# Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume
## 1 Introduction  
Arches are frequently erected in many engineering fields because of their beautiful appearance and excellent structural function. The horseshoe arch is an arch shape in which the length of the longitudinal secant line increases from the peak of the arch, reaches its maximum value on the horizontal axis, and decreases to the span length of the lower opening [1]. In particular, the horseshoe arch is widely used as a framed structure in the fields of civil, architecture, aerospace, and ship-building engineering because it can utilize a wide range of working space under the arch member [2]. The arch can be graded in cross-section in the axial direction; meanwhile, in order to meet the aesthetic and economic cost (or minimum weight) demands of the arch, the non-prismatic arch may be employed in the form of tapered, piecewise, and continuous segmented member [3]. Figure 1 shows an example of a horseshoe arch erected as an entrance portal that goes well with a modern building.  
Exciting the dynamic load acting on the structure creates an element that weakens the structure function due to the resonance phenomenon of the free vibration characteristics. Therefore, free vibration is one of the important research tasks of structural analysis, from the design stage of the structure to monitoring the soundness of the structure in public use [4].  
Over the last several decades, many researchers, such as [PERSON] [5], [PERSON] [6], and [PERSON] and [PERSON] [7], have published free vibration behavior, including arch structures. Considering the research trends of the free vibration of the arch, it is as follows. The arch's free vibration solutions can be divided into the closed and approximate. The closed form solution [8, 9] is very complicated, so there are few research results. The approximate form solution is obtained from the numerical method that calculates mode shapes by numerically integrating the governing differential equation, the eigenvalues of natural frequencies by the determinant search method, and most of research works included here [1, 2, 10, 11, 12, 13, 14, 15].  
In the design of the arched structure, it is most important to choose the suitable arch shape, so various linear shapes such as circular [1, 8, 9, 12, 13, 15], parabolic [10], elliptical [2, 11], catenary [16], and elastica [17] have been studied. In addition, research topics include the variable cross-section [8, 13, 14], material properties [9, 12, 15], multi-span arch [18], secondary effects on frequencies [19], and frequency optimization [20]. As mentioned above, the free vibration studies of arches are still being actively studied, from the linear shape to frequency optimization of the arch.  
In the open literature, only two works [1, 2] that directly dealt with the problem of free vibration of a horseshoe arch are available, where the circular and elliptic arches with uniform cross-sections, but not tapered, are covered. High-rise arches, especially horseshoe arches, are vulnerable to stability, so tapered members should be used properly to improve the stiffness of the arch. In this regard, this paper focuses on the free vibration of a tapered horseshoe arch.  
In this study, differential equations governing in-plane free vibrations of tapered horseshoe circular arches are derived and numerically solved to compute natural frequencies with their mode shapes. In particular, the effects of a cross-sectional shape, taper ratio, rotatory inertia couple, and volume ratio are included, in combination or separation, in the governing differential equations, implying the novelty of this study. Parametric studies affecting natural frequencies and mode shapes have been conducted and discussed extensively.