# Dynamic Response Analysis of RC Frame against Progressive Collapse Based on Orthogonal Test
## 1 Introduction
The results showed that the deformation of the structure was underestimated when it showed upward displacement and initial velocity at the beginning of the progressive collapse. [PERSON] et al. [PERSON] et al. (2018) established a numerical model through LS-DYNA to study the influence of the dynamic response of the reinforced concrete beam and plate substructure on the initial damage that occurred after the sudden removal of the middle column. The results showed that the damage caused by the sudden removal of the column reduced the progressive collapse resistance of the substructure. Only single-factor analyses have been conducted on the residual structural dynamic response without studying the combined factor analysis and degree of influence of different factors on the structural dynamic response. In this paper, firstly, a dynamic response analysis of the initial displacement and initial velocity single factor was carried out under the working condition of side column failure. Then, five levels of initial velocity, initial displacement, and failure time were selected for each of the three working conditions of side column, corner column, and inner column failure by orthogonal test to arrange the combination reasonably. The data from the orthogonal tests were analyzed using the extreme difference method, and the most unfavorable combination could be derived from the maximum average degree of influence. The primary and secondary factors were derived from the magnitude of the extreme difference values. The orthogonal test method, also known as the orthogonal test design method, is widely used in chemical, mechanical, agricultural, and other fields [PERSON] et al. (2018); [PERSON] et al. (2018). The orthogonal test method is a scientific and reasonable arrangement of the multi-factor test method. When studying more complex problems, there will be multiple factors, and the different state of each factor is called the level. Various factors at numerous levels require an orthogonal test table to arrange the combination reasonably. The orthogonal table test arrangement scheme is representative and can fully reflect the influence of each factor on the index. Therefore, the orthogonal test method can reduce the number of tests. The test data obtained on this basis can analyze and draw correct conclusions consistent with the results obtained by comprehensive experiments. The orthogonal table is Ln(tq), where q represents the number of factors, t represents the number of levels, and n represents the number of trials. According to the orthogonal table test protocol, the test results were obtained, and the average influence degree of each factor in the i-level state was calculated _ki._  
\[k_{i}=\sum_{j=1}^{n}s_{j_{i}} \tag{1}\]  
(_Sji_ is the maximum vertical displacement of a factor in the _j_th test at the i level.) \[k_{i}=\frac{1}{n}K_{j} \tag{2}\]  
(n represents the number of levels.) The extreme difference, \(R\), for each factor, was obtained by the difference between the maximum value \(k_{imax}\) and the minimum value \(k_{imin}\) in each factor's average degree of influence in state _i._  
\[R=k_{imax}-k_{imin} \tag{3}\]  
For each factor, a level of maximum average influence _ki_ is chosen to determine the most unfavorable combination of factors that produce the maximum dynamic response of the structure. The extreme difference value \(R\) shows the degree of influence of each factor. The greater the extreme difference value, the greater the degree of influence of the factor, and the factors' order of priority can be derived.