# Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume
## 4 Numerical Examples and Discussions  
Table 2 shows the effect of the rotatory inertia couple \(T\) on the frequency parameter \(C_{i}\). The input arch parameters are square cross-section, \(\alpha=1.5\pi\) and \(m=1.5\). Considering the effect of \(T\), \(C_{i}\) decreases compared to the case where \(T\) is not considered. When considering \(T\) in the dynamic analysis of a structure, the natural frequency decreases since the deformation increases under the same energy. The effect of \(T\) on \(C_{i}\) is greater in the higher order mode than in the lower order mode. For example, in the clamped-clamped arch, under the same \(\kappa=0.3\), the reduction rate of \(i=1\) is 0.19% (\(0.17660/0.17626~{}=~{}1.0019\)), but the larger reduction rate of \(i=4\) is 0.57% (\(0.90445/0.89934~{}=~{}1.0057\)). The effect of \(T\) is greater for larger volume ratio than for smaller volume ratio. In the above clamped-clamped arch, under the same \(i=4\), the reduction rate of \(\kappa=0.9\) is 1.43% (\(1.03688/1.02230~{}=~{}1.0143\)), which is larger than the reduction rate 0.57% of \(\kappa=0.3\). Therefore, when a high frequency dynamic load is applied to a tapered horseshoe arch with a large volume ratio, the influence of \(T\) should be considered to accurately analyze the dynamic behavior. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**End Condition**} & \multirow{2}{*}{\begin{tabular}{c} **Geometry of Arch with** \\ **Non-Dimensional Parameters** \\ \end{tabular} } & \multirow{2}{*}{\(i\)} & \multicolumn{2}{c}{**Angular Frequency \(\omega_{i}\) (rad/s)**} \\ \cline{3-4}  & & & **ADINA** & **This Study** \\ \hline \multirow{4}{*}{\begin{tabular}{c} Hinged-hinged \\ \(D_{c}=0.2\) m (\(\alpha=1.2\pi\), \(m=0.5\), \(\kappa=0.108\)) \\ \end{tabular} } & 1 & 443.2 & 441.0 \\  & \begin{tabular}{c} Square : \(D_{a}=0.1\) m and \\ \(D_{c}=0.2\) m (\(\alpha=1.2\pi\), \(m=0.5\), \(\kappa=0.108\)) \\ \end{tabular} & 2 & 1428.0 & 1434.5 \\  & \begin{tabular}{c} 3 \\ 4 \\ \end{tabular} & 2932.3 & 2968.5 \\  & \begin{tabular}{c} 4 \\ \end{tabular} & 4147.6 & 4259.5 \\ \hline \multirow{4}{*}{\begin{tabular}{c} Hinged-clamped \\ \(\alpha=1.4\pi\), \(m=1\), \(\kappa=0.176\)) \\ \end{tabular} } & 1 & 595.0 & 591.0 \\  & \begin{tabular}{c} Uniform cross-section \\ \(D_{a}=D_{c}=0.2\) m \\ \((\alpha=1.4\pi\), \(m=1\), \(\kappa=0.176\)) \\ \end{tabular} & 2 & 1574.3 & 1584.6 \\  & \begin{tabular}{c} 3 \\ 4 \\ \end{tabular} & 3040.8 & 3045.5 \\  & \begin{tabular}{c} 4 \\ \end{tabular} & 4 & 4323.9 & 4363.9 \\ \hline \multirow{4}{*}{\begin{tabular}{c} Clamped-clamped \\ \(D_{c}=0.1\) m (\(\alpha=1.6\pi\), \(m=2\), \(\kappa=0.0737\)) \\ \end{tabular} } & 1 & 375.8 & 382.4 \\  & \begin{tabular}{c} Circular : \(D_{a}=0.2\) m, \\ \(D_{c}=0.1\) m (\(\alpha=1.6\pi\), \(m=2\), \(\kappa=0.0737\)) \\ \end{tabular} & 2 & 841.1 & 852.0 \\  & \
\begin{tabular}{c} 3 \\ 4 \\ \end{tabular} & 1513.8 & 1534.7 \\  &
\begin{tabular}{c} 4 \\ \end{tabular} & 2313.0 & 2375.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of angular frequency \(\omega_{i}\) between ADINA and this study. Figure 5: Convergence analysis. Table 3 shows the effect of the cross-sectional shape on frequency parameter \(C_{i}\). The input arch parameters are \(\alpha=1.5\pi\), \(m=1.5\) and \(\kappa=0.25\)  with the rotatory inertia couple. The \(C_{i}\) of the square cross-section is larger than those of the circular cross-section, regardless of the end condition and mode number. Its effect is larger in lower mode than in higher mode. For example, in clamped-clamped end conditions, the effect is 2.28% (0.16130/0.15771 = 1.0228) for the first mode, while the effect is 0.081% (0.87560/0.86855 = 1.0081) for the fourth mode. In the following discussion, the effect of \(T\) was included in numerical examples. Figure 6 shows the natural frequency curve of the frequency parameter \(C_{i}\) according to the change in the opening angle \(\alpha\). The input arch parameters are square cross-section, \(m=1.5\) and \(\kappa=0.25\). As \(\alpha\) increases, \(C_{i}\) decreases.