Civil Engineering Reference
In-Depth Information
5.24.4 Dispersion of the prestress
The third area of reinforcement is often called equilibrium steel, or secondary bursting
steel. As the prestressing forces spread out and adopt their elastic distribution, transverse
tensile stresses are set up which may require substantial reinforcement.
The slab shown in Figure 5.24 (a) is stressed by two tendons located near its edges.
At some distance from the anchor face, the force of the two anchors will be reacted
by a uniform load spread across the width of the slab. Figure 5.24 (a) also shows the
lines of force, and Figure 5.24 (b) shows the idealised strut-and-tie diagram. It is clear
that a substantial tie is required at the end of the deck. The force in this tie depends on
the angle assumed for the struts. Generally, it is satisfactory to assume that the angle
of the steepest line of force is no greater than 30°. Steeper angles result in excessive
reinforcement with no benefi t. However, the designer must be alert to special situations
where the angle of the struts is affected by the geometry of the member. If cracking
is to be avoided, the tie should consist of reinforcing bars working at not more than
250 MPa or of prestressing tendons.
Another way of understanding the need for this tie is to imagine the slab cut down
its axis. Each half slab then behaves like a column loaded eccentrically by the prestress
anchor, Figure 5.24 (c). The defl ection of the two columns away from each other is
the mechanism causing the slab to crack and the tie force is that necessary to pull the
two halves together again. This model also demonstrates that if the reinforcement was
omitted, the structure would fi nd an alternative equilibrium, albeit with wide cracks.
If the longitudinal prestressing cables are themselves angled in plan, they create
additional transverse forces which must be considered. For instance, if the tendons were
angled to follow the mean lines of force, Figure 5.24 (d), they would generate transverse
forces which would cancel out the dispersion forces shown in Figure 5.24 (c).
Figure 5.25 shows the same slab with the two tendons anchored near the axis. Here
the lines of force spread outwards from the anchors, creating a transverse compression
between the anchors, and a zone of tension some distance into the slab. The main
practical difference to the previous example is that here the tension zone may be
assumed to be quite deep, while the tension in the previous example is concentrated at
the slab end. Consequently, it is likely that existing transverse slab reinforcement that
is underused, such as reinforcement on the compression face of the slab, may make up
a considerable part of the tie force required.
From these simple examples, it should be clear that the arrangement of tendons at
a beam end needs thought to minimise the additional reinforcement required, with its
attendant costs and congestion.
Particular problems can occur when the transverse tensions caused by the spreading
out of the prestress force combine with other tension forces. For instance, in a bridge
built by free cantilever erection, it is common practice to anchor the prestressing
tendons in the webs, Figure 5.26. These tendons give rise to an upwards shear force,
which causes a principal tensile stress that is approximately perpendicular to the line
of the prestressing ducts. As the prestress force spreads out from the anchor, it adds
its transverse tensile stress to the principal tensile stress due to shear, and there is an
enhanced risk of cracking along the weak plane constituted by the presence of the
ducts. In this case, conservative assumptions need to be made in the design of the
dispersion steel in the webs. Several bridges known by the author have cracked during
construction in this way.
