Flexural Strengthening of Beams and Slabs - Strengthening Design of Reinforced Concrete with FRP - page 108

Q 3

4

3 2max 3 1 2max 2max

BQ = 27 ε +εε−ε−Ψ− +Ψ εε +Ψεε 6 Q 9 Q (1 )3 4 3 Q

2 c 2 c fu 1 c f f c fu f 1 c fu

Q

( )

1

= 9(3 QQ − (1 −Ψ ε+ − ))

3

3 2 Q 1 −Ψ−Ψεε 3

2max

(5.68)

2 1 f c 2 f f cfu

4

Q

( )

1

3 2max 3 2max max2

DQ =− ε− εε +ε−Ψ + 9 3 Q 9 Q 1 6 + Ψεε+Ψεε 3 6

2 c 2 c fu 1 c f f c fu fcfu

4

6
4

2max 2max

+Ψεε −Ψεε 3 Q

fc fu 1 fc fu

Q

( )

1

3 2max max2

D9 (1 = Q −Ψ− ε+ −Ψ +Ψ−ε ε+Ψεε ) Q 3 (1 )6 Q 6

1 f 2 c f f 2 cfu f c u

2

(5.69)

Q

( )

1

3 2max max2 max3 2max

EQ =− 31 −Ψ ε− +Ψ εε −Ψεε −Ψ ε −Ψε ε 3 3 6 6

1 f c f c fu fcfu f fu f cfu

4

7
4

3
4

max2 2max

−Ψεε +Ψεε Q

fcfu 1 fc fu

Q 31

4

( ) ( )

1

3 2max max2 max3

EQ =− 31 −Ψ ε− −Ψ +Ψ εε −Ψεε −Ψ ε 3 1 5

1 f c f f cfu f c u f fu

4

(5.70)

7
4

1
4

2max max2 max3

F =Ψεε +Ψεε +Ψε 3 (5.71)

fc fu fcfu f fu

This equation is not practical to solve in design. There are two alternatives to
use in design. The first one is an approximate solution for d f , and the second one
is an almost exact statistically correlated linear equation (Rasheed and Motto 2010;
Saqan, Rasheed, and Hawileh 2013).

5.4.3.3 Approximate Solution for Singly Reinforced Rectangular Sections
If ε′ c is approximated using the typical value of 0.002, which is accurate for nor-
mal strength concrete, the fifth-degree polynomial is reduced to a cubic polynomial
(Rasheed and Motto 2010).
The α and β expressions will be extremely simplified by the substitution of
ε′ c = 0.002:

2

α= ε− ε
cf

500 83333 (5.72)

cf

0.33 − 41.67 ε

cf

β=

(5.73)

1 − 166.67 ε

cf