

reinforced concrete beams and for post-tensioned concrete T-beams.

The finite element modelling of experimental specimens using ANSYS Ver.12 is presented

here. The ultimate purpose of a finite element analysis is to recreate mathematically the

behaviour of an actual engineering system. Three and two dimensional nonlinear finite

element analysis is used here to simulate the performance of the experimental model.

The model in this study comprised three materials (aluminium, CFRP, and adhesive). By

taking advantage of the symmetry of the model, a half of the full section was used for

modelling. This approach reduced computational time and computer disk space requirements

significantly. The half of the entire model is shown in Figure 12.

Figure 12 Use of a half (aluminium model). (a) Three-dimension view. (b) Modified view for the

materials

In the FE models, SOLID45 elements were used to model the aluminium section, as shown in

Figure 12. SHELL63 elements represent the CFRP sheet. The material properties which were

supplied by the material tests and the manufacturer details Weber [8] were used to model the

CFRP sheet as an orthotropic shell element with the following orthotropic properties:

(a)

(b)

Aluminium section

adhesive (0.1 mm)

CFRP (0.167 mm)

15

The elastic modulus in the fibre direction Ey= 240 GPa

The elastic modulus in perpendicular direction to the fibre Ex= Ez = 31 GPa

The Poisson’s ratios and .

Figure 13 shows a representation of the adhesive which was used to bond the CFRP sheet

with the aluminium section. A three dimensional grid of COMBIN39 elements was used to

simulate the 0.1 mm thickness of adhesive layer. The experimental results of the bond test

between the aluminium and CFRP were used in the finite element model (input as real

constants of the COMBIN39 elements). Eight COMBIN39 elements in all were used to

simulate the bond of one SHELL63 element to an aluminium SOLID45 element. Four

COMBIN39 elements were connected orthogonally (toward the direction of the adhesive)

from the four nodes of the SHELL63 element to another four nodes on the aluminium

section. The other four COMBIN39 elements were arranged diagonally to connect the same

four nodes of the SHELL63 element to a central node on the aluminium body, located under

the centre of SHELL63 element as shown in Figure 13(b). This way of simulating the bond

between the CFRP and the aluminium section provided comparable theoretical behaviour to

actual physical behaviour and is addressed later in this section. The result of the bond test

(see Figure 6) between the CFRP sheet and aluminium section were used for the real constant

of the COMBIN39 elements. In addition, the CONTA178 elements bond the two pieces of

aluminium to provide the boundary conditions at the sloping interface.

Figure 13 Grid of COMBIN39 elements. (a) Top view of the simulated system of CFRP and adhesive

(sixteen SHELL63 elements and forty-one COMBIN39 elements) . (b) 3D view of individual

SHELL63 element with eight COMBIN39 elements.

16

Nonlinear analysis was invoked in these FE models by the nature of the elements used. The

numerical results of the aluminium model were then compared with the physical model. It

was then verified that the proposed numerical model can be used to analyze the two pieces of

aluminium which are bonded together by the CFRP sheets. Figure 14 shows the comparison

of deflection (at gauge No. V2, see Figure 11) with theoretical results from ANSYS.

Locations of the LVDT gauges are shown in Figure 8.

Figure 14 Load-displacement relationships for comparison between the experimental and analytical

results for CFRP oriented at (a) with 90o. (b) with 0

o (c) with 45

o to the longitudinal axis.

In general, the load-displacement plots for all cases of orientation from the FE analysis agree

exceptionally well with experimental data. The numerical results of ANSYS show similar

behaviour to experimental results, especially in the case of the horizontal orientation of

CFRP.

To focus solely on the effect of CFRP orientation, a simple FE model was created comprising

BEAM3 elements. This model is a 2-D model and the BEAM3 elements form the two parts

of the aluminium sections (total number of BEAM3 is 267). These two groups of BEAM3

elements are bonded together by four SHELL63 elements. Also the boundary conditions for

17

the sloping interface were provided using five CONTA178 elements in addition to the

supports. Near the boundary surface the BEAM3 elements were arranged in a novel way to

provide for different orientations of CFRP, as shown in Figure 15.

Figure 15 Simple FE models with different orientation of CFRP (a) with 90o, (b) with 0

o (c) with 45

o,

(d) with 55o.

Figure 16 shows the comparison of the results for different orientations of CFRP. Full

connection between SHELL63 and BEAM3 was assumed, hence the linear behaviour seen

from the graphs. It is easy to observe that the behaviour of CFRP sheet is influenced by the

orientation of the fibre. The strengthening with an orientation angle of 55o (between the

principal CFRP fibres direction and the longitudinal axis of the aluminium section) was

greater than the strengthening with 90o, 45

o, and 0

o orientation angles. So the angle of 55

o

provided the greatest capacity, even though the boundary surface between the two pieces of

aluminium (simulating a crack between them) was at a 45o angle. This infers that the

behaviour of CFRP does not just depend on the angle of shear crack but also on the

differential slip between the two parts of the beam which are on either side of this crack. This

conclusion illustrated that the existing idealizations deal with CFRP in an analogous way to

steel reinforcement behaviour. Therefore, the design equations of CFRP are not accurate,

especially for orientation.

(a) (b)

(d) (c)

18

Figure 16 Load-displacement with different orientation of CFRP

(a) FE models strengthening with 90o degree oriented CFRP after failure

(b) Load-displacement for the horizontal and vertical displacement

Figure 17 Comparison between the horizontal and vertical displacement for same point (v2)

Figure 17 shows the comparison between the horizontal and vertical displacement at the

bottom of the aluminium section for the vertical orientation case (at the same point as gauge

V2, as shown in Figure 8). Figure 17 (b) illustrates that the difference between the horizontal

19

and vertical displacement is limited. Therefore the addition of a horizontal strip at the bottom

of the section, as shown in Figure 18(a), led to an increase of the stiffness, as shown in Figure

18(b).

(a) Additional horizontal strip at the bottom of the section

(b) Comparison of different techniques.

Figure 18 Vertical orientation with additional horizontal strip (new technique)

6 Conclusion

The conceptual model presented, through physical testing, has provided promising insight

into the contribution of FRP to shear strengthening in beams. The incorporation of a

boundary plane to represent a ’pre-cracked’ condition has provided parameter isolation

which is necessary to identify the strength contribution solely of the FRP. This work

provides initial research into the shear contribution of the FRP to reinforced concrete

beams, and the model has been used successfully both in physical experiments conducted

on RC beam samples and analytical (FE) modelling.The numerical work has specifically

shown that the behaviour of CFRP is dependent on the angle of the shear crack and also

20

on the differential slip between the two segments of the beam which flank the crack. It is

envisaged that the FE modelling techniques employed will be further developed for future

work.

References:

1. Al-Rousan R., Issa M., Fatigue performance of reinforced concrete beams strengthened with

CFRP sheets, Construction and Building Materials 25 (2011) 3520–3529.

2. Lu XZ, Chen JF, Teng JG, Ye LP, Jiang JJ Rptter JM. RC beams shear-strengthened with

FRP: Stress distributions in the FRP reinforcement. Construction and Building Materials 23

(2009) 1544–1554.

3. Taljsten B. Strengthening concrete beams for shear with CFRP sheets. Constr Build Mater

(2003)15–26.

4. Khallifa A, Nanni A. Rehabilitation of rectangular simply supported RC beams with shear

deficiencies using CFRP composites. Construction and Building Materials, Volume 16, Issue

3, April 2002, 135-146.

5. ACI Committee 440.2R-02. Guide for the Design and Construction of Externally Bonded FRP

Systems for Strengthening Concrete Structures, American Concrete Institute, Farmington

Hills, Mich. 2003.

6. Triantafillou TC. Shear strengthening of reinforced concrete beams using epoxy-bonded FRP

composites. ACI Struct J 1998;95(2):107 –15.

7. Chen, J.F., Teng, J.G. Shear capacity of FRP-strengthened RC beams: FRP debonding.

Construction and Building Materials.2003 17, 27–41.

8. Weber Saint-Gobain, http://www.netweber.co.uk/(25/06/2011).

9. Lawrance KL., ANSYS Tutorial Release 7.0 and 6.1, SDC Publications, Canonsburg, 1.1-

2.25, (2002).

10. Barbosa AF, Riberio GO. Analysis of reinforced concrete structures using ANSYS nonlinear

concrete model. Comput. Mech. 1(8): 1-7, (2004).

11. Hemmaty Y, De Roeck G, Vandewalle L. Finite element modelling of corner joints in

reinforced concrete frames: International conference concrete 2000, economic and durable

construction through excellence, UK. University of Dundee, Concrete Technology Unit,

Dundee. pp. 465-472. (1993).

12. Lu XZ, Chen JF, Teng JG, Ye LP, Jiang JJ, Rptter JM. RC beams shear-strengthened with

FRP: Stress distributions in the FRP reinforcement. Construction and Building Materials 23

(2009) 1544–1554.

21

13. Huyse, L., Hemmaty, Y., and Vandewalle, L., Finite Element Modeling of Fibre Reinforced

Concrete Beams, Proceedings of the ANSYS Conference, Vol. 2, Pittsburgh, Pennsylvania,

May 1994.

14. Santhakumar, R. and Chandrasekaran, E.. Analysis of Reinforced Concrete Shear Beams

Using Carbon Fiber Composites. Electronic Journal of Structural Engineering. Vol. 4. (2004)

15. Al-Mahaidi, R., Lee, K. Taplin, G. "Behavior and Analysis of RC TBeams Partially Damaged

in Shear and Repaired with CFRP Laminates. Structures Congress, ASCE, Washigton DC

(2001).

16. Fanning, P. Nonlinear Models of Reinforced and Post-tensioned Concrete Beams, Electronic

Journal of Structural Engineering, Vol. 2, 2001, pp. 111-119.