# Progressive Collapse Performance of Prefabricated Step-Terrace Frame Structures in Mountain Cities
## 2 Nonlinear Alternative Load Path Method  
The alternative load path method does not focus on the causes of initial failure but, instead, on the mechanical performance of the remaining structure following the initial failure so that it can prevent or control the spread of local failure after the internal force redistribution. Because this method does not consider the causes of collapse, it is widely used in the analysis of structural performance and progressive collapse resulting from various reasons. The progressive collapse of structures is a complex, nonlinear process that involves significant deformation and internal force redistribution. The linear analysis method cannot accurately simulate the nonlinear process, so it is not suitable for the progressive collapse performance. Consequently, this paper will employ static nonlinear and dynamic nonlinear alternative load path methods to assess the progressive collapse performance. In the static nonlinear analysis, the suggestions from DoD2016 [15] were considered comprehensively; we adopted a full-span loading mode, applying a uniform and progressively increasing vertical static load of \(\alpha\)\(\times\) (1.2 DL + 0.5 LL) to all parts of the structure, as shown in Figure 1, where \(\alpha\) is the load amplification factor, DL is the dead load, and LL is the live load. The static nonlinear push-down analysis method adopts a load-displacement curve to evaluate the structure's performance to resist progressive collapse, where displacement refers to the vertical displacement at the selected failure point in the structure, and load pertains to the load amplification factor \(\alpha\). The load combination for dynamic nonlinear analysis was selected from DoD2016 as (1.2 DL + 0.5 LL) [1], which means that the full span of load with a load amplification  
Figure 1: Loading scheme of frame structure static nonlinear analysis. factor \(\alpha=1\) in static nonlinear analysis is arranged in all areas of the structure. The specific loading steps for dynamic nonlinear analysis are as follows:  
1. Applying a static load to the undamaged original structure and solving the internal force \(P_{0}\) of the column that is to be removed. 2. Removing the objected column, and the reaction force \(P_{0}\) of the internal force of the objected column is applied to the remaining structure, so that the static equivalent of the original complete structure model is formed. 3. Applying a load P which varies in size with time, and is opposite to \(P_{0}\) on the top of the equivalent model failure column, making \(P_{0}\) equal to P and completing column removal. The column removal time is 0.1 times the vertical natural period of the remaining structure according to the General Service Administration (GSA) [22]. Taking the C2K2 frame structure as an example, we can have the vertical natural period of the remaining structure, which is 0.281 s, so the removal time is 0.0281 s in this frame structure (1/10 of the vertical natural period of remaining structure), that is, 28.1 ms, and it is the same for others, so that the dynamic nonlinear analysis results of the structure were obtained. The loading scheme is shown in Figure 2.  
In this paper, the Rayleigh damping was used for nonlinear dynamic analysis; the Rayleigh damping assumes that the damping matrix is proportional to a combination of the mass matrix and the stiffness matrix, so we can have  
\[\mathbf{c}=a_{0}\mathbf{m}+a_{1}\mathbf{k} \tag{1}\]  
We can assume the damping ratio of control frequency \(\xi_{m}=\xi_{n}=\xi\); in this situation, the coefficient can be simplified as  
\[ \begin{pmatrix}a_{0}\\ a_{1}\end{pmatrix} =\frac{2\xi}{\omega_{m}+\omega_{n}} \begin{pmatrix}\omega_{m} \omega_{n}\\ 1\end{pmatrix}  \tag{2}\]  
We took the first two periods or frequencies of the structure, and \(\xi_{1}=\xi_{2}=0.05\) for calculations in this paper.