Wave Function Expansion Method Research Articles

Study on deformation patterns of tunnel isolation layers and seismic response of a shield tunnel

Tunnel seismic isolation is an effective measure to reduce the seismic response of tunnels; however, the unclear mechanism limits the popularization and application of this measure. In this study, the deformation patterns of tunnel seismic isolation were investigated using theoretical analyses and shaking-table experiments. The analytical solution for the dynamic response of the seismic isolation tunnel under SH-wave incidence was derived using the wave-function expansion method. A novel loading device based on the reaction-displacement method was used to study the mechanism of the tunnel seismic isolation layer. It was found that the seismic isolation layer helped reduce and spread the deformation from the outer surface to the inner surface, effectively minimizing the deformation affecting the lining. The DSCF response of the lining is subsequently reduced. Moreover, the installation of seismic isolation did not result in a significant alteration in the acceleration response of the tunnel. These results provide a reference for the seismic isolation design of tunnels.

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Analytical solution to 2D dynamic soil-tunnel-building interaction under seismic SH-wave excitation considering the influence of the stiffness of tunnel and building's foundation

This investigation examines the structure-soil-structure interaction (SSSI) problem between a tunnel and nearby aboveground building, considering the impact of the stiffness of tunnel and building's foundation. A two-dimensional (2D) model is utilized, which includes a flexible circular tunnel and a three-story building anchored by a flexible semi-circular foundation embedded in a homogeneous half-space. The response of the system to incident plane SH-wave excitation is solved analytically using the wave function expansion method. The analysis focuses on how the system response is affected by the stiffness of the tunnel and the building's foundation. Tunnel stiffness plays a leading role in tunnel response analysis. The model with a rigid tunnel and rigid foundation can offer a reasonable system response overall, but it fails to accurately represent tunnel deformation and stress distribution. Depending on the stiffness of the building's foundation, the maximum value of the inter-story displacement of the first floor can be overestimated by around 70 %, with even greater overestimations for higher floors. The results help understand observed seismic behaviors and provide insight into the effects of foundation and tunnel stiffness on the tunnel-building interaction system.

Efficient Analysis of Unlined Elliptic Tunnels: An Approximate Method for Dynamic Response to Incident Plane SH Waves

ABSTRACT Dynamic analysis of underground structures, such as tunnels, is crucial to ensure their safety and stability during seismic loading. This paper proposes a more general method of large elliptic arc assumption-based wave function expansion, which is attributed to the important role of the large circular arc assumption-based wave function expansion method in the elastic wave scattering problem. The study is conducted within elliptic coordinate systems, employing the Mathieu function and Mathieu function addition theorem. The applicability and limitations of this method are analysed by comparing it with the existing exact analytical solutions obtained through the mirror-based wave function expansion method.

Kinematic response of a pile within a soil slope to SH wave excitation

It is widely recognized that the topography of a slope can significantly enhance the magnitude of seismic waves. Nevertheless, the mechanism behind the topographic amplification on the kinematic response of a pile within a soil slope remains unclear. Consequently, a newly rigorous method is employed to investigate the kinematic response of a pile within a soil slope. The proposed method which combines the Wave Function Expansion (WFE) method and the Beam on Dynamic Winkler Foundation (BDWF) model advances the rigorous theoretical study for the kinematic response of a pile on slope from the pseudo-static analysis to the frequency-domain analysis. The proposed solution well captures the kinematic response of a piles within a slope. The parametric analysis focuses on investigating the influence of slope topographic amplification effects on pile vibration under different incident frequencies, slope gradients, and pile-to-soil modulus ratios. The results indicate that steep slopes and high pile-to-soil modulus ratios lead to higher topographic amplification effects, with specific frequencies significantly amplifying the kinematic response of pile within a slope. Nonetheless, as the pile located beyond twice the length of the slope, the amplification effect notably diminishes.

Influence of Incident Orientation on the Dynamic Response of Deep U-Shaped Cavern Subjected to Transient Loading

In deep rock engineering, caverns are often disturbed by engineering loads from different directions. To investigate the dynamic response of deep U-shaped caverns under different incident orientations, a theoretical solution of the dynamic stress concentration factor along the cavern boundary was derived based on the wave function expansion and conformal mapping method, and the failure characteristics around the cavern were further investigated by PFC2D (Particle Flow Code in two dimensions). As the incident orientation increases from 0° to 90°, the dynamic compressive stress concentration area transforms from both the roof and the floor to the sidewalls, and the peak dynamic stress concentration factor of the roof decreases from 2.98 to −0.20. The failure of the floor converts from dynamic compression shear failure to dynamic tensile failure. Compared to a stress wave incident from the curved boundary, a stress wave incident from the flat boundary causes severer damage. When the stress wave is incident from the sidewall, the cavern with a larger height-to-width (h/w) ratio exhibits severer damage. Conversely, the cavern with a smaller h/w ratio tends to fail as the stress wave is incident from the floor. This paper provides a basic understanding of dynamic responses of the deep U-shaped cavern.

Linear analytical model for the seismic response of a straight‐jointed shield tunnel

AbstractIn this study, the segment joint and ring joint of the lining of the straight‐jointed tunnel are simplified as an equivalent open cylindrical shell with a small central angle and an equivalent closed cylindrical shell with a small width, and the lining of the straight‐jointed tunnel is thus simplified as an equivalent continuous periodic shell (ECPS) described by the cylindrical shell theory. Since the ECPS lining is periodic along the longitudinal direction, the tunnel‐soil system can thus be treated as a periodic system, which is referred to as the periodic tunnel‐soil system (PTSS) in this study. Based on the proposed ECPS lining model, an analytical method for the tunnel with the ECPS lining (ECPS tunnel) under seismic waves is established in this study. By employing the periodicity condition for the PTSS as well as the wave function expansion method, the representation for the wave field in the soil is established. With the aforementioned cylindrical shell theory and Fourier series expansion method, the convolution type constitutive relation for the ECPS lining and the Fourier space equations of motion are derived. By using the soil‐lining continuity condition and aforementioned formulations for the ECPS lining and soil, the coupled Fourier space equations of motion for the PTSS are established, with which the response of the ECPS lining and scattered waves in the soil can be determined.

Scattering of SH waves by elliptical cavity and type-III crack in deep anisotropic geology

Most of the underground rock formations are anisotropic, and the study of the dynamic response of anisotropic geology containing internal defects is necessary for the development of underground tunnels or caverns. In this paper, the wave function expansion method, the complex function method, and the Green's function method are used to solve the problem of scattering of SH waves in anisotropic rock in the presence of elliptical cavity and type-III crack. Boundary conditions for solving the problem are given using the conformal transformation method, and crack in the medium is constructed using the "crack cutting" technique. The scattered wave field generated by the crack is constructed based on the Green function method, and the displacement and stress fields in the model are obtained by the wave field superposition principle in the case of SH wave incidence. The dynamic stress concentration factor (DSCF) on the boundary of the elliptical cavity and the dynamic stress intensity factor (DSIF) at the crack tip are derived. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, the effects of the anisotropic strength of the rock mass, the dimensions of the cavern, and the incidence angle and wave number of the SH wave on the DSCF and DSIF are analyzed by numerical calculations in both the frequency and time domains. The results show that the anisotropy of the rock mass, the shape of the cavern and the defects of the surrounding medium have a great influence on the dynamic response of the underground structures, which should be given enough attention in the development and design of deep underground caverns or tunnels.

Dynamic inverse plane characteristics of circular holes in a strip-shaped medium under line source excitation

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of circular holes in the strip domain under line source excitation were studied. The total wave field in the strip domain is constructed using the concept of infinite mirroring, which eliminates the mutual impact of the wave field under the action of the upper and lower limits of the strip domain. To determine the overall wave field, solve the wave field's unknown coefficients in accordance with the boundary conditions. The right number of mirror images is found using the dynamic stress distribution of a circular hole in the band-shaped domain as a specific calculation example. The effect of altering the incident frequency and band-shaped domain width on the dynamic stress concentration coefficient surrounding the circular hole is also explored. As a consequence of the findings, the dynamic stress concentration coefficient around the circular hole in the zone domain at low frequencies is substantially bigger than it is at medium and high frequencies. Under identical conditions, the dynamic stress concentration coefficient will exhibit periodic extreme values and converge as the distance increases between the lower boundary and the circular hole’s center.

Dynamic response of semi-cylindrical depression, cylindrical cavity and type-III crack to SH wave in half-space anisotropic media

In this study, the anti-plane dynamic response of an elastic half-space anisotropic medium containing surface semi-cylindrical depressions and internal cylindrical cavity and type-III crack is solved analytically. The wave function expansion method, the complex function method and the Green's function method can be used to effectively construct the free wave field equation and the scattered wave field equation in the case of SH wave incidence, in which the scattered wave field generated by the crack is constructed by the Green's function method and the "crack cut" method. For the scattered wave field generated by a circular cavity is constructed using the mirror method, where the positional offset coefficient in the mirror method of a half-space anisotropic medium is introduced, which well solves the problem of the asymmetry of the scattered wave field. Finally, the effects of anisotropic strength of the medium, SH wave incidence angle and dimensionless incident wave number on the surface displacement amplitude (|W|), dynamic stress concentration factor (DSCF) at the boundary of the circular cavity and dynamic stress intensity factor (DSIF) at the crack tip are investigated in both frequency and time domains under different working conditions, and conclusions different from those of isotropic medium are obtained, which provide some theoretical references for the earthquake-resistant design of the underground structure with canyon topography and the surface building.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

Preliminary design and evaluation method for the seismic isolation layer of a shield tunnel

Theoretical analyses and numerical simulations, proposed to increase the efficiency of tunnel seismic isolation measures, are too complex for solving practical problems. Herein, a simplified design method for a tunnel isolation layer is developed for preliminary calculations of the required physical parameters based on the surrounding rock conditions and seismic isolation performance. The internal force response of a shield tunnel in homogeneous ground before and after the installation of a seismic isolation layer is determined using the proposed method, considering the coupled effect of the isolation layer and surrounding rock by calculating the equivalent elastic modulus. The effectiveness of the derived method is verified using wave function expansion and finite element numerical simulation methods. The results obtained using the simplified design formulae are consistent with those obtained using the wave function expansion method. Furthermore, the effects of the seismic wave frequency, surrounding rock type, elastic modulus, and thickness of the seismic isolation layer on the seismic isolation effect are examined via parametric analysis. The proposed method could be applied to preliminary design and evaluation of seismic isolation layers for shield tunnels with different surrounding rock types.

Vibration isolation of infinitely periodic pile barriers for anti-plane shear waves: An exact series solution

The application of periodic pile barriers for isolating ambient vibrations has emerged as a significant research topic, which is essentially the problem of wave scattering by periodically distributed structures. This study presents an analytical series solution for anti-plane shear (SH) wave scattering by infinitely periodic pile barriers using the wave function expansion method, incorporating the wave field characteristics of infinitely periodic distributed scatterers subjected to plane waves. From the perspective of periodic theory, the wave fields around each scatterer are completely identical, differing only by a phase shift (frequency domain) or a time difference (time domain). As a result, as long as the boundary condition of one scatterer is satisfied, boundary conditions of the remaining scatterers can be satisfied simultaneously. This novel approach allows the selection of an arbitrary single pile as a reference in the infinitely periodic model to be solved, yielding accurate results without truncation errors and significantly reducing computational and storage requirements. A FORTRAN code is developed to achieve numerical evaluation of the theoretical formulation, followed by verification of the solution's accuracy using two degenerate examples. Numerical examples are conducted, focusing on the influence of pile stiffness, pile spacing, and pile type on the vibration isolation. The results demonstrate that decreasing the spacing between piles can effectively increase the bandgap width and reduce the displacement response. The screening effectiveness of solid piles is better than that of hollow pipe piles and filled pipe piles. In vibration mitigation design for engineering projects, the larger the shear stiffness ratio of the pile-soil is not always better, an economical and reasonable pile stiffness should be selected.

Safety Criterion for Hollow Pipe Piles Under Blasting Vibration Based on Wave Function Expansion Method

When the drilling and blasting method is used to excavate urban subway tunnels, the blasting seismic waves will inevitably produce vibration effects on the pile foundations under the surface buildings, which will seriously threaten the safety of surrounding buildings and the environment. Based on the transmission and reflection laws of stress waves at the interface between soil medium and pile medium and the displacement–stress continuous assumption, the wave function expansion method is used to analyze the whole process mechanism of vibration effect of hollow pipe piles under the action of incident [Formula: see text] waves. In addition, the theoretical model of safety criterion for hollow pipe piles under blasting vibration is proposed and the reliability of the model is verified. Furthermore, the impacts of the elastic modulus, density, and pile section parameters of soil and piles on the safety criterion for the blasting vibration effect of hollow pipe piles are analyzed. The results show that as the elastic modulus and density of the soil increase, the critical peak particle velocity (PPV) curves of hollow pipe piles move to the high-frequency and low-frequency bands, respectively, resulting in a decrease in the overall safety of piles. This indicates that hollow pipe piles should not be buried in soil layers with high elastic modulus and density. It is also shown that the larger the hollow area of hollow pipe piles, the lower their safety. Compared to solid circular piles, hollow pipe piles have a higher overall seismic safety reserve. The study results have guidance and reference value for effectively evaluating and controlling the harm of blasting vibration effects of hollow pipe piles.

Surface motion of P/SV wave scattering by a semicircular canyon in saturated half-space by complex function and conformal mapping

Based on the complex function theory of elastic dynamics, a complex function solution of P/SV-wave scattering by a semicircular canyon in an elastic porous saturated half-space is obtained. Firstly, through Biot theory and Helmholtz decomposition, the wave potential functions with unknown coefficients in the model are obtained based on the wave function expansion method. Secondly, the complex function expressions of stress and displacement in the model are obtained by using the complex function method and conformal mapping. According to the boundary conditions of the horizontal ground surface and semicircular canyon surface, this boundary value problem can be converted into a series of algebraic equations and solved numerically by truncation. Then, the correctness of the complex function solution in this paper is verified by comparing it with the published results through degenerating the scattering problem in the saturated porous elastic half-space in this paper into that in the elastic half-space. Finally, the influence of the boundary drainage conditions, the medium porosity, and the incident angle and frequency of the incident wave on the ground surface displacement is investigated. Although the P/SV-wave scattering problem is a traditional problem, there is currently no accepted analytical solution that strictly satisfies the zero-stress condition on the horizontal ground surface. This paper is an attempt at this problem, which may provide insight into the search for the strictly analytical solution for P/SV-wave scattering.

Dynamic response of water-rich tunnel subjected to plane P wave considering excavation induced damage zone

The stability analysis of a deep buried tunnel subjected to dynamic disturbance is an important issue. In this study, the transient response has been obtained by establishing a water-rich tunnel model considering excavation damage zone (EDZ). Based on Biot’s two-phase dynamic theory and wave function expansion method, the analytical solution of dynamic response around the water-rich tunnel containing EDZ subjected to P wave is derived. Moreover, Fourier transform and Duhamel’s integral technique is introduced to calculate the transient response, and the equivalent blasting curve is adopted to input excitation function. The dimensionless parameters thickness N and shear modulus ratio μ∼ are defined to characterize the degree of damage in the surrounding rock, investigating the influencing factors, such as the parameters and the incident source frequencies. The results indicate that the dynamic stress concentration factor (DSCF) gradually decreases as the dimensionless parameters increase. Additionally, it is observed that the DSCF is more sensitive to changes in the thickness parameter N. Finally, the influence of the waveform parameters has been taken into account in the analysis of transient response, and the stress state and transfer process in each time stage of the EDZ are analyzed. This study establishes a theoretical foundation for comprehending the mechanical behavior and support design considerations associated with a deep-buried water-rich tunnel containing EDZ.

Analytical method for a tunnel with periodically jointed lining in the poroelastic soil under seismic waves

In this study, a longitudinal equivalent periodic shell (LEPS) model and corresponding continuous joint model are proposed for the periodically jointed lining (PJL) tunnel. Based on the LEPS model for the PJL tunnel, an analytical method for the PJL tunnel under seismic waves is established. The surrounding soil of the PJL tunnel is assumed to be the poroelastic medium, and described by the Biot's theory. The lining of the PJL tunnel is assumed to be composed of a series of lining rings linked periodically by ring joints along the longitudinal direction, which are simplified as a LEPS and described by the cylindrical shell theory. Due to the periodicity of the LEPS lining, the tunnel-soil system is treated as a periodic structure in this study. Based on the periodicity condition for the tunnel-soil system as well as the wave function expansion method, the representation for the scattering wave field in the soil is established. By using the aforementioned cylindrical shell theory and Fourier series expansion method, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining are derived. By using the lining-soil continuity conditions, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining as well as the representation for the wave field in the soil, the coupled equations for the tunnel-soil system are established, with which the Fourier components of the displacements of the LEPS lining and wave function coefficients for the scattering waves in soil can be solved for. Numerical results indicate that the presence of the ring joint in the LEPS lining of the tunnel will affect the distribution of the internal forces in the lining considerably, and it will also enhance the maximum internal forces occurring in the lining.

A theoretical calculation method of seismic shear key pounding based on continuum model

The evolution of shear key design for bridges is accompanied by research on structural earthquake resistance. However, the vast majority of pounding forces, responses, and corresponding data for the study and design of shear keys have been based on expensive experimentalism and imprecise empiricism approaches for decades. Hence, strengthening theoretical study on seismic performance of shear key is essential. In this paper, a “Beam-Spring-Beam + Concentrated Mass” continuum dynamic model is proposed. Meanwhile, the transient wave function expansion method and the mode superposition method are applied to determine the analytical expression of the dynamic response from the girder and pier system (pier and cap beam). Furthermore, the combined transient internal force method and Duhamel integration method are introduced to assess the elastic pounding process. Through programming and numerical analysis, a series of pounding response data related to the shear key under various working circumstances will be explored. As mentioned above, the proposed theoretical method can optimize shear key design and boost the reliability of seismic limiting devices in the future.•Establishing a feasible “Beam-Spring-Beam + Concentrated Mass” continuum model of girders and piers based on a two-span continuous girder bridge.•Deriving the analytical solutions of responses by conducting the response equations under horizontal seismic excitation (containing orthonormality verification).•Simulating the pounding process by embedding elastic pounding calculation methods into Continuum Model.

Transient Response of Dynamic Stress Concentration around a Circular Opening: Incident SH Wave

The present study aims to investigate the transient response of stress concentration around a circular opening. The study focuses on the composition of the shockwave, which consists of SH waves of multiple frequencies. The wave equation, expressed by the displacement function, is transformed into the Helmholtz equation through the Fourier transform method. The spectral function can be obtained by employing analytic continuation and Fourier transform of the incident wave field. An analytical expression for the dynamic stress around the aperture can be derived using the wave function expansion method and by considering the boundary conditions. The influence of the aperture on the transient response is discussed based on the distribution of the dynamic stress concentration coefficient and stress peak coefficient under different aperture sizes. The results show that the peak of the dynamic stress concentration coefficient changes with the aperture. In contrast, the stress peak coefficient is primarily concentrated in the early stages of the transient response. Furthermore, it is observed that larger radii can induce alternating stress in the material, which may lead to fatigue failure. This strategy provides a solution for addressing similar challenges.