Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In several applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded. The physics of waves in open waveguides significantly diers from closed waveguides. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. Leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main diculty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This diculty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method and a perfectly matched layer technique. In this paper, two-dimensional stratified waveguides are considered. The numerical eigenvalue spectrum of the method is analyzed. Numerical dispersion curves are then compared to analytical results.