Background Knowledge - Strengthening Design of Reinforced Concrete with FRP - page 12

0.35

M n f y

f´ c

f y
f c

0.30

= ρ ρ

1-0.59

bd 2 f ´ c

0.25

0.20

M n
bd 2 f c

0.15

0.10

0.05

0

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

f y
ρ f´ c

FIGURE 2.4 Tests from 364 beams governed by tension. (Courtesy of Portland Cement
Association, 2013 .)

Substituting = a 1 from the force equilibrium, Equation (2.3),

M
bd

f
f

Af
bdf

n y sy

=ρ 1 −

2

f 2*0.85

c c c

M
bd

f
f

f
f

n y y

=ρ −ρ 1 0.59 (See Figure 2.4) (2.9)

2

f

c c c

)

(2.10)

R =ω −ω (1 0.59

where

f
f

M
bd

y

n

ω=ρ , R =

2

f

c c

2

R =ω−ω 0.59 (AnalysisEquation)

(2.11)

2

0.59 ω−ω+ = R 0

ω= −− R

112.36
1.18

(Design Equation)

(2.12)

2.2.4 C onStitutive r elationShipS
Concrete is strong in compression and weak in tension because of structural crack-
ing at a relatively low level of stress. Therefore, concrete needs to be reinforced with