Retrofitting of Reinforced Concrete Beams with CARDIFRC Farshid Jandaghi Alaee1 and Bhushan Lal Karihaloo, F.ASCE2 Abstract: A new retrofitting technique based on a material compatible with concrete is currently under development at Cardiff University. It overcomes some of the problems associated with the current techniques based on externally bonded steel plates and FRP 共fiber-reinforced polymer兲 laminates which are due to the mismatch of their tensile strength and stiffness with that of the concrete structure being retrofitted. This paper will describe briefly the technology necessary for preparing high-performance fiber-reinforced concrete mixes 共HPFRCC兲, designated CARDIFRC. They are characterized by high tensile/flexural strength and high energy-absorption capacity 共i.e., ductility兲. The special characteristics of CARDIFRC make them particularly suitable for repair, remedial, and upgrading activities 共i.e., retrofitting兲 of existing concrete structures. The promising results of several studies using CARDIFRC for retrofitting damaged concrete flexural members will be presented. It will be shown that damaged reinforced concrete beams can be successfully strengthened and rehabilitated in a variety of different retrofit configurations using precast CARDIFRC strips adhesively bonded to the prepared surfaces of the damaged beams. To predict the moment resistance and load-deflection response of the beams retrofitted in this manner an analytical model will be introduced, and the results of the computations will be compared with the test results to evaluate the accuracy of the model. DOI: 10.1061/共ASCE兲1090-0268共2003兲7:3共174兲 CE Database subject headings: Retrofitting; Beams; Steel fibers; Bonding; Composite materials.

Introduction Existing concrete structures may, for a variety of reasons, be found to perform unsatisfactorily. This could manifest itself by poor performance under service loading, in the form of excessive deflections and cracking, or there could be inadequate ultimate strength. Additionally, revisions in structural design and loading codes may render many structures previously thought to be satisfactory, noncompliant with current provisions. In the present economic climate, rehabilitation of damaged concrete structures to meet the more stringent limits on serviceability and ultimate strength of the current codes, and strengthening of existing concrete structures to carry higher permissible loads, seem to be a more attractive alternative to demolishing and rebuilding. The performance of current techniques of rehabilitation and strengthening 共the collective term retrofit, which implies the addition of structural components after initial construction, captures both rehabilitation and strengthening兲 using externally bonded steel plates and fiber-reinforced plastic 共FRP兲 laminates has been extensively investigated 共Ahmed and Gemert 1999; El-Refaie et al. 1999; Fanning and Kelly 1999; Yagi et al. 1999兲. The technique of retrofitting using externally bonded steel plates has gained widespread popularity, being quick, causing minimal site disruption, and producing only minimal change in section size. However, several problems have been encountered with this tech1

PhD Candidate, School of Engineering, Cardiff Univ., Queen’s Buildings, P.O. Box 925, Cardiff CF24 0YF, Wales, UK. 2 Professor, School of Engineering, Cardiff Univ., Queen’s Buildings, P.O. Box 925, Cardiff CF24 0YF, Wales, UK 共corresponding author兲. E-mail: [email protected] Note. Discussion open until January 1, 2004. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on November 27, 2001; approved on July 2, 2002. This paper is part of the Journal of Composites for Construction, Vol. 7, No. 3, August 1, 2003. ©ASCE, ISSN 1090-0268/2003/3-174 –186/$18.00.

nique, including the occurrence of undesirable shear failures, difficulty in handling heavy steel plates, corrosion of the steel, and the need for butt joint systems as a result of limited workable lengths 共Jones et al. 1988; Ziraba et al. 1994; Hussain et al. 1995兲 共Fig. 1兲. FRP materials as thin laminates or fabrics would appear to offer an ideal alternative to steel plates. They generally have high strength to weight and stiffness to weight ratios and are chemically quite inert, offering significant potential for lightweight, cost effective and durable retrofit 共Nanni 1995; Bu¨yu¨ko¨ztu¨rk and Hearing 1998兲. Retrofitting using FRP is also vulnerable to undesirable brittle failures due to a large mismatch in the tensile strength and stiffness with that of concrete 共Fig. 2兲. The key advantage of CARDIFRC mixes for retrofitting is that unlike steel and FRP, their tensile strength, stiffness, and coefficient of linear thermal expansion are comparable to that of the material of the parent member. Several studies have previously been undertaken at Cardiff into the feasibility of using CARDIFRC for the rehabilitation and strengthening of damaged RC flexural members 共Karihaloo et al. 2000, 2002; Alaee et al. 2001a,b兲. This paper, without repeating the results reported in those papers, expands on those studies, applying this technique on different types of beam 共with and without shear reinforcement兲 and introducing an analytical model. First, the material selection resulting from a rheological study, conducted recently at Cardiff, is outlined, and the application of these materials for retrofitting of beams is then discussed. Following that, to predict the behavior of the beams retrofitted with this technique, an analytical model is introduced. Finally, the results of the computations are compared with the test results, and the accuracy of the model is evaluated.

CARDIFRC A rheological study was recently carried out in Cardiff to optimize high-performance fiber-reinforced concrete mixes. The aim

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Table 1. Mix Proportions for Optimized CARDIFRC Mix I and

Mix II 共per m3兲 Constituents 共kg兲

Mix I

Mix II

Cement Microsilica Quartz sand 9–300 ␮m 250– 600 ␮m 212–1,000 ␮m 1–2 mm Water Superplasticizer Fibers ⫺6 mm ⫺13 mm

855.00 214.00

744.00 178.00

470.00 470.00 — — 188.00 28.00

166.00 — 335.00 672.00 149.00 55.00

390.00 78.00

351.00 117.00

0.22 0.18

0.20 0.16

Water/cement Water/binder

Fig. 1. Failure of beams retrofitted with steel plates: 共a兲 by plate debonding and 共b兲 by ripping off of the concrete cover 共after Ziraba et al. 1994兲

was to achieve good workable mixes with a very low water/ binder ratio and a high volume fraction of steel fiber, in order that the resulting material, in its hardened state, will be very ductile with a relatively high tensile strength. As a result of many trial mixes and testing, the mixes shown in Table 1 are the optimized ones. Two different mixes 共designated CARDIFRC, Mix I and Mix II兲 of high-performance fiber reinforced concrete differing mainly by the maximum size of quartz sand used in the mix have been developed using novel mixing and fiber dispersion procedures. These procedures are described in the patent application GB 0109686.6. Brass-coated steel fibers diameter 0.16, 6, or 13 mm long are used to prevent corrosion. The optimized grading of quartz sands leads to a considerable reduction in the water demand without loss in workability. All materials used in Table 1 are available commercially. A volume fraction of 6% short and long fibers is used, comprising 5% short fibers and 1% long fibers for Mix I, and 4.5% short fibers and 1.5% long fibers for Mix II. The specimens were hot-cured at 90°C for seven days. The strengths attained have been found to be the equivalent of standard 28-day water curing

Fig. 2. Failure modes in FRP retrofitted concrete beams: 共a兲 steel yield and FRP rupture; 共b兲 concrete compression failure; 共c兲 shear failure; 共d兲 debond of layer along rebar; 共e兲 delamination of FRP plate; and 共f兲 peeling due to shear crack 共after Bu¨yu¨ko¨ztu¨rk and Hearing 1998兲

Table 2. Typical Material Properties of CARDIFRC Mix I and

Mix II Material properties

Mix I

Mix II

Indirect tensile strength 共MPa兲 Fracture energy 共J/m2兲 Compressive strength 共MPa兲

24 12,210 207

25 12,380 185

Fig. 3. Internal reinforcement and load configuration in Stage II

at 20°C. Table 2 shows the material properties of the optimized mixes. The Young modulus of CARDIFRC is around 50 GPa.

Test Beams Two types of beams differing only by the reinforcement were used for Stages I and II of the experimental program. The beams in Stage I were reinforced in flexure only with a single 12 mm rebar, whereas in the beams tested in Stage II, stirrups— consisting of 6 mm deformed steel bars placed at 65 mm spacing—were also provided in the shear spans of the beams 共Fig. 3兲. As no shear reinforcement was provided in the beams tested in Stage I, both modes of failure, i.e., shear and flexural were expected. However, the beams tested in Stage II were designed in such a manner to fail in flexure. All the beams were made from a standard concrete mix and were 1,200 mm long, 100 mm wide, and 150 mm deep. The beams were removed from their molds after one day and water cured at ambient temperature 共20°C兲 for a minimum of 28 days. The mechanical properties of concrete and steel can be found in Table 3.

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Table 3. Parameters Assumed for Modeling the Behavior of Concrete and Steel Concrete In compression Stage

In tension

Steel

f cm 共MPa兲

E c 共GPa兲

E c1 共GPa兲

␧ c1 共—兲

␧ cu 共—兲

f ctm 共MPa兲

G F 共N/mm兲

W 1 共mm兲

W c 共mm兲

f y 共MPa兲

E s 共GPa兲

45 47

35.6 36.1

20.7 21.4

0.0022 0.0022

0.00315 0.00330

4.0 3.5

0.0725 0.0675

0.017 0.015

0.128 0.128

544 500

205 203

I II

Stage I Of the 32 beams used in Stage I, four were tested to failure as control beams to compare with the performance of those retrofitted with CARDIFRC strips. They were loaded in three-point bending over a span of 1,100 mm. Ten transducers 共five on each side兲 were used to record the deflection of the beams at various points along its span. The transducers were SOLATRON type ACR-25 and DCR-15 LVDTs. An aluminum frame 共yoke兲 was designed with two bars and a total of ten slots, to accommodate the transducers. One bar was placed on each side of the beam at midheight, as shown in Fig. 4. As expected, two control beams failed in shear, one in flexure and the fourth in a combination of shear and flexure modes. The average failure load was 29.48 kN. The remaining 28 beams were preloaded to approximately 75% of the above failure load to induce flexural cracking. In addition to parameters such as the material 共Mix I or II兲 and thickness of retrofit strips 共16 or 20 mm兲, four different configurations of retrofitting were investigated. Retrofitting with: • One strip bonded on the tension face 关Fig. 5共a兲兴, • Three strips 共one bonded on the tension face and the others on the vertical sides兲 关Fig. 5共b兲兴, • One strip bonded on the tension face and four rectangular strips on the vertical sides 关Fig. 5共c兲兴, and • One strip bonded on the tension face and four trapezoidal strips on the vertical sides 关Fig. 5共d兲兴. In total, ten different combinations of retrofitting were achieved with the 28 damaged beams tested in Stage I, as detailed in Tables 4 and 5.

mately 75% of the failure load 共31 kN兲. To improve the flexural behavior of the damaged beams three configurations of retrofitting strips were investigated in this stage. Retrofitting with: • One strip bonded on the tension face 关Fig. 6共a兲兴, • One strip bonded on the tension face and four short strips on the vertical sides covering the supports and the ends of the tension strip 关Fig. 6共b兲兴, and • One strip bonded on the tension face and four short and two continuous strips on the vertical sides, fully covering the supports and the tension strip 关Fig. 6共c兲兴. It should be mentioned that the last configuration 关Fig. 6共c兲兴 can be realized by bonding a strip to the tension face and two longer continuous strips on the vertical sides covering fully the supports and the sides of the tension strip. The solution chosen here was dictated by the fact that the precast strips were shorter 共1,030 mm long兲 than the overall span of test beams 共1,200 mm兲. Only Mix I was used as the retrofitting material in Stage II. As in the previous stage, this material was used for retrofitting the beams in two different thicknesses, i.e., 16 and 20 mm. In total, six different combinations of retrofitting were achieved in this stage, as detailed in Table 6.

Stage II Of the 14 beams produced for Stage II, three were tested without any repair as control beams. These beams were tested to failure under four-point bending over a span of 1,100 mm 共Fig. 4兲. The spacing between the applied loads was 400 mm. As expected, all the control beams in this stage failed in pure flexure and their average failure load was 42.03 kN. The remaining 11 beams were preloaded in the same manner as the control beams to approxi-

Fig. 4. Arrangement for testing beams: 共a兲 side view, 共b兲 cross section, and 共c兲 plan

Fig. 5. Configurations of retrofitting in Stage I: 共a兲 beam retrofitted with one strip on tension side; 共b兲 beam retrofitted with one strip underneath and two side strips; 共c兲 beam retrofitted with one strip underneath and four rectangular side strips; and 共d兲 beam retrofitted with one strip underneath and four trapezoidal side strips

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Table 4. Test Results 共Three-Point Bending兲 and Analytical Model Predictions of Stage I Beams

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Table 5. Test Results 共Four-Point Bending兲 and Analytical Model Predictions of Stage I Beams

Casting of Strips The retrofit materials, CARDIFRC Mix I and Mix II were cast as flat strips in 1,030 mm long and 100 mm wide steel molds with a well-oiled base and raised border whose height could be adjusted to give 16 or 20 mm thick plates. The molds were filled on a vibrating table at 50 Hz frequency and smoothed over with a float. To ensure a uniform thickness 共within 1 mm兲 a glass panel was located on top of the raised border. The strips were left to cure in the molds for 24 h at 20°C before demolding. The retrofit strips were then hot-cured at 90°C for a further nine days 共including one day for raising and one day for lowering the temperature兲.

The short rectangular and trapezoidal side strips, were cut from the long cast strips to the required size using a diamond saw.

Adhesive Bonding To improve the bond between the retrofit strips and the damaged beams, all contacting surfaces were carefully cleaned and roughened. An angle grinder was used to create a grid of grooves approximately 3 mm deep at a spacing of 50 mm on the contacting surfaces of the damaged beams. The retrofit strips were bonded to the prepared surfaces of the damaged concrete beams with a commercial thixotropic epoxy adhesive. The two parts of the adhesive were thoroughly mixed and applied to the tension side of the damaged beam with a serrated trowel to a uniform thickness of 3 mm. The strips were placed on the adhesive and evenly pressed. To ensure good adhesion, pressure must be applied to the strips during the hardening of the adhesive 共24 h兲 in accordance with the manufacturer’s recommendation. For the retrofitted beam with more than one strip, the beam was turned on its side to which the strip was bonded in the same manner as above. After another 24 h, this procedure was repeated on the other side of the damaged beam. In practice, to ensure good adhesion between the strips and the damaged beam pressure can be applied using G-clamps.

Test Results Stage I Fig. 6. Configurations of retrofitting in Stage II: 共a兲 one strip bonded on tension face; 共b兲 one strip bonded on tension face and four short strips on vertical sides covering supports and ends of tension strip; and 共c兲 one strip bonded on tension face and four short and two continuous strips on vertical sides, fully covering supports and tension strip sides

Of the 28 beams retrofitted in Stage I, 22 beams were tested in the same manner as the corresponding control beams, i.e., in threepoint bending over a span of 1,100 mm. The remaining six beams, i.e., the beams retrofitted with one continuous and four 20 mm thick rectangular strips were tested over the same span, but in four-point bending. The spacing between the applied loads was 400 mm. In all cases, the load was controlled by the movement of

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Table 6. Test Results 共Four-Point Bending兲 and Analytical Model Predictions of Stage II Beams

the actuator 共stroke control兲. The rate of loading was 0.01 mm/s at the beginning of all tests, but it was increased to 0.02 mm/s when the midspan deflection exceeded about 3 mm. In some cases, the beams were deliberately unloaded after the attainment of the maximum load and then reloaded. Therefore, the stiffness of the beam during reloading could be compared with the initial stiffness and the damage accumulated in the beam could be evaluated. Tables 4 and 5 show the test results of control and retrofitted beams. Of the seven beams retrofitted with one strip only on the tension face, four beams failed in flexure, two in shear, and one in a combination of flexure and shear. All the beams failed at loads at least equal to the average failure load of the control beams. The six beams retrofitted with three 16 mm strips all failed in pure flexure. Their failure was characterized by the formation and opening of a single flexural crack around the midspan of the beam 关Fig. 7共a兲兴. This configuration of retrofitting not only increased the load carrying capacity by more than 60% over that of the control beams, but also improved significantly the serviceability of the beams in terms of a significant reduction in the number and the width of the cracks. For instance, the midspan deflection of the retrofitted beams at a load level of 20 kN was only about 14%

Fig. 7. Flexural cracking in beam retrofitted with: 共a兲 three 16 mm thick strips and 共b兲 four trapezoidal strips on sides and continuous strip on tension face

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Fig. 8. Typical load-deflection response of beams retrofitted with three continuous strips, compared with that of control beams

of that of the control beams. Fig. 8 compares the typical loaddeflection response of the beams retrofitted with three continuous strips with that of the control beams. Only one beam was retrofitted by three 20 mm thick strips. Due to the over-strengthening of the midspan by the retrofit strips, this beam failed suddenly in shear. The energy released by cracking was so large that the beam split into two parts. To prevent shear failure of the beams retrofitted with three 20 mm strips, two systems of repair with one continuous strip and four rectangular/trapezoidal strips 关Figs. 5共c and d兲兴 were adopted. As with the other retrofitting configurations, the continuous strip was bonded to the tension face, whereas the rectangular/ trapezoidal strips partly covered the sides of the beam closest to the supports. The main objective of using trapezoidal strips was to check whether or not the gradual change in the cross section of the side strips improves the behavior of the beams and decreases the stress concentration. Two damaged beams were repaired using one continuous and four trapezoidal 20 mm thick strips. These beams failed in flexure with the opening of a flexural crack in the middle of the beam 关Fig. 7共b兲兴. Of the 12 damaged beams retrofitted by one continuous and four rectangular strips, six beams 共with 16 mm thick strips兲 were tested in three-point bending and the remaining six 共with 20 mm thick strips兲 in four-point bending. The typical load-deflection response of the beams tested in three-point bending can be found in Fig. 9共a兲. It should be noted however, that these beams will have a deflection capacity of only 3 mm under an accidental overload. The dominant cracks in four of these beams were nearly vertical and formed in the middle of the beams. These observations are typical of a flexural failure. In the other two beams, the dominant crack formed in the middle third of the beams, initiating where the side retrofit strip stopped and extending up to the point of loading. There were signs that the flexural and shear stresses influenced the dominant crack, so that the beams failed in a combined shear-flexure mode. In three-point bending, the shear force in all the sections of a beam is the same as at the supports. If the failure under the influence of the shear stress occurs in the middle third of the beam, then the failure will be ductile. However, if it occurs at the supports where there are no retrofitting strips, then it will be a brittle failure. The difference between the behavior of the beams retrofitted with trapezoidal and rectangular strips was found to be negligible. To confirm that the shear-flexural failure of the two beams retrofitted with one continuous and four rectangular 16 mm thick strips was because they were tested in three-point bending, the beams retrofitted with the same configuration but with 20 mm thick strips, were tested in four-point bending. For this configu-

Fig. 9. Load-deflection response of retrofitted beams: 共a兲 Stage I beams retrofitted with one continuous and four rectangular 16 mm thick strips under three-point bending and 共b兲 Stage II beams retrofitted with only one 20 mm thick strip on tension face under fourpoint bending

ration of loading, five beams failed in pure flexure with opening of pre-existing cracks and the appearance of some vertical cracks in the middle third of the beams. Only one beam failed in shear at its left support where there was no side strip to strengthen the section in shear.

Stage II The beams retrofitted in Stage II were tested in the same manner as their corresponding control beams, i.e., in four-point bending over a span of 1,100 mm. The four beams that were retrofitted with one strip only on the tension face all failed in flexure. However, in most of the beams 共three out of four兲 some signs of shear distress in the form of tiny diagonal cracks were observed at the end of the strips near the supports. These cracks propagated towards the nearest loading point and caused a local drop in the load. The load-deflection response of the beams retrofitted with 20 mm thick strips can be found in Fig. 9共b兲. It should be noted again, that these beams will have a deflection capacity of around 3 mm under an accidental overload. To overcome this problem, the anchorage area of the tension strip was strengthened in shear by covering the sides of the beams at the supports and near the ends of the strip by additional short

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Fig. 10. Stress-deformation diagrams assumed in model for: 共a兲 steel; 共b兲 concrete in compression; 共c兲 concrete in tension; and 共d兲 CARDIFRC in tension. Constant A depends on aspect ratio and volume fraction of fiber and fracture toughness of cementitious matrix.

strips. As mentioned before, two types of retrofitting strips were investigated. Four beams were retrofitted with one strip bonded on the tension face, and four short strips on the vertical sides, covering the supports and the ends of the tension strip 关Fig. 6共b兲兴; and three beams were retrofitted with one strip bonded on the tension face and four short and two continuous strips on the vertical sides, fully covering the supports and the tension strip sides 关Fig. 6共c兲兴. In fact in the second configuration, further improvement in flexural behavior of the beams was also expected. All seven beams failed in pure flexure with the opening of a pre-

existing crack after it had penetrated into the retrofit strip in the middle third of the beam. No shear cracks or drop in the load were observed thus confirming the usefulness of the covering the sides of the beams.

Analytical Model To predict the moment resistance and the load-deflection behavior of the control and retrofitted beams an analytical model has been

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Table 7. Parameters Assumed for Modeling the Tensile Behavior of

CARDIFRC CARDIFRC in tension Stage I and II

E

ft 共MPa兲

f tp 共MPa兲

共GPa兲

GF 共N/mm兲

w1 共mm兲

␧ max 共—兲

18

7.2

50

13

0.5

7.6⫻10⫺4

developed. In this model the strain hardening as well as tension softening of both concrete and CARDIFRC in tension have been taken into account. The stress-strain relationships of materials were assumed to be according to the test results or the proposed stress-deformation diagrams of the Model Code CEB-FIP 共1993兲. Based on this code, steel is assumed to be perfectly elasto-plastic 关Fig. 10共a兲兴, whereas a parabolic relation is used for concrete in compression 关Fig. 10共b兲兴. Table 3 shows the values of the relevant parameters assumed in the model. For concrete, the compressive strength was measured experimentally, and the remaining parameters were calculated from the relations proposed by CEB-FIP. The yield stress f y and the modulus of elasticity E s of steel were obtained from tension test on rebars. Tensile failure of concrete and CARDIFRC is always a discrete phenomenon. Therefore, to describe this behavior a stressstrain and a stress-crack opening relation should be used for the uncracked and cracked sections, respectively. For normal concrete in tension the stress-deformation behavior proposed by CEB-FIP was assumed 关Fig. 10共c兲兴, whereas the behavior of CARDIFRC in tension was modeled based on the theory of fracture mechanics and a few available test results 关Fig. 10共d兲兴. The parameters assumed for modeling the behavior of concrete and CARDIFRC in tension can be found in Tables 3 and 7, respectively. For concrete, the direct tensile strength f ctm was estimated from the splitting test results and the remaining parameters were again calculated from the relations proposed by CEB-FIP. For CARDIFRC the tensile strength of the matrix f tp was estimated from the splitting test results of the mix without fibers. However, the specific fracture energy G F and the modulus of elasticity E were directly measured using the notched beam and prism specimens, respectively. The remaining parameters in Table 7 were obtained from a few direct tension tests on dog-bone shape specimens. The moment resistance of a section retrofitted by CARDIFRC can be calculated based on the distribution of stresses caused by bending. To determine the strain distribution along the height of the section the following assumptions are made: • Plane sections remain plane after bending. In other words, the distribution of strain through the full height of the beam is linear 共Bernoulli hypothesis兲 and • The bond between the retrofit strips and the original beam is perfect and there is no sliding at the interface 共deformation compatibility兲. This assumption was fully validated by tests. The stress distribution in concrete and CARDIFRC strips cannot be assessed directly from the value of strain after cracking, as the constitutive relations are expressed in terms of stress-crack opening rather than stress-strain. Using the following assumptions, the evaluation of the crack opening from the strain distribution becomes possible: • The crack opening at the tension retrofit strip 共w兲 is the product of the strain at this level (␧ f ), and an effective length of retrofit strip (L eff) and • The dominant flexural crack tip is located at the level of the neutral axis. The faces of this crack open in a linear manner 关Fig. 11共a兲兴.

In fact, the strain over the effective length of retrofit strip (L eff) is released in the form of a local crack. To determine L eff , the length of the strain-free part of the retrofit strip should be calculated. If the tensile stress carried by the cracked strips is ignored in comparison with the tensile stress transferred by the reinforcement, the shear stress at the interface is dependent on the shear stress applied by the reinforcement, as shown in Fig. 11共b兲. Assuming the shear stress at the level of reinforcement is distributed at 45°, a length of retrofit strip (L eff) is stress-free and consequently strain-free. The deformation of this length of strip is localized in the crack opening. Therefore, to calculate the crack opening of the tension retrofit strip, the strain at this level (␧ f ) can be multiplied by this effective length (L eff), i.e., twice the distance between the reinforcement and the tension strip. It can be seen that by using this method the stress distribution in the repair material can also be worked out from the strain distribution. Due to the fact that the crack opening displacements 共i.e., crack widths兲 of the test beams were too small for accurate measurement, the crack openings calculated from the above method could not be compared directly with measured values. However, the consequences of the above assumptions to the calculation of the moment resistance and the load deflection response of the beams will become clear when we compare the model and test results later in this paper. To evaluate the moment resistance of the control beams and the beams retrofitted with different configurations of CARDIFRC strip, which fail in flexure, a program was written. Fig. 12 illustrates the flowchart of this program. First, a strain in the top concrete fiber and a neutral axis depth are assumed. Then, the linear strain distribution along the height of the beam is defined in terms of these assumed values. The depth between the top compression fiber and the neutral axis is divided into ten sections. The average strain over each section is calculated assuming piecewise linear fiber strain. The compressive stress can now be found using the concrete stress-strain relation. Multiplying this by the area of the section gives the compressive force. A similar calculation is made to determine the tensile forces in the concrete in tension, in the retrofit strips, and in the reinforcing steel. As mentioned before, to determine the tensile stress of cracked concrete and retrofit strips the crack opening should also be calculated.

Fig. 11. 共a兲 Modeling of flexural crack in middle of beam strengthened with three strips and 共b兲 effective length of strip for calculation of crack opening

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Fig. 12. Flowchart of program for calculating moment resistance of beams

Having calculated all the forces the neutral axis is adjusted until the sum of the compressive forces equals the sum of the tensile forces. When this is achieved, the moment is determined by summing the compressive and tensile forces times their moment arms about a single point. In addition, the curvature of the beam can also be easily worked out using the strain in the top concrete fiber and the neutral axis depth. This process is repeated for different assumed strains in the top fiber of concrete. The maximum moment resistance of the section occurs when either

the moment reduces for an increase in the top fiber strain, or the top fiber concrete strain exceeds the ultimate strain of concrete in compression (␧ cu). Fig. 13 shows the relation between the moment resistance of the section and the curvature of the beam for the beams in Stage I. Due to the fact that all the beams tested are statically determinate, their bending moment diagrams at any stage of loading are uniquely defined. This information can be combined with the moment-curvature diagram of sections to produce the curvature

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Fig. 13. Relation between moment resistance of sections and curvature in Stage I

diagram of the beam at different load levels by dividing the beam span into a number of intervals. The moment-area method is then used to compute the slope and the deflection of any point of the beam.

Model Results Stage I Tables 4 and 5 compare the maximum moment resistance of the beams predicted by the analytical model with the three- and fourpoint bend test results of Stage I, respectively. This comparison is also made in Fig. 14. It should be emphasized that the present model is only applicable to beams which fail in flexure. The load-deflection response of the control beams is compared in Fig. 15 with three model predictions. In the first model prediction 关Fig. 15共a兲兴, the tensile capacity of concrete is completely ignored. As a result the initial stiffness of the test beams is much higher than the predicted value. In the second model prediction 关Fig. 15共b兲兴, the tensile capacity of concrete up to the peak load is taken into account but its postpeak tension softening is again ignored. As a result the predicted initial response is much closer to the recorded response, but there are still some differences beFig. 15. Comparison of load-deflection response of control beams predicted by model with test results in three cases: 共a兲 when tensile capacity of concrete is completely ignored; 共b兲 when strain hardening of concrete is taken into account while tension softening is ignored; and 共c兲 when tension softening and strain hardening of concrete are considered

Fig. 14. Comparison of moment resistance of Stage I beams with predictions of analytical model

tween the test and model results after the concrete has cracked but before the steel has yielded. In the third model prediction, the complete tensile response of concrete including the tension softening is considered. Fig. 15共c兲 shows that the entire loaddeflection curve predicted by the model is now very close to that recorded in the tests. It can be seen that the load-deflection response of the beams is not accurately predicted, unless the full constitutive behavior of all the contributing materials, including the tension softening behavior of normal concrete is properly taken into account.

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Fig. 16. Comparison of typical load-deflection response of retrofitted beams with that predicted by model 共Stage I兲: 共a兲 beams retrofitted with one continuous and four rectangular 20 mm strips under fourpoint bending and 共b兲 beams retrofitted with three continuous 16 mm strips under three-point bending

Fig. 15 also shows that the model slightly under-estimates the load carrying capacity of the control beams. It can be due to the under-estimation of the yield stress of steel. Sensitivity analysis on the model shows that if the yield stress of steel is increased by 10%, the moment resistance of the beam is increased by 9.5%. Fig. 16共a兲 compares the typical load-deflection response of the beams retrofitted with one continuous and four rectangular 20 mm strips 关Fig. 5共c兲兴 with the model predictions. The same comparison is made in Fig. 16共b兲 for the beams retrofitted with three 16

Fig. 18. Comparison of load-deflection response of Stage II beams with model predictions: 共a兲 control beams and 共b兲 retrofitted beams. Of four beams tested, two 共1-16-S-1 and 1-16-S-2兲 were retrofitted with only one 16 mm strip on tension face, while remaining two had additionally short retrofit strips bonded on each vertical side near supports.

mm continuous strips 关Fig. 5共d兲兴. It can be clearly seen that the model predictions are in very good agreement with the test results, especially before the attainment of the maximum load.

Stage II

Fig. 17. Comparison of load carrying capacity of Stage II beams with that predicted by analytical model

Table 6 and Fig. 17 compare the maximum load carrying capacity of the Stage II beams 共with shear reinforcement兲 with the model predictions. It can be seen that the model predictions are again in good agreement with the test results. Of course, the failure load of some beams retrofitted with 20 mm strips is lower than that predicted by the model. This is likely to be the result of the poor quality of some 20 mm strips used for retrofitting the beams. The load-deflection response of the control beams in Stage II is compared in Fig. 18共a兲 with the model predictions. It can be seen that up to the load level corresponding to the cracking of concrete in tension, the response predicted by the model and that recorded in the test are identical. However, when concrete cracks in tension, the model predicts a larger deflection than the measured value. This is because the model assumes the cracked section condition for all sections in the region of the maximum moment. Although this local increase in the deflection was not observed in the tests, the stiffness of all control beams had de-

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creased considerably at this load level. As the load is increased, the model predictions again approach the test results, and finally a ductile failure with the yielding of steel is observed in both the model and the test results. The maximum load carrying capacity of the control beams is slightly higher than the predicted value. In addition, a local drop in the load is observed just after the attainment of the maximum load. This is due to the local instability induced by the yielding of steel across the existing flexural cracks which is not included in the model. Fig. 18共b兲 compares the load-deflection response of the Stage II beams with model predictions. Of the four beams tested, two 共1-16-S-1 and 1-16-S-2兲 were retrofitted with only one 16 mm strip on the tension face, while the remaining two had additionally short retrofit strips bonded on each vertical side near the supports 关see, Figs. 6共a and b兲兴. It can be seen that the load-deflection predicted by the model is close to the test results, especially before the maximum load is reached.

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

maximum fiber length in CARDIFRC; load; maximum load; deflection; strain; strain of concrete at maximum stress; strain of concrete in tension; ultimate strain of concrete in compression; stress; stress of concrete in tension; ultimate compressive stress of concrete in compression; ␻ ⫽ crack opening; ␻ 1 ⫽ crack opening of concrete at knee of tension softening diagram; and ␻ c ⫽ maximum crack opening.

L fmax P Pu ␦ ␧ ␧ c1 ␧ ct ␧ cu ␴ ␴ ct ␴ cu

References Conclusions The new technique using the CARDIFRC strip bonding system is a promising method for improving the flexural and shear behavior, as well as the serviceability of damaged concrete beams. It does not suffer from the drawbacks of the existing techniques, which are primarily a result of the mismatch in the properties between the concrete and the repair material. The mechanical properties of CARDIFRC Mixes I and II are very similar, therefore there is no real difference in the behavior of the beams retrofitted with either of these mixes. The moment resistance and load-deflection response of the beams retrofitted using this technique can be predicted analytically, providing that the strain hardening and tension softening response of concrete and CARDIFRC are properly taken into account. The technique described in this paper may be used when there is a need to improve the durability of existing concrete structures, as CARDIFRC mixes are very durable because of their highly dense microstructure. Research is currently being undertaken to study the fatigue, shrinkage, and creep properties of CARDIFRC and the performance of concrete structures retrofitted with CARDIFRC under dynamic, thermal, and hygral loads.

Acknowledgment This work is supported by U.K. EPSRC Grant No. GR/R11339.

Notation The following symbols are used in this paper: E ⫽ modulus of elasticity; E c ⫽ modulus of elasticity of concrete; E c1 ⫽ secant modulus of elasticity of concrete; E s ⫽ modulus of elasticity of steel; f cm ⫽ compressive strength of concrete; f ctm ⫽ tensile strength of concrete; f t ⫽ tensile strength of CARDIFRC; f tp ⫽ tensile strength of CARDIFRC matrix 共i.e., mix without fibers兲; f y ⫽ yield stress of steel; G F ⫽ specific fracture energy;

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