This work studies time-periodic solutions for two Hamiltonian PDE's on a finite interval \([0,\pi]\): the nonlinear wave equation \(u_{tt}-u_{xx}=\pm u^3\), with Dirichlet boundary conditions, and the nonlinear beam equation \(u_{tt}+u_{xxxx}=\pm u^3\) on a finite interval with Navier boundary conditions (the function and its second derivative vanishes at the endpoints). Such equations are known to be challenging because of small-denominator phenomena. For the nonlinear wave equation, existence of solutions with certain rational frequencies is proved, while for the nonlinear beam equation existence of solutions with a positive-measure set of frequencies is proved. This is done by means of computer-assisted proofs, whereby estimates required to prove the existence of solutions which are close those of finite-dimensional, using functional analysis, are verified as part of the computation. An advantage of this approach is that, along with existence, it provides precise information about the properties of the periodic solutions. This information is displayed in bifurcation diagrams given in the paper, which reveal various bifurcations along the solution curves.