Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 209

speci ed period during which creep, shrinkage and relaxation have occurred.

The additional internal forces produced by the live load are assumed to pro-
duce instantaneous changes in stress and strain and also cause cracking
which reduces the e

fi

ective area of the section. The instantaneous changes in
stress and strain are calculated but no time-dependent e

ff

ects are considered.
It is believed that these assumptions are not too restrictive and they represent
most practical situations. Other assumptions adopted in the analysis are
stated in the following section.
If the load which produces cracking is sustained, the e

ff

ects of creep and
shrinkage which occur after cracking are the same as for a reinforced concrete
section without prestressing.

ff

7.2 Basic assumptions
Concrete in the tension zone is assumed to be ine

ff ective in resisting internal

forces acting on a cracked cross-section. The e ective area of the cross-

section is composed of the area of the compressive zone and the area of
reinforcement.
Plane cross-sections are assumed to remain plane after the deformation
and strains in concrete and steel are assumed to be compatible. These two
assumptions are satis

ff

ed by using in the analysis the area properties of a
transformed fully cracked section composed of: A c , the area of the compres-
sion zone and

fi

E s / E c ; E s is the modulus of elasticity of the
reinforcement. E c is the modulus of elasticity of concrete at the time of
application of the load when the analysis is concerned with instantaneous
stress and strain. When creep and shrinkage are considered, E c is the
age-adjusted modulus (see Section 1.11).
Due to creep and shrinkage, the depth of the compression zone changes;
thus, A c is time-dependent. In the analysis of stress and strain changes due to
creep and shrinkage during a time interval, A c is considered a constant equal
to the area of the compression zone at the beginning of the time interval. This
assumption greatly simpli

α A s where α =

fi es the analysis, but involves negligible error.

7.3 Sign convention
A positive bending moment M , produces compression at the top bre (Fig.

7.1(a) ). The axial force, N , is positive when tensile. N acts at an arbitrarily
chosen reference point O. The eccentricity e

fi

= M / N and the coordinate

y of any bre are measured downward from O. Tensile stress and the

corresponding strain are positive. Positive M produces positive curvature

fi

.
The above is a review of some of the conventions adopted throughout this
book (see Section 2.2).

ψ