## 2 Numerical Modeling  
### Modeling of the Tested Structures  
In this study, according to the structural design code of China [29], a model for RC conventional frame structures with five layers, three spans, and five bays is designed. The floor height is 3.3 m. The structural model has a longitudinal span of 7.2 m and a transverse span of 6 m. Component dimensions and reinforcement information are listed in Table 1.  
Symmetrical three-dimensional building models are established with the OpenSEES program [30]. The Concrete02 material model is selected as the constitutive model of concrete. The model considers the tensile strength of the material and simplifies the tensioned softening branch to a linear curve, which ensures the accuracy of the calculation model and has good convergence along with high computational efficiency. The skeleton curve of the concrete in the constitutive model adopts the modified Kent-Park model [31], as shown in Figure 1. This study selected the Steel02 material model as the constitutive model of reinforcement. The Steel02 material model considers the steel tensile strength, in which the tensile softening branch is simplified as a linear processing. The model considers the effect of material strain hardening, which has been put forward by [PERSON] [32] and modified by [PERSON] [33]. The material model also considers the Bauschinger effect. Figure 2 shows the Steel02 material monotonic envelope curves. This study used a nonlinear beam column element and fiber section in addition to the torsional section. The symmetrical building model is a three-dimensional numerical model. Each node has six degrees of freedom. The floor slab adopts a three degrees of freedom rigid diaphragm beam in OpenSEES. The floor mass is concentrated at the geometric center of the floor. Figure 1: Material envelope curve models of concrete. Design load represents the combination of permanent loads and variable loads. The permanent loads include the structural self-weight, decorations, fixed equipment, and long-term storage. The variable loads include the weight of the distributed population, removable partitions, changing pipelines, and the weight of furniture placed on the floor. The combination of permanent loads and variable loads is calculated by using the following formula:  
\[S_{\text{d}}=\mu_{\text{G}}S_{\text{G}}+\mu_{\text{QI}}S_{\text{QI}}+\mu_{ \text{Q}}S_{\text{Q}} \tag{1}\]  
where \(S_{\text{d}}\) is the designed load value; \(S_{\text{G}}\) and \(S_{\text{Q}}\) are permanent loads and variable loads, respectively; \(S_{\text{QI}}\) is the main controlled variable load; \(\mu_{\text{G}}\) and \(\mu_{\text{Q}}\) are combined factors for permanent loads and variable loads, which are 1.2 and 0.7, respectively; \(\mu_{\text{QI}}\) is the combined factor for main controlled variable load, which is 0.98. On the basis of the symmetric structural model, mass eccentric structural models are developed. Generally, the eccentric structural models are divided into \"mass eccentricity models\" and \"rigidity eccentricity models\". In this study, the mass eccentricity model is chosen to characterize structural eccentricity, which is used to represent a wide range of situations including probable distribution of furniture load, personnel distribution, equipment installation, construction error, and so on. To achieve different eccentricities in the structural models, the mass concentration point (i.e., floor mass center) of each floor is continuously shifted along the horizontal direction. The floor rigidity center remains unchanged. Then, the models will have different eccentricities with the movement of the mass concentration point. Moreover, with the eccentricity adjustments, there are no changes in other building parameters, including floor mass and floor stiffness distribution. Thus, the only variable that affects the torsional response is eccentricity.