Viscous-spring Boundary Research Articles

Implementation of the non-reflection boundary for the seismic response analysis in the framework of non-local general particle dynamics

A thorough understanding of the seismic response of geotechnical structures has always been an aspiration for all engineers. Recently, the non-local general particle dynamics (NLGPD), which considers the variation of material density, has exhibited excellent promise in modelling extremely large deformation and dynamic problems. To extend the capacity of NLGPD to carry out seismic response analysis, four non-reflection boundaries are newly added in the framework of NLGPD in this paper. The viscous boundary allows wave energy to be absorbed by dashpot, in which a stress input method is proposed to input the seismic motion. The viscous-spring boundary can capture the elasticity recovery ability of the far-field medium. The free-field boundary achieves the free-field motion at lateral boundaries in the process of absorbing wave, and the corresponding relation between the points of the main model and the free-field column is determined. The static-dynamic unified boundary is introduced to model the transformation process from fixed boundary conditions in a static state to free field boundary conditions in dynamic analysis. Several representative numerical examples are solved to validate the feasibility of the proposed method. The numerical results show that the NLGPD method provides robustness and broad applicability for the seismic response analysis.

Seismic performance of base-isolated structures for swimming pool reactors under different foundation conditions.

The swimming pool reactor (SPR) is an innovative and environmentally friendly heating source. An SPR building was selected as the research subject, and a 3D dynamic interaction model incorporating the liquid sloshing effect was created using ANSYS software and the secondary development characteristics of user-programmable features (UPFs). Energy dissipation from scattered waves was accounted for using viscous-spring boundary elements, while the dynamic hydraulic effect was modeled via the Housner equivalent mechanical model. Considering soil-structure interaction (SSI) effects, this study examines the impact of isolation measures on the structure's seismic mitigation performance. It investigates how varying foundation conditions affect the seismic resistance of the isolated structure. Results indicate that seismic isolation ratios for acceleration, floor response spectra, displacement, and base shear diminish as site stiffness decreases. However, regarding sloshing wave height, seismic isolation amplified the height under all conditions but remained within safe limits. These findings offer valuable insights for seismic design across different SPRs.

Dynamic responses of a bituminous concrete core rockfill dam on layered overburden under seismic wave input

The dynamic nonlinearity, overburden stratification characteristics and oblique incidence of seismic waves in soil greatly increase the difficulty of determining the free field of a dam foundation, which restricts seismic input accuracy. Based on the potential function theory, this paper constructs a transfer relationship between the wave amplitude matrix at the top layer of a dam foundation with overburden and the wave amplitude matrix at any layer and obtains the shear strain of the soil through Fourier inverse transformation. Then, the equivalent linearization method is adopted to depict the nonlinearity of the soil. An analytical calculation method for the free field of a dam foundation with layered overburden under inclined seismic waves is proposed. This method can not only forward the free field according to the incident wave in the deep part of the foundation but also directly invert the free field according to the seismic motion on the ground surface, and the inversion efficiency is high. Using a nonlinear viscous-spring boundary element developed by the authors, combined with the free field, a wave input method for a dam foundation with a layered overburden under inclined seismic waves is established. The maximum error between the numerical solution under wave input and the analytical solution is only 5%. The computational precision of the method is high. Finally, the dynamic responses of a bituminous concrete core rockfill dam on a layered overburden are studied. The results show that an upstream buoyancy and oblique incidence are the main reasons for a larger acceleration on the upstream side of the dam than on the downstream side. As the incident angle increases, the differential distribution becomes more significant, and the amplification coefficient of the anti-seepage system also increases. When the incident angle is 25°, the horizontal and vertical amplification coefficients at the top of the core wall increase by 12.8% and 25.2% compared to the vertical incidence, respectively. The accelerations of the dam are overestimated under uniform seismic input, and the amplification coefficients of the dam crest are overestimated by 52.3% and 39.8% compared with the wave input, respectively. The wave input reasonably solves the problem of seismic input for a dam foundation with layered overburden, considers the inclined seismic waves, and can reflect the seismic waves propagation characteristics. The wave method provides a solid foundation for the seismic safety of rockfill dams with layered overburden.

STUDY ON EARTHQUAKE DESTRUCTION MODE OF THE LARGEST CANAL CROSSING HIGHWAY BRIDGE BASED ON IEM BOUNDARY IN SOUTH-TO-NORTH WATER DIVERSION

To study the dynamic failure mechanism and damage development law of highway bridge structure under the boundary effect in the process of seismic dynamic duration, the Wenchang Highway Bridge with the largest canal crossing in the South-to-North Water Diversion is taken as an example for seismic design analysis. Based on the finite element and infinite element coupling theory, the infinite element method boundary is introduced, the concrete damage plasticity is introduced, and the half-space free field model is established to study the energy dispersion phenomenon of waves in the boundary and the absorption effect of the infinite element method boundary on wave energy is verified. Under different peak acceleration intensities, the seismic response analysis of the bridge structure was carried out. The results show that: Under the action of selected artificial waves, the damage location of the bridge mainly concentrated in the junction of the box girder supported by the pier, the bottom of the pier and the junction of the pier and beam. The damage tends to develop downward near the bottom of the box girder. The damage at both ends of the beam extends from both ends to the middle. And the bottom and top of the pier have penetrating damage. These are weak points in seismic design. At a horizontal peak acceleration of 0.6g, in addition to damage to the pier column, damage also occurred to the bottom of the box girder. Therefore, when the horizontal peak acceleration of the seismic wave is greater than 0.6g, the failure of the bottom of the box girder is paid attention to. Moreover, the IEM boundary has a good control effect on the far-field energy dissipation of the wave, which is simpler and more efficient than the viscous–spring boundary.

A numerical framework for underground structures in layered ground under inclined P-SV waves using stiffness matrix and domain reduction methods

A numerical framework was proposed for the seismic analysis of underground structures in layered ground under inclined P-SV waves. The free-field responses are first obtained using the stiffness matrix method based on plane-wave assumptions. Then, the domain reduction method was employed to reproduce the wavefield in the numerical model of the soil–structure system. The proposed numerical framework was verified by providing comparisons with analytical solutions for cases involving free-field responses of homogeneous ground, layered ground, and pressure-dependent heterogeneous ground, as well as for an example of a soil–structure interaction simulation. Compared with the viscous and viscous-spring boundary methods adopted in previous studies, the proposed framework exhibits the advantage of incorporating oblique incident waves in a nonlinear heterogeneous ground. Numerical results show that SV-waves are more destructive to underground structures than P-waves, and the responses of underground structures are significantly affected by the incident angles.

Investigation on artificial boundary problem in the combined finite-discrete element method (FDEM)

The combined finite-discrete element method (FDEM) is an advanced finite and discrete element coupling method, which can commendably simulate the large deformation and fracture process of materials. The explicit solution method in FDEM is naturally suitable for solving dynamic problems, and the key to dynamic analysis is to establish artificial boundary conditions that can be prepared to simulate infinite domain motion. At present, there is only the viscous boundary condition in the original FDEM, but this is only applicable for solving the internal source wave problems (e.g., blasting, mechanical vibration), and there are no artificial boundary conditions for solving the external source wave problems (e.g., earthquake). Therefore, three artificial boundary conditions have been developed in the present study to enhance the ability of FDEM to conduct dynamic response analysis. Firstly, the basic principle of FDEM and the existing viscous boundary condition are briefly introduced. Then, three newly added boundary conditions are introduced in sequence: (1) viscous-spring boundary condition, which can absorb stress wave energy at the boundary and restore the residual displacement to meet the actual engineering; (2) free-field boundary condition, which is coupled with free-field motion to absorb scattered waves at the lateral boundaries of the model, and the algorithms of mesh matching retrieval and coupling calculation are also introduced; (3) static-dynamic unified boundary condition, which can accurately transform the fixed boundary under the quasi-static analysis into the non-reflective boundary condition under the dynamic analysis. Finally, several classic models are built to verify the feasibility of the newly added boundary conditions. The comparison with the shaking table test also shows that the improved FDEM can be used for seismic response analysis under actual earthquakes.

A Time-Domain Artificial Boundary for Near-Field Wave Problem of Fluid Saturated Porous Media

The near-field wave problem of the saturated soil involves the energy radiation effect of the truncated infinite media. A viscous spring boundary is proposed for the fluid-saturated porous media. Based on the process of wave propagation under internal point source, the stress and flow velocity boundaries are constructed by reasonable assumptions of outgoing waves and Green’s function, respectively. Without the permeability assumption, the proposed boundary avoids the low accuracy caused by the assumption of zero permeability that is widely used in the existing methods. The boundary simultaneously has a simple form, clear physical meaning, and less computational cost due to its local character. Meanwhile, a completely explicit integration algorithm considering the damping is constructed to solve the finite element equations of saturated porous media with the proposed boundary. The accuracy and high computational efficiency of the wave numerical method are verified in the examples.

Dynamic response analysis of high earth-rockfill dam subjected to P wave with arbitrary incoming angles

The dynamic response analysis of high earth-rockfill dams mostly only considers the vertical incidence of seismic waves. However, due to the randomness and uncertainty of earthquakes, for near-field earthquakes, oblique incident waves generally dominate. Therefore, understanding the nonlinear dynamic response of high earth-rockfill dams under the oblique incident seismic wave is of great significance to the seismic safety evaluation of the dams. For this purpose, the seismic wave input method of P wave with arbitrary angle that considers near-field oblique incidence is proposed, which is realized through the Fortran program, and the validity and accuracy of the method and the program are verified by two examples. The proposed method is applied to study the dynamic response analysis of the Nuozhadu earth core rockfill dam under oblique incident P wave. In addition, the three-dimensional interaction system of the dam-foundation-canyon is considered based on the viscous-spring boundary model, which reflects as much as possible the environmental conditions of the high earth-rockfill dam when the actual earthquake occurs. The numerical results reveal that the incident angle of P wave significantly affects the dynamic response of the dam. For acceleration amplification factor and residual deformation, the most damaging incident angle is the incident angle α = 60°, rather than in the vertical direction. Besides, when the seismic wave is incident obliquely, the value of residual deformation along the transverse direction is greater than that along the stream direction. It is recommended that the influence of oblique incident P wave should be considered in the seismic design of high earth-rockfill dams.

An Arc-Consistent Viscous-Spring Artificial Boundary for Numerical Analysis of Seismic Response of Underground Structures

A new arc-consistent viscous-spring artificial boundary (ACVAB) was proposed by changing a traditional flat artificial boundary based on the theory of viscous-spring artificial boundaries. Through examples, the concept underpinning the establishment and specific setting of the boundary in the finite element software were described. Through comparison with other commonly used artificial boundaries in an example for near-field wave analysis using the two-dimensional (2D) half-space model, the reliability of the ACVAB was verified. Furthermore, the ACVAB was used in the numerical analysis of the effects of an earthquake on underground structures. The results were compared with the shaking table test results on underground structures. On this basis, the applicability of the ACVAB to a numerical model of seismic response of underground structures was evaluated. The results show that the boundary is superior to common viscous-spring boundaries in terms of accuracy and stability, and therefore, it can be used to evaluate radiation damping effects of seismic response of underground structures and is easier to use.

Numerical study on P-wave propagation across the jointed rock masses by the combined finite-discrete element method

The combined finite-discrete element method (FDEM) is an advanced finite and discrete element coupling method. This study further applies it to the problem of stress wave propagation in the jointed rock mass. Firstly, the fundamental theory of FDEM is briefly described and the methods commonly used in FDEM to characterize structural planes are comprehensively analyzed. Then, based on the viscous boundary condition (VBC), the viscous-spring boundary condition (VSBC) is added to absorb the reflected wave at the artificial boundary and restore the residual displacement to meet the actual engineering. In addition, the application range of the VBC and VSBC is verified, which indicates that the VBC and VSBC in the improved FDEM can be well applied not only to the continuum range but also to the case of the new fracture formed. Finally, several classic models are solved to verify the stress wave propagation across different types of joints by FDEM. The results reveal that results from FDEM agree well with analytical solutions based on the displacement discontinuity model and numerical solutions from universal distinct element code (UDEC), which means that the improved FDEM can capably and accurately simulate wave propagation in the jointed rock mass.

A hybrid boundary method for seismic wave propagation problems in slopes

The seismic behavior of a slope site has a critical impact on its stability and the seismic response of structures located on or near the slope. In this paper, a hybrid boundary method(HBM) is proposed for the site response analysis of slopes. Viscous and viscous-spring boundaries are applied in the HBM and equivalent loads are calculated separately on the bottom and the lateral boundary. A horizontally extended model is used to verify the HBM. The results show that the maximum error of the HBM is only 2.010%. On the ground surface of the slope site, amplification occurs at the crest side and de-amplification occurs at the toe side under various cases. The ratio of the maximum and minimum PGA on the ground surface could exceed 2. Therefore, great attention should be paid to the spatial variability of ground motion on slope sites. The HBM is also compared with four conventional approaches. The maximum errors of all the conventional approaches exceed 70%. These examples demonstrate that the HBM has much higher accuracy in practice than conventional approaches.

Ground motion duration effect on responses of hydraulic shallow-buried tunnel under SV-waves excitations

Although intensive research of the influence of ground motion duration on structural cumulative damage has been carried out, the influence of dynamic responses in underground tunnels remains a heated debate. This study attempts to highlight the importance of the ground motion duration effect on hydraulic tunnels subjected to deep-focus earthquakes. In the study, a set of 18 recorded accelerograms with a wide-range of durations were employed. A spectrally equivalent method serves to distinguish the effect of duration from other ground motion features, and then the seismic input model was simulated using SV-wave excitation based on a viscous-spring boundary, which was verified by the time-domain waves analysis method. The nonlinear analysis results demonstrate that the risk of collapse of the hydraulic tunnel is higher under long-duration ground motion than that of short-duration ground motion of the same seismic intensity. In a low intensity earthquake, the ground motion duration has little effect on the damage energy consumption of a hydraulic tunnel lining, but in a high intensity earthquake, dissipation of the damage energy and damage index of concrete shows a nonlinear growth trend accompanied by the increase of ground motion duration, which has a great influence on the deformation and stress of hydraulic tunnels, and correlation analysis shows that the correlation coefficient is greater than 0.8. Therefore, the duration of ground motion should be taken into consideration except for its intensity and frequency content in the design of hydraulic tunnel, and evaluation of seismic risk.

Earthquake response analysis of tunnels at a complex saturated site

Only a seismic analysis of tunnels at a single-phase site can be performed using the commonly used finite element software owing to the lack of a dynamic two-phase element. The seismic analysis of tunnels at a saturated site is primarily based on OpenSees or self-compiled programmes, the majority of which are limited to uniform or horizontal-layered sites owing to the related issues of the inconvenience of modelling and the limited computation scale. In this study, based on ABAQUS, a finite element model for the earthquake response analysis of twin tunnels at a complex saturated site is established by calling the user-defined element, which is developed for the dynamic analysis of two-phase media. The wave propagation towards infinity is simulated by the viscous-spring boundary, the seismic wave is inputted in the form of the equivalent nodal force, and the soil nonlinearity is considered via the equivalent linear method. Differences between the earthquake responses of tunnels at a complex saturated site and a complex single-phase site as well as those between tunnels at a complex saturated site and a horizontal-layered saturated site are investigated to emphasise the effects of dynamic solid-fluid coupling and complex topography and geology.

Seismic performance and failure modes of the Jin'anqiao concrete gravity dam based on incremental dynamic analysis

To investigate the potential failure modes of concrete gravity dams under seismic loads, a typical non-overflow dam section of the Jin'anqiao roller compacted concrete (RCC) gravity dam is taken as a case study. A compressive and integral analysis model of the dam body–foundation–reservoir water interaction system is established, in which the viscous–spring boundary and its corresponding free field input mechanism is employed to account for the dam–foundation interaction and fluid–solid coupling method is used to account for the dam–reservoir interaction. Based on the incremental dynamic analysis (IDA) method, IDA curves are generated with the relative displacement angle on the x-axis (damage measure, DM) and peak ground acceleration (PGA) on the y-axis (intensity measure, IM). The damage processes and potential failure modes for the Jin'anqiao gravity dam under different peak accelerations are investigated. It is concluded that under ground motion loadings of the Jin'anqiao dam, the locations at which functional failure may occur are mostly found in the stress concentration of the dam slope, the boundary of the rolling section, the junction between the dam and dam foundation, and the top of the corridor. Therefore, strengthening the seismic protection in these areas is conducive to improving the overall aseismic level of the dam.

A modified domain reduction method for numerical simulation of wave propagation in localized regions

A modified domain reduction method (MDRM) that introduces damping terms to the original DRM is presented in this paper. To verify the proposed MDRM and compare the computational accuracy of these two methods, a numerical test is designed. The numerical results of the MDRM and DRM are compared using an extended meshed model. The results show that the MDRM significantly improved the computational accuracy of the DRM. Then, the MDRM is compared with two existing conventional methods, namely Liao’s transmitting boundary and viscous-spring boundary with Liu’s method. The MDRM shows its great advancement in computational accuracy, stability and range of applications. This paper also discusses the influence of boundary location on computational accuracy. It can be concluded that smaller models tend to have larger errors. By introducing two dimensionless parameters, φ1 and φ2, the rational distance between the observation point and the MDRM boundary is suggested. When φ1>2 or φ2>13, the relative PGA error can be limited to 5%. In practice, the appropriate model size can be chosen based on these two parameters to achieve desired computational accuracy.

Seismic stability of jointed rock slopes under obliquely incident earthquake waves

Seismic stability of slopes has been traditionally analyzed with vertically propagated earthquake waves. However, for rock slopes, the earthquake waves might approach the outcrop still with a evidently oblique direction. To investigate the impact of obliquely incident earthquake excitations, the input method for SV and P waves with arbitrary incident angles is conducted, respectively, by adopting the equivalent nodal force method together with a viscous-spring boundary. Then, the input method is introduced within the framework of ABAQUS software and verified by a numerical example. Both SV and P waves input are considered herein for a 2D jointed rock slope. For the jointed rock mass, the jointed material model in ABAQUS software is employed to simulate its behavior as a continuum. Results of the study show that the earthquake incident angles have significance on the seismic stability of jointed rock slopes. The larger the incident angle, the greater the risk of slope instability. Furthermore, the stability of the jointed rock slopes also is affected by wave types of earthquakes heavily. P waves induce weaker responses and SV waves are shown to be more critical.

Earthquake input for finite element analysis of soil-structure interaction on rigid bedrock

When using the finite element method to analyze seismic soil-structure interaction (SSI) problem on rigid bedrock, several methods have been developed to input earthquake on the lateral boundary of finite element model. These boundary conditions include the free boundary, the viscous-spring (VS) boundary, the tied degrees of freedom (TDOF) boundary and the free-field loading combined viscous-spring (FFL-VS) boundary. Their accuracy and location should be investigated. In this paper, the formulations and ABAQUS implementations of these boundaries are given. The accuracy properties of these boundaries are then compared by numerical examples including the free-field and SSI problems. The comparison studies indicate that the free and VS boundaries failed to reproduce the free-field and SSI responses when a relatively small size SSI model is employed. The TDOF and FFL-VS boundaries can simulate the exact free-field response and have good performances in seismic SSI analyses. The FFL-VS boundary is more accurate than the TDOF boundary. The appropriate locations for the TDOF and FFL-VS boundaries depend on the SSI model. A simplified approach is developed for appropriate boundary location. This approach is applicable to the FFL-VS boundary but not to the TDOF boundary due to the former more accurate than the latter.

Input Seismic Motion Based on the Viscous-Spring Boundary: Analytical Solution and Numerical Simulation

Practical geotechnical engineering problems are usually solved within an infinite half-space domain, and a finite domain is necessary to analyze the dynamic response of the medium by the finite element method (FEM). How to deal with the boundary is vital for obtaining accurate results in numerical simulation. This study obtains the analytical solution of input stress at the boundary based on the viscous-spring boundary and develops a procedure for the two-dimensional artificial viscous-spring boundary application in the seismic analysis. The analytical solution based on the viscous-spring boundary is verified using other three artificial boundaries. Furthermore, the simulation accuracy of the procedure is carried out by establishing a two-dimensional model for different element sizes and modified coefficients. The results show that the element size must be in a certain range to avoid the loss of main frequency of seismic motions. When the modified coefficient is less than the suggested value, obvious shifts appear on the calculation results. On the opposite, when the modified coefficient is greater than that, the result will fluctuate dramatically before convergence. All the above factors can lead to a distortion of simulation accuracy of artificial boundary model. Compared with the viscous boundary, viscous-spring boundary has higher simulation accuracy.

A Seismic Wave Oblique Incidence Method and the Application in Gravity DAM Seismic Research

The incidence of seismic wave is generally assumed to be vertical in seismic analysis of large-scale hydraulic structures, but during some near-field earthquakes, the direction of seismic waves is at certain angle to the ground. The non-uniform motion produced by oblique incidence can significantly affect the structures. In this paper, based on the theories of wave motion an oblique P-wave and SV-wave incidence method based on viscous-spring boundary was introduced into finite element analysis, which can accurately simulate the incidence at different oblique angles. Then the proposed method was applied in seismic analysis of a gravity dam located in southwest China. The seismic responses under oblique incidence of P-wave and SV-wave at different oblique angles were discussed. The results indicated that the seismic response was obviously influenced by the oblique incidence, and the traditional vertical incidence was unsuitable for some near-field earthquakes. The oblique incidence should be considered in the seismic analysis of high dams as well as other large-scale water-retaining structures located in seismic regions.

A viscous-spring transmitting boundary for cylindrical wave propagation in saturated poroelastic media

Based on the u–U formulation of Biot equation and the assumption of zero permeability coefficient, a viscous-spring transmitting boundary which is frequency independent is derived to simulate the cylindrical elastic wave propagation in unbounded saturated porous media. By this viscous-spring boundary the effective stress and pore fluid pressure on the truncated boundary of the numerical model are replaced by a set of spring, dashpot and mass elements, and its simplified form is also given. A u–U formulation FEA program is compiled and the proposed transmitting boundaries are incorporated therein. Numerical examples show that the proposed viscous-spring boundary and its simplified form can provide accurate results for cylindrical elastic wave propagation problems with low or intermediate values of permeability or frequency content. For general two dimensional wave propagation problems, spuriously reflected waves can be greatly suppressed and acceptable accuracy can still be achieved by placing the simplified boundary at relatively large distance from the wave source.