Page 296 - Handbook of Civil Engineering Calculations, Second Edition

PRESTRESSED CONCRETE 2.81

one at each end and one at the deflection point above the interior support caused by the
change in direction of the prestressing force.
The horizontal component of the prestressing force is considered equal to the force it-
self; it therefore follows that the force acting at the deflection point has no horizontal
component.
Since the three forces that the tendons exert on the concrete are applied directly at the
supports, their vertical components do not induce bending. Similarly, since the forces at A
and C are applied at the centroidal axis, their horizontal components do not induce bend-
ing. Consequently, the prestressing system having the trajectory shown in Fig. 53 does
not cause any prestress moments whatsoever. The prestress moments for the beam in the
present instance are therefore identical with those for the beam in the previous calculation
procedure.
The second method of analysis is preferable to the first because it is general. The first
method demonstrates the equality of prestress moments before and after the linear trans-
formation where the trajectory is parabolic; the second method demonstrates this equality
without regard to the form of trajectory.
In this calculation procedure, the extremely important principle of linear transforma-
tion for a two-span continuous beam was developed. This principle states: The prestress
moments remain constant when the trajectory of the prestressing force is transformed lin-
early. The principle is frequently applied in plotting a trial trajectory for a continuous
beam.
Two points warrant emphasis. First, in a linear transformation, the eccentricities at the
end supports remain constant. Second, the hypothetical prestressing systems introduced
in step 5 are equivalent to the true system solely with respect to bending stresses; the axi-
al stress F i /A under the hypothetical systems is double that under the true system.

CONCORDANT TRAJECTORY OF A BEAM

Referring to the beam in the second previous calculation procedure, transform the trajec-
tory linearly to obtain a concordant trajectory.

Calculation Procedure:
1. Calculate the eccentricities of the concordant trajectory
Two principles apply here. First, in a continuous beam, the prestress moment M p consists
of two elements, a moment F i e due to eccentricity and a moment M k due to continuity.
The continuity moment varies linearly from zero at the ends to its maximum numerical
value at the interior support. Second, in a linear transformation, the change in F i e is off-
set by a compensatory change in M k , with the result that M p remains constant.
It is possible to transform a given trajectory linearly to obtain a new trajectory having
the characteristic that M k 0 along the entire span, and therefore M p F i e. The latter
is termed a concordant trajectory. Since M p retains its original value, the concordant tra-
jectory corresponding to a given trajectory is found simply by equating the final eccen-
tricity to M p /F i .
Refer to Fig. 50, and calculate the eccentricities of the concordant trajectory. As before,
e a 0.40 in. ( 10.16 mm) and e c 0.60 in. ( 15.24 mm). Then e d 4704(12)/
96,000 0.588 in. ( 14.9352 mm); e b 9792(12)/96,000 1.224 in. ( 31.0896
mm); e e 5024(12)796,000 0.628 in. (15.9512 mm).