## 2 The Proposed Seismic Design Procedure for Plan-Asymmetric RC Frame Buildings
\[e_{x,k}=\big{|}x_{cr,k}-x_{cm,k}\big{|}. \tag{4}\]  
Accordingly, \(\gamma_{j,k}\) values are chosen so that \(e_{x,k}\) will be zero. To minimize strength difference between frames and result in insignificant torsional rotation, \(m-2\) frames are assigned arbitrary \(\gamma_{j,k}\) values that range from 0.7 to 1.2. Thus, torsion will not impose high displacement demand to frames  
Figure 2: \(k\)-th story plan view. far from the supported mass center compared to those near it. One-frame relative stiffness value for story \(k\), \(\gamma_{1,k}\), is calculated by setting Equation (5) to zero and, can have a value other than a value between 0.7 to 1.2. If the origin is taken from the \(x\)-coordinate of the center of supported mass, \(e_{x,k}\) will be equal to \(x_{\sigma,k}\). Setting \(x_{\sigma,k}\) to zero as in Equation (5) and letting \(x^{\prime}_{j}\) be the \(x\)-coordinate from the center of supported mass to the frame \(j\), \(\gamma_{1,k}\) can be determined. \[\frac{\sum\limits_{j=1}^{m}\gamma_{j,k}K_{ref,k}x^{\prime}_{j}}{\sum\limits_ {j=1}^{m}\gamma_{j,k}K_{ref,k}}=0\Rightarrow\frac{\sum\limits_{j=1}^{m}\gamma _{j,k}x^{\prime}_{j}}{\sum\limits_{j=1}^{m}\gamma_{j,k}}=0\Rightarrow\sum \limits_{j=1}^{m}\gamma_{j,k}x^{\prime}_{j}=0. \tag{5}\]  
For frames closer to the center of supported mass, lower values of the range can be used and vice versa. When a negative value is obtained in calculating the \(\gamma_{1,k}\), the range can be adjusted until a positive value is found. The relative stiffness values can vary along the height following the variation of the center of supported mass location. In that case, average relative stiffness values can be used for values within a ten percent variation in a given consecutive story range. Typically, some frames have fewer columns and beams than other frames in a given story. For example, in Figure 2, frame A has two columns and one beam while frame J has three columns and two beams. In this case, an adjustment is needed. The \(k\)-th-story's lateral stiffness for the \(j\)-th frame and the reference frame can be equal if factors \(\eta_{lk}\) and \(\eta_{ck}\) are used for the \(j\)-th frame as follows:  
\[\frac{12E_{c,ref}/s_{k}^{2}}{\frac{1}{\sum\limits_{i=1}^{\frac{1}{k_{ref}}} \left(I_{gb,ref}/I_{bref}\right)_{i}}+\frac{1}{\sum\limits_{i=1}^{\frac{1}{k_{ ref}}}\left(I_{gc,ref}/s_{k}\right)_{i}}}=\frac{\frac{12E_{c,j}/s_{k}^{2}}{ \left(I_{gb,ref}/I_{gb}\right)_{i}}+\frac{1}{\eta_{ck}\sum\limits_{i=1}^{\frac {1}{k_{c}}}\left(I_{gc,j}/s_{k}\right)_{i}}} \tag{6}\]  
where \(I_{gb,ref}\) and \(I_{gc,ref}\) are the gross moment of inertia for beams and columns of the reference frame at story \(k\), respectively. \(I_{gb,ref}\) and \(I_{gc,j}\) are the gross moment of inertia for beams and columns of the \(j\)-th frame at story \(k\), respectively. \(I_{b,ref}\) and \(I_{b,j}\) are the beam span for the reference and \(j\)-th frame at story \(k\), respectively. \(b_{ref}\) and \(c_{ref}\) represent the number of beams and columns in story \(k\) for the reference frame, respectively. \(b_{j}\) and \(c_{j}\) represent the number of beams and columns in story \(k\) for the \(j\)-th frame, respectively. If \(E_{c,ref}=E_{c,j}\), then  
Figure 2.0 (a) Reference frame (Frame A). (b) Frame B. (c) \(J\)-th frame. Figure 2.1  
Keeping the equality of the left and right sides of Equation (7) and after simple algebraic manipulation,  
Figure 2.2 Figure 2.3  
Determine the factors \(\gamma\), \(\eta_{b}\) and \(\eta_{c}\) at all stories for all frames in the \(y\)-direction. Frames in the \(x\)-direction are unchanged. There is no need to calculate the actual value of the story's secant stiffness for any of the elements. The factors \(\alpha_{b}\) and \(\alpha_{c}\) along with \(\gamma\), \(\eta_{b}\) and \(\eta_{c}\) can be input as section property modifiers in the analysis software, in this case, SAP 2000 V.18. Steps 1 and 2 significantly reduce torsional responses and result in predominantly translational motion of the plan-asymmetric buildings.