Tunnels are now an integral part of the infrastructure in major cities around the world. For many reasons, these tunnels have horseshoe-shaped cross-sections with round top and flat bottom. This paper presents some improvements to the use of the Hyperstatic Reaction Method-HRM for analysing tunnels with horseshoe-shaped cross-sections when these tunnels operate under the influence of earthquakes, particularly in cases when the tunnel lining is a continuous lining. The analysis used parameters of a tunnel from the Hanoi metro system, as well as parameters of the strongest earthquake that may occur in the central Hanoi area in the improved HRM and 2D numerical methods using the ABAQUS software. On the basis of the results obtained, the paper gives conclusions about the HRM methodology when it is used to calculate tunnels that have horseshoe cross-sections operating under the influence of earthquakes.

中文翻译：

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地震作用下马蹄形断面隧道超静力学反应计算

如今，隧道已成为全球主要城市基础设施不可或缺的一部分。由于许多原因，这些隧道具有马蹄形的横截面，其圆形顶部和底部平坦。本文介绍了在地震作用下，特别是在隧道衬砌为连续衬砌的情况下，采用超静力反应法-HRM分析具有马蹄形横截面的隧道的方法。该分析使用了来自河内地铁系统的隧道参数，以及使用ABAQUS软件改进的HRM和2D数值方法中可能发生在河内中部地区的最强烈地震的参数。根据获得的结果，