Chapter 6: Two-way Slab-Column Connections
7.3. Fracture M echanics M odel 1. M aterial m odel o f concrete
7.3.2. Practical applications o f Flillerborg model
A typical sectional analysis w as p erfo rm ed based on the prop o sed com bined stress-strain relation o f concrete, show n in Fig. 7.1 (c), for rein fo rced or p re stressed concrete beam s as w ell as for a prestressed concrete b eam strengthened w ith p re stressed C FR P sheets, as show n in Figs. 7.1 (d), (e), and 7.2. D oubly-rein fo rced concrete beam s w ere no t taken into consideration in this study because such beam s had not been tested. T he practical flexural response o f each beam subjected to load is describ ed as follow s:
202 Y ail J. K im , P .E ng., P h.D . T hesis
C h ap ter 7: F racture M echanics for S trengthening w ith P restressed C F R P Sheets
7.3.2.1. R einforced concrete beam (Kim et al. 2006)
• P hase I:
P hase I indicates th at the stress o f concrete is less than the peak stress ( f \ ) . A linear
Fig. 7.1 (b). B ased on force equilibrium , the resistin g m om ent (M ,) o f a reinforced concrete beam w ith a re ctan g u lar cro ss-sectio n is o btained as:
w here Ts, A s, and E s are the tensile force, cross-sectional area, and elastic m odulus o f steel reinforcem ent, respectively, s cand es are the concrete and steel strains, respectively, e v is the yield strain o f steel, and o v is the yield strength o f steel. T he m u ltip lier (£) is calculated based on the force equilibrium o f a given cross-section.
P hase II represents that the concrete strain exceeds the peak-stress strain (so); thus softening b ehaviour, accom panying the strain localization, is expected. N ote that so = 0.002 is com m only assum ed in this study as reco m m en d ed by Flillerborg (1990). P hase II is fu rth er d ivided into tw o segm ents such as before and after yield in g o f steel reinforcem ent.
response is applied according to the assum ed com bined stress-strain relation, as show n in
' y = A s
• P hase II:
203 Y ail J. K im , P .E ng., P h.D . T hesis
B ased on the force equilibrium o f a rectan g u lar section, the follow ing is obtained:
P .4 ] -
j c bdf'c
w here f ’c is the p eak concrete strength and b is the w idth o f a beam . T he concrete stress at the top fibre o f a beam section (07) and the lo catio n at the peak stress, m easu red from the n eutral axis, are expressed based on Fig. 7.1 (c):
r . 5 ] <7, = / ; 1 ( g . . - g 0) and t)
e x - a x{ex - s ^ ) / f ' c
w here <?/ is the size-dependent p aram eter that contributes to the u p p er lim it o f concrete strain, as show n in Fig. 7.1 (b). S ubstituting Eq. 7.5 into 7.4 gives the follow ing im plicit expression:
[7.6] J - A E S ( i - £ )
b d fl e - £ d )
w here J - — \ £ -
2 1 -
/,
(^“<0 V<r
= 0 (ec > so and es < sy)
e £ ~
204 Y ail J. K im , P .E ng., P h.D . T hesis
C hapter 7: F racture M echanics fo r S trengthening w ith P restressed C F R P Sheets
ii) A fte r yield in g o f steel
Eq. 7.4 is sim plified as the steel yields considering the assum ed elastic-p erfectly -p lastic b eh av io u r o f the steel reinforcem ent, as show n in Fig. 7.3. Thus,
. . c r
[7.7] J ---- :—7- = 0 (sc > so and ss > sy)
T he follow ing resistin g m o m en t (M r) is calcu lated based on the n ecessary term s by solving the above equations.
[7.8] M r b d 2 f c
J c
(Sc > S 0)
N ote that Eq. 7.8 is the analogous exp ressio n obtained by Flillerborg (1990).
7.3.2.2. Prestressed concrete beam
• P hase 1:
Phase I is conceptually the sam e w ith that o f a rein fo rced concrete beam except for the initial prestress effect due to p restressin g strands, as show n in Fig. 7.1 (d). B ased on force equilibrium , the resistin g m om en t (M ,) is obtained:
P-9] M , = T r [ \ - ^ \ j d J ^ A r E r ( s r, + £ ,,) + A r E r ^ - & i
2 0 5 Y ail J. K im , P .E ng., P h.D . T hesis
o f p restressin g strands, respectively, and epe and ece are the steel strain and con crete strain at the level o f steel strands due to prestressin g , respectively. It is assum ed th at the prestressin g strands have not yield ed at this stage.
• P hase II:
Phase II includes the stress-softening b eh a v io u r o f concrete, and consists o f tw o segm ents as in the case o f the reinforced concrete beam . It is assum ed that the stress-strain response o f prestressing strands is bilinear; thus, the transition strain (epv) is called the yield strain herein, Fig. 7.3 (b).
i) B efore yielding o f steel
As in the case o f the rein fo rced concrete beam , the follow ing is obtained:
P .1 0 ] i
\ A E (s
P _ P v p e + £ + £
p p \ p e c e p ,
/ ) M f
w here sp is the strain in the prestressing strand at an arbitrary load level. T he strain (sp) is obtained from the strain diagram as show n in Fig. 7.1 (c):
[7.11]
Substituting Eqs. 7.5 and 11 into Eq. 7.10 yields:
206 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 FR P Sheets
s
= 0
(ec > so and spe+sce+£P < spy)
Iterations are required, until the force equilibrium o f a given section is satisfied, to obtain the neutral axis that also p ro v id es the m u ltip lier (£).
ii) A fte r y ield in g o f steel
(sc > £0 and spy< spe+sce+sp < spil)
w here apy, spu, and Ep ’ are the yield stress, ultim ate strain, and m odulus after yield in g o f steel strands, as show n in Fig. 7.3 (b). T he sam e resisting m om ent (M ,) is o btained as in Eq. 7.8, based on force equilibrium .
1 3 .2 3 . Prestressed concrete beam strengthened with prestressed CFRP sheets
S trengthening usin g C FR P sheets is an effective rehabilitation technique. T he b enefits are briefly m en tio n ed in Sec. 7.2. T he strengthening effect is enhanced by applying p restress to the C FR P sheets such as the active load-carrying m echanism , and im proved serviceability and load-carrying capacity. M ore details on this technique are described elsew here (W ight et al. 2001, E l-H a ch a et al. 2001, K im et al. 2005) and outlined in Ch. 2.
= 0
207 Y ail J. K im , P .E ng., P h.D . T hesis
P hase I is consistent in represen tin g the flexural b ehaviour until the concrete stress exceeds the p eak stress. In the case o f a prestressed concrete beam stren g th en ed w ith prestressed C FR P sheets, the co ntribution o f the C F R P sheets is in clu d ed to calculate the resisting m om ent (M,):
w here 7), Af, and E f are the tensile force, cross-sectional area, and m odulus o f elasticity o f C FR P sheets, respectively, and Sfe and scf e are the initial C FR P strain and concrete strain at the bottom o f the beam due to prestressed C F R P sheets, respectively. It is again assum ed that the p restressin g strands h ave not yielded yet and the C F R P sheets have not ruptured in this phase.
• P hase II:
Phase II o f a prestressed concrete beam strengthened w ith prestressed C FR P sheets is expressed as follow s:
= K M v + V ..)+ A„Er s, (1 - | J r f
+ U , E r f /. + )+ A, E , ( > < ' & - Y U - f
+ £ ) + A pE p£c{ i - ^ / ^ - ^ y
( s c < So)
208 Y ail J. K im , P .E ng., P h.D . T hesis
C h ap ter 7: F racture M echanics for S trengthening w ith P restressed C FR P Sheets
i) B efore yield in g o f steel
T aking into consid eratio n o n the prestress effect in the C FR P sheets, Eq. 7.10 is m odified as follow s:
[7 .'5 ] - ( T 0l A p E p (£ pe + 8 c e + 8 p )+ A f E f { s fe+ e cfe + £ j )
v J c j
= 0
w here Sf is the strain in the C F R P sheets at an arbitrary load level. T he strain {ef) is also obtained in a sim ilar m an n er as in Eq. 7.11:
[ 7 .1 6 ] E f = A - ,
& j n
S ubstituting Eqs. 7.5 and 7.16 to Eq. 7.15 yields,
[ 7 . 1 7 ]
j - T C L + £ ) + ( k _ d
b d fc j 8 \
AfE, ■ ( cr.
bdf'c
8 ^ ~ Eo£)
J c
(w eg, £pe~\"Sce~^Sp ^ £py? and £fe~\~ScfeJr£f ''' £ff)
w here Sf, is the ru pture strain o f C FR P sheets.
209 Y ail J. K im , P .E ng., P h.D . T hesis
J [7.18]
— 1 cr + E b d fc ' pv p A f E f
~ W c
(£ p e + £ J - { h / d - Z )
0 - 4 0
£ \ ~ (£ \ ~ £ ^ ) J c
0
(<■( ':> '-Vj. S p y <:' £pu-, and i:fc—<;cfL, r.j <:' <pu)
The resisting m om ent ( Mr) is obtained based on force equilibrium :
[7.19] M r = b d 2f c £ ( 3 - 2 £ + /7) + y K £ - 7 7 ) ( 3 - £ + /7) + T f ( h ~ d ) (ec >£())
O nce the sum m ation o f C FR P strains {£fe + scfe + sj) exceeds the ultim ate strain o f C FR P sheets (e/„), the contribution o f the C F R P is ignored. N ote that the ru pture o f C FR P sheets before yield in g o f the steel can be avoided by adequate strengthening design as suggested by C S A S806-02 (2002), for exam ple. S ince C FR P rupture before yielding is an undesirab le failure m ode, it is no t considered in this analysis.