An analytic solution of stress distribution around a shallow buried hollow cylinder karst cave in limestone strata

: The chief objective of the article is to learn the spatial characteristics of stress distribution around a shallow buried cylinder karst cave in limestone strata. Firstly, taking into account the geometry of limestone formations, and the characteristics of karst geomorphology in China, a spatial axial-symmetrical hollow model was established. Concurrently, combing available work and the concept of elasticity, the boundary conditions are determined. Subsquently, Love displacement method was introduced, in addition to the expressions of stress components were gained. The diagram characteristics of each stress component are summarized, which are affected by various influencing factors. Finally, in order to prove the rationality of the general solution, a numerical simulation was carried out on the basis of 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 cylinder karst cave in limestone strata.

problem in mantled Karst region, and brings great loss to people's life and property. According to incomplete statistics, the annual economic loss caused by karst ground collapse in 24 provinces (cities and districts) in China amounts to more than 120 million yuan. The losses are related to agricultural engineering, highway engineering, railway engineering, mining engineering, industrial and civil construction engineering, etc 5-8 . Therefore, the prevention and control of ground collapse is an important requirement for national security and economic development.
Commonly, carbonate rocks are softened by faintly acidic waters 9-11 . Then, a series of buried karst caves formed near subsurface. Karst caves may provide transport channel or storage space for overlying rock and soil mass, and cause the change of spatial stress distribution. Furthermore, affected by natural and human factors, ground collapse may happen. In order to investigate the effect on spatial stress distribution ,caused by buried Karst caves, a lot of research work was performed. Goodier 12 presented that there is a concentration around void or defect. In order to achieve quantitative expression of stress distribution, Howland et al. 13 simplified the problem into a thin plate containing circular holes, and inverse method was used to solved the problem. Taking into account the variety of Karst caves and the complexity of loading conditions, Rao et al. 14 analysed the stresses of surrounding rock containing a tubular filled elliptical karst cave, and the analytic general formula for the component of stresse were solved. Li et al. 15 established the plane mechanical models with different stress boundary conditions in each direction, and the exact general solution was obtained. Shi et al. 16 summarized the stress distribution around rectangular cavity. Considering the spatial geometric characteristics of Karst caves, Liao et al. 17 provided the extreme value of critical point on the wall of an elliptical spherical cavity under triaxial stress.
To sum up, A large number of studies on the stress distribution characteristics of surrounding rock containing shallow buried Karst caves has been made. However, complex function theory was mainly used, and the method is relatively single. In addition, spatial geometry of strata is neglected. Therefore, this paper is to study the spatial stress distribution characteristics around a shallow buried cylinder karst cave in limestone strata. Firstly, taking into account the geometry of limestone formations, a spatial axial-symmetrical model was established. Concurrently, on the basis of 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.

Ⅱ. MECHANICAL MODEL
According to the characteristic of karst geomorphology in China, karst caves are usually shallow buried, and affected by external load and stress field in the Earths crust (Fig. 1). The mechanical model can be simplified into a spatial axial-symmetrical model in this study (h<2.5D, the hidden depth of the karst cave is h, the span of the karst cave is D), and the cylindrical coordinates (r, θ, z) is chosen as the coordinate system (Fig. 2). The parameters are γ (unit weight), μ (poisson's ratio) and E (elastic modulus) respectively. The effect generated by external loads and the gravity of overlying limestone are simplified into vertical uniform distributed loads. In addition, stress field in the Earths crust will cause vertical stress on the bottom and horizontal stress surrounding the karst cave (the side force coefficient k0=μ/(1-μ)). Combining available work and the concept of elasticity, the boundary conditions are as follows: . THEORETICAL ANALYSIS A. The basic theory 18 Taking into account the influence of gravity, the equilibrium differential equations are expressed as 0 0 r r zr z zr zr Love displacement method is an effective way, the stress components are as follows Where  2 is Laplace operator
Matching the equivalent coefficients of both sides of Equation (11) Substituting boundary conditions (4) and (5) into Equation (4) and (7) Combining Equation (8)- (10) and Equation (12)-(16), the undetermined coefficients are solved Substituting the expressions of the undetermined coefficients into Equation (4)-(7) yields For radial pressure and tangential pressure, there are two influencing factors， one is the radius (r), another is the depth (z). The figures of circumferential stress and radial stress are comparable to a paraboloid (Fig. 3). Nevertheless, the tendency of value alteration is the opposed. Furthermore, to discuss the effect of the depth (z) as well as the radius (r), the variation curves under the influence of single factor are drawn when depth (z=3m) and radius (r=1m) are constant respectively (Fig. 4). Fig. 4(a) shows that the circumferential stress and radial stress has a reverse trend with the increase of radius value, but the change ratio is consistent. In the meanwhile, with the increase of depth, the values of the circumferential stress and radial stress increase commonly. and for the change ratio, the circumferential stress is more obvious (Fig. 4(b)). . 4. The distribution of radial stress and circumferential stress effected by single influencing factor (a) depth is constant (z=3m), radius varies from 0.5m to 5.5m (b) radius is constant (r=1m), the depth varies from 0m to 6m Fig. 5 and Fig. 6 are the diagrams of vertical stress and shear stress respectively. For vertical stress, depth (z) is the influencing factor, in addition to the curve of vertical stress is linear (Fig. 5). For shear stress, radius (r) is the influencing factor, in addition to the curve of shear stress is nonlinear (Fig.  6).

B. Validation test of general solution
So as to confirm the justification of general solution, a mathematical model was conducted, and rectangular coordinate system was employed. According to ground penetrating radar survey and laboratory analysis of rock in Chongqing, The parameters of buried karst cave and thick limestone are shown in Table 1.The dimension and mesh of numerical model are shown in Fig. 7. Through mathematical model, constraints were applied to the bottom of the model in the upright 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 p2=8.833.5z (z is vertical coordinate value), which is caused by stress field in the Earths crust (Fig. 8). The facts on the line L1 were recycled to monitor the horizontal as well as vertical stress (Fig. 9), in addition to the monitoring results are revealed in Table 2    Owing to theoretical analysis and numerical simulation were carried in different coordinate system, the monitoring statistics was converted into the standards in cylindrical coordinates using Equations (21)-(23). Fig. 10 presents the monitoring data and calculation value of stress component together. For vertical stress, the theoretical calculation value is consistent with monitoring facts. For radial stress as well as tangential stress, the extreme fault is not more than 5.0%, which meets the proposal standard necessity in practice.

Ⅴ. DISCUSSION
In summary, the expressions of stress components are composed of power function, which are beneficial to the application in practice. Nevertheless, there is a fault (the maximum fault is not more than 5.0%) for radial stress as well as tangential stress, it's necessary to optimize the analytic solution in the future. On the one hand, increase the type of Love displacement function component may be an effective way. On the other hand, various method could be introduced.

Ⅵ. CONCLUSION
Taking into account the geometry of limestone formations, and the characteristics of karst geomorphology in China, a spatial axial-symmetrical hollow model was generated. A general solution of stress components was gained using Love displacement method .
For radial stress and tangential stress, there are two influencing factors，one is the radius (r)， another is the depth (z). The diagrams of the radial stress as well as tangential stress are comparable to a paraboloid, in addition to the values tend to a common constant. Nevertheless, the tendency of value variation is the contradictory. When the value of depth (z) is consistent, the radial stress and circumferential stress has a reverse trend with the increase of radius value, but the change ratio is consistent. In the meanwhile, when the value of radius (r) is consistent, the values of the circumferential pressure and radial pressure growth commonly with the increase of depth. For the change ratio, the circumferential stress is more obvious. The depth (z) as well as the radius (r) are influencing factor for vertical stress and shear stress respectively. The curve of the vertical stress is linear, in addition to the diagram of shear stress is nonlinear.
Based on ground penetrating radar survey and laboratory analysis of rock in Chongqing, a mathematical model was conducted, and the judgement in the middle of monitoring data and calculation value of stress component was carried out. For vertical stress, the calculation value using general solution is consistent with the monitoring data. For radial stress as well as tangential stress, the maximum fault is not more than 5.0%, which meets the proposal standard necessity in practice. As a result, the analytical solution could represent the spatial characteristics of stress distribution around a shallow buried cylinder karst cave in limestone strata.