The paper presents a systematic method for analyzing the out-of-plane dynamic behaviours of non-circular curved beams the governing equations of which take into account the effects of shear deformation, rotary inertia, and viscous damping. The procedure consists of formulating an analytical solution for the transformed governing equations in the Laplace transform domain and computing the responses in the time domain by means of numerical Laplace inversion. As key elements in the dynamic stiffness method, the first known transformed dynamic stiffness matrix and equivalent nodal loading vector for non-circular curved beams subjected to distributed external loading are established from the analytical solution developed using the famous Frobenius method. With a simple modification, the formulation of the solution is also suitable for free vibration analysis, which results in an exact solution. As numerical examples for transient analysis, time responses of displacement components and stress resultants are found for a two-span elliptic beam subjected to a rectangular impulse. The behaviours of the responses due to the variation of the ratio of the long axis to the short one are investigated.