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As is well known, the safe travel velocity of high-speed rail must remain below a well-defined limit commonly referred to as the critical speed . This upper bound depends, in turn, on the speed at which waves propagate in the ground. But as models for trains in motion have increased in complexity, over the years the concept of critical speed has grown in obscurity. It was not until late in the twentieth century that it was realized that the critical speed was somehow connected to the so-called dispersion spectrum for the complete system, but until now, the justification for its application in practice has remained empirical and lacking in rigorous mathematical explanations. This task is taken up in this article, where for the first time a most general proof is provided for the problem at hand that is applicable to any layered soil configuration when one such system is subjected to one or more loads in motion.

This study evaluates the ballast shear strength from the testing of 20 specimens, performed in a very large direct shear box that has a shearing plane of 1 m by 1 m and can hold specimens with a thickness up to 80 cm. Tests comprise two different types of ballast (mylonitic and andesitic) and two different specimen preparation techniques (with and without compaction). The normal stresses used cover a wide range from 10 kPa to 400 kPa. Geometrical and physical properties of both ballasts were obtained according to EN-13450, Aggregates for Railway Ballast, showing that both ballasts are similar, can be ranked as high-quality class ballast and fulfill the recommended limiting values set by the American Railway Engineering and Maintenance-of-Way Association. The values obtained for the Mohr–Coulomb failure criterion were: (a) for noncompacted ballast—a friction angle of 39° combined with an “apparent cohesion” of 25 kPa, and (b) for compacted ballast—a friction angle of 50° and cohesion between 10 kPa and 20 kPa. Test results were also interpreted according to a parabolic model that proved the nonlinear nature of ballast shear envelope. The results also showed the decrease of secant friction angles with an increase of normal stress. It is worth remarking that, for very low normal stress (10 kPa), secant friction angles are around 70°, while for large stresses (400 kPa), they decrease to values around 40°. Finally, the use of direct shear boxes of two different sizes (30 cm by 30 cm and 1 m by 1 m) made it possible to analyze the scale effect in these tests. Test results show that the use of the smallest direct shear boxes produces an increase in the shear strength between 40 % and 60 %. Results obtained in this study agree well with those in the literature.