Simplified method for estimating the impact of the underlying tunnel excavation on existing tunnel

Tunneling underneath is bound to induce the free displacement of surrounding soil, which will further significantly affect the overlying tunnel's deformation. Based on this, a simplified method for calculating the deformation of the overlying tunnel due to shield tunnel excavation is proposed. Firstly, the Loganathan formula is selected to solve the additional stress along the pre-existing tunnel induced by tunneling underlying, then the overlying tunnel is further simplified into an infinitely Euler-Bernoulli beam resting on the Pasternak foundation. The influence of the lateral soil reaction on the existing tunnel is introduced, then adopting the energy method to describe the soil and tunnel response. The minimum potential energy principle is utilised to solving the variational energy solutions due to new tunnel excavation underlying and further obtain the explicit solutions of tunnel response. The proposed method's effectiveness is confirmed through comparisons with centrifuge tests and field case studies obtained from prior research. In comparison with the Pasternak foundation model degenerated from the proposed method, the outcomes produced using the proposed method align more closely with the measurement data. Further parameter studies show that skew angle between tunnels, depth of existing tunnel, and volume loss rate are significant factors affecting the tunnel behaviors due to tunneling underneath.