T he com m ercial softw are pack ag e A N S Y S (2003) w as em ployed to p re d ic t the flexural beh av io u r o f the experim ental beam s. T he com p u tatio n al m odel co n sisted o f tw o different elem ents: com posite 3-D solid elem ents (S O L ID 65) to m odel the concrete, and 3-D spar elem ents (L IN K 8) to rep resen t the steel strands and the C FR P sheet, as typically show n in Fig. 2.1 (d). T he unid irectio n al shell-like C FR P sheet w as sim plified
20 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
to spar elem ents to save com putational effort. N o te that the p restressin g effect o f the C FR P sheet w as applied to the m odel by p ro v id in g an initial strain and the num erical prestressin g o f the steel strands and C FR P sheets w as applied sim ultaneously. Flollaw ay and L eem ing (1999) reported that contribution o f the adhesive to the global structural p erfo rm an ce o f a finite elem ent analysis (F E A ) m odel w as insignificant; correspondingly, the epoxy adhesive w as no t included in the m odel. P erfect bond b etw een m aterials w as assum ed and p rem atu re delam in atio n failure o f the C FR P sheet w as no t consid ered in accordance w ith the laboratory observation. T im e-d ep en d en t properties o f the m aterials (i.e., creep, drying shrinkage, and relaxation) w ere no t included in the m odels. N o n lin ear solutions w ere p erform ed using the m odified N ew to n -R ap h so n m ethod w ith a constant initial stiffness predictor. D etailed descriptions o f the F E A m odel including m aterial m od ellin g are available in K im et al. (2006).
2.6.2. N onlinear iterative analytical model
A sim plified n o n lin ear iterative m odel w as d eveloped to predict the flexure o f tested beam s (i.e., load-strain, and crack w idth and depth progressions). T his analytical m odel is directly co n n ected to the p red ictio n o f crack w id th based on the C E B -FIP code (C E B -F IP
1993). B asic assum ptions for the m odel are:
• L in ear strain d istributions throu g h o u t the cross section o f the beam
• P erfect b o n d b etw e en m aterials; thus slip induced by the adhesive is ignored
• B ilin ear stress-strain response o f p re stressin g strand and linear response o f C FR P
• T hickness o f the C FR P sheets is ignored w h en re sistin g m om ent is calculated
21 Y ail J. K im , P .E ng., P h.D . T hesis
such failure
• T he overall failure is by concrete crushing (ec > ecu = 0.0038) w h ich ty p ically occurs after yield in g o f p restressin g strands or ru pture o f C FR P sheets (8f > EfU = 0.023)
• M icro dam age o f m aterials is no t co n sid ered for crack p rogression
• C rack b ridging and b ranching due to ag gregates are no t sim ulated
A typical cross section at m id-span o f a stren g th en ed b eam is show n in Fig. 2.2, including strain d istributions, actual stresses, and sim plified stresses. To save the m od ellin g effort, the actual com p ressiv e b eh a v io u r o f co n crete is sim plified to equivalent rectan g u lar stress blocks appropriate to the given stress level (C ollins and M itchell 1987):
[2 . 1] a = •
V £ c-0 J
> /P and p 4 - £ c / £c(
6~ 2£„ / £,.
w here a and /? are the co efficients for the eq u iv alen t concrete stress block, and ec and eco are the concrete strain at an arbitrary load and concrete strain at the sp ecified concrete strength { fc ) , respectively. U nlike o th er g enerally u sed m odels, tensile contribution o f concrete w as included in the m odel. A ctual ten sio n stress develops linearly from the neutral axis until the m axim um ten sile stress ind icated b y the m odulus o f rupture (/();
then decreases exponentially to the tip o f a crack; this decrease is called stress softening (S hah et al. 1995). F or this m odel, only the lin ear p o rtio n o f the tension in concrete is tak en into account. F urther, concrete cracks are assum ed to prop ag ate w hen the
22 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
calculated tensile stress exceeds the m odulus o f rupture (C S A A 23.3 1995, 2006). T he load is iterativ ely applied until the concrete crushes (or the C FR P sheet ru p tures) and co rresp o n d in g crack p rogressions are noted. A b rie f flow chart o f the iterative m odel is show n in Fig. 2.3. To com pute the crack w idth, the follow ing equation is u sed (C E B -F IP 1993):
w here w m is the average crack w idth, £Cf is the b o tto m concrete strain, s m is the average crack spacing, c, s, ki, fo, and ch, are the clear concrete cover, spacing b etw een the reinforcem ent, co efficient for bo n d pro p erties o f rebar, coefficient to acco u n t for strain gradient, and re b ar diam eter, respectively, and p ef is the ratio betw een rein fo rcem en t and its effective em bedm ent zone.