The Karst Protection Foundations Design

Abstract

In this chapter, two karst protection methods are analyzed: the structural karst protection method, when designed structures of the underground part prevent the development of the bearing structures strains while karst deformations occur; the geotechnical karst protection method, when the design of a foundation includes a protection geotechnical screen in the base, which excludes or essentially decreases the negative influence of karst development on the bearing structures of the building. The results of numerical calculations are analyzed for different types of foundations. The advantages of the model of the variable coefficient of subgrade reaction are discussed. It is proposed to determine the coefficients of subgrade reaction (stiffness of the piles) around the karst cavity by decreasing these coefficients in relation to the same coefficients, calculated using standard methods (i.e., without karst deformations). Analytical solutions for different types of foundations are presented. Analytical solutions for designing bridge support pile foundations in karst areas are proposed. The correct design model and criteria for stability evaluation of the arch over the cavity are selected and the method for estimating the thickness of an effective karst protection geotechnical screen is proposed.

1. Introduction

The methods for analysis and design of the foundation of buildings and structures in karst areas depend on the complexes of the karst protection measures used. Two possible protection options are suggested:

• the creation of such a constructive scheme of the underground part of a building or structure that will not allow the forces of the bearing structures to exceed the permissible values;

• the installation of a protective geotechnical screen, either at the base of the foundation or above the karstic soils, will eliminate or substantially reduce the negative influence of karst development on the bearing structures.

The choice of protection against karst deformations is determined by the level of karst danger. In Russian Codes (SP 22.13330.2011), the two indicated options are assigned to the corresponding groups of measures of karst protection (structural and geotechnical, respectively) and the requirements for analysis of the karst protection foundations are significantly different.

As the results of the investigations of the karst deformations regularities [1, 2, 3, 4], and in accordance with the accepted classification the deformations can be divided into the holes and subsidences (Figure 1).

Figure 1.

Karst deformation types: a – Karst hole; b – Surface subsidence.

The foundation calculation, providing karst deformations in a base, is the most advisable with due regard for the building and base interaction by the numerical modeling. The most simple decision in the given case is the modeling of the karst hole under the foundation base according to the hole dimensions. In places of the formation of the hole, the soil “goes out” from the foundation base and the load is redistributed to the adjacent parts where the contact of the foundation with the soil is provided. When modeling the building and base interaction with the karst hole, the choice of the base model and the determination of its initial parameters are of great importance.

The more important question is the bridge support pile foundation in the karst areas design. The design of the support pile foundation can be carried out in accordance with the comparison of the load-bearing capacity of the piles and the load transferred to the pile. Studies of the pile behavior during the formation of a karst hole in the base [5, 6, 7, 8] show that additional vertical and horizontal loads are transferred to the piles. The additional vertical loads are considered as “negative friction” and it is indicated that they must be taken into account. However, so far there have been no proposals to define such additional loads either in the normative or scientific literature.

The numerical studies for defining additional loads on the bridge support foundation piles during the karst cavity formation in the soil under the pile bottoms, depending on the distance to the karst soil and the karst cavity predicted size, are effective. As a result of the studies performed, the regularities of changes in the additional load transferred to the pile, depending on the variable parameters, are established and formulas for calculating the support foundation piles above the karst cavity are proposed.

2. Requirements for analysis of the foundation as a constructive measure of karst protection

The purpose of constructive measures of karst protection is to prevent the destruction of the structure when karst deformations occur at the base of the foundation. These measures are designed on the basis of the analyses that ensure a sufficient load-bearing capacity of the foundation and above-foundation structures to accept the additional loads that arise when karst deformations occur in the base. This is usually achieved in two ways:

• by carrying out analysis of the foundation in conjunction with the above-foundation structures for the case of the karst deformation occurrence with the specified parameters;

• by cutting through the karst soils and supporting the foundation on monolithic rocks.

Examples from the practice of design and construction show that the foundations designed for the karst deformations occurrence protects the building or structure from destruction when the karst processes in the base are activated.

However, the inclusion of a foundation that provides effective karst protection of a building can only be guaranteed if it is designed on the basis of calculated positions and initial data corresponding to the nature of the development of karst deformations. The main initial data, in this case, are the design parameters of karst deformations. The design parameters of karst deformations are determined (predicted) depending on their type.

There are three types of karst and suffusion deformations development:

• “Hole”, when the karst cavity develops in karst soils and “floats” under the foundation base;

• “Subsidence” as the result of the karst and suffusion processes development in the cover mass;

• “Local subsidence”, when the karst cavity develops in karst soils or the cover layer, but does not “float” under the foundation base.

The decision which kind of karst deformations is critical is determined by the soil conditions and design features of the projected building or structure. The most dangerous variant of the development of deformations is accepted for design.

For shallow buildings or structures, it is advisable to perform calculations for the occurrence of a karst hole under the foundation base (the design diameter of the karst hole is taken as the design parameter of the karst deformation) or for the formation of a cauldron with the specified parameters.

For buildings or structures with the underground part, the most dangerous can be a karst deformation of the type “local subsidence”, since the foundation is approaching karst soils and the growth of the cavity in them, even if the stability of the arch is maintained, can cause significant additional forces in the bearing structures of the underground part. At the same time, the size of the karst cavity can be adopted as the design parameter of karst deformation, for which its arch is stable. Figure 2 shows an example of determining the size of such a cavity. In this case, the mathematical modeling of the karst cavity growth is performed using the finite element calculation with elastic–plastic model of the soil by eliminating the weakened zones (zones of the local loss of stability) around the karst cavity while maintaining the constant control of the equilibrium conditions of the arch. The growth of the cavity occurs before the maximum value of its diameter is attained, at which the equilibrium condition of the system is satisfied in the pre-limit state of the cover mass soil. Figure 2 shows the lines of equal soil shear strains with the cavity width increase from value b₁ to b₃ in karst soils.

Figure 2.

The zones of equal shear strains: a, b, c – The width of the cavity is b₁, b₂, b₃, respectively (b₁ ˂ b₂ ˂ b₃); d- the cavity width (b₃) due to occurrence of the equilibrium condition of the arch.

To design reliable and economical foundations, it is important to take the effect of the occurrence of karst deformations on the stress–strain state of the base and bearing structures into account. Taking into consideration the fact that the geometric dimensions of karst cavities in karstic rocks are not strictly defined, and the modeling of karst occurrence at the foundation base of a building or structure cannot guarantee the reliability of the results of the foundation analysis, the simplest solution is to model a karst hole under the foundation base in accordance with the dimensions determined by the statistical - probabilistic methods. At the same time, in the places of the formation of the hole, the ground “leaves” from under the foundation base, and the load is redistributed to adjacent areas, in which there is a contact of the foundation with the base. The modeling of the base behavior when karst deformations occur under the foundation base is possible using both elastic–plastic models of the base and the contact model.

The contact base model or the model of the variable coefficient of subgrade reaction, compared with the other base models is the simplest and most understandable for the practicing engineer. It allows both the heterogeneity of the base and its real distribution ability to be taken into account. The use of this model for the foundation in numerical modeling of the building (structure) and the base interaction also allows to reduce the order of the system of equations compared to the elastic and elastoplastic models of the base and, accordingly, the analysis errors.

Practical design experience of Russian engineers-researchers confirms the efficiency of a combined approach used in the foundation’s analysis while karst deformations occurrence is simulated using an elastoplastic soil model for calculating stresses, deformations and coefficient of subgrade reaction of the base. The most effective way to determine the coefficients of subgrade reaction is to use the lowering coefficients with respect to the coefficients of subgrade reaction defined by standard methods without taking karst deformation into account [5].

With karst deformations of a “hole” or “subsidence” type, the compliance of the base is reduced due to the de-compaction of the soil around them with the load increase on these areas in the first case, and with the weakening of the base and unloading of the neighboring stronger sections in the second one. Therefore, it is suggested to determine the coefficients of subgrade reaction (pile stiffness coefficients) for the areas around the karst hole K_h by taking into account the decreasing coefficients ξ with respect to the coefficients of subgrade reaction (pile stiffness coefficients) K defined by standard methods without taking karst deformation into account:

$$K_h=\raisebox{1ex}{$K$}\!\left/ \!\raisebox{-1ex}{$\xi $}\right..$$E1

Based on the results of numerical and field studies, methods for determining of the coefficients ξ for raft, pile-raft, and pile strip foundations have been developed.

Analysis of the raft foundation on the karsted area is usually performed for the karst deformation of a “hole” type when the diameter of the karst hole is taken as the design parameter. In this case, the coefficient of subgrade reaction within the boundaries of the karst hole is equated to zero, and outside these boundaries, it decreases with approaching the hole.

For a building or structure with a developed underground part, such an approach may be erroneous and lead to unpredictable deformations of the base and stresses in the foundation sections, since the karst cavity in the karst soils may be of a larger diameter than the karst hole “floating” as a result of the cavity arch failure. At the same time, due to the proximity of the foundation base to the karst soils, the local subsidence of the base above the cavity will provoke greater forces in the foundation sections than the karst hole under the foundation base of a smaller diameter. Therefore, in this case, it is suggested to take the diameter of the karst cavity in the karst soils (d_p) as the design parameter of karst deformations. It is the maximum diameter of the karstic cavity when the soil cover mass is stable and the cavity does not “float” to the surface in kind of a hole, but there occurs local subsidence of the base above the cavity [9]. As a result of 3D finite element calculation with the elastic–plastic model of the soil, a method for analysis of the coefficient of subgrade reaction for the raft foundation base of a buried building, has been developed. This method allows to determine the decreasing coefficient ξ with respect to the coefficient of subgrade reaction determined without regarding cavity occurrence by any known methods:

$$\xi =\frac{h_k-h_f-\alpha \left(d_p-3\right)}{h_k-h_f+\beta \left(d_p-3\right)},$$E2

where h_k is the depth of the karsting soils; h_f - deepening of the foundation; d_p is the cavity diameter; coefficient α = 0,871-0,0261∙t; coefficient β = 1.2691–0.4163∙t; t is the thickness of the foundation slab; all units are given in meters.

As shown in Figure 3 the subgrade reaction coefficient and pressures under the raft base for the occurrence of karst cavity of the design diameter d_p decrease. The radius of the zone for reducing the coefficient of subgrade reaction (R) is determined by the formula:

Figure 3.

A schematic of the subgrade reaction coefficient and pressures under the raft base for the occurrence of karst cavity of the design diameter d_p.

$$R=\sqrt[4]{\frac{16{\mathit{Et}}^3\beta \left(d_p-3\right)S}{3P\left(5+\mu \right)\left(1-\mu \right)\left(h_k-h_f\right)}},$$E3

where E and μ are the deformation modulus and the Poisson’s ratio of the raft concrete, respectively, P is the pressure in the raft base; S – the settlement under the raft base center, defined before the karst cavity occurrence.

Analysis of the pile-raft foundation in karsted areas is usually performed for karst deformations of a “hole” type. The stiffness ratio of the piles is equated to zero within the boundaries of the karst hole, and outside these boundaries, it is assumed to be constant and is determined by the standard methods, that is, without taking into account the formation of a karst hole.

Due to the peculiarities of the pile-raft foundation behavior, namely, the effect of pile pre-stressing in the soil, a situation is possible when the soil mass, stabilized with piles, accepts stresses of karst deformations and the karst cavity under the pile tips does not develop to the foundation base. In this case, karst deformations should be considered as “local subsidence”. In this case, the forces in the raft sections and, accordingly, the reinforcement of the raft, can be significantly reduced. Considering these features of the pile-raft foundation behavior, a method was developed for the analysis of the stiffness coefficient of the pile foundation above the karst cavity located under the pile bottoms. Analytical solutions were obtained to determine the pressures in the base and the settlements of the raft above the karst cavity [5, 8]. By the results of the analytical investigations using the linear-elastic approach, the method of calculation of the pile deformability ratio above the karst cavity is developed. The stressed-deformed state of the base with the full design column load is analyzed at the moment of the karst cavity formation under the pile bottoms. The pile compression in soil and the extra radial stresses along the pile shaft due to adjacent pile loads are taken into account.

As it is shown in Figure 4 the pile design scheme above the karst cavity is characterized by the radial stresses σ_r and the friction force f along the lateral surface. The axisymmetrical problem of the radial stresses σ_r distribution is solved when each pile in a pile field is loaded with a load P.

Figure 4.

Pile design scheme above karst cavity.

$${\sigma }_r=\frac{1}{r}\frac{\mathit{d\phi }}{\mathit{dr}},$$E4

where φ is defined by the equation of the deformations compatibility, the general integral of which is the function:

$$\mathrm{\phi }=C_1+C_2\mathit{Inr}+C_3{r}^2+C_4{r}^2\mathit{Inr}\mathrm{.}$$E5

The coefficients С₁, С₂, С₃ (Eq.5) are defined by the equilibrium of forces around the piles with the total number of piles m and the distance to the neighbor piles b_i:

$$\underset{\mathrm{L}}{\int }{\mathrm{T}}_{\mathrm{\sigma }}\mathrm{d\sigma }-\sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{b}}_{\mathrm{i}}\underset{\mathrm{L}}{\int }{\mathrm{\phi }}_{\mathrm{i}}\left(\mathrm{\sigma }\right)\mathrm{d\sigma }=0\mathrm{.}$$E6

The values of T_σ are defined according to R. Mindlin solution

$${\mathrm{T}}_{\mathrm{\sigma }}\left(\mathrm{r}=\raisebox{1ex}{$\mathrm{a}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=\frac{\mathrm{P}\left(\mathrm{L}-\mathrm{z}\right)}{\mathrm{L\pi }\left(1-\mathrm{\upsilon }\right)}\left[\frac{4\mathrm{z}\left(1-2\mathrm{\upsilon }\right)}{{\mathrm{R}}^3}+\frac{2\left(1-\mathrm{\upsilon }\right)\left(1-2\mathrm{\upsilon }\right)}{\mathrm{R}\left(\mathrm{R}+2\mathrm{z}\right)}+\frac{6{\mathrm{z}}^3\left(1-4\mathrm{\upsilon }\right)}{{\mathrm{R}}^5}-\frac{3{\mathrm{r}}^6}{{\mathrm{R}}^7}\right],$$

$$R=\sqrt{r^2-4z^2}.$$

As the result of the solution (Eq. 6), the coefficients C (Eq. 5) are defined as the functions of pile length, pile spacing, pile cross-section, and distance Z_i from the soil surface. With the coefficients, C the solution for the evaluation of the radial stresses σ_r from unit loads on the pile is obtained (Eq. 4, Eq. 5).

The condition, when the piles do not “move” in soil and karst deformations should be considered as “local subsidence”, is evaluated by the expression:

$$P_{\mathit{lim}}<\left(\sum _{i=1}^n{\tau }_{i,\mathit{lim}}U{h}_i\right)-{\gamma }_{\mathit{av}}L{a}^2,$$E7

where P_lim is the pile limit load above the karst cavity and is evaluated as P_lim = pa²; p is the pressure transmitted to a raft base; a is the pile spacing; U is the pile perimeter; h_i is the length of the ith section; n is the number of sections by the pile length; L is the pile length; γ_av is the weighted average value of soil density; τ_i,_lim is the soil specific resistance by Coulomb accounting the stress σ_r and the friction force f

$${\tau }_{i,\mathit{lim}}=c_i+\mathit{tg}{\phi }_i\left(P{\sigma }_{\mathit{ri}}+{\gamma }_i{z}_i\beta \right),$$E8

where P is the given pile load (by the calculation of the pile field in conditions of normal operation when the karst holes are not formed); c_i, φ_i, γ_i are the specific cohesion, angle of inner friction, and the soil density of the considered layer, respectively; σ_ri is the stress of pile-soil compression due to the unit loads at the distance _Zi from the soil surface; $\beta$ is the lateral pressure coefficient.

To define the pressure (p) transmitted to a raft base, a problem for the foundation piled raft is solved for the case of a karst cavity under the pile bottoms. The foundation raft is considered as the plate of the infinite radius on the combined base with the karst cavity of r_k radius.

The solutions of Russian scientist Korenev B.G. are used to evaluate the pressure in the raft base and the settlements of the raft base in Bessel functions [5]:

$$p_r=\frac{N}{2\pi }\underset{0}{\overset{\infty }{\int }}\frac{\gamma J_0\left(\mathit{\gamma r}\right)\mathit{d\gamma }}{1+\frac{D}{k_0}{\gamma }^4+\mathit{cD}{y}^4},$$E9

$${\omega }_r=\frac{N}{2\pi }\underset{0}{\overset{\infty }{\int }}\frac{\left(\frac{\gamma }{k_0}+\mathit{c\gamma }\right)J_0\left(\mathit{\gamma r}\right)\mathit{d\gamma }}{1+\frac{D}{k_0}{\gamma }^4+\mathit{cD}{y}^4},$$E10

as well as Hankel conversion for the function c evaluation:

$$c=2\pi \underset{0}{\overset{\infty }{\int }}\mathit{rK}\left(r\right)J_0\left(\mathit{\gamma r}\right)\mathit{dr},$$E11

where k₀,k - pile deformability ratio (k₀), the bed coefficient of the elastic half-space (k); $J_0\left(\mathit{\gamma r}\right)$- the Bessel integral; D- plate cylindrical stiffness Eh²/12 (1-ν²); E-concrete deformation modulus; h-plate thickness; ν- Poisson’s ratio, γ- is defined from the boundary conditions due to the karst cavity of r_k radius formation.

The function $K_{\left(r\right)}$ corresponding to the settlement surface when karst cavity formation under the pile bottoms is taken as:

$$K_{\left(r\right)}=\frac{B}{2\mathit{\pi r}}exp\left(-\mathit{\delta r}\right).$$E12

The parameters B, δ are defined from the boundary conditions due to the karst cavity of r_k radius formation.

By the results of the calculations of the improper integrals of Bessel’s function, the pressures in the raft base (p_r), raft settlements above the karst cavity (ω_r) due to the unit load depending on the radius of the karst cavity under the pile bottoms (r_k), raft thickness (h), and the pile deformability ratio (k₀) are defined.

Using the analytical dependence of the pressure in the raft base on the pile deformability ratio above the hole and solving the inverse problem, the pile deformability ratio (k₀) is calculated that corresponds to the given pressure (p_r).

Analysis of the pile strip foundation with karst deformations of the “hole” type is usually performed by mathematical modeling of the foundation on an unevenly deformed base. For modeling of the base and foundation, in this case, it is advisable not to complicate the calculation model, but, on the contrary, to apply the simplified models. Such a calculation model of the pile strip foundation base when a karst hole occurs is the contact base model, according to which the piles around the karst hole behavior is modeled by constraints of finite stiffness, defined as the stiffness ratio of the piles. The piles under the karst hole are excluded. Based on the results of the experimental and theoretical studies of the stress–strain state of the “pile strip foundation - base” interaction, the regularities of the change in pile behavior around the karst hole were obtained. As a result of 3D finite element calculation with the elastic–plastic model of the soil the analytical solution was developed to determine the stiffness ratio of piles [6]. The analytical solution allows to determine the decreasing stiffness ratio ξ with respect to the stiffness ratio, determined without taking the karst hole into attention. The decreasing stiffness ratio ξ is determined depending on the pile length (L), the hole depth (H), the distance from the pile to the hole boundary (B):

$$\xi =1+\frac{0.041H^4}{L^2\left(B^2+0,04H^2\right)}.$$E13

3. Requirements to analysis as a geotechnical measure of karst protection

One of the most effective geotechnical karst protection measures is the cementation of the cover soils above the karst soils. Schemes of karst protection cementation of the foundation are developed on the basis of the Russian Code (SP 22.13330.2011), which recommends cementation of the cavities and the entire thickness of the karst soils. However, in practice, these strata often reach considerable sizes (from 15 to 20 m), and their cementation to the full depth to monolithic rock soils where karst cavities do not form is not possible due to a significant rise in the cost of construction and technological problems of cementation and control its quality at great depths.

Investigations of the stress–strain state of the artificially strengthened foundation base over the karst cavity [10] made it possible to establish that the most efficient method was the cementation of the soil mass in the roof of the karst soils. In this case, the additional deformations in the foundation base are minimal when cavities occurrence in karst soils. Also, soil collapse into the karst cavity is not allowed if the height of the probable collapse area above the cavity does not exceed the thickness of the artificially strengthened soil layer.

The design forecast of the possibility of soil collapse into the karst cavity is based, as a rule, on the classical view of the distribution of stresses and the mechanism of the arches formation above the karst cavities. The arch above the cavity in the equilibrium state (up to the moment of its collapse) can be considered as an area of increased stresses and deformations, the size of which is determined by the strength and deformation characteristics of the soils. When cementing a soil layer of a given thickness over the karst soils in which the cavity growth is predicted, this area depends on the thickness and characteristics of the cemented soils, as well as on the maximum predicted cavity size for the standard operating life of the building and on the building loads.

The state of the collapse process characterizes the excess of the boundary values of the tensile and compressive stresses around the cavity that can be obtained from Mohr’s circle of stress. Therefore, the boundaries of the region of increased stresses and the formation of shear strains can be determined using the strength condition according to Mohr-Coulomb failure criteria, taking it as the boundary condition of rock flow and its collapse. Thus, the boundaries of a possible collapse area are defined as the locus of points at which the Mohr-Coulomb failure criteria are met.

Determination of additional load on the bridge foundation pile under karst deformation.

During the formation of a karst hole in the base of the bridge support pile foundation the additional loads are transferred to the piles. The additional vertical loads can be taken into account as the tangential stresses on the lateral surface of the piles, directed to the pile base.

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:

• the soil mass under the rocky karst soil is represented by firm clay with the characteristics specified in Table 1;

• a square-shaped grillage combining 36 piles with a diameter of 1.2 m and a length of 33 m.;

• the size of the calculated area L (along X and Y axes) was determined by the condition that it did not affect the results of the calculation, the nodes at the boundaries of the area were fixed;

• the size of the calculated area H (along Z-axis) was limited by the roof of the rocky karst soils, the nodes at the boundaries of the area were fixed, except for the predicted karst cavity;

• the predicted karst cavity was represented by the absence of anchoring of the nodes (along Z-axis) within it;

• the pile-soil contact was taken into account with the help of special interface elements.

• The Finite element model section and scheme of the calculated foundation is shown in Figure 5.

Characteristic name	Clay
Density, kN/m^3	18.0
Deformation modulus, MPa	25.0
Angle of internal friction, degree	20
Cohesion, kPa	80.0

Table 1.

Physical and mechanical characteristics of soils.

Figure 5.

Finite element model (section).

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;

• estimated cavity size during operation (B): 3 m, 5.5 m, 7.8 m, 10 m;

• distributed load over the grillage top: 400 kN/m2, 550 kN/m² (corresponds to the load on the pile of 2800 kN and 3900 kN).

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:

• The initial stress–strain condition of the soil mass was determined;

• The pile foundations of the supports were calculated for the design loads under normal operating conditions and the tangential stresses on the lateral surface of the piles were determined;

• The cavity size growth in karst soils located at a given distance from the bottom of the piles was determined step-by-step and the tangential stresses on the lateral surface of the piles were defined.

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 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.

Figure 7.

Mohr-coulomb points above cavity.

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, 12, 13, 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 P₁, kN, was determined by the formula:

Figure 8.

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

$$P_1=u·\sum ∆{\tau }_{z,i}·h_i,$$E14

where: u is the perimeter of the pile, m; $∆{\tau }_{z,i}$ is the change in the shear stress value on the pile lateral surface in the considered ith layer in comparison with the design phase under normal operating conditions, kN/m²; h_i is the thickness of the ith 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 ${\tau }_z$ decreased or its direction changed.

The proportion of the increase in the load on the pile ∆P=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 ∆P 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.

4. Analysis of finite element study results

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

• the position of the Coulomb-Mohr points above the karst cavity to assess the size of the “collapse arch” and control the development of that arch to the bottom of the piles (Figure 7);

• change of the tangential stresses on the lateral surface of the pile (Δτ_z kN/m²) during the growth of the karst cavity (Figure 8).

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 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 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 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.

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/m², 550 kN/m².

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:

Figure 11.

Dependence of P/P1 on b/B ratio.

$$P/P_1=2,1e^{1,7\bullet \frac{b}{B}}.$$E15

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):

$$P_1=P/2,1e^{1,7\bullet \frac{b}{B}}.$$E16

5. Conclusions

1. To design the reliable and economical foundations, the practical design experience of Russian engineers-researchers confirms the efficiency of a combined approach used in the foundation’s analysis while karst deformations occurrence based on the use of the finite-element method with elastoplastic soil model for calculating stresses, deformations, and coefficient of subgrade reaction of the base.

2. The most effective way to determine the coefficients of subgrade reaction is to use the lowering coefficients with respect to the coefficients of subgrade reaction defined by standard methods without taking karst deformation into account.

3. Based on the results of numerical and field studies with the elastoplastic soil model, methods for determining the lowering coefficients for pile-raft, pile strip, and raft foundations have been developed [5, 6, 9].

4. One of the most effective geotechnical karst protection measures is the grouting of the cover mass above the karst soils. In this case, the soil collapse into the karst cavity will not occur if the height of the probable collapse area above the cavity does not exceed the thickness of the artificially strengthened soil layer. The boundaries of a possible collapse area can be defined as the locus of the Mohr-Coulomb failure criteria points [10].

5. On the basis of numerical calculations for bridge support pile foundation, it was shown that there were additional loads on the piles under karst deformations at the base of the foundation. The value of those loads for clay cover layers over karst soils was determined.

6. For bridge support pile foundation it was established that additional loads largely depended on the distance from the bottom of the piles to the karst soils roof and the cavity diameter in the karst soils. The method for calculating the additional loads on the bridge support piles depending on the size of the cavity and the distance from the bottom of the piles to the karst soil roof has been developed.

Acknowledgments

The author expresses gratitude to his graduate students Kayumov M., Davletyarov D., Evdokimov A., together with whom the research was carried out and the solutions presented in this paper were obtained.