Chapter 6: Two-way Slab-Column Connections
7.7. Analysis o f Flexural Response 1. Effect o f the size-dependent param eter
O ne o f the sign ifican t investigations o f the p ro p o sed fracture m echanics m odel is the use o f the size-dependent param eter (£/). H illerb o rg (1990) conceptually reported the effect o f tw o size-d ep en d en t param eter values such as £/ = 0.002 and 0.008 by constructing a m o m ent-curvature diagram o f a re inforced concrete beam . H ow ever, the theoretical assum ption w as no t valid ated by any m eans, w hich m ay lim it the u se o f the p ro p o sed 213 Y ail J. K im , P .E ng., Ph.D . T hesis
investigations on the effect o f the size-dependent p aram eter (<?/) w ith an additional p aram eter o f s/ - 0.005. T he effect o f the size p aram eter on the brittleness o f the b eam s w as significantly influenced by the am ount o f reinforcem ent. T he b eh a v io u r o f R C B eam A, h av in g a reinforcem ent ratio ip) o f 0.03, indicated various flexural responses depending u p o n the values o f the size param eter, including the load variatio n s after the p eak loads (i.e., 12.6 % , 1.4 % , and 2.3 % d ifferences for the cases o f s/ = 0.002, 0.005, and 0.008, respectively). O n the o ther hand, the effect o f the size-p aram eter to R C B eam B w ith p = 0.015 w as in significant (i.e., 0.2 %, 1.3 %, and 0.9 % differences for £/ = 0.002, 0.005, and 0.008, respectively). T he contribution o f the size p aram eter to the brittlen ess o f PC B eam A w ith A p = 99 m m 2, including initial steel strain o f 0.006, w as insignificant;
how ever, that o f PC B eam B, including additional reinforcem ent o f A f v = 25 m m 2 w ith initial C FR P strain o f 0.008 (i.e., about 50 % o f the ultim ate fibre strain) in addition to the sam e reinfo rcem en t o f PC B eam A, w as consid erab ly affected by the size-p aram eter (i.e., the ductile and b rittle responses in PC B eam s, as show n in Fig. 7.5 (c), (d)). The sensitivity on the size p aram eter increased as the contribution o f reinforcem ent in the beam s increased (i.e., increase o f internal or external reinforcem ent). A lthough the size p aram eter did no t contribute to the brittlen ess o f the low reinforced beam s, as com pared in Fig. 7.5 (a) and (b) for exam ple, the p aram eter contributed to the d evelopm ent o f strains in the concrete and the steel, as show n in Fig. 7.5.
7.7.2. Stress-strain relation o f concrete in bending
A uniax ial stress-strain response o f concrete, based on the constitutive b eh a v io u r o f concrete obtained from concrete cy lin d er tests, is com m only u sed in conventional 214 Y ail J. K im , P .E ng., P h.D . T hesis
C hapter 7: F racture M ech an ics for S trengthening w ith P restressed C F R P Sheets
rein fo rced or p restressed beam design. T he stress-strain relation o f co n crete in com p ressio n is size-dependent, including concrete beam s in bend in g (Flillerborg 1990, K im et al. 2001). T he stress-strain resp o n se o f concrete in each beam , b ase d on the fracture m echanics m odel, is show n in Fig. 7.6. T he post-peak b eh av io u r o f concrete w as significantly affected by the size-d ep en d en t p aram eter (e/). In ord er to p rovide a com parison to the uniax ial response, tw o w ell-k n o w n em pirical equations, developed by H ognestad (1951) and T odeschini (1964) from M ac G reg o r (1997), w ere also presen ted in Fig. 7.6. T he fundam ental differen ce betw een the uniaxial and flexural stress-strain relations is found in th eir strain profiles: the uniax ial stress-strain relation has no strain variations across the cross-section o f a com pression m em ber (i.e., concrete cylinder), but the b ending causes variable profiles (i.e., e= 0 at the neutral axis and e = m axim um at the extrem e fibre). In Fig. 7.6, the u n iax ial resp o n se w as stiffer p rio r to the p eak stress than the lin ear response o btained from the fracture m echanics m odel. T he p o st-p eak b eh a v io u r in bending w as significantly d ifferent from that in uniaxial com pression, as g raphically com pared in Fig. 7.6. A n analogous p attern on the p o st-p ea k b eh a v io u r w as experim entally observed by K im et al. (2001) w ho conducted an experim ental w ork to investigate the b eh a v io u r o f concrete in co m pression under bending. The stress-strain response in bending, based on the size p aram eter o f £/ = 0.005, exhibited the best agreem ent w ith the uniax ial resp o n ses based on H og n estad (1951) and T odeschini (1964).
It m ay be therefore co ncluded that the u niaxial com pression curve o f concrete can be reasonably used as long as the size p aram eter o f Et = 0.005 p rovides the b est p redictions o f the flexural b eh a v io u r o f the re in fo rced and prestressed concrete beam s. N o te that the u ltim ate concrete strains o f the fracture m echanics m odel are m uch h ig h er than the strains obtained by the uniax ial m odels, b ecause the fracture m echanics m odel includes full 215 Yail J. K im , P .E ng., P h.D . T hesis
p redictions o f beam s.
7.7.3. Com parison o f beams in flexure
T he load-strain diagram , based on the selected size-dependent p aram eter o f si = 0.005, is com pared to the experim ental, iterative m odel, and F E A results in Fig. 7.7. A ll o f the responses agreed w ell. B ased on the co m prehensive com parisons, the size-dependent param eter o f ey = 0.005 is reco m m en d ed rath er than the values studied (e/ = 0.002 and 0.008) by H illerborg (1990). T he m om en t-cu rv atu re response is com p ared in Fig. 7.8.
N ote that the u p graded iterative strength-based m odel for RC B eam s A and B im proved the m o m ent-curvature response by ap p roxim ately 5 % , close to the experim en tal data (i.e., com parisons o f the slopes befo re y ielding o f the steel), by co nsidering the tensile contribution o f concrete as com pared to the prelim in ary rep o rt (K im et al. 2006). The fracture m echanics m odel pro v id ed b etter p redictions than the iterative strength-based m odel; fo r exam ple, the average errors o f p rediction on the curvature w ere 15.3 % and 33.7 % by the fracture m echanics and strength-based m odels, respectively, at 60 % o f the experim ental yield m om ents in the case o f reinforced concrete beam s. S im ilar trends w ere observed in the prestressed concrete beam s. A rem arkable difference b etw een PC B eam s A and B w as found in ductility. T he fracture m echanics m odel indicated m ore ductile responses since the strain-softening h ad b een fully included, resu ltin g in a delayed crushing failure o f the concrete beam s. O n the o th er hand, in the iterative m odel, the iterations stopped w h en the concrete strain reach ed the m axim um usab le strain (;;cll = 0.0035) as p er C SA A 23.3-94 (1995). O verall, the F E A pro v id ed the best p redictions; for exam ple, the average error on the pred icted curvature at 60 % o f the yield m om ents w as 216 Y ail J. K im , P .E ng., P h.D . T hesis
C h ap ter 7: F racture M ech an ics for S trengthening w ith P restressed C FR P Sheets
0.7 % in the rein fo rced concrete beam s. N evertheless, the F E A m odels did no t exhibit ductile b eh a v io u r due to num erical d ivergence beyond the p ea k m om ent. T he strength based on the conventional design m eth o d w as also show n in Fig. 7.8. A cracked-sectional analysis and strain com patib ility approaches w ere used to calculate the design strength o f the tested beam s w ith all re d u ctio n factors bein g unity. F lexural responses are sum m arized in T able 7.2. N ote that the cracking loads in the fracture m echanics m odel w ere not calculated since the focus o f this study w as on the b eh av io u r bey o n d the service state as in H illerborg (1990).
7.7.4. Change o f the neutral axis
The size-dependent p aram eter (e/) is a function o f the neutral axis for a beam in bend in g (H illerborg 1990) as show n in E qs. 7.1 and 7.2, and Fig. 7.1. T he p red ictio n o f the v ariation o f a n eu tral axis is an im portant factor for evaluating the size effect such as different depth and length o f rein fo rce d concrete beam s (K im et al. 2001). T he contribution o f the size p aram eter to vario u s geom etric conditions, including d ifferent reinforcem ent effects, has already show n in Fig. 7.5. The change o f the n eutral axis, based on the fracture m echanics m odel w ith a selected size p aram eter o f £;= 0.005, w as com pared to that o btained from o th er analysis m ethods in Fig. 7.9. T he neutral axis w as m easured from the to p fibre o f the beam . T he experim ental neutral axis w as calcu lated based on the strain record in g s w ith an assum ption that plane sections rem ain plane.
T ypical b ilin e ar curves w ere o b serv ed in the conventional design m ethod for a reinforced or prestressed concrete beam . T he fracture m echanics m odel and the iterative m odel show ed good ag reem ent w ith the d esign m ethod, as graphically com pared in Fig. 7.9. As the applied load increased, the d epth to the neutral axis rem ained also constant in the 217 Y ail J. K im , P .E ng., P h.D . T hesis
observed in the prestressed concrete beam s due to the p restressin g effect. Som e discrepancies, despite good agreem ent w ith the trend, w ere o b served b etw een the experim ent and the predictive m odels, as show n in Fig. 7.9. T his is p o ssib ly attributed to the facts that the initial assum ption (i.e., linear strain diagram ) w as no t fully correct w hen the experim ental neutral axes w ere calculated, and the slip o f b o n d o f the strain m easuring device (i.e., displacem en t-ty p e strain gauge transducers), b o n d ed o n the w eb o f the beam s at the sam e level as the rebar, affected the strain readings as w ell.