Analytical Solution of Terzaghi’s Ground Arch Model for Loads on Circular Tunnels

Abstract

To date, the important problem of Terzaghi’s ground arch model in 1946 has not been solved for loads on circular tunnels. This study comprehensively presented a general analytical solution of Terzaghi’s ground arch model for loads on circular tunnels by the limit equilibrium method, considering the complete boundary conditions of the loosening zone, the effects of water pressure, and the stiffness of the lining and ground. The proposed solution indicates the relationship between the load on circular tunnels and that on trapdoors, and there is a negative correlation between the lateral and vertical earth pressures in the ultimate state. Model tests and field measurements were used to verify the proposed solution. The results indicated that the proposed solution agrees well with the experimental results of circular tunnels in dense sand, loose sand, saturated sand, and saturated clay; the solution is significantly less than the design load from Terzaghi’s formula in good ground conditions. Moreover, the solution demonstrated an interesting rule: the distribution of total loads (including water pressure) on nonrigid circular tunnels is nearly uniform in saturated granular soil. A particular solution was obtained for circular tunnels in saturated granular soil. In addition, the vertical pressure on a circular tunnel in sand increases and the lateral pressure decreases as the groundwater table increases within the tunnel range; the case where the groundwater table is located at the crown is proved to be a critical condition for shield tunnels in sand. Previous studies indicate that the load on shield tunnels in good ground conditions would be overestimated by Terzaghi’s formula, which was proposed in 1943 and is widely adopted in tunnel design nowadays. In this study, a general analytical solution for loads on circular tunnels was established based on Terzaghi’s ground arch model, which was improved by Terzaghi in 1946 for arched tunnels but has not been solved. For various overburden depths, the proposed solution agrees well with the experimental results of circular tunnels in dense sand, loose sand, saturated sand, and saturated clay, but Terzaghi’s formula based on trapdoors fails to explain the test results of circular tunnels in sand. The proposed solution is significantly less than the design load from Terzaghi’s formula in good ground conditions. The solution is applicable for both shallow and deep circular tunnels, and it will be very helpful for the economical and rational design of shield tunnels normally in good ground conditions, such as sandy soil, crushed rock, and stiff or hard clayed soil.