A Pile-Soil-Structure Coupled Model For A Defected Periodic Viaduct Subjected To Moving Loads

Viaduct is an important construction form for high-speed railway. Hence, the research about the vibration of the high-speed railway viaduct under moving loads of the train and the corresponding control is of fundamental significance for engineering practice.This paper focuses on the problem of dynamic response of the defected periodic viaduct to moving loadings. The variation of the dynamic property of the ordered periodic viaduct due to the defect is investigated via theoretical analyses and numerical calculations in this study. The results obtained in this paper are useful for optimization design and vibration control of viaduct. The main researches conducted in this study are as follows:(1) The dynamic response of a rigidly-supported periodic viaduct(RDPV) to a moving point load. Using the Fourier transform method, the representation of the moving load in the frequency-wavenumber domain(FWD) is obtained. Thus, the original problem is reduced to determining the response of the RDPV to a unit FWD harmonic load. Using the FEM equations for the beams and pier of each span, the joint condition at the beam-beam-pier junction and the periodicity condition for the span of the viaduct subjected to the FWD harmonic load, the global FEM equation for one span of the viaduct is established, whereby the dynamic response of the RDPV is obtained. Using span FEM equation, the eigenvalue equation for the ROPV is derived. The presented numerical results indicate that axial displacement and force of the beam due to the horizontal moving loading are several orders of magnitude greater than those due to the vertical moving loading, while for the vertical displacement, rotation angle, shear force and bending moment, the opposite scenario occurs. The resonance peak numbers are few when the speed of the moving load is large. For a vertical moving load, the resonance peaks of the responses of the RDPV may appear in the first, second and third passbands of the viaduct and the dominant one usually locates in the first or second passband. However, for a horizontal moving load, all resonance peaks occur in the first passband, meaning that the energy of the response is confined in the low frequency range.(2) The dynamic response of a rigidly-supported defected periodic viaduct(RDPV) to a moving point load. Similar to the case of the ROPV, the problem of determining the dynamic response of the RDPV to a moving point load can be reduced to the problem of determining the dynamic response of the RDPV to a FWD harmonic load. The latter can be further converted into three problems for finding the dynamic responses of the left and right semi-infinite periodic viaducts as well as the defected span via the FEM. The wave fields in the left and right semi-infinite periodic viaducts can be decomposed into the free wave fields which are the wave fields in the left and right semi-infinite periodic viaducts in the absence of the defected span and the scattered wave fields which arise due to the presence of the defected span. Applying the FEM to the left and right semi-infinite periodic viaducts as well as the defected span yields the free and scattered wave fields. Numerical results show that the presence of the defected span may generate defect state frequencies. The locations of resonance peaks associated with the ordered periodic viaduct are unaffected by the types and degrees of the defects. Because of the highly decaying property of the scattered characteristic waves, the influence of the defected span is confined to the spans near the defected span, while the influence is insignificant for the spans far from the defected span.(3) The dynamic responses of a pile-supported periodic viaduct(POPV) and a pile-supported defected periodic viaduct(PDPV) to a moving point load. In order to account for the pile-soil-structure interaction, a wavenumber domain boundary element method(WDBEM) model for the pile-soil system is employed to determine the compliances of the pile foundation, by which the superstructure of the viaduct and the pile foundations are coupled. The FEM treatment of the superstructures of the POPV and PDPV are similar to that of the rigidly-supported case except that the coupling of the bottom of the pier and the pile foundation should be considered here. Comparing the numerical results with and without consideration of the pile-soil-structure coupling, one may find that the locations of the resonance peaks of the ROPV and POPV are the same, but the magnitudes of the latter are smaller than the former. The amplitudes of the time domain responses of the ROPV and POPV are comparable, but the latter attenuates faster than the former. The reason for the differences may be that wave radiation occurs in the POPV, making the damping of its characteristic waves lager than that of the characteristic waves of the ROPV. Vibration is also localized around the defected span of the PDPV. In comparison with the RDPV, for the PDPV, the number of the defected state frequencies is smaller and in some cases, obvious additional peaks are not observed.(4) Establishment of the resonance and cancellation conditions for the defected periodic viaduct subjected to multiple equidistant moving loadings. The conditions are generally applicable to the straight periodic structure. When the time interval between two neighboring moving loadings is equal to the integer or half-integer multiples of the period of certain dominant peak of the response due to a corresponding single moving loading, resonance or cancellation phenomena will occur. Only when the resonance peaks of the frequency domain response of the viaduct to the corresponding single moving loading are few, can the resonance and cancellation phenomena of the viaduct become obvious.(5) The representations of the powers done by the vertical or horizontal moving loads and the corresponding drag forces are presented. Analyzing the variations of the powers and drag forces versus the velocity shows that below 300m/s, for v=170 m/s and v=240 m/s, the power for a vertical or horizontal moving load is relatively larger. The trends of the variations of the power and averaged drag force versus velocity of a vertical or horizontal moving load are similar for the ROPV and POPV. Below 150m/s, the power and averaged drag force for a vertical moving load covering the defected span of the PDPV are slightly smaller than those of the RDPV, while the power and averaged drag force of a horizontal moving load covering the defected span of the PDPV are larger than those of the RDPV.