Wave Function Expansion Method Research Articles

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. In this paper, 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 wave field 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 medium. Then, combining the free boundary conditions with a Fourier series expansion method we solve for 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| w1|( w), the dynamic stress concentration factor (DSCF) and the displacement amplitude| w2|. The results show 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 can also provide a basis for seismic design of underground structures.

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.

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.

Safety criterion for hollow pipe piles under blasting vibration based on wave function expansion method

Safety criterion for hollow pipe piles under blasting vibration based on wave function expansion method

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.

Scattering of plane compressional waves by a variable thickness circular lined tunnel

The determination of the dynamic stress concentration factor (DSCF) associated to a circular lined tunnel with variable thickness liner incident upon by compressional P-waves is of particular interest. Due to the lack of symmetry, the treatment of such a problem via the traditional wave-function expansion method is not possible. To circumvent this dilemma it is proposed to employ the dynamic equivalent inclusion method (DEIM). As it will be shown, this method which is based on the notions of the eigenstress and eigenbody-force fields provides an accurate solution for the problem under consideration. In the simple special case where a tunnel with a concentric liner is subjected to plane compressional P-waves the exact wave-function expansion solution given in the literature is recovered. The effect of liner thickness variability on the static and dynamic stress concentration along the interior surface of the liner and along the liner-matrix interface is studied in detail through numerical examples. The maximum value of the DSCF for the concrete and steel liners with positive and negative values of eccentricity is obtained and compared with those of the corresponding concentric cases. It is observed that the eccentricity of the liner results in different behaviors of the stress concentration for the hard (steel) and soft (concrete) liners.

Surface Motion for P-Wave Scattering by an Arbitrary-Shaped Canyon in Saturated Half-Space

This paper obtained a semianalytical solution for the P-wave scattering problem by an arbitrary-shaped canyon in a saturated half-space by using Biot’s theory, the wave function expansion method, and the moments method. Firstly, based on the Biot fluid-saturated porous media theory and the wave function expansion method, the wave potentials which automatically satisfy the zero-stress boundary condition on the surface of the half-space are obtained. Then, the boundary value problem is transformed into an algebraic problem by the method of moments according to the boundary conditions, and then solved numerically by truncation. By adjusting the parameters, the saturated medium in the original model approximately degenerates into a single-phase elastic medium, and the correctness of the proposed method is verified by comparing it with published results. Finally, the effects of the canyon shape, the porosity of the soil, and the incidence angle and frequency of the incident wave on the amplitude of ground surface motion are investigated. The results show that the incidence angle has a significant effect on the ground surface motion, while the porosity of the soil has little influence on the amplitude of surface motion. The influence of canyon shape on surface motion is mainly reflected in the shielding effect on the incident wave. Although P-wave scattering by a canyon is a traditional problem, most of the analytical solutions are limited to solving the scattering of seismic waves in geometric regular canyons. However, the actual shape of the canyon is not regular, which limits the application of closed analytical solutions in practical engineering. In this paper, the scattering of P-waves in arbitrary-shaped canyons is successfully solved by using a semianalytical method combined with a numerical method-moments method, which provides a possibility for engineering application.

Theoretical study on dynamic stress redistribution around circular tunnel with different unloading paths

This study presents a solution for stress redistribution induced by dynamic excavation of a rock tunnel in hydrostatic and non-hydrostatic stress states. Considering rock as an elastic medium, the analytical solution of dynamic excavation unloading was derived in the Laplace transform space utilizing the wave function expansion method and mode decomposition, and transformed into the time domain using the approximate numerical inversion method. The effects of the unloading path (linear, cosine, and exponential), unloading rate, and initial stress state on stress redistribution were analyzed. The analytical results indicated that a high unloading rate led to a high stress concentration and oscillations around the tunnel, and instantaneous dynamic unloading caused 20–30% stress concentration. When dynamic unloading is complete, the stress state around the tunnel converges to the Kirsch solution. The 3D numerical results indicated that deformation of the tunnel boring face increased as the initial stress increased. Moreover, the accumulation and release of strain energy and stress state change paths were analyzed and discussed. The results of this study provide a theoretical basis for understanding the stress adjustment and failure of surrounding rock induced by excavation of deep-buried openings in hydrostatic and non-hydrostatic stress states.

Dynamic response of shallow-buried circular composite lining tunnel with imperfect interface under SV waves

In this paper, the dynamic response of lined shallow-buried composite lining tunnel with imperfect interface in semi-infinite elastic medium under plane SV waves is studied, and the spring model is introduced to describe the imperfect interface relationship among the surrounding rock, the isolation layer, the primary support layer and main support layer. By using the wave function expansion method, the analytical solutions of displacements and stresses are presented. The interacting effects of the interface properties and the buried depth under different dimensionless frequencies are discussed. The results show that the peak values of the dynamic stress in the linings gradually shift to the upper part of the tunnel with the increase of dimensionless frequency, and the maximum dynamic stress in the linings also tends to decrease. The dynamic stress around the main and primary support layers increases when the interfaces are weakly bonded, and the effect of transverse springs is greater than that of normal springs. The tunnel with diminutive buried depth has a larger dynamic stress under the low-frequency SV waves. Reducing the stiffness in the isolation layer will lead to the increase of dynamic stress in the surrounding rock, while the dynamic stress in the main and primary support will be weakened.

Dynamic characteristics of rockbolt anchorage structure under radial cylindrical P wave

The anchorage structure formed of rockbolt, anchoring agent and surrounding rock plays a significant role in supporting underground engineering, however, there are still damage and failure cases of rockbolt supporting systems under blasting load. Therefore, it is important to clarify the mechanical behavior of rockbolt anchorage structure subjected to dynamic load. In this paper, we derive the analytical solution of dynamic response around anchorage structure subjected to radial cylindrical P wave based on the wave function expansion method, and the transient response is obtained by using Fourier transform. Two cases with and without anchoring agent are taken into consideration, and the elastic-slip model is introduced into describing the interaction between the surrounding rock and rockbolt without anchoring agent. The influence of multiple parameters such as thickness, stiffness and distance from the disturbance source on dynamic response is investigated in detail. The steady-state results demonstrates that the alternating tensile and compressive stress concentration is generated with the increase of wavenumber. The dynamic stress concentration can be reduced by appropriately decreasing the thickness and stiffness of anchoring agent, and the optimal values are obtained. The dynamic stress distribution in rock is opposite to that in anchoring agent under transient incidence and the tensile damage in rock along the incident direction and that in anchoring agent at the vertical direction of propagation must be alert. The novelty of the manuscript consists in studying the dynamic mechanical behaviors of anchorage structure subjected to radial dynamic load. The implications of these investigations for practical engineering will serve as a reference to design supporting methods and improve anchoring agent parameters.

EFFECTS OF NEAR-FAULT ON DYNAMIC RESPONSE OF A STRUCTURE EXCITED BY EARTHQUAKE WAVES

The goal of this study is to concentrate on the near-fault effects on the structure behavior of underground excitation. To figure out the exact behavior of the structure, a simple model is chosen. During the solution process, dynamic soil-structure interaction is taken into account. The physics of the problem is very complicated owing to the reflection of the waves from the free surface and fault boundary, and the scattering of the waves from the foundation and transmission of the energy into the structure. The materials of the soil and the structure are considered as homogeneous and isotropic, and the foundation is assumed as a rigid body. Response of the structure in the frequency domain is derived using Wave Function Expansion Method and Image Method. Numerical results show that when the structure approaches the fault, the amplitudes of the foundation and relative displacements of the structure are getting bigger. The incident angles of the waves have a big influence on the response of the structure close to the fault, therefore the standard design code should not be applied in the analysis of the structures. It is concluded that near-fault effects should be considered carefully.

Dynamic Analysis for a Shallow Isosceles Trapezoid Depression Impacted by SH Waves

This paper presents a series solution of shear horizontal (SH) wave scattering by a shallow isosceles trapezoid depression in half-space. An appropriate multiregion-matching technique was adopted to divide the analytical model into three parts, which is necessary to meet the conditions of applying the wave function expansion method. Based on the complex function method, the wave field expressions, satisfying stress-free boundary conditions, were constructed. In addition, introducing complex variables avoided the redundant formula derivation for the implementation of coordinate transformation. The unknown coefficients in the wave field expressions were solved by Fourier series expansion in a complex field, and the convergence analysis was carried out to determine the number of traction terms of the series solution. Frequency-domain and time-domain results were presented to analyze the wave motion amplification and propagation process. The proposed method not only provides a benchmark for numerical simulation results but also makes up for the limitations of the theoretical research method of trapezoidal depression topography.

Visco-Elastic Interface Effect on the Dynamic Stress of Symmetrical Tunnel Embedded in a Half-Plane Subjected to P Waves

A semi-analytical method is developed to study the visco-elastic interface effect on the dynamic stress of a non-circular symmetrical tunnel in a half-plane subjected to P waves, and the interaction between the visco-elastic interface around the tunnel and the traction free boundary at the half-plane is analyzed. The complex variable function method combined with the wave function expansion method is introduced to obtain the theoretical expressions of waves around the non-circular tunnel. To satisfy the traction-free boundary condition at the half-plane, the large circle assumption is applied. The expanded coefficients are determined by using appropriate boundary conditions with stiffness parameters and viscosity coefficients of the interface. Selected numerical results are presented, and the interface effect on the dynamic stress under different embedded depths is discussed. It is found that the interacting effect of the visco-elastic interface and the traction-free boundary is quite related with the wave frequency. Comparison with existing results is also given to validate the semi-analytical method.