CAPITULO UNICO

The modification of the natural frequencies is the most sensible way of resolving vibration problems in a construction. However, in order to modify the natural frequencies of a construction significantly, it is very often necessary to carry out extensive structural modifications so as to increase the stiffness of the construction.

Most of the time, a way of increasing the natural frequencies is sought, so that the first mode, and thus all the following modes, are outside the range of risk. In certain cases, when the frequency of the first mode is low, but within the range of risk, and that of the second mode is sufficiently high, it may be advantageous to reduce the frequencies so as to bring the first mode below the range at risk, provided that the second mode remains above that range.

However, this is not very satisfactory. In addition, by reducing the rigidity of the construction, it becomes more flexible and static deflection is increased.

Of the ways of increasing the natural frequencies of the footbridge, the following can be quoted:

4.5.1.1 Vertical vibrations

Let us consider, for example, the case of the vertical vibrations of a deck formed from a steel box girder. If the depth of the box girder can be increased, its stiffness can thus be increased without increasing its mass. It is sufficient to retain the thicknesses of the top and bottom flanges and to reduce the thicknesses of the webs in proportion to the increase in their depth. But, in a great number of cases, it is not possible to increase the depth of the box girder, or not by enough, for functional (height of passage under the footbridge) or architectural reasons. If the thicknesses of the flanges and of the webs of the box girder are increased, the inertia increases in proportion to the thickness, but the self-weight also increases, which reduces the overall effect. In this case there is no solution for increasing the natural frequencies, other than by modifying the static diagram of the construction: creating recesses in the piers, adding cable stays, etc.

In the case of a deck with a solid-core mixed steel-concrete beam, an increase in the thickness of the bottom steel flange is more effective in increasing the frequency of vibration. The self- weight does not increase as quickly, as it comprises a major part, due to the mass of the concrete slab, which does not increase.

In the case of a lattice deck, the inertia varies according to the square of the depth, whereas the section of the flanges (therefore the mass of the flanges) varies inversely to the depth. It is therefore advantageous to increase the depth in order to increase the frequency of vibration. In the case of a concrete deck, an increase in the strength of the concrete enables its modulus, and thus the stiffness of the deck, to be increased, without increasing its mass, but in reduced proportions, as the modulus only increases according to the cube root of the compressive strength. Another traditional way of increasing stiffness without increasing the mass consists of replacing a rectangular section with an I-section. A deck formed by a box girder will thus have a higher frequency of vibration than a deck of the same thickness formed from rectangular beams.

Normal concrete can also be replaced by lightweight concrete in order to reduce the mass (with a slight reduction in stiffness) and thus increase the frequency of vibration.

In the case of a cable-stayed deck, an increase in the sections of the stays generally allows the stiffness to be increased without increasing the mass by much. This solution is effective, but it is not economical, as the quantity of stays has to be increased, with no offset. A fan arrangement of stays is stiffer than a harp arrangement. Taller pylons also lead to an increase in the stiffness without increasing the mass, and thus to an increase in the frequency of vibration.

In the case of a suspended deck, the frequency of vibration increases according to the square root of the tension of the cables divided by the linear density of the cables and of the deck. There is therefore no advantage in simply increasing the section of the cables. Their deflection must, in particular, be reduced.

The vertical stiffness of a deck can also be increased by making the balustrades participate in the stiffness.

4.5.1.2 Torsional vibrations

The torsional vibrations of the deck cause it to move vertically, away from the longitudinal axis of the construction. The values of the torsion vibration frequencies of the deck must also therefore be considered when the vertical vibrations felt by pedestrians are being studied. The

frequency of torsional vibration is proportional to the square root of the stiffness in torsion and inversely proportional to the square root of the polar density of the deck. There is therefore every advantage in designing a deck that is rigid in torsion.

There are several ways of increasing the frequency of the torsion vibrations of a deck. One of these, of course, is to increase the torsional inertia. A box girder deck thus has greater torsional inertia than a deck formed with lateral beams. The torsional inertia can be increased yet more by increasing the cross-sectional area of the box girder. The addition, to a deck formed from filler joists supported by lateral beams, of a bottom horizontal lattice wind-brace connecting the bottom flanges of the two beams, will also allow the torsional stiffness to be increased, but by a smaller amount than a box girder.

In the case of a cable-stayed bridge with lateral suspension, the deck of which is formed using two lateral beams, anchoring the cable stays into the axial plane of the construction (on an axial pylon or to the top of an inverted Y or inverted V pylon), and not into two independent lateral pylons, will allow the torsional frequency to be increased by a factor of close to 1.3 (Ref. [52]). This is what was done for the footbridge for the Palais de Justice in Lyon.

4.5.1.3 Horizontal vibrations

The frequency of horizontal vibration is proportional to the square root of the horizontal stiffness and inversely proportional to the square root of the mass of the deck.

One obvious way of increasing horizontal stiffness is to increase the width of the deck. But at a cost that is no less obvious.

At a given width, one way of increasing horizontal stiffness consists of providing resistant elements on the edges of the deck: for example, two S-section lateral beams can be used, rather than four S/2-section beams equally spaced under the deck.

In the case of cable-stayed or suspended footbridges that are very narrow in relation to their span, lateral cables can be used to stiffen the construction. This is the case for the suspended footbridge at Tours, over the Cher.