## 1 Introduction  
The issue of continuous welded rail (CWR) track stability, both straight and curved, has been the subject of many scientific and technical studies. Track stability loss can take place on two planes--vertical and horizontal. Rail bending stiffness value along the \(x\) axis is much bigger, and dead weight of the rail-sleeper structure is also important. Therefore, track deformations most frequently take place laterally to the axis [1]. The buckling wave usually is from 0.3 to 0.5 m, and it appears in a section of 8-20 m, although there have been cases recorded of it reaching even 1 m. Before buckling, smaller track deformations may also appear, the length of which is much smaller [2, 3]. By formulating the criteria of track stability taking into consideration the required level of safety factors, it is possible to perform a pre- and post-buckling analysis of the track. The analysis of railway track stability was presented, i.a., in articles [4, 5]. The issue of CWR track stability generally is related to the horizontal plane. The analysis of track stability on the vertical plane is discussed in articles [6, 7], and the analysis of the impact of vertical loads on the stability is presented in articles [8, 9, 10]. Experimental studies are related to the determination of ballast resistances--lateral and longitudinal in relation to track axis, less often to artificial track buckling. The experiments are also related to the studies aimed at assessing the effectiveness of applying various methods of structural strengthening to increase track stability. However, theoretical analyses, including the improvement of calculation models, are necessary for assessing the degree of stability of a used CWR track. This is related specifically to the assessment of various methods of decreasing the risk of losing stability [11]. Railway track is a complex mechanical system. If a vehicle-railway track system, in which the vehicle is moving with constant speed, is included in the deliberations, a mechanical system with multiple degrees of freedom can be obtained [12]. In such a system, due to dynamic stability, there are many critical configurations and speeds. When analysing a CWR track as a rail beam on an elastic foundation loaded with a concentrated force system (these forces reflect the pressure from the wheels of a train), the interaction of railway track structure with the foundation during train passage is taken into consideration. Directly under the wheels, the lateral resistance is increased as a result of applying vertical force. It is different in the mid-section between the wheels, in which the track is elevated and lateral resistances are decreased. As a result, the track in the area with the train is more vulnerable to deformations. This may cause the loss of stability of the railway track's structure, and buckling. This means that vertical load from vehicles has a significant influence on the analysis of track stability in the lateral dimension [4]. A semi-analytical approach for the analysis of track stability in the horizontal plane is based on two concepts:  
1. Analysis of a substitute beam, where rail-sleeper structure bending stiffness is described by \(\mathrm{EI}_{\mathrm{z}}\) (E--[PERSON]'s modulus of rail steel) and the substitute moment of inertia \(\mathrm{I}_{\mathrm{z}}\) is determined by experiments (physical or numerical, e.g., [5] presented paper);
2. Analysis of a substitute beam, where rail-sleeper structure stiffness is equal \(\mathrm{2 EI}_{\mathrm{st}}\) (\(\mathrm{I}_{\mathrm{st}}\)--moment of inertia of one rail on the horizontal plane) and unit stiffness of rotation in fasteners s* decrease thermal force in rails (e.g., [7; 13]).