Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 259

Substitution in Equation (7.20) and solution by trial or use of Table
7.1 gives the depth of the compression zone (Fig. 7.13(d) ):

c = 0.263 m.

The transformed section to be used here is composed of the area of
concrete in compression plus α times the area of all reinforcements;

where 6.667.

The transformed area, its

α = E s / E c ( t ) = 200/30 =

rst and second moments about an axis
through the reference point O are (Tables 7.3 and 4 may be used for this
purpose):

fi

A = 0.1736 m 2 B = − 25.484 × 10 −3 m 3 I = 13.270 × 10 −3 m 4 .

Changes in axial strain and in curvature produced by M 2 and N 2
(Equation (2.15) with E ref = 30 GPa) are

( ∆ ε O ) 2 = 68 × 10 −6 ( ∆ ψ ) 2 = 1714 × 10 −6 m −1 .

The distributions of strain and stress changes are shown in Fig.
7.13(e).
The changes in stress in the bottom reinforcement and in the
prestress steel are:

200 × 10 9 (68 1714 × 0.547)10 −6 ( ∆ σ ns ) 2 = + = 201.1 MPa (29.17 ksi)

200 × 10 9 (68 1714 × 0.447)10 −6

( ∆ σ ps ) 2 = + = 166.8 MPa (24.19 ksi)

(e) Strain and stress immediately after cracking
The stress diagram in Fig. 7.13(e), obtained by multiplying the strain
diagram in the same fi gure by the value E c ( t ) = 30 GPa, represents the

fi nal stress in the reinforce-

ment may be obtained by summing up the stress values calculated above
in steps (a) to (d). Thus, the stress in the bottom non-prestressed steel is

nal stress in concrete after cracking. The fi

− 5.6 − 16.8 − 12.3 + 201.1 = 166.4 MPa (24.13 ksi).

The stress in the prestressed steel is

1030.5 − 124.5 − 5.8 + 166.8 = 1067.0 MPa (155 ksi).