## 3 Experimental Results
### Interlayer Slip Conditions of Composite Rock Layers  
The experiment was simplified to a composite, simply supported beam model with a concentrated force \(P\) acting on the top (Figure 13). The dimensions and material properties of beam 1 and beam 2 were exactly the same. Compared with other analogic models, this model is a basic mechanical model of rock beams. The focus of the model is when the interlayer shear slip occurs rather than when the rock beams will fracture. To simplify the analysis, it was assumed that the interface was exactly located at the neutral axis of the composite, simply supported beam. According to the theory of material mechanics, a constant shear stress \(\tau\) will be generated between the layers:  
\[\tau=\frac{3P}{2 bh\Big{(}1+\sqrt{\frac{E_{t}}{E_{c}}}\Big{)}}, \tag{2}\]  
Figure 12: Relationships between the loading curve and the (**a**) acoustic emission duration and (**b**) amplitude. Figure 13: Simple beam model of composite sandstone layers. where \(h\) is the thickness of the beam; \(b\) is the width of the beam section; \(E_{t}\) is the Young's modulus; and \(E_{c}\) is the bulk modulus. When \(\tau\) is greater than the shear strength between the layers, the combined rock layer will experience interlayer shear sliding during bending. If the friction between the upper and lower layers of the rock is defined as the shear strength between layers, then, in addition to \(\tau\), the interlayer shear sliding is also related to the contact force \(q(x)\) between the layers and the friction coefficient \(\mu\). The starting condition for interlayer shear sliding is  
\[\tau\geq\frac{\mu}{b}q(x)+S_{coh}, \tag{3}\]  
where \(\mu\) is the friction coefficient; \(q(x)\) is the contact force between the layers; and \(S_{coh}\) is the cohesion force of the interface. The interlayer contact force \(q(x)\) is related to the bending stiffness of beams 1 and 2. Under the experimental conditions, the effect of the self-weight of the rock specimen on the deformation of the specimen was neglected, and the mechanical parameters of beams 1 and 2 were the same. The composite beam model shown in Figure 12 was decomposed into two independent, simply supported beam models (Figure 14). Considering the symmetry of the beam, the study range was only selected as \(0\leq x\leq l/2\). The deflection curves of beam 1 and beam 2 under independent conditions are  
\[w_{1}(x)=\frac{P}{48E_{t}I}(4x^{3}-3l^{2}x)\ (0\leq x\leq\frac{l}{2}), \tag{4}\]  
\[w_{2}(x)=0\hskip 56.905512 pt(0\leq x\leq\frac{l}{2}), \tag{5}\]  
where \(I\) is the moment of inertia of the beam, \(I=bl^{3}/12\). Assuming that \(q(x)\) is proportional to the deflection difference \(w_{1}(x)-w_{2}(x)\) between beams 1 and 2, then the interlayer contact force function \(q(x)\) is  
\[q(x)=kb[w_{1}(x)-w_{2}(x)]=k\frac{P}{48E_{t}I}(4x^{3}-3l^{2}x)\ (0\leq x\leq \frac{l}{2}), \tag{6}\]  
where \(k\) is an undetermined coefficient. The support reaction forces of beam 1 and 2 are \(F_{RA1}\) and \(F_{RA2}\). \[F_{RA1}=\frac{P}{2}+\int_{0}^{\frac{l}{2}}-q(x)\mathrm{d}x=\frac{P}{2}+\frac{ 5 kPl^{4}}{768E_{t}I}, \tag{7}\]  
Figure 14: Decomposed model of (**a**) beam 1 and (**b**) beam 2. \[F_{RA2}=\int_{0}^{\frac{l}{2}}q(x)\mathrm{d}x=-\frac{5 kPl^{4}}{768E_{t}I}. \tag{8}\]  
The bending moments, \(M_{1}(x)\) and \(M_{2}(x)\), of beams 1 and 2 on any section are  
Figure 12:0  
Figure 12:1  
According to the principle of virtual work, when a unit load acts on the mid-span of a simply supported beam, the bending moment of the beam is  
Figure 12:2  
The deflections of beam 1 and beam 2 in the mid-span are  
Figure 12:3  
Figure 12:4  
According to the deformation compatibility condition, the deflections of beams 1 and 2 at the mid-span location of \(x=l/2\) are the same, that is, \(w_{1}=w_{2}\).