The pedestrian bridge in Radonice nad Ohří was built in 1984 and, from a structural standpoint, is a single-span prestressed suspension bridge anchored at its end supports.
A stress ribbon is a special type of suspension structure that does not use hangers. The deck is attached directly to the supporting suspension cable, which gives the structure its characteristic parabolic shape.
The bridge is straight, with a variable longitudinal slope. The free width between the railings is 3,00 m, and the bridge width is 3,80 m. The length of the stress ribbon is 63,36 m. The sag and the length of the span are variable and depend on the load and temperature. The designed deck sag at a temperature of 10°C without variable loads was 1,20 m.The variation in span length is made possible by placing the bridge on supports via elastomeric bearings not connected to the supporting structure, and the span ranges from 57,73 m to 63,36 m.
The long-term monitoring has been made by the team under the guidance of Radim Nečas, Associate professor at Institute of Concrete and Masonry Structures, Faculty of Civil Engineering, Brno University of Technology.
The evaluation has been performed by Kristína Bezručová.
In 2017, a pedestrian bridge in Prague’s Trója collapsed. From a structural standpoint, it was a similar design—a three-span stress ribbon.
Following the accident, several similar structures in the Czech Republic (Nymburk – 3 spans, Kroměříž – 1 span, Radonice nad Ohří – 1 span) were inspected.
The footbridge in Nymburk was demolished after an inspection and replaced with a new one.
The footbridge in Kroměříž was successfully renovated.
Photo: The footbridge in Kroměříž after renovation
On the footbridge in Radonice nad Ohří, the diagnostic survey was accompanied by dynamic measurements, and since 2020, several dynamic tests have been conducted on it in various seasons.
The aim of the long-term measurement was to demonstrate the reliability of the structure. In the case of failure to the load-bearing or prestressing reinforcement, the pressure reserve in the bridge deck would be depleted. This would result in the opening of joints between segments and thus a decrease in the bending stiffness of the structure. The decrease in stiffness would lead to a decrease in the natural frequency of the footbridge. Changes in stiffness would also be accompanied by a change in geometry – an increase in the sag. In the limit case, the action of the structure would change to cable action.
No changes in natural modes and frequencies associated with structural failure were detected by dynamic measurements on the footbridge in Radonice. Nevertheless, significant seasonal behavior related to temperature and the structure’s geometry was observed.
The effect of thermal loads on structures generally depends on their ability to deform. If the structure can deform freely, thermal loads will cause it to deform without changing the internal forces. Conversely, any constraints that prevent free deformation will induce normal or bending stresses. In curved structures, deformation (change in length) can occur through a change in span. Constant heating of the structure along its entire length leads to its elongation and thus an increase in sag. Conversely, constant cooling leads to its shortening and thus a decrease in sag.
In stress ribbons, temperature-induced vertical deformation of the deck results in high bending moments near the supports. Mitigating the bending stress in these areas can be achieved in several ways; one of them is to allow the deck to roll off the saddle.
The abutment thus does not provide permanent support for this part of the deck, but rather support of variable length. This depends on the applied load; in the event of downward deformation of the deck, the structure rests on the abutment. Upward deformation results in the deck being lifted. This contact is most commonly achieved via elastomeric bearings. Using this solution means that the individual spans of the structure do not have a constant span length, but rather a span length that depends on the applied load and the current geometry.
Frequency - loci* are values of natural frequencies for selected natural modes related to a selected parameter affecting the natural frequency value. Typically, parameters such as stiffness, weight of the structure or its part, support type, or parametric change of its geometry can be used. In relation to the parametrization of natural mode shapes and frequencies, two phenomena are mainly mentioned in curved structures – crossing and veering.
Crossing is a phenomenon where frequency - loci* intersect. When the chosen parameter is changed, the natural frequencies approach each other until they cross, where the structure has the same frequency for two natural modes. After reaching the crossing point, further changes to the parameter increase the difference between the frequencies, changing the order of the natural modes without affecting the shape of the mode. Crossing occurs when the analyzed system exhibits high symmetry.
If the symmetry of the structure is broken (differences in mass distribution, stiffness, or asymmetrical geometry), there doesn't occur crossing, just an approaching, known as veering. The change in the order of eigenmodes proceeds continuously in the so-called hybridization region, where the mode shapes exchange their characteristics or mix with each other. In the hybridization range of eigenmodes, both modes exhibit partial similarity to two modes that exchange their order when the parameter is changed.
Crossing and veering are often associated with arched and curved structures, as these structures exhibit a high degree of dependence of their mode shapes on geometric and mechanical parameters.
Following long-term monitoring of the footbridge in Radonice, it was possible to observe the dependence of natural frequencies and modes on temperature, which directly affects the geometry (length and sag) of the structure. An approach of natural frequencies and hybridization of natural modes was observed in the summer months.
ARTeMIS Modal was used for evaluating and tracking the mode shapes in dependence to season of the year and the temperature. The automatic SSI-UPCX method was used for the evaluation of the mode shapes and frequencies.
For tracking the modes, Analysis History module was used. The reference behavior of the structure was set from the measurements made in spring – measurements made in February and March. The reference state represents behavior of the structure by low temperatures, the structure is sufficiently cooled after winter.
The dependence of the natural frequencies for the first 4 mode shapes is shown in the following graphs.
The horizontal axis presents the measurements according to the month of the year (not available measurements in all months) as follows: February, March (three times), June, July, September, October, November.
The change of the order of the modes with one and two halfwaves is visible in the summer months, while increasing the temperature of the structure and its sag accordingly.
The veering phenomena is visible, the change of the mode shapes is not performed clearly while crossing the frequency-loci*, but there is a visible hybridization region in June and September. The first two mode shapes lose their symmetry according the middle of the span.
These mode shapes cannot be tracked by ARTeMIS Modal automatically, when reasonable MAC value (0,8) is set. The lack of symmetry of these hybrid modes causes low similarity (low MAC values) when comparing with behavior in winter/summer.
The typical singular value graphs for summer and winter behavior are shown below, completed with the evaluated mode shapes.
There is also shown the singular value graph for behavior in June, where the proximity of two modes is visible.
Using SSI-UPCX method provides more accurate evaluation of close modes in the hybridization region in comparison with the methods based on FDD.
Dynamic response of concrete footbridges
Monitoring and Diagnostics of Bridges Using Operational Modal Analysis
Cable-stayed bridges, diagnostic survey of stays
The phenomenon of veering in the assessment of r.c. arch bridges
Frequency-veering and mode hybridization in arch bridges
In-plane vibrations of slightly curved beams: two models compared
Palle Andersen
Managing Director, Ph.D.
Niels Jernes Vej 10
9220 Aalborg East, Denmark
