**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

**Figure 1.** *Karst deformation types: a – Karst hole; b – Surface subsidence.*

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

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:


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:


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

**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.*

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.

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

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 Kh 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:

$$K\_h = {}^\mathbb{K}\!/\!\_\xi. \tag{1}$$

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 (dp) 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:

$$\xi = \frac{h\_k - h\_f - a(d\_p - 3)}{h\_k - h\_f + \beta(d\_p - 3)},\tag{2}$$

where *hk* is the depth of the karsting soils; *hf* - deepening of the foundation; *dp* 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 *dp* 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* dp*.*

$$R = \sqrt[4]{\frac{16 \operatorname{Et}^3 \beta \left(d\_p - 3\right) \mathbf{S}}{3P(\mathbf{5} + \mu)(\mathbf{1} - \mu)\left(h\_k - h\_f\right)}},\tag{3}$$

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 t*he 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<sup>r</sup> and the friction force *f* along the lateral surface. The axisymmetrical problem of the radial stresses σ<sup>r</sup> distribution is solved when each pile in a pile field is loaded with a load *P*.

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

**Figure 4.** *Pile design scheme above karst cavity.*

$$
\sigma\_r = \frac{1}{r} \frac{d\rho}{dr},
\tag{4}
$$

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

$$\Phi = \mathbf{C}\_1 + \mathbf{C}\_2 I m r + \mathbf{C}\_3 r^2 + \mathbf{C}\_4 r^2 I m r. \tag{5}$$

The coefficients С1, С2, С<sup>3</sup> (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 bi:

$$\int\_{\mathcal{L}} \mathbf{T}\_{\sigma} \mathbf{d} \sigma - \sum\_{i=1}^{m} \mathbf{b}\_{i} \int\_{\mathcal{L}} \boldsymbol{\uprho}\_{i}(\sigma) \mathbf{d} \sigma = \mathbf{0}.\tag{6}$$

The values of T<sup>σ</sup> are defined according to R. Mindlin solution

$$\mathbf{T}\_{\sigma}(\mathbf{r} = \omega\_{2}) = \frac{\mathbf{P}(\mathbf{L} - \mathbf{z})}{\mathbf{L}\pi(\mathbf{1} - \mathbf{u})} \left[ \frac{4\mathbf{z}(\mathbf{1} - 2\mathbf{u})}{\mathbf{R}^{3}} + \frac{2(\mathbf{1} - \mathbf{u})(\mathbf{1} - 2\mathbf{u})}{\mathbf{R}(\mathbf{R} + 2\mathbf{z})} + \frac{6\mathbf{z}^{3}(\mathbf{1} - 4\mathbf{u})}{\mathbf{R}^{5}} - \frac{3\mathbf{r}^{6}}{\mathbf{R}^{7}} \right],$$

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

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 Zi from the soil surface. With the coefficients, *C* the solution for the evaluation of the radial stresses σ<sup>r</sup> 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:

$$P\_{lim} < \left(\sum\_{i=1}^{n} \tau\_{i,lim} \, \mathbf{U} h\_i \right) - \chi\_{av} \mathbf{L} a^2,\tag{7}$$

where Plim is the pile limit load above the karst cavity and is evaluated as Plim = pa<sup>2</sup> ; *p* is the pressure transmitted to a raft base; *a* is the pile spacing; *U* is the pile perimeter; *hi* is the length of the *i*th section; n is the number of sections by the pile length; *L* is the pile length; γav is the weighted average value of soil density; *τ*i,lim is the soil specific resistance by Coulomb accounting the stress σ<sup>r</sup> and the friction force *f*

$$\pi\_{i,lim} = \mathfrak{c}\_i + \mathfrak{tg}\,\rho\_i(\mathcal{P}\sigma\_{ri} + \chi\_i\mathbf{z}\_i\boldsymbol{\beta}),\tag{8}$$

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); *ci, φi, γ<sup>i</sup>* are the specific cohesion, angle of inner friction, and the soil density of the considered layer, respectively; *σri* is the stress of pile-soil compression due to the unit loads at the distance Zi from the soil surface; *β* 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 *rk* 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]:

$$p\_r = \frac{N}{2\pi} \int\_0^\infty \frac{\gamma J\_0(\gamma r) d\gamma}{1 + \frac{D}{k\_0}\gamma^4 + cD\gamma^4},\tag{9}$$

$$\rho\_r = \frac{N}{2\pi} \int\_0^\infty \frac{\left(\frac{\nu}{k\_0} + c\gamma\right) I\_0(\chi r) d\chi}{1 + \frac{D}{k\_0}\chi^4 + cD\chi^4},\tag{10}$$

as well as Hankel conversion for the function *c* evaluation:

$$\mathcal{L} = 2\pi \int\_0^\infty rK(r)J\_0(\mathcal{yr}) dr,\tag{11}$$

where k0,k - pile deformability ratio (*k0*), the bed coefficient of the elastic halfspace (k); *<sup>J</sup>*0ð Þ *<sup>γ</sup><sup>r</sup>* - the Bessel integral; D- plate cylindrical stiffness Eh<sup>2</sup> /12 (1-ν 2 ); E-concrete deformation modulus; h-plate thickness; ν- Poisson's ratio, γ- is defined from the boundary conditions due to the karst cavity of rk radius formation.

The function *K*ð Þ*<sup>r</sup>* corresponding to the settlement surface when karst cavity formation under the pile bottoms is taken as:

$$K\_{(r)} = \frac{B}{2\pi r} \exp\left(-\delta r\right). \tag{12}$$

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

By the results of the calculations of the improper integrals of Bessel's function, the pressures in the raft base (*pr*), raft settlements above the karst cavity *(ωr*) due to the unit load depending on the radius of the karst cavity under the pile bottoms (*rk*), 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 (*pr*).

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):

$$\xi = 1 + \frac{0.041H^4}{L^2(B^2 + 0, 04H^2)}.\tag{13}$$
