# Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume
## 2 Problem Formulation  
### Geometry of Arch  
Figure 2 shows the geometry of a horseshoe circular arch with radius \(r\) and the opening angle \(\alpha\)\((\pi\leq\alpha\leq 2\pi)\). The arch axis is defined by the angular coordinates \((r,\theta)\) where the center of the circle is O. Both ends \(a\)\((\theta=0)\) and \(b\)\((\theta=\alpha)\) of the arch are supported by the hinged or clamped end constraint. The arch member is tapered, and its depth, area, and second moment of plane area at \((r,\theta)\) are denoted by \((D,A,I)\) respectively, which are defined in Figure 3. When the arch vibrates, each of the radial deformation \(w\), tangential deformation \(v\), and angular rotation \(\psi\) occurs in the cross-section at \((r,\theta)\). In this study, the free vibration assumes a harmonic motion in which each dynamic coordinate is proportional to \(\sin\omega_{i}t\). For example, the dynamic radial deformation \(w_{r,t}\) is expressed as \(w_{r,t}=w\sin\omega_{i}t\), where \(w\) is the radial amplitude (deformation), \(\omega_{i}\) is the angular frequency, \(i(=1,2,3,\cdots)\) is the mode number, and \(t\) is the time. As a result, the dynamic axial force \(N\), shear force \(Q\) and bending moment \(M\) occur in the cross-section of the arch due to deformations \((w,v)\).  
Figure 1: An example of a horseshoe circular arch as an entrance portal to a hospital at Wonkwang University in Iksan-si, Jeollabuk-do, Korea (Photo by [PERSON]).  
Shown in Figure 3 is the axially functionally tapered arch, from which the cross-sectional properties of \(A\) and \(I\) can be established. The cross-sectional shape is square or circular with the varying arch depth \(D\) at \((r,\theta)\) as shown in Figure 3a. Here, \(D\) is the height for the square and the diameter for the circular cross-sections, respectively. In Figure 3b, the tapering fashion is represented by a function of a single variable \(\theta\). At the support \(a\) (\(\theta=0\)) and \(b\) (\(\theta=\alpha\)), \(D\) is depicted by \(D=D_{a}\) and at the mid-arc \(c\) (\(\theta=2/2\)), \(D\) is depicted by \(D=D_{c}\). The taper function of \(D\) is practically used in terms of the linear, polynomial and periodic function [3; 19; 20; 21]. In this study, the tapering function is chosen as a quadratic function with respect to \(\theta\).  
To determine the quadratic function \(D\), the following taper ratio \(m\) is defined:  
\[m=\frac{D_{a}}{D_{c}} \tag{1}\]  
Using Equation (1), the quadratic function \(D\) in terms of a univariate variable \(\theta\) can be expressed as follows.  
\[D=D_{c}f,\;\;f=4(m-1)\left(\frac{\partial^{2}}{\alpha^{2}}-\frac{\theta}{ \alpha}\right)+m \tag{2}\]  
where the equation \(f\) is a convex taper when \(m>1\), a uniform prismatic when \(m=1\) and a concave taper when \(m<1\), (see Figure 3b).  
Using Equation (1), \(A\) and \(I\) at \((r,\theta)\) are expressed by the definition:  
\[A=c_{1}D^{2}=c_{1}D_{c}^{2}f^{2} \tag{3}\]  
\[I=c_{2}D^{4}=c_{2}D_{c}^{4}f^{4} \tag{4}\]  
Figure 3: (**a**) Cross-sectional shape of arch and (**b**) variation function of depth \(D\)  
Figure 2: Geometry of tapered horseshoe circular arch.  
where, \((c_{1}=1,c_{2}=1/12)\) for square and \((c_{1}=\pi/4,c_{2}=\pi/64)\) for cross-section.  
The volume \(V\) of the arch is obtained as follows.  
\[V=\int_{0}^{\alpha}Ard\theta=c_{1}rD_{c}^{2}\int_{0}^{\alpha}f^{2}d\theta=c_{1 }c_{3}\alpha rD_{c}^{2},\ \ c_{3}=\frac{1}{15}\Big{(}3m^{2}+4m+8\Big{)} \tag{5}\]  
in which it is particularly noted that \(V\) is constant in free vibration problems considered in this study.  
The above Equation (5) is rearranged against the depth \(D_{c}\) at the mid-arc, or  
\[D_{c}=\sqrt{\frac{V}{c_{1}c_{3}\alpha r}} \tag{6}\]  
Using Equation (6) together with Equations (3) and (4) yields  
\[A=c_{1}D^{2}=\frac{V}{c_{3}\alpha r}f^{2} \tag{7}\]  
\[I=c_{2}D^{4}=\frac{c_{2}V^{2}}{c_{1}^{2}c_{3}^{2}\alpha^{2}r^{2}}f^{4} \tag{8}\]