Civil Engineering Reference
In-Depth Information
5.21 Anchoring the shear force
In the equivalent truss model for shear force, described in
3.10
, a proportion of
the main tension reinforcement in the bottom boom of the beam must be anchored
beyond the centre line of the end support to resist the horizontal component of the
last inclined compression strut. This tie must provide a force that is equal to the shear
force. This is equally valid in a prestressed beam, and requires that suffi cient tendons
be anchored close to the bottom fi bre to anchor the net shear force, that is the applied
ULS shear force less the ULS prestress shear force.
In our example, the ultimate net shear force is 1.78 MN. The ultimate strength of
the 12/13 mm tendons adopted is 1,860 × 12 × 100 ×10
-6
/ 1.15 = 1.94 MN. (1.15
is the material factor corresponding to BS5400: Part 4: 1990.) Thus one tendon must
be anchored near the bottom of the end face of the beam. Alternatively, the net shear
force may be anchored using an appropriate section of passive reinforcing steel.
5.22 Defl ections
5.22.1 General
In many structures, the control of defl ections is a critical aspect of their performance.
The cancellation of dead load defl ections is one of the main benefi ts provided by
prestressing. A typical example is the design of fl oors for standard offi ce blocks. The
thickness of the fl oor for a reinforced concrete scheme is governed by its long-term
dead load defl ections. Excessive defl ections give problems to fl oor fi nishes, partitions
and facades. Consequently fl oor slabs have a span/depth ratio that typically does not
exceed 25. Prestressing the slab allows the dead load defl ections to be completely
cancelled, allowing span/depth ratios of 40 and above. In a multi-storey block, this
may allow a lower building height, or more fl oors.
Reinforced concrete bridge decks must also be relatively stocky to limit their dead
load defl ections, while prestressed decks tend to defl ect upwards under permanent
loads and consequently may be much more slender. This will be demonstrated using
the beam described in Figure 5.1. Generally, the slenderness of prestressed concrete
decks is limited only by considerations of vibration under the effect of live loads, by
the strength of the concrete, or by cost, as slenderness is expensive.
5.22.2 Approximate calculation of prestress defl ections
The defl ection of a simply supported beam subject to uniform loads may be calculated
from the formula
= 5
Ml
2
/48
EI
where
M
is the bending moment at mid-span. As
described in
5.18
, a parabolic prestressing cable of force
P
with eccentricity zero at
the beam ends and
e
at mid-span, applies an equivalent uniformly distributed load of
8
Pe
/
L
2
, and a moment at the beam centre of
Pe
. Thus the defl ection due to prestress may
be calculated by substituting
Pe
for
M
in the above equation. The mid-span defl ection
is not sensitive to minor deviations of the prestressing cable away from the parabola. If
the prestress has an eccentricity at the beam ends, an additional defl ection due to end
moments will need to be added.
When it is considered necessary to calculate the defl ections or beam end rotations
due to prestress with more precision, the calculation must be carried out from fi rst
principles, using the exact shape of the centroid of the prestress cables. However, it
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