Mechanical analysis of circular tunnels supported by steel sets embedded in primary linings

• Analytical solutions of circular composite liners by using shell theory.
• Specific support characteristics of the steel sets embedded in the shotcrete.
• Applicability of the classical “equivalent section” approach in calculating the support characteristics.

A series of analytical studies have been performed to investigate the support characteristics of the steel sets embedded in the tunnel primary lining. The analytical models for various modes of loading on an infinite cylinder carrying uniform line load, carrying uniform surface load, reinforced by a single circular ring and reinforced by equidistant rings are proposed respectively. Comparing the deformation characteristics of the infinite cylinder carrying uniform line load and uniform surface load reveals that for a practical tunnel application, the difference between the deformations induced by line load and equivalent surface load is less than 1.5%. The steel set reinforcements in the primary lining can therefore be represented by plane rings in the cylindrical shell. A study of the infinite cylinder reinforced by a single ring reveals that the most prominent effect of a steel set is attributed to its role as a cantilever fixing. It helps carry a certain length of the surrounding ground both ahead and behind before shotcreting and during shotcrete hardening. A subsequent study of the infinite cylinder reinforced by equidistant rings suggests that if the steel set spacing is smaller than a threshold value, the differences of the total stiffness calculated by the proposed method at the locations of the steel sets and other parts of the liner are negligible. The results obtained by the proposed method and the classical method have negligible differences. If the steel set spacing is larger than a threshold value, the classical method should be used with caution. Besides, the proposed method is more accurate than the classical method for calculating the support characteristics of the composite liner.