Civil Engineering Reference
In-Depth Information
5.8 Calculation of the prestress force
5.8.1 General
It will be assumed initially that the design is carried out to Class 1, that is the prestress
must be sized so that no tensile stresses arise under the effects of the applied bending
moments.
5.8.2 Calculation using the concepts of central kern and centre of pressure
For greatest effectiveness, the tendons will be placed as low as possible in the section,
but until the number of tendons has been calculated, their exact arrangement and
proximity to the bottom fi bre is not known. Assume that the centroid of the tendons is
at 0.2 m from the bottom of the section. The essential geometry of the design section
is shown in Figure 5.5 (a).
When the beam is considered to be weightless and subject to no bending moment,
the centre of pressure will be coincident with the centroid of the prestressing tendons.
When an external sagging bending moment
M
, that will tend to compress the top
fi bres is applied to the beam, the centre of pressure moves upwards by a distance of
M/P
, where
P
= prestress force.
For the stress on the bottom fi bre of the beam to be zero when the maximum
bending moment due to external loads is applied to the beam, the centre of pressure
of the prestress must be located at the upper limit of the kern. The distance between
the centroid of the tendons and the upper limit of the kern is:
(
y
b
- 0.2) +
a
t
or
1.472 - 0.2 + 0.488 = 1.76 m
see Figure 5.5 (b).
The distance the centre of pressure may move under the maximum moment at a
section is normally referred to as
l
p
, the prestress lever arm.
l
p
=
M
max
/
P
or
P
=
M
max
/
l
p
.
Thus in this case, where the maximum moment = 16.51 MNm, and
l
p
= 1.76 m, the
prestress force required is
P
= 16.51 / 1.76 = 9.38 MN.
The distance of the prestress tendons from the neutral axis is known as
e
, the
eccentricity of the prestress. Here
e
= 1.472 - 0.2 = 1.272 m. The bending moment
applied to the beam by the prestress force is
Pe
.
It is also necessary to check that when the beam is subjected to the minimum
bending moment there are no tensile stresses on the top fi bre. For this condition to be
satisfi ed the centre of pressure must be at or above the bottom limit of the kern, under
the actions of prestress force plus the minimum bending moment. For this example,
the minimum bending moment exists when the beam is subjected only to its own self
weight, in addition to the prestress.
The distance between the assumed prestress centroid and the bottom kern limit is:
