Coal pillar group-roof system chain type instability space mechanics mechanism determination method

Abstract

The invention belongs to the technical field of coal mine room and pillar mining, and particularly relates to a coal pillar group-roof system chain type instability space mechanics mechanism determination method. Comprising the following steps: simplifying a roof of a room-type goaf into a rectangular thin plate, assuming a coal pillar group as an elastic foundation model, and constructing an elastic foundation thin plate mechanical model of a coal pillar group-roof system; the coal pillar group-roof system is subjected to the effect of the uniform load q of the overlying strata; establishing a coal pillar group-roof system instability control equation based on a virtual work principle; and solving based on a displacement interpolation function, and determining the deflection and the average stress borne by the single coal pillar at any position. According to the method, the chain type instability space mechanical response characteristics of the coal pillar group and the roof system are determined, a two-dimensional analysis framework of a traditional mine ground pressure theory is broken through, a theoretical basis is provided for constructing a multi-scale evaluation system for the stability of the coal pillar group-roof system, brand-new mechanical interpretation is provided for goaf disaster prevention and control, and the goaf stability evaluation method has good application prospects. And support guarantee is provided for optimizing and upgrading coal mining technology and equipment and re-entering the goaf to recover the coal pillars.

Description

Technical Field

[0001] The present invention belongs to the technical field of coal mining in room-and-pillar method, and particularly relates to a method for determining the spatial mechanical mechanism of chain instability of coal pillar group-roof system. Background Art

[0002] While promoting economic and social development, coal resource development has also left a large number of underground goafs within the mining area. These goafs not only have an adverse impact on the efficient production of coal mines but also pose a huge safety hazard to the surface stability. Among various goafs, room goafs account for a certain proportion. After the coal rooms are mined, a large number of coal pillars exist in the goaf in the form of clusters to form a coal pillar group. Within a certain time period, the stable coal pillar group can effectively support the roof of the goaf. Except for the caving of the weak false roof, the rest of the roof does not break and collapse. At this time, the stable coal pillar group and the stable roof constitute a coal pillar group-roof system, which plays an important role in bearing the load of the overlying strata and maintaining the safety of the goaf and even the entire stope to the surface. However, over time, under the influence of many factors, some coal pillars become unstable first and break the stable equilibrium stress state in the overlying strata above them, thereby triggering a series of "domino" - type chain reactions, resulting in the instability of more coal pillars and large-scale roof caving. A series of disasters such as rock bursts, water inrush and sand bursting, and coal and gas outbursts may be triggered during this period, threatening the safe mining of horizontally adjacent and overlying or underlying coal seams. In addition, the overall instability of the coal pillar group-roof system will also cause surface subsidence and damage the surface ecological environment of the mining area. Therefore, it is of great significance to study the chain instability characteristics of the coal pillar group-roof system. However, existing research mostly simplifies the room goaf into a two-dimensional structure for research, resulting in a deviation between the calculation results and the actual situation, and unable to truly and accurately reflect the spatial mechanical mechanism of chain instability of the three-dimensional coal pillar group-roof system. Summary of the Invention

[0003] In order to solve the above technical problems, the present invention proposes a method for determining the spatial mechanical mechanism of chain instability of coal pillar group-roof system, including the following steps:

[0004] a. Simplify the roof of the room goaf into a rectangular thin plate, assume the coal pillar group as an elastic foundation model, and construct an elastic foundation thin plate mechanical model of the coal pillar group-roof system; the coal pillar group-roof system is subjected to the action of a uniform load q of the overlying strata; the xy plane bisecting the thickness of the thin plate is the middle plane of the plate, x is the length direction, y is the width direction, and the points on the middle plane satisfy z = 0;

[0005] b. Calculate the roof deflection

[0006]

[0007] Where: ω (xi,yi) represents the deflection of the roof at the coal pillar i; n GPIndicates the number of nodes of the roof along the x-axis and y-axis, and at the same time indicates the number of coal pillars along the x-axis and y-axis. The coal pillars correspond one by one to the nodes; both s1 and t1 are index variables for traversing the interpolation nodes, and the self is excluded in the continuous multiplication to achieve the characteristics of the basis function; both s and t are indices of specific nodes, used to identify specific interpolation nodes, corresponding to x s and x t ; x represents the independent variable, which is any given input point at which the function approximation needs to be calculated during the interpolation process; x s and x t both represent a certain specific node among the interpolation nodes; x s1 represents any node in the interpolation node set except x s ; x t1 represents any node in the interpolation node set except x t ; A represents the surface area of the roof; N i(xi,yi) is the interpolation function; h r represents the thickness of the roof; N i,xx represents the second partial derivative of the interpolation function with respect to x; N i,yy represents the second partial derivative of the interpolation function with respect to y; N i,xy represents the first partial derivatives of the interpolation function with respect to x and y respectively; D0 represents the elastic matrix of the roof; n′ p represents the total number of coal pillars; k pi represents the elastic foundation coefficient of coal pillar i; k γ represents the rotational stiffness around the perimeter of the roof.

[0008] Preferably,

[0009]

[0010] In the formula: E r is the elastic modulus of the roof; μ r is the Poisson's ratio of the roof.

[0011] Preferably, when k γ takes 0, it means that the perimeter of the roof is in a simply supported constraint state, and when k γ takes infinity, it means that the roof is in a fixed constraint state on all four sides.

[0012] Preferably, it further includes c. calculating the average stress borne by coal pillar i at any position

[0013]

[0014] In the formula: σ pi represents the average stress borne by a single coal pillar i at any position; x i1 and x i2 respectively represent the starting and ending coordinates of coal pillar i on the x-axis; y i1 and yi2 respectively represent the starting and ending coordinates of coal pillar i on the y-axis, S pi represents the cross-sectional area of coal pillar i at any position.

[0015] Beneficial technical effects: The chain instability of the coal pillar group-roof system is a three-dimensional spatial dynamic process. By determining the spatial mechanical response characteristics of the chain instability of the coal pillar group-roof system, the present invention breaks through the two-dimensional analysis framework of traditional mine pressure theory, provides a theoretical basis for constructing a multi-scale evaluation system for the stability of the coal pillar group-roof system, provides a new mechanical explanation for the prevention and control of goaf disasters, and provides support and guarantee for the optimization and upgrading of coal mining technologies and equipment to reclaim coal pillars in the goaf again. Brief Description of the Drawings

[0016] Figure 1 is the thin plate on elastic foundation mechanical model of the coal pillar group-roof system. Detailed Embodiment

[0017] The detailed embodiments of the present invention will be introduced below with reference to the drawings.

[0018] The method for determining the spatial mechanical mechanism of chain instability of the coal pillar group-roof system proposed by the present invention includes the following steps:

[0019] S1: Construct the thin plate on elastic foundation mechanical model of the coal pillar group-roof system

[0020] S11: Construct the thin plate model of the roof

[0021] After the coal rooms in the room-and-pillar goaf are mined out, the roof is supported by the remaining coal pillar group, and a three-dimensional spatial structure is formed between the coal pillar group and the roof. The instability deformation of the roof can be analyzed by plate theory; the roof of the room-and-pillar goaf is simplified into a Figure 1 rectangular thin plate model as shown. The xy plane bisecting the thickness of the thin plate is the middle plane of the plate (x is the length direction, y is the width direction), and the points on the middle plane satisfy z = 0, with the downward direction being positive, and a coordinate system is established accordingly;

[0022] S12: Construct the elastic foundation model of the coal pillar group

[0023] The Winkler foundation model believes that the foundation is composed of several vertical springs, and the force received by each spring is positively correlated with its settlement deformation; there are a large number of remaining coal pillars in the room-and-pillar goaf. Before these coal pillars become unstable, they are in direct close contact with the roof and play a supporting role for the roof. At this time, the coal pillars in the coal pillar group can be assumed to be several vertical springs; at the same time, under the action of the roof load, the vertical deformation of the coal pillar is positively correlated with the roof force received by it; in summary, the coal pillar group can be assumed to be an elastic foundation model;

[0024] S13: Construct the thin plate on elastic foundation mechanical model of the coal pillar group-roof system

[0025] Assume that the coal pillar group-roof system is subjected to the uniform load q of the overlying strata, and the mechanical model of the elastic foundation thin plate of the coal pillar group-roof system is established as shown in Figure 1 follows; the roof is stable under the pressure of the overlying load and the support of several coal pillars and boundary coal pillars. When the underlying coal pillars become unstable, the roof will deform and fail;

[0026] S2: Instability control equation of coal pillar group-roof system based on the principle of virtual work

[0027] The principle of virtual work is one of the most basic and widely used principles in mechanics. It can be applied to any system in a state of equilibrium. For a balanced system, the sum of the virtual work done by all forces on the virtual displacement is equal to zero. It can provide the most general method for solving the structural equilibrium problems of rigid bodies, elastic bodies, plastic bodies, etc.; among them, the virtual displacement is a hypothetical, arbitrary, and infinitesimal displacement that satisfies the constraint geometric equation and displacement boundary conditions. The work done by the external force on the virtual displacement is the virtual work; for an elastic body, if the elastic body is in a state of equilibrium under the action of an external force, then when the elastic body undergoes an arbitrary and infinitesimal virtual displacement that satisfies the constraint geometric equation and displacement boundary conditions, the virtual work done by the external force is equal to the virtual strain energy of the elastic body. Based on the principle of virtual work, for the internal and external forces acting on the roof, the control equation for the chain instability of the coal pillar group-roof system is derived as follows:

[0028] The elastic virtual strain energy generated by all internal forces in the roof doing work on the corresponding virtual strains is:

[0029]

[0030] In the formula: δ represents the virtual variation, which is used to describe a hypothetical infinitesimal change of a function and reflects the concept of virtual and infinitesimal changes in the variational method; δU1 is the elastic virtual strain energy generated by all internal forces in the roof doing work on the corresponding virtual strains, J; is the internal force of the roof, MPa; is the virtual strain generated by the action of the internal force of the roof, %, V represents the volume of the roof, m 3 ;

[0031] The elastic potential energy generated by the overall supporting force of the coal pillar group doing work on the roof is:

[0032]

[0033] In the formula: δU2 is the elastic potential energy generated by the overall supporting force of the coal pillar group doing work on the roof, J; n′ p represents the total number of coal pillars; k pi represents the elastic foundation coefficient of a single coal pillar i at any position; i represents the coal pillar number; ω (xi,yi) represents the deflection of the roof at the coal pillar i, m; δω(xi,yi) Denotes the virtual deflection of the roof at coal pillar i, m;

[0034] For the elastic foundation coefficient of the coal pillar, for the convenience of solution, it is temporarily considered that the overall initial size of the coal pillar can effectively support the roof, that is, the initial cross-sectional size of the coal pillar is its effective bearing size; in addition to bearing the load of the overlying rock above, each coal pillar i also needs to bear the load transmitted from the surrounding roof; taking the square coal pillar with the same spacing and row spacing as an example, the elastic foundation coefficient of a single coal pillar i at any position can be obtained according to the subordinate area method as:

[0035]

[0036] In the formula: E pi Denotes the elastic modulus of a single coal pillar i at any position, GPa; S pi Denotes the cross-sectional area of a single coal pillar i at any position, m 2 ; h pi Denotes the height of a single coal pillar i at any position, m; l pi Denotes the width of a single coal pillar i at any position, m, l fi Denotes the width (coal pillar spacing) of the coal roadway at a single coal pillar i at any position, m;

[0037] The elastic potential energy generated by the roof constrained by the boundary coal pillar is:

[0038]

[0039] In the formula: δU3 represents the elastic potential energy generated by the roof constrained by the boundary coal pillar, J; Γ represents the integration region, which here refers to the geometric boundary of the boundary coal pillar; k γ Denotes the rotational stiffness around the roof, N·m / rad. When k γ takes 0, it means that the roof is in a simply supported constraint state around. When k γ takes infinity, it means that the roof is in a fixed constraint state on all four sides; ω γ Denotes the deflection generated by the roof constrained by the boundary coal pillar, m; n represents the normal direction of the boundary Γ;

[0040] The work done by the overlying rock load on the roof is:

[0041] δW = ∫∫ A qδωdxdy (5)

[0042] In the formula: δW represents the virtual work done by the overlying rock load on the roof, J; A represents the surface area of the roof, m 2 ; q represents the overlying rock load, MPa; δω represents the virtual deflection generated by the roof under the action of the overlying rock load, m;

[0043] It can be known from the equilibrium condition of the virtual work principle of the elastic body that:

[0044] δW = δU1 + δU2 + δU3 (6)

[0045] That is:

[0046]

[0047] The stress-strain relationship of any point in the roof slab is:

[0048]

[0049] In the formula: σ x represents the normal stress of the roof slab in the x direction, MPa; σ y represents the normal stress of the roof slab in the y direction, MPa; τ xy represents the shear stress in the roof slab, MPa; E r is the elastic modulus of the roof slab, GPa; μ r is the Poisson's ratio of the roof slab; ε x represents the strain of the roof slab in the x direction; ε y represents the strain of the roof slab in the y direction; γ xy represents the shear strain between the x and y directions in the roof slab;

[0050] Substituting formula (8) into formula (7), we can get:

[0051]

[0052] Among them:

[0053]

[0054] In the formula: D0 represents the elastic matrix of the roof slab; h r represents the thickness of the roof slab, m;

[0055] Set nodes corresponding to each coal pillar on the basic roof, calculate the displacement difference of the nodes on the basic roof, and the product of the interpolation function and the interpolation node displacement can describe the deflection change of the entire basic roof through the nodes. The specific equation is as follows:

[0056]

[0057] In the formula: ω (xi,yi) represents the deflection of the roof slab at node i. Since the coal pillar and the node are in one-to-one correspondence, it also represents the deflection of the roof slab at coal pillar i, m, and is also the deflection of the roof slab at any single coal pillar i; N i(xi,yi) represents the shape function for interpolating the nodes on the roof slab, that is, the interpolation function; x i and y i respectively represent the abscissa and ordinate of any node on the basic roof; Denote the displacement at the interpolation nodes of the roof slab;

[0058] Substitute Equation (11) into Equation (9), and since is arbitrary, thus can be eliminated, and further simplified to obtain:

[0059]

[0060] In the formula: N i,xx denotes the second partial derivative of the displacement interpolation function with respect to x; N i,yy denotes the second partial derivative of the displacement interpolation function with respect to y; N i,xy denotes the first partial derivatives of the displacement interpolation function with respect to x and y respectively;

[0061] Rearranging Equation (12) gives the displacement at the interpolation nodes of the main roof:

[0062]

[0063] S3: Solution based on the displacement interpolation function

[0064] Assume that the projection of node i in the roof slab on the x-axis corresponds to the s-th node, and the projection on the y-axis corresponds to the t-th node. Then the two-dimensional Lagrange interpolation function of node i is:

[0065] N i(x，y) = N s(x) ·N t(y) (14)

[0066] In the formula: N i(x,y) is the two-dimensional Lagrange interpolation function. In this paper, the one-dimensional Lagrange interpolation function is extended to a two-dimensional interpolation function by using the method of Lagrange product;

[0067] Assume that there are n GP nodes along both the x-axis and y-axis directions on the roof slab (the number of coal pillars in the length direction and width direction is the same). N s(x) denotes the interpolation function of the s-th node in the one-dimensional Lagrange interpolation function with respect to x1 to x nGP (x1, x2,..., x nGP ), N t(y) denotes the interpolation function of the t-th node in the one-dimensional Lagrange interpolation function with respect to y1 to y nGP (y1, y2,..., y nGP ), and N s(x) and N t(y) are respectively expressed as:

[0068]

[0069] Then, the two-dimensional Lagrange interpolation function can be written as:

[0070]

[0071] In the formula: n GP represents the number of nodes of the roof along the x-axis and y-axis respectively; s and t are both indices of specific nodes, used to identify specific interpolation nodes, corresponding to x s and x t respectively, to construct the basis function of this node; s1 and t1 are both index variables for traversing the interpolation nodes, excluding itself in the continuous multiplication to achieve the characteristics of the basis function; x represents the independent variable, which is any given input point at which the approximate value of the function needs to be calculated during the interpolation process; x s and x t both represent a specific node among the interpolation nodes (known data points), which are the key known points for constructing the interpolation basis function; x s1 represents any node in the interpolation node set except x s ; x t1 represents any node in the interpolation node set except x t ;

[0072] Substituting formulas (17) and (13) into formula (11), the final solution expression for the roof deflection at a single coal pillar i at any position can be obtained:

[0073]

[0074] Meanwhile, it can be obtained that the average stress borne by a single coal pillar i at any position is:

[0075]

[0076] In the formula: σ pi represents the average stress borne by a single coal pillar i at any position, MPa; x i1 and x i2 represent the abscissa range of any single coal pillar (representing the starting and ending coordinates of coal pillar i on the x-axis respectively); y i1 and y i2 represent the ordinate coordinate range of any single coal pillar (representing the starting and ending coordinates of coal pillar i on the y-axis respectively), S pi represents the cross-sectional area of any single coal pillar, m 2 .

[0077] Formulas (18) and (19) can be calculated using MATLAB software.

[0078] The present invention is not limited to the above-described preferred embodiments. Any person can derive other various forms of methods under the inspiration of the present invention. However, any technical solutions that are the same as or similar to the present application fall within the protection scope of the present invention.

Claims

1. A method for determining the spatial mechanical mechanism of chain instability of a coal pillar group-roof system, characterized in that: include: a. The roof of the room-type goaf is simplified into a rectangular thin plate, the coal pillar group is assumed to be an elastic foundation model, and the elastic foundation thin plate mechanical model of the coal pillar group-roof system is constructed; the coal pillar group-roof system is subjected to the uniform load q of the overlying rock strata; the xy plane that divides the thickness of the thin plate is the mid-plane of the plate, x is the length direction, y is the width direction, and the points on the mid-plane satisfy z = 0; b. Calculate the deflection of the roof at coal pillar i Where: n GP It indicates the number of nodes of the roof along the x-axis and y-axis directions, and also indicates the number of coal pillars along the x-axis and y-axis directions. The coal pillars correspond to the nodes one by one. s1 and t1 are index variables for traversing interpolation nodes. They exclude themselves in the multiplication to realize the basis function characteristics. s and t are the indexes of specific nodes, which are used to identify specific interpolation nodes, corresponding to x s and x t ; x represents the independent variable, which is any given input point where the approximate value of the function needs to be calculated during the interpolation process; x s and x t Each represents a specific node in the interpolation node; x s1 Indicates that the interpolation node set is excluding x s Any node outside x t1 Indicates that the interpolation node set is excluding x t Any node outside; A represents the surface area of ​​the top plate; N i(xi,yi) is the interpolation function; h r Indicates the thickness of the top plate; N i,xx Indicates that the interpolation function takes two partial derivatives with respect to x; N i,yy Indicates that the interpolation function takes two partial derivatives with respect to y; N i,xy represents the partial derivative of the interpolation function with respect to x and y respectively; D0 represents the elastic matrix of the top plate; n′ p Indicates the total number of coal pillars; k pi represents the elastic foundation coefficient of coal pillar i; k γ Represents the rotational stiffness around the top plate.

2. The method for determining the spatial mechanical mechanism of chain instability of the coal pillar group-roof system according to claim 1 is characterized in that: Where: E r is the elastic modulus of the top plate; μ r is the Poisson's ratio of the top plate.

3. The method for determining the spatial mechanical mechanism of chain instability of the coal pillar group-roof system according to claim 1 is characterized in that: When k γ When k is 0, it means that the top plate is in a simply supported state. γ When it is infinite, it means that the top plate is in a state of fixed constraints on four sides.

4. The method for determining the spatial mechanical mechanism of chain instability of the coal pillar group-roof system according to claim 1 is characterized in that: Also includes: c. Calculate the average stress on coal pillar i at any position Where: pi represents the average stress borne by a single coal pillar i at any position; x i1 and x i2 Respectively represent the starting and ending coordinates of coal pillar i on the x-axis; y i1 and i2 They represent the starting and ending coordinates of coal pillar i on the y-axis, S pi Represents the cross-sectional area of ​​coal pillar i at any position.

Images