Flexural Strengthening of Beams and Slabs - Strengthening Design of Reinforced Concrete with FRP - page 93

ε cu

C ´
C

d ´

a

ε s

C

A s

d f d

M u

A s

T s

ε s >ε y

T f

A f

ε f

b

FIGURE 5.8 Doubly reinforced cross section with strain distribution and force profile.

5.4.1.2 Flexural Strengthening of a Doubly Reinforced Section
For the ductile concrete crushing failure mode, ε cu of 0.003 is reached after the
yielding of tension reinforcement. The effective strain in the FRP is given by
Equation (10.3) of ACI 440.2R-08:

dc
c

− f

ε= 0.003 −ε ≤ε

(5.21)

fe bi fd

See Figure 5.8.
From strain compatibility:

cd
c

−

ε= 0.003

(5.22)

s

Invoking force equilibrium:
+=+ CC TT s

f

0.85 fb cAf

1 β+ =+

Af Af

(5.23)

c s s s y f fe

f
f

f
f

a
d

f
f

f
f

()

c
fe

y
fe

s
fe

s
fe

ρ= 0.85 −ρ +ρ =ρ +ρ

(5.24) f s s f s

sin gly

Using moment equilibrium:

MTjd = + TjdCjd + u s s f f s

− β

c

− β

c β c 1 1 1

MAfd =φ +φΨ Af d +φ Af − d

(5.25)

u s y f f e f ss

2 2 2

2
φ fb cf , Dividing Equation (5.25) by

f
f

M
fbd

d
d

a f

f

d
d

a f

f

d
d

a

u y

ff fe
cf

s

=ρ −+Ψρ d d −+ρ − d

(5.26) s f s

2 2 2 2

φ 2 2 2

cf cf cf