**3.1 Finite element research technique and justification of the accepted computational model**

To develop an engineering calculation method, complex numerical studies were performed by means of mathematical modeling of the bridge support pile foundation in various geological conditions. The calculation method was based on the analysis of the design documents for bridge crossings at Moscow-Kazan HSL section. As a result of the analysis, a variable finite element calculation model with the following parameters was compiled:



#### **Table 1.**

*Physical and mechanical characteristics of soils.*

*The Karst Protection Foundations Design DOI: http://dx.doi.org/10.5772/intechopen.103100*


As shown in **Figure 6** the calculations were performed by varying the following parameters:

• the distance to the rock roof (*b*): 6 m, 10 m, 14 m, 18 m, 22 m;

#### **Figure 5.**

*Finite element model (section).*

**Figure 6.** *The scheme of calculated foundation.*


Finite element calculations were made in a three-dimensional representation with Midas GTS NX Software. Soil, grillage, and piles were modeled by three-dimensional elements. A linear-elastic model was used to model concrete. The elastic–plastic Mohr-Coulomb model was used for soil modeling with 3-dimensional finite elements. Using the strength criterion implemented in the model, it was possible to estimate the "collapse arch" size in the cover layer of the soil above the karst cavity. In this way, the "subsidence" deformation type and "failure" deformation type can be realized. The possibility of using that strength criterion was confirmed by the convergence of the calculation results with the model experiment data of the "collapse arch" formation above the cavity [10].

The calculation was performed in the following sequence:


During calculations, the growth of the "collapse arch" above the karst cavity was monitored. **Figure 7** shows the predicted "collapse arch" under pile bottom with

**Figure 7.** *Mohr-coulomb points above cavity.*

*The Karst Protection Foundations Design DOI: http://dx.doi.org/10.5772/intechopen.103100*

**Figure 8.**

*Tangential stresses on the lateral surface of the pile (τz, kN/m2): а – Before cavity formation, b – After cavity formation.*

Mohr-Coulomb points above karst cavity. Assuming the possibility of the arch development not higher the bottom of the piles, the additional load on the pile, realized at the time of the cavity formation, was determined.

The additional load was transferred to the pile at the time of the cavity formation due to the occurrence of "negative friction" on the lateral surface of the piles in their lower part. When modeling the formation of a cavity in karst soils, the occurrence of "negative friction" was determined by changing the tangential stresses on the lateral surface of the piles in comparison with the calculated ones in normal operating conditions.

Under normal operating conditions, tangential stresses on the lateral surfaces of piles increased with depth, while on the extreme and corner piles the growth began from the top of the pile (the pile was included in work entirely). In the central piles, tangential stresses developed in the lower part of the pile (due to the "compression" effect, the side surface friction of the central piles was not fully realized). Similar results of experimental and theoretical studies of piles behavior in the group were obtained in Russian and abroad [11–14].

When a cavity was formed, the soil of the cover layer subsided, which led to a change in the nature of the pile lateral surface work: the tangential stresses on the lateral surface in the lower part decreased, but along the rest of the pile length they increased. **Figure 8** shows tangential stresses on the lateral surface of the pile before cavity formation (a) and after cavity formation (b). That indicated the occurrence of the "negative friction" effect in the lower part of the piles and the inclusion of the most part of its lateral surface at the time of the cavity formation. The additional load on pile *P*1, kN, was determined by the formula:

$$P\_1 = u \cdot \sum \Delta \tau\_{x,i} \cdot h\_i,\tag{14}$$

where: *u* is the perimeter of the pile, m; Δ*τz*,*<sup>i</sup>* is the change in the shear stress value on the pile lateral surface in the *considered i*th layer in comparison with the design phase under normal operating conditions, kN/m<sup>2</sup> ; hi is the thickness of the *i*th soil layer in contact with the lateral surface of the pile, m. Thus, when calculating (Eq. 14) only those layers were taken into account where *τ<sup>z</sup>* decreased or its direction changed.

The proportion of the increase in the load on the pile *ΔP=P/P1*, where *P* was the load on the pile under normal operating conditions, was determined. With these data, the graph for the dependence of the value *ΔP* on the ratio *b/B* was plotted. So, the additional load on the pile can be determined, having the values of the load on the pile in normal operation *P*, the distance from the roof of the karst soils to the bottom of the piles b, and the calculated diameter of the karst cavity *B*.
