Mathematical model for single-pile vibration displacement induced by tunnel construction vibrations based on the Pasternak model

Abstract

Shield tunneling in hard rock strata generates intense vibrations at the cutterhead, inducing accompanying vibrations in nearby foundation structures and ultimately reducing their bearing capacity. Numerous experiments and simulations were conducted to access the dynamic response of pile under shield tunneling vibrations, but theoretical explanations are not sufficiently reported. This study derived a single pile motion equation in the form of the Pasternak model based on Hamilton's principle. Using Lame's solution from the theory of pressure tunnels, the additional vibrational load on a single pile adjacent to a harmonically vibrating tunnel was derived and incorporated into the pile's motion equation as an excitation. The general solution was obtained using the separation of variables method. The motion equation was validated using FDM. Parameter analysis was conducted to examine the effects of pile-tunnel distance and source dynamics on the pile response mechanism. The results indicated that the motion equation accurately predicted pile deformation pattern and peak horizontal displacement, meeting design requirements. The parameter analysis indicated that the peak horizontal displacement and pile-tunnel distance relationship could be modeled using a Gaussian function, while the peak horizontal displacement exhibited a strong linear correlation with the source amplitude. The proposed pile vibration displacement model provides a theoretical reference for analyzing the dynamic response of adjacent foundations to tunnel construction vibrations.