Mechanical model of artificial roof overlay deformation elastic thin plate

After presupporting of roof, overlying roof bending deformation law has undergone greatly changed. In order to accurately calculate the top roof of artificial roof movement law after the roof pre-control top. Based on the analysis of the top moving characteristics of the roof and the theory of elastic foundation thin plate, the mechanics model of pre-control top-mining filling body-artificial roof-roof is established. The basic deflection, stress equations and the critical conditions of roof breaking are derived. The impact of roof bending deformation of the key factors affects the artificial roof thickness and elastic modulus. Combination of Wo Hu Shan specific engineering conditions, the calculation equation of tensile stress and sinking displacement of lying tiger hill is deduced and the critical roof thickness is 1.87m. To retain a certain safety factor, the artificial roof thickness of 2.5m is used. The monitoring results show that the maximum roof displacement is 15.80mm, which is consistent with the theoretical calculation.


Introduction
Under unstable conditions when mine rock is broken, the stope is prone to collapse, and leading to the loss of mine resources, which increase the difficulty in dealing with the mined area and stope mining around the caving area. Some studies have proposed that the roof can be reinforced after pre-cutting mining [1], that is, "pre-control top" to support roof rock without self-supporting capacity. When the original layered joints of the stope or the rock on the top of the stope are developed with cracks, severe structural influences, strong water absorption, and poor water permeability, if no control measures are taken, the roof of the stope may explode for several days or hours. "Pre-controlling roof" is the use of anchor cables or anchor nets to reinforce surrounding rock and construct an artificial roof, which can effectively support the roof and reduce the displacement of the roof, thereby improving the working environment and accelerating the work cycle [2]. However, the laws of movement and deformation of the stope roof at the top of the pre-control roof are still unknown.
At present, the research methods for the rock movement law mainly include the key layer theory, masonry beam theory, shatter block theory, and continuous beam theory, etc. It is very effective for understanding the movement law of the upper rock strata in the stope, but for the pre-control of the top, breaking the roof has a limited effect [3]. This paper deduces the theory to modify the elastic foundation slab model, calculates the rules of the movement of the pre-control roof covering, and validates with the numerical simulation and the actual measurement results, and provides theoretical guidance for the pre-control roof technology.

Pre-controlled top cover movement characteristics
For rock masses of Class III, IV, and V in underground rock masses, the self-stabilization capability of the chamber with a span of more than 10 m is extremely poor, and rock formations will loosen and deform within a few days. As China's mines gradually enter the deep, the stress and tectonic stress of the stope increase gradually, and the probability of loose deformation of the cracked rock mass increases. In underground mines, after the excavation of the stope, the surrounding rock will be directly exposed to the air. The originally broken rock will soon loosen and deform under the influence of the overburden pressure and air weathering erosion. The top plate fell off and the working face was in danger. The ground pressure of the overlying strata is not controlled in time. With the advancement of the working face, the exposed area gradually increases, the original loose deformation will cause the roof to fall, and the overlying strata will move in large scale, causing geo logical disasters and waste of resources.
After the stope is excavated, the broken roof shall be timely reinforced with shotcrete and anchor nets, and the concrete in the extremely broken area shall be reinforced by advancement [4], that is, "pre-control roof" technology, to construct an artificial roof and close the roof to prevent the roof from being exposed, weathered and corroded. At the same time, the artificial false top and the overlying roof act as active supports, and the artificial roof acts as a permanent support, bearing the pressure of overlying rock formations. The overlying rock formations will only undergo certain bending deformation, and thus slow down. The speed of the overlying strata will reduce the loose deformation of overlying strata and prevent the overburden destruction.

Assume the pre-control roof-top plate mechanical model
(1) Basic assumption of elastic sheet. Pre-control roof can ensure the roof does not break and collapse, and in the absence of large structural stress, the roof meets the basic assumption of the elastic sheet.
(2) Assumption of elastic foundation [5]. The artificial roof belongs to the reinforced concrete structure, which is combined with the action of the suspension and composite beam of the upper cover plate through the anchor rod, and is closely combined with the deformation of the overlying rock layer. The artificial roof and the overlying roof can be approximated. Think of a flexible foundation system.
(3) The elastic sheet shape assumption. According to the theory of the elastic sheet [6], the elastic sheet will satisfy the following conditions: Where h is the thickness of the sheet, m; a is the width of the sheet, m.
The stope of underground mines is arranged along the strike or vertical direction. The width is generally 15~30m, the length is 30~60m, the pre-control top thickness is 1~3m, and the middle section is divided into more than ten stopes along the strike. The thickness of the pre-control roof and the single stope or the entire middle section of the mining area all conform to the assumption of the elastic sheet shape and can be regarded as an elastic sheet [7].

Pre-control roof-top plate mechanical model
The upper part of the roof is subjected to uniformly distributed overburden loads q , the lower part is supported by an elastic foundation, and the artificial roof supporting force is kw , and the anchoring force of the anchor rod can be directly transmitted to the ceiling through an artificial false ceiling [8]. Therefore, the top plate can be regarded as an upper elastic plate with four sides fixedly supported by the elastic roof artificial roof. The mechanical model of the elastic plate is shown in Figure 1.  shown in Figure 1. Among them, b is the length of the middle pre-control roof, a is the width of the pre-control roof, q is the uniform load of the overlying rock on the roof, k is the elastic foundation coefficient, and w is the deflection of the roof.

Top plate bending deformation analysis
According to the principle of equivalent replacement [9], the system with four sides fixedly clamped is simplified as a simplified system that uses four sides to simply support and four sides receive the bending moments According to the common method used to solve the plate problem in mechanics, the double sine series adopted by Navier is used as the bending surface equation of the plate: Again, we see the top plate and the pre-control top elastic foundation as a whole system [10].
The uniform load on the overburden q , the fixed edge () Mx , () My and the virtual work performed T  are: The sum in the above equation of In the equations of equation (6), in each special case, the Fourier coefficients 1 3 5 7 , , , E E E E  and 1 3 5 7 , , , F F F F  can be obtained from these two equations by the successive approximation method.
According to the relationship between elastic modulus, stress, strain and elastic foundation coefficient [12] In the formula, w -roof deflection  From the deformation coordination conditions of composite materials in material mechanics, the elastic modulus of the artificial roof-filled body combination can be obtained as follows: In the formula: z E is the artificial roof elastic modulus; h E is the filling body elastic modulus; z A is the artificial roof cross-sectional area; h A is the filling body cross-sectional area;  is the proportion of cross-sectional area for artificial roof.

Case studies
The above deduced the deflection equation, bending moment equation and maximum stress of the artificial roof, and the main factors affecting the deformation and stress of the prestressed roof are the length, elastic modulus and bending stiffness of the roof along the inclination and strike. , Overlying uniform load, elastic modulus, cross-sectional area ratio of filling material and artificial roof, and stope height.
It is known that the length, breadth and height of the stope in the middle section of the 40-foot section of Wohushan Mine are respectively 40 12.5 16 m m m


.Stope vertical layout. Twenty stope filling and filling operations have been completed. The length is 250m, the width is 40m, mining height (recovery height and pre-control roof thickness) is 16m, pre-control roof height is less than 3.5m, filling body height is more than 12.5m, the elastic modulus z

Tensile stress analysis
The first two terms of the Fourier coefficient representing the fixed bending moment are approximated by m E and n F . Substituting the above data into equation (6), calculate . The relationship between the maximum tensile stress of the top plate and the thickness and elastic modulus of the artificial roof is obtained by bringing  Figure 2(a) and (b) below. As can be seen from the figure, with the increase of the artificial roof thickness, the maximum tensile stress in the roof gradually decreases, and the decreasing speed of the maximum tensile stress gradually decreases as the thickness of the artificial roof increases, and when the value increases to a certain value, the maximum tensile stress in the roof Gradually tends to be stable, indicating that the top plate plays the most significant supporting role when the thickness is small, and it can well control the stress in the top plate. The critical point where the roof does not break in both cases has been marked in the figure.
Maximum tensile stress in X Maximum tensile stress in Y  (12) yields the relationship between the maximum subsidence of the top plate max w and the thickness and elastic modulus of the artificial top plate. Since the relational form is too complex, it is not listed here. The theoretical calculation value of the maximum roof subsidence under different artificial roof thicknesses is shown in Figure 3. Here, the polynomial fitting theory calculation result of  As can be seen from the figure, the settlement displacement of the roof also gradually decreases with the increase of the thickness of the artificial roof, and the speed is getting slower and slower. When the thickness reaches a certain critical value, the settlement displacement of the roof is stable near a certain value, indicating that the artificial roof is opposite to the roof The control of the settlement displacement has a good effect, and the displacement of the roof can be controlled by adjusting the thickness of the artificial roof and the elastic modulus (the ratio of the concrete, etc.).
The relationship between the artificial roof thickness and the settlement displacement of the roof can be calculated. When the artificial roof thickness is the critical value of 1.87m, the settlement displacement of the roof is 22.85mm.

Project overview
The ore section basically adopts the method of pre-controlling the top-to-the-minute sub-section miner's backfill and mining method. The vertical ore body of the mine is oriented towards the height of 60m and is divided into 4 subsections. The span of the mine house is 12.5m, the height is 15m, and the length is the ore body. Thickness, roof support in each mine roof, namely pre-control roof, precontrol ceiling height 3m.
The spatial relationship of the stope-artificial roof-artificial roof is shown in the figure 4 below. Each roof of the stope is provided with a monitoring point to monitor the sinking displacement of the roof.

Pre-control top height
After the above theoretical calculation, the thickness of the artificial roof is 1.87m, the roof will theoretically not break, and the sinking displacement is 22.85mm. After numerical simulation and engineering practice, the mine reserves a certain safety factor and does not change the condition of the concrete used for filling and artificial false ceilings. The artificial roof used at present has a thickness of 2.5 m. At this time, the maximum tensile stress of the roof is 2.5 MPa. The sinking displacement is 16.17mm.