## 5 Application  
After the response spectrum surface is calculated by the program, it can be used to analyze the time-domain response of the building structure. Previous studies only used the maximum value of the acceleration response [7; 8; 9; 10; 11], and could not use the response spectrum for time-domain analysis, which is reflected in the seismic codes of various countries that use the response spectrum curve. Therefore, this study used the response spectrum for time-domain analysis of building structures after extending the response spectrum to the time domain. The standard spectrum can only calculate the maximum response of the building structure within the earthquake time history, which is the difference between, and the advantage of, the response spectrum surface over the standard response spectrum curve. With regard to the response spectrum matrix, this section introduces the specific calculation method of the dynamic time-history analysis of building structures by using the response spectrum matrix through mode decomposition (response spectrum method of time-domain mode decomposition). Some simplification is needed when performing a dynamic analysis of structures. Structures with a significant mass concentration, high stiffness in the upper part, and small inter-story displacement are simplified to single-mass-point systems for seismic response analysis. Structures with a scattered mass distribution, low stiffness in the upper part, and large inter-story displacement are reduced to multiple-mass-points systems and subjected to seismic response analysis using the time-domain seismic response spectra specific to the multiple-mass-points systems. Before deriving the standard values of the seismic response in the time domain, the multiple-mass-points system's basic vibration modes must first be determined. On this basis, the natural vibration periods of different vibration modes and the displacement of each mass point for each vibration mode can be determined [12; 13; 14; 15]. During the basic vibration modes calculation, the mass points undergo purely translational motion with no rotation. Therefore, the shear stiffness is used to calculate the natural vibration period \(T_{i}\) and the displacement \(X_{ji}\) for each vibration mode. The seismic response of each mass point within the seismic duration is calculated by the time-domain mode decomposition reaction spectroscopy method [16; 17; 18; 19; 20]. For a multiple-mass-point system, the \(i\)-th mass point's standard value for the \(j\)-th vibration mode is expressed as follows:  
\[F_{ji}=\alpha_{tx}\,j\gamma_{j}X_{ji}G_{i} \tag{9}\]where \(\alpha_{tx\;j}=\left\{ \begin{array}{c}\alpha_{t1\;j}\\ \alpha_{t2\;j}\\....\\ \alpha_{tm\;j}\end{array} \right\}\) is the seismic influence coefficient matrix in the time domain for the \(j\)-th vibration mode under the corresponding natural vibration period, \(\gamma_{j}\) is the modal participation factor of the \(j\)-th vibration mode, \(X_{ji}\) is the displacement of the \(i\)-th mass point for the \(j\)-th vibration mode, and \(G_{i}\) is the representative value of the gravity load concentrated at the \(i\)-th mass point. Hence, the seismic response of each point in the time domain has the following form:  
\[F_{ji}=\left\{ \begin{array}{c}\alpha_{t1\;j}\\ \alpha_{t2\;j}\\....\\ \alpha_{tm\;j}\end{array} \right\}\gamma_{j}X_{ji}G_{i}=\left\{ \begin{array}[] {c}F_{t1\;ji}\\ F_{t2\;ji}\\....\\ F_{tm\;ji}\end{array} \right\} \tag{10}\]  
From the above, the standard value of the seismic shear force at time \(t_{x}\) at the \(i\)-th mass point for the \(j\)-th vibration mode is derived as follows:  
\[V_{ji\;tx}=\sum_{y=i}^{p}F_{jy\;tx} \tag{11}\]  
The standard value of the horizontal seismic force at \(t_{x}\) at the \(i\)-th mass point is obtained by extracting the square root of the sum of squares, which yields \(V_{i\;tx}\). Here, the physical meaning of the horizontal seismic forces combination is more explicit than in the conventional mode decomposition.