## 2 The Working Principle of Rocking Walls  
In the rocking wall structure, the bottom constraints are relaxed, allowing the rocking walls to sway during an earthquake. This reduces the seismic response of the structure and effectively prevents concentrated deformation failure in frame structures under the \"strong column-weak beam\" scenario. Additionally, dampers can be installed at the rocking wall's motion points to increase energy dissipation. This mitigates concentrated deformation in the structure, enhances energy absorption, and protects structural safety. Energy-dissipating components can be quickly replaced after an earthquake, restoring the structure's functionality. The working principle of rocking walls can be illustrated through a simple frame-rocking wall structure. The rocking wall is connected to the floors using connectors that only transfer shear forces. Rigid links are used instead of connectors, with the shear force transferred by the connectors being replaced by the axial force of the rigid links. The frame is simplified using a continuous method, considering only shear deformation, not bending deformation, and modeled as a shear beam with constant shear stiffness. Conversely, the rocking wall is simplified to consider only bending deformation and not shear deformation, modeled as a bending beam with constant flexural stiffness. It is assumed that the simplified beams of the frame and the rocking walls are tightly connected, with axial forces continuously distributed along the interface. The schematic diagram for the calculation is shown in Figure 3.  
Figure 1: Seismic damage of mountainous structures. Figure 2: Distribution of plastic hinges in low-cycle repeated loading tests for step-terrace frame structures. Let the lateral displacement of the frame and rocking wall axes be \(y(x)\), where \(x\) is the vertical direction along the simplified beam. The flexural stiffness of the rocking wall is \(E_{w}I_{w}\). The shear stiffness of the frame is \(K\), \(p(x)\) is the external load distribution, and the continuously distributed internal force between the rocking wall and the frame is \(p_{F}(x)\). From the equilibrium of forces on the rocking wall, it can be obtained that:  
\[E_{w}I_{w}\frac{d^{4}y}{dx^{4}}=p(x)+K\frac{d^{2}y}{dx^{2}} \tag{1}\]  
Introducing dimensionless parameters \(\lambda\) and \(\zeta\):  
\[\lambda=H\sqrt{\frac{K}{E_{w}I_{w}}} \tag{2}\]  
\[\zeta=\frac{x}{H} \tag{3}\]  
Derive the general solution:  
\[y=C_{1}+C_{2}\zeta+A\sinh\lambda\zeta+B\cosh\lambda\zeta+y_{0} \tag{4}\]  
where \(sinh\ x\) is the hyperbolic sine function, \(cosh\ x\) is the hyperbolic cosine function, and \(y_{0}\) is a particular solution of the differential equation:  
\[sinh\ x=\frac{e^{x}-e^{-x}}{2}cosh\ x=\frac{e^{x}+e^{-x}}{2} \tag{5}\]  
Write the expression for \(y(x)\). Based on this expression, derive the bending moment and shear force of the rocking wall, as well as the shear force of the frame:  
\[M_{w}=E_{w}I_{w}\frac{d\theta}{dx}=\frac{E_{w}I_{w}}{H^{2}}\frac{d^{2}y}{d \zeta^{2}} \tag{6}\]  
\[V_{w}=-E_{w}I_{w}\frac{d^{2}\theta}{dx^{2}}=-\frac{E_{w}I_{w}}{H^{3}}\frac{d^{ 3}y}{d\zeta^{3}} \tag{7}\]  
Figure 1:0  
Figure 1:1 Structural model of the frame-rocking wall: (**a**) flat model (arrows represent horizontal seismic forces) (**b**) computing model. When the external load distribution is uniform \(p(x)=q\), the boundary conditions of Equation (1) can be derived from the above formulas, as follows:  
When \(x=H(\zeta=1)\), \(V=V_{w}+V_{F}=0\). When \(x=H(\zeta=1)\), \(M_{w}=0\). When \(x=0(\zeta=0)\), \(M_{w}=0\). When \(x=0(\zeta=0)\), \(y(0)=0\). The lateral displacement of the structure is obtained as follows:  
Figure 1:2  
The bending moment of the rocking wall is:  
Figure 1:3  
The shear force of the rocking wall is:  
Figure 1:4  
The shear force of the frame is:  
Figure 1:5  
From Equations (11) and (12), it can be seen that the shear force shared by the frame becomes more uniformly distributed along the height as \(\lambda\) decreases.