Derivation of Analytical Equations for the Fundamental Period of Framed Structures Using Machine Learning and SHAP Values

The fundamental period is one of the most important parameters for the design of new structures as well as for estimating the capacity of existing ones. Thus, to estimate it, various design codes and researchers have adopted several approximate analytical equations based on a number of key structural parameters. To this end, the present study introduces a novel methodology for deriving the analytical equations for the fundamental period of reinforced concrete structures. The methodology is based on machine learning explainability techniques, specifically the so-called SHapley Additive exPlanations values. These values are commonly employed as an explainability tool. However, in the proposed novel approach they are employed as a basis to fit analytical curves, which allows the resulting equations to be constructed sequentially and in an informed manner while controlling the balance between accuracy and complexity. An extended dataset consisting of 4026 data points is employed, on which a Gradient Boosting Machine model is fitted. The model achieves excellent accuracy, with a coefficient of determination R2≈0.99, while the equations derived from the proposed formulation achieve an R2≈0.95 and Mean Absolute Error ≈0.12. This demonstrates the potential applicability of the proposed methodology in a wide array of similar engineering challenges.