## 2 Multi Rigid Block Failure Mechanism
### Establishment of the Permissible Velocity Field  
To do an upper-bound analysis of strip footing under vertical uniform load, a flexible permitted velocity field must be created for each rigid block. The allowable velocity field can be determined using the triangular geometric relationship of the allowable velocity field. The absolute and relative velocity vectors of three types of rigid blocks are analyzed and solved, respectively:  
In Equations (9)-(33), the internal friction angle of the upper soil is \(\varphi_{1}\). The internal friction angle of the soil is \(\varphi_{2}\). \(\theta\) is the bottom angle of rigid block \(ABC\). \(\sum_{i=1}^{i-1}\alpha_{i}\) is the sum of \(i-1\) triangle the rigid block's top angle \(\alpha_{1}\) before rigid block \(i\). \(\beta_{i}^{u}\) is the bottom angle \(\beta\) of rigid block \(i^{u}\). \(\theta_{i}^{u}\), \(\theta_{i^{u}}^{v}\) and \(\theta_{i^{u}}^{v}\) represents the absolute velocity vector direction angle of rigid block \(i\), rigid block \(i^{u}\), and rigid block \(i^{u}\), where \(\theta_{i}^{v}\) is determinable. \(v_{i}\), \(v_{i^{u}}\) and \(v_{i^{u}}\) represents the absolute velocity vector of rigid block \(i\), rigid block \(i^{d}\), and rigid block \(i^{u}\). The relative speed between the two rigid blocks, \(i-1\) and \(i\), is represented by \(v_{i-1,i}\). The expression of relative velocity between the other rigid blocks is similar, so it will not be repeated. For rigid block: Type IWhen the two-layered strip foundation fails, rigid block \(ABC\) of Type I is designed to travel vertically downward at a speed of \(v_{0}\). For rigid block: Type IIDuring the analysis of Type II, due to the uncertainty of the relative motion direction between rigid blocks, there are two cases of compatible velocity relationship between rigid blocks \(i\) and rigid blocks \(i-1\), as shown in Figure 6.  
Figure 5: Calculation Diagram of Type III of Failure Mechanism (The bottom edge of the rigid block is above the soil layer). Figure 6: Two different cases of permissible velocity field between rigid blocks \(i-1\) and \(i\): **(a)** The relative velocity \(v_{i-1,i}\) between rigid blocks \(i\) and \(i-1\) face upwards along the velocity discontinuity; **(b)** The relative velocity \(v_{i-1,i}\) between rigid blocks \(i-1\) and \(i\) face downward along the velocity discontinuity. According to the compatible velocity vector triangle in two cases, we can calculate \(v_{i-1,i}\) using the Equations (9) and (10) or the Equations (11) and (12). Case: II-1: (Figure 6a)  
For Figure 6a, the relative velocity \(v_{i-1,i}\) between rigid blocks \(i\) and \(i-1\) face upwards along the velocity discontinuity and form an angle \(\varphi_{1}\). \[v_{i}=\frac{\sin\left(\frac{\pi}{2}\!+\!\varphi_{1}\!+\!\theta_{i-1}^{v}- \theta-\sum_{i=1}^{i-1}\alpha_{i}\right)}{\sin\left(-\frac{\pi}{2}-\varphi_{1} -\theta_{i}^{v}\!+\!\theta\!+\!\sum_{i=1}^{i-1}\alpha_{i}\right)}v_{i-1} \tag{9}\]  
\[v_{i-1,i}=\frac{\sin\left(\theta_{i}^{v}-\theta_{i-1}^{v}\right)}{\sin\left(- \frac{\pi}{2}-\varphi_{1}-\theta_{i}^{v}\!+\!\theta\!+\!\sum_{i=1}^{i-1}\alpha _{i}\right)}v_{i-1} \tag{10}\]  
Equation (11) applies to case: II \(-1\): (Figure 6a):  
\[ \begin{array}{c}\theta_{i}^{v}>\theta_{i-1}^{v}\\ -\frac{\pi}{2}\!<\varphi_{1}\!+\!\theta_{i-1}^{v}-\theta-\sum\limits_{i=1}^{i- 1}\alpha_{i}<\frac{\pi}{2}\\ \frac{\pi}{2}<-\varphi_{1}-\theta_{i}^{v}\!+\!\theta\!+\!\sum\limits_{i=1}^{i- 1}\alpha_{i}<\frac{3\pi}{2}\end{array}  \tag{11}\]  
Case: II-2: (Figure 6b)  
In the second case in Figure 6b, the relative velocity \(v_{i-1,i}\) between rigid blocks \(i-1\) and \(i\) face downward along the velocity discontinuity \(\varphi_{1}\).