Displacements of cracked members - Concrete Structures: Stresses and Deformations - page 272

Figure 8.4 A reinforced concrete member in flexure.

where M r is the value of the bending moment that produces rst cracking; W 1

is the section modulus in state 1. Thus, W 1 is calculated for the cross-section
area of concrete plus

fi

α times the cross-section area of steel. f ct is the tensile

strength of concrete in exure (modulus of rupture).

For a bending moment M > M r , cracking occurs and the steel stress along
the reinforcement varies from a maximum value at the crack location to a
minimum value at the middle of the spacing between the cracks. Assuming
that the concrete between the cracks has the same e

fl

ect on the mean strain in
steel as in the case of axial force, Equation (8.10) can be adopted. Thus,

ff

ε sm = (1 − ζ ) ε s1 + ζ ε s2 (8.16)

where

2 2

σ sr
σ s2

M r
M

ζ = 1 − β 1 β 2 = 1 − β 1 β 2 (8.17)

σ s2 are the steel stresses calculated for M r and M , with
assumption that the section is fully cracked.
For spacing between cracks s rm , the width of one crack can be calculated by
Equation (8.14), which is repeated here:

Here σ sr and

w m = s rm ζε s2 (8.18)

The curvature at an uncracked or a cracked section can be expressed in
terms of the bending moment and

fl exural rigidity or in terms of strains as

follows:

M
EI

ψ = (8.19)