A cantilever beam under axial flow, confined or not, is known to develop self-sustained oscillations at sufficiently large flow velocities. In recent decades, the analysis of this archetypal system has been mostly pursued under linearized conditions, to calculate the critical boundaries separating stable from unstable behavior. However, nonlinear analysis of the self-sustained oscillations ensuing flutter instabilities are considerably rarer. Here we present a simplified one-dimensional nonlinear model describing a cantilever beam subjected to confined axial flow, for generic axial profiles of the fluid channels. In particular, we explore how the shape of the confinement walls affects the dynamics of the system. To simplify the problem, we consider symmetric channels with plane walls in either converging or diverging configurations. The beam is modeled in a modal framework, while bulk-flow equations, including singular head-loss terms, are used to model the flow-structure coupling forces. The dynamics of the system are first analyzed through linear stability analysis to assess the stabilizing/destabilizing effects of the channel walls configuration. Subsequently, we develop a systematic nonlinear analysis based on the continuation of periodic solutions. The harmonic balance method is used in conjunction with the asymptotic numerical method to calculate branches of periodic solutions. The continuation-based methods are used to investigate bifurcations with respect to both the reduced flow velocity and the channel slope parameter. From the results presented, we illustrate how continuationbased approaches and bifurcation analysis provide an efficient tool to analyze the nonlinear behavior of flow-induced vibration problems, particularly when reduced/simplified models are available.