Structural Concrete - Hassoun

19.4 Analysis of Flexural Members 747
e = K b = Lower kern
e′ = K t = Upper kern
Figure 19.5
Kern points: (a) lower,(b) upper, and (c) central.
compressive stress at the extreme bottom fibers. The stress at the top fibers is
σ t =− F i
A + (F ie)y t
= 0
I
e = K b = lower Kern =
I
(19.17)
Ay t
Similarly, if the prestressing force is applied at an eccentricity e ′ above the centroid such that
the stress at the bottom fibers is equal to 0, that prestressing force is considered acting at the upper
Kern point (Fig. 19.5). In this case the eccentricity e ′ is denoted by K t , and the stress distribution
is triangular, with maximum compressive stress at the extreme top fibers. The stress at the bottom
fibers is
σ b =− F i
A + (F ie ′ )y b
= 0
I
e ′ = K t = upper Kern =
I
(19.18)
Ay b
The Kern limits of a rectangular section are shown in Fig. 19.5.
19.4.3 Limiting Values of Eccentricity
The four stress equations, Eqs. 19.13 through 19.16, can be written as a function of the eccentricity
e for the various loading conditions. For example, Eq. 19.13 can be rewritten as follows:
(f i e)y t
I
σ ti =− F i
A + (F ie)y t
− M Dy t
I I
≤ f ti + F i
A + M Dy t
I
I
(
Fi
F i y t A + M )
Dy t
+ f
I ti
e ≤
≤ f ti
(19.19)