Equivalent loads and linear transformation

The equivalent loading due to prestress can generally be found by simple equilibrium of forces. For example, for the externally prestressed bridge illustrated in Fig. 2.11(a), equilibrium of vertical forces gives an upward force at B of:

As the angle, θ, is generally small, this can be approximated as:

(2.7)

It also follows from the small angle that the horizontal force is P cosθ ≈P. Finally, as the forces are eccentric to the centroid at the ends, there are concentrated

Fig. 2.11 Prestressed concrete beam with external post-tensioning: (a) elevation showing tendon; (b) equivalent loading due to prestress

moments there of magnitude (Pcosθ)e2≈Pe2. Hence the total equivalent loading due to

prestress is as illustrated inFig. 2.11(b). It can be shown that the equivalent loading due to prestress is always self-equilibrating.

A parabolically profiled prestressing tendon generates a uniform loading which again can be quantified using equilibrium of vertical forces. A small segment

of such a profile is illustrated inFig. 2.12(a). At point 1, there is an upward vertical component of the prestress force of:

(2.8)

As the angles are small:

(2.9)

where x1is the X coordinate at point 1. This force is upwards when the slope is positive.

Similarly the vertical component of force at 2 is:

(2.10)

where F2is downwards when the slope is positive. The intensity of uniform loading on this

segment is:

(2.11)

The equivalent loads on the segment are illustrated inFig. 2.12(b).

Example 2.3: Parabolic profile

The beam illustrated inFig. 2.13is prestressed using a single parabolic tendon set out according to the equation:

(2.12)

where s is referred to as the sag in the tendon over length l as indicated in the figure. It is required to determine the equivalent loading due to prestress.

Fig. 2.13 Beam with parabolic tendon profile: (a) elevation; (b) equivalent loading due to prestress

Differentiating equation (2.12) gives:

(2.13)

As θAis small:

For a positive slope, the equivalent point load at A would be upwards and of magnitude P(eB−eA−4s)/l. However, in this case, the slope is negative and the force is downwards of

magnitude P(−eB+eA+4s)/l.

As B is on the right-hand side, this force is downwards when positive. Hence, the equivalent point loads are as illustrated inFig. 2.13(b). The intensity of uniform loading is given by equation (2.11) where the second derivative is found by differentiating equation (2.13):

(2.14)

This too is illustrated in the figure.

Example 2.3illustrates the fact that the intensity of equivalent uniform loading due to a parabolic tendon profile is independent of the end eccentricities. A profile such as that illustrated inFig. 2.13(a)can be adjusted by changing the end eccentricities, eAand eBwhile

keeping the sag, s, unchanged. Such an adjustment is known as a linear transformation and will have no effect on the intensity of equivalent uniform loading as can be seen from equation (2.14). This phenomenon is particularly useful for understanding the effect of prestressing in continuous beams with profiles that vary parabolically in each span.

Example 2.4: Qualitative profile design

A prestressed concrete slab bridge is to be reinforced with 10 post-tensioned tendons. The preliminary profile for the tendons, illustrated in Fig. 2.14(a), results in insufficient compressive stress in the top fibres of the bridge at B. It is required to determine an

amendment to the profile to increase the stress at this point without increasing the prestress force.

In a determinate structure, stress at the top fibre can be increased by moving the prestressing tendon upwards to increase the eccentricity locally. This increase in tendon eccentricity, e, increases the (sagging) moment due to prestress, Pe, which increases the compressive stress at the top fibre. However, in an indeterminate structure, the response of a structure to such changes is not so readily predictable. In the structure ofFig. 2.14, increasing the eccentricity locally at B without changing the sags, as illustrated in Fig. 2.14(b), does little to increase the compressive stress at the top fibre at that point. This is because the eccentricity at B has been increased without increasing the tendon sag in the spans. As was seen above, the equivalent uniform loading due to prestress is a function only of the sag and is, in fact, unaffected by eccentricity at the ends of the span. Thus, the change only results in adjustments to the equivalent point loads at A and B and to the equivalent loading near B. As these forces are at or near supports, they do not significantly affect the distribution of bending moment induced by prestress. A more appropriate revision is illustrated inFig. 2.14(c)where the profile is lowered in AB and BC while maintaining its position at the support points. This has the effect of increasing the tendon sag which increases the intensity of equivalent uniform loading. Such a uniform upward loading in a two-span beam generates sagging moment at the interior support which has the desired effect of increasing the top-fibre stress there.

Fig. 2.14 Adjustment of tendon profile: (a) original profile; (b) raising of profile at B by linear transformation; (c) lowering of profile in AB and BC to increase sag

Most prestressing tendons are made up of a series of lines and parabolas and the equivalent loading consists of a series of point forces and segments of uniform loading. This can be seen in the following example.

Example 2.5: Tendon with constant prestress force

A three-span bridge is post-tensioned using a five-parabola symmetrical profile, half of which is illustrated in Fig. 2.15(a). It is required to determine the equivalent loading due to prestress assuming that the prestress force is constant throughout the length of the bridge.

Fig. 2.15 Tendon profile forExample 2.5:(a) partial elevation showing segments of parabola; (b) equivalent loading due to prestress

Similarly, the intensities of loading in the second and third parabolas are respectively:

and

The point load at the end support is the vertical component of the prestress force.

Differentiating the equation for the parabola gives the slope, from which the force is found to be:

All of the equivalent loads due to prestress are illustrated inFig. 2.15(b). Verifying that these forces are in equilibrium can be a useful check on the computations.

Note that in selecting the profile, it has been ensured that the parabolas are tangent to one another at the points where they meet. This is necessary to ensure that the tendon does not generate concentrated forces at these points.