# Method for Determining the Susceptibility of the Track
## 2 Experimental Testing on a Real Object
### Stiffness Calculations in the Turnout  
The second part of the study concerned the determination of stiffness parameters and moments of inertia for the turnout. This task was carried out in two ways. In the first, calculations were made on the basis of analytical relationships. In the second part, the quantities were tested on the actual object. In order to determine the stiffness values analytically, it was necessary to determine the moment of inertia of the turnout in the area of the crossing. The moment of inertia of the rail is used to describe the rotational movement of the rail vehicle-track system around the longitudinal axis of the track, usually denoted as OX. The course of the cross-rail moment of inertia variation was determined. This waveform is shown in Figure 7.  
The moment of inertia is necessary when determining the stiffness along the turnout. Comparative calculations of the static deflections of a cross-beam loaded by a two-axle bogie with a load of 20 \(\rm kN/axle\) were carried out, assuming that:  
1. the cross-beam is a beam of real length with variable moment of inertia and cross-section (Figure 7 curve 1) supported on elastic supports (turnout sleepers),  
Figure 7: Course of variation of the moment of inertia of a turnout cross member: 1—crossbuck as a monolithic unit \(\rm\dot{\gamma}_{T}=9.6\cdot 10^{-5}m^{4}\), 2—crossbuck bow as an element independent of the wing rails \(\rm J_{y}=3.03\cdot 10^{-5}m^{4}\)[15]. 2. the cross-beam is a beam of infinite length supported on a continuous elastic support; the moment of inertia of the cross-section can be taken as constant (averaged). Keeping the parameters the same, it turned out that at the points of force application, the difference in deflection results obtained by the two methods does not exceed 0.5%, at the other points of the cross-beam the difference is not greater than 3.5%. With the above assumptions, the deflection of the cross member is:  
\[\mathrm{z=}\frac{\mathrm{k}}{2\mathrm{U}}\sum_{\mathrm{i=1}}^{\mathrm{n}} \mathrm{P_{i}\eta_{i}}, \tag{2}\]  
where  
\[\mathrm{k=}\sqrt[4]{\frac{\mathrm{U}}{4\mathrm{EJ}}},\ \mathrm{U=C\frac{4 \mathrm{lb}}{2\mathrm{a}}} \tag{3}\]  
E = 2.1 \(\times\) 10\({}^{5}\) MN/m\({}^{2}\)--Young's modulus of the rail steel,  
J = 10 \(\times\) 10\({}^{-5}\) m\({}^{4}\)--average moment of inertia of the cross section,  
P\({}_{i}\)--load from the next wheel,  
\(\eta_{i}\)--ordinates of the influence line of settlement of an infinitely long beam resting on a continuous elastic foundation,  
U--support factor of the rail,  
C--ground coefficient,  
\(\alpha\) = 0.8--ratio of the average settlement of the subgrade to its settlement under the rail,  
l = 2 m--half of the average length of the turnout sleeper,  
b = 0.26 m--width of the subgrade base,  
a = 0.6 m--axial spacing of the turnout. When loaded with a single force P, the deflection of the rail is:  
\[\mathrm{z=}\frac{\mathrm{k}}{2\mathrm{U}}\mathrm{P}, \tag{4}\]  
The vertical stiffness coefficient of the rail and subgrade is thus:  
\[\mathrm{k_{tz}=}\frac{\mathrm{P}}{\mathrm{z}}=\frac{2\mathrm{U}}{\sqrt[4]{ \frac{\mathrm{U}}{4\mathrm{EJ}}}}, \tag{5}\]  
As can be seen, the dependence of the \(\mathrm{k_{tz}}\) coefficient on the rail support coefficient U or the substrate coefficient C is ground coefficient. Calculations were performed for two ground coefficients equal to C = 50 MN/m\({}^{3}\) and C = 200 MN/m\({}^{3}\). These results are shown in Figure 8.  
After calculating the deflection of the cross member loaded by a single concentrated force P, the following was obtained: \(\mathrm{k_{tz}=50.8\ MN/m}\), he analogous value for S-60 rail is 27.6 MN/m and so the stiffness of the cross-rail is about 80% greater than that of a single rail assuming C = 50 MN/m3. The influence of the length of the turnout gear on the value of the \(\mathrm{k_{tz}}\) coefficient is significant.