## A Study of some Factors in Mechanistic Railway Track Design

##### Abstract

This thesis is composed of three main parts: The first part that uses classic track models as a basis for further developments, the second part that deals with constitutive behaviour of granular materials and the third part that describes the development of a new triaxial cell apparatus and the testing of a ballast material using this apparatus.
The description of classic track models is focused on the beam-on-elastic-foundation model (abbr. BOEF model), which make use of the Winkler foundation, and a simple beam element model with linear discrete support. The shortcomings of the BOEF model is discussed: It assumes a continuous foundation, a continuously welded track, the weight of the track ladder is not incorporated, linear support which imply prediction of tension in the uplift regions, no shear deformation in the rails is taken into account, it cannot predict stresses and strains within the granular layers. While some of the shortcomings may easily be incorporated others are not: Especially to remove tension in the uplift zones, and to calculate stresses and strains in the granular layers. The latter actually requires a continuum approach. A track model that approximately eliminates the tension in the uplift regions has been developed for a single axle load. As expected, the model shows that the length of the uplift zone and the amount of uplift have higher values than predicted by the BOEF model. The model may be useful when considering contact problems in the track, for instance in a buckling-of-rails analysis.
For the BOEF model a tool that makes use of dimensionless sensitivity diagrams has been developed. The method will in an easy way provide the new maximum track reactions when one or more track parameters are changed. It is hoped that this tool will prove very helpful in a design process, at least as a first step. Dimensionless sensitivity diagrams have been worked out for rail deflection, rail moment, rail seat load, tensional rail base stress and vertical stress between sleeper and ballast. The parameters considered are the design wheel load, rail moment of inertia, position of neutral axis in the rail, sleeper spacing, sleeper width and the length of the sleeper that carries the vertical load. The dimensionless sensitivity diagrams for the BOEF model may be used both for a single axle load and for a double axle load. Also for a beam element model with linear discrete support the dimensionless sensitivity diagrams may be used, but only for a single axle load which is located directly above one of the supports, i.e. a sleeper. For the beam element model the diagrams for the rail deflection, rail seat load and vertical stress between sleeper and ballast are almost identical to the ones for the BOEF model, while the diagrams for the rail moment and tensile rail base stress are somewhat different.
A beam element model with Euler-Bernoulli beam elements resting on nonlinear discrete supports was developed for a single axle load. The discrete supports, which were located at the sleeper positions, were modelled by a two-parameter power function. The model takes advantage of a measured load-deflection relationship, which is also modelled by a two-parameter power function. These latter parameters are generally found by regression of the measured data, while the two parameters for the discrete supports are found as part of the overall solution to the problem. The present version of the model only takes into account a short track section and further development of the model is therefore needed. The track ladder weight and a no tension option in the uplift region are not incorporated in the present version. The model is useful when the BOEF model cannot be used because of nonlinear track response.
Regarding constitutive behaviour it is argued that the plastic strain per load cycle in a well functioning railway track must be very small and normally below 1/100 000 of the elastic strain per load cycle. If also the hysteresis of the material during a load cycle is small, then an elastic approximation could be justified when it comes to calculating the stresses. The plastic strains may then be detached from the stress-strain calculation and modelled separately on the basis of laboratory or field measurements.
Several elastic constitutive models are described: The Hooke's law generalised to three dimensions, the cross anisotropic elastic model, two versions of the k-model, and two hyperelastic models.
The general elasto-plastic framework with isotropic hardening is also described. The basics of repeated loading of a frictional system is described by analogy to a simple model with springs and frictional sliders. This model can be viewed as the basis for the pure kinematic multisurface model by Mróz and Iwan. Through energy considerations in cyclic loading of the frictional system the concept of reclaimed plastic strain is rejected.
The concept of initial stresses and strains is discussed. It is argued that initial stresses cannot be large in the upper part of a road or railway embankment. The main reason for this is that granular materials cannot self equilibrate stresses through tension.
The development and construction of triaxial equipment for testing railway ballast in its original grading is described. The specimens are 300 mm by 600 mm (diameter by height). A new and direct way of applying the confining load was developed, which allowed faster variation of the confining stress. A new instrumentation concept was invented where instrumentation rings are fastened to material particles instead of being attached to the outer membrane or to plugs embedded in the material. This arrangement measures the horizontal deformation. The vertical deformation has to be measured over the whole specimen length as resilient particle rotations prevented on-sample instrumentation.
A test series on Vassfjell railway ballast was conducted to evaluate the feasibility of the new apparatus and to characterise the ballast material. The overall performance of the apparatus was found to be good with a reliable repeatability, but some modifications were suggested to improve the loading procedure in the beginning of the load steps.
The test series on Vassfjell ballast was rather limited and no advanced modelling of the results was found to be appropriate. Instead an isotropic linear elastic approach was followed. Moisture was added, to the natural retention capacity, to some of the specimens. It was found that the added moisture only slightly affected the mechanical behaviour of the material. A somewhat denser grading was also tested, but the observed effect on the material properties was limited.