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<p>Second-order (P–Δ) effects critically control the global stability of coupled shear walls (CSWs), particularly in slender, tall buildings where geometric nonlinearity substantially amplifies lateral displacements and internal forces. This paper presents, for the first time, closed-form analytical expressions for the second-order analysis of single-bay symmetric and asymmetric CSWs based on a three-field CTB continuous beam. Compared with classical continuous models, the formulation explicitly incorporates two additional deformation mechanisms—axial extensibility and local shear deformation of the wall—allowing the continuous model to represent the full spectrum of behavior from shear-dominated to bending-dominated responses and thus extending applicability across arbitrary slenderness ratios. Closed-form expressions are derived for the stability coefficient α and the amplification factor <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\gamma }_{z}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>z</mi> </msub> </math></EquationSource> </InlineEquation> using the continuum method and a subsystem-decomposition approach. These two parameters are widely used in design practice and are explicitly required by the Brazilian standard NBR 6118, which currently lacks valid formulations for CSWs. New dimensionless parameters are introduced to quantitatively demonstrate that the existing NBR 6118 provisions are overly conservative and should be updated in light of the proposed expressions. A comprehensive parametric study of a building composed of seven identical CSWs (heights 5–30 stories) shows that the proposed formulas are essentially insensitive to load distribution and agree closely with finite element simulations (maximum error &lt; 5%). These findings confirm the accuracy, simplicity, and practical applicability of the proposed closed-form solutions, providing structural engineers with reliable analytical tools for evaluating P-Δ effects in CSWs and support targeted updates to NBR 6118.</p>

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Second-Order Effects (P-Δ) in Single-Bay Coupled Shear Walls: Exact Analytical Solutions Using the Three-Field CTB Beam

  • 期刊论文

  • 发表日期：

    2026年1月23日

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  • Mao Cristian Pinto-Cruz

摘要

Second-order (P–Δ) effects critically control the global stability of coupled shear walls (CSWs), particularly in slender, tall buildings where geometric nonlinearity substantially amplifies lateral displacements and internal forces. This paper presents, for the first time, closed-form analytical expressions for the second-order analysis of single-bay symmetric and asymmetric CSWs based on a three-field CTB continuous beam. Compared with classical continuous models, the formulation explicitly incorporates two additional deformation mechanisms—axial extensibility and local shear deformation of the wall—allowing the continuous model to represent the full spectrum of behavior from shear-dominated to bending-dominated responses and thus extending applicability across arbitrary slenderness ratios. Closed-form expressions are derived for the stability coefficient α and the amplification factor \({\gamma }_{z}\) γ z using the continuum method and a subsystem-decomposition approach. These two parameters are widely used in design practice and are explicitly required by the Brazilian standard NBR 6118, which currently lacks valid formulations for CSWs. New dimensionless parameters are introduced to quantitatively demonstrate that the existing NBR 6118 provisions are overly conservative and should be updated in light of the proposed expressions. A comprehensive parametric study of a building composed of seven identical CSWs (heights 5–30 stories) shows that the proposed formulas are essentially insensitive to load distribution and agree closely with finite element simulations (maximum error < 5%). These findings confirm the accuracy, simplicity, and practical applicability of the proposed closed-form solutions, providing structural engineers with reliable analytical tools for evaluating P-Δ effects in CSWs and support targeted updates to NBR 6118.