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Closed-form solutions for the stability coefficient and amplification factor in P–Delta (P–Δ) analysis: Double-Beam systems Timoshenko

Pinto-Cruz M.C. · Mechanics Based Design of Structures and Machines (2026) · DOI: 10.1080/15397734.2025.2594203

Resumen

The Brazilian code NBR 6118 evaluates global second-order effects produced by the combined action of wind and gravity loads through two control parameters: the stability coefficient ((Formula presented.)) and the amplification factor ((Formula presented.)). The determination of these parameters, in their conventional formulation, requires detailed structural models, incremental and iterative analyses, and considerable computational effort. Since second-order effects amplify both internal forces and lateral displacements, it is pertinent to develop analytical expressions that allow their accurate assessment without resorting to computationally intensive procedures. This study presents exact closed-form solutions for calculating the stability coefficient (Formula presented.) and the amplification factor (Formula presented.) The expressions are derived using the continuous method, which enables the incorporation of (Formula presented.) effects through a single first-order analysis. Unlike the classical model based on the sandwich beam, the proposed formulation stems from the parallel coupling of two Timoshenko beams, a configuration capable of coherently reproducing the four fundamental deformation mechanisms: global bending, global shear, local bending, and local shear. The developed expressions are applicable to planar structural systems and torsion-free buildings. Numerical results show that the maximum deviations from reference solutions are +2.39% for the stability coefficient (Formula presented.) and −2.84% for the amplification factor (Formula presented.) These results confirm the accuracy and consistency of the proposed solutions, which can be reliably used in structural engineering practice as well as in preliminary stages of building analysis and design.

Instituciones: Universidad Nacional de Ingeniería