Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 237

30×10 9 383.2 × 10 3

1

ψ =

31.73 × 10 −3

= 403 × 10 −6 m −1 (10.2 × 10 −6 in −1 ).

Stress at the top fi bre = 30 × 10 9 [ − 87 + 403( − 0.229)]10 −6 = − 5.38 MPa.

Stress in bottom steel 60.8 MPa.

The strain and stress distributions are shown in Fig. 7.7(c).

= 200 × 10 9 [ − 87 + 403 × 0.971]10 −6 =

7.5 Effects of creep and shrinkage on a reinforced
concrete section without prestress
Consider a cross-section cracked due to the application of a positive bending
moment M and an axial tensile (or compressive) force N at an arbitrarily
chosen point O (Fig. 7.1(a) ). The internal forces M and N are assumed to
have been introduced at age t 0 . The instantaneous strain and stress distribu-
tions immediately after application of M and N are assumed to be available
(see Section 7.4). It is required to nd the changes in strain and in stress due

to creep and shrinkage occurring between t 0 and t , where t > t 0 .
In a fully cracked section, only the part of the concrete area subjected to
compression is considered e

fi

ective in resisting the internal forces. Creep and
shrinkage generally result in a shift of the neutral axis towards the bottom of
the section. Thus, to be strictly consistent, the e

ff

ff ective area of the cross-

section must be modi ed according to the new position of the neutral axis.

However, this would hamper the validity of the superposition involved in the
analysis. To avoid this di

fi

ective area of the cracked section is
assumed to be unchanged by creep or shrinkage. The error resulting from this
assumption can be assessed at the end of the analysis and corrected by iter-
ation procedure. But, because the error is usually small, the iteration is hardly
justi

culty, the e ff

ed.
With the above simpli

fi

cation, the analysis for the changes in axial strain
and in curvature and the corresponding stresses can be done by the procedure
given in Section 2.5.2. The resulting equations are given in Section 3.4 and
repeated here.
A reference point O is chosen at the centroid of the age-adjusted trans-
formed section , composed of the area of the compression zone plus

fi

( t , t 0 )

t i mes the area of steel (Figs. 7.8 and 7.9(a) ); where E s / E c ( t , t 0 ), with

E c ( t , t 0 ) the age-adjusted modulus of elasticity of concrete (see Equation
(1.31) ). Creep and shrinkage produce the following changes in axial strain at
O, in curvature and in stresses:

( t , t 0 ) =

∆ ε O = η [ φ ( t , t 0 )( ε O + ψ y c ) + ε cs ( t , t 0 )] (7.26)