Exact Closed-Form Solutions for Static Lateral Deflection of Buildings Considering Local Shear Deformation via Subsystem Decomposition of the Double-Timoshenko-Beam Systems

Continuous beam models for the lateral deflection of multi-story buildings have systematically neglected local shear deformation in walls, a simplification shown to introduce non-negligible errors in practical configurations. The Double-Beam Systems Timoshenko (MSB) model overcomes this limitation by coupling two Timoshenko beams in parallel, simultaneously capturing global bending, global shear, local bending, and local shear deformation mechanisms. However, its sixth-order coupled governing equation has remained analytically intractable. This paper derives, for the first time, exact closed-form expressions for the static lateral deflection of regular buildings using the MSB model, explicitly incorporating local wall shear deformation.

The sixth-order governing differential equation is rigorously decomposed, exactly and without approximation, into three independently solvable classical subsystems: pure bending (EBB), pure shear (SB), and coupled bending–shear (CTB) beams. The exactness of this decomposition is formally established by verifying strict identity of boundary conditions between the original coupled system and the superposed subsystems under all load cases, confirming full mathematical equivalence. Closed-form solutions are derived for uniform, triangular, and top-concentrated lateral loads. A dimensionless static correction factor η emerges analytically from the decomposition, height-independent, acting selectively on shear and interaction components only, providing a theoretically exact enhancement to classical formulations. Validation is performed against SAP2000 finite element analyses on symmetric and asymmetric benchmark buildings ranging from 10 to 30 stories, covering shear-dominated, interaction-dominated, and bending-dominated structural regimes.

The total lateral displacement is explicitly decomposed into bending, shear, and bending–shear interaction contributions. The interaction component is consistently negative and reaches magnitudes comparable to the shear term in interaction-dominated configurations, making it structurally significant and non-negligible. For symmetric buildings, errors relative to finite element results range from −7.39% to +9.02% depending on the adopted stiffness formulation. For asymmetric buildings, errors remain bounded within −8.98% for lateral displacement, −5.76% for rotational displacement, and between −7.31% and +1.51% for total displacement, confirming stable and predictable accuracy across all examined structural regimes.

This framework provides the first exact analytical solution for static deflection of the MSB beam, founded on a mathematically exact subsystem decomposition that is fully equivalent to the original coupled formulation. The analytically derived factor η constitutes a rigorous, height-independent corrector directly applicable to existing classical design expressions without increasing model complexity. The proposed formulations are well-suited for preliminary design, independent FEM verification, and future extensions to nonlinear and code-oriented analyses.