## 2 Analysis of Overburden Load Transfer Mechanical Model
### Roof Breaking and Load Transfer Process
#### 2.1.1 Mechanical Analysis before Roof Breaking  
The working face begins to advance from the open-off cut position. In the initial period, the hanging area of the basic roof is small, and the structural stability load capacity is strong. The overlying strata load is mainly concentrated in the bottom boundary position of the basic roof along the coal wall to the deep rock mass. At this time, the load is mainly transferred through the beam structure of the lower rock layer of the basic roof. This stage can be regarded as the bending of the beam on the elastic foundation (Figure 5).  
Figure 4: Full mining.  
Figure 5: Beam’ bending model before roof collapse.  
Figure 3: Periodic weighting.  
Assuming that the initial stress field acting on the rock mass is the original rock stress field, there is  
\[q=\sum\limits_{i=1}^{n}g_{i}h_{i} \tag{1}\]  
In the formula: \(g_{i}\), \(h_{i}\) are the bulk density and thickness of the \(i\)-th rock mass.  
Due to the difference between rock stratification and rock mass properties, the actual force acting on a rock beam can be judged and calculated by the following formula:  
\[q_{n}^{(x)}=\frac{E_{i}h_{i}^{3}\sum\limits_{i=1}^{n}g_{i}h_{i}}{\sum\limits _{i=1}^{n}E_{i}h_{i}^{3}} \tag{2}\]  
In the formula: \(E_{i}\), \(g_{i}\), \(h_{i}\) are the elastic modulus, bulk density, and thickness of the \(i\)-th layer rock mass; \(n\) is the number of rock strata acting on the beam.  
The shear force and bending moment of any section of the fixed beam are  
\[Q_{x}=f_{1}-qx,\ M_{x}=f_{1}x-\frac{qx^{2}}{2}+M_{1} \tag{3}\]  
Since the 'beam' is symmetrical under uniform load, therefore, the reaction forces at both ends of the beam: \(f_{1}=f_{2}=qa\), bending moment: \(M_{1}=M_{2}=-\frac{qa^{2}}{3}\).  
Therefore:  
\[Q_{x}=q(a-x),\ M_{x}=\frac{q}{6}(6 ax-3x^{2}-2a^{2}) \tag{4}\]  
It can be seen from Equation (4) that the maximum shear force is obtained at both ends of the fixed beam: \(Q_{\text{max}}=qa\), ultimate bending moment: \(M_{\text{max}}=-\frac{1}{3}qa^{2}\) (\(0<x<2a\)).  
Therefore, the maximum tensile force at the ultimate bending moment is  
\[\sigma_{\text{max}}=\frac{2 qa^{2}}{h^{2}} \tag{5}\]  
When \(\sigma_{\text{max}}=R_{t}\), the maximum pulling force at this point reaches the tensile strength of the rock stratum, the rock stratum will break at this point, and the limit span when the fixed beam breaks is  
\[S=h\sqrt{\frac{2R_{t}}{q_{1}}} \tag{6}\]  
In the formula: \(h\) is the single thickness of the main roof; \(R_{t}\) is basic roof tensile strength; \(q_{1}\) is basic top uniform load.  
From the above formula, it can be seen that the beam is the main structure of the load before the first weighting of the main roof, and its bending moment and pulling force are closely related to the hanging length of the roof. When the working face advances further, the length of the suspended roof increases, and the beam reaches its ultimate pulling force and breaks. The beam structure is transformed into a plate structure, and the distance between the breaking position and the open-off cut position is the length of the beam.