# Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume
## 4 Numerical Examples and Discussions
As \(\alpha\) increases, the span length of the arch decreases, while the height increases and structural stiffness of the arch decreases, resulting in a decrease in \(C_{i}\). In the hinged-hinged arch, when the arch becomes a complete circle of \(\alpha=2\pi\) marked with \(\bullet\), \(C_{i}\) vanishes, i.e., \(C_{i}=0\) and the arch becomes unstable. The hinged-hinged arch with \(\alpha=2\pi\) means that the whole structure becomes 'unstructured' because it cannot resist the lateral load. This is consistent with the results of the free vibrations of elastica arch [17]. Meanwhile, the hinged-clamped and clamped-clamped arches with \(\alpha=2\pi\) can maintain a stable state because it can resist the lateral load at the clamped end. For the hinged-hinged and clamped-clamped arch, there is a double frequency in the coordinates marked with \(\blacktriangle\). It is a well-known phenomenon that the mode shape can be changed from symmetric to anti-symmetric or vice versa [19; 20; 21]. On the other hand, in the case of the hinged-clamped arch, there is no double frequency because the end condition is asymmetric. This is also a well-known fact, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**End Condition**} & \multirow{2}{*}{**\(\kappa\)**} & \multirow{2}{*}{\(T\,\)Index\({}^{*}\)**} & \multicolumn{4}{c}{**Frequency Parameter \(C_{i}\)**} \\ \cline{3-6}  & & & \(i\)=1 & \(i\)=2 & \(i\)=3 & \(i\)=4 \\ \hline \multirow{3}{*}{Hinged-hinged} & \multirow{3}{*}{0.03} & 0 & 0.02628 & 0.08804 & 0.18147 & 0.30340 \\  & & 1 & 0.02628 & 0.08799 & 0.18124 & 0.30272 \\ \cline{2-6}  & & 0 & 0.04794 & 0.16045 & 0.33027 & 0.54985 \\ \cline{2-6}  & & 1 & 0.04791 & 0.16014 & 0.32886 & 0.54594 \\ \hline \multirow{3}{*}{Hinged-clamped} & \multirow{3}{*}{0.1} & 0 & 0.07270 & 0.19714 & 0.37902 & 0.60448 \\  & & 1 & 0.07265 & 0.19674 & 0.37736 & 0.60025 \\ \cline{2-6}  & & 0 & 0.12521 & 0.33693 & 0.64030 & 0.90027 \\ \cline{2-6}  & & 1 & 0.12497 & 0.33502 & 0.63295 & 0.89397 \\ \hline \multirow{3}{*}{Clamped-clamped} & \multirow{3}{*}{0.3} & 0 & 0.17660 & 0.39766 & 0.71535 & 0.90445 \\  & & 1 & 0.17626 & 0.39542 & 0.70764 & 0.89934 \\ \cline{1-1} \cline{2-6}  & & 0 & 0.29736 & 0.62380 & 1.00197 & 1.03688 \\ \cline{1-1} \cline{2-6}  & & 1 & 0.29611 & 0.61738 & 0.99030 & 1.02230 \\ \hline \hline \end{tabular} \({}^{*}\)\(T\) index = 0: without rotatory inertia couple; index = 1: with.  
\end{table} 
Table 2: Effect of rotatory inertia couple \(T\) on frequency parameter \(C_{i}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**End Condition**} & \multirow{2}{*}{**Cross-Section**} & \multicolumn{4}{c}{**Frequency Parameter \(C_{i}\)**} \\ \cline{3-6}  & & \(i\)=1 & \(i\)=2 & \(i\)=3 & \(i\)=4 \\ \hline \multirow{2}{*}{Hinged-hinged} & Square Circular & 0.07555 & 0.25147 & 0.51290 & 0.82607 \\  & Circular & 0.07384 & 0.24586 & 0.50174 & 0.81136 \\ \hline \multirow{2}{*}{Hinged-clamped} & Square Circular & 0.11427 & 0.30716 & 0.58290 & 0.86026 \\  & Circular & 0.11171 & 0.30045 & 0.57070 & 0.84994 \\ \hline \multirow{2}{*}{Clamped-clamped} & Square Circular & 0.16130 & 0.36368 & 0.65471 & 0.87560 \\  & Circular & 0.15771 & 0.35597 & 0.64163 & 0.86855 \\ \hline \hline \end{tabular} \({}^{*}\) See text for the input arch parameters.  
\end{table} 
Table 3: Effect of cross-sectional shape on frequency parameter \(C_{i}\) *a so-called veering phenomenon, that in the case of an asymmetric arch, the two frequency curves are merely approached, but not met as shown in this figure [25]. Figure 7 shows the natural frequency curve of the frequency parameter \(C_{i}\) according to the change in the taper ratio \(m\). The input arch parameters are square cross-section, \(\alpha=1.5\pi\) and \(\kappa=0.25\). As \(m\) increases, \(C_{i}\) increases. If \(m\) is larger, the tapered arch is more rigid at the two support ends comparing the mid-arc.