METHOD FOR QUANTIFYING BEARING CAPACITY OF FOUNDATION CONTAINING SHALLOW-HIDDEN SPHERICAL CAVITIES

Abstract

The present disclosure discloses a method for quantifying a bearing capacity of foundation containing shallow-hidden spherical cavities, comprising: in Step 1, constructing a spatial axisymmetric calculation model for stability analysis of the foundation containing shallow-hidden spherical cavities; in Step 2, solving the model to obtain a general solution which reflects the spatial stress distribution of surrounding rock containing shallow-hidden spherical cavities; in Step 3, obtain a mathematical expression by derivation for calculating the bearing capacity of the foundation containing shallow-hidden spherical cavities; and in Step 4: completing the determination of the foundation bearing capacity. Benefits: This method has many advantages such as comprehensive consideration, high accuracy and reliability of calculation results, and may provide the scientific basis for the development of prevention and control against the instability of the foundation containing shallow-hidden cavities. This method is easy to operate and feasible to become popular in actual engineering projects.

Claims

1. A method for quantifying bearing capacity of foundation containing shallow-hidden spherical cavities, comprising: In Step 1, by comprehensively considering a combined effect of spatial three-dimensional characteristics of a shallow-hidden spherical cavity, a self-weight stress of an overlying rock-soil layer, a virgin rock stress, and an external vertical load, a spatial axisymmetric calculation model is constructed for stability analysis of the foundation containing shallow-hidden spherical cavities; In Step 2, by selecting a spherical coordinate system and introducing the Laplace displacement method, the model is solved through theoretical analysis to obtain a general solution which reflects the spatial stress distribution of surrounding rock containing shallow-hidden spherical cavities; In Step 3, by combining the general solution obtained through theoretical analysis in Step 2, a damage mode of ground collapse, and the Mohr-Coulomb strength theory, a mathematical expression for calculating the bearing capacity of the foundation containing shallow-hidden spherical cavities is derived; In Step 4, by substituting data of a vertical load of stratum, a gravity density of a rock-soil layer, a Poisson's ratio of rock-soil mass, a thickness of an overlying stratum, and a radius of a spherical cavity that are measured in actual engineering projects into the mathematical expression in Step 3 for calculation, the bearing capacity of the foundation may be determined.

2. The method for quantifying bearing capacity of foundation containing shallow-hidden spherical cavities according to claim 1, wherein: the spatial axisymmetric calculation model in Step 1 and related parameters are as follows: p.sub.z—external load borne by the ground; p.sub.0—horizontal load of virgin rock stratum, p.sub.0=μ[p.sub.z+γ(h+z)]/(1−μ); γ—gravity density of rock-soil layer, in kN.Math.m; μ—Poisson's ratio of rock-soil mass; h—thickness of overlying stratum, in m; R—distance from any point to spherical cavity center, in m.

3. The method for quantifying bearing capacity of foundation containing shallow-hidden spherical cavities according to claim 1, wherein: the calculation formulas of the general solution in Step 2 are as follows: $\begin{array}{cc}{\sigma }_R=\left\{4\left[16\left(1-\mu \right)A_2-A_4\right]{\sin }^2\mathrm{\phi cos\phi }+8\left[\left(2-\mu \right)A_4\right]{\cos }^3\phi \right\}R+\left\{2\left[3\mu A_3+\left(2\mu -1\right)A_5\right]{\sin }^2\phi +22\left[\left(1-\mu \right)A_3+2\left(2-\mu \right)A_5\right]{\cos }^2\phi \right\}+A_9\left[15{\cos }^6\phi +15{\sin }^4{\mathrm{\phi coscos}}^2\phi +30{\sin }^2{\mathrm{\phi cos}}^4\phi -3{\sin }^4\phi -6\left(1+\mu \right){\cos }^4\phi -3\left(3+2\mu \right){\sin }^2{\mathrm{\phi cos}}^2\phi +\left(1-2\mu \right){\sin }^2\phi +\left(2\mu -1\right){\cos }^2\phi \right]\frac{1}{R^3}& \left(1.1\right)\\ {\sigma }_{\theta }=\left\{4\left[\left(2\mu -1\right)A_4\right]\cos \phi \right\}R+2\left[3\mu A_3+\left(2\mu -1\right)A_5\right]+A_9\left[3\left(2\mu -1\right){\cos }^2\phi +\left(1-2\mu \right)\right]\frac{1}{R^3}& \left(1.2\right)\\ {\tau }_{R\phi }=\left\{4\left[8\left(1-\mu \right)A_2+\left(3\mu -5\right)A_4\right]{\sin }^2{\mathrm{\phi cos}}^2\phi -4\left[8\left(1-\mu \right)A_2-\mu A_4\right]{\sin }^3\phi \right\}R+[6\left(2\mu -1\right)A_3+2\left(4\mu -5\right)A_5]\sin \mathrm{\phi cos\phi }+2A_9\left(1+\mu \right)\frac{1}{R^3}\sin \mathrm{\phi cos\phi }& \left(1.3\right)\\ A_2=\frac{\left(6\mu -1\right)}{64\left(1-\mu \right)\left(1-2\mu \right)}\gamma A_3=-\frac{B_5}{6B_2}{A}_4=\frac{\mu }{4\left(1-\mu \right)\left(2\mu -1\right)}\gamma A_5=-\frac{B_6}{2B_2}{A}_9=-\frac{B_7}{B_2}{B}_5=2\left(2\mu -1\right)h^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi ]-\mu \left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]-4\mu h^3\left(2-\mu \right)\left(2\mu -1\right)\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]+\left(2\mu -1\right)R_1^3\left[15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi \right][\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }]B_2=\left(1-\mu \right)\left(2\mu -1\right)\left\{R_1^3\left(1-2\mu \right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+R_1^3\mu \left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+4\mu h^3\left(2-\mu \right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]+2h^3\left(1-\mu \right)\left(1-2\mu \right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]\right\}{B}_6=-2\mu h^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi ]+\mu \left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+2\mu h^3\left(1-\mu \right)\left(2\mu -1\right)\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]-\mu \left[15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi \right][\left(6\mu -1\right)\left(1-\mu \right)R_1^4{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1^4{\mathrm{\gamma cos}}^3\phi +\mu R_1^4{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }]B_7=h^3\left\{\left(1-2\mu \right)R_1^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi ]+\mu R_1^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi ]+2\mu \left(2-\mu \right)\left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right]-\mu \left(1-\mu \right)\left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right]-2\mu \left(2-\mu \right)R_1^3\left[\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }\right]+\left(1-\mu \right)\left(2\mu -1\right)R_1^3\left[\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }\right]\right\}& \left(1.4\right)\end{array}$ Related symbols in the formulas are defined as follows: p.sub.z—external load borne by the ground; φ—vertical angle at a random point in the space, in rad; γ—gravity density of rock-soil layer, in kN.Math.m; μ—Poisson's ratio of rock-soil mass; h—thickness of overlying stratum, in m; R—distance from any point to spherical cavity center, in m.

4. The method for quantifying bearing capacity of foundation containing shallow-hidden spherical cavities according to claim 1, wherein: the mathematical expressions in Step 3 are as follows: $\begin{array}{cc}{p}_{z3}=\left[\frac{6\mathrm{\mu \gamma }h\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}+\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\right]+\sqrt{\begin{array}{c}[\frac{6\mathrm{\mu \gamma }\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}h+\\ {\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)]}^2-\frac{12\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}\frac{27{\mu }^2{\gamma }^2{h}^2}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}-4\\ {\left[\frac{6\mu -1}{2\left(2\mu -1\right)}\gamma R_1+\frac{{\mu }^2}{\left(1-\mu \right)\left(2\mu -1\right)}\gamma R_1\right]}^2-\frac{24\mathrm{\mu \gamma }}{\left(1-\mu \right)\left(2\mu -1\right)}\mathrm{hc}\tan \\ \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)+16c^2{\tan }^2\left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\}\end{array}}/\frac{6\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}& \left(2.1\right)\\ p_{z4}=\left[\frac{6\mathrm{\mu \gamma }h\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}+\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\right]-\sqrt{\begin{array}{c}[\frac{6\mathrm{\mu \gamma }\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}h+\\ {\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)]}^2-\frac{12\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}\frac{27{\mu }^2{\gamma }^2{h}^2}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}-4\\ {\left[\frac{6\mu -1}{2\left(2\mu -1\right)}\gamma R_1+\frac{{\mu }^2}{\left(1-\mu \right)\left(2\mu -1\right)}\gamma R_1\right]}^2-\frac{24\mathrm{\mu \gamma }}{\left(1-\mu \right)\left(2\mu -1\right)}\mathrm{hc}\tan \\ \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)+16c^2{\tan }^2\left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\}\end{array}}/\frac{6\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}& \left(2.2\right)\end{array}$ Related symbols in the formulas are defined as follows: φ.sub.1—internal friction angle of rock-soil mass, in rad; c—cohesion of rock-soil mass, in kPa; γ—gravity density of overlying rock-soil layer on the top surface of columnar cavity, in kN.Math.m; μ—Poisson's ratio of rock-soil mass; h—thickness of overlying stratum, in m; R.sub.1—sectional radius of a spherical cavity, in m.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] FIG. 1 is a process diagram of the method described according to the present disclosure.

[0036] FIG. 2 is a schematic diagram of the calculation model described according to the present disclosure.

[0037] FIG. 3 is a schematic diagram of the P-S curve at the key point during the test according to the present disclosure.

DETAILED DESCRIPTION

[0038] Please refer to FIGS. 1-3, wherein:

[0039] The present disclosure provides a method for quantifying the bearing capacity of foundation containing shallow-hidden spherical cavities, including steps described as follows:

[0040] In Step 1, by comprehensively considering a combined effect of spatial three-dimensional characteristics of a shallow-hidden spherical cavity, a self-weight stress of an overlying rock-soil layer, a virgin rock stress, and an external vertical load, a spatial axisymmetric calculation model is constructed for stability analysis of the foundation containing shallow-hidden spherical cavities;

[0041] In Step 2, by selecting a spherical coordinate system and introducing the Laplace displacement method, the model is solved through theoretical analysis to obtain a general solution which reflects the spatial stress distribution of surrounding rock containing shallow-hidden spherical cavities;

[0042] In Step 3, by combining the general solution obtained through theoretical analysis in Step 2, a damage mode of ground collapse, and the Mohr-Coulomb strength theory, a mathematical expression for calculating the bearing capacity of the foundation containing shallow-hidden spherical cavities is derived;

[0043] In Step 4, by substituting data of a vertical load of stratum, a gravity density of a rock-soil layer, a Poisson's ratio of rock-soil mass, a thickness of an overlying stratum, and a radius of a spherical cavity that are measured in actual engineering projects into the mathematical expression in Step 3 for calculation, the bearing capacity of the foundation may be determined.

[0044] Various parameters and formulas in the spatial axisymmetric calculation model in Step 1 are as follows:

[0045] p.sub.z—external load borne by the ground;

[0046] p.sub.0—horizontal load of virgin rock stratum, p.sub.0=μ[p.sub.z+γ(h+z)]/(1−μ);

[0047] γ—gravity density of rock-soil layer, in kN.Math.m;

[0048] μ—Poisson's ratio of rock-soil mass;

[0049] h—thickness of overlying stratum, in m;

[0050] R—distance from any point to spherical cavity center, in m.

[0051] The calculation formulas of the general solution in Step 2 are as follows:

[00003]$\begin{array}{cc}{\sigma }_R=\left\{4\left[16\left(1-\mu \right)A_2-A_4\right]{\sin }^2\mathrm{\phi cos\phi }+8\left[\left(2-\mu \right)A_4\right]{\cos }^3\phi \right\}R+\left\{2\left[3\mu A_3+\left(2\mu -1\right)A_5\right]{\sin }^2\phi +22\left[\left(1-\mu \right)A_3+2\left(2-\mu \right)A_5\right]{\cos }^2\phi \right\}+A_9\left[15{\cos }^6\phi +15{\sin }^4{\mathrm{\phi coscos}}^2\phi +30{\sin }^2{\mathrm{\phi cos}}^4\phi -3{\sin }^4\phi -6\left(1+\mu \right){\cos }^4\phi -3\left(3+2\mu \right){\sin }^2{\mathrm{\phi cos}}^2\phi +\left(1-2\mu \right){\sin }^2\phi +\left(2\mu -1\right){\cos }^2\phi \right]\frac{1}{R^3}& \left(1.1\right)\\ {\sigma }_{\theta }=\left\{4\left[\left(2\mu -1\right)A_4\right]\cos \phi \right\}R+2\left[3\mu A_3+\left(2\mu -1\right)A_5\right]+A_9\left[3\left(2\mu -1\right){\cos }^2\phi +\left(1-2\mu \right)\right]\frac{1}{R^3}& \left(1.2\right)\\ {\tau }_{R\phi }=\left\{4\left[8\left(1-\mu \right)A_2+\left(3\mu -5\right)A_4\right]{\sin }^2{\mathrm{\phi cos}}^2\phi -4\left[8\left(1-\mu \right)A_2-\mu A_4\right]{\sin }^3\phi \right\}R+[6\left(2\mu -1\right)A_3+2\left(4\mu -5\right)A_5]\sin \mathrm{\phi cos\phi }+2A_9\left(1+\mu \right)\frac{1}{R^3}\sin \mathrm{\phi cos\phi }& \left(1.3\right)\\ A_2=\frac{\left(6\mu -1\right)}{64\left(1-\mu \right)\left(1-2\mu \right)}\gamma A_3=-\frac{B_5}{6B_2}{A}_4=\frac{\mu }{4\left(1-\mu \right)\left(2\mu -1\right)}\gamma A_5=-\frac{B_6}{2B_2}{A}_9=-\frac{B_7}{B_2}{B}_5=2\left(2\mu -1\right)h^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi ]-\mu \left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]-4\mu h^3\left(2-\mu \right)\left(2\mu -1\right)\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]+\left(2\mu -1\right)R_1^3\left[15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi \right][\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }]B_2=\left(1-\mu \right)\left(2\mu -1\right)\left\{R_1^3\left(1-2\mu \right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+R_1^3\mu \left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+4\mu h^3\left(2-\mu \right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]+2h^3\left(1-\mu \right)\left(1-2\mu \right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]\right\}{B}_6=-2\mu h^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi ]+\mu \left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right][15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi ]+2\mu h^3\left(1-\mu \right)\left(2\mu -1\right)\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi -\left(1+\mu \right){\sin }^2\phi \right]-\mu \left[15{\cos }^7\phi -6\left(1+\mu \right){\cos }^5\phi +\left(2\mu -1\right){\cos }^3\phi \right][\left(6\mu -1\right)\left(1-\mu \right)R_1^4{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1^4{\mathrm{\gamma cos}}^3\phi +\mu R_1^4{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }]B_7=h^3\left\{\left(1-2\mu \right)R_1^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi ]+\mu R_1^3\left[\left(1-\mu \right)\left(2\mu -1\right)p_z+2\mu \left(2-\mu \right)\gamma h\right][2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi ]+2\mu \left(2-\mu \right)\left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[{\mathrm{\mu sin}}^2\phi +\left(1-\mu \right){\cos }^2\phi \right]-\mu \left(1-\mu \right)\left(2\mu -1\right)R_1^3\left(p_z+\gamma h\right)\left[2\left(2-\mu \right){\cos }^2\phi +\left(2\mu -1\right){\sin }^2\phi \right]-2\mu \left(2-\mu \right)R_1^3\left[\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }\right]+\left(1-\mu \right)\left(2\mu -1\right)R_1^3\left[\left(6\mu -1\right)\left(1-\mu \right)\gamma R_1{\sin }^2\mathrm{\phi cos\phi }-2\left(2-\mu \right)R_1{\mathrm{\gamma cos}}^3\phi +\mu R_1{\mathrm{\gamma sin}}^2\mathrm{\phi cos\phi }\right]\right\}& \left(1.4\right)\end{array}$

[0052] Related symbols in the formulas are defined as follows:

[0053] p.sub.z—external load borne by the ground;

[0054] φ—vertical angle at a random point in the space, in rad;

[0055] γ—gravity density of rock-soil layer, in kN.Math.m;

[0056] μ—Poisson's ratio of rock-soil mass;

[0057] h—thickness of overlying stratum, in m;

[0058] R—distance from any point to spherical cavity center, in m.

[0059] The mathematical expressions in Step 3 are as follows:

[00004]$\begin{array}{cc}{p}_{z3}=\left[\frac{6\mathrm{\mu \gamma }h\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}+\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\right]+\sqrt{\begin{array}{c}[\frac{6\mathrm{\mu \gamma }\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}h+\\ {\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)]}^2-\frac{12\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}\frac{27{\mu }^2{\gamma }^2{h}^2}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}-4\\ {\left[\frac{6\mu -1}{2\left(2\mu -1\right)}\gamma R_1+\frac{{\mu }^2}{\left(1-\mu \right)\left(2\mu -1\right)}\gamma R_1\right]}^2-\frac{24\mathrm{\mu \gamma }}{\left(1-\mu \right)\left(2\mu -1\right)}\mathrm{hc}\tan \\ \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)+16c^2{\tan }^2\left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\}\end{array}}/\frac{6\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}& \left(2.1\right)\\ p_{z4}=\left[\frac{6\mathrm{\mu \gamma }h\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}+\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\right]-\sqrt{\begin{array}{c}[\frac{6\mathrm{\mu \gamma }\left({\mu }^2-7\mu +1\right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}h+\\ {\frac{8}{1-\mu }c\tan \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)]}^2-\frac{12\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}\frac{27{\mu }^2{\gamma }^2{h}^2}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2\left(2\mu -1\right)}-4\\ {\left[\frac{6\mu -1}{2\left(2\mu -1\right)}\gamma R_1+\frac{{\mu }^2}{\left(1-\mu \right)\left(2\mu -1\right)}\gamma R_1\right]}^2-\frac{24\mathrm{\mu \gamma }}{\left(1-\mu \right)\left(2\mu -1\right)}\mathrm{hc}\tan \\ \left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)+16c^2{\tan }^2\left(\frac{\pi }{4}+\frac{{\phi }_1}{2}\right)\}\end{array}}/\frac{6\mu \left(2-\mu \right)}{{\left(1+\mu \right)}^2{\left(1-\mu \right)}^2}& \left(2.2\right)\end{array}$

[0060] Related symbols in the formulas are defined as follows:

[0061] φ.sub.1—internal friction angle of rock-soil mass, in rad;

[0062] c—cohesion of rock-soil mass, in kPa;

[0063] γ—gravity density of overlying rock-soil layer on the top surface of columnar cavity, in kN.Math.m;

[0064] μ—Poisson's ratio of rock-soil mass;

[0065] h—thickness of overlying stratum, in m;

[0066] R.sub.1—sectional radius of a spherical cavity, in m.

[0067] Implementations of the present disclosure will be further described with reference to figures and a certain engineering project below.

[0068] According to a preliminary survey report of a certain project, extremely thick limestone is the basic component among the survey field lithology. At the same time, the existence of spherical Karst caves identified by geophysical technologies such as drilling machines and geological radar also matches the research background of the present disclosure. According to relevant parameters provided in survey reports and interpretation results of 2D GPR (ground penetrating radar) images detected by field GPRs, physical and mechanical parameters of a study area are determined as shown in Table 1:

TABLE-US-00001 TABLE 1 Physical mechanical parameters of survey region Parameter Natural Internal Internal gravity Elasticity friction Spherical filling density modulus Poisson's angle radius Cohesion pressure Material (kg/m.sup.3) (GPa) ratio (°) (m) (MPa) (MPa) Limestone 2650 35 0.25 42.3 0.5 7.8 0

[0069] By substituting related coefficients in Table 1 into (2.1)-(2.2), we get pz3=88.16 MPa, and pz4=37.94 MPa. By selecting a positive and smaller value as the roof bearing capacity, the roof bearing capacity is finally determined as 37.94 MPa.

[0070] As shown in FIG. 3, when the applied vertical load reaches 36 MPa and then increases by 0.1 MPa more, the vertical displacement of the key point will show a great abrupt change. On the other hand, it can be seen from the change of a plastic zone obtained by numerical simulation that the key point experienced a whole process: no plastic zone occurs at first, then both sides of the cave start to yield, and finally a sliding face is formed which reaches and penetrates through the ground. When the applied vertical load reaches 36 MPa, a sliding face penetrating the ground is also formed. By combining the above two conclusions comprehensively, 36 MPa at the abrupt change point is finally determined as the roof bearing capacity.

[0071] It can be seen from a comparison between the theoretical calculation results and the numerical simulation results that the calculation results of the two methods are quite consistent, and the numerical simulation results deviate within 5%, which are highly consistent.