More Track, Structure and Interaction
So, nearly four months later: Here comes the second part of my introduction to Track Structure Interaction (or TSI, for fun). I’ll just continue rambling on, so it might make sense to pop back to the original article if you need an introduction.
As with all something/something-interaction models, the interactors themselves are secondary.
As such, the track itself can be modelled as a single line representing all the rails on the structure, and all the furniture (sleepers, clips, etc.) can be ignored. Any expansion joints (typically 50mm), must be modelled, however, as this they will allow the rails to move. Note that joints will relieve stress in the rails, at the expense of increased structural deflection.
For those of you who have better things to do than know the structural properties of a piece of rail track:
- Structural Area: 15.36 x10-3 m2
- Moment of Inertia: 61 x10-6 m4
The ballast is a different story. This connection between track and structure is modelled as a linear-plastic spring; emulating how the ballast bed provides increasing stiffness until the granular fill ‘locks’. In the UK the track-bed resistance is considered fully mobilised after 2mm deflection, however the maximum stiffness achieved depends on loading and maintenance. As you would expect, vertical loading increases resistance; as does well maintained, and thus tamped, ballast. At this point its worth noting that the current Network Rail standards refer to the worse maintenance condition as ‘modern’, I suspect it’s a misprint of ‘moderate’- but it’s something that makes me chuckle.
The original research (provided in the UIC guide) outlines the development of the ballast interface model; including ‘frozen’ and concrete beds, which are not often applicable in the UK. In this discussion it notes that the linear-plastic relationship between stiffness and deflection is likely to be an underestimate at small (read < 0.5mm) deflections.
The track model should extend until boundary conditions do not influence the solution (i.e. axial load in the rails tends to zero). As a rule, expect to need 75 – 150m of track either side of the structure, fixed through the ballast interface onto fixed points (as the ground is unlikely to deflect laterally under loading). It’s important to note, however, that if this field of analysis includes another structure the model should be extended to include that and the additional length for dispersion. Interfaces between substantially different structures can be critical points on a TSI analysis.
Rotation is especially important for bridge piers (which are normally taller than 5m) where small rotations at the base will translate into large deflections for the structure.
Once again the structure itself is almost extraneous to the proceedings. A single line, with the general properties of the structure placed at the neutral axis will suffice. Note that for complex bridges, like trusses, this could mean that the bridge elements are placed above the rail. Once again any movement joints need to be modelled (how else will the structure be able to deflect?).
The final step is probably the most important, and its omission from the title is unfortunate. It is the soil/sub-structure interaction. TSI is ultimately an analysis about relative stiffness and restraint: when all is said and done, it is the sub-structure that dictates how much deflection occurs in a TSI model.
How precisely each element of the sub-structure need be modelled depends on the fixity- typically bridge foundations are significantly stiffer vertically than laterally. A common assumption is, therefore, that piers without longitudinal fixity need only be modelled as vertical restraint. For longitudinally fixed bearings, however, it is best to model the complete pier down the the foundation. Where intermediate piers consist of two longitudinally fixed bearings it will be necessary to model the complete pier head. There is provision to allow for the resistance of the bearings- but it seems fair (and is allowable) to omit this.
Although it is not necessary to go as far as modelling the complete foundation structure (e.g. piles), a Winkler spring ground model is essential. It’s important to remember that, unlike Young’s Moduli, the Subgrade Modulus is not a fundamental parameter of the soil- or even the foundation. Changing the magnitude of the load, or it’s nature, will have an effect on the accuracy of your foundation stiffness. Typically this means that some iteration and sensitivity (think ±40%) will be required.
The axial, lateral and rotational stiffness of the foundations needs to be modelled. Rotation is especially important for bridge piers (which are normally taller than 5m) where small rotations at the base will translate into large deflections for the structure. As a guide, transient actions (i.e. traction and breaking) will likely be the dominant actions for movement; therefore dynamic soil properties are arguably appropriate.
The above example demonstrates the key parts of a TSI model, with the true structure in grey behind it. The red line shows the structural members placed at the neutral axis, with a gap at the movement joint. Orange ballast springs connect to the continuous green track member. The red structure lines are propped (purple arrow) where the bearing is free, and connected to the cross-head and pier (blue) where it is fixed longitudinally. Finally the piled foundation is modelled as purple springs in all degrees of freedom. Obviously colour is important…
In the third (and hopefully final) part of this series, early next the year, I’ll discuss the analysis process, as I understand it, and provide a few pointers that I’ve picked up from my experience with TSI.
Note that the bare facts of these articles are quoted from Network Rail (typically NR/L3/CIV/020) and research by the UIC (774-3R). The rest is interpretation and notes from a graduate’s experience. As always, your attention is drawn to the disclaimer at the bottom of every page.