This article describes a mathematical model and two solution methodologies for efficiently predicting the equilibrium paths of an arbitrarily shaped, precurved, clamped beam. Such structures are common among multistable microelectromechanical systems (MEMS). First, a novel polynomial-based solution approach enables simultaneous solution of all equilibrium configurations associated with an arbitrary mechanical loading pattern. Second, the normal flow algorithm is used to negotiate the particularly complex nonlinear equilibrium paths associated with electrostatic loadingand is shown to perform exceptionally well. Overall, the techniquespresented herein provide designers with general and efficient computational frameworks for studying the effects of loading, shape, and imperfections onbeam behavior. Sample problems motivated from switch and actuator applications in the literature demonstrate the methodologies' utility in predicting the nonlinear equilibrium paths for structures of practical importance.