Flexural members - Reinforced Concrete with FRP Bars: Mechanics and Design - page 114

Frosch also noted that the crack width at the level of the reinforcement,
w c , depends on the strain level in the reinforcement, ε , and the crack spacing.
Therefore, w c can be written as

= σ wS S

c

(4.68) =ε⋅ ⋅

c c

E

Substituting the expression for S c in Equation (4.68) and assuming the case of
maximum crack spacing ( ψ s = 2.0), Frosch obtained the expression in Equation
(4.66). The coefficient β was introduced to account for the strain gradient
and was defined as the ratio between the strain at the bottom of the member
and the strain at the level of the reinforcement.
Frosch's equation can be rearranged to solve for the permissible bar spacing,
s, as a function of the permissible maximum crack width, w c :

2

+

wE
fk

s

cf

d

2

(4.69) = c

2

β

f b

Given the difficulty and uncertainty in determining crack width, a design
approach based on maximum bar spacing appears more logical. Frosch [23]
also developed a simplified equation that was modified and adopted in ACI
318 as a new crack control equation to evaluate the maximum bar spacing for
the 1999 revision of the building code:

−

40, 000 40, 000

s 15 2.5 2 c (4.70)

= ≤

c

f f s s

−

280 280

in SI units]

[or s

=

380 2.5 c

≤

300 c

f f s s

where f s is the steel reinforcement stress at service level in psi (MPa) and c c
is the clear concrete cover.

Equation (4.70) represents the starting point for the development of the
new proposed model by Ospina and Bakis [25] for indirect flexural crack
control of one-way concrete flexural members. Normalizing the first term
in the left-hand side of Equation (4.70) by the crack width and the elas-
tic modulus ratios, using 0.017 in. and 29,000 ksi (4 mm and 200 GPa)