Abstract:

The shifts between the foundations of a long span bridge could cause serious problems to the structure. The effects could be traumatic when large shifts are caused by very strong earthquakes, as in the case of the Akashi Kaikyo Bridge in Kobe, where the January 1995 earthquake caused a permanent displacement between the towers of about 1 m. At that moment the bridge was in construction and only the two pylons had been erected and the main ropes had been positioned.

Slow shifts can be controlled, but they certainly act on the completed structure, interesting pylons, cables and deck. The problem could be very delicate for railway bridges, as in the case of the Messina Strait Bridge.

As a matter of fact, recent studies carried out by ENEA pointed out the existence of relative displacements between the costs of Sicily and Calabria, at the zones where the foundations of the pylons of the Messina Strait Bridge should be constructed. Both relative vertical and horizontal displacements have been pointed out. For the cost in Calabria an uplift rate of about 2 mm/year has been estimated, while Sicilian costs are rising up with the rate of 0.5 mm/year. The difference, equal to 1.5 mm/year, represents the present relative uplift rate. Besides, geodesic space studies by means of GPS systems in South Italy, pointed out the presence of relative horizontal displacements with a rate of about 10.0 mm/year. Of course, in these cases detailed studied are needed to control the displacement of the costs by means a monitoring system.

In this paper the results of a study on the effects of distortions between the foundations, i.e., tower and anchorages, of a suspension bridge are shown. The analysis is carried out by referring to a general model, considering any kind of displacement. In the numerical investigation the most interesting cases are analysed.

This study is relative to very long-span suspension bridges, therefore the following hypotheses are made about the model. The deck is suspended to one or more parallel steel rows. These are  anchored at the soil at their ends and can move with reference to the towers in the direction orthogonal to B-B' and C-C', respectively. The deck can be suspended to the main cables along all the three spans; the beam is fixed in the transversal direction at the pylons, but it can move, with reference to them in vertical and longitudinal directions. Spherical hinges are between the main span and the side spans, so the relative rotations with reference to both the vertical and the horizontal axes are allowed. Simple supports at the end of the girder allow the longitudinal displacements at these sections.

The main span is subjected to a uniform load w, which represent essentially the total dead load, which is much higher than the travelling load in long-span bridges. The side spans are subjected to the loads wa and wb respectively. If the deck is suspended in the main span only, then loads wa and wb are related to the self weight of the cables only. If this is negligible when compared to load w, then cables subject to high axial forces at their ends assume an almost straight line configuration. It is worth nothing that in very long-span bridges the cable self weight is a high ratio of the total load.

The displacements of joints A, B, C and D are independent one of the others and so are the displacements of the corresponding joints A', B', C' and D'. The pylons are able to react, at joints B, B', C and C', with a vertical force and with an horizontal force acting in the directions B-B' and C-C', respectively.

It is worth reminding that, with these hypotheses, the structural behaviour is very similar to the behaviour of a couple of simple cables, the stiffening contribution of the girder being negligible [3, 4]. Therefore, the study is carried out by referring to the model of a three span simple cable and the effects on the girder are evaluated later.

Fig. 1 Suspension bridge model

The general methodology is based on the usual equilibrium equations and on the compatibility condition, which imposes the initial length of the cables to be unique. The relation ships relative to the general case are first shown. Then the most interesting cases, which could happen in practice are analysed in details. These are relative to longitudinal shift of joint D only and longitudinal shift of both C and D, vertical shifts of D and vertical shifts of C and D, transversal shift of joint D and Both C and D, rotation of tower B around its vertical axis. The effects on cable, in terms of sag ratio and horizontal force variations, and on the girder are analysed. In the numerical investigation the typical sag ratio value oif 1/11 has been considered, while the ratio between the tension in th ecable under load w and its Young’s modulus has been assumed equal to 0.05. For the load ratios wa/w and wb/w have been considered the two limit values of 0 and 1. 

Relative large shifts between the foundations in suspension bridge can be very dangerous. Actually the usual values of tectonic displacements are very low, so the structural elements will not suffered dangerous increases of forces. As a matter of fact the behaviour is linear at least up to shifts equal to 0.001, which is suggested as the limit value for safe of very long-span bridges. The variation of the sag ratio and the horizontal force are limited. For higher values, actually not real the variations both in terms of sag ratio and horizontal force are very high. Obviously, the allowable displacement values relative to the tower rotation are much smaller.