Civil Engineering Reference
In-Depth Information
5.12 Sign convention
A simply supported beam is a very simple example of the calculation of prestress
force and position, as the bending moments due to applied loads are all of the same
sign. However, there are more complex examples, such as continuous beams where
any section is subjected to moments of different sign, or portals and frames where the
meaning of positive and negative moments must be defi ned carefully. For this reason,
it is useful to set out the sign convention that is adopted for this topic, and that is used
widely in the industry:
•
•
Compression is positive, tension negative.
A positive moment causes tension on the bottom fi bre of a beam, or the inside
fi bre for a portal.
The bending moment (
Pe
) due to tendons situated below the neutral axis stresses
the top fi bre in tension. Consequently eccentricity
e
is negative
below the neutral
axis, and positive above. However, the sign for eccentricity is frequently omitted
when there is no ambiguity about its location, such as for statically determinate
beams when the cables are always below the neutral axis.
•
In the author's experience, there is no effective substitute to thinking about the
physical reality, understanding in which direction the centre of pressure moves when
a moment is applied and drawing diagrams similar to those shown in Figure 5.5 to
visualise the limits of the cable zone.
5.13 Arrangement of tendons at mid-span
The prestress force of 9.38 MN now needs to be split up into individual tendons. A
useful rule of thumb for preliminary design is to assume that the working stress for
strand is 1,000 MPa, which means that modern 13 mm strand which has an area of
100 mm
2
,
may be assumed to have a force of 100 kN per strand, and 15 mm strand,
with an area of 150 mm² per strand, 150 kN. (Clearly, for detailed design or for
estimating quantities, it is necessary to carry out a calculation of the prestress losses
and hence to determine more accurately the long-term prestress force in each strand.)
Hence, 9.38 MN represents 63 of the larger strands or 94 of the smaller size. As
standard anchorages are available for 12 or 19 strands, a reasonable choice would be
to adopt 8 tendons each consisting of 12 of the 13 mm strand, providing 96 strands.
These tendons will be housed in ducts with internal/external diameters of 65 mm/
72 mm. (If plastic ducts are adopted they will be somewhat larger.) (See also Chapter
9 for the determination of web thickness and methods of compacting the concrete.)
The force provided will thus be 96 × 0.1 MN = 9.6 MN, slightly more than
required. These 8 tendons need to be arranged at mid-span in such a way that fi lling
the heel of the beam with concrete is facilitated, and then the eccentricity used for this
sizing exercise should be checked. Figure 5.6 shows a possible arrangement that is in
accordance with the rules of the British Standard, and which maximises the eccentricity
of the tendon group. The centroid of the force is 0.136 m from the soffi t. Thus the
value of 0.2 m adopted for the calculation was conservative, and a small reduction
in prestress force would be achieved by increasing the eccentricity adopted for the
