Civil Engineering Reference
In-Depth Information
δ =
(1.875 × 100)/30,000 = 0.00625 m.
The new length of the catenary calculated from the force is thus
100.10667 + 0.00625 = 100.11292 m.
However, using Equation 2, the length of a catenary with a sag of 2.5 m should be
100.16667 m. As this is higher than the fi gure calculated from the stress, the guessed
sag is too great.
Try sag = 2.2 m.
T = (0.042 × 100
2
)/8 × 2.2 = 23.86 MN,
the change in stress is (23.86 - 18.75)/1.2 = 4.26 MPa.
The extension of the catenary is 0.01420 m and its new length is thus
100.10667 + 0.01420 = 100.12087 m.
Calculating the length with a sag of 2.2 m gives 100.12907 m. The guessed sag is
still slightly too great.
Try sag = 2.14 m.
T
= 24.53 MN;
change in stress = (24.53 - 18.75)/1.2 = 4.82 MPa; extension = 0.01605 m.
New length of catenary calculated from force is
100.10667 + 0.01605 = 100.12272 m.
The length calculated from geometry is 100.12212 m. The guessed sag is now
slightly too small, but the results for the sag and the force in the ribbon are suffi ciently
accurate.
The effect of creep, shrinkage and temperature change can be found by trial and
error in the same way.
It is important that the sag of the catenary at the end of construction is not too
small; a value of span/40 to span/45 is typical. As the deck shortens due to temperature
drop, shrinkage and creep, the shortening is taken up by a reduction in the sag, by
the extension of the deck under the increased tension, and in some cases by an elastic
response of the abutments to the changes in tension. If a smaller initial sag is chosen, a
greater proportion of the shortening must translate into an increased tension force. At
the limit, for a straight slab with no sag and rigid abutments, all the shortening would
translate into tension following the expression:
strain = stress/
E
.
The difference between the length of the ribbon and the chord length is the main
driver of the behaviour of these bridges. The sensitivity of a concrete stressed ribbon
to sag and to changes in length may be illustrated by the following fi gures. A 100 m
long chord with a sag of 2 m (1/50) has a ribbon length of 100.10667 m, so the ribbon
is 107 mm longer than the chord, while for a sag of 1.43 m (1/70) the length becomes
100.05453 m; the ribbon length would now be only 55 mm longer than the chord.
The shortening of the ribbon due to temperature drop, shrinkage and creep would
typically be of the order of 50 mm for a 100 m length, using mainly precast concrete
elements for the deck, which minimises shrinkage and creep. It should be intuitively
