ABSTRACT
Figure 12.3 shows a typical parabolic tendon profile for a simple or a continuous span. The tendon exerts on the concrete a uniform load whose intensity is:
where Pm is the absolute value of the mean prestress force, an average of the values before and after the loss due to creep, shrinkage and relaxation; l is the span; f0 is the distance between the chord joining the tendon ends over the supports and the tendon profile at mid-span; this distance is measured in the direction of the normal to the centroidal axis of the beam. The value of Pm is assumed to be constant within each span; at a simply supported end the tendon has zero eccentricity; the value (qprestress/q) is assumed to be the same in all spans; thus, when the spans are unequal, f0 is assumed to vary such that q p r e s t r e s s is the same for all spans (Equation (12.4)). The intensity q of the per¬ manent or quasi-permanent load is assumed to be constant and equal in all spans. With these assumptions, the balanced deflection factor, jjD is the same as the balanced load factor; thus,
(12.3)
(12.6)
prestressing force Pm and due to the sustained load q, respectively. The permanent stress at any fibre due to Pm and q combined is:
where A and I are the area and the second moment of area about centroidal axis of the gross concrete section; y is the coordinate of the fibre considered, measured downward from the centroidal axis (Fig. 12.2). Substituting Equa¬ tion (12.5) to (12.7) in Equation (12.2) and solving for the balanced deflection factor gives:
1 - CTa, 1 + I/(8aqAf0 y)
(12.8)
where CTq is a hypothetical value of the stress that would occur at the fibre considered if q were applied without prestressing and the section is homogeneous noncracked:
(12.9)
aq is a dimensionless coefficient defined as:
The mean value of the prestressing force required is (by Equations (12.4) and (12.5)):
Pm = j l q l 2
' 8f (12.11)
For a simple span, Equation (12.8) is to be applied at the section at midspan to give jD and the result substituted in Equation (12.11) to give P m . In a continuous beam the two equations should be applied for critical sections over the interior supports and at (or close to) mid-spans. The largest Pm thus obtained should be adopted in design.
