CN112528490B - Method for calculating limit load of independent foundation in karst foundation - Google Patents
Method for calculating limit load of independent foundation in karst foundation Download PDFInfo
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Abstract
The invention discloses a method for calculating limit load of an independent foundation in a karst foundation, which takes the karst foundation consisting of the independent foundation, a covering layer and a collapsed rock mass at the top of a karst cave as a research object; carrying out stress analysis on the collapsed rock mass at the top of the karst cave; establishing an additional stress condition on a rock-soil interface of a collapsed rock body at the top of the karst cave; establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation; and solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation to obtain the limit load of the independent foundation, so that the obtained limit load of the independent foundation is more accurate than the result obtained by the traditional method.
Description
Technical Field
The invention relates to a method for calculating limit load of an independent foundation in a karst foundation, and belongs to the technical field of foundation bearing capacity analysis.
Background
In the limestone area, karst caves with certain scales often appear in karst foundations, and the upper parts of the karst caves are covered with a fourth series loose soil layer with certain thickness. An independent foundation is usually used as a foundation type of an upper structure on the surface of the covering layer, and a cavern at the lower part of the covering layer is usually collapsed under the action of a load on the independent foundation, so that potential safety hazards are brought to construction and operation of a project. The independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave form a complex karst foundation system, and if the limit load of the independent foundation can be accurately determined, theoretical basis is provided for the design and operation of the karst foundation. Therefore, it is necessary to study the calculation method of the limit load of the independent foundation on the top of the covering layer of the cavern.
Numerous scholars and engineers conduct research on karst cave collapse mechanisms, stability evaluation methods, critical thickness of karst cave top plates and the like, but the following defects exist in the aspect of ultimate bearing capacity of karst foundations: (1) The load on the independent foundation is transmitted to a rock-soil interface through the covering layer, so that the cave of the karst cave is caused, which is a very complicated mechanical problem, and the related research on a mechanical model of an independent foundation-covering layer-cave top collapsed rock mass system is lacked at present; (2) The research on the stress limit state of the collapsed rock mass at the top of the karst cave under the action of the limit load on the independent foundation is insufficient; (3) Solving the ultimate load of the independent foundation is a maximized mathematical programming problem, but the current research results all make some artificial assumptions on the foundation system, convert the hyperstatic problem into the statically determinate problem and solve the statically determinate problem by using a mathematical iteration method, so that the obtained ultimate load error is large. Therefore, it is necessary to establish a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation to obtain a more accurate solution.
Disclosure of Invention
The invention provides a method for calculating the limit load of an independent foundation in a karst foundation so as to obtain the limit load of the independent foundation in the karst foundation.
The technical scheme of the invention is as follows: a method for calculating limit load of an independent foundation in a karst foundation takes the karst foundation consisting of the independent foundation, a covering layer and collapsed rock mass at the top of a karst cave as a research object; analyzing the stress of the collapsed rock mass at the top of the karst cave; establishing an additional stress condition on a rock-soil interface of a collapsed rock body at the top of the karst cave; establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation; and solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation to obtain the limit load of the independent foundation.
The method comprises the following steps:
step 1, drawing up basic parameters for calculating the limit load of an independent foundation in a karst foundation;
step 2, analyzing the stress of the collapsed rock mass at the top of the karst cave;
step 3, establishing additional stress conditions on a rock-soil interface of the collapsed rock body at the top of the karst cave;
step 4, establishing a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation according to the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the ultimate state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition and the additional stress condition of the collapsed rock mass at the top of the karst cave;
and 5, solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation by using a dual simplicity method to obtain the limit load of the independent foundation.
Basic parameters for calculating the ultimate load of the independent foundation in the karst foundation comprise:
1. determining geometric parameters of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave aiming at the karst foundation consisting of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave;
2. and determining physical and mechanical parameters of a covering layer soil body and a collapsed rock body at the top of the karst cave.
The geometrical parameters of the independent foundation, the covering layer and the collapsed rock body at the top of the karst cave comprise: the width b of the independent foundation, the length L of the independent foundation, the thickness H of the covering layer soil body, the height H of the collapsed rock body at the top of the karst cave, and the top of the karst cave is a hemisphere with the diameter d; the physical and mechanical parameters of the covering layer soil body and the collapsed rock body at the top of the karst cave comprise: stress spread angle theta of covering soil mass, bulk density gamma of covering soil mass s (ii) a Cohesive force c of collapsed rock mass at top of karst cave r Inner angle of friction of collapsed rock mass at top of karst cave
Bulk density gamma of collapsed rock mass at top of karst cave r 。
The stress analysis of the collapsed rock body specifically comprises the following steps:
ultimate load F acting on independent basis z The soil is transmitted to a rock-soil interface of a collapsed rock body at the top of the karst cave through diffusion of a covering layer soil body, the collapsed rock body at the top of the karst cave is a cylinder, the lower part of the collapsed rock body at the top of the karst cave is a hollow semi-sphere, the diameter of the collapsed rock body at the top of the karst cave is d, the height of the collapsed rock body at the top of the karst cave is h, the fracture surface of the collapsed rock body at the top of the karst cave is a cylindrical surface with the diameter of d and the height of h, the diameter of the hollow semi-sphere at the lower part of the cylinder is d, and the rock-soil interface of the collapsed rock body at the top of the karst cave is acted with additional stress sigma generated by the self weight of the covering layer soil body f And ultimate load F of the independent foundation z Additional stress sigma generated F (ii) a The positive stress sigma acts on the fracture surface of the collapsed rock body at the top of the karst cave r And shear stress τ r (ii) a The centroid of the collapsed rock mass at the top of the karst cave acts with dead weight G r 。
The additional stress condition on the rock-soil interface of the collapsed rock body at the top of the karst cave is established specifically as follows:
1. establishing an additional stress condition generated by the self weight of the covering layer soil body on a rock-soil interface, which specifically comprises the following steps:
σ f ＝γ s H
in the formula: sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body by the dead weight of the covering layer soil body; h is the thickness of the overburden body; gamma ray s Is the volume weight of the overburden body;
2. establishing an additional stress condition generated by the ultimate load of an independent foundation on a rock-soil interface, and calculating according to the stress diffusion principle and the following formula:
in the formula: f z Is the ultimate load of the independent foundation; sigma F Ultimate load F being an independent basis z Additional stress generated on a rock-soil interface of the collapsed rock body; b is the width of the independent foundation and L is the length of the independent foundation; theta is the stress spread angle of the overburden body.
The establishment of the linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation specifically comprises the following steps:
1. establishing an objective function
And setting the limit load of the independent foundation as an objective function, wherein the objective function is as follows:
Maximize:F z
in the formula: f z Is the ultimate load of the independent foundation; maximize means "Maximize";
2. establishing a balance equation of a collapsed rock mass at the top of the karst cave, which specifically comprises the following steps:
in the formula: d is the diameter of the collapsed rock mass at the top of the karst cave; h is the height of the collapsed rock mass at the top of the karst cave; tau. r The shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent basis z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave;
is the dead weight of the collapsed rock mass at the top of the karst cave, gamma r The volume weight of the collapsed rock mass at the top of the karst cave;
3. establishing limit state constraint conditions of the stress of the collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: sigma r Is the normal stress acting on the fracture surface of the collapsed rock body at the top of the karst cave;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave,
is the internal friction angle of the collapsed rock mass at the top of the karst cave;
4. establishing yield conditions of the collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
5. establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation:
integrating the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the extreme state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition of the collapsed rock mass at the top of the karst cave and the additional stress condition to obtain a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation as follows:
in the formula: gamma ray s Is the volume weight of the covering layer soil body; h is the thickness of the overburden body; b is the width of the independent foundation and L is the length of the independent foundation; theta is the stress spread angle of the overburden body.
The linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation specifically comprises the following steps: the parameters d, H, H, L, theta, c are known r 、
γ s 、γ r Substituting a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation to obtain the limit load F of the independent foundation z As an objective function, of
σ f 、σ F 、σ r 、τ r The linear mathematical programming model is solved for the decision variables by using a dual simplex method, and the ultimate load F of the independent foundation in the karst foundation is solved z And decision variables
σ f 、σ F 、σ r 、τ r The calculation result of (2); wherein d is the diameter of the collapsed rock mass at the top of the karst cave; h is the height of the collapsed rock mass at the top of the karst cave; h is the thickness of the overburden body; l is the length of the independent foundation; theta is the stress spread angle of the overburden body; c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
is the internal friction angle of the collapsed rock mass at the top of the karst cave; gamma ray s Is the volume weight of the overburden body; gamma ray r Is the collapse of the top of the cavernThe volume weight of the rock mass;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent foundation z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave; sigma r Is the normal stress of the fracture surface action of the collapsed rock mass at the top of the karst cave; tau is r Is the shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave.
The invention has the beneficial effects that:
(1) The method establishes a mechanical model for calculating the limit load of the independent foundation in the karst foundation, and establishes an additional stress condition generated by the limit load of the independent foundation on a rock-soil interface according to a stress diffusion principle; establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation by taking the limit load of the independent foundation as a target function; the independent foundation ultimate load obtained by the method is more accurate than the result obtained by the traditional method.
(2) The method is rigorous in theory and simple in programming, can be applied to recheck check calculation of the limit load of the karst cave foundation independent foundation, and provides a theoretical basis for the design of the karst foundation.
Drawings
FIG. 1 is a technical flow diagram of the present invention;
FIG. 2 is a three-dimensional view of a computational model of ultimate loads of an independent foundation of a karst foundation;
figure 3 is a cross-sectional view of the karst foundation along the vertical mid-plane of the collapsed rock mass at the top of the cavern;
FIG. 4 is a schematic illustration of a collapsed rock mass at the top of a cavern;
figure 5 is a cross-sectional view of the vertical mid-plane of a collapsed rock mass at the top of a cavern.
Detailed Description
Example 1: as shown in fig. 1-5, a method for calculating the ultimate load of an independent foundation in a karst foundation takes the karst foundation consisting of the independent foundation, a covering layer and collapsed rock mass at the top of a karst cave as a research object; carrying out stress analysis on the collapsed rock mass at the top of the karst cave; establishing an additional stress condition on a rock-soil interface of a collapsed rock body at the top of the karst cave; establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation; and solving the linear mathematical programming model of the limit load of the independent foundation in the karst foundation to obtain the limit load of the independent foundation.
Further, the method steps may be arranged as follows:
step 1, drawing up basic parameters for calculating the limit load of an independent foundation in a karst foundation;
step 2, analyzing the stress of the collapsed rock mass at the top of the karst cave;
step 3, establishing additional stress conditions on a rock-soil interface of the collapsed rock body at the top of the karst cave;
step 4, establishing a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation according to the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the ultimate state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition and the additional stress condition of the collapsed rock mass at the top of the karst cave;
and 5, solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation by using a dual simplicity method to obtain the limit load of the independent foundation.
Further, basic parameters for the ultimate load calculation of the independent foundation in the karst foundation can be set to include:
1. determining geometric parameters of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave aiming at the karst foundation consisting of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave;
2. and determining physical and mechanical parameters of a covering layer soil body and a collapsed rock body at the top of the karst cave.
Further, the geometrical parameters of the independent foundation, the covering layer and the collapsed rock body at the top of the karst cave can be set to comprise: the width b of the independent foundation is 3m, the length L of the independent foundation is 3m, and the thickness of the covering soil bodyThe degree H is 5m, the height H of a collapsed rock body at the top of the karst cave is 3m, the diameter d of the karst cave body is 5m, and the top of the karst cave is a hemisphere with the diameter d; the physical and mechanical parameters of the covering layer soil body and the collapsed rock body at the top of the karst cave comprise: the stress diffusion angle theta of the covering soil body is 22 degrees, and the volume weight gamma of the covering soil body s Take 19kN/m 3 (ii) a Cohesive force c of collapsed rock mass at top of karst cave r Taking 300kPa, the internal friction angle of the collapsed rock mass at the top of the karst cave
Taking 30 degrees, the volume weight gamma of the collapsed rock mass at the top of the karst cave r Taking 25kN/m 3 。
Further, the stress analysis of the collapsed rock mass can be set, specifically:
a three-dimensional calculation model of a karst foundation consisting of an independent foundation, a covering layer and a karst cave collapsed rock body is shown in figure 2, wherein the independent foundation is positioned right above the karst cave, the bottom surface of the independent foundation is the surface of the covering layer, and a limit load F acting on the independent foundation z The soil body of the covering layer is diffused and transferred to a rock-soil interface of the collapsed rock body at the top of the karst cave; as shown in FIG. 4, the collapsed rock mass at the top of the karst cave is a cylinder, the lower part of the collapsed rock mass at the top of the karst cave is a hollow semi-sphere, the diameter of the collapsed rock mass at the top of the karst cave is d, the height of the collapsed rock mass at the top of the karst cave is h, the fracture surface of the collapsed rock mass at the top of the karst cave is a cylindrical surface with the diameter of d and the height of h, the diameter of the hollow semi-sphere at the lower part of the cylinder is d, the sectional view of the karst foundation along the vertical middle surface of the collapsed rock mass at the top of the karst cave is shown in FIG. 3, the stress state is shown in FIG. 5, and the additional stress sigma generated by the self-weight of the soil mass is acted on the rock-soil interface of the collapsed rock mass at the top of the karst cave f And ultimate load F of the independent foundation z Additional stress sigma generated F (ii) a The positive stress sigma acts on the fracture surface of the collapsed rock body at the top of the karst cave r And shear stress τ r (ii) a The centroid of the collapsed rock mass at the top of the karst cave acts with dead weight G r 。
Further, additional stress conditions on the rock-soil interface of the collapsed rock body at the top of the karst cave can be set, and the additional stress conditions are as follows:
1. establishing an additional stress condition generated by the self weight of the covering layer soil body on a rock-soil interface, which specifically comprises the following steps:
σ f ＝γ s H
in the formula: sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body by the dead weight of the covering layer soil body; h is the thickness of the overburden body; gamma ray s Is the volume weight of the overburden body;
2. establishing an additional stress condition generated by the ultimate load of an independent foundation on a rock-soil interface, and calculating according to the stress diffusion principle and the following formula:
in the formula: f z Is the ultimate load of the independent foundation; sigma F Ultimate load F being an independent basis z Additional stress generated on a rock-soil interface of the collapsed rock body; h is the thickness of the overburden body, b is the width of the independent foundation, and L is the length of the independent foundation; theta is the stress spread angle of the overburden body.
Further, the establishing of the linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation may be specifically:
1. establishing an objective function
And setting the limit load of the independent foundation as an objective function, wherein the objective function is as follows:
Maximize:F z
in the formula: f z Is the ultimate load of the independent foundation; maximize means "max";
2. establishing a balance equation of a collapsed rock mass at the top of the karst cave, which specifically comprises the following steps:
in the formula: d being the top of the cavernThe diameter of the collapsed rock mass; h is the height of the collapsed rock mass at the top of the karst cave; tau. r The shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent basis z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave;
is the dead weight of the collapsed rock mass at the top of the cavern, gamma r The volume weight of the collapsed rock mass at the top of the karst cave;
3. establishing limit state constraint conditions of the stress of the collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: sigma r Is the normal stress acting on the fracture surface of the collapsed rock body at the top of the karst cave;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave,
is the internal friction angle of the collapsed rock mass at the top of the karst cave;
4. establishing yield conditions of the collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: sigma r Is the normal stress of the fracture surface action of the collapsed rock mass at the top of the karst cave; tau is r The shear stress is acted on the fracture surface of the collapsed rock body at the top of the karst cave; c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
is the internal friction angle of the collapsed rock mass at the top of the karst cave;
5. establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation:
integrating the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the extreme state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition of the collapsed rock mass at the top of the karst cave and the additional stress condition to obtain a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation as follows:
in the formula: gamma ray s Is the volume weight of the covering layer soil body; h is the thickness of the overburden body; b is the width of the independent foundation and L is the length of the independent foundation; theta is the stress spread angle of the overburden body.
Further, a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation can be set, specifically: the known parameters d, H, H, L, theta, c r 、
γ s 、γ r Substituting a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation to obtain the limit load F of the independent foundation z As an objective function, of
σ f 、σ F 、σ r 、τ r The method is used for solving the linear mathematical programming model for decision variables by using a dual simplex method to obtain the independent foundation ultimate load F in the karst foundation z Is 2305kN, and a decision variable
σ f 、σ F 、σ r 、τ r The calculation results are detailed in table 1; wherein d is the diameter of the collapsed rock mass at the top of the karst cave; h is the height of the collapsed rock mass at the top of the karst cave; h is the thickness of the overburden body; l is the length of the independent foundation; theta is the stress spread angle of the overburden body; c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
is the internal friction angle of the collapsed rock mass at the top of the karst cave; gamma ray s Is the volume weight of the overburden body; gamma ray r The volume weight of the collapsed rock mass at the top of the karst cave;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent basis z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave; sigma r Is the normal stress of the fracture surface action of the collapsed rock body at the top of the karst cave; tau. r Is the shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave.
TABLE 1 statistical table of calculation results of examples
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.
Claims (5)
1. A method for calculating the limit load of an independent foundation in a karst foundation is characterized by comprising the following steps: taking a karst foundation consisting of an independent foundation, a covering layer and a collapsed rock mass at the top of the karst cave as a research object; carrying out stress analysis on the collapsed rock mass at the top of the karst cave; establishing an additional stress condition on a rock-soil interface of a collapsed rock body at the top of the karst cave; establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation; solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation to obtain the limit load of the independent foundation;
the additional stress condition on the rock-soil interface of the collapsed rock body at the top of the karst cave is established specifically as follows:
1. establishing an additional stress condition generated by the self weight of the covering layer soil body on a rock-soil interface, which specifically comprises the following steps:
σ f ＝γ s H
in the formula: sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body by the dead weight of the covering layer soil body; h is the thickness of the overburden body; gamma ray s Is the volume weight of the overburden body;
2. establishing an additional stress condition generated by the ultimate load of an independent foundation on a rock-soil interface, and calculating according to the stress diffusion principle and the following formula:
in the formula: f z Is the ultimate load of the independent foundation; sigma F Ultimate load F being an independent foundation z Additional stress generated on a rock-soil interface of the collapsed rock body; b is the width of the independent foundation and L is the length of the independent foundation; theta is the stress spread angle of the overburden body;
the establishment of the linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation specifically comprises the following steps:
1. establishing an objective function
And setting the limit load of the independent foundation as an objective function, wherein the objective function is as follows:
Maximize:F z
in the formula: f z Is the ultimate load of the independent foundation; maximize means "max";
2. establishing a balance equation of a collapsed rock mass at the top of the karst cave, which specifically comprises the following steps:
in the formula: d is the diameter of the collapsed rock mass at the top of the karst cave; h is the height of the collapsed rock mass at the top of the karst cave; tau is r Is the shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent foundation z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave;
is the dead weight of the collapsed rock mass at the top of the karst cave, gamma r The volume weight of the collapsed rock mass at the top of the karst cave;
3. establishing limit state constraint conditions of the stress of the collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: sigma r Is the normal stress acting on the fracture surface of the collapsed rock body at the top of the karst cave;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave,
is the internal friction angle of the collapsed rock mass at the top of the karst cave;
4. establishing yield conditions of a collapsed rock mass at the top of the karst cave, which specifically comprise the following steps:
in the formula: c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
5. establishing a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation:
integrating the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the extreme state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition of the collapsed rock mass at the top of the karst cave and the additional stress condition to obtain a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation as follows:
in the formula: gamma ray s Is the volume weight of the overburden body; h is the thickness of the overburden body; b is the width of the independent foundation and L is the length of the independent foundation; theta is the stress spread angle of the overburden body;
the linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation specifically comprises the following steps: the parameters d, H, H, L, theta, c are known r 、
γ s 、γ r Substituting into a linear mathematical programming model for solving the limit load of the independent foundation in the karst foundation to obtain the limit load F of the independent foundation z As an objective function, of
σ f 、σ F 、σ r 、τ r The method is used for solving the linear mathematical programming model for decision variables by using a dual simplex method to obtain the independent foundation ultimate load F in the karst foundation z And decision variables
σ f 、σ F 、σ r 、τ r The calculation result of (2); wherein d is the diameter of the collapsed rock mass at the top of the karst cave; h is the height of the collapsed rock mass at the top of the karst cave; h is the thickness of the overburden body; l is the length of the independent foundation; theta is the stress spread angle of the overburden body; c. C r Is the cohesive force of the collapsed rock mass at the top of the karst cave;
is the internal friction angle of the collapsed rock mass at the top of the karst cave; gamma ray s Is the volume weight of the overburden body; gamma ray r The volume weight of the collapsed rock mass at the top of the karst cave;
is the static soil pressure coefficient of the collapsed rock mass at the top of the karst cave; sigma f The additional stress is generated on the rock-soil interface of the collapsed rock body at the top of the karst cave by the self weight of the covering layer soil body; sigma F Ultimate load F being an independent basis z Additional stress is generated on a rock-soil interface of a collapsed rock body at the top of the karst cave; sigma r Is the normal stress of the fracture surface action of the collapsed rock mass at the top of the karst cave; tau. r Is the shear stress acting on the fracture surface of the collapsed rock body at the top of the karst cave.
2. The method for calculating the ultimate load of an independent foundation in a karst foundation according to claim 1, characterized in that: the method comprises the following steps:
step 1, drawing up basic parameters for calculating the limit load of an independent foundation in a karst foundation;
step 2, analyzing the stress of the collapsed rock mass at the top of the karst cave;
step 3, establishing an additional stress condition on a rock-soil interface of a collapsed rock body at the top of the karst cave;
step 4, establishing a linear mathematical programming model for solving the ultimate load of the independent foundation in the karst foundation according to the objective function, the balance equation of the collapsed rock mass at the top of the karst cave, the ultimate state constraint condition of the stress of the collapsed rock mass at the top of the karst cave, the yield condition and the additional stress condition of the collapsed rock mass at the top of the karst cave;
and 5, solving a linear mathematical programming model of the limit load of the independent foundation in the karst foundation by using a dual simplicity method to obtain the limit load of the independent foundation.
3. The method for calculating the ultimate load of an independent foundation in a karst foundation according to claim 2, characterized in that: basic parameters for calculating the ultimate load of the independent foundation in the karst foundation comprise:
1. determining the geometric parameters of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave aiming at the karst foundation consisting of the independent foundation, the covering layer and the collapsed rock mass at the top of the karst cave;
2. and determining physical and mechanical parameters of a covering layer soil body and a collapsed rock body at the top of the karst cave.
4. The method for calculating the limit load of an independent foundation in a karst foundation according to claim 3, wherein: the geometrical parameters of the independent foundation, the covering layer and the collapsed rock body at the top of the karst cave comprise: the width b of the independent foundation, the length L of the independent foundation, the thickness H of the covering layer soil body, the height H of the collapsed rock body at the top of the karst cave, and the top of the karst cave is a hemisphere with the diameter d; the physical and mechanical parameters of the covering layer soil body and the collapsed rock body at the top of the karst cave comprise: stress spread angle theta of covering soil mass, bulk density gamma of covering soil mass s (ii) a Cohesive force c of collapsed rock mass at top of karst cave r Internal friction angle of collapsed rock mass at top of karst cave
Bulk density gamma of collapsed rock mass at top of karst cave r 。
5. The method for calculating the ultimate load of an independent foundation in a karst foundation according to claim 1 or 2, characterized in that: the stress analysis of the collapsed rock body specifically comprises the following steps:
ultimate load F acting on independent basis z Is transferred to the solution by diffusion of the covering soil bodyOn the rock-soil interface of the collapsed rock mass at the top of the cave, the collapsed rock mass at the top of the cave is a cylinder, the lower part of the collapsed rock mass at the top of the cave is a hollow semi-sphere, the diameter of the collapsed rock mass at the top of the cave is d, the height of the collapsed rock mass at the top of the cave is h, the fracture surface of the collapsed rock mass at the top of the cave is a cylindrical surface with the diameter of d and the height of h, the diameter of the hollow semi-sphere at the lower part of the cylinder is d, and the rock-soil interface of the collapsed rock mass at the top of the cave is acted with additional stress sigma generated by the self weight of a covering layer soil body f And ultimate load F of the independent foundation z Additional stress sigma generated F (ii) a The positive stress sigma acts on the fracture surface of the collapsed rock body at the top of the karst cave r And shear stress τ r (ii) a The centroid of the collapsed rock mass at the top of the karst cave acts with dead weight G r 。
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