Verifications of Design Equations of Beams Externally Strengthened with FRP Composites

Transcription

1 Verifications of Design quations of Beams xternally Strengthened with FRP Composites Houssam Toutanji, P.., F.ASC 1 ; Liangying Zhao 2 ; and ugene Anselm 3 Abstract: Based ohe test information available ihe literature since 1990, a comprehensive database is assembled for an extensive survey of existing studies ohe flexural behavior of reinforced concrete beams externally strengthened with fiber-reinforced polymer FRP composites. Beam dimensions, material properties concrete, steel reinforcement, FRP composites, etc., and corresponding flexural responses such as failure modes, moment capacities, and so on, are collected ihis database. The purpose of this database is to verify the design formulas presented in ACI 440.2R-02, Guide for the Design and Construction of xternally Bonded FRP Systems for Strengthening Concrete Structures. The performance of some other simple strength design models is investigated based ohe same database and compared with that of the ACI model, which is found to have the least scattered prediction compared to others. Finally, a modified maximum strain FRP equation is recommended. DOI: / ASC :3 254 C Database subject headings: Databases; Design; Fiber reinforced polymers; Concrete; Reinforced; Beams. Introduction One of the techniques developed during the last decade for flexural strengthening of reinforced concrete RC beams is the use of fiber-reinforced polymer FRP plate externally bonded to the tension face. With the addition of FRP reinforcement, beams exhibit several unique failure modes under flexural loading: rupture of a FRP plate, concrete cover separation, and plate end and intermediate interfacial debonding of a FRP plate, as shown in Fig. 1 arrows indicate direction of crack propagation. These failure modes are reported ihe literature along with conventional failure modes such as crushing of concrete and concrete shear failure. Concrete cover separation is initiated by the occurrence of a shear crack close to the FRP end and ended by the shearing of the concrete cover along the level of the steel reinforcement. The interfacial debonding of a FRP plate can be induced either by interfacial normal and shear stress concentration reaching a critical value, or by propagation of an intermediate crack toward the FRP end, as shown in Figs. 1 c and d, respectively. Interfacial debonding and concrete cover separation are generally defined as premature failure because neither the FRP nor the 1 Professor, Dept. of Civil and nvironmental ngineering, Univ. of Alabama in Huntsville, Huntsville, AL corresponding author. -mail: touanji@cee.uah.edu 2 Ph.D. Candidate, Graduate Research Assistant, Dept. of Civil and nvironmental ngineering, Univ. of Alabama in Huntsville, Huntsville, AL Graduate Research Assistant, Dept. of Civil and nvironmental ngineering, Univ. of Alabama in Huntsville, Huntsville, AL Note. Discussion open until November 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASC Managing ditor. The manuscript for this paper was submitted for review and possible publication on March 18, 2005; approved on September 21, This paper is part of the Journal of Composites for Construction, Vol. 10, No. 3, June 1, ASC, ISSN /2006/ / $ concrete reaches its full strength capacity under ultimate beam load. From experimental observation, over 63% of test beams failed in a premature manner Bonacci and Maalej The failure initiated at the FRP end Figs. 1 b and c usually results in a huge reduction of the member bearing capacity. Among the premature failures, 44% of test beams failed by concrete cover separation and 56% by interfacial debonding of FRP Zhao and Toutanji For the significance of premature failure in FRP strengthening, many analytical and empirical models have been proposed to predict the premature failure strength. Smith and Teng 2002a,b reviewed 12 debonding strength models and evaluated them in a database of 59 test beams. Their study shows that most of these models do not provide a prediction of ultimate strength that is sufficiently safe for design use. To standardize the use of the FRP strengthening technique, the American Concrete Institute ACI published a Guide for the Design and Construction of xternally Bonded FRP Systems for Strengthening Concrete Structures ACI Committee 440.2R-02 ACI A design method for predicting the premature debonding strength of an RC beam with FRP reinforcement is presented. This ACI design method is of great importance in producing a safe design for the application of the FRP strengthening technique, but the current ACI 440 design recom- Fig. 1. FRP-related failure modes of RC beams strengthened by FRP plate 254 / JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006

2 mendations was recently claimed to be potentially unsafe for structures strengthened with FRP plates Reed et al Ina further study by Pham and Al-Mahaidi 2004, the ACI debonding strain equation generally predicts higher failure loads, especially for beams that failed through end cover separation. This paper therefore aims to verify the suitability of this method by applying different experimental data. A database is carefully assembled based on 115 test beams from an extensive survey of existing studies ohe flexural behavior of RC beams externally reinforced with FRP plate Triantafillou and Plevris 1992; Nakamura et al. 1996; Takeda et al. 1996; Spadea et al. 1997, 1998; Rahimi and Hutchinson 2001; Deng 2002; Grace et al. 2002; Smith and Teng 2002b; Zhang 2002; Breña et al. 2003; Pornpongsaroj and Pimanmas 2003; Valcuende et al. 2003; Zhang et al The database is used to verify the accuracy of the ACI 440 strength design equations. c = c h c db 0.003; s = c d c c ; c s = d c c c 3 The depth of the concrete compression block c can be calculated from q. 4a or q. 4b, depending on whether or not the tension steel has yielded 0.85f c b c + A s s s = A s s s + A f f db when s a 0.85f b c c + A s s s = A s f y + A f f db when s b where is defined in section of ACI The bending moment of the beam M u can be obtained from q. 5a or q. 5b M u = 0.85f c b c c 2 c + A s s s c d c + A s s s d c + A f f db h c 5a Overview of Strength Design Models Several debonding strength design models established by different concepts are selected and reviewed. The ACI 440 strength model is based ohe maximum usable FRP strain, the shear capacity-based model relates the beam strength to ultimate shear capacity at the plate end, and the maximum steel reinforcement ratio model determines the ultimate strength of the unstrengthened beam with a maximum steel reinforcement ratio. ACI 440 Model ACI Committee 440 proposed a design equatioo predict the maximum usable FRP strain. This implies that the ultimate rupture strain of a FRP plate, either reported by the manufacturers or obtained ihe laboratory, cannot be directly used in finding the ultimate moment capacity. An empirical reduction factor k m, given by q. 1, which is a function of the stiffness nt f f N/mm and the rupture strain of FRP fu, has to be imposed on fu to give the maximum usable FRP strain db, thus limiting the tension force developed in FRP in calculating the debonding strength of FRP-bonded RC beams. Then by applying the strain compatibility method and equilibrium equations to the reinforced section, the ultimate strength could be found ACI 2002 = 1 nt f f 0.90 for nt 60 fu 1 k m 360,000 f f 1 60 fu 90,000 nt f f 0.90 for nt f f 180,000 where n number of plies; t f thickness for each ply mm ; and f tensile modulus of FRP N/mm 2. So the maximum usable strain in FRP reinforcement db is given by q. 2 db = k m fu and the strains ihe extreme fiber of concrete, compression steel, and tension steel can be determined by q M u = 0.85f c b c c 2 c + A s s s c d c + A s f y d c + A f f db h c 5b where =0.85. The ACI model is only applicable to beams with intermediate span debonding failure. Shear Capacity-Based Models The rationale of the shear capacity-based models is that the premature failure strength is related to the shear strength of the concrete with no contribution of the steel shear reinforcement. The debonding strength is generally given ierms of the shear force at the plate end. One advantage of this model is that the interfacial stress betweehe plate and the beam does not need to be evaluated and little calculation is required to predict the strength. Jansze s Model Jansze 1997 proposed a plate end debonding strength model for steel plated beams; information about this model is found in Smith and Teng 2002a. The model was based ohe initiation of shear cracking in an RC beam without the contribution of shear reinforcement. The critical shear force ihe RC beam at the plate end to cause debonding V u,end is given as follows: where shear strength is and V u,end = bd 3 = d a s f v d c 1 s a v = 4 2 da 3 8 s where b and d width and effective depth of beam section; s steel tension reinforcement ratio; f concrete c compressive strength; a distance betweehe cutting point of the FRP plate to the nearest support; a v shear span; and a modified v shear span as given by q. 8. Ifa v as defined by q. 8 is greater than the actual shear span a v of the beam, thehe modified shear span should be given by a v +a /2. v Jansze s model appears to be invalid for soffit plates terminated at the support as a v becomes zero 6 7 JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006 / 255

3 Table 1. Database of RC Beams Strengthened with FRP Reinforcement Debonding Cover separation Wet layup Pultruded plate Wet layup Pultruded plate References Beam References Beam References Beam References Beam c Breña et al A3 b Garden 1997 a Ahmed and David 1999 a B1 Van Gemert 1999b a c Deng O-1 3O-1 3O-2 4O-1 b Hau 1999 a 4 5 c Nakamura et al P 24-2P 65-1P 65-2P c Takeda et al b Täljsten 1999 a FB1 b Garden et al a 2Au 3Au 3Bu 3Cu B3u,1.0 B4u,1.0 B5u,1.0 B1u,4.5 Ahmed and Van Gemert 1999a a DF.2 DF.3 DF.4 AF3 CF2-1 CF3-1 CF4-1 b Grace et al C-3 Beber et al a VR5, VR6 VR7, VR8 VR9, VR10 b Juvandes et al a B7 C1 C2 D1 D2 Hau 1999 a b Quantrill et al a B3 Pornpongsaroj and Pimanmas 2003 b Ross et al a 1B, 1C 2B, 2C 2D, 3B 3C, 3D Tumialan et al a A-420-P B-200-P A3 A8 C2 Garden et al a Garden et al a Nguyen 2001 a Quantrill et al a Rahimi and Hutchinson 2001 P2 P3 P4 P5 1Au, 1Bu 1B2u, 1Cu 2Bu, 2Cu B1u,1.0 B2u,1.0 B1u,2.3 A950 A1100 A1150 B2 B4 B6 B3, B4 B5, B6 B7, B8 B Ritchie et al a C, D, G, I, M c Triantafillou and 4, 5, 6, 7, 8 c Saadatmanesh and Plevris 1992 hsani 1991 a c Valcuende et al A-S1, A-S2 b Spadea et al A1.1 B-S1, B-S2 c Zhang 2002 OR-3L-A b Spadea et al A3.1 OR-3L-B OR-4L c Zhang et al A-AT A-AK B-AT B-AK B-C1 b Täljsten 1999 a SB1 SB2 SB3 MB1 HB1 a Beam information is collected from Smith and Teng 2002b. b Beam failed by plate end debonding. c Beam failed by intermediate debonding. and q. 8 predicts that debonding is never possible. Jansze 1997 s model can be applied only to those beams with concrete cover separation failure and with a soffit plate length less thahe beam span. Blaschko s Model Blaschko 1997 suggested a simple design method where the acting shear force is limited to the modified concrete capacity without shear reinforcement, as shown in q. 9 V u,end = kbd l 0.18 f 1/3 c 9 where k takes size effect into account: k=1.6 d 1, d is in meters, and l is given by q. 10 f l = A s + A f 10 s / bd where A f, f, and s cross-section area, modulus of elasticity of FRP reinforcement, and the modulus of elasticity of tension steel reinforcement, respectively. As ihe ACI model, the Blaschko 1997 model is only applicable to beams with intermediate span debonding failure. Matthys s Model Matthys 2000 developed another similar shear capacity-based model that predicts the ultimate strength of RC beams reinforced by FRP. The shear capacity V u,end is given by q. 11 V u,end = bd and from his experiment, shear strength is = l and l is the same as q. 10. As ihe ACI and Blaschko models, the Matthys model is also only applicable to beams with intermediate span debonding failure. 256 / JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006

4 Table 2. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Debonding Wet Layup+Pultruded Table 4. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Debonding Pultruded Plate Only Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of Smith and Teng s Model Smith and Teng 2002b proposed a simple design-oriented, shear capacity-based model. The debonding strength is given by q. 13 V u,end = V c 13 where V u,end ultimate shear capacity at the plate end; V c shear capacity of the concrete ihe RC beam alone, without the contribution from the shear reinforcement; =1.5; and V c can be calculated according to different design codes. For ACI ACI 2000, q. 14 is specified to calculate the shear capacity of the RC beam V c = 1.9bd fc + 2,500 s bd 3.5bd fc 14 For the Australian concrete code Standards 1988, q. 15 is giveo calculate the shear capacity of the beams V c = 1.4 d/2,000 bd s f 1/3 c 15 The above expression for V c requires 1.4 d/2, Naaman s Model The maximum steel reinforcement ratio model is suggested by Naaman He estimated that, compared to the unstrengthened RC beam, it is reasonable to desighe strength increments for the same RC beam bonded with FRP as approximately 20% of the ultimate strength of the same RC beam, calculated assuming a steel reinforcement ratio equal to max, which could be expressed as follows: M u = M uns + 0.2M max 16 Thus M u and M uns ultimate moment capacity of strengthened and unstrengthened beams, respectively, and M max bending strength of the same RC beam calculated assuming a maximum steel reinforcement ratio without FRP reinforcement. Both M uns Table 3. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Debonding Wet Layup Only ACI Blaschko Matthys Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of and M max could be calculated based on ACI , considering the section is singly reinforced. Like the shear capacity-based model, a lot of calculation work could be saved compared to the ACI method, since apparently q. 16 does not include any information about FRP reinforcement. Database To assess the above design methods, a comprehensive database is constructed for flexural tests on RC beams externally strengthened with FRP composites. The criteria enforced in collecting the database are 1 all beams have rectangular sections and are underreinforced, regardless of the FRP plate; 2 all beams are simply supported under three- or four-point loading; 3 all beams failed either by FRP debonding or cover separation; 4 all beams were statically loaded until failure and none were preloaded; and 5 no anchorage on FRP plate has been used on any beam, such as the bolted end of FRP or extension under support. Smith and Teng s 2002b database meets all the above requirements, and 58 beams from it are included ihe current database, with only one overreinforced beam eliminated. The data are categorized with respect to different failure modes interfacial debonding or cover separation and FRP curing methods wet layup or pultruded, as shown in Table 1. It is interesting to note that most beams with intermediate debonding failure are prepared by the wet layup technique, and most beams with plate end debonding failure are strengthened by pultruded plate. Results and Discussions Verification of Models Using Database Statistical tools are used in analyzing the performance of the three strength models. The ratio of experimental nominal strength to predicted results is adopted to access each model. Tables 2 7 show the statistical values of the experimental-to-predicted Table 5. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Cover Separation Wet Layup+Pultruded Jansze Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006 / 257

5 Table 6. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Cover Separation Wet Layup Only Jansze Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of Fig. 2. Verification of ACI design model strength ratio. Average, standard deviation, coefficient of variation, and percentage of exceedence are calculated. The percentage of exceedence is defined as the percentage of the number of tests with an experimental-to-predicted ratio less than 1. The greater is this ratio, the less conservative the model. Only beams with intermediate debonding failure were used for verifying the ACI, Blaschko, and Matthys models, and only beams with cover separation failure were used for checking Jansze s model. Tables 2 7 are for beams that failed through plate debonding and cover separation, respectively. Tables 2 and 5 are for both types of FRP plate, wet layup and pultruded plate; Tables 3 and 6 are for wet layup plate beams only; and Tables 4 and 7 are for pultruded plate beams only. Verifications have also been shown in Figs. 2 8 for all models by plotting the predicted strength against experimental values. In each figure, a straight line with a 45 angle is drawn, indicating the exact accurate prediction. From Table 3 and Fig. 2, the ACI model yields the most accurate and least scattered prediction for intermediate debonding failure, but more than half of its predictions are unconservative, with a percentage of exceedence of 64%, as shown in Table 3 and Fig. 2. This conclusion was confirmed by Reed et al. 2004, showing that the ACI 440 strength design equation is potentially unsafe. Blaschko s 1997 model yields conservative average strength for beams with intermediate debonding failure, while the average strength given by Matthys 2000 is less safe than Blaschko 1997, as in Table 3 and Figs. 3 and 4. Jansze s model gives conservative average for wet layup beams Table 6 and an unconservative average for pultruded plate beams Table 7, and this is also graphically shown in Fig. 5. As for Smith and Teng s 2002b model, when shear capacity of RC beams is calculated based ohe Australian AS code, the average prediction is more conservative than whehe ACI code is used, as shown in Tables 2 to 7. Whehe Australian code is used, Smith and Teng s model is more conservative for beams that failed by concrete cover separation rather than by debonding. This is close to Smith and Teng s conclusion iheir model, since most of the beams iheir database failed by concrete cover separation. But Smith and Teng s 2002b model cannot provide a safe design for beams with debonding failure ihe current database, no matter whether the Australian or the ACI code is used for shear capacity of RC beams, as shown in Figs. 6 and 7. Naaman s 2003 model produces safe average predictions, of which about 80 to 90% are conservative, as shown in Tables 2 and 5 and Fig. 8. Note that Naaman s method is totally conservative for pultruded plate beams that failed because of cover separation, as can be seen in Table 7. Among these models, one may conclude that the ACI 440 model is most accurate in predicting ultimate beam strength and requires the most calculation compared to other models, since a quadratic equation needs to be solved to find the depth of the neutral axis c. But the shortcoming of the ACI model is purely empirical so that it does not have any theoretical background. Moreover, the maximum usable strain equation given by ACI 440 does not involve information ohe substrate where FRP is bonded, which is another imperfection of that model ACI Table 7. xperimental-to-predicted Debonding Strength Ratio for Beams Failed by Cover Separation Pultruded Plate Only Jansze Smith ACI Smith AS Naaman =percentage of specimens with unsafe prediction; and =number of Fig. 3. Verification of Blaschko et al. s model 258 / JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006

6 Fig. 4. Verification of Matthys s model Fig. 6. Verification of Smith and Teng s model ACI shear capacity equation As stated in ACI 440, further development of the equation will likely account for the stiffness of not only the laminate but also of the member to which the laminate is bonded. All shear capacitybased models are more scattered in strength prediction, and less calculation work is needed than ihe ACI model; they are potentially unsafe for strength design. Naaman s model is the most conservative but can only be used for preliminary desigo estimate whether or not the requirement of strength increase can be met whehe amount of FRP is not yet specified. From a design perspective, however, none of these models qualifies for yielding safe design strength for the test beams ihe current database. Consequently, it is suggested that the ACI 440 model be revised to be reasonably safe while maintaining accuracy. Modification of ACI 440 Model The ACI 440 model was developed from a number of experiments and assumes the failure is always due to delamination of the FRP reinforcement ACI Assuming that a major crack occurs ihe vicinity of one of the loading points, as shown in Fig. 1 d, the FRP reinforcement withihe shear span of the beam is presumably under shear stress. Such a mechanism can be simulated by a simple shear test to find out the bond capacity of a specific FRP-to-concrete joint. A number of simple shear tests have been conducted by different researchers Chajes et al. 1996; Lorenzis et al and the theoretical solution of ultimate bond capacity was first derived by Täljsten 1996 using fracture mechanics P f = b f 2Gf nt f f 17 where G f fracture energy per unit area of the joint. Dividing q. 17 by b f nt f f, the tensile strain, db,inthe loaded end of the fiber sheet at bond failure is as follows db = 2G f 18 nt f f For different fracture paths withihe FRP concrete bond, fracture energy, G f, has different expressions: if the fracture path Fig. 5. Verification of Jansze s model Fig. 7. Verification of Smith and Teng s model AS shear capacity equation JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006 / 259

7 is within concrete, G f function of concrete strength, and if the fracture occurs ihe interface and no concrete is sheared off from the substrate, the G f could be a constant and related to the specific epoxy used. Assuming that 2Gf is equal to F, Ulaga et al summarized the test results where concrete failure controls, F is given as F f c = 0.30f 0.33 c 19 Savoia et al analyzed the test results from Chajes et al and presented the following equation for F in a mixed failure mode concrete shearing and interfacial failure F f c = 1.02f c 20 Lorenzis et al found that for an interfacial failure, F is no longer a function of concrete strength, and for the epoxy used in their experiment, F turned out to be 1.69, as in q. 21 Fig. 8. Verification of Naaman s model F f c = Considering the effect of concrete strength on a FRP concrete bond, the modified ACI 440 debonding strain equation caake the form of q. 22 Fig. 9. Determination of debonding strain of FRP Case / JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006

8 Fig. 10. Debonding FRP strain versus FRP stiffness intermediate debonding failure Fig. 12. Determination of function F f c based on regression of database db = F f c G nt f f = f c nt f f 22 which can only be applied to predict the FRP failure strain when an intermediate debonding failure occurs. The coefficients,, and will be determined by the database, and the procedure is as follows: 1. Find the debonding strain db with respect to two cases: Case I: Whehe FRP debonding strain db is available in references, use it directly; and Case II: If FRP debonding strains are not available ihe references, use the chart in Fig. 9. The theoretical background of the chart is based on beam theory. As stated by Pham and Al-Mahaidi 2004, beam theory is able to predict the full composite action of beams strengthened with FRP no delamination occurs. Therefore, to find out the debonding FRP strain db, beam theory is also applicable givehe information from the database. 2. Plot the debonding strain db against the FRP stiffness nt f f for each beam ihe same figure, as shown in Figs. 10 and 11. It can be clearly seehat as the FRP stiffness increases, the debonding strain decreases. This trend is found to be the same as the ACI 440 debonding strain equation. Fig. 10 is for wet layup beams with intermediate debonding failure. For concrete strength between 31.5 and 51.2 MPa, function F f c remains 11.3, and for concrete strength of 23.5 MPa, F f c is 8.0. Function G nt f f takes the form of nt f f There is no harm in applying the same procedure for beams with plate end debonding failure to look for a similar trend between FRP stiffness and debonding strain. Fig. 11 is for pultruded plate beams with plate end debonding failure and shows that such a trend also exists for plate end debonding failure. For a concrete strength range of 35 to 56 MPa, F f c is 2.51 and G nt f f takes the form of nt f f Based ohe information obtained in step 2, one can assume that F f c increases with the concrete strength f c, starting from 0, and when f c reaches a certain value, F f c remains a constant. For simplicity, the ascending piece of F f c is assumed to be a straight line. The function F f c is plotted against concrete strength f c in Fig. 12. Based ohe above analysis, the modified FRP debonding strain equations for the ACI design model are as follows: For wet layup beams intermediate debonding failure db = 0.35f nt c f f 0.65, f c nt f f , f c 31.5 MPa For pultruded plated beams plate end debonding failure db = 0.08f nt c f f 0.5, f c nt f f , f c 31.5 MPa where f c is in mega Pascals, t f is in millimeters, and f is in mega Pascals. The complete modified debonding strength equation is Fig. 11. Debonding FRP strain versus FRP stiffness plate end debonding failure where M u = 0.8 k 1 k 2 f bc 2 c + A s s s c d c + A s f y d c + A f f db h c 25 k 1 = 500 c c 26 JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006 / 261

9 performance of other simple strength design models is investigated and compared with that of ACI. The following conclusions can be drawn from this study: 1. ACI gives the most accurate and least scattered strength prediction compared to the shear capacity-based models and the maximum reinforcement ratio method. 2. The maximum reinforcement ratio model is the most conservative, and after some modification, can be used for preliminary strength design. 3. The modified debonding strain equation, including the effect of concrete strength, has more theoretical background than the original equation ihe ACI 440 document and is an improvement over the current ACI 440 document. 4. The modified ACI debonding strain equation can be combined with beam theory to give more accurate design strength, and in addition, this new model yields safe prediction of ultimate beam strength. Fig. 13. Verification of new model and k 2 = c c 27 c = c h c db The determination of the depth of the neutral axis c is given by where Ac 2 + Bc + C =0 A = k 1 f c b B = c A s s + A f f A s f y C = c A s s d c + A f f h 32 The derivation of qs. 25 to 32 can be found in Zhang Compared to the ACI design model, the modified debonding equation is simple to use and has more theoretical background. The modified debonding model is applied to all the beams in the current database, and the verification of the modified ACI model is shown in Fig. 13. Note that the model predicts an average value of 0.8 experimental strength and a range of 0.6 to 1.0 experimental strength. Note also that the modified debonding equation is derived based ohe beams with debonding failure, but it can also predict very well the strength for beams that failed due to concrete cover separation. Thus, the modified ACI debonding strain equation and beam theory can be combined to give the most conservative and accurate design strength. Conclusions Based ohe test information available ihe literature since 1990, a comprehensive database is assembled for an extensive survey of existing studies of the flexural behavior of reinforced concrete RC beams externally strengthened with fiberreinforced polymer FRP composites. The purpose of this database is to verify the design formulas presented in ACI 440.2R-02, Guide for the Design and Construction of xternally Bonded FRP Systems for Strengthening Concrete Structures ACI The Acknowledgment The authors would like to acknowledge the financial support of National Science Foundation Grant CMS Notation The following symbols are used ihis paper: A coefficient; A f cross-section area of FRP reinforcement; A s total cross-section area of tension steel; A s total cross-section area of compression steel; a distance between cutting point of FRP plate to nearest support; a v shear span; a v modified shear span in Jansze s model; B coefficient; b width of beam section; b f width of FRP reinforcement; C coefficient; c depth of concrete compression block; d effective depth of beam section; d c distance between concrete surface and centroid of compression steel; f Young s modulus of FRP; s Young s modulus of steel; F function of concrete compressive strength; f c concrete compressive strength; f f stress in FRP; f s stress iension steel; f s stress in compression steel; f y yielding strength of steel; G function of FRP stiffness; G f fracture energy per unit area of FRP-concrete bond; h depth of beam section; k size effect factor in Blaschko s model; k 1 coefficient; k 2 coefficient; k m FRP strain reduction factor; L clear beam span; L f length of FRP reinforcement; M exp experimental maximum moment; 262 / JOURNAL OF COMPOSITS FOR CONSTRUCTION ASC / MAY/JUN 2006

10 M uns ultimate moment capacity of unstrengthened beam; M max ultimate moment capacity of unstrengthened beam with maximum steel reinforcement ratio; M u predicted ultimate moment of strengthened beam; n number of plies of FRP; P f ultimate load capacity of FRP-concrete bond; thickness of each ply of FRP; t f V c shear capacity of concrete in RC beam alone; V u,end critical shear force at plate end to cause debonding; coefficient; coefficient defined in section of ACI ; coefficient; c cu db f fu s compressive strain in extreme fiber of concrete; ultimate compressive strain of concrete; debonding strain in FRP; FRP strain; ultimate tensile strain of FRP reinforcement; strain iension steel; s strain in compression steel; y yielding strain of steel; coefficient; l equivalent reinforcement ratio in Blaschko and Matthys s models; s reinforcement ratio of tension steel; shear strength; and reduction factor ohe contribution of FRP to beam strength. 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