A railway bridge does not respond to longitudinal actions as an isolated bridge deck.
The rails continue across the bridge, are connected to sleepers and the deck, and extend into the approach track on the embankments. Bearings and substructures restrain the bridge with finite stiffness. Temperature changes, braking forces and deck deformation can therefore generate forces that are shared between the rails, the bridge and the surrounding track.
This is the essence of track–bridge interaction, or TBI. It is also why seemingly minor modelling decisions can influence calculated rail stresses, support reactions and practical design decisions, such as whether a rail expansion joint is required.
A 2025 master’s thesis from Lund University, written by Moa Magnusson and Rakel Nilsson, investigated this interaction through two Swedish bridge cases: the existing bridge over the Brynge River and the planned bridge over the Sege River. The authors used sensitivity studies to examine which modelling assumptions had the greatest influence on the results.
In a conventional isolated bridge model, longitudinal actions may be viewed simply as loads applied to the bridge, with the resulting forces resisted by bearings, foundations or abutments.
For continuously welded rail, however, the physical system extends beyond the bridge itself. The rails can carry longitudinal forces across the bridge and transfer part of those forces into the adjacent track. The relative movement between the rails and the deck depends on the rail-to-deck connection, while the stiffness of the substructure controls how freely the bridge can expand and contract. The length of the approach track determines how much of the surrounding system must be included before the model reaches a boundary that is effectively unaffected by the bridge.
These mechanisms are strongly coupled.
Changing the support stiffness can alter the relative movement between the deck and the rails. This, in turn, affects rail stresses and the forces transferred through the support system. Similarly, changing the rail-to-deck interaction law can influence local slip and the distribution of longitudinal forces. If the approach track is truncated too close to the bridge, the boundary condition may artificially constrain the model.
These characteristics make TBI a natural application for sensitivity analysis. Instead of assuming that one particular representation is correct, the engineer can vary uncertain or idealised model components and assess how strongly the design quantities respond.
The thesis, Track-Bridge Interaction – Sensitivity Studies on the Bridge across Brynge River and TBI Analysis on the Bridge across Sege River, used finite-element modelling to address two different questions.
For the Brynge River bridge, the authors systematically varied a number of modelling parameters, including:
The railway bridge across the Brynge River
For the Sege River bridge, the focus was on assessing TBI for a planned bridge rather than on conducting a broader parameter study.
A finite-element environment such as BRIGADE/Plus can be used to represent the bridge, supports and idealised interaction components, making it possible to compare the effects of different modelling assumptions in a structured way.
Rail-to-deck interaction is commonly represented using force–displacement relationships. A more sophisticated nonlinear or bilinear formulation may initially appear preferable because it more closely resembles an idealised physical slip response.
However, increased model complexity is only beneficial if it changes the quantities relevant to the engineering decision.
For the Brynge River case, the authors reported that a linear vertical stiffness representation of the rail-to-deck connection produced results without significant loss of accuracy compared with more complex bilinear alternatives.
This should not be interpreted as a general rule that linear springs are always sufficient. Its value lies in demonstrating how a simplification can be assessed and justified for a specific application.
A robust engineering workflow is:
This is a more defensible approach than simplifying a model solely because the simpler version is easier to run.
The same principle is important in parametric studies and production models. An interaction law that is easier to define, verify and reproduce can improve the robustness of the overall workflow, provided the sensitivity study demonstrates that the additional sophistication does not materially affect the results.
A TBI model cannot extend indefinitely into the adjacent track. At some point, the finite model must be terminated.
If the model is too short, the boundary condition may influence the region around the bridge and distort the calculated rail forces. If it is unnecessarily long, the model becomes larger without improving the accuracy of the result.
For the Brynge River case, the thesis reports that an embankment length of approximately 300 metres was sufficient to capture the additional stresses in the system investigated.
This value is specific to the studied bridge and should not be treated as a universal modelling requirement. The broader lesson is that the model extent should be checked for convergence.
The engineer can progressively increase the represented approach-track length until the quantities of interest in the bridge region no longer change significantly. Once this condition has been reached, adding further track length provides little analytical benefit.
This is analogous to mesh convergence, but applied to model extent rather than element size.
The Sege River analysis provides another important insight. According to the thesis, nearly half of the total longitudinal forces in the studied system were carried by the rails and transferred into the embankments.
This illustrates why treating the rails merely as passive recipients of bridge deformation can be misleading. The rails are also load-carrying components in the longitudinal system.
For bridge engineers, this affects how the results should be interpreted. If a significant proportion of the longitudinal force is transferred through the track, checking only the reactions in the bridge substructure provides an incomplete picture. Conversely, assuming that every longitudinal action must be resisted entirely by the bridge may lead to unnecessarily conservative results.
The relevant load path is:
bridge movement → rail-to-deck interaction → longitudinal rail force → transfer into the approach track and embankment
A TBI model is particularly valuable when it makes this load path visible and allows it to be quantified.
The Sege River results led the authors to discuss whether simplified Eurocode approaches could be applicable to bridges whose lengths exceed the limits that would normally indicate the need for a more detailed TBI analysis.
This conclusion requires careful interpretation. A single thesis case cannot redefine a code requirement, and the applicable standards may also have changed since the study was completed. The thesis compared the Swedish implementation in force at the time with a draft of the next generation of rules that was available during the project.
The broader lesson is therefore not that long bridges can automatically be assessed without a detailed TBI analysis. Rather, detailed modelling can help explain why a simplified method works—or where its underlying assumptions cease to be valid.
This is one of the most valuable applications of advanced finite-element analysis. It provides insight into the structural behaviour behind a design rule, rather than simply producing another set of design forces.
For future projects, the same approach can be used to investigate:
For the Sege River bridge, the calculated rail stresses met the criteria assessed under both the rules in force at the time and the draft future provisions considered in the thesis. The authors therefore concluded that the bridge was expected to be feasible without a rail expansion joint.
That conclusion applies specifically to the bridge studied and to the standards considered in the thesis. Nevertheless, it demonstrates how TBI analysis can support practical project decisions.
Rail expansion joints are not merely code-check outputs. They affect construction, maintenance and the long-term performance of the railway system. If an analysis demonstrates that an expansion joint is unnecessary, the resulting benefit may extend well beyond a reduction in calculated rail stress.
Such a decision depends on the complete force-transfer system. This makes transparency in the modelling assumptions essential. Support stiffness, rail-to-deck interaction and the representation of the approach track should be documented as engineering inputs rather than hidden inside the model.
Rail expansion joint
One of the strongest aspects of the thesis is that the sensitivity analysis was used not only to produce a research result, but also to increase confidence in the modelling strategy.
A similar approach can be applied in production projects:
Start with the dominant mechanisms. Build a model that represents the structural behaviour most relevant to the engineering question.
Identify uncertain assumptions. Typical TBI parameters include support stiffness, interaction stiffness, nonlinear slip behaviour, approach-track length and boundary conditions.
Vary the assumptions systematically. Monitor rail stresses, support reactions, deck movements and other governing output quantities.
Investigate sensitive parameters in greater detail. If a parameter has little influence on the results, a simplified representation may be adequate. If the results are highly sensitive to the parameter, its physical basis should be examined and refined.
Document the robustness of the conclusion. A design that remains acceptable across a realistic range of assumptions is more defensible than one that satisfies the criteria only for a single calibrated value.
This is a general finite-element analysis principle—not one limited to railway bridges.
Track-Bridge Interaction – Sensitivity Studies on the Bridge across Brynge River and TBI Analysis on the Bridge across Sege River was written by Moa Magnusson and Rakel Nilsson at the Division of Structural Engineering, Lund University. It was published in 2025 as Report 25/5309 (LUTVDG/TVBK/25/5309).
The complete thesis is available through the Lund University LUP Student Papers repository.
The thesis received the 2026 Structorstiftelsen and Brosamverkan Award for Best Master’s Thesis in Bridge Engineering. More information is available here.
Model of the bridge over Brynge river
BRIGADE/Plus can be used to build finite-element models of bridges with the structural and interaction representations needed for systematic sensitivity studies and project-specific assessments.
Talk to our BRIGADE team to discuss modelling strategies for railway bridges and coupled structural behaviour.