## 2 Materials and Methods
### Response Spectrum Analysis  
This method is based on the combination of the excitations of each vibration mode of the structure. The response spectrum with the parameters from the previous section is used. With this method, maximum displacements in X and Y for seismic loading are analysed using the Finite Element Moment (FEM) defined in the structural model. Once the response spectrum type is defined, different loads are assigned. The contribution of each vibration mode must be considered for each case (X or Y axis). The NCSE-02 standard specifies that at least all modes with periods greater than \(T_{A}\) must be considered. The response spectrum characteristic period \(T_{A}\) is dependent on the contribution coefficient (K) and the geotechnical characteristics of the ground. It is determined in accordance with NCSE-02. According to modal analysis, the structural model with FEM ultimately employed the first six modes to ensure movement of at least 93% of the mass. Regarding the combination rule of the results for different modes, generally, the maximum response is not aggregable through arithmetic summation. The square root of the sum of squares rule, which is a geometric sum, was used.  
The different modes of vibration, are affected by the adopted damping ratio. This value is theoretically not constant, it varies according to the mode of vibration, mass and stiffness. If the mass and stiffness of the structure are maintained, it can be taken as constant. The modes of vibration which move most of the mass will be affected by the adopted damping ratio, but will vary according to the considered axis X or Y.  
The influence of ductility on the modification of the seismic response spectrum has a significant influence on the response of the structure according to the spectral superposition model. As can be seen, the difference in ductility is the most evident. A structure with higher ductility will have a higher damping and, therefore, better earthquake behaviour. Deformations will be smaller, and stresses will be lower. Additionally, the ductile behaviour of the structure will be varied to check its effect on displacements. Ductility will be varied through the elastic response spectrum. Regarding ductility, it can be varied either according to the option allowed by NCSE-02, which defines a ductility behaviour coefficient (\(\mu\)) that affects the components of the acceleration vector, depending on the vibration mode and degree of freedom, or as in the present analysis, affecting ductility by varying the viscous damping ratio (\(\chi\)).  
In the model implementation, a seismic response spectrum according to the location, NCSE-02 standard, and current Eurocodes applied in Spain [29] will be adopted. The site parameters, corresponding to the design of the current structure for a residential building of the considered type, are \(a_{c/g}=0.10\) m/s\({}^{2}\) and a contribution coefficient \(K=1.3\)[35]. The current required value for the structure at the considered location is \(a_{c/g}=0.13\) m/s\({}^{2}\)[11]. However, the structure was not renovated according to this value, so the previous value was considered.  
It was determined that the spectrum parameters for spectral modal analysis are \(T_{A}=0.208\) s, \(T_{B}=0.832\) s, and \(T_{D}=2\) s (in accordance with [32]), where \(T_{A}\) and \(T_{B}\) are the characteristic response periods of the structure, and \(T_{D}\) is the value that defines the start of the response spectrum range in which displacement begins to be constant. It is defined by the Eurocode EC-8(2004) [36]. Load combination was carried out according to a concrete structure in compliance with the Spanish structural code standard (CE) that implements Eurocode recommendations [32; 35]. Particular attention was paid to the superposition of accidental actions, including the seismic effect.