## 3 The Fundamental Principles of Vulnerability Analysis  
Seismic vulnerability describes the probability of a structure reaching a certain limit state and experiencing damage, establishing the relationship between the degree of structural damage and the intensity of seismic motion. Structural seismic vulnerability can be expressed using Equation (13):  
\[P_{f}(x)=P\langle D>C|IM=x\rangle \tag{13}\]  
In the equation, \(P_{f}(x)\) represents the exceedance probability function, \(D\) denotes the structural damage indicator, DM(\(\theta_{max}\)), C represents the limit state, and IM denotes the seismic intensity measure. In civil engineering, seismic vulnerability is often represented using vulnerability curves, with the seismic intensity measure (_IM_) plotted on the horizontal axis and the structural damage indicator (_DM_) on the vertical axis. By analyzing the various limit state points on the vulnerability curve, the probability of experiencing ultimate damage can be determined. As an effective method for evaluating the probability of a structure's resistance to collapse, seismic vulnerability analysis methods have garnered significant attention [17]. [PERSON] et al. [18] conducted research on the collapse resistance of reinforced concrete frame structures at different fortification levels based on structural collapse vulnerability curves, using collapse reserve coefficients as evaluation indicators. [PERSON] et al. [19] further studied the influence of three-dimensional seismic input on collapse vulnerability curves. [PERSON] et al. [20,21] conducted research on probability seismic risk theory and applications based on seismic vulnerability analytical functions. This study explores the influence of the number of stories and spans on the vulnerability of mountain landslide frame structures with energy dissipation rocking walls from various aspects, including IDA percentile curves, seismic probability models, vulnerability functions, probability of damage states, vulnerability indices, and collapse resistance. Seismic demand probability model:  
Specify that the relationship between the demand measure (_DM_) and intensity measure (_IM_) follows an exponential distribution:  
\[DM=\alpha(IM)^{\beta} \tag{14}\]  
Then, it can be inferred that:  
\[\theta_{max}=\alpha(PGA)^{\beta} \tag{15}\]  
Taking the natural logarithm of both sides of Equation (15) yields the following expression:  
\[ln(\theta max)=A+B\ ln(PGA)^{\beta} \tag{16}\]  
In the equation, \(A=ln\alpha\) and \(B=\beta\). The values of \(A\) and \(B\) are obtained through logarithmic linear regression using structural response data obtained from incremental dynamic analysis. The established probabilistic seismic demand model provides a basis for subsequent vulnerability analysis. Plotting vulnerability curves:  
According to the literature, under a specified intensity measure (_IM_), both the demand function, \(D\), and the capacity function, \(C\), follow a lognormal distribution. The probability of failure, \(P_{f}\), is:  
\[P_{f}(x)=\Phi\left[\frac{-ln\left(\hat{C}/\hat{D}\right)}{\sqrt{{\beta_{c}}^{2 }+{\beta_{d}}^{2}}}\right] \tag{17}\]  
In the equation, \(\hat{C}\) represents the median of the limit state indicator, \(\beta_{c}\) is the corresponding logarithmic standard deviation, \(\hat{D}\) denotes the median of the structural damage indicator, and \(\beta_{d}\) is the corresponding logarithmic standard deviation. Substitute Equation (15) into Equation (16) and simplify to obtain:  
\[P_{f}(x)=\Phi\left[\frac{ln\left(\alpha(PGA)^{\beta}/m_{c}\right)}{\sqrt{{ \beta_{c}}^{2}+{\beta_{d}}^{2}}}\right] \tag{18}\]  
The values of \(\beta_{c}\) and \(\beta_{d}\) are determined based on the HAZU99 [22] guidelines. When the spectral acceleration, \(S_{a}\), is used as the intensity measure (_IM_), \(\sqrt{\beta_{c}^{2}+\beta_{d}^{2}}\) is taken as 0.4. When the peak ground acceleration (PGA) is used as the intensity measure (IM), \(\sqrt{\beta_{c}^{2}+\beta_{d}^{2}}\) is taken as 0.5.