The present study deals with the nonlinear forced vibration of an axially functionally graded beam subjected to an electromagnetic actuator, moving load, and Casimir force, considering the curvature of the beam and it resting on a nonlinear elastic Winkler–Pasternak foundation. The presence of an electromagnetic actuator and Casimir force besides the presence of mechanical impact (moving load) and nonlinear elastic foundation is a characteristic of a real system, but this has not been studied in this form until now, currently representing a remaining gap. The governing differential equations of motion in the considered system are based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. The material properties are expressed according to a power law function through the thickness direction. We point out that the present study is the first to consider the curvature in combination with electromagnetic actuation, Casimir force, an elastic foundation, and moving load. Unlike in other works, axial inertia is not assumed to be negligible in our investigation. The Optimal Homotopy Asymptotic Method is employed to obtain an approximate analytical expression for the nonlinear dynamic response and the nonlinear frequency. The solutions obtained are very accurate in comparison with numerical solutions, and our procedure is simple and easy to implement for nonlinear problems. The local stability near the primary resonance and internal resonance is analyzed by means of the variable expansion method, the homotopy perturbation method, equilibrium points, the Jacobian matrix, and the Routh–Hurwitz criterion.