Structural Concrete - Hassoun

746 Chapter 19 Introduction to Prestressed Concrete
19.4 ANALYSIS OF FLEXURAL MEMBERS
19.4.1 Stresses Due to Loaded and Unloaded Conditions
In the analysis of prestressed concrete beams, two extreme loadings are generally critical. The first
occurs at transfer, when the beam is subjected to the prestressing force, F i , and the weight of the
beam or the applied dead load at the time of transfer of the prestressing force. No live load or
additional dead loads are considered. In this unloaded condition, the stresses at the top and bottom
fibers of the critical section must not exceed the allowable stresses at transfers, f ci and f ti , for the
compressive and tensile stresses in concrete, respectively.
The second case of loading occurs when the beam is subjected to the prestressing force after
all losses F and all dead and live loads. In this loaded condition, the stresses at the top and bottom
fibers of the critical section must not exceed the allowable stresses, f c and f t , for the compressive
and tensile stresses in concrete, respectively.
These conditions can be expressed mathematically as follows:
1. For the unloaded condition (at transfer):
• At top fibers,
• At bottom fibers,
σ ti =− F i
A + (F ie)y t
− M Dy t
I I
≤ f ti (19.13)
σ bi =− F i
A − (F ie)y b
+ M Dy b
≥ −f
I I ci (19.14)
2. For the loaded condition (all loads are applied after all losses):
• At top fibers,
• At bottom fibers,
where
σ t =− F A + (Fe)y t
I
σ b =− F A − (Fe)y b
I
− M Dy t
I
+ M dy b
I
− M Ly t
I
+ M Ly b
I
≥ −f c (19.15)
≤ f t (19.16)
F i , F = prestressing force at transfer and after all losses
f ti , f t = allowable tensile stress in concrete at transfer and after all losses
f ci , f c = allowable compressive stress in concrete at transfer and after all losses
M D , M L = moments due to dead load and live load
y t , y b = distances from neutral axis to top and bottom fibers
In this analysis, it is assumed that the materials behave elastically within the working range
of stresses applied.
19.4.2 Kern Limits
If the prestressing force is applied at the centroid of the cross section, uniform stresses will develop.
If the prestressing force is applied at an eccentricity, e below the centroid such that the stress at
the top fibers is equal to 0, that prestressing force is considered acting at the lower Kern point
(Fig. 19.5). In this case e is denoted by K b , and the stress distribution is triangular, with maximum