Displacements of cracked members - Concrete Structures: Stresses and Deformations - page 276

The strain in steel in state 2 is

ε s2 = ψ 2 y s = 1530 × 10 −6 (0.75 − 0.191) = 856 × 10 −6 .

The width of a crack (Equation (8.14) ) is

w m = 300 × 0.88 × 856 × 10 −6 = 0.23 mm (0.0091 in).

8.5 Curvature due to a bending moment combined
with an axial force
Fig. 8.6 shows a reinforced concrete member subjected to a bending moment
M and an axial force N at the centroid of the transformed uncracked section.
The values of M and N are assumed to be large enough to produce cracking
at the bottom bre.

The use of the equations of Section 8.4 will be extended to calculate the
mean steel strain and the mean curvature in a cracked member subjected to N
and M .
The eccentricity of the axial force is:

fi

e = M / N (8.25)

Our sign convention is as follows: N is positive when tensile and M is
positive when it produces tension at the bottom bre. It thus follows that e

is positive when the resultant of M and N is situated below the centroid of
the transformed uncracked section (Fig. 8.6).
Without change in eccentricity, we can

fi

fi nd the values of N r and the corre-

sponding M r that produce at the bottom fi bre a tensile stress f ct , the strength

of concrete in tension:

1
A

e
W bot

− 1

N r = f ct + (8.26)

1

M r = eN r (8.27)

where A 1 and W 1 are the area and section modulus with respect to the bottom
fi

bre of the transformed uncracked section.
Equation (8.26), of course, does not apply when the bottom bre is in

compression. This occurs when the resultant normal force on the section is
tensile, acting at a point above the top edge of the core of the transformed
uncracked section ( e

fi

( W bot / A ) 1 ). This occurs also when the resultant
normal force is compressive acting within the core ( ( W top / A ) 1

−

e − ( W bot /