This work investigates the mechanical behavior of a clamped-clamped microbeam modeled within the framework of the strain-gradient elasticity theory. The governing equation of motion gives proper account of both the effect of the nonlinear midplane stretching and of an applied axial load. An electric-voltage difference, introducing into the model a further source of nonlinearity, is considered, including also a correction term for fringing field effects. The electric force acting on the microbeam is rearranged by means of the Chebyshev method, verifying the accuracy of the proposed approximation. The results show that a uniform error on the whole domain can be achieved. The static solution is obtained by a numerical differential quadrature method. The paper looks into the variation of the maximal deflection of the microbeam with respect to several parameters. Study of the pull-in limit on the high-order material parameters introduced by the nonclassical approach and a comparison with respect to the classical beam theory are also carried out. The numerical simulation indicates that the static response is larger, affected by the use of a nonclassical theory near the pull-in instability regime. The dynamical problem is, finally, analyzed, deriving the multi degree-of-freedom problem through a Galerkin-based approach. The study on the single degree-of-freedom model enables us to note the large influence of the nonlinear terms.