Key Takeaways & Executive Findings
- •• The improved Pasternak foundation beam model incorporates generalized shear force to account for soil continuity beyond beam ends, eliminating anomalies in existing umbrella arch analytical models. • Analytical solutions for finite-length umbrella arches under arbitrary loads were derived using the initial parameter method and validated against existing test data. • Dimensionless characteristic parameters representing relative stiffness between arch and soil were proposed; soil parameters significantly affect arch deformation and internal forces. • Tunnel excavation height and advance have contrasting influences on unsupported and buried umbrella arch sections, informing design optimization.
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.
1. Introduction
The umbrella arch method (UAM), characterized by its simplicity in craftsmanship, ease of construction, and cost-effectiveness. It has been widely applied to ensure the safety of construction in tunnels, underground pipe gallery and subway stations [1−3], especially in areas with complex terrain and adverse geological conditions at tunnel entrances [4, 5]. Calibrating design parameters in umbrella arch structures is crucial for controlling tunnel deformation and optimizing costs. Thus, understanding the mechanical behavior of the umbrella arch is essential for a solid theoretical foundation in support system design. Many studies have focused on the stress mechanisms of tunnel umbrella arch support, with a significant focus on elastic foundation beam models. The Winkler elastic foundation beam model, valued for its simplicity and computational ease, is widely used. It accounts for spatial variations in stress distribution along the entire length of the umbrella arch. ZHENG et al [6] delved into the systematic distribution patterns of foundation coefficients preceding the working face, thereby elucidating a Winkler foundation umbrella arch analytical model under the variability of foundation coefficients.
In recent years, the integration of dual-parameter foundation models into umbrella arch computational analyses has become more prominent. GOU et al [7] developed an elastic fixed-end dual-parameter foundation beam model, deriving equations for deflection, stress, and strain in the umbrella arch. This approach has led to the widespread use of dual-parameter foundation beam models for understanding stress dynamics and deformation in umbrella arches. WANG et al [8] used the Pasternak elastic foundation beam theory to analyze deflections and internal force distributions in the umbrella arch, highlighting its effectiveness in load transmission and stress management at the working face. SONG et al [9] created a structural analysis
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CHEN Lei, JIA Chao-jun, LEI Ming-feng, HE Yan-chun, SHI Cheng-hua, LI Ao (2025). An improved model of the Pasternak foundation beam umbrella arch considering the generalized shear force. Journal of Central South University. https://doi.org/10.1007/s11771-024-5777-2
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Frequently Asked Questions
What is the umbrella arch method (UAM)?
The umbrella arch method is a tunnel support technique that uses a series of steel pipes or arches installed ahead of the excavation face to improve ground stability and control deformation, especially in weak or complex geological conditions.
What does the improved Pasternak foundation model address?
The improved model incorporates the generalized shear force to account for the influence of soil beyond the beam ends, overcoming limitations of traditional elastic foundation beam models that ignore soil continuity, thereby predicting more realistic deformation and internal forces.
How was the analytical model verified?
The model's reliability was confirmed by comparing its predictions with existing test data, showing good agreement and correcting anomalies such as abnormal upward deformation and improper maximum deflection locations.
What are the key findings regarding soil characteristic parameters?
Soil characteristic parameters significantly affect umbrella arch behavior; within common soil ranges, maximum deformation and internal forces decay nearly linearly with decreasing parameters in semi-logarithmic coordinates, providing a design reference.
How do tunnel excavation dimensions influence the umbrella arch?
Excavation height has minor impact on unsupported sections but more significant on buried parts, while the influence of tunnel excavation advance is opposite.
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