## 4 Energy Equilibrium as Identification Criterion  
A series of experiments allowing for the identification of the two basic parameters (B\({}_{\rm z}\) and r\({}_{\rm 0}\)) were performed with the application of the described numerical model. Based on energy equilibrium equations, the work of force P (kN) resulting from the maximum displacement was determined, calculating energy of the system as a sum of the bending energy of the substitute beam and the work of ballast deformation in the lateral direction. On this basis, the substitute stiffness of the rail-sleeper structure with a given material solution (B\({}_{\rm z}\)) and the ballast resistance (r\({}_{\rm 0}\)) were determined. In the equilibrium, the longitudinal track resistance parameters were neglected and only three elements were taken into account. The energy equation takes the following form:  
\[{\rm W_{p}=E_{zg}+E_{sp}} \tag{18}\]  
where  
\[{\rm E_{zg}=\frac{1}{2}\cdot B_{z}\int_{-\infty}^{+\infty}\;y^{\prime\prime}\; dx} \tag{19}\]  
\[{\rm E_{sp}=r_{0}\int_{-\infty}^{+\infty}ydx} \tag{20}\]  
Figure 6: Lateral resistance of ballast and sleeper were described in the chart below [14]. Figure 7: The whole rail-sleeper structure used for calculations (P—external force). \[\mathrm{W_{P}}=\frac{\mathrm{P\cdot y_{max}}}{2} \tag{21}\]  
E\({}_{\mathrm{zg}}\)--bending energy (N*m);  
E\({}_{\mathrm{sp}}\)--ballast deformation work in lateral direction (N*m);  
B\({}_{\mathrm{z}}\)--substitutes stiffness of rail-sleeper structure (N*m2);  
r\({}_{0}\)--substitute lateral resistance of ballast (N/m);  
W\({}_{\mathrm{P}}\)--work of the external force P with the lateral displacement y (N*m);  
P--force (kN);  
y\({}_{\mathrm{max}}\)--maximum lateral displacement (mm). For the assumed interval of admissible ranges of parameters B\({}_{\mathrm{z}}\) and r\({}_{0}\), the values of the left and right sides of the Equation (18) are comparable, and the differences are minimised. The energy equilibrium is formulated for a few (three to four) external forces. The lateral displacement function (20) and the second derivative (19) were calculated with the use of numerical data. The analyses were performed for two types of tracks and two levels of ballast stabilisation. First, the basic rail-sleeper structure was analysed, i.e., rail, concrete sleepers, and the elastic fastening system. Afterwards, the reinforced rail-sleeper structure was analysed. The reinforcement was the introduction of steel slabs welded to flat bars and fastened to the sleepers with screws anchored in the sleepers, used in the type of sleepers for fastening guard rails on bridges (PS-94/M and PS-93/M sleepers). Flat bars approximately 12 mm thick and 400 mm width were welded to concrete sleepers in their lowered parts. Afterwards, after careful placement of the sleepers and track adjustment in the vertical and horizontal planes, slabs approximately 12 mm thick and 0.25 m wide were welded to the flat bars [18]. Figure 8 presents the reinforced rail-sleeper structure. Additionally, both the loose ballast and the stable ballast were taken into consideration in the analyses [9]. For calculations the following Bz and r\({}_{0}\) parameter intervals were assumed:  
(1) For loose ballast:  
(a) Track without reinforcement:  
(b) Track with reinforcement:  
(c) Lateral resistance of ballast r\({}_{0}\) (6.5 \(\div\) 8 kN/m);  
(d) Substitute stiffness of rail-sleeper structure B\({}_{\mathrm{Z}}\) (10,000 \(\div\) 30,000 kNm2);  
(e) For medium-stabilised and stable ballast:  
(e) The total number of tracks in the ballast is 125,000