This article aims at developing a semi-analytic approach for studying the free and forced vibrations of a pseudoelastically behaving shape memory alloy beam. Based on the Euler–Bernoulli beam theory, equations of motion were derived through Hamilton principle, and the obtained partial differential equations were decomposed by applying the Galerkin approach and were solved using Newmark integration method. A three-dimensional phenomenological model of shape memory alloy, which is capable of identifying the main properties of the shape memory alloy, was employed to model the behavior of the shape memory alloy beam. A closed-form numerical algorithm was introduced to simulate the governing kinetic equations of the shape memory alloy beam coupled with transformation strain. The presented novel solution approach is simple, flexible, and time-saving. Stability analysis was performed using phase state trajectories to show dynamic characteristics of the shape memory alloy beam. Due to hysteric behavior of the shape memory alloy, energy dissipation was clearly observed in early stages of the free vibration and within the transient regions of the forced vibration. The numerical results showed that, due to the hysteric induced damping effect, the vibration amplitude is smaller in comparison to an equivalent elastic beam, and consequently, the shape memory alloy beam exhibits more stable behavior at the resonant frequencies. This property can potentially find applications in energy damping applications and vibration control. Moreover, an interesting phenomenon called jumping was observed in the results of frequency response analysis. At jumping frequency, the amplitude of the frequency response has two distinct levels. This jumping frequency is as a result of the hysteresis behavior of the shape memory alloy, and it is a function of the exciting amplitude. © 2018, © The Author(s) 2018.