## 2 The Proposed Seismic Design Procedure for Plan-Asymmetric RC Frame Buildings  
A seismic design procedure is proposed for plan-asymmetric buildings to alleviate the shortcomings of DDBD. The main tasks undertaken are the following:  
1. Assigning moment of inertia reduction factors for members of frames so that the center of rigidity and center of supported mass coincide. 2. Scaling the moment of inertia reduction factors so that the allowable story displacements, drifts, and plastic hinge rotations are not exceeded. 3. Determining elastic higher-mode force effects that are combined with DDBD-obtained force effects. The buildings' stiffness modified according to tasks (1) and (2) is used for modal analysis. Design can then be performed for the modal superposed lateral force effects, P-\(\Delta\) forces, and gravity loading. Before applying the proposed procedure, member dimensions are assumed and gravity loads are determined. A given performance objective is chosen, which the building must fulfill under specified seismic hazard levels anticipated at the building site. With the above data obtained, the steps to be followed are given below. A flowchart is provided to show a brief outline of the steps as given below (Figure 1). In this study, uni-directional ground motions in the \(y\)-direction (Figure 2) are applied and responses checked. Figure 1: Flowchart for the proposed seismic design procedure accounting for higher-mode and torsional effects. (Step 1) Consider the plan view of a given building at the \(k\)-th story as shown in Figure 2. Using Equation (1), calculate the \(x\)-coordinate's location of the \(k\)-th story center of supported mass, \(x_{cm,k}\). \[x_{cm,k}=\frac{\sum\limits_{i=k}^{n}m_{i}x_{mi}}{\sum\limits_{i=k}^{n}m_{i}} \tag{1}\]  
where \(m_{i}\) is the mass at story \(i\), \(x_{m,i}\), the \(x\)-coordinate of the center of mass at story \(i\), and \(n\) represents the total number of stories. (Step 2) Equation (2) is developed by [PERSON] et al. [36] to determine the lateral stiffness of a given frame at story \(k\), \(K_{k}\). \[K_{k}=\frac{12E_{c}/s_{k}^{2}}{\frac{1}{\sum\limits_{l=1}^{\frac{n}{l}}\left(l _{gb}/l_{b}\right)_{i}}+\frac{1}{\sum\limits_{l=1}^{\frac{n}{l}}\left(l_{gc}/s _{k}\right)_{i}}} \tag{2}\]  
where \(E_{c}\) is the young modulus of the concrete material, \(s_{k}\) is the \(k\)-th-story height, and \(l_{b}\) is the beam's span length. \(b\) and \(c\) represent the number of beams and columns in story \(k\), respectively. \(l_{gb}\) and \(l_{gc}\) are the gross moment of inertia for beams and columns, respectively. To determine the \(x\)-coordinate of the center of rigidity at story \(k\) of a building in nonlinear range, \(x_{cr,k}\), Equation (3) is used. \[x_{cr,k}=\frac{\sum\limits_{j=1}^{m}K_{jk}x_{j}}{\sum\limits_{j=1}^{m}K_{jk}}= \frac{\sum\limits_{j=1}^{m}\gamma_{jk}K_{ref,k}x_{j}}{\sum\limits_{j=1}^{m} \gamma_{jk}K_{ref,k}} \tag{3}\]  
where \(K_{jk}=\gamma_{j,k}\)\(K_{ref,k}\). \(K_{ref,k}\) is the secant lateral stiffness of a reference frame at story \(k\). A reference frame is preferably chosen arbitrarily, which rises to the top of the building roof. Figure 2 shows that frame A is a reference frame while its elevation view is depicted in Figure 3. The \(k\)-th story secant stiffness for the reference frame is determined by substituting \(l_{gb}\) with \(\omega_{b}l_{gb}\) and \(l_{gc}\) with \(\alpha_{c}l_{gc}\) (for the first-story column only). \(\omega_{b}\) and \(\alpha_{c}\) are the moment of inertia reduction factors for beams and columns, respectively. \(x_{j}\) is the \(x\)-coordinate for frame \(j\). \(\gamma_{jk}\) is a factor assigned to the \(j\)-th frame to relate the frame to the reference frame secant stiffness at story \(k\). \(\gamma_{ref,k}\), is clearly 1, and \(m\) stands for the number of frames at story \(k\). To eliminate the out-of-plane torsional floor rotation at story \(k\), the in-plane stiffness eccentricity at story \(k\), \(e_{x,k}\), given by Equation (4) below, is set to zero.