An Analytical Solution for 2D Dynamic Structure-Soil-Structure Interaction for Twin Flexible Tunnels Embedded in a Homogeneous Half-Space

The interaction between subway tunnels is investigated by using a 2D analytic model of a twin tunnels system embedded in a homogenous half-space. The closed-form analytical solution for tunnel displacement response is derived through the wave function expansion method and the mirror method, and the correctness of the solution is verified through residuals convergence and comparison with the published results. The analysis focuses on the effects of tunnel relative stiffness on tunnel–soil–tunnel interaction. Tunnel relative stiffness has a great influence on tunnel displacement response. For small tunnel relative stiffness, tunnel displacement amplitude can be enlarged by 3.3 times that of single rigid tunnel model. The response of the tunnel–soil–tunnel interaction system depends not only on the distances between tunnels but also on the frequency of the incident wave and the incident angle. The strength of the interaction between the tunnels is highly related to the tunnel spacing distance. The smaller the distance between tunnels, the stronger the interaction between them. When the distance between tunnels reaches s/a = 20, the interaction between tunnels can be ignored. It is worth pointing out that the seismic design of underground tunnels should consider the interaction between tunnels when the tunnel distance is small.

Rayleigh wave equations with couple stress: modeling and dispersion characteristic

In elastostatics, the scale effect is a phenomenon in which the elastic parameters of a medium vary with specimen size when the specimen is sufficiently small. Linear elasticity cannot explain the scale effect because it assumes that the medium is a continuum and does not consider microscopic rotational interactions within the medium. In elastodynamics, wave propagation equations are usually based on linear elasticity. Thus, nonlinear elasticity must be introduced to study the scale effect on wave propagation. In this work, we introduce one of the generalized continuum theories—couple stress theory—into solid earth geophysics to build a more practical model of underground medium. The first-order velocity-stress wave equation is derived to simulate the propagation of Rayleigh waves. Body and Rayleigh waves are compared using elastic theory and couple stress theory in homogeneous half- space and layered space. The results show that couple stress causes the dispersion of surface waves and shear waves even in homogeneous half-space. The effect is enhanced by increasing the source frequency and characteristic length, despite its insufficiently clear physical meaning. Rayleigh waves are more sensitive to couple stress effect than body waves. Based on the phase-shifting method, it was determined that Rayleigh waves exhibit different dispersion characteristics in couple stress theory than in conventional elastic theory. For the fundamental mode, the dispersion curves tend to move to a lower frequency with an increase in characteristic length l. For the higher modes, the dispersion curves energy is stronger with a greater characteristic length l.

Assessment of AASHTO Load-Spreading Method for Buried Culverts and Proposed Improvement

AASHTO’s ad hoc method (AAM) for predicting free-field soil stress under a rectangular loading area is a simple and very useful tool for the analysis of buried culverts subject to vehicular wheel loads. AAM assumes the surface load spreads with soil depth into an ever-increasing rectangular area whose dimensions are controlled by a constant spread angle θ usually taken as 30°, denoted as AAM-30°. Both simplified and comprehensive culvert analysis procedures utilize AAM predictions for adjusting pressure distributions acting on the culvert periphery. Also, AAM-30° is routinely used to determine the two-wheel soil interaction depth, in which the combined effect of both axial wheels need to be considered. To date, a thorough accuracy analysis of AAM-30° has not been published in the open literature. This paper provides a unique and rigorous evaluation of AAM-30° using an exact solution from an elasticity-based model (EBM) of a homogeneous half-space with rectangular surface load. One key discovery is the depth parameter called y*, which is the soil depth at which AAM-30° peak-stress prediction exactly matches the exact EBM solution. Moreover, it is shown that y* may be determined by a simple, yet accurate formula that only depends on the square root of the load area. However, the investigation reveals that AAM-30° significantly underestimates peak stress in the shallow-depth zone 0 < y < ½ y* by as much as 31.3% of the applied surface pressure. As this is a large nonconservative error it cannot be ignored. Accordingly, a very simple modification is introduced called AAM-θ*, in which θ* is a spread angle that linearly increases to 30° at soil depth ½ y* and thereafter θ* remains constant at 30°. An accuracy evaluation of AAM-θ* reveals an order of magnitude increase in accuracy in which the small residual error is conservative, not nonconservative. The paper concludes with discussions on applying AAM-θ* to the analysis of buried culverts when using either simple or finite element model solution procedures.

Mathematical Modeling of Vibration Dampers of Vibration-Insulated Structures

Vibration dampers are installed on the machine foundations in order to reduce the vibration level. Such technological solutions are most expedient in the case of a harmonic load with a low instability of the vibration frequency. Unfortunately, dampers do not provide such a large reduction in the dynamic effect on the base, as vibration isolation, but in some cases their efficiency turns out to be quite sufficient with a relatively simple implementation and low manufacturing cost. The use of dynamic vibration dampers gives a great effect when an increased vibration of foundations occurs during the operation of equipment in metallurgical production, for example, when processing materials by pressure, reconstructing enterprises and replacing heavy equipment. During the operation of heavy forging equipment and manipulators for various purposes, the foundations of these devices can be considered as a rigid body. The model soil on which this foundation is installed can be considered a homogeneous elastic isotropic half-space. When calculating with such mathematical models, one can use solutions of the corresponding dynamic contact problems. A comparative analysis of the effectiveness of damping foundation vibrations using different foundation models, including the model of an elastic, homogeneous half-space and a system of semi-infinite rods, the modulus of elasticity of which increases with depth according to the quadratic law, shows a fairly close agreement.

Exact closed-form solutions for Lamb’s problem—II: a moving point load

SUMMARY In this paper, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb’s problem. Starting with the integral solutions of Bakker et al., we followed the method developed by Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker et al., which confirms the correctness of our formulae. The solution obtained in this paper may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.