Paper 1Stiffness Extrema of Proportionally Loaded Structures: Coupling Matrix, Energy Balance, and Unified Criteria (new version)

Stiffness extrema in proportionally loaded structures — load levels at which structural stiffness reaches local maxima or minima — can be characterized through multiple independent frameworks: Mang's variational criterion (the inflection condition d²χ₁/dλ² = 0 on a special eigenvalue function), sensitivity analysis (the amplification factor β), and energy balance (the ratio ρE = |UG|/UE). While each framework has proven effective, their mutual relationships have remained unclear. This paper introduces the coupling matrix C = I + ∂KG/∂KE, which encapsulates the nonlinear interdependence between section stiffness and geometric stiffness through the equilibrium chain. We prove that the coupling singularity det (C) = 0 and the sensitivity reversal β = −1 are algebraically equivalent, and that the coupling singularity and Mang's stiffness extremum condition are driven by the same physical mechanism — the amplification of geometric effects through the dominant structural mode — with their proximity controlled by a computable alignment cosine cos θ. From the energy perspective, we derive the critical section stiffness EIcrit = (PL²/π²) ·f (λ, ν) and introduce the critical strain index CI as an element-level diagnostic. Five criteria — det (C) = 0, β = −1, dχ₁/dλ = 0, ρE = 1, and CI → 1 — are shown to form a layered hierarchy: the first two are exactly equivalent, the third is related through the alignment cosine, and the last two are energy-space and strain-space projections. We further demonstrate that three families of existing code provisions — AISC compression member slenderness limits, cable sag-to-span requirements, and arch rise-to-span ratios — are thresholds on the energy ratio ρE, providing a unified theoretical basis for rules that developed independently over decades. Illustration on two benchmark structures from Mang and Aminbaghai (2025) confirms the theoretical predictions.