Analytical Solutions Of The Dynamic Responses Of Convey Tunnel Induced By Earthquake Waves

:Long distance water diversions projects are constructed to alleviate the unbalanced distribution of water resources in China. However, it is inevitable that the long distance water diversions projects need through the seismic regions. Therefore, the study on the seismic response of water diversions projects has a vital significance to the projects construction and operation. This paper investigates the analytical solutions about the dynamic responses of convey tunnel, a commonly used structure in long distance water diversions projects, in a series of special fields.Firstly, this paper presents an auxiliary plane method for harmonic function transformation between two skew cylindrical coordinate systems. By creating a series of auxiliary plane corresponded with spatial point, the3-D harmonic wave function is transformed from cylindrical coordinate system to polar coordinate. Then the wave function under a cylindrical coordinate system is transformed into another which is skewed with the former by using the Graf’s addition formula on the auxiliary plane method. Thus the problem is transformed from3-D to2-D. At last, the coordinate transformation formula between two skew cylindrical coordinate systems is given.Secondly, it is given that the plane and3-D analytical solutions of the dynamic responses of water-filled convey tunnel induced by plane P waves and SV waves by using the wave function expansion method in this paper. The example analysis shows the effects of the incident wave parameters, the tunnel dimension, the material of tunnel lining, and the Poisson’s ratio of half space on the seismic response of the water-filled convey tunnel. It is proved that the dynamic responses of water-filled diversion tunnel are proportional to the amplitude of incident wave. And the maximums of the dynamic displacements of tunnel lining are proportional to the tunnel radius, but the maximums of the dynamic stresses are irrelevant to the radius. With the increasing of the tunnel lining thickness, the maximums of the dynamic stresses of tunnel are decrease, but the maximums of the dynamic displacements are increase. Moreover, the elastic modulus of the lining and the Poisson’s ratio of the soil have a significant effects on the dynamic responses of water-filled convey tunnel.Thirdly, this paper establishes a model of underwater water-filled convey tunnel, and gives the plane analytical solution of the dynamic responses of the model caused by the incident P waves and SV waves. The example analysis shows the effects of the incident wave parameters, the depth of surface water, and the buried depth of underwater tunnel on the seismic response of the tunnel lining. It can be found from the results that the dynamic responses of underwater convey tunnel is increase with the increasing of the surface water depth when the P waves incidences. Moreover, with the increasing of the buried depth of underwater convey tunnel, the maximums of the dynamic stresses of tunnel lining caused by incident P waves with a low frequency are decrease, but the maximums of the dynamic displacements are increase. While, when the incident wave is SV waves, the dynamic responses of underwater convey tunnel is decrease with the increasing of tunnel buried depth.Finally, this paper gives the analytical solution of the dynamic responses of a complicated partial field in which a convey tunnel is beneath and pass through a subway tunnel caused by incident P waves and SV waves. Through the example analysis, the interaction between the convey tunnel and the subway tunnel is investigated. The results show that the lower convey tunnel acts as a barrier to the subway tunnel, and it makes the dynamic responses of the upper subway tunnel reduced around the intersection. Inversely, the upper subway tunnel makes the dynamic responses of the lower convey tunnel magnified around the intersection.