When performing a dynamic analysis based on modal methods – for example, spectrum analysis or modal time-history analysis – it is common to calculate only a limited number of eigenmodes. In practice, however, a structure has an almost infinite number of eigenmodes. To account for the higher modes that are not explicitly included, residual modes are used. This method is well established in modern dynamics analysis and provides an efficient way to improve the mass representation in the analysis. But is it well established in you team when carrying out dynamic bridge analyses?
That is a question many bridge engineers ask themselves when carrying out dynamic analyses for bridges that will be trafficked by high-speed rail traffic. It is in principle the same as asking: What is a suitable cut-off frequency for my analysis? In practice, many refer to the following threshold values when vertical accelerations, deformations, or rotations around supports are studied:
This means that the cut-off frequency (fmax) should be the highest of 30 Hz, 1.5 times the first mode frequency (f1), or the frequency of the bridge’s third mode (f3). For other design criteria, however, the cut-off frequency is limited by the convergence of the calculated quantity, e.g., bending moment or shear force. Quite often, the section forces resulting from High Speed Load Models (HSLM) - as defined in EUROCODE - are lower than those obtained from static design loads, but not always. In cases where section forces from dynamic loads must be studied, the bridge designer therefore needs to make a judgment and select an appropriate cut-off frequency for the analyses, adapted to the specific bridge under investigation.
In theory, the analysis becomes increasingly accurate as the cut-off frequency is raised, i.e., as more and more eigenmodes are included. The problem, however, is that the more modes that are included, the longer the analysis time becomes. Thus, including more modes may reassure the engineer that the results are more reliable, but at the same time, it may make analysis times unreasonably long.
Across Europe, new high-speed railways/routes are being built. In projects such as Rail Baltica – where more than 870 km of railway and hundreds of bridges are planned for speeds of up to 249 km/h with a total budget of about €5.8 billion – calculation costs and delivery times are critical. There, as in several other projects, it is therefore of interest to create a deeper understanding of this issue and investigate whether there is a smart way forward to shorten analysis times.
Residual modes are based on replacing the effect of the omitted eigenmodes with an approximate mode shape. In this way, accuracy is maintained while keeping the analysis manageable.
The master’s thesis by Lundin and Mårtensson demonstrates a method for creating suitable residual modes and applications in bridge analysis. The method presented is a guideline for ensuring that a sufficient portion of the excited mass is included in the calculation – without having to analyse every single high-frequency mode.
Higher reliability: You know that the analysis model better accounts for the mass that is truly excited (set into vibration).
Shorter analysis times: Residual modes reduce the number of modes that need to be included in the simulation, compared to including as many natural modes as would otherwise be required to reach the same total excited mass.
Better resource utilization: In large projects with many parallel analyses, both CPU time and license capacity are freed up.
In BRIGADE/Plus, ready-to-use tools and recourses are available for this type of analysis. In Workshop 3 – Dynamic response of high-speed trains, a detailed workflow is shown for building a bridge model for dynamic analyses, including how to extract eigenfrequencies and link them to HSLM load cases. Guidance is also provided on how to select the cut-off frequency. A shorter tutorial on how to think about residual modes in bridge analyses is available in the BRIGADE/Plus tutorial Eigenfrequency analysis. You can find both the workshop and the tutorial here, among all other TECHNIA recourses.
An example is made with a Reinforced Concrete (RC) frame bridge. In the example below, a gravity load applied to the entire structure in the vertical direction was used as a pre-step to the eigenfrequency step where the residual mode is calculated. By including this residual mode, a higher share of the effective mass in the vertical direction is represented in the analysis. In cases where high-speed trains cross the bridge, this preloading can be refined further by replacing the general gravity load with a series of point loads along the rail lines, see Lundin and Mårtensson.
An example of a RC frame bridge (not the same as used in the analysis example)
In short, the following workflow can be used to include residual modes in the analysis:
Static “pseudo-mode” with a distributed load on the deck that is used to create the residual mode.
Natural eigenmode, with a big effective mass in the vertical direction.
Residual mode, also with a high portion of effective mass in the vertical direction.
For those who want to take the next step, TECHNIA offers complete training in Bridge Dynamics with BRIGADE/Plus, where residual modes, Eurocode requirements, and best practices for dynamic analysis are covered in depth. You can read more about various best-practice aspects of dynamic analyses for bridges subjected to high-speed rail traffic here: Best Practices for Dynamic Analysis of Bridges
Or why not take our one day course on Dynamic analyses i BRIGADE/Plus. Read more about our courses here: BRIGADE training
As Europe’s high-speed network expands – from Rail Baltica in the east to new lines in France, Spain, and Italy in the west – hundreds of bridges will be designed for trains traveling at 200–300 km/h. Saving analysis time without compromising safety thus becomes a strategic advantage.
👉 Learn more about BRIGADE here: technia.se/software/brigade