Unified Design Method for Flexure and Debonding in FRP Retrofitted RC Beams

Transcription

1 Unified Deign Method for Flexure and Debonding in FRP Retrofitted RC Beam G.X. Guan, Ph.D. 1 ; and C.J. Burgoyne 2 Abtract Flexural retrofitting of reinforced concrete (RC) beam uing fibre reinforced polymer (FRP) plate i a common way to increae the flexural capacity. There i a lack of rational deign method to determine where the trengthening plate can afely be curtailed. A a reult, retrofitted beam commonly fail by debonding of the FRP plate, which occur well before the target flexural capacity. Debonding i brittle failure and thu ductility of the retrofitted beam need to be enured by debonding prevention. With ufficient ductility the ultimate trength can continuouly increae beyond teel yield with the elatic behaviour of the FRP trengthening plate. Debonding prevention ha been accounted-for empirically in mot deign approache o far. The Global Energy Balance Approach (GEBA) uing fracture mechanic ha been propoed to determine the debonding load of an FRP-RC beam that i affected by the plate curtailment location. The GEBA reult for variou FRP-RC beam can be ummaried uing debonding contour on plot of moment capacity againt the afe plate curtailment location, and the debonding contour contructed in thi way for the beam with the ame ratio of depth to fracture energy are virtually the ame. Thi paper how how GEBA can be incorporated into the deign proce to prevent premature debonding of the FRP plate. The method make ue of the debonding contour and derive from thee implified deign chart that could be made available to deigner. The retrofitting deign conideration and the theoretical background of thi unified deign method are firt explained, followed by the derivation of the conceptual deign chart. Numerically correct deign chart are then contructed for a wide range of 1

2 Introduction FRP plate retrofitting conideration FRP plate can be ued to enhance the capacity of under-reinforced RC beam but ha the effect of reducing ductility. It i relatively eay to decide how much FRP i needed to achieve a certain trength, but it i harder to enure ufficient ductility and retrofitted beam are known to uffer from premature debonding at load below their deign trength. Thi paper how how to deign beam for trength, ductility and debonding prevention. Figure 1 how typical load deflection relationhip for three beam. Curve (A) applie to an untrengthened under-reinforced beam; it ha a relatively long plateau at virtually contant load a the teel yield before the concrete cruhe. Curve (B) how the effect of adding a moderate amount of trengthening; the beam yield at a higher load becaue of the preence of the CFRP, and continue to reit more load after the teel yield becaue the CFRP remain elatic. However, final cruhing of the concrete occur at a lower deflection becaue the neutral axi i deeper. The limiting cae i hown in Curve (C) for a balanced ection, where the concrete cruhe at the ame time a the teel yield. The original ervice load i hown a P u- while the original ultimate load capacity i P u-u. There are everal limit on the amount of flexural trengthening. 1. In order to prevent brittle failure, mot RC beam are under-reinforced. Thu a beam hould not be over trengthened to become over-reinforced, o the correponding balanced ection deign limit the amount of FRP to be added (Curve C). 2. It i undeirable for the beam to undergo platic deformation under normal ervice load to avoid accumulation of platic deformation, o the retrofitted ervice load limit (P -y ) i calculated from the moment that caue firt yield of the original teel 3

3 a place where the flexural train are mall. Strain criteria are thu not relevant. A tudy baed on fracture analyi of concrete, which relate the change in the train energy in the beam and the potential energy of the load to the energy that i releaed in the concrete when the fracture propagate ha been ued to predict when debonding would occur. Thi i known a the Global Energy Balance Approach (GEBA) (Achintha and Burgoyne 2008; Gune et al. 2009; Carpinteri et al. 2009; Guan (2012); Guan and Burgoyne 2012). The key comparion i between the Energy Releae Rate G R with the Fracture Energy of Concrete G f. The energy releae rate i calculated by comparing the energy difference per unit area for the ame beam under both preumed debonded and intact tate with moment-curvature model for beam energy etimate. The particular value that hould be choen for G f will be dicued later but i normally within the range 0.05 to 0.3 N/mm (Shah and Carpinteri 1991; Bazant and Becq- Giraudon 2002). A parametric tudy of GEBA ha been preented by the author in Guan (2012), where it i hown that a debonding contour (DBC) plot can be ued a the PE debonding criterion: G R varie a a function of the loading tate at which debonding occur and a a function of where the fracture take place. G R i determined from M-κ model and the DBC can be plotted on a 2 graph of normalied moment capacity ( M /( f c ' bd ) ) againt curtailment location L / L ), which allow the trength deign and the debonding deign to be combined, a ( cur hear hown in Fig. 2. The DBC i where the G R urface intercept the horizontal plane that i defined by G f. The DBC varie for beam with different depth, reinforcing teel, FRP material etc. A detailed dicuion of the DBC i found in Guan (2012) where it wa hown that a normalied 5

4 define a limiting maximum curtailment length. If a beam i deigned uch that the beam tate point lie above thi dahed line, premature debonding occur before the beam flexural capacity i reached. It i axiomatic that the untrengthened beam wa under-reinforced, o debonding prevention i going to depend heavily on the tenion teel ratio ( ) a well a the FRP ratio ( ). To f trengthen a particular RC beam, i fixed but the deigner can change, wherea when f conidering the deign of new beam alo varie. The effect of thee two change are hown eparately in Fig 4 (a) and (b); adding either type of reinforcement alway move YL to the right (YL1 to YL2), but it ha different effect on the DBC. Increaing f mean that debonding occur more eaily o the debonding line move upward. In contrat, with a larger, debonding i le likely o the debonding contour move downward. The maximum curtailment (L cur-max ) change correpondingly. In order to make the above deign chart cover a wide range of deign cae, ay for the practical range for from 0.4 to 2.0% and f from 0.1 to 1.5%, a very large number of DBC and YL would be needed and the chart would be very complicated. However, it i noted that the critical point i the interection point of the DBC and YL. Any deigned beam tate point below and to the left of thi point i afe, which lead to the impler deign chart decribed below. Two chart will be contructed; one for the pre-yielding tage which will be hown to depend on the amount of teel, and the other will cover the pot-yield condition and will depend on the amount of FRP plate. 7

5 Although the ervice load mut occur while the teel i till elatic, it ha been noted above that the beam may have to carry load above yield in order to provide ufficient reerve of trength. It i eential that the FRP doe not debond before the ultimate trength of the beam i reached. Thi can be accomplihed by mean of a different et of curve, which are contructed in the ame way a the STI curve, thi time varying FRP-ratio track of intercept (FTI) (Fig.6(a)). f only. The reult i the A with the STI curve, one FTI curve replace multiple DBC. For any particular deign, the FTI line alway lie below the DBC line once the applied moment exceed the yield moment (Fig 6(b)). Thu, uing the FTI line to determine where the plate hould be curtailed i conervative for the pot-yield condition. If the reerve of trength that i required beyond yield i high, it i poible that a negative curtailment i predicted a hown in Fig.6(b) (no poitive intercept for the FTI at 1.5M y ). Thi indicate that an anchor i required in addition to the bond. Since increaing make debonding le likely while an increae of f make it more likely, the FTI and STI curve have a imilar trend but different inclination and for a particular combination of (, ) they cro each other at the yielding tate. f Unified deign procedure It i now poible to produce a unified deign procedure, making ue of the STI and FTI curve. 1. It i aumed that an exiting beam i being trengthened, o the area of teel i known, fixing. 9

6 The chart given below are contructed for beam with cylindrical concrete trength f c = 37 MPa, with teel yield trength f y = 530 MPa and Young modulu E = 200 GPa, and with FRP elatic modulu E f = 165 GPa. The STI and FTI are the locu of interection of YL and DBC. Here the YL are contructed auming the tenion teel yield at the train f y /E, the FRP plate behave elatically, and the concrete in compreion follow an unfactored Hognetad-type parabolic tre-train relationhip (Hognetad et al. 1955). When conidering DBC, the mot important parameter i the ratio h/g f (MPa -1 ). It wa hown in Guan (2012) that DBC for beam with the ame h/g f value are virtually identical. Thu the DBC and the reulting STI and FTI chart below apply to all the beam having the ame h/g f. Contruction of detailed STI deign chart The STI curve can give a conervative deign curtailment in the pre-yielding tage, o that they are ued to conider debonding prevention for the ervice tate. Fig. 8(a) how the contruction of a typical STI deign chart that relate to a 400 mm deep beam, with G f taken a 0.15 N/mm, o h/g f = MPa -1. It ha been produced by keeping f contant (at 0.5%) and varying continuouly. The family of thin curved line are the DBC for different value of, while the different vertical dahed line are the correponding YL line. The darker curved line i the STI which goe through the interection of the correponding pair of DBC and YL. The darker olid (vertical) line relate to ρ = 1.0%. One STI curve cover the retrofitted deign of a beam with a certain depth and f value, but variou value. A imilar et of curve i hown in Fig 8(b) for a beam of depth 800 mm and fracture energy 0.15 N/mm, o h/g f = MPa -1 11

7 Contruction of detailed FTI deign chart The FTI curve can be contructed in a imilar way. They give a conervative deign curtailment in the pot-yielding tage and are thu adopted to enure that debonding doe not occur up to the ultimate capacity. They are by keeping contant and varying continuouly. Figure 11(a) relate to a beam of 400 mm deep, with G f taken a 0.15 N/mm; a imilar et of curve i hown in Fig 11(b) for a beam with depth of 800 mm. f By repeating the proce for different, a family of FTI i given to cover all the deign cae for 400 mm deep beam with h/g f value a MPa -1 (Fig 12). A with the STI Band figure, the FTI Band figure are ued to conider deign for beam with the ame h/g f value but different depth. The FTI Band chart for h/g f = MPa -1 i a hown in Fig. 13, where the 1000 and 200 mm depth give the left and right boundarie repectively. The deign procedure uing FTI Band figure i the ame a uing STI Band figure. Significance of the implified deign A pair of STI and FTI Band chart i required for one deign to conider both ervice tate and ultimate tate. Since h/g f typically ha a value in the range MPa -1 for beam ranging from 200 to 1000 mm deep, and having a G f ranging from 0.05 to 0.30 N/mm, in total, about ten pair of STI and FTI Band chart are able to cover mot deign cenario. Thu, the implified deign with STI and FTI Band chart provide a convenient way for practical engineering. 13

8 determine which et of deign chart to ue i h/g f, which i 400/0.15 = 2.67 MPa -1 in thi cae. Fig. 16(a) i a reproduction of Fig. 10, but with the relevant line highlighted. For the ervice tate, a vertical line i firt drawn at M/(f c bd 2 )= which repreent the moment capacity required at ervice after retrofitting. Second, the maximum curtailment for beam with depth of 200 mm and 1000 mm are given by it interection with the STI curve at ρ f = 0.7%, a L cur /L hear = 19.5% and 22.5% repectively. Then the maximum curtailment for the 400 mm beam in the problem i obtained from linear interpolation a L cur /L hear = 20.3%. When conidering the ultimate tate, the FTI band chart are ued for debonding prevention; Figure 16(b) i a reproduction of Fig. 13, again with the relevant line highlighted. Following the ame procedure, the maximum curtailment for the beam i etimated a L cur /L hear = 16.0%. Conequently, the ultimate tate govern the debonding prevention, and the FRP plate hould be curtailed le than 16.0% of the hear pan away from the upport. A explained in Section 2 & 3 above, the curtailment predicted from the STI and FTI band chart are conervative. The exact maximum curtailment obtained from DBC chart (exact deign) i alo provided here (Fig. 17). The maximum value of L cur /L hear given by the exact deign at ervice and ultimate tate are 24.8% (intercepting M - ) and 16.8% (intercepting 1.5M - ) repectively, which are greater than thoe predicted by implified deign above. Furthermore, if the FRP plate i curtailed at 17.8% of the hear pan, it debond when the tenion teel yield. If the FPR plate i curtailed to 16.8%, debonding and cruhing of compreive concrete almot occur imultaneouly, ince in thi cae M u- (= 0.266) i cloe to M - (= 0.264). Thee value all exceed the value of 16.0% given by the implified deign chart. It hould be noted that the DBC curve would not generally be available to deigner, wherea it i uggeted that implified STI and FTI band chart could be provided. 15

9 Thi new method provide a way of incorporating a fracture mechanic approach to debonding in a conventional beam deign. APPENDIX. Flexural capacity deign for the beam in the worked example STEP 1 Aement of original capacity The original deign flexural capacity correpond to the firt yield of the ection, a hown in Fig. 18. The concrete compreion i calculated uing an equivalent rectangular tre ditribution according to ACI318-08, where α 1 and β 1 are taken a 0.85 and 0.77 repectively, and the reult are in Fig. 18(b). From force equilibrium: f ' bx bde bdf (1) c p p y where f / E x d p and y p d x Subtituting value and olving give: x = 77.5 mm; F cc = kn; F c = 17.2 kn; F t = kn. Since the beam i under-reinforced, the contribution of the nominal compreion teel i negligible. The moment capacity of thi untrengthened ection (at yield) i thu: M u u M u y Ft ( d x) ( ) kNm (2) STEP 2 Deign of amount of trengthening The beam i now to be trengthened o that M - = knm, and M -u > 1.5 M - o i knm. After ome trial and error it i found that FRP plate having a cro-ectional area equal to 0.7% of the beam ection (ρ f = 0.7%) will provide the neceary trengthening. Thi amount of FRP i ufficient, by checking the new ervice and ultimate condition, a hown in Fig

10 P -, P -y, P -u -- trengthened ervice/yield/ultimate load capacity P u-, P u-y, P u-u -- untrengthened ervice/yield/ultimate load capacity t f -- Thickne of the FRP trengthening plate t a -- Thickne of the adheive layer -- Strain at tenion teel p -- Strain at compreion teel f -- Strain at the centre of FRP trengthening plate c -- Strain at top concrete fibre -- Tenion teel ratio (A /(bd)) -- Compreion teel ratio (A p p /(bd)) -- FRP trengthening material ratio (A f f /(bd)) -- Curvature Reference ACI Committee 440. (2008). Guide for the deign and contruction of externally bonded FRP ytem for trengthening concrete tructure. Farmington Hill, MI, USA. ACI Committee 318. (2008). Building code requirement for tructural concrete and commentary. Farmington Hill, MI, USA. Achintha, P.M.M., and Burgoyne, C.J. (2008). Fracture mechanic of plate debonding. J. Compo. Contr. 12(4): Bazant, Z.P., and Becq-Giraudon, E. (2002). Statitical prediction of fracture parameter of concrete and implication for choice of teting tandard. Cem. Concr. Re. 32:

11 Lit of Figure Caption Figure 1 Conideration in FRP retrofitting deign Figure 2 Determination of DBC 21

12 Figure 6 (a) Conceptual deign chart with FTI (b) Comparion between FTI deign and the exact deign Figure 7 Comparion of the exact deign, and deign baed on STI and FTI Figure 8 (a) Contruction of STI for beam having h = 400 mm and G f = 0.15N/mm (b) Contruction of STI for beam having h = 800 mm and G f = 0.15N/mm 23

13 Figure 11 (a) Contruction of FTI for beam having h = 400 mm and G f = 0.15N/mm (b) Contruction of FTI for beam having h = 800 mm and G f = 0.15N/mm Figure 12 Numerically correct FTI for 400 mm deep beam (h/g f = MPa -1 ) 25

14 Figure 16 Determination of the (L cur /L hear ) max from STI and FTI band chart Figure 17 Debonding prevention deign from DBC chart (exact deign) Figure 18 (a) Section analyi according to ACI; (b) Original RC beam at jut yield (ultimate tate) 27