Second-order effects (P– \(\Delta \) and P– \(\delta \) ) play a central role in the stability assessment of steel frames and require nonlinear analysis procedures that remain reliable near critical states. This paper presents a certified second-order framework in which mathematically guaranteed two-sided bounds for the smallest eigenvalue of the global tangent stiffness are embedded in a corotational stability-function formulation. The upper bound is obtained from a Rayleigh–Ritz approximation on low-dimensional trial spaces, whereas the lower bound is constructed using Temple/Kato-type estimates together with residual-based certification and enforced spectral separation. The resulting certified interval \([\mu -\delta ,\mu ]\) provides a rigorous measure of the margin to loss of positive definiteness and remains stable with respect to solver tolerances and moderate roundoff effects. These certificates are combined with a goal-oriented, drift-aware load-stepping strategy and a standard backtracking line search within the Newton procedure. Numerical examples involving portal and multi-story moment frames demonstrate reliable path tracing and tight eigenvalue certificates with modest subspace dimensions. The associated computational overhead is quantified in the numerical study and remains small relative to the nonlinear equilibrium solution cost. The formulation retains the exact axial-force coupling inherent in stability functions, is consistent with Direct Analysis Method terminology, and offers a practical basis for certified stability assessment and adaptive step-size control in steel frame analysis.
