Curved beams are more efficient in transfer of loads than straight beams because the transfer is effected by bending, shear and membrane action. The analysis of curved beams involves the solution of a sixth order ordinary differential equation derived by Rao and Sundararajan [1] which includes the effects of shear deformation and rotatory inertia. The same equation was given by Archer [2] without considering the effects of shear deformation and rotatory inertia but including the damping effect. The six constants involved in the solution of this differential equation are to be determined from the boundary conditions. There are three at each end; two of them are the same as those used in the analysis of beams and the third is that used in the study of axial members. For beams with hinged and clamped ends solutions of the differential equation are available [1,2]. Extension of this method of solution to analyze beams having free edges and also for beams of variable cross section is difficult. For such cases approximate methods have to be resorted to. Raleigh-Ritz methods [3] have been used to study static and free vibration behaviour of beams with hinged and clamped ends.
The finite element method is a versatile method for solving structural mechanics problems and curved beam problems have been solved by this method by various authors. In this method of solution, two shape functions are chosen separately: one for in-plane displacement and the other for lateral displacement. Ashwell and Sabir [4] carried out a static analysis of beams choosing three different combinations of shape functions for the lateral and in-plane displacements, respectively: cubic-linear, cubic-cubic, and trigonometric (Cantin-Clough). Arches with fixed supports as well as sliding supports were analyzed. From the studies, the third choice was found to yield the best results. Strain functions were also used to solve the problem [5]. Petyt and Fleischer [6] used the same three shape functions to determine the free vibration behavior of simply supported, hinged-hinged, and clamped beams and found that the cubic-cubic shape function yielded the best results. Sabir and Ashwell [7] extended previous work [4,5] from static to vibration problems using four different shape functions; the first was that of Cantin and Clough, the second was due to Bogner, Fox and Schmit, the third function was a reduced form of the second, and the last was based on a simple strain function. They studied the convergence pattern for thick as well as thin rings and found that the shape function based on the strain function converged faster. Bickford and Storm [8] sought an exact solution for the in-plane and out-of-plane vibrations of arbitrarily shaped curved bars, including the effects of shear deformation and rotatory inertia, by a vector/transfer matrix approach, using exact solutions of the differential equations. One of the straight beam element models giving results for all thickness values is due to To [9]. He derived explicit expressions for the stiffness and mass matrices for a linearly tapered straight beam element, including the effects of shear deformation and rotatory inertia. Issa, Wang and Hsiao [10] analytically found the dynamic stiffness matrix for circular curved members, including the effects of shear deformation,