Predicting maximum deflection of N-Edged thin-shelled hyperbolic-Paraboloid umbrella using machine learning techniques

Abstract

Many thin-shelled hyperbolic paraboloid (hypar) umbrella forms have been built in the last 60 years as roof coverings. While the stresses in these forms remain relatively low, the deflections are a critical design parameter, and one that must be considered by architects and engineers in the conceptual design phase, meaning the design stage when the scale and form is given to the structure. To-date, there is no closed-form solution that can predict the maximum deflection of umbrellas due to their highly varying and complex geometry. The best option for predicting deflection is via finite element analysis, which is time-consuming for conceptual (preliminary) design purposes. In response, this paper uses machine learning via genetic programming (GP) and gene expression programming (GEP) to develop closed-form equations that predict the maximum corner deflection of N-edged hypar umbrellas – where N = 3, 4, 5, 6, 7, and 8. For a given a boundary condition and material, geometry is the most significant parameter influencing a shell's stiffness; thus, elastic finite element (FE) models use geometric properties as input variables (N, projected area, normalized rise, and shell thickness). The maximum corner deflection is recorded as an output and the FE analyses generate a large dataset of 53,754 results. It is observed that both GP and GEP can effectively parameterize the maximum deflection of N-edged hypar umbrellas, with GEP producing more concise, but relatively less accurate, equations than GP. While the formulations are trained using concrete material, a material factor multiplier transforms the results to other material properties within the assumption of elastic limits. The results of the study can be used to assist with conceptual design of hypar umbrellas and to validate complex FE models of hypar umbrellas. This research also illustrates the use of machine learning techniques as applied to the conceptual design of structures with highly varying geometries.