Generalized Shear Correction Factor for Non-Homogeneous Beam Cross-Sections with an Embedded Steel Core

In this study, an energy-consistent analytical–numerical framework is proposed to determine the effective shear correction factor ks for non-homogeneous cross-sections within the Timoshenko beam theory, such as a porous cementitious matrix (e.g., perlite-based material) combined with an embedded steel I-section. The formulation enforces equivalence between the real heterogeneous shear strain energy, governed by a spatial shear modulus field G(y,z), and its beam-theory representation based on ks(GrefA). A pixel/voxel discretization is introduced to evaluate the generalized shear-energy integral and to quantify the deviation of ks from classical homogeneous benchmarks. The results demonstrate that shear stiffness may be controlled by localized energy concentrations near weak matrix regions and phase interfaces, which can lead to non-negligible errors in deflection predictions when standard shear correction factors are adopted. The proposed framework provides a transparent and computationally efficient tool to support reliability-driven stiffness identification, model updating, and health monitoring strategies for heterogeneous and hybrid beam components. The study shows that even a small volumetric fraction of high-modulus steel may significantly increase the area-averaged reference modulus while leaving the shear response matrix-dominated due to compliance-driven energy localization. Consequently, the classical shear correction factor may substantially overestimate the effective shear stiffness in heterogeneous hybrid members. These findings have direct implications for serviceability assessment, stiffness identification, and monitoring-based durability evaluation of lightweight eco-material systems.