## 4 Results and Discussion
### Objective 1: Results in Terms of Displacements
Figure 8: Mean values of \(R_{D}\), 20-story building: (**a**) \(NS\) direction, (**b**) \(EW\) direction. Objective 2: Results in Terms of Resultant Stresses: Interstory Shears, Axial Loads, and Bending Moments  
The structural demands in terms of resultant stresses are now discussed. The ratios \(R_{V}\), \(R_{A}\), and \(R_{M}\), calculated according to Equations (3)-(5) given below, are used to compare the responses of the buildings for the two column depths in terms of interstory shears, axial loads, and bending moments, respectively. \[R_{V} =\frac{V_{M}}{V_{L}} \tag{3}\] \[R_{A} =\frac{A_{M}}{A_{L}}\] (4) \[R_{M} =\frac{M_{M}}{M_{L}} \tag{5}\]  
In the above equations, \(V_{M}\) and \(V_{L}\), \(A_{M}\) and \(A_{L}\), or \(M_{M}\) and \(M_{L}\) have a similar meaning as that of \(D_{M}\) and \(D_{L}\) in Equation (1), except that interstory shears, axial loads, and bending moments are now considered. Only the mean values of these ratios are given and discussed. The mean values of \(R_{V}\) are shown in Figures 9-11 for the three-, nine-, and twenty-story  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Sa/g**} & \multirow{2}{*}{**\(R_{DT}\)**} & \multirow{2}{*}{**EXT**} & \multirow{2}{*}{**INT**} & \multirow{2}{*}{**EXT**} & \multirow{2}{*}{**EXT**} & \multirow{2}{*}{**INT**} \\  & & **(1)** & & **(2)** & & **(3)** & **(4)** & **(5)** & **(6)** \\ \hline \hline \multirow{8}{*}{1} & \multirow{8}{*}{_NS_} & 0.2 & 1.21 &  
It can be observed that the mean values of \(R_{V}\) are smaller than unity in all cases, indicating that the interstory shears are larger for the models with deep columns. This was expected since the shear stiffness of deep columns is larger than that of medium columns; for example, the design shear strength, \(\phi_{V}V_{n}\), of the exterior base column of the three-story building (W14 \(\times\) 257) is 572 kips (2545 kN), while that of the corresponding equivalent deep column (W27 \(\times\) 258) is 852 kips (3791 kN). The values \(\phi_{V}\) and \(V_{n}\) are the shear reduction factor and the nominal shear strength, respectively. A little or moderate variation is also observed in the mean values of \(R_{V}\) with the seismic intensity, with the story number, or with the model height; the values range from 0.75 to 0.96, from 0.77 to 0.96, and from 0.79 to 0.94 for the three-, nine-, and twenty-story buildings, respectively. Results of Table 6 indicate that the \(R_{A}\) mean values for the three-story building are very close to unity, tending to slightly decrease with the seismic intensity; the values range from 0.99 to 0.91. For the other two models, they are essentially equal to unity; this is explained as follows: the axial load in the columns of the buildings with medium columns is produced by three sources: the horizontal seismic component (A\({}_{\rm{MH}}\)), the vertical seismic component (A\({}_{\rm{MV}}\)), and the gravity load (A\({}_{\rm{MG}}\)). The corresponding parameters for the buildings with deep columns are A\({}_{\rm{LH}}\), A\({}_{\rm{LV}}\), and A\({}_{\rm{LG}}\). The axial loads produced by the combined effect of the vertical seismic component and the gravity load for the models with medium columns (A\({}_{\rm{MV}}\) + A\({}_{\rm{MG}}\)) are observed to be quite similar to that of the models with deep columns (A\({}_{\rm{LV}}\) + A\({}_{\rm{LG}}\)). In addition, in both cases (medium or deep columns) the above-mentioned combined effect is much larger than the corresponding effect of the horizontal component (A\({}_{\rm{MH}}\) or A\({}_{\rm{LH}}\)) in such a way that values of \(R_{A}\) very close to unity are expected. In all cases, the values are quite similar for the \(NS\) and \(EW\) directions as well as for the interior and exterior columns. Hence, it can be stated that the axial load demands are essentially the same for the models with medium and deep columns.