Background <p>This paper investigates the stability and bifurcation behavior of a modified Euler–Bernoulli beam with periodically varying mass, where inertia modulation acts as a parametric excitation mechanism that can alter both instability thresholds and post-critical responses. The study is motivated by the limited availability of explicit analytical transition relations for beam-derived reduced-order models with nonlinear restoring effects.</p> Methods <p>To address this gap, the governing beam equation is reduced to an El Borhamy–Rashad–Sobhy-type oscillator with Duffing nonlinearity, providing a physically motivated reduced model that retains the essential effects of inertia modulation and cubic stiffness. An inductive Lindstedt-Poincaré method is then employed to derive closed-form expressions for the transition curves and equilibrium states associated with the first and second instability tongues.</p> Results <p>The analytical results show that crossing these boundaries causes the trivial equilibrium to lose stability, changing from a center-type state to a saddle-type state, and leads to the emergence of symmetric nontrivial equilibria through supercritical or subcritical pitchfork bifurcations, depending on whether the nonlinearity is hardening or softening. Numerical integration and Poincaré maps confirm the analytical predictions and further show that the softening case exhibits a wider instability region and richer nonlinear dynamics than the hardening case, especially near the second instability tongue, where additional subharmonic motions appear. </p> Conclusions <p>These findings provide an explicit analytical and physically grounded framework for understanding parametric instability and post-buckling-type transitions in beam-like systems with inertia modulation.</p>