Dynamically Scaled Suspension Bridge Model

The New Little Belt (Lillebelt) Bridge, opened in 1970, crosses the Little Belt strait in Denmark. It has a 600-meter main span with 240-meter side spans, 113-meter tall towers, and a distance between the main cables of 28 meters. 

Operational Modal Analysis of the Full-Scale Structure

Physical measurements of the Lillebelt Bridge were collected by Silas Christensen, Michael Andersen, and Anders Brandt, and published at the 2018 International Modal Analysis Conference (IMAC). The research team instrumented the structure with 45 geophones and collected velocity measurements with a sample rate of 1000 Hz, ultimately working with 60-minute data blocks. Nine mode shapes were identified, including the three that were measured on the original dynamically scaled model, reported below. 

Phone-Based Measurement of the Dynamically Scaled Model Structure

The dynamically scaled model of the Lillebelt Bridge that was constructed during the design phase is now housed at Bauhaus University in Weimar, Germany. It is perhaps the only extant dynamically scaled suspension bridge model in the world. 

Only three relevant modes of vibration can be measured based on the detailing of this model: 

Results and Discussion

Reported here are the frequencies from the original model structure, the full-scale bridge, and more recent measurements of the model structure using the iPhone and methods described above. 

1. The first symmetrical vertical bending mode 

   • 0.156 Hz model structure [Ostenfeld et al. 1966]

   • 0.156 Hz full structure [Christensen et al. 2018]

   • 0.154 Hz (2.20 Hz measured) [Riley 2023]

2. The first antisymmetric vertical bending mode 

   • 0.153 Hz model structure [Ostenfeld et al. 1966]

   • 0.171 Hz full structure [Christensen et al. 2018]

   • 0.141 Hz (2.02 Hz measured) [Riley 2023]

3. The first symmetrical torsional mode 

   • 0.500 Hz model structure [Ostenfeld et al. 1966]

   • 0.523 Hz full structure [Christensen et al. 2018]

   • 0.524 Hz (7.49 Hz measured) [Riley 2023]

Each result has fairly good agreement with the original model measurements, as well as those conducted on the full-scale structure. The first symmetrical vertical bending mode is the dominant mode of vibration and is the easiest to excite. All results for this mode are within 2% of eachother. The first antisymmetric vertical bending mode occurs at a lower frequency, but the most recent measurement is nearly 8% lower than the original model result, while the full-scale structure has a higher frequency. This is perhaps an artifact of the scaled dynamic model that has a discontinuity between the deck and towers on each side of the towers, while the actual physical structure has a continuous deck. The first symmetrical torsional mode occurs at a much higher frequency than the other two, which is observable in the animated gif above. The measured results are each within 5% of eachother. 

The iPhone measurement approach offers a particular benefit in this case due to the relative lightness and flexibility of the model; more accurate results will be obtained with a non-contact vibration measurement that avoids the added-mass effect. 

It is worth noting that 7.49 Hz, measured for the torsional mode, is the likely maximum frequency measurable and appears to be at the temporal limit of the LiDAR sensor's ability (given its 15-Hz sampling rate). Thus, the accuracy of this result may be in question based on the limited sampling rate of the sensor. 

Ultimately, the ability of the iPhone LiDAR to capture the natural frequencies of this very unique structure model appears to be quite good. This bodes well for using this method to explore dynamically scaled models in the classroom. 

Numerical Modeling in Sofistik

The team at Bauhaus University has a finite element model developed in SOFiSTiK for the smaller dynamically scaled model so students can adjust structural parameters and modeling assumptions to explore their effect on the dynamic properties of this unique model. 

References

Ostenfeld,C., Haas,G., Frandsen,A.G.: Motorway bridge across Lillebelt: model tests for the superstructure of the suspension bridge. Presented at Symposium on Supension Bridges, Lisbon (1966).

Christensen, S.S., Andersen, M.S. and Brandt, A., 2019. Dynamic characterization of the little belt suspension bridge by operational modal analysis. In Dynamics of Civil Structures, Volume 2: Proceedings of the 36th IMAC, A Conference and Exposition on Structural Dynamics 2018 (pp. 17-22). Springer International Publishing.

Addis, Bill, Karl-Eugen Kurrer, and Werner Lorenz. Physical Models: Their historical and current use in civil and building engineering design. John Wiley & Sons, 2020. 

https://structurae.net/en/structures/little-belt-bridge-1970 

https://en.wikipedia.org/wiki/Little_Belt_Bridge_(1970) 

https://www.researchgate.net/publication/387996009_Efficient_dynamic_modal_load_reconstruction_using_physics-informed_Gaussian_processes_based_on_frequency-sparse_Fourier_basis_functions