# 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  
The force chain arch formed by releasing particles from the bottom under biaxial compression load is analyzed in the following section. The basic assumptions are as follows: the force chain arch ACB is an arch under plane strain, and its thickness has no impact on the arch; the overlying load \(q_{y}\) is a vertical uniformly distributed load, and the left lateral load \(q_{x}\) represents a horizontal uniformly distributed load, with the bottom and right sides being fixed, also with uniformly distributed loads \(q_{y}\), \(q_{x}\), respectively. As shown in Figure 4a, points A and B are the left and right arch foot, respectively; point C is the arch apex; and D is any point on the arch. The arch span is \(2l\), and the arch height is \(h\). First, the left half arch ADC is analyzed. The horizontal reaction force at the arch foot is \(F_{H}\); the vertical reaction force is \(F_{V}\); and the reaction force at the arch apex is \(F_{m}\). Based on the static equilibrium conditions at the arch apex (point C)  
\[\left\{ \begin{array}{l}\sum F_{x}=0\\ \sum F_{y}=0\\ \sum M_{C}=0\end{array} \right. \tag{1}\]  
Figure 4: Bottom particle flow: (**a**) Schematic of the force chain arch equilibrium; (**b**) Schematic of the forces at any point on the force chain arch. Figure 3: Force chain arch after particle release from the bottom: (**a**) Identified via autocorrelation algorithm; (**b**) Quadratic function fitting. which leads to  
\[\left\{ \begin{array}{l}F_{H}+q_{x}h-F_{m}=0\\ F_{V}-q_{y}l=0\\ F_{H}h+q_{y}l\frac{1}{2}-F_{V}l+q_{x}h\frac{l}{2}=0\end{array} \right. \tag{2}\]  
Solving the equations, the boundary conditions are obtained:  
\[\left\{ \begin{array}{l}F_{H}=-\frac{1}{2}q_{x}h+\frac{1}{2H}q_{y}l^{2}\\ F_{V}=q_{y}l\\ F_{m}=\frac{1}{2}q_{x}h+\frac{1}{2H}q_{y}l^{2}\end{array} \right. \tag{3}\]  
Any point D on the arch, with coordinates (\(x\), \(y\)), satisfies the interface trace equation. As shown in Figure 4b, the horizontal distance from point D to the left arch foot is \(l(x)\); its height is \(h(x)=y(x)\); and the angle between the central axis of the force chain arch and the \(x\)-axis, \(\angle\)OO\({}_{1}\)D = \(\theta(x)\). The normal and tangential forces are \(F_{N}\) and \(F_{T}\), respectively, where \(l(x)\), \(h(x)\), \(\theta(x)\) are functions uniquely determined by the abscissa \(x\), with \(q_{x}=q_{y}=66.7\) kN/m being the known uniformly distributed load. The equilibrium conditions for any point D(\(x\),\(y\)) on the arch are calculated in the following equations:  
\[\left\{ \begin{array}{l}\sum F_{x}=0:F_{H}-F_{T}sin\theta(x)-F_{N}cos\theta( x)+q_{x}h(x)=0\\ \sum F_{y}=0:F_{V}+F_{N}sin\theta(x)-F_{T}cos\theta(x)-q_{y}l(x)=0\end{array} \right. \tag{4}\]  
which leads to  
\[ \begin{bmatrix}cos\theta(x)&sin\theta(x)\\ sin\theta(x)&-cos\theta(x)\end{bmatrix}\begin{bmatrix}F_{N}\\ F_{T}\end{bmatrix} = \begin{bmatrix}q_{x}h(x)+F_{H}\\ q_{y}l(x)-F_{V}\end{bmatrix}  \tag{5}\]  
Therefore  
\[ \begin{bmatrix}F_{N}\\ F_{T}\end{bmatrix} = \begin{bmatrix}cos\theta(x)&sin\theta(x)\\ sin\theta(x)&-cos\theta(x)\end{bmatrix} ^{-1} \begin{bmatrix}q_{x}h(x)+F_{H} \\ q_{y}l(x)-F_{V}\end{bmatrix}  \tag{6}\] \[= \begin{bmatrix}(q_{y}l(x)-F_{V})sin\theta(x)+(q_{x}h(x)+F_{H}) cos\theta(x)\\ -(q_{y}l(x)-F_{V})cos\theta(x)+(q_{x}h(x)+F_{H})sin\theta(x)\end{bmatrix} \]  
Substituting Equation (3) to determine the force at any point D(\(x\), \(y\)) on the arch yields  
\[\left\{ \begin{array}{l}F_{N}=q_{y}[l(x)-l]sin\theta(x)+\begin{bmatrix}-\frac {1}{2}q_{x}h+\frac{1}{2H}q_{y}l^{2}+q_{x}h(x)\end{bmatrix}cos\theta(x)\\ F_{T}=q_{y}[l-l(x)]cos\theta(x)+\begin{bmatrix}-\frac{1}{2}q_{x}h+\frac{1}{2H} q_{y}l^{2}+q_{x}h(x)\end{bmatrix}sin\theta(x)\end{array} \right. \tag{7}\]  
Since \(x\) and \(\theta\) have a one-to-one correspondence, the terms related to \(l\) and \(h\) in Equation (7) are represented as functions of \(\theta\) to derive a more general conclusion.