Civil Engineering Reference
In-Depth Information
y
b
-
a
b
- 0.2, or 1.472 - 0.867 - 0.2 = 0.405 m.
Under the effect of the self-weight moment, which is 4.74 MNm, the centre of
pressure moves up from the centroid of the tendons by 4.74 / 9.38 = 0.505 m,
Figure 5.5 (c). As this is above the bottom kern limit, no tensile stresses will occur
under combined permanent prestress force and self weight alone. The addition of the
deck fi nishes will further compress the top fl ange.
Figure 5.5 illustrates these calculations graphically. This graphical representation is
extremely useful, and a diagram based on Figure 5.5 should be sketched
to scale
and
referred to whenever this type of calculation is being done. Use of this fi gure is much
more effective than a sign convention for clarifying the direction of movement of the
centre of pressure, in particular when moments of different sign are applied to the
section, or for complicated structures such as portals and frames.
5.8.3 Calculation by direct consideration of stresses
Finding the required prestress force using the direct calculation of stresses operates as
follows. The function of the prestress is to just cancel out the tensile stresses created
on the bottom fi bre of the beam by the maximum applied moment. This tensile stress
is shown in Table 5.1, and is -23.02 MPa.
Assume that a prestress force of 1 MN is applied 0.2 m above the soffi t. The stresses
caused by that force acting alone would be as for the column described above,
P
/
A
±
M
/
z
, where
M
= the moment due to prestress =
Pe
. As the section is not symmetrical,
z
is different for the top and bottom fi bres, and is given in the list of section properties
in
5.3
, together with
A
.
Eccentricity
e
= 1.272 m, and
Pe
= 1 × 1.272 MNm.
The stresses on the extreme fi bres due to unit prestress acting alone are:
To p fi bre
1 / 1.47 - 1×1.272 / 1.274 = 0.680 - 0.998 = -0.318 MPa
Bottom fi bre 1 / 1.47 + 1×1.272 / 0.717 = 0.680 + 1.774 = +2.454 MPa
Consequently, in order to counteract the bottom fi bre tensile stress of -23.02 MPa
the prestress force must be 23.02 / 2.454 = 9.38 MN. This is of course identical with
the former calculation.
The stresses on the extreme fi bres due to 9.38 MN of prestress alone are thus:
To p fi bre
9.38 × -0.318 = -2.98 MPa
Bottom fi bre 9.38 × +2.454 = +23.02 MPa
