INTERNAL EQUILIBRIUM

If a vertical cut is taken along the beam shown in Fig. 1.15 and the member is separated into free bodies, one containing the steel tendon and one containing the concrete, the forces acting on the free bodies are as shown in Fig. 1.18(a). The respective forces in the concrete and steel are a compressive force P and a tensile force T.

The location of the compressive force in the concrete must, in order to maintain equilibrium, be at the location of the tendon. This may seem obvious for this simple example, but in Section 1.5 a more general treatment is given, where the location of the concrete compressive force is not so obvious. Thus, considering the composite member of steel and concrete, internal equilibrium of forces is maintained, and there is no net internal moment, the two forces P and T being equal and opposite, and coincident. This is to be expected from consideration of overall equilibrium, since there is no external axial load on the member.

If, as in Fig. 1.16, the member is mounted on simple supports and a uniform load is applied, there is now a bending moment M at midspan, which can be found by considering equilibrium of the beam as a whole. The resultants of the steel and concrete stresses across the midspan section form an internal resisting moment which must balance the bending moment M. Since the force in the tendon is fixed in position, defined by the location of the tendon at midspan, it must be the force in the concrete which moves in order to provide an internal resisting couple with lever arm z (Fig. 1.18(b)). The locus of the concrete force

Figure 1.18 Internal equilibrium.

Figure 1.19

along a member is often referred to as the line of pressure, a concept which will be useful in dealing with statically indeterminate structures (Chapter 11).

If the section were analysed under the action of a force P acting at a lever arm z from the tendon location, the resulting stress distribution would be that shown in Fig. 1.16(b).

The relationship between prestress force, lever arm and applied bending moment described above is valid right up to the point of collapse of a member and will be used to find the ultimate strength of sections in Chapter 5.

Example 1.1 ■■

A simply supported beam with a cross-section as shown in Fig. 1.19 spans 15 m and carries a total uniform load, including self weight, of 50 kN/m. If the beam is prestressed with a force of 2000 kN acting at an

eccentricity of 400 mm below the centroid, determine the stress distribution at midspan.

Maximum bending moment at midspan=50×152/8=1406.3 kNm. Section properties:

Zb=Zt=70.73×106 mm3

Ac=2.9×105 mm2.

Thus the stresses at midspan are: Top:

Bottom:

The resulting stress distribution is shown in Fig. 1.20.

The alternative approach is to consider the location of the line of pressure in the concrete. From Fig. 1.18(b)

z=M/P=1406.3/2000=0.703 m.

Thus the location of the force in the concrete is (703−400)=303 mm above the centroid. The stresses may be determined by considering the

section under the action of an axial load of 2000 kN and a moment of 2000×0.303=606.0 kNm.

The stresses at midspan are thus:

■■

It is useful to consider what happens at the supports. The bending moment there is zero, and therefore the stress distribution is as shown in Fig. 1.21. A large net tensile stress is produced at the top of the beam. It is thus undesirable to have the same eccentricity at the ends of the member as at the midspan section. This can be overcome by reducing the eccentricity near the supports, as described in the next section. For pretensioned members an alternative is to destroy the bond between the steel and the surrounding concrete by greasing the tendons, or by providing sleeves around them in the form of small tubes, or an extruded plastic coating.

The preceding description of the internal equilibrium of a prestressed concrete member still applies when the prestress force is not actually applied within the section.

Figure 1.22(a) shows a beam with a prestressing tendon situated such that, at midspan, it is outside the concrete section. Nevertheless, the free-body diagram for the concrete in one half of the beam, shown in Fig. 1.22(b), indicates that the concrete section would still behave as though a prestressing force were acting on it with an eccentricity e.

The situation shown in Fig. 1.22 is known as ‘external prestressing’ and has been used in several modern bridge projects. In practice the

Figure 1.22 Externally prestressed beam (a) elevation (b) free-body diagram.

tendons are usually contained within the space inside a box-section bridge deck (Fig. 1.5), so that they are protected from the outside environment but still available for inspection and, if necessary, replacement.

For sections near to the end of the beam in Example 1.1, the tendon is still in the same location in the section, but the bending moment is smaller than at the midspan. Thus the value of the lever arm z must change in order to provide a reduced internal resisting moment. The pressure line for a member with no applied load must be coincident with the tendon in order to satisfy internal equilibrium. Once a load is applied, however, the pressure line must move away from the tendon location in order to provide the internal couple necessary to resist the applied bending moment.