An improved model of the Pasternak foundation beam umbrella arch considering the generalized shear force

Abstract

The existing analytical models for umbrella arch method (UAM) based on elastic foundation beams often overlook the influence of the surrounding soil beyond the beam edges on the shear stresses acting on the beam. Consequently, such models fail to adequately reflect the continuity characteristics of soil deformation. Leveraging the Pasternak foundation-Euler beam model, this study considers the generalized shear force on the beam to account for the influence of soil outside the beam ends on the shear stress. An analytical model for the deformation and internal forces of finite-length beams subjected to arbitrary loads is derived based on the initial parameter method under various conditions. The mechanical model of the elastic foundation beam for advanced umbrella arch under typical tunnel excavation cycles is established, yielding analytical solutions for the longitudinal response of the umbrella arch. The reliability of the analytical model is verified with the existing test data. The improved model addresses anomalies in existing models, such as abnormal upward deformation in the loosened segment and maximum deflection occurring within the soil mass. Additionally, dimensionless characteristic parameters reflecting the relative stiffness between the umbrella arch structure and the foundation soil are proposed. Results indicate that the magnitude of soil characteristic parameters significantly influences the deformation and internal forces of the umbrella arch. Within common ranges of soil values, the maximum deformation and internal forces of the umbrella arch under semi-logarithmic coordinates exhibit nearly linear decay with decreasing soil characteristic parameters. The impact of tunnel excavation height on the stress of unsupported sections of the umbrella arch is minor, but it is more significant for umbrella arch buried within the soil mass. Conversely, the influence of tunnel excavation advance on the umbrella arch is opposite.

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