# Mechanics and Stability of Force Chain Arch in Excavated Granular Material
## 3 Evolution of Force Chain Networks during the Solid-Fluid Transition Process
### Force Analysis of the Force Chain Arch Formed by Bottom Particle Flow
Thus, the force at any point D(\(x\)(\(\theta\)), \(y\)(\(\theta\))) on the arch is dependent on \(\theta\), expressed as Equation (8). The normal and tangential forces at various points on the arch are shown in Figure 5a,b, respectively. \[\left\{ \begin{array}{l}F_{N}=q_{y}[l(\theta)-l]sin\theta+\begin{bmatrix}- \frac{1}{2}q_{x}h+\frac{1}{2H}q_{y}l^{2}+q_{x}h(\theta)\end{bmatrix}cos\theta \\ F_{T}=q_{y}[l-l(\theta)]cos\theta+\begin{bmatrix}-\frac{1}{2}q_{x}h+\frac{1}{2 H}q_{y}l^{2}+q_{x}h(\theta)\end{bmatrix}sin\theta\end{array} \right. \tag{8}\]  
The normal force \(F_{N}\) on the arch is zero at \(\theta=\pi/2\), increases in absolute value toward the arch feet on both sides and attains a maximum value of approximately 1.5 kN at the arch feet. The tangential force \(F_{T}\) reaches the maximum of 20.75 kN at the arch feet and the arch apex, and the local minimum of 18.34 kN between the arch apex and the arch feet. Considering the stability of the arch shoulders, the relationship between the normal force and tangential force in equilibrium state is as follows:  
\[F_{N}=F_{T}tan\alpha \tag{9}\]  
network, or the particle stress may change. Thus, when the system attains a new equilibrium, the force chain network becomes updated. 2. When the particle system is subjected to a new stress disturbance, i.e., at least one of the boundary loads \(q_{y}\) or \(q_{x}\) increases, this causes a change in the forces at various points on the arch, and thus, the breaking of the stress arch and redistribution of the force chain network. The following scenario considers the critical conditions for the instability of the force chain arch under changes in boundary loads \(q_{y}\) or \(q_{x}\) and arch span \(L\): 1. Keeping the lateral load \(q_{x}\) constant and increasing or decreasing the overlying load \(q_{y}\), the resulting normal and tangential forces are shown in Figure 6a,b, respectively, and the ratio of normal to tangential force is shown in Figure 6c. When \(q_{y}\) decreases, the absolute value of normal force at each point increases and exceeds the value in the original equilibrium state (curve defined by green symbols). The normal force at the foot becomes the maximum value, and the critical point occurs at the arch foot. The tangential force decreases as a whole, and the value at the arch apex reaches the maximum. When \(q_{y}\) increases, the normal force at each point also rises and exceeds the maximum value in the original equilibrium state, with the arch feet being the critical points. The tangential force at each point increases along with \(\mathrm{q_{y}}\) increases. For one particular curve, for example, \(q_{y}=33.33\) kN/m, the tangential force decreases from the arch foot to the arch apex, defining the arch foot as the critical section. In summary, when the lateral load remains constant and the overlying load varies, the potential failure modes include normal buckling failure at the arch feet and tensile failure at the arch apex. When the overlying load increases, the potential failure modes include normal buckling failure and tensile failure at the arch feet. 2. Keeping the overlying load \(q_{y}\) constant and changing the lateral load \(q_{x}\), the normal force, tangential force and their ratio are shown in Figure 7a-c.  
Figure 6: Bottom particle flow with the changes in forces at various points on the arch under variation in the overlying load \(q_{y}\): (**a**) Normal force \(F_{N}\); (**b**) Tangential force \(F_{T}\); (**c**) The ratio of normal force to tangential force \(F_{N}\)/\(F_{T}\). When \(q_{x}\) decreases, the normal force at each point increases and exceeds the maximum value in the original equilibrium state (curve defined by green symbols), with the critical section located at the arch feet. The tangential force at each point decreases, remaining in a safe state, and the ratio of normal force to tangential force rises, with the critical section located at the arch feet.