Strengthening of Columns for Confinement - Strengthening Design of Reinforced Concrete with FRP - page 170

7.2.1 l am anD t eng m oDel
The equations that describe the Lam and Teng (2003a) model are given as follows:

( )

2

EE
f

−

c 2 2

E ε− ε 0 ≤ ε≤ε

cc c c t

4

f =

c

(7.3)

c

f +ε E ε≤ε≤ε

c 2 c t c ccu

f − f

cc c

E = (7.4)

2

ε

ccu

f
EE

2

c

ε= − (7. 5)

t

c 2

f =+Ψ× ×κ f 3.3 f (7.6)

cc c f al

2 nt E

D

ε

f f fe

f = (7.7)

l

where Ψ f = 0.95 is an additional reduction factor added by ACI 440.2R-08, κ a is a
strength efficiency factor to account for the section geometry (κ a = 1.0 for a circular
section, while its computation for a noncircular section is discussed later in this
chapter), and ε fe = κ ε ε fu = 0.586ε fu as averaged by Lam and Teng (2003a). Others
confirmed this reduction by obtaining κ ε from experiments in the range of 0.57-0.61
(Carey and Harries 2005). Pessiki et al. (2001) attributed this strain reduction to
the multiaxial state of stress that the FRP is subjected to in this application. D is
the diameter of the circular section or the diagonal of the noncircular section, as
discussed in Section 7.2.2. According to Lam and Teng (2003a,b), the minimum
confinement ratio ( f l / f c ) should exceed 0.07 in order for the confined axial stress-
strain diagram of circular columns to have an ascending second branch, as seen in
Figure 7.2. For noncircular sections, the ratio ( f l / f c ) is multiplied by (κ a ) with the
product to exceed 0.07 in order to have an ascending second branch. On the other
hand, ACI 440.2R-08 increased this minimum limit to 0.08 to guarantee the out-
come of an ascending curve only while equally applying ( f l / f c ) to circular and non-
circular sections. Abd El Fattah (2012) allowed this ratio ( f l / f c ) to drop below 0.08, in
which case a descending branch of the stress-strain curve is accounted for by using
a Mander constitutive model (Mander et al. 1988). The ultimate axial column strain
is found by the empirical formula suggested by ACI 440.2R-08.

0.45

f
f

ε
ε

l
c

fe
c

ε=ε 1.50 + 12 κ (7.8)

ccu c b

ε≤
ccu

0.01 (7.9)

where κ b is a strain efficiency factor to account for the section geometry (κ b = 1.0 for
a circular section, while its computation for a noncircular section is discussed later).
The maximum axial compressive strain (ε ccu ) is limited by ACI 440.2R-08 to a value