Temperature effects in three dimensions

When the temperature of a particle of material in a bridge is increased, the particle tends to expand in all three directions. Similarly, when a differential distribution of temperature is applied through the depth of a bridge slab, it tends to bend about both axes. If there is restraint to either or both rotations, bending moment results about both axes as will be illustrated in the following example.

Example 3.12: Differential temperature

The slab bridge ofFig. 3.41is articulated as shown inFig. 3.41(a)to allow axial expansion in both the X and Y directions. However, for rotation, the bridge is two-span

Fig. 3.41 Slab bridge ofExample 3.12: (a) plan showing directions of allowable movement at bearings; (b) section A-A; (c) imposed temperature distribution in deck (section 1−1); (d) imposed temperature distribution in cantilever (section 2–2)

longitudinally and is therefore not able to bend freely. Further, there are three bearings transversely at the ends so that it is not able to bend freely transversely either. The deck and cantilevers are subjected to the differential temperature increases illustrated in Figs. 3.41(c)

and (d)respectively. It is required to determine the equivalent loading and the associated BMD due to this temperature change. The coefficient of thermal expansion is 9×10−6/°C and the modulus of elasticity is 32×106kN/m2.

The specified temperature distributions are different in the cantilevers and the main deck of this bridge. However, for longitudinal bending, the bridge will tend to act as one unit and bending will take place about the centroid. The location of this centroid is:

below the top surface. The bridge deck is divided into parts as illustrated in Fig. 3.42

Fig. 3.42 Cross-section with associated distribution of imposed stress: (a) deck; (b) cantilevers

changes are converted into stresses. Taking moments about the centroid gives a longitudinal bending moment per metre on the main deck of:

The corresponding bending moment per metre on the cantilever is:

These equivalent longitudinal moments are illustrated inFig. 3.43.

The transverse direction is different from the longitudinal in that the cross-section is rectangular everywhere. In the cantilever region, bending is about the centroid of the

Fig. 3.44 Resolution of imposed stress in cantilever into axial and bending components

cantilever. The applied stress distribution is resolved into axial and bending components as illustrated inFig. 3.44. The axial expansion is unrestrained while the bending stress

distribution generates a moment of:

Fig. 3.45 Associated BMDs: (a) plan showing section locations; (b) section B–B; (c) section A–A; (d) section C–sC

In the main deck, the differential distribution is applied to a 0.8 m deep rectangular section giving a moment about the centroid of:

As M3is applied to the outside of the cantilever, only (M4−M3) needs to be applied at the

deck/cantilever interface as illustrated inFig. 3.43. As these applied moments generate distributions of longitudinal and transverse moment, there are two associated BMDs as illustrated inFig. 3.45. As for the previous example, the problem is completed by analysing the slab (by computer) and subtracting the associated BMDs from the solution.

3.6 Prestress

The effects of prestress in bridges are similar to the effects of temperature and the same analysis techniques can be used for both. However, there is one important distinction. An unrestrained change in temperature results in a change in strain only and no change in stress. Prestress, on the other hand, results in changes of both stress and strain. For example, if a beam rests on a sliding bearing at one end, it can undergo axial changes in temperature without incurring any axial stress. However, prestressing that beam does (as is the objective) induce a distribution of stress. When the movements due to prestressing are unrestrained, the stress distributions are easily calculated and analysis is not generally required. However, there are many bridge forms where the effects of prestress are restrained to some degree or other and where analysis is necessary.

Example 3.13: Frame subject to axial prestress by moment distribution

The frame ofFig. 3.16, reproduced here asFig. 3.46(a), is subjected to a prestressing force along the centroid of the deck, ABC, of magnitude, P. It is required to determine the net prestress force in the deck and the resulting BMD. The frame is analysed by moment distribution.

Step 1: The system of fixities used inExample 3.5is used again here as illustrated inFig. 3.46(b). The BMD due to applied ‘loading’ on the fixed structure is zero everywhere as the prestress forces are applied at fixing points.

Step 2: The effects of inducing rotations or translations at the fixing points are the same as forExample 3.5. The normalised versions are presented here in Figs.3.47(a)and(b)(unit discontinuity in moment) and in Figs.3.47(c)and(d)(unit discontinuity in force).

Fig. 3.46 Frame subjected to prestress force: (a) geometry and loading; (b) system of fixities

Fig. 3.47 Effect of displacements at fixing points: (a) normalised BMD due to rotation; (b) normalised forces due to rotation; (c) normalised BMD due to translation; (d) normalised forces due to translation

Fig. 3.48 Effect of prestress force: (a) BMD after correction for force equilibrium; (b) internal forces after correction for force equilibrium; (c) BMD after correction for moment equilibrium; (d) internal forces after correction for moment equilibrium

Step 3: The translational fixity is released first to apply the prestress force. This consists simply of factoring Figs.3.47(c)and(d)by P. It can be seen in the results, illustrated in Figs.

3.48(a) and(b), that equilibrium of forces at A and C is then satisfied. The discontinuity of moment which results is removed by factoring Figs.3.47(a)and (b)by 0.0178Pl and adding to give Figs. 3.48(c)and (d).

Step 4: As force equilibrium inFig. 3.48(d)is satisfied to a reasonable degree of accuracy, no further iteration is deemed necessary.

Example 3.13serves to illustrate the ‘loss’ of prestress force that occurs in a frame due to the restraint offered by the piers. In this example, about 5% of the applied force is lost as shear force in the piers. It is also of importance to note the bending moment that is inadvertently induced by the prestress. Interestingly, this bending moment is independent of the elastic modulus and is therefore unaffected by creep. In a concrete frame, a prestressed deck will continue to shorten with time due to creep. However, the bending stresses induced by this shortening are also relieved by creep with the result that creep has little net effect on the bending moment due to prestress.

Example 3.14: Analysis for eccentric prestressing

The beam illustrated inFig. 3.49is prestressed with a straight tendon at an eccentricity, e, from the centroid with a prestress force, P. It is required to determine the induced

distributions of axial force and bending moment.

The method of equivalent loads is applicable to prestress just as it is to temperature. The only difference is that, as prestress generates stress as well as strain, it is not appropriate to deduct the associated stresses from the analysis results as was necessary in temperature analysis. In this example, the prestress force is applied at an eccentricity to the centroid. This is equivalent to applying a moment alongside the force as illustrated in Fig. 3.50(a). The axial force diagram is clearly as illustrated inFig. 3.50(b). To determine the bending moment diagram, however, is not so straightforward as the beam is not free to lift off the supports at B and C. The analysis to determine the BMD will be carried using moment distribution.

Fig. 3.49 Beam subjected to eccentric prestress force

Fig. 3.50 First stage in equivalent loads method: (a) equivalent loads; (b) axial force diagram due to prestress

Step 1: The beam is fixed as illustrated inFig. 3.51(a). The BMD in the fixed structure due to the equivalent loading is as illustrated inFig. 3.51(b).

Step 2: The moments required to induce unit rotation at B and C are illustrated inFig. 3.52(a), the resulting BMD inFig. 3.52(b)and the normalised BMD inFig. 3.52(c).

Step 3: The discontinuity of bending moment evident inFig. 3.51(b)is removed by

factoring Fig. 3.52(c)by Pe/2 and adding. The result is illustrated inFig. 3.53. As there is no further discontinuity, this is the final BMD due to prestress.

Fig. 3.51 First step in analysis by moment distribution: (a) system of fixities; (b) fixed BMD

Fig. 3.52 Effect of rotation of fixing points: (a) moments required to induce unit rotation; (b) BMD associated with unit rotation; (c) normalised BMD

Fig. 3.53 Final BMD due to eccentric prestress force

It is interesting to note fromExample 3.14that the effect of the tendon below the centroid is to generate sagging moment in the central span. In a simply supported beam, a tendon below the centroid generates hogging moment.

Example 3.15: Profiled tendons

In most post-tensioned bridges the tendons are profiled using a combination of straight portions and parabolic curves. For preliminary design purposes, the actual profiles are sometimes approximated by ignoring the transition curves over the internal supports as illustrated inFig. 3.54. For this beam, it is required to find the BMD due to a prestress force, P.

A parabolic profile generates a uniform loading, the intensity of which can be determined by considering equilibrium of forces at the ends of the parabola. (This was covered in greater detail inChapter 2.) For the parabola in Span AB, the slope is found by differentiating the equation as follows:

Fig. 3.55 Equivalent loading due to profiled tendon: (a) equivalent forces in span AB; (b) summary of all equivalent forces on beam

Fig. 3.56 Equivalent loads method: (a) system of fixities for analysis by moment distribution; (b) equivalent loads and BMDs due to prestress in fixed structure; (c) BMD after

At A, x=0 and the slope becomes −0.08. From Fig. 3.55(a)it can be seen that the vertical component of the prestressing force at A is P sin θ1≈P tan θ1=0.08P. Similarly, at x=l, the

slope is 0.12 and the vertical component of prestress is 0.12P. Hence, equilibrium of vertical forces requires a uniform loading of intensity:

In BC, the vertical components can be found similarly. They are both equal to 0.1P and the intensity of loading is, coincidentally, wBC=0.2P/l. In CD, the intensity is, by symmetry,

wCD=wAB=0.2P/l. Thus, the complete equivalent loading due to prestress is as illustrated in

Fig. 3.55(b). The beam is analysed for this loading using moment distribution.

Step 1: The symmetrical system of fixities is illustrated inFig. 3.56(a)and the associated BMD (Appendix A) is given inFig. 3.56(b).

Step 2: The BMD associated with simultaneous rotations at B and C is identical to that derived forExample 3.14and illustrated inFig. 3.52(c).

Step 3: To remove the moment discontinuity of 0.00833Pl inFig. 3.56(b),Fig. 3.52(c)is factored by this amount and added toFig. 3.56(b). The result is illustrated inFig. 3.56(c). As there is no further discontinuity, this is the final BMD due to prestress in this beam.

Example 3.15serves to illustrate that the effect of profiled prestressing tendons can be quite similar to the effect of self weight in that it applies a uniform loading throughout the beam. The obvious difference is that typical prestress loading is in the opposite direction to loading due to self weight.