Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 77

FRP
rupture

ε cu

f c = 2(0.85f c ´)(ε c /ε o )
1 + (ε c /ε o ) 2

α 1 f cd

C

C

β 1 c

c=c b

ξ

N. A . ( ne u tr a l a x is )

d

h d - β 1 c/2

d - c

Concrete
Crushing

ε fd

T T

A f

b (a) Strain

(b) Equivalent
rectangular
stress-block
for ε c = ε cu

(c) Todeschini
stress-block
approach
for ε c < ε cu

Figure 4.2 Failure mode regions for FRP RC flexural members.

geometry, stiffness and boundary conditions, and the applied loads; the
term “factored” means that the calculated bending moment associated to a
specified loading condition has been amplified by the safety factors related
to the acting loads. M u comes from the structural analysis performed on
the system being studied.

4.5.1 Failure mode and flexural capacity
Balanced failure. Both concrete crushing and FRP rupture are possible
failure modes. When the failure mode is controlled by the simultaneous
occurrence of concrete crushing and FRP rupture, it is termed “balanced
failure.” The neutral axis position for balanced failure, c = c b , can easily be
determined from strain compatibility as follows (Figure 4.2):

ε
ε+ε

cu

c = d

(4.21)

b

cu fu

where ε fu is the design tensile strength of the FRP reinforcement.
The position of the neutral axis corresponding to balanced failure is used
as the basis to establish the member failure mode. When the position of
the neutral axis at ultimate, c, is larger than c b , the failure is controlled by
the crushing of the concrete; conversely, when c is less than c b , the failure is
initiated by rupture of the FRP reinforcement.

COMMENTARY

Traditional steel reinforced flexural members are designed to display con-
crete failure when the strain in the steel has passed its yielding limit. In this
way, the member is said to be “under-reinforced.” Such behavior corresponds
to a ductile failure mode with signs of the incipient collapse in the form of
extensive cracking and large deflections visible on the flexural member.