<p>This paper derives closed-form solutions for the static analysis of coupled shear walls with an arbitrary number of stiffening beams at arbitrary locations and with arbitrary properties, explicitly incorporating local wall shear deformation, for single- and multiple-bay configurations. Coupled shear walls derive their lateral efficiency from the interaction among four deformation mechanisms: Global bending and global shear of the system, and local bending and local shear of the walls, whose relative contribution governs both the total response and the effectiveness of the stiffening beams employed in tall buildings. Existing analytical formulations model the walls as Euler–Bernoulli beams, suppressing local shear; this assumption introduces systematic errors that grow with the cross sectional depth of both the walls and the coupling beams, while simultaneously restricting closed-form solutions to the single stiffening beam case, with no possibility of systematic extension to the general case. This work demonstrates that both limitations share a common root and are resolved simultaneously. By modeling the walls as Timoshenko beams through a physics-based derivation, the resulting governing equation is fourth-order, structurally consistent with the classical sandwich beam model, from which it recovers as a particular case upon suppression of the local shear stiffness. This equation admits an exact decomposition into three independent subsystems: A pure bending beam, a pure shear beam, and a bending–shear beam, whose linear combination exactly reproduces the total response under any static load profile, and which allows the lateral displacement and story drift to be decomposed into their three physical components: bending, shear, and interaction. The decomposition establishes two results that had not been analytically demonstrated: The stiffening beams act exclusively on the interaction component, leaving the bending and shear components strictly invariant regardless of their number, location, or stiffness; and local shear acts exclusively on the shear component, without affecting the interaction component. Since only the bending–shear subsystem carries the discontinuities introduced by the stiffening beams, the general problem reduces to a linear algebraic system of four equations per stiffening beam, from which closed-form solutions are derived for lateral displacement, interstory drift, additional axial force, shear flow, and degree of coupling under uniform, triangular, and concentrated loads. Validation against finite element models confirms errors below 1.2% in lateral displacement and interstory drift across all cases analyzed, including up to five stiffening beams with non-uniform cross sections and locations.</p>

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Exact decomposition of coupled shear walls with multiple stiffening beams: closed-form solutions including local wall shear deformation

  • Mao Cristian Pinto-Cruz

摘要

This paper derives closed-form solutions for the static analysis of coupled shear walls with an arbitrary number of stiffening beams at arbitrary locations and with arbitrary properties, explicitly incorporating local wall shear deformation, for single- and multiple-bay configurations. Coupled shear walls derive their lateral efficiency from the interaction among four deformation mechanisms: Global bending and global shear of the system, and local bending and local shear of the walls, whose relative contribution governs both the total response and the effectiveness of the stiffening beams employed in tall buildings. Existing analytical formulations model the walls as Euler–Bernoulli beams, suppressing local shear; this assumption introduces systematic errors that grow with the cross sectional depth of both the walls and the coupling beams, while simultaneously restricting closed-form solutions to the single stiffening beam case, with no possibility of systematic extension to the general case. This work demonstrates that both limitations share a common root and are resolved simultaneously. By modeling the walls as Timoshenko beams through a physics-based derivation, the resulting governing equation is fourth-order, structurally consistent with the classical sandwich beam model, from which it recovers as a particular case upon suppression of the local shear stiffness. This equation admits an exact decomposition into three independent subsystems: A pure bending beam, a pure shear beam, and a bending–shear beam, whose linear combination exactly reproduces the total response under any static load profile, and which allows the lateral displacement and story drift to be decomposed into their three physical components: bending, shear, and interaction. The decomposition establishes two results that had not been analytically demonstrated: The stiffening beams act exclusively on the interaction component, leaving the bending and shear components strictly invariant regardless of their number, location, or stiffness; and local shear acts exclusively on the shear component, without affecting the interaction component. Since only the bending–shear subsystem carries the discontinuities introduced by the stiffening beams, the general problem reduces to a linear algebraic system of four equations per stiffening beam, from which closed-form solutions are derived for lateral displacement, interstory drift, additional axial force, shear flow, and degree of coupling under uniform, triangular, and concentrated loads. Validation against finite element models confirms errors below 1.2% in lateral displacement and interstory drift across all cases analyzed, including up to five stiffening beams with non-uniform cross sections and locations.