Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 258

σ O ( t ) = − 3.490 MPa γ ( t ) = 9.737 MPa/m.

The decompression forces (Equations (7.40) and (7.41) ) are

3.490 × 10 6 ) 8.734 × 10 −3 × 9.737 × 10 6 ]

N 1 = − [0.2981( − +

= 0.955 × 10 6 N

M 1 = − [8.734 × 10 −3 ( − 3.490 × 10 6 ) + 26.74 × 10 −3 × 9.737 × 10 6 ]

= − 229.9 × 10 3 N-m.

The changes in strain at O and in curvature (Equations (7.45) and
(7.46) ) are

3.490 × 10 6
30×10 9

116 × 10 −6 ( ∆ ε O ) 1 = =

9.737 × 10 6
30×10 9

( ∆ ψ ) 1 = − = − 325 × 10 −6 m −1 .

The changes in stress in the bottom reinforcement and in the
prestressed steel are:

( ∆ σ ns ) 1 = 200 × 10 9 (116 − 325 × 0.547)10 −6 = − 12.3 MPa

200 × 10 9 (116 325 × 0.447)10 −6 ( ∆ σ ps ) 1 = − = − 5.8 MPa.

The changes in strain and in stress distributions in the decompression
stage are shown in Fig. 7.13(c).

(d) Changes in strain and stress in the cracking stage
Internal forces producing cracking (Equations (7.33) and (7.34) ) are

M 2 = 400 × 10 3 − ( − 229.9 × 10 3 ) = 629.9 × 10 3 N-m

N 2 = 0 − 0.955 × 10 6 = − 0.955 × 10 6 N.

Eccentricity of the resultant of M 2 and N 2 measured from the bottom
reinforcement

629.9 × 10 3
−

e s =

0.955 × 10 6 −

0.547 = − 1.206 m.