**4. Analysis of finite element study results**

After performing the variable calculations, the following results were obtained and processed:


As the calculation result of Coulomb-Mohr points location shown in **Figure 7**, the curves of the ratio *h/b* ("arch collapse" height/distance from the karts soil roof to the bottom of the pile, respectively) dependence on the ratio *b/B* (distance from the karts soil roof to the bottom of the pile/cavity diameter, respectively) were plotted. The defined parameters are shown in **Figures 6** and 7. The graphs for two loads over the grillage top are shown in **Figure 9**. As can be seen from **Figure 9** the possibility of

#### **Figure 9.**

*Dependence of the relative distance from the predicted top of the collapsed vault to the bottom of the piles (*h/b*) on the ratio of the distance from the top of karst soils to the bottom of the piles to the maximum cavity diameter (*b/B*): A, b – Distributed load over the grillage top: 400 kN/m2 , 550 kN/m2 .*

**Figure 10.**

*Dependence of the additional load on the pile (*P1*, kN) on the diameter of the cavity (*B*, m) and the distance to the karst soil roof (*b*, m): A, b – Distributed load over the grillage top: 400 kN/m<sup>2</sup> , 550 kN/m2 .*

developing the arch above the cavity not higher than the bottom of the piles was determined by the condition *h/b* ≤ *1*. This condition could be met at *b/B >* 1, which limited the scope of the results of this study and the proposed solution for determining the additional load on the pile.

**Figure 10** shows graphs of the dependence of the additional load on the pile (*P1*, kN) on the diameter of the cavity (*B*, m) and the distance to the karst soil roof (*b*, m). These graphs are based on variable Finite element 3D calculations of the bridge support pile foundation under the karst cavity in karst soils and show that the *P1* increases with the extension of the cavity (*B)* and decreases of the distance to the karst soil roof (*b)*. However, such calculations are complicated and laborious and require specialized Software to make them.

To determine the additional load on the pile (*P1*) to the load on the pile under normal operating conditions (*P*) before the formation of the cavity, statistical processing of the calculation results was performed. It can be indicated from **Figure 11** that the relationship between *Р/P1* and *b/B* has the form of the power function. As a result of statistical analysis the analytical dependence *Р/P1* from *b/B* was obtained:

$$P/P\_1 = 2, 1e^{1.7\frac{b}{\pi}}.\tag{15}$$

The value of a reliable approximation was R2 = 0.9194. Thus, the additional load on the pile *P1* was determined in accordance with the design load transferred to the pile under normal operating conditions (*P*) before the formation of the cavity, the size of the cavity (*B*), and the distance from the bottom of the piles to the karst soil roof *(b*):

$$P\_1 = P/2, \mathbf{1}e^{\mathbf{1}\cdot\mathbf{7}\frac{b}{\mathbf{5}}}.\tag{16}$$
