A ll o f the tested beam s show ed a typical ductile failure m ode (i.e., internal reinfo rcem en t yielded first; follow ed by crushing o f concrete or ru pture o f a C FR P sheet). T he flexural beh av io u r o f each b eam is show n in Fig. 2.4 and a sum m ary o f significant values in flexure is given in T able 2.3. A s the initial prestress w as applied to the beam s, upw ard cam ber w as o b served in each b eam and h erein only F E A values are available since the beam s w ere no t instrum ented d uring the p re stressin g operation in the laboratory. Thus, the lo ad-displacem ent resp o n se o f each b eam in Fig. 2.4 starts from zero displacem ent.
[2 .2]
23 Y ail J. K im , P .E ng., Ph.D . T hesis
to the experim ental data.
T he predictions obtained from the F E A show that the load-carrying capacity o f the strengthened beam (B -3) w as h ig h er th an that o f the u n d am aged beam (B -l). T he initial cracking loads (T able 2.3) in the labo rato ry w ere found to be 57 kN , 34 kN , and 54 lcN for the undam aged, dam aged, and stren g th en ed beam s, respectively, w h ich show s that the strengthened b eam alm ost reco v ered the cracking strength o f the u n d am ag ed beam . The cracking load o f the dam aged b eam w as appro x im ately 60 % o f that o f the un d am ag ed control beam . T he co rresponding dow n w ard deflection at cracking o f the strengthened beam w as 82 % o f that o f the un d am ag ed beam . T hese phen o m en a indicate that the strengthened b eam can re co v er the full cracking capacity w ith ap p roxim ately the sam e deflection as the u n d am aged beam , w hich is a b enefit from a serviceability p o in t o f view . A fter the initial cracking loads, the beam s w ere fully unloaded and the residual deflections w ere m easured; h o w ever, this p rocess w as not sim ulated in the FEA . T he residual deflections w ere no t significantly d ifferent from each o th er and this m ay be due to the robust elastic resp o n se o f the beam s.
W hen the beam s w ere loaded again, linear load-d eflectio n responses w ere m onitored up to cracking. A t cracking, the load-d eflectio n curves changed slope and then rem ain ed linear until yield in g o f the internal reinforcem ent. T he strengthened b eam fully reco v ered the level o f the u n d am aged condition; w hereas, the d am aged beam re ach ed only 63 % o f the undam aged condition. T he co rresponding deflection o f the strengthened beam at the
24 Y ail J. K im , P .E ng., Ph.D . T hesis
C h ap ter 2: D uctility and C racking o f PC B eam s w ith P restressed C FR P Sheets
yield load w as slightly less th an that o f the un d am ag ed beam in the experim en tal results.
A s the applied loads reached the p eak loads, the strengthened beam sustained m uch h ig h er loads: alm ost tw o tim es h ig h er w ith resp ect to the u nstrengthened beam ; h o w ever, the corresponding deflectio n w as very sim ilar to that o f the unstren g th en ed beam due to the hig h er p o st-p eak stiffness for the strengthened beam (T able 2.3).
The stiffness befo re the first crack in itiated w as m ainly governed by the con crete strength;
rather th an by the internal reinforcem ent. T his phen o m en o n is clearly show n in the laboratory and F E A results b ecause the stiffness w as approxim ately the sam e for all beam s before cracking. F o r the stiffness b etw een the cracking and y ield loads, a significant increase w as o b served in the strengthened beam w ith respect to the dam aged and u n d am aged beam s in the laboratory; n evertheless, the stiffness o f the strengthened beam did no t reach that o f the u n dam aged beam in the F E A , w hich can be explained by the fact that the stiffness o f the internal rein fo rcem en t m ight be u n d erestim ated in the F E A m odel due to the b ilin e ar constitutive m od ellin g (Fig. 2.2): corresp o n d in g ly the m odel show ed very sim ilar yield loads (98.6 kN and 97.5 kN for the experim ental and F E A yield loads, respectively), but w ith h ig h er deflection (23.7 m m and 28.9 m m , respectively) w ith re sp ect to the experim ental results. T he strengthened beam show ed considerably h ig h er stiffness as com pared to the other tw o beam s after the internal reinforcem ent had yielded. N o te that the strengthened beam has resisted fu rth er loads after the p eak load, as show n in Fig. 2.4 (e). A gradual rupture o f the C FR P sheet w as clearly observed in Fig. 2.4 (e) by sudden load-drops. A plateau w as o b served after com plete rupture o f the C FR P sheet, indicating yield o f the internal reinforcem ent.
25 Y ail J. K im , P .E ng., P h.D . T hesis
referred to K im et al. (2006).
2.7.2. Structural ductility
D uctile b eh a v io u r o f a beam strengthened w ith p re stressed C FR P sheets is o f im portance because o f the natu re o f a sudden and brittle failure o f the C FR P. A flexural response in accordance w ith the level o f prestress to C F R P sheets is significant in a strengthened beam . A s m entioned in the p rev io u s section on ductility, currently several m ethods to quantify the structural ductility h ave been p roposed, and each m eth o d has its ow n positive and n egative aspects (G race et al. 1998, S padea et al. 2001). In this chapter, tw o p opular m ethods w ere chosen to evaluate the ductile resp o n se o f the tested and analytical beam s: energy ductility and the p erfo rm an ce facto r based on deform ability recom m ended by C H B D C (2000). A lthough the p e rfo n n a n c e factor should be u sed for F R P bars or grids, this m eth o d w as selected b ecau se o f a lack o f code p rovisions fo r the externally bonded C FR P sheets.
• D uctility index f.i ~ ^
E uviM
P erform ance factor J = -
^<>.001 •A).(
w here fj,e is the ductility index based on the energy concept, Uyjeid and Uuu are the energy absorbed until the yield and ultim ate loads, respectively, M t,u is the ultim ate m om ent capacity o f the section, (puit is the curvature at M„it, A f o.ooi is the m om en t w hen the
26 Y ail J. K im , P .E ng., P h.D . T hesis
C h ap ter 2: D uctility and C racking o f PC B eam s w ith P restressed C F R P Sheets
m axim um con crete com pressive strain reaches 0.001, and (pco.ooi is the curv atu re at Afo.ooi- D etailed com parisons o f the undam aged, dam aged, and strengthened beam s are m ade in T able 2.4, in cluding the param etric study results o btained from the FEA . A dd itio n al load- deflection responses o f the sim ulated beam s are show n in Fig. 2.5.
In the energy approach, the un d am ag ed beam ( B - l) show ed the low est ductility index, w hich can be explained b y the fact th at a structure m ay h ave m ore ductile b eh a v io u r i f it is m ore lightly reinforced. T he general trend on this energy approach in the strengthening application w ith prestressed C FR P sheets show s that ductility o f a strengthened b eam is inversely prop o rtio n al to the level o f p restress in the C FR P sheets, despite the in significant discrep an cy (i.e., B -3 -4 0 and B -3-30), as show n in T able 2.4. T he energy absorption after the p ea k load o f the strengthened beam w as no t tak en into acco u n t to p rovide a fair co m p ariso n b etw een the strengthened and u n stren g th en ed beam s, considering th eir d ifferent failure characteristics. G race et al. (1998) p ro p o sed a sim ple theoretical m ethod to p re d ic t the unlo ad in g curves after the peak load, bu t the accuracy still rem ains as a co n sid erab le m atter. C orrespondingly, the ductility values in T able 2.4 are conservative fo r system s that allow redistrib u tio n o f load.
E lastic energy acco u n ted fo r 70 % , 42 % , and 31 % o f the energy absorbed fo r the undam aged, dam aged, and strengthened beam s in the laboratory, and the F E A show ed sim ilar results, as show n in Fig. 2.4. T his o bservation indicates that inelastic p o rtio n o f the flexural b eh a v io u r increases as the internal reinforcem ent ratio d ecreases, although the external rein forcem ent has contributed to the inelastic behaviour. T he elastic energy
27 Y ail J. K im , P .E ng., Ph.D . T hesis
20 20
% o f the ultim ate fibre strain, as show n in T able 2.4. T hus, the ductility index increased for these lo w er prestress levels. T hese p h en o m en a support that the stren g th en ed beam w ith prestressed C FR P sheets effectively contributes to the structural ductility by experiencing m ore inelastic behaviour, i f the p restress level in the C FR P sheets is less than 40 % o f the ultim ate fibre strain. T he b eam strengthened w ith a p re stressed C FR P sheet show ed a h ig h er value o f the ductility index w h en com pared to the u n d am ag ed control beam .
F or the d eform ability approach based on C H B D C (2000), the u n d am aged beam show ed the highest p erform ance factor, 27 % h ig h er th an that o f the dam aged beam , and the strengthened beam reco v ered up to 94 % o f the p erform ance facto r o f the u n d am aged beam , based on the laboratory investigation. A cco rd in g to C H B D C (2000), the suggested p erform ance factors are 4.0 and 6.0 fo r a re ctan g u lar section and a T -section, respectively.
N ew h o o k et al. (2002) suggested that the factor should no t depend on the geom etry o f a structure, and recom m ended a value o f 4.0 fo r both rectan g u lar and T -sections. T o satisfy these recom m ended values, the u p p er lim it fo r the level o f p restress for the external strengthening should be ap p roxim ately 25 % o f the u ltim ate fibre strain for the beam s investigated in this study.
2.7.3. Crack patterns
T he flexural cracks expanded tow ards each support as the curvature o f the b eam s increased. T ypical crack patterns o f each test beam are show n in Fig. 2.6, including
28 Y ail J. K im , P .E ng., Ph.D . T hesis
C h ap ter 2: D uctility and C racking o f PC B eam s w ith P restressed C FR P Sheets
com parisons w ith the num erical crack patterns. In the dam aged beam (B -2), no distinct cam b er cracks w ere observed because o f the relativ ely low level o f p restress; how ever, significant ca m b er cracks w ere found in the un d am ag ed ( B - l) and the strengthened (B -3) beam s both in the laboratory and FEA . T he ca m b e r cracks w ere evenly d istributed along the entire span o f the beam s due to the p ure bend in g effect during the p re stressin g operation. A s the beam s w ere subjected to flexural loads, dam age localizatio n w as o bviously inv estig ated in the flexural region o f the dam aged beam ; how ever, m uch less localized dam age w as observed in the strengthened beam , as show n in the num erical crack patterns in Fig. 2.6. Fig. 2.6 (c) show s that the cracks w ere evenly spread along the entire span in the case o f the strengthened beam and the level o f d am age w as low er than observed for the o th er beam s. T his o bservation supports the conclusion that prestressed C FR P sheets can effectively redistribute the applied stress. S im ilar experim ental observations w ere m ade by E l-H acha et al. (2004).
2.7.4. Load-crack response
B efore the initial unloading, three L V D T s w ere installed across cracks at the m ost visible crack locations to m easure the co rresponding crack w idths. T he crack w idths at m id-span w ere also investigated in the F E A m odels. It is w o rth notin g that the crack w id th in the F E A w as obtained from the strain d evelopm ent o f an elem ent located at m id -sp an since there is no p h y sical m eth o d to investig ate the crack w id th in a sm eared crack m odel. T he finite elem ent size should be about 1 to 3 tim es the m axim um aggregate size to obtain a unifo rm crack w id th (K w ak and F ilip p o u 1990: from T eng et al. 2004). In this study, the elem ent lengths w ere 30 to 50 m m , w hich w ere 1.2 to 2.5 tim es the m ax im u m aggregate
29 Yail J. K im , P .E ng., P h.D . T hesis
w here the cracks w ill initiate and th eir m ax im u m w id th are not readily achieved (C H B D C 2000). T ypical com parisons am ong m easurem ents, theory, and F E A are m ade in Fig. 2.7.
T hree apparent phases in the crack w idth pro g ressio n s w ere m onitored. Initially, the crack did no t exist before the tensile stress ex ceeded the m odulus o f rupture. A s further loading occurred, the crack w idth grew linearly w ith in the service state and then beyond the service state, the p ro g ressio n rate increased at a m u ch faster rate. P o st-p eak b ehaviour o f the u n d am ag ed beam w as som ew hat d ifferent in com parison to the strengthened beam s because the C FR P sheet ruptured; thus releasing all the elastic energy stored in the C FRP.
O ne or tw o L V D T s m easured very w ide cracks in the laboratory, bu t such cracks w ere not pred icted in the F E A m odels b ecause the m odels could not include any uncertainties.
A n analytical m odel based on C E B -FIP p re d ic ted relatively w ell the w id er cracks because o f the em pirical nature o f the equation. N evertheless, the accuracy o f the C E B - FIP m odel decreases w hen a beam is externally strengthened w ith C FR P s. This m ay be explained b y the fact that the m odel is based on the strain d ev elopm ent o f internal reinforcem ent, and the contribution o f external rein fo rcem en t (i.e., confining effects and stress redistribution) is not sufficiently acco u n ted for. B ased on the analytical m odel, despite the m in o r deficiency, a param etric study w as co nducted on the level o f p restress in C FR P sheets. T he crack w idth grow th is inversely p roportional to the level o f prestress in the sheet, as show n in Fig. 2.7 (d). T he crack w idths based on the F E A are not show n in the figure since the analytical crack w idths are m ore critical. T he service state is typically 50 % to 60 % o f yield load fo r estim ating crack w idths and co rrespondingly the
30 Y ail J. K im , P .E ng., P h.D . T hesis
C h ap ter 2: D uctility and C racking o f PC B eam s w ith P restressed C FR P Sheets
codes (A A S H T O L R F D 1994; 2003, C SA A 23.3 1995; 2006, C H B D C 2000) reco m m en d the acceptable crack w idths to be from 0.15 m m to 0.40 m m . A cco rd in g to these re com m endations, the b eam strengthened w ith prestressed C FR P sheets adequately satisfied the requirem ents as show n in Fig. 2.7 (c).
Fig. 2.8 show s load -crack depth responses o f each beam . G ood ag reem ent b etw een the laboratory and both the F E A and analytical m odel w as obtained, as g raphically show n in Fig. 2.8 (a) to (c). A fte r initiation o f the cracks, the crack depth pro p ag ated in a stable m an n er and b ecam e alm ost constant b ey o n d the service state. This is explained by the fact that, initially, the internal energy resistin g the crack grow th w as larger th an the energy accelerating crack grow th; h ow ever, as further loading occurred, the resisting energy w as sm aller (i.e., increase o f dam age) and eventually the beam failed w ith constant crack depth. F rom a fracture m echanics p erspective, the rate o f resistan ce is greater than that o f the energy release rate during stable crack grow th; how ever, the beam fails w hen the energy release rate is g reater than th at o f the resistance. A param etric study on the level o f prestress levels in C FR P sheets show ed that hig h er levels o f prestress in the sheet induced h ig h er cracking loads and low er initial crack depths, as show n in Fig.
2.8 (d).
2.7.5. Contribution o f tension in concrete
T ension in concrete after cracking is u su ally ignored in design for convenience, w hich indicates a co nservative design; how ever, the ten sio n in concrete is investigated for better predictio n in this study. T he contribution o f the tensile force in concrete at different load
31 Y ail J. K im , P .E ng., P h.D . T hesis
levels is show n in Fig. 2.9. A ppro x im ately 18 % o f the total tensile force is attributed to the tension in concrete im m ediately after the crack in g load. A lthough the po rtio n o f tension in concrete decreases ex ponentially as the load increases, the tensile force in concrete is not negligible at typical service levels. It w as also found that the level o f p restress in C FR P sheets w as no t correlated w ith the p o rtio n o f tensile force in duced by the concrete; how ever, the p restress level significantly influ en ced the initial cracking load, as show n in Fig. 2.8 (d). It is therefore re co m m en d ed that tension in concrete should be taken into account w hen serviceability is consid ered for im proved p redictions; w hereas, this portion m ay still be ignored in design fo r conservative and conv en ien t design.