## 2 Multi Rigid Block Failure Mechanism  
### Two-Layered Strip Foundation Failure Mode  
In terms of the different soil characteristics of a two-layered strip foundation, this paper makes the following assumptions before analyzing the bearing capacity of the foundation:  
1. The two-layered soils are assumed as elastic-perfectly plastic material, and the soils above the base level are simplified as overload q on both sides of the foundation. 2. Two-layered soils are subjected to the associated flow rule and Mohr-Coulomb yield criterion. 3. The foundation is assumed to be relatively still, and its base surface is assumed to be completely rough. 4. The foundation moves downward at an assumed vertical speed \(v_{0}\). Based on the above assumptions, a planar kinematically permissible multi-block failure mechanism under vertical uniform load is established (Take the right half for a diagram), as shown in Figure 1a. \(B_{0}\) is the width of the strip footing, \(H\) is the thickness of the upper soil, \(P_{u}\) is the vertical uniformly distributed load, \(q\) is the overload on both sides, and \(v_{0}\) is the vertical downward speed of the strip foundation. \(\varphi_{1}\), \(\gamma_{1}\), \(c_{1}\) and \(\varphi_{2}\), \(\gamma_{2}\), \(c_{2}\) are the internal friction angle, soil weight, and cohesion of the upper and lower soil, respectively. In this model, the base angle of the isosceles triangle is marked as \(\theta\), and base edge length is \(B_{0}\). Starting from the rigid block \(ABC\), the remaining rigid blocks are numbered to the nth block in turn from the left to right. For each rigid block, its angle variable is recorded as \(a_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), \(n\)), \(\beta_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), \(n\)), and the side length is recorded as \(l_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), \(n\)), \(d_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), \(n\)). The side length can be calculated through the geometric relationship of triangles, and the area of each rigid block is recorded as \(S_{ABC}\), \(S_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), \(n\)). It should be mentioned that, due to the uncertain position of the soil layer boundary, the multi-rigid block failure mechanisms for the two-layered strip foundation are discussed with three cases herein. The soil layer boundary passes through the stiff block \(ABC\) in the first case, which is depicted in Figure 1a. The second case is that the soil layer boundary line does not pass through rigid block \(ABC\) but passes through some subsequent rigid blocks, which is depicted in Figure 1b. The third case is that the soil layer boundary is completely placed under the foundation failure mechanism, as shown in Figure 1c. Since the weight of the upper and lower soil layers is different, it is necessary to calculate the gravity of the rigid block of different soil layers. Thus, the superscripts \(u\) (up), \(m\) (middle), and \(d\) (down) are used to differ the variables of the rigid block in the upper and lower soil layers, respectively. Figure 1: _Cont._  
The rigid blocks in the multi-rigid block failure mechanism of the two-layered strip foundation are divided into three types to calculate their area. The rigid block \(ABC\) is Type I. The failure mechanism of a stiff block of Type II only penetrates the top layer of soil, while that of Type III penetrates two soil layers. Figure 1: Three Cases of Two–Layered Strip Foundation Failure Mechanism: (**a**) Failure mechanism of the soil layer passing through the rigid block ABC; (**b**) Failure mechanism of the soil layer not passing through the rigid block ABC but through the subsequent rigid block; (**c**) Failure mechanism with the soil layer completely under the rigid block. (1) Type I  
When calculating Type I, there are two cases, that is, whether Type I passes through the upper and lower soil layers at the same time, as shown in Figure 2a,b. It is necessary to determine whether the rigid block \(ABC\) passes through the two layers soil by Equation (1).