Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 214

and its rst moment and second moment about an axis through the

reference point O (Fig. 7.1). In this case the section is uncracked and use
of Equation (7.10) is not needed.

fi

(2) When the section is made of plain concrete, without reinforcement,

Equation (7.10) applies only when resultant force is compressive and
situated within the height of the section; that is when,

M

y t

N

y b (7.13)

(3) When the section has two or more reinforcement layers and the normal

force N is tensile, Equation (7.10) applies only when:

Σ I s − y t Σ B s

M Σ

I s − y b Σ B s

N

(7.14)

Σ B s − y t Σ A s Σ B s − y b Σ A s

where Σ A s , Σ B s and Σ I s are sum of cross-sectional areas of reinforcement

layers and their rst and second moments about an axis through the

reference point, O (Fig. 7.1). This inequality gives lower and upper limits
of a range of ( M / N ) within which Equation (7.10) does not apply. The
lower and the upper limits of the range are respectively equal to the third
and the

fi

rst terms in Equation (7.14). When ( M / N ) is equal to the lower
limit or to the upper limit, the neutral axis coincides with the bottom or
top

fi

bres, respectively. In other words, when ( M / N ) is within this range,
Equation (7.10) has no solution for y n that lies between y t and y b . In this
case the resultant tensile force is resisted entirely by the reinforcement;
the strain and stress in any reinforcement layer can be determined by
Equations (2.19) and (2.20), substituting

fi

Σ A s , Σ B s and Σ I s for A , B and I ,

respectively.

(4) When the section has only one reinforcement layer and the normal force

N is tensile, the compression zone is at top fi bre and Equation (7.10)

applies when ( M / N ) y s ; where y s is the y -coordinate of the reinforce-

ment layer. But when ( M / N ) < y s the compression one is at the bottom;
the direction of the y axis must be reversed; the coordinate of the
reinforcement layer becomes (

− y s ) before Equation (7.10) can be applied.

7.4.2 Neutral axis position in a T or rectangular fully
cracked section
The equations of the preceding section are applied below for a T section
reinforced by steel layers A ns and A

bres (Fig.
7.2). The section is also assumed to have one layer of prestress steel A ps
situated anywhere in the tension zone. Presence of A ps simply adds an area

′ ns near the bottom and top fi