This article presents the fundamentals of an analytical method for determining the stress–strain state of a railway subgrade reinforced with geosynthetic material. The reinforcement described is a combined system where the geosynthetic material forms an open shell containing a layer of compacted crushed stone. The overall stress–strain state is proposed to be viewed as a superposition of two states of the subgrade. The stresses and displacements in the first state refer to the unreinforced subgrade (matrix), while the stress–strain state of the reinforcement element is determined using analytical constructs from composite theory. The dependencies of the overall stress–strain state are applied in a numerical analysis, which confirms the positive effect of reduced subgrade deformations. A small-scale experimental model further validates the accuracy of the analytical approach.

The current task today is the development of theoretical and methodological approaches, as well as practical recommendations for determining the technical feasibility of creating high-speed railway (HSR) lines in the European context. The development of railways in individual countries, followed by the creation of a pan-European high-speed railway network, has raised questions about the compatibility of the technical systems of individual national HSRs. This paper addresses these issues using the example of Ukraine. The research is based on an analysis of scientific developments related to the design of HSR lines and the synthesis of European experience in identifying priority route directions in Ukraine. For comparing various scenarios for the development of railway connections, the authors have developed a forecasting and efficiency assessment model based on the Net Present Value (NPV) indicator. It has been demonstrated that considering the population attracted to HSR and the volume of transit passenger transportation alone is insufficient to achieve the normative investment payback. This situation can only be rectified by implementing mixed traffic involving high-speed passenger trains and accelerated freight trains. However, mixed traffic of passenger and freight trains on high-speed rail lines may face numerous issues and constraints that require careful planning and coordination.

In developing a mathematical model of a railway track, the question of determining the dimensions of the modeling domain inevitably arises. If the modeling area is too small, boundary effects may significantly influence the results, reducing their accuracy. Conversely, excessively large areas can increase computational complexity without substantial improvements in accuracy. An optimal choice of dimensions enables the balancing of computational costs and accuracy. Solving this problem is non-trivial, as it depends on numerous factors, primarily the type of mathematical model and the problem being addressed. In most cases, preference is given to minimal domain sizes that ensure the approach’s adequacy. The aim of this study is to justify the dimensions of the modeling domain by addressing such tasks as load scaling, introducing additional boundary conditions, and making relevant assumptions. The main object of the study is the minimum adequate longitudinal length of the track for the spatial model. The research is based on the analytical application of modern approaches in the theory of elasticity. The results are analyzed using mathematical methods, such as modeling the railway track through the propagation of elastic waves and finite element modeling. These findings can be applied to a wide range of problems related to the mathematical modeling of the stress–strain state of railway tracks.