Displacements of cracked members - Concrete Structures: Stresses and Deformations - page 278

N r
N

2

ζ = 1 − β 1 β 2 (8.31)

The symbols in Equations (8.28) and (8.29) are de fi ned below:

ε s1 and ε s2 = strain in the bottom steel due to M combined with N , on a section

in states 1 and 2, respectively

σ s2 = stress in the bottom steel due to M and N on a section in state 2

σ sr = stress in the bottom steel due to M r and N r on a section in state 2

It is to be noted that in a fully cracked section, the position of the neutral
axis depends on the eccentricity e M / N , not on the separate values of M and

N . Because e is assumed to be unchanged, ( M / N )

=

= ( M r / N r ) and

σ sr
σ s2

M r N r

N

=

M

= (8.32)

Assuming that the cracks are spaced at a distance s rm , the width of a crack

w m = s rm ζ ε s2 (8.33)

The mean curvature in the cracked member

ψ m = (1 − ζ ) ψ 1 + ζ ψ 2 (8.34)

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

and an axial force N , with the assumptions that the section is in states 1 and 2,
respectively.

ψ 1 and

Example 8.3 Rectangular section subjected to M and N
Calculate the mean curvature for the reinforced concrete section of
Example 8.2 subjected to M

= 250 kN-m (184 kip-ft) combined with an

axial force N 45 kip) at mid-height. All other data are the

same as in Example 8.2. Assuming spacing between cracks, s rm

= − 200 kN ( −

=

300 mm, nd the width of a crack.

The area of the transformed section in state 1

fi

0.336 m 2 . A 1 =

The centroid of A 1 is very close to mid-height; the eccentricity is
considered to be measured from mid-height: