## Materials and Methods
### Experimental Equipment and Strength Calculation Method  
The equipment used in this experiment mainly includes C64.106/1000 kN electro-hydraulic servo universal testing machine (Figure 2) and C-CAST experimental system (Figure 3). Figure 3a,b show the physical diagram and 3D structure diagram of C-CAST system, respectively. The C-CAST system is mainly composed of the top adapter, the bottom adapter, the limiting groove and the connecting components. The top and bottom adapters have the same structure and are symmetrically arranged in the experiment, consisting of the base and the locating dial [40; 41]. The experimental system can realize the inclined loading of coal and rock samples, with an inclination range of 0\({}^{\circ}\)-30\({}^{\circ}\) and an interval of 5\({}^{\circ}\). The installation process of the experimental system can be divided into four steps. Firstly, the upper and lower locating dials are rotated to the specified angle according to the experimental requirements, and fixed on the corresponding adapter by bolts; secondly, the prefabricated standard coal samples are placed in the circular groove (the design depth is 10 mm) of the locating dial, and then the supporting accessories are installed into the limiting groove to prevent the slippage extrusion failure; thirdly, the top adapter installed with the sample is fixed on the loading end of the electro-hydraulic servo universal testing machine through a rigid fixed ring, so as to realize the servo control during the loading process of the sample; lastly, through the fine-tuning testing machine, the sample is slowly inserted into the circular slot of the locating dial of the bottom adapter. After the above installation, experiments can be carried out according to the loading scheme. The specific structural assembly of C-CAST system is shown in Figure 3.  
Figure 3: Structure chart of the combined compression and shear test (C-CAST) system: (**a**) The physical diagram of C-CAST system; (**b**) The 3D structure diagram of C-CAST system. Figure 2: C64.106/1000 kN electrohydraulic servo universal experimental system: (**a**) Loading system; (**b**) Control system. Figure 4 shows the deformation and stress condition of coal samples under traditional uniaxial compression and inclined uniaxial compression, respectively. As shown in Figure 4, when the load applied by the testing machine to the end of coal samples is perpendicular to the axial direction of the sample, the axial stress and axial strain can be expressed as follows:  
\[\sigma=F/A, \tag{1}\]  
\[\varepsilon=d/l, \tag{2}\]  
where \(\sigma\) and \(\varepsilon\) are the axial stress and axial strain of coal samples, respectively; \(F\) is the vertical pressure applied by the testing machine to the end face of the sample; \(A\) is the cross-sectional area of the coal sample; \(d\) is the total vertical displacement of the testing machine after coal samples are loaded and damaged, namely the vertical compression displacement of the sample under the traditional uniaxial compression; and \(l\) is the initial height of the sample. When the vertical load applied by the testing machine to the end face of coal samples is distributed at an angle with the axial direction of coal samples, additional shear stress will be generated at the end face of coal samples. The vertical load applied by the testing machine on the end face of coal samples is decomposed into the compressive stress along the axial direction of coal samples and the shear stress parallel to the end face of coal samples.