## 3 Knife Handle-Type Working Face Overlying Rock Breaking and Inducing Mechanism
### Stope Roof Model  
Thin plate theory is used to analyze the stability of the working surface roof [27; 28], and the bending moment distribution state of the knife handle-type working surface roof is studied. Taking the I010206 working face as an example, the back of the I010206 working face is the gob; there is a 15 m protective coal pillar on the side of tailgate. The degree of plastic damage in the protected coal pillar is high, resulting in its limited support capacity for the roof, and the overlying rock of the I010202 working face has been broken; therefore, the lower side and left side of the roof are set as simply supported boundaries, and the headgate and front of the roof of the working face are solid coal areas, which are in a stable supporting state. The upper and front sides of the stope roof are set as the fixation boundaries.  
According to the above roof boundary characteristics, the front of the coal wall of the working face and the tailgate trough protective coal pillar are fixedly supported boundaries, the goat area behind the working face and the protective coal pillars of the two working faces are simply supported boundaries, and the cyclic failure mechanics model of the continuous working face is as follows: The structure is fixedly supported on both sides and simply supported on both sides. The top plate is subject to a uniform load \(q\). The mechanical model is shown in Figure 2.  
Figure 1: Layout of I010206 working face.  
Figure 2: Mechanical model of periodic breakage of working face roof.  
Its boundary conditions are:  
\[w_{x=0}=0,\left(\frac{\partial^{2}w}{\partial^{2}x}\right)_{x=0}=0,w_{x=a}=0, \left(\frac{\partial w}{\partial x}\right)_{x=a}=0 \tag{1}\]  
\[w_{y=0}=0,\left(\frac{\partial^{2}w}{\partial y^{2}}\right)_{y=0}=0,w_{y=b}=0, \left(\frac{\partial w}{\partial y}\right)_{y=b}=0 \tag{2}\]  
Assume that the deflection function is:  
\[w=Cxy\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{2}\right)^{2} \tag{3}\]  
According to [PERSON]'s law:  
\[\
abla^{4}w=8C\left[15 xy\left(x^{2}-a^{2}\right)^{2}+15 xy\left(y^{2}-b^{2} \right)^{2}+4\left(5x^{3}-3a^{2}x\right)\left(5y^{3}-3b^{2}y\right)\right] \tag{4}\]  
\[\int_{0}^{a}\int_{0}^{b}q\varphi(x,y)=\frac{qa^{6}b^{6}}{36} \tag{5}\]  
\[\int_{0}^{a}\int_{0}^{b}D\
abla^{4}w\varphi(x,y)=CD\frac{8192a^{7}b^{7}\left( 45a^{4}+44a^{2}b^{2}+45b^{4}\right)}{1091475} \tag{6}\]  
Calculated as:  
\[C=\frac{121275q}{32768 ab\left(45a^{4}+44a^{2}b^{2}+45b^{4}\right)D} \tag{7}\]  
\[w=\frac{121275 qxy\left(x^{2}-a^{2}\right)^{2}\left(y^{2}-b^{2}\right)^{2}}{32 768 ab\left(45a^{4}+44a^{2}b^{2}+45b^{4}\right)D} \tag{8}\]  
Then, the bending moment value of the plate structure with fixed support on both sides and simple support on both sides is obtained:  
\[M_{y}=-\frac{121275q\left[v\left(5x^{3}-3a^{2}x\right)y\left(y^{2}-b^{2} \right)^{2}+x\left(x^{2}-a^{2}\right)^{2}\left(5y^{3}-3b^{2}y\right)\right]}{ 8192 ab\left(45a^{4}+44a^{2}b^{2}+45b^{4}\right)} \tag{9}\]  
The energy \(U_{(x)}\) at any position of the short fixed support edge of the working surface roof can be obtained from the bending moment function, and its expression is:  
\[U_{(x)}=\int\frac{M^{2}\left(y\right)}{2 EI}dx \tag{10}\]