Flexural Strengthening of Beams and Slabs - Strengthening Design of Reinforced Concrete with FRP - page 103

Strengthened beam analysis:
It is known that the failure mode is brittle crushing (prior to yielding of steel), so

0.85 fb cAf 1 β= + Af c s s f fe

dc
c

− dc

c

−

f

0.85 fb cAE 1 β= 0.003 + AE 0.003 −ε

c s s f f bi

( ) ( )

2

46.24 c = 313.2 10.19 − c + 100.08 11.8 − c − 26.69 c

2

46.24 c + 439.97 c − 4372.45 = 0

= −

439.97 + 439.97

2

+× × 4 46.24 4372.45

c = 6.07 " a = 5.16"

2 × 46.24

0.003

( )

ε= −= <ε = dc 0.00204 0.00207

s y

c

0.003

( )

ε= dc −−ε= 0.00203 <ε

fe f bi fd

c

4000

ε= 0.083 = 0.00406

fd

3 × 13900000 × 0.04

Thus, the brittle crushing failure is confirmed.

a a

MMAf =φ =φ −+φΨ d Af d −

u n ss f f fe f

2 2

5.16
2

= 0.65 × 3.629,000 × × 0.00204 × 10.19 − + 0.65 × 0.85 × 2.4

5.16
2

× 13,900 × 0.00203 × 11.8 − = 1053.62 + 345.01 = 1398.63 k-in.

So for a large amount of CFRP, the section is strengthened by only 1.3%. This
example is intended to show the ineffectiveness of this failure mode that is not
admissible unless the section is already heavily reinforced to start with. To reinforce
this idea even further, the following design example (Example 5.9) is presented.

Example 5.9: Design

For the section analyzed in Example 5.8, design for the required area of FRP A f
and determine the failure mode involved using the ACI 440.2R-08 approach if

M = 1380.60 k − in

u

M = 1518.66 k − in

u

(i.e., 10% strengthening level)
Use dh f

== 12"