Closed‐Form Solutions for the Second‐Order (P−Δ$$ P-\varDelta $$) Analysis of Regular Buildings: Inclusion of Local Shear Deformation of Walls

ABSTRACT The current Brazilian standard NBR 6118, along with similar codes, evaluates global second‐order effects due to wind actions through a stability coefficient () and an amplification factor (), both of which require detailed structural modeling, incremental iterations, and significant computational cost. Given the significance of second‐order effects—which amplify internal forces, lateral displacements, and interstory drifts—this paper addresses both conceptual inconsistencies and computational burdens by proposing exact, closed‐form solutions for the calculation of the stability coefficient and the amplification factor . The proposed solutions are derived using the continuous method approach, modeling the structure as an equivalent continuous generalized sandwich‐type beam—an extension of the classical sandwich beam obtained by its serial coupling with a shear beam. This specific serial coupling allows for an accurate reproduction of the four fundamental deformation mechanisms: global bending, global shear, local bending, and local shear. First, an analytical expression is proposed for computing the stability coefficient , which modifies the classical equation by introducing a dimensionless parameter . This parameter accounts for the influence of the four deformation mechanisms and may either reduce or amplify the value proposed by the NBR 6118 code. Subsequently, two analytical expressions are proposed for computing the amplification factor . The first is derived directly from a static analysis in which the second‐order moment is obtained through a series of incremental moment applications, loading the structure successively from its previously displaced configuration. The second is based on a dynamic analysis that relates second‐order effects to the fundamental period of the structure. The proposed analytical expressions enable the incorporation of second‐order effects through a single‐step first‐order analysis and are applicable to planar structural systems and regular symmetric buildings. Numerical applications show that the proposed expressions exhibit maximum errors of −6.26% for the stability coefficient and +1.65% for the amplification factor , confirming their reliability for safe and practical use in structural engineering analysis and design.