The investigations were made on the rails which are maintained in the underground conditions. They essentially distinguish from their work in the main way. Therefore the requirements to their reliability should answer conditions of traffic safety of trains. The existing normative base on continuous replacement of rails for undergrounds was applied, as well as for a main way, without consideration curvature of a railway track. On the basis of use of the theory of reliability and the theory of probability the mathematical model for the description of gradual refusals of rails in real conditions of underground which takes into account the passed tonnage, curvature of a railway track and a root-mean-square rejection as function of curvature of a way and the passed tonnage is offered. The normal distribution of withdrawal of rails because of defects and damages is set in the basis of the model. The usege of this model allows to develop scientifically - substantiated recommendations for definition of the normative tonnage for continuous replacement of rails. It depends on established for railways scale of their exit taking info account operational factors of a railway.

A waveform wear, nascent on rails during their exploitation, is described by successive alternation of smooth geometrical roughness of three groups – short, middle and long. The two component model of waveform wears and methods of depth and length determination of roughness are offered. The methods we the principles of correlation analysis within the offered model.

Rails –are the basic constructive element of the top structure of a railway which directly perceive dynamic loading from a rolling stock. During their operation in rail metal there are the irreversible processes lowering their carrying capacity. Thus carrying capacity of rails is provided under the conditions that working loading for all period of their operation should not exceed admitted probabilities. The developed mathematical model takes into account casual rails load process load rails and carrying capacity, and also non-stationary character of reduction strength properties of rails. It allows to investigate all range of operational work of rails in a track at the given strength.