2.5D formulation and analysis of a half-space subjected to internal loads moving at sub- and super-critical speeds

https://doi.org/10.1016/j.soildyn.2020.106550Get rights and content

Highlights

  • 2.5D response of half-space to an internal point load moving is studied analytically and numerically.

  • Both sub- and super-critical velocities of the load relative to Rayleigh's wave velocity are considered.

  • Dissipation of the moving load is considered by adjusting the multiplying factor.

  • 2.5D solutions for a point load moving extremely fast are same as those for 2D solutions for a line load.

  • Attenuation of half-space above the load level is reduced by dissipation of the load.

Abstract

The 2.5D dynamic response of a half-space to an internal point load moving at sub- and super-critical speeds is studied in Cartesian coordinates both analytically and numerically. Firstly, the partial differential equations of waves are converted to the ordinary differential equations by the Fourier transformation. Then, a multiplying factor is derived and added to the stiffness matrix to account for the dissipation of the load moving at different velocities, with or without self-frequency. Finally, the displacements of the half-space induced by the waves propagating upward and downward are obtained analytically for the specified boundary conditions. For comparison, the half-space is also analyzed by the 2.5D finite/infinite element method. The findings of the paper include: (1) the 2.5D solutions for a point load moving extremely fast are same as those for the line load by the 2D approach, (2) the dissipation of the moving load can be considered by adjusting the multiplying factor, (3) the attenuation of the half-space above the load level is reduced by the dissipation of the moving load, and (4) the displacement of the half-space increases with the self-frequency of the moving load.

Introduction

The problem of a half-space subjected to sources in different forms is firstly solved by considering the wave propagation in an isotropic and elastic medium. Sommerfeld [1] dealt with the theory of motion in elastic media and derived the equations for wave motions. According to Ewing et al. [2], the wave equations induced by sources vary in the form of radiation, which are mainly divided into plane and spherical types and solved by different integral functions in Cartesian and polar coordinates, respectively. Rayleigh [3] provided the theoretical solution for the waves propagating on the free surface of a semi-infinite elastic solid. Knott [4] derived the general equations for reflections and refractions of elastic waves at plane boundaries. Lamb [5] studied the disturbance generated in a semi-infinite medium subjected to a line load or point load on the surface. He also provided the solutions for internal sources in the form of integrals. For the surface source, the force is applied normal to the free surface, from which the waves propagate vertically downward. For the internal source, the waves propagate cylindrically outward from the internal line source. Lapwood and Jeffreys [6] discussed the problem of an internal line source of compressional waves in detail. Layered soil is a specific problem in soil dynamics, since the formulation is greatly simplified by the assumption of isotropic layers, for which the stiffness matrices for analyzing wave propagation was presented by Kausel and Roësset [7].

In addition to the above analytical works, numerical approaches such as the finite element method has been adopted to solve the problem of wave propagation in the half-space. To deal with the loss of energy for waves traveling to the infinite domain, Ungless [8] and Bettess [9] presented an effective concept of using the infinite element to simulate the far field. The infinite element concept has been extended to other fields by researchers, including the hydrodynamic problems [10,11], contaminant transport problems [12], static and elastodynamic problems [[13], [14], [15], [16], [17], [18]], and others. Yang et al. [19] used the dynamic condensation method to automatically generate the impedance matrices for waves of varying frequencies, starting from the highest frequency considered. Using the same approach, Yang et al. [20] analyzed a 2D semi-infinite solid containing a buried circular cavity subjected to harmonic P and SV waves, and obtained solutions that are in agreement with those of Luco and De Barros [21]. Recently, Yang and Li [22] solved the problem of the liquid-solid layers subjected to an internal line load both analytically and numerically, and compared their results with Ewing [2].

The research of vibration in the half-space gradually extends from 2D to 3D problems for the moving loads, mainly due to the global construction of rapid transit systems and high speed railways. Eason [23] studied the 3D steady-state problem for a uniform half-space subjected to point loads or loads distributed over a circular or rectangular area. Alabi [24] performed a parametric study on the 3D ground-borne vibrations induced by rail traffic, considering the effects of train speed, train distance, and ground depth. Krylov and Ferguson [25] used the Green's function to study the generation of elastic waves by sleepers of the track subjected to the wheel axles' loads. This approach was extended to cover the condition for train speeds greater than the Rayleigh waves velocity, for which very large vibration may occur [26].

For simulation of wave transmission in a half-space, Yang and Hung [27,28] first proposed the 2.5D finite/infinite element approaches that adopts two in-plane degrees of freedom (DOFs) and one out-of-plane DOF for each element. The approach was also adopted to study the problem of layered soils under surface moving loads [29]. More researches on railway-induced vibration were presented by Yang and co-workers concerning the railway irregularity [30], seismic excitation of underground tunnels [31], and reduction of vibrations in railway-side buildings by trenches [32].

For 2D and 3D problems subjected to surface or internal sources, Kausel [33] proposed the direct stiffness matrix method (SMM). Ba et al. [34] combined the SMM with the inverse Fourier transform to obtain the 3D dynamic response of a layered half-space to moving point loads. They also derived the 2.5D Green's functions for moving distributed loads acting along an inclined line in a layered half-space [35], with the effect of pore pressure considered [36]. Noori et al. [37] extended the SMM to obtain the Green's function for 2.5D elastodynamic problems with layered soils. Kausel [38] worked further on problems of 2.5D or 3D sources that are neither plane nor axisymmetric. The methods presented above work well, but are also limited to layered soils. For problems with variations in materials or geometry, due to existence of tunnels or adjacent buildings, which cannot be treated as uniform or layered soils, recourse should be had to methods such as the finite/infinite element method to be presented herein.

The idea of the 2.5D approach continues to grow recently. Ghangale et al. [39] used the 2.5D finite and boundary element method to evaluate the vibration energy flow radiated by underground railway tunnels. Ba et al. [40] combined the 2.5D approach with indirect boundary element method to obtain the 3D seismic response of a multi-layered half-space under obliquely incident P-, SV- and SH-waves. Through the various applications, the idea of the 2.5D approach has been proved to be effective.

In this paper, the 2.5D approach will be adopted to deal with a half-space subjected to an internal point load moving at different velocities, both analytically and numerically. Firstly, from the equations of motion for the solid in Cartesian coordinates, the partial differential equations of waves are derived in Section 2. Then, by using the field functions and triple Fourier transformation, the partial differential equations are converted to the ordinary differential equations. In Section 3, the different forms of radiation of the internal and surface sources will be discussed. To simulate the dissipation of the internal moving point load, a multiplying factor is proposed and added to the stiffness matrix. With this, a procedure is presented in Section 4 to obtain the vibration of the upper and lower parts of the half-space. The treatment of internal source in the Cartesian coordinates in Sections 3 Internal source vs. surface source for the half-space, 4 Half-space under an internal moving load is believed to be new in this paper. In Section 5, the 2.5D finite/infinite elements approach is briefed. For the internal load depth reduced to d = 0 m, the solution obtained will be compared with that for a surface load. In addition, a vast range of examples will be presented for verification of the analytical and numerical solutions, concerning the load moving speeds and self-frequencies, and for exploiting the symmetry of displacements along the vertical axis. The last section is the conclusions.

Section snippets

Governing equations

As shown in Fig. 1, the half-space is uniform along the z-axis. To simulate the soil vibrations induced by an underground moving train, an internal point load P is considered moving along the z-axis with speed c at depth d. The black arrows indicate the directions of waves propagating from the loading point in the medium, and the gray ones the reflected waves. For the solid considered, the equations of motion is(λ+μ)u+μ2u+ρf=ρu¨where λ and μ are Lamé’s constants, u and f the displacement

Internal source vs. surface source for the half-space

In general, the waves generated by a point surface source are of the plane type in Cartesian coordinates, and they propagate and attenuate in the downward direction, as was treated by Ewing et al. [2]. In contrast, for an internal source, the waves generated are of the spherical or cylindrical type in polar coordinates [2], which propagate and attenuate radially in all directions. To deal with a half-space subjected to an internal source in Cartesian coordinates, the medium is divided into two

Half-space under an internal moving load

Using the above multiplying factor Mu, the response of a half-space to an internal moving load in Fig. 1 is solved analytically in this section. Consider a vertical harmonic point load Pd=(0,Pd,0) with self-frequency ω0 acting at depth d and moving along the z-axis, i.e.,Pd(x,y,z,t)=δ(x)δ(yd)δ(zct)exp(iω0t)Twhere δ is Dirac's delta function and T the magnitude of the load. By applying the triple Fourier transformation in Eq. (6a) to Eq. (26), one can obtain Pˆd asPˆd(y=d)=T(2π)2cδ˜(kz)δ(ωω0c

Numerical analysis by 2.5D finite/infinite elements approach

As shown in Fig. 4, the 2.5D finite and infinite elements are used for the near and far fields of the half-space, i.e., by the 8-node quadrilateral (Q8) finite element and degenerated (Q5) infinite element, respectively. For accuracy of solution, the element size and mesh range should be chosen to meet the guidelines in Ref. [27]. In this paper, a half-space with a free surface and subjected to a moving load P at depth d is considered. Due to symmetry of the problem, only the left half of the

Verification of present approach

In this section, an internal point load applied at the depth d=0m is considered, for which the results obtained will be compared with those of the surface load solved analytically by Ref. [27], using the same properties as the latter. For the moving load with velocity c=90m/s, the analytical and numerical results solved by the present approach were plotted as curves with rectangles and triangles in Fig. 7(a). As can be seen, good agreement has been achieved with the analytical ones (curves in

Concluding remarks

In this paper, the 2.5D approach is adopted to obtain the dynamic response of a half-space to an internal point load moving at sub- and super-critical speeds analytically and numerically. Based on the assumptions and data adopted in the modeling and the results obtained, several conclusions are drawn for the present study:

  • (1)

    For a point load moving at an extremely high velocity, the displacements obtained by the 2.5D approach (both analytical or numerical) appear to be the same as those for the

CRediT authorship contribution statement

Y.B. Yang: Conceptualization, Methodology, Validation, Writing - review & editing, Supervision. P.L. Li: Software, Formal analysis, Investigation, Writing - original draft. W. Chen: Visualization. J. Li: Data curation. Y.T. Wu: Resources, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Endowment of the Fengtay Chair Professorship to the senior author is greatly appreciated. Agencies that have sponsored this research include: National Natural Science Foundation of China (Grant No. 51678091), Special Program for Scientific and Technological Talents (Grant No. cstc2018jcyj-yszxX0013), Chongqing Municipal Natural Science Foundation (Grant No. cstc2017zdcy-yszxX0006), Chongqing Municipal Natural Science Foundation (Grant No. cstc2018jcyj-yszx0012), and Science, Technology Research

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