Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 100

where ρ f = A f / bd and ρ b represent the FRP reinforcement ratio and the FRP
reinforcement ratio at balanced failure condition, respectively. The latter may
be expressed as A fb / bd, where A fb represents the area of flexural reinforcement
producing balanced failure. In such conditions, the result of compressive and
tensile forces in both concrete and FRP reinforcement can be calculated as

C = 0.85 f ′β cb

c 1 b

(4.47)

TAf = = ρ bdf

f u b fu

b

where c b is the neutral axis depth at balance. Replacing the expression of c b
from Equation (4.21) into Equation (4.47) and equating C = T, the follow-
ing relationship for ρ b can be determined (note that ε fu has been replaced
with f fu / E f ):

′

f
f

ε
ε+ε

c
fu

cu

ρ= 0.85 1 β (4.48)

b

cu fu

For bond-critical sections, a ϕ-factor of 0.55 is recommended.

COMMENTARY

The reasoning behind the development of Equation (4.46) is presented here. If
ρ f ≤ ρ b failure of the concrete cross section is initiated by the rupture of the FRP
reinforcement; if ρ f > ρ b (or better, ρ f > 1.4 ρ b ), failure is controlled by crushing
of the concrete. The corresponding ϕ -factors are 0.55 and 0.65, respectively,
and there is a linear transition between the two failure modes. The value
of 1.4 was adopted to account for the variability of the concrete compres-
sive strength in ensuring the predicted failure (higher than specified concrete
strength would produce FRP rupture). Such value was determined as 1/0.75 =
1.333, rounded to 1.4, where the 0.75 coefficient represented the threshold
set by ACI 318 prior to the 2002 edition of the code to ensure concrete crush-
ing after extensive steel yielding. The 0.65 value of the strength-reduction
factor has been derived from the steel RC tradition to ensure the same struc-
tural reliability of under-reinforced systems [12]. The value of 0.55 is based
on ACI 440 Committee's consensus and represents a further reduction to
penalize the less “ductile” behavior shown by FRP rupture.

Similarly to ACI 318-11, in order to introduce a unified strength-reduc-
tion factor applicable to any member subject to flexure (e.g., columns
for which ρ b is not used), the failure mode is linked to the strain in the
extreme tensile layer of reinforcement (ε f in this study as opposed to ε t in