Displacements of cracked members - Concrete Structures: Stresses and Deformations - page 273

Figure 8.5 Moment versus curvature in a reinforced concrete member in flexure.

or

ε s −

( ε c ) top

d

ψ = (8.20)

where is the curvature; E is the modulus of elasticity; I is the moment of

inertia of the section;

ψ

ε s is the strain in steel reinforcement and ( ε c ) top is the

strain at the extreme bre of the compression zone and d is the distance

between steel in tension and the extreme compression

fi

fi bre (Fig. 8.4). Assume

that cracking has an e ect on the strain in

axial tension. Thus, the mean curvature is expressed in this form:

ff ect on curvature similar to its e ff

ψ m = (1 − ζ ) ψ 1 + ζψ 2 (8.21)

where ψ 2 are the curvatures corresponding to a bending moment M ,

with the assumptions that the section is in states 1 and 2, respectively.
Thus, the coe

ψ 1 and

is employed to interpolate between the curvatures in
states 1 and 2 to obtain the mean curvature. This is illustrated in the moment-
curvature graph in Fig. 8.5. The cracked member has a mean

cient ζ

fl exural rigidity

given by:

M
ψ m

( EI ) m = (8.22)

The curvatures ψ 1 and ψ 2 are given by: