Abstract A numerical approach, the Hyperstatic Reaction Method (HRM), is presented in this paper to study the behavior of two kinds of U-shaped tunnel linings (with and without an inverted arch). The developed numerical HRM model is validated by a comparison with the results obtained from finite difference numerical models using FLAC3D. The results of this paper show that the behavior of U-shaped tunnel supports could be effectively estimated by the proposed HRM model. Using the HRM model, the effect of the shear springs and the toe constraint on the behavior of U-shaped tunnel linings without an invert are studied. In addition, the results of two kinds of U-shaped tunnels obtained by the HRM are compared with those of a circular-shaped kind. A parametric investigation is conducted considering the changes in the stiffness of the springs, the loads on the supports, the geometries, and the bending stiffness of the support structures.

中文翻译：

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使用超静力反应法的 U 形隧道衬砌设计——倒置的影响

摘要 本文提出了一种数值方法，即超静力反应法（HRM），用于研究两种 U 形隧道衬砌（带和不带倒拱）的行为。开发的数值 HRM 模型通过与使用 FLAC3D 的有限差分数值模型获得的结果进行比较来验证。本文的结果表明，所提出的 HRM 模型可以有效地估计 U 形隧道支护的行为。使用 HRM 模型，研究了剪切弹簧和趾部约束对无倒置 U 形隧道衬砌行为的影响。此外，将 HRM 获得的两种 U 形隧道的结果与圆形隧道的结果进行了比较。考虑弹簧刚度的变化进行参数研究，