Karst caves in karst areas are mainly composed of carbonate rocks. Because of geological structure, joints, cleavage or calcite veins often develop in rock mass, which can be regarded as weak structural plane, and its shear strength is much lower than that of surrounding rock. The weak structural plane in rock mass is the main factor affecting the stability of foundation. The stability evaluation of this foundation can be analyzed by limit equilibrium method and elastic-plastic theory.

2.4. 1 limit equilibrium analysis method

When there are two or more groups of weak structural planes on the inner wall of the cave, the stability of the cave is in a state of limit equilibrium (Figure 2-8).

Figure 2-8 Stability of Cave Wall Block and Cave Roof Block

Figure 2-8 Stability of Rock at Cave Wall and Cave Roof

(1) Stability coefficient fs of tunnel wall block;

fs =(w2 cosαTGφ 1+c 1l 4)/(w2 sinα)(2- 10)

Where: φ is L4 internal friction angle (degree) of structural plane; C 1 is the cohesion of structural plane L4 (kPa); A is the inclination angle (degree) of structural plane L4; W2 is the gravity (kN) of the block.

(2) Stability coefficient Fs of the roof block:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

Where: c 1 is the cohesion of structural plane L 1 (kPa); C2 is the cohesion of structural plane L2 (kPa); α is the dip angle (degree) of structural plane L 1; β is the inclination angle (degree) of structural plane L2; γ is the severity of rock mass (kN/m3).

When Fs≥2, the block is stable; When fs < 2, the block is unstable.

2.4.2 Elastic-plastic theoretical analysis method

When there is a weak structural plane in the rock mass around the karst cave, the stress state at the weak structural plane can be obtained by elastic-plastic theory, and then its stability can be judged by Coulomb-Molar strength criterion.

According to the knowledge of material mechanics, the formula for finding the normal stress σ and shear stress τ on any inclined section at an angle β with the large principal stress action surface is:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

In this stress state, the stability of karst cave foundation mainly depends on the strength of the structure and structural plane in the rock mass around the cave. Under the control of structural plane, the stability of rock mass mainly depends on the anti-sliding stability of structural plane, and its reasonable failure criterion is Coulomb equation, that is,

τ=σtgφ+c (2- 13)

Its stability conditions are:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

Where: k is the anti-sliding stability coefficient of structural plane; T is the sliding force acting on the structural plane; σ is the normal stress acting on the structural plane. Known:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

As shown in Figure 2-9, at any point in the surrounding rock of the cave wall (such as point B):

β=90 -α

Figure 2-9 Schematic diagram of stress of karst cave foundation with structural plane

Figure 2-9 Stress of Cave Foundation with Structural Plane

The normal stress and shear stress acting on the structural plane are respectively [34]:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

The position of structural plane is fixed (its dip angle and dip angle are unchanged), while the stress and radial angle α acting on different positions of structural plane are variable (for example, from point B to point A of structural plane, the radial angle changes from α to α0). From the geometric relationship in Figure 2-9, we can get:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

Where: R is the distance from point O to point B in the center of the cave.

The stress state formula around the karst cave is:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

Where: A is the distance from point O to point A; P0 is the initial stress of the original rock.

Substitute the stress state formula (2- 18) and formula (2- 17) around the cave into formula (2- 16) to obtain:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

After completing the above formula, you can get:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

According to formula (2-20), it can be checked whether the rock mass at point B is damaged. According to the limit equilibrium condition τ=σtgφ+c, we get:

Influence of Karst Cave and Soil Cave on Building Foundation in Karst Area

Using formula (2-2 1), it is found that the rock mass within radius r will be destroyed by trial and error algorithm.