We investigate in terms of Poisson-Boltzmann theory the contribution of electrostatic double layers to the bending elasticity of charged symmetric bilayers, considering in detail three special problems: first, the coupling of the monolayers by the electric field traversing a curved bilayer; second, the dependence of the elastic moduli on the position of the monolayer's neutral surface; and third, the role of the mechanical stresses balancing the electric ones. According to the results the tendency of charged membranes toward spontaneous vesiculation may be stronger than hitherto expected.