Flexural Strengthening of Beams and Slabs - Strengthening Design of Reinforced Concrete with FRP - page 112

2.20

y = 0.7815x + 1
R ² = 0.9973 2.00

1.80

1.60

1.40

1.20

FRP e � ective force
Linear (FRP e � ective force)

1.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4

(ρ f f f /ρ s f y ) × ( d f /d)

FIGURE 5.15 Linear correlation of design data derived after the work of Saqan, Rasheed,
and Hawileh (2013).

()
2

β c

1 un

MAfd = − (5.88)

n s y

Dividing M n from Equation (5.87) by M n from Equation (5.88),

− β

c

− β

c

1 1

Af d +ψ Af d

sy f f f f

M
M

2 2 n

= (5.89)

()
2

β c

1 un

n

Af d −

sy

Substituting the steel ratio ρ s = A s / bd and the FRP ratio ρ f = A f / bd in Equation
(5.89) and rearranging, the following expression is derived:

− β

c
d
c
d

− β

c
d
c
d

1

1

1

1

2

M
M

+ψ ρ
ρ

f
f

d
d

2

f

n f f f

= (5.90)

f

()
2

β ()

2

β

1 un 1 un

n sy

1 − 1 −

In the limit when M n approaches M n , the first term of Equation (5.90) approaches 1.
The quotient in the second term

( ) (

)

− β β is found to have very little
variation. Accordingly, Equation (5.90) closely represents a linear relationship. This
relationship may be directly used in design irrespective of the failure mode admitted.
Accordingly, it will be used to determine the FRP reinforcement ratio ρ f directly by
substituting =

c c

d

( )

1

1

1

1 un

2 d 2

f

f 0.9 f in Equation (5.86).

f fu