Critical load deterministic interval estimation for variable-stiffness rods using the initial parameter method

This paper presents a modified initial parameter method for deterministic interval estimation of critical loads in axially compressed variable-stiffness rods. In the present study, the term "interval" refers to the range bounded by lower and upper estimates obtained from inscribed and circumscribed piecewise-constant geometric approximations of continuously varying cross-sections. The method approximates continuously varying cross-sections by inscribed and circumscribed piecewise-constant segments, providing reliable upper and lower bounds without requiring closed-form solutions. The formulation is applicable to both conservative and follower-load problems under various boundary conditions. Numerical examples for conical and smoothly varying rods demonstrate monotonic convergence of the estimates, with relative deviations below 1-3% for refined discretization. The proposed framework provides a computational tool for stability analysis of structural elements with variable stiffness and establishes rigorous deterministic lower and upper bounds that may serve as a baseline for subsequent uncertainty or reliability analyses. The present study is restricted to linear elastic rod theory; extension to elastoplastic behaviour is considered as a possible direction for future research.