Theoretical analysis on stress distribution characteristics around a shallow buried spherical Karst cave containing fill materials in limestone strata

The chief objective of the article is to learn the spatial characteristics of stress distribution around a shallow buried spherical Karst cave containing fill materials in limestone strata. First, considering the effect of external load, stress field in the Earth’s crust, internal filling and Karst geomorphology characteristics in China, a spatial axisymmetrical model was established. Concurrently, combing available work and the concept of elasticity, the boundary conditions are determined. Subsequently, the Love displacement method was introduced, and stress component expressions were obtained. The diagram characteristics of each stress component are summarized, which are affected by various influencing factors. Finally, to prove the rationality of the general solution, a numerical simulation was carried out based on practical engineering, and the maximum error is less than 5%. Thus, the analytical solution could represent the spatial characteristics of stress distribution around a shallow buried spherical karst cave containing fill materials in limestone strata.


Introduction
Karst geomorphology exists widely in the world, and the Karst cave is one of the typical representatives. Up to now, various patterns of Karst cave has been found, including columnar, alter shaped, spherical, funnel shaped and so on (Gutierrez et al. 2014;Xie et al. 2018a), and filled by fill materials (such as clay, water, air and so on). Unlike abroad (Zhao et al. 2012), most of this area is shallow depth. Due to the rapid urban development and growing terrestrial occupation, the dimensions and speed of engineering construction have significantly accelerated (Xie et al. 2018b). Ground collapse is a common engineering problem in the mantled Karst region, and seriously losses people's lives and property. The losses are related to agricultural engineering, highway engineering, railway engineering, mining engineering, industrial and civil construction engineering, etc. (Scotto et al. 2018;Kaufmann et al. 2018;Wan et al. 2018;Harris et al. 2018). Therefore, the prevention and control of ground collapse is an important essential requirement for national security and economic development.
Generally, carbonate rocks are dissolved by faintly acidic waters (Zhang et al. 2007;Williams et al. 2008;Piccini et al. 2009), Karst caves may provide transport channels or storage space for overlying rock and soil mass, and cause the change of spatial stress distribution. Many research works were performed to investigate the effect on spatial stress distribution, caused by buried Karst caves. Goodier (1933) presented that there is a concentration around void or defect. To achieve quantitative expression of stress distribution, Howland et al. (1939) simplified the problem into a thin plate containing circular holes, and the inverse method was used to solved the problem. Considering the variety of Karst caves and the complexity of loading conditions, Rao et al. (2015) analyzed the stresses of surrounding rock containing a tubular filled elliptical Karst cave, and the general analytic formula for the stress components was solved. Li et al. (2014) established the plane mechanical models with different stress boundary conditions in each direction, and the exact general solution was obtained. Shi et al. (2014) summarized the stress distribution around the rectangular cavity. Considering the spatial geometric characteristics of Karst caves, Liao et al. (2010) provided the extreme value of the critical point on an elliptical spherical cavity wall under triaxial stress.
To sum up, many studies on the stress distribution characteristics of surrounding rock containing shallow buried Karst caves have been made. However, complex function theory was mainly used, and the method is relatively single. In addition, the spatial geometry of strata and internal filling are neglected. Therefore, this paper will study the spatial stress distribution characteristics around a shallow buried spherical Karst cave containing fill materials in limestone strata. First, taking into account the geometry of limestone formations, a spatial axial-symmetrical model was established. Concurrently, based on the theory of elasticity, the expressions of stress components were obtained. Finally, a numerical simulation was carried out to prove the rationality of the general solution.

Mechanical model
According to the characteristic of Karst geomorphology in China, Karst caves are usually affected by internal filling, external load, and stress field in the Earth's crust (Fig. 1). To facilitate analytical analysis, the basic assumptions are as follows: ① the limestone strata can be simplified into a spatial axial-symmetrical model. ② the spherical Karst cave containing fill materials are shallowly buried in limestone strata (h < 2.5D, the hidden depth of the Karst cave is h, the diameter of the Karst cave is D). ③ The limestone strata is homogeneous, continuous, and isotropic. In addition, the parameters are γ (unit weight), μ (Poisson's ratio), and E (elastic modulus), respectively.
According to the above assumptions, the spherical coordinates (r, θ, φ) are chosen as the coordinate system (Fig. 2). Thus, the effect generated by external loads and the gravity of overlying limestone is simplified into vertical uniform distributed loads. In addition, the stress field in the Earth's crust will cause vertical stress on the bottom and horizontal stress surrounding the Karst cave (the side force coefficient The parameters in Figs. 1 and 2 are: p z -external vertical load; p 0 -horizontal stress caused by external vertical load and field in the Earth's crust, p 0 = k 0 [p z + γ(h + z)]; p i -radial stress caused by fill materials; h-The vertical distance from the top of the limestone strata to the center of the spherical Karst cave; R-the radius of the buried spherical Karst cave; k 0 -the side force coefficient, k 0 = μ/(1 − μ).

Boundary conditions
Combining available work and the concept of elasticity, the boundary conditions are: 1. z = −h , z = p z is the vertical stress generated by external vertical load; 2. r → ∞ , r = [p z + (h + z)] ∕(1 − ) is the radial stress considering external vertical load and the gravity of the medium of the strata; 3. r → ∞ , = [p z + (h + z)] ∕(1 − ) is the tangential stress considering external vertical load and the gravity of the medium of the strata; 4. R = R 1 , R = p i is the internal pressure considering the gravity of fill materials of the buried spherical Karst cave.

Theoretical analysis
The basic theory (Timoshenko et al. 1965) Taking into account the influence of gravity, the equilibrium differential equations are expressed as The Love displacement method is an effective way, and the stress components are as follows: where ∇ 2 is Laplace operator.
The relationship of the stress components between spherical coordinates and cylindrical coordinates are as follows: (1)

The general solution
To obtain the general solution of stress components using Love displacement method mainly includes three parts: established the Love displacement function, the undetermined coefficients of each stress components expressions were solved, and the result was processed (Fig. 3).
(2) yields Substituting the expressions of the stress components into Eq. (3) yields [15 cos 6 + 15 sin 4 cos 2 + 30 sin 2 cos 4 − 3 sin 4 − 6(1 + ) cos 4 − 3(3 + 2 ) sin 2 cos 2 + (1 − 2 ) sin 2 − 6(1 + ) cos 4 − 3(3 + 2 ) sin 2 cos 2 Then, the general solution of the stress components are obtained, which can consider the common effect of the external load effects, stress field in the Earth' s crust, internal filling, and the Karst landform characteristics in China. In general, the value of is definite for a specific site. Therefore, the radius (R) and the angle (φ) are two influencing factors for the characteristics of the stress distribution in the strata.
When an angle is constant (φ = ℼ/2), the circumferential stress and radial stress curves are non-linear. Nevertheless, the tendency of value alteration is the opposed to the increase of radius value (Fig. 4a). Figure 4b indicates that tangential stress value increases with the increase of radius value, and tends to a constant value. In addition, the relationship between shear stress and radius is linear (Fig. 4c).
Furthermore, to discuss the effect of angle (φ), the curves of each stress component are drawn in Fig. 5 (R = 2 m). Figure 5 shows the curves of each stress component are symmetrical, and φ = ℼ/2 is the axis of symmetry except shear stress (angle (φ) varies from 0 to 2ℼ). In addition, the value of shear stress is positive, as well as the other stress component contains a positive and negative value.
In summary, the result is agreed with the previous research (Goodier et al. 2007). In the meanwhile, the characteristics of spatial stress distribution was presented through above discussion, which will provide scientific evidence for bearing capacity determination of foundation containing shallow buried Karst cave in the further research.

Validation test of the general solution
To verify the validation of the proposed analytical solution, a numerical simulation was conducted, and a rectangular coordinate system was employed.
To understand the features of the stratum structure and Karst cave, a GPR survey is conducted along these survey lines in Chongqing. Owing to the different attributes of the media it penetrates, a part of the signals emitted is reflected by the interface between the various materials. The reflected signals are received, magnified, and digitized by a receiver and then relayed to the mainframe for storage. The signals in the GPR profiles are enhanced via data processing, and the higher accuracy and better visual geophysical signatures are represented. Detailed information is provided by images interpretation, including information of the embedded depth, boundary of the Karst caves, and the thickness of the soil and inter-layer. A spatial database in coordinate system relative to the geological map could be easily positioned. Finally, based on this spatial database, a series of two dimensional geological cross-sections are obtained, and a spatial model was generated. Geological survey records are required as the sources of the geological information. Timely data are collected during geological mapping or other field works, consisting of borehole, elevation, and outcrop descriptions along with the lithological information. Laboratory tests are used to obtain the parameters of the rock and soil materials. The dimension of shallow buried Karst cave and thick limestone parameters are shown in Table 1. Considering the model symmetry, a 1/2 geometry model was selected, and a numerical model are shown in Fig. 6. Through mathematical model, constraints were applied to the bottom of the model in the vertical and horizontal direction, and the excavation of void is used to simulate the formation of natural Karst cave. Horizontal stress surrounding the Karst cave is p 2 = 8.833.5z (z is vertical coordinate value), caused by stress field in the Earth's crust (Fig. 7). The red facts were recycled to monitor the horizontal as well as vertical stress (Fig. 8).
Theoretical analysis and numerical simulation were done in different coordinate systems, and the monitoring statistics were converted into the standards in spatial coordinates using Eqs. (22)-(24). Figure 9 presents the monitoring data and calculation value of the stress component together. The theoretical calculation value is consistent with monitoring facts, and the extreme fault is not more than 5.0%, which meets the proposal standard necessity in practice.

Discussion
In summary, the expressions of stress components are composed of power functions, which are beneficial to the application in practice. Nevertheless, there is a fault (the maximum fault is not more than 5.0%) for the stress component. Therefore, it is necessary to optimize the analytic solution in the future. On the one hand, increasing the type of Love (22) r = x cos 2 + y sin 2 + 2 xy sin cos (23) = x sin 2 + y cos 2 − 2 xy sin cos Fig. 4 The distribution of stress component affected by a single influencing factor ((φ = ℼ/2, and radius (R) varies from 0.5 m to 5.5 m). a The distribution of circumferential stress and radial stress, b the distribution of tangential stress, c the distribution of shear stress On the other hand, the various method could be introduced. The characteristics of spatial stress distribution could be presented through using the general solution relatively well. In the further research, the formula of foundation bearing capacity calculation in the hidden Karst region could be obtained, combining failure mechanism of ground collapse and the expressions of stress components.

Conclusion
Considering the effect of external load, stress field in the Earth's crust, internal filling and the characteristics of Karst geomorphology in China, a spatial axial-symmetrical model was generated. A general solution of stress components was gained using the Love displacement method.