Out-of-plane dynamic analysis of beams with arbitrarily varying curvature and cross-section by dynamic stiffness matrix method

The ®rst known dynamic sti€ness matrix for noncircular curved beams with variable cross-section is developed, with which an exact solution of the out-of-plane free vibration of this type of beam is derived. By using the Laplace transform technique and the developed dynamic sti€ness matrix and equivalent nodal force vector, the highly accurate dynamic responses, including the stress resultants, of the curved beams subjected to various types of loading can be easily obtained. The dynamic sti€ness matrix and equivalent nodal force vector are derived based on the general series solution of the di€erential equations for the out-of-plane motion of the curved beams with arbitrary shapes and cross sections. The validity of the present solution for free vibration is demonstrated through comparison with published data. The accuracy of the present solution for transient response is also con®rmed through comparison with the modal superposition solution for a simply-supported circular beam subjected to a moving load. With the proposed solution, both the free vibration and forced vibration of non-uniform parabolic curved beams with various ratios of rise to span are carried out. Nondimensional frequency parameters for the ®rst ®ve modes are presented in graphic form over a range of rise-to-span ratios (0.05 R h/l R0.75) with di€erent variations of the cross-section. Dynamic responses of the ®xed±®xed parabolic curved beam subjected to a rectangular pulse are also presented for di€erent rise-to-span ratios.