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Article

Dynamic Response Analysis of Wedge-Shaped Rock Slopes under Harmonic Wave Action

by
Yihan Du
,
Wenzhi Xu
,
Wei Han
*,
Bo Huang
,
Hui Liu
and
Xuze Du
School of Architecture and Civil Engineering, Anhui Polytechnic University, Wuhu 241000, China
*
Author to whom correspondence should be addressed.
Buildings 2023, 13(10), 2623; https://doi.org/10.3390/buildings13102623
Submission received: 30 August 2023 / Revised: 1 October 2023 / Accepted: 10 October 2023 / Published: 18 October 2023
(This article belongs to the Special Issue Monitoring, Prevention and Control of Dynamic Disasters in Underground Space Engineering)
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Abstract

:
In dynamic disasters involving rock slopes, wedge failures formed by complex structural discontinuities are more predominant, and the dynamic response associated with them remains a classic concern in rock slope engineering. To address this concern, this paper utilized refined modeling to analyze a wedge-shaped rock slope by inputting horizontal harmonics as loads. We conducted dynamic response analyses by varying the inclination of the structural surface on the wedge-shaped rock slope, the axial offset angle, the friction coefficient, and the configuration of the single sliding surface. The results in this paper indicate that for wedge-shaped and single-sliding-surface configurations of rock slopes, with an increase in the structural surface inclination angle, the dynamic response of the sliding body, stress distribution, excellent frequency, and spectrum values all increase. Furthermore, wedge-shaped rock slopes’ dynamic responses are more significant than those of single-sliding-surface rock slopes. For wedge-shaped rock slopes, increases in the axial offset angle and structural surface friction coefficient reduce the dynamic response, excellent frequency, and spectrum values. Meanwhile, within the context of the axial offset angle conditions in wedge-shaped rock slopes, the dynamic response, excellent frequency, and spectrum values are better than the variations in the structural surface friction coefficient. Under the influence of these factors, stress concentration occurs at the sliding fronts of rock slopes.

1. Introduction

The western region of China features numerous mountains and valleys harboring a significant number of rock slopes. Simultaneously, China is situated between the Circum-Pacific Seismic belt and the Mediterranean–Himalayan Seismic Belt. The instability of rock slopes goes beyond arc sliding and planar sliding. Specifically, it includes wedge sliding caused by complex structural discontinuities. The dynamic disaster issues associated with wedge-shaped rock slopes pose a significant challenge to advancing underground spatial engineering. Consequently, to avert the dynamic disasters posed by such slopes and ensure the secure construction and operation of infrastructure like highways, railways, and underground spatial projects, a thorough investigation into the dynamic response of wedge-shaped rock slopes has become particularly crucial [1,2,3,4,5,6].
In dynamic disasters, the instability of rock slopes primarily encompasses arc sliding, planar sliding, and wedge sliding. A substantial number of scholars have researched the mechanisms of the arc and planar sliding failure modes, as exemplified by the work of Fan [7], who used the Hilbert–Huang transform signal processing technique to analyze the time–frequency–amplitude characteristics of earthquake waves on rock slopes. By calculating the instantaneous seismic safety factor, it was determined that the upper part of a slope is more susceptible to damage. Wang [8] employed refined modeling and shake table experimentation techniques to investigate the dynamic response characteristics of steep loess slopes subjected to seismic loading conditions. Zang [9] established a physical model based on the similarity ratio to analyze rock slopes’ seismic responses and progressive failure characteristics through shaking table tests. In addition, many scholars have conducted numerical simulations to study rock-slope failure mechanisms and processes [10,11,12,13,14]. Simultaneously, the influence of seismic waves on rock slopes has been analyzed. Hassan [15] used the discrete element method to study the dynamic responses of slopes under different seismic amplitude and frequency conditions, summarizing the essential characteristics of seismic wave propagation, motion amplification factors, and resonance phenomena. Chen [16] conducted vibration table tests on a physical model to simulate the stability and failure process of anti-dip rock slopes under different seismic loads. It was found that a low frequency and a high PGA resulted in greater degrees of damage, and the aspect ratio influenced the transfer of influence forces. The sequence of motion for anti-dip rock slopes under seismic loads was also determined. Wang [17] analyzed the dynamic responses of underground chambers within rock formations to seismic quality factors and P-wave incidence angles. In addition, research has also been conducted on surface terrain dynamics using the Boundary Element Method (BEM). For example, Panji [18] utilized the BEM approach to analyze the surface motion of arbitrarily shaped impact craters subjected to incident planar shock waves. Panji [19] also employed the BEM method to perform a dynamic analysis of inclined SH-waves propagating over a semi-sine-shaped alluvial hill above a subsurface circular cavity. The application of continuity boundary conditions to analyzing examples showed that material properties and underlying spatial inhomogeneities significantly affect surface responses.
Regarding the failure modes and the stability analysis of wedge failures, Zhang [20] studied the stability of wedge rock wedges and tension cracks during the dynamic evolution of groundwater through model experiments and a case analysis. Mao [21] conducted finite element studies on the failure mechanism of slope wedge failures. It was revealed that the rear edge of a slope experienced tensile loss. In contrast, the front edge and lateral sliding surfaces underwent shear collapse, with the main sliding surface’s continuity and strength determining the wedge slope’s stability. Zhang [22] investigated the influence of freeze–thaw cycles and hydraulic conditions on wedge landslides. However, research concerning the seismic aspects of wedge-shaped rock slopes is scarce. Xu [23] conducted an investigation into the behavior of slope wedges under both static and seismic conditions using refined modeling techniques, indicating that with an increase in the reduction coefficient, wedge-shaped slopes tend toward instability and failure.
Based on the above research, a common scenario has emerged in dynamic rock-slope disasters where wedge-shaped failures are often the result of complex structural surface interactions. Consequently, dynamic disasters involving these wedge-shaped rock slopes (Figure 1) have become a matter of more concern in rock-slope management. Nevertheless, conventional research predominantly centers on the instability of rock slopes formed with a single sliding surface. Simultaneously, there is a dearth of research concerning the instability of wedge-shaped formations resulting from the cutting of complex structural surfaces and the dynamic responses of such slopes. This study established a refined model of wedge-shaped rock slopes. Dynamic response analyses were carried out by systematically varying parameters such as the structural surface inclination, axial offset angle, friction coefficient, and single-sliding-surface configuration on a wedge-shaped rock slope. This research holds significance for mitigating dynamic disasters on wedge-shaped slopes and provides insights for similar engineering designs and studies.

2. Establishment of Dynamic Analysis Models

2.1. Model Establishment

The foundational scenario is outlined as follows by employing refined modeling in conjunction with an illustrative case of rock slopes on both sides of a highway in Jiangbei District, Chongqing. The geological composition consists mainly of loose fill, silty clay, and sandstone. Most of it is covered by Quaternary deposits, with some areas showing bare rock formations. The roadbed along the route is in good condition and has no adverse geological conditions. The terrain exhibits slopes ranging from 0° to 65°, with occasional sections inclining up to 75°, resulting in an average slope gradient of approximately 60°. Due to the influence of the roadbed, one side of the road has higher rock slopes that can reach up to a height of 50 m.
According to the site’s terrain features, lithology, and structural characteristics, a refined wedge-shaped rock slope model was established using Abaqus 2021 [24] for a slope with a height of 50 m and a slope angle of 60°. As seen in Figure 2, a model of wedge-shaped, single-sliding-surface, and axial offset wedge-shaped rock slopes was established. For static boundary conditions, the model’s bottom boundary was fixed with Ux = Uy = Uz = 0, and the lateral boundaries were constrained with Uz = 0. The model’s boundary conditions for dynamic analyses included a free surface at the top, the release of constraints in the x-direction at the bottom boundary, and lateral boundaries set to Uy = Uz = 0 [15,25]. Harmonic loading with acceleration time history was applied, and mechanical damping was implemented using bulk viscosity. This helped reduce numerical oscillations and enhanced the accuracy of the computed results. Therefore, the parameters were set as a linear bulk viscosity parameter of 0.06 and a quadratic bulk viscosity parameter of 1.2 [24]. After conducting geological surveys and analyzing the geotechnical layer properties, it was determined that sandstone is the predominant rock material in the surrounding area. The specific parameters are outlined in Table 1 [26]. The site exhibits two structural surfaces as cracks, simplified as through-going joint planes in the model analysis. Based on the physical properties of the rock mass, the mechanical parameters for these surfaces were as follows: cohesion of 2.87 MPa, an internal friction angle of 20°, and a friction coefficient of tan 20°. The refined model used structural optimization techniques to adopt an elastic–plastic constitutive model, a Drucker–Prager failure criterion, and mesh division [27]. Simultaneously, this model performed a sensitivity analysis on element types to assess the suitability of a tetrahedral mesh that offers increased automation, efficiency, reliability, and geometric versatility. This paper mainly analyzed the dynamic response by varying the inclination of the structural surface on the wedge-shaped rock slope, the axial offset angle, the friction coefficient, and the configuration of the single sliding surface. Detailed parameters are available in Table 2.

2.2. Selection of Harmonic Waves

Seismic waves possess immense energy and can severely damage rock slopes. Based on prior research experience, it is generally observed that the horizontal shear strength values of buildings are lower than their vertical shear strength values, and horizontal seismic waves tend to induce more significant structural damage [28]. Therefore, this study solely focused on horizontal seismic waves. Furthermore, harmonic waves with the same peak value and frequency contain more energy and are more intense than seismic waves, rendering their calculated outcomes more secure and reliable [29,30]. Thus, this study adopted a horizontal harmonic wave with a frequency of 6 Hz as the seismic load, as depicted in Figure 3.

3. Analysis of the Dynamic Response in Rock Slopes

3.1. Analysis of Displacement and Acceleration Response Characteristics

As shown in Figure 4a, the inclination angle of the structural surface of the slide body increased, and the displacement of each point on the slide gradually increased. The displacement increased from 0.26 m to 3.6 m, with its growth rate increasing and decreasing as it reached the dividing line at 45°. The displacement at the front of the slide, i.e., at monitoring point 15, was slightly higher than at the other monitoring points. In Figure 4b, the acceleration increased with an increase in the inclination angle. Before monitoring point 12, there was a slight upward amplification trend in the acceleration along the slope surface. At the front end of the sliding body, specifically at monitoring point 15, due to the vulnerability within the sliding body and the tip effect, the acceleration experienced a sudden rise. The acceleration increased from 334 m/s2 to 1310 m/s2, with the angle reaching 55°, marking the maximum acceleration. It is evident that as the inclination angle of the structural surface increased, the displacement and acceleration of the sliding body also increased, leading to an amplification of its dynamic response.
As shown in Figure 5a, with the increase in the inclination angle of the structural surface of the single-sliding-surface body, the displacement of the sliding body gradually increased. The displacement value increased from 0.38 m to 3.60 m, and 45° was a demarcation line of the growth rate for first increasing and then decreasing. However, the growth rate was slightly smaller than for the wedge-shaped body, and the displacement of the front end of the sliding body had an increasing trend and was also smaller than that of the wedge-shaped body. Figure 5b shows that the acceleration increased with the increase in the inclination angle of the single-sliding-surface body. Small fluctuations caused the accelerations at each point before monitoring point 8. They were slightly amplified as the slope increased. However, the acceleration at the front end of the sliding body increased abruptly from 256 m/s2 to 834 m/s2 due to structural weakness and the tip effect. The acceleration response at the front end of the sliding body was more significant than that in the upper part of the slope. Energy accumulated at the front end, resulting in a peak value at monitoring point 15.
Wedge-shaped and single-sliding-surface rock slopes had similar dynamic responses with consistent displacement responses. However, the wedge-shaped bodies’ acceleration responses were greater than those of single-sliding-surface bodies. The structural form at the front end of a wedge-shaped body is more susceptible to damage. Therefore, it can be seen that under the action of harmonic waves, the degree of destruction for wedge-shaped bodies is more significant than that for single-sliding-surface bodies.
Figure 6a shows that as the wedge-shaped body’s axial offset angles increased, the displacement gradually decreased, and the trend accelerated. The displacement decreased from 2.50 m to 2.06 m. As the offset angle increased, the displacement of the front end of the slide accelerated. The displacement on the slope surface was smaller than that of the front end of the body, and all maximum displacements occurred at the front end. As shown in Figure 6b, the acceleration at each point decreased gradually in fluctuation as the angle of axial offset increased in the wedge-shaped body. However, it showed an amplification trend at the top of the slope. After monitoring point 12, the acceleration rapidly increased to reach its peak value, which then decreased from 1340 m/s2 to 523 m/s2. Energy convergence was visible at the front of the slide body, causing the front end to be the first to sustain damage.
Figure 7a demonstrates that displacement gradually decreased as the friction coefficient increased. The displacement ranged from 3.8 m to 2.1 m. Notably, the sliding front exhibited a more significant displacement response than the slope. Furthermore, as the friction coefficient increased, the displacement response of the sliding front gradually intensified, with a maximum rise of 0.1 m. As seen in Figure 7b, the acceleration before monitoring point 12 fluctuated between 50 m/s2 and 180 m/s2 and decreased with the increase in the friction coefficient. The overall amplification trend was identified at monitoring point 1 at the top of the slope. At the front of the slide, the acceleration increased abruptly due to the weak point at the edge and the tip effect, reaching up to 680 m/s2. It can be seen that the friction coefficient of the structural surface had a cause that affected the stability of the wedge-shaped body.
Under the axial offset and structural surface friction coefficient conditions, the dynamic response decreased as the offset angle and friction coefficient increased. The dynamic response at the front end of the slider slightly increased. Still, the displacement and acceleration dynamic responses at the front end of the slider due to axial offset were greater than those due to the friction coefficient.

3.2. Slip-Body Structural Surface Stress

Continuing the analysis, we examined the stress distribution on the structural surface of the sliding body when subjected to harmonic waves. Additionally, we investigated how changes in parameters such as the inclination angle of the wedge-shaped body, the single-sliding-surface body’s structural surface, the axial offset of the wedge-shaped body, and the friction coefficient affected the stress cloud on this surface.
As depicted in Figure 8, the stress distribution on the structural surface changed with variations in the inclination angle of the wedge-shaped body. As the inclination angle of the structure’s surfaces increased, the stress on both surfaces gradually increased. The maximum stress values were between 43 MPa and 68.3 MPa, and the stress was mainly concentrated at both ends of the intersection line between the two structural surfaces. With an increasing inclination angle, the front-end stress spread to both sides and gradually expanded its range. Eventually, the stress was mainly concentrated at the edges and the front end of the wedge-shaped body. Please refer to Table 3 for specific parameters. As the inclination angle of the wedge-shaped body increased, stress concentration occurred at the front end and both side edges of the wedge-shaped body, and the range gradually expanded. Therefore, it was more likely for the front end and both side edges of the wedge-shaped body to experience damage.
As depicted in Figure 9, the stress distribution on the structural surfaces of the single-sliding-surface body changed with varying inclination angles. As the inclination angle increased, the stress values gradually increased by between 1.1 MPa and 75.2 MPa. The maximum stress values were between 44 MPa and 75.2 MPa, as detailed in Table 4. Furthermore, the maximum stress values were mainly distributed at the lower end of the structural surface, and both the stress magnitude and range increased gradually with an increasing inclination angle. When the inclination angle of the structural surface exceeded 30°, stress concentration occurred at the middle position of the lower end. Figure 9e depicts it spread to both sides. It can be seen that with an increasing inclination angle, the front end of the sliding body experienced stress concentration first, and the stress concentration spread to both sides.
With the increases in the inclination angles in the wedge-shaped bodies and single-sliding-surface bodies, the stress gradually increased and the range expanded. However, in the case of a single-sliding-surface inclination angle, due to different structural forms, stress was concentrated at the midpoint of the front end of the sliding body. On the other hand, for wedge-shaped bodies, stress was mainly focused on both sides of the front end and along the edges of the structural surface.
As depicted in Figure 10, the stress distribution on the structural surfaces changed when altering the axial offset angle of the wedge-shaped body. When the offset was 5°, the maximum stress on the structural surface reached 114 MPa at the front end of the sliding body. As shown in Table 5, as the offset angle increased, the maximum stress decreased and stabilized around 64 MPa. The maximum values of stress on the structural surface were between 63 MPa and 114 MPa, mainly concentrated at the lower end of the axis and the edges of the sliding body. As the axis deviated, the maximum stress value moved towards the lower end of the axis and the left edge of the structural surface, with stress concentration occurring at the lower end of the axis. As the wedge-shaped body’s axial offset angle increased, the stress range on the structural surface increased, and stress concentration occurred at the front end of the sliding body. The rock mass at the axis hindered the movement of the sliding body.
As depicted in Figure 11, the stress distribution on the structural surfaces of the wedge-shaped body varied with changes in the friction coefficient. As the friction coefficient increased, the stress values fluctuated between 1.3 MPa and 76.4 MPa. The maximum stress values ranged from 59.2 MPa to 76.4 MPa, with a distribution mainly concentrated at the lower end of the sliding body axis and both sides of the front edge of the sliding body. With an increase in the friction coefficient, the range of the stress distribution gradually expanded, as shown in Table 6. It can be seen that changing the friction coefficient on the structural surface had a relatively small impact on the stress magnitude of the structural surface. Simultaneously, stress concentration occurred at the front end of the sliding body, making it highly susceptible to damage.
When comparing the effects of the axial offset angle and friction coefficient on the structural surface, it was observed that the stress experienced by the structural surface was slightly higher than that associated with the friction coefficient under the axial offset parameters. Under the axial offset parameters, the stress concentration occurred along the axis at the front end of the sliding body. Conversely, when considering friction coefficient parameters, stress was primarily concentrated at the front end and the edges of the sliding body on the structural surface.

3.3. Characterization of Slip-Body Acceleration Spectra

In this study, we employed the Fourier transform [31,32] to analyze the dynamic responses of several parameters, including the inclination angle of a wedge-shaped body and a single-sliding-surface body, the axial offset of a wedge-shaped body, and the friction coefficient. The analysis was based on acceleration time history data obtained from sliding bodies subjected to harmonic excitation. Its specific focus was on analyzing acceleration data collected at monitoring point 1, situated at the crest of the slope.
In Figure 12, the Fourier spectra of the acceleration at monitoring point 1 is depicted for various inclination angles of the wedge-shaped body. The high-frequency range mainly concentrated on frequencies ranging from 0 to 200 Hz. At an inclination angle of 30°, the excellent frequencies at monitoring point 1 were approximately 26 Hz, 67 Hz, and 130 Hz. The spectrum value reached a peak of 23 m/s2 in the frequency band centered around 67 Hz. More detailed parameters can be found in Table 7. As the wedge-shaped body’s inclination angle increased, the excellent frequency and the amplitude value within the Fourier spectrum of the acceleration response gradually increased. However, this increase was followed by a subsequent decrease.
In Figure 13, the Fourier spectra of acceleration are depicted for a single-sliding-surface body at different inclination angles. At an inclination angle of 20°, the excellent frequencies observed at monitoring point 1 were primarily concentrated around 37 Hz, 64 Hz, and 101 Hz. The measured spectrum value at 101 Hz was recorded as 12 m/s2. For specific parameters, please refer to Table 8. As the inclination angle of the single-sliding-surface body increased, there was a gradual increase in the excellent frequency of the model’s acceleration response in the Fourier spectrum. However, when the inclination angle reached 50°, the excellent frequency and spectrum value decreased. This decrease was attributed to the fact that a bigger degree of inclination angle reduced the thickness and mass of the sliding body, resulting in a diminished impact on the propagation of harmonics within it.
The excellent frequency at the top of the slope changed significantly when using a seismic wave with a frequency of 6 Hz. As the inclination angle of the structural surface increased, there was a more significant fluctuation in the excellent frequency in the Fourier spectrum of the sliding body. The spectrum values gradually shifted from low frequencies to high frequencies, resulting in varying degrees of fluctuations for both types of bodies. After comparing Figure 12 and Figure 13, as well as Table 7 and Table 8, it can conclude that single-sliding-surface bodies exhibit fewer minor frequency fluctuations compared to wedge-shaped bodies.
In Figure 14, the plots display the Fourier spectra for various axial offset angles. The high-frequency components were mainly concentrated in the 0–250 Hz frequency range. At an axial offset of 5°, the excellent frequencies at monitoring point 1 were primarily centered around 24 Hz, 96 Hz, and 157 Hz. The spectrum value at 157 Hz reached its highest point at 26 m/s2 and had the most significant proportion within the frequency band of 96 Hz. For specific parameters, please refer to Table 9. As the angle of axial offset increased, both the excellent frequency and spectrum values decreased in the Fourier spectrum analysis of the wedge-shaped body’s acceleration response. This finding confirmed our conclusion regarding its displacement variation.
The Fourier spectra in Figure 15 demonstrate that the main high-frequency components were concentrated between 0 and 250 Hz. As the friction coefficient of the contact surface increased, the excellent frequency of the Fourier spectrum response gradually decreased. As seen in Table 10, at a friction coefficient of tan5° for the contact surface, the excellent frequencies at monitoring point 1 were primarily around 89 Hz, 143 Hz, and 197 Hz. The spectrum value reached its maximum at 143 Hz, measuring 21 m/s2. It held the highest proportion within this frequency range. It is evident that as the friction coefficient of the contact surface increases, both the excellent frequency and maximum spectrum values decrease for the acceleration response in wedge-shaped bodies.
When the axial offset angle and friction coefficient of the wedge-shaped body were altered, significant changes occurred in the frequency and spectrum values of the rock slope when subjected to harmonic excitation. As these parameters increased, higher spectrum values gradually transitioned from high to low frequencies, decreasing the sliding body’s excellent frequency band and spectrum values. After comparing Figure 14 and Figure 15, along with the data in Table 9 and Table 10, it is apparent that the fluctuations in the Fourier spectrum responses were more noticeable for the axial offset angle than for the structural surface friction coefficient.

4. Conclusions and Discussion

This study aimed to perform dynamic response analyses on various wedge-shaped rock slopes, exploring structural surface inclination angles, axial offset angles, and friction coefficients. It also examined structural surface inclinations in the context of single-sliding-surface rock slopes. The results in this paper indicate the following:
(1)
For wedge-shaped rock slopes, the sliding body’s displacement and acceleration increase as the inclination angle of the structural surface increases within a specific range. At the front end of the sliding body, a tip effect leads to a sudden acceleration increase. The stress on the structural surface gradually rises, resulting in stress concentration at the front end of the sliding body. Concurrently, the frequency of excellence at the summit of the slope rises, and the spectrum value increases.
(2)
Single-sliding-surface rock slopes within a specific range of structural surface inclination angles exhibit behavior similar to wedge-shaped rock slopes. Nonetheless, the displacement, acceleration response, structural surface stress, excellent frequency, and spectrum values at the crests of wedge-shaped rock slopes slightly surpass those of single-sliding-surface rock slopes. Consequently, under identical conditions, the stability of wedge-shaped rock slopes is marginally inferior to that of single-sliding-surface rock slopes.
(3)
In the context of wedge-shaped rock slopes, it was observed that within a certain range of axial offset angles, an increase in the axial offset angle of the wedge-shaped body results in a reduction in the displacement and acceleration dynamic responses of the sliding body, along with a decrease in the stress applied to the structural surface. This phenomenon is attributed to the hindrance caused by the movement of the sliding body as the offset angle increases along the axis, leading to stress concentration at the central axis of the sliding body’s front end. Consequently, this condition leads to a notable reduction in the frequency and a decrease in spectrum values.
(4)
Within a particular range of friction coefficients on the structural surfaces of wedge-shaped rock slopes, an increase in the friction coefficient results in a decrease in the dynamic response of the wedge-shaped body in terms of displacement and acceleration. The stress variation on the structural surface remains relatively modest, with the primary stress concentration occurring along the axis of the structural surface. Additionally, the value of the frequency reduction spectrum decreases accordingly. In this scenario, it becomes evident that the dynamic response of the axial offset angle in the wedge-shaped rock slope surpasses that of the structural surface’s friction coefficient.
Despite having obtained valuable research findings in this study, there are still some deficiencies. This paper focused on examining the dynamic responses of wedge-shaped rock slopes subjected to harmonic waves using numerical simulations. To enhance optimization and conduct a more comprehensive analysis, future research should investigate the dynamic responses of wedge-shaped rock slopes under the influence of harmonic waves and their changing behavior. This can be achieved by combining physical model tests and relevant field engineering practices.

Author Contributions

Conceptualization, Y.D., W.H. and B.H.; software, W.X.; validation, Y.D., W.H. and H.L.; investigation, W.X. and X.D.; data curation, Y.D., W.X. and B.H.; writing—original draft preparation, W.X. and X.D.; writing—review and editing, Y.D., W.H., B.H. and H.L.; visualization, W.H. All authors have read and agreed to the published version of the manuscript.

Funding

This study was funded by the Natural Science Research Project of Colleges and Universities in Anhui Province (grant No. 2022AH050963); the National Natural Science Fund Pre Project of Anhui Polytechnic University (grant No. 2019yyzr08); the Start Fund of Talent Introduction of Anhui Polytechnic University (grant No. 2023YQQ016); and the National Innovation and Entrepreneurship Training Project for College Students (grant No. 202210363107).

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Schematic diagram of wedge-shaped rock slope failure.
Figure 1. Schematic diagram of wedge-shaped rock slope failure.
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Figure 2. Rock slope models and boundary conditions: (a) wedge-shaped rock slope, (b) single-sliding-surface rock slope, and (c) axial offset angle rock slope.
Figure 2. Rock slope models and boundary conditions: (a) wedge-shaped rock slope, (b) single-sliding-surface rock slope, and (c) axial offset angle rock slope.
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Figure 3. Acceleration histories of the harmonic wave.
Figure 3. Acceleration histories of the harmonic wave.
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Figure 4. Dynamic responses of the wedge-shaped body inclination angles: (a) displacement and (b) acceleration.
Figure 4. Dynamic responses of the wedge-shaped body inclination angles: (a) displacement and (b) acceleration.
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Figure 5. Dynamic responses of the single-sliding-surface body inclination angles: (a) displacement and (b) acceleration.
Figure 5. Dynamic responses of the single-sliding-surface body inclination angles: (a) displacement and (b) acceleration.
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Figure 6. Dynamic responses of the wedge-shaped body with axial offset angles: (a) displacement and (b) acceleration.
Figure 6. Dynamic responses of the wedge-shaped body with axial offset angles: (a) displacement and (b) acceleration.
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Figure 7. Dynamic responses of the wedge-shaped body for the friction coefficients of structural surfaces: (a) displacement and (b) acceleration.
Figure 7. Dynamic responses of the wedge-shaped body for the friction coefficients of structural surfaces: (a) displacement and (b) acceleration.
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Figure 8. Stress fields of the inclination angles of structural surfaces in the wedge-shaped body: (a) 20°, (b) 30°, (c) 45°, (d) 50°, and (e) 55°.
Figure 8. Stress fields of the inclination angles of structural surfaces in the wedge-shaped body: (a) 20°, (b) 30°, (c) 45°, (d) 50°, and (e) 55°.
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Figure 9. Stress fields of the inclination angles of structural surfaces for the single-sliding-surface body: (a) 20°, (b) 30°, (c) 45°, (d) 50°, and (e) 55°.
Figure 9. Stress fields of the inclination angles of structural surfaces for the single-sliding-surface body: (a) 20°, (b) 30°, (c) 45°, (d) 50°, and (e) 55°.
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Figure 10. Stress fields of the axial offset angles of structural surfaces for a wedge-shaped body: (a) 5°, (b) 10°, (c) 15°, (d) 20°, and (e) 25°.
Figure 10. Stress fields of the axial offset angles of structural surfaces for a wedge-shaped body: (a) 5°, (b) 10°, (c) 15°, (d) 20°, and (e) 25°.
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Figure 11. Stress fields of the friction coefficients of structural surfaces for the wedge-shaped body: (a) tan5°, (b) tan10°, (c) tan15°, (d) tan20°, and (e) tan25°.
Figure 11. Stress fields of the friction coefficients of structural surfaces for the wedge-shaped body: (a) tan5°, (b) tan10°, (c) tan15°, (d) tan20°, and (e) tan25°.
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Figure 12. The Fourier spectra of acceleration for the wedge-shaped body inclination angles: (a) 30°, (b) 45°, (c) 50°, and (d) 55°.
Figure 12. The Fourier spectra of acceleration for the wedge-shaped body inclination angles: (a) 30°, (b) 45°, (c) 50°, and (d) 55°.
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Figure 13. The Fourier spectra of acceleration for single-sliding-surface body inclination angles: (a) 20°, (b) 30°, (c) 45°, and (d) 50°.
Figure 13. The Fourier spectra of acceleration for single-sliding-surface body inclination angles: (a) 20°, (b) 30°, (c) 45°, and (d) 50°.
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Figure 14. The Fourier spectra of acceleration for the wedge-shaped body with axial offset angles: (a) 5°, (b) 10°, (c) 15°, and (d) 25°.
Figure 14. The Fourier spectra of acceleration for the wedge-shaped body with axial offset angles: (a) 5°, (b) 10°, (c) 15°, and (d) 25°.
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Figure 15. The Fourier spectra of acceleration for the wedge-shaped body with friction coefficients of structural surface: (a) tan5°, (b) tan10°, (c) tan20°, and (d) tan25°.
Figure 15. The Fourier spectra of acceleration for the wedge-shaped body with friction coefficients of structural surface: (a) tan5°, (b) tan10°, (c) tan20°, and (d) tan25°.
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Table 1. Mechanical parameters of the homogeneous slope [26].
Table 1. Mechanical parameters of the homogeneous slope [26].
ParametersDensity (kg/m3)Cohesion (MPa)Friction
Angle (°)		Young’s
Modulus (GPa)		Poisson’s Ratio
Value2.472.87402.90.25
Table 2. Parameters under various influencing factors.
Table 2. Parameters under various influencing factors.
CategoriesWedge-Shaped Body Inclination AnglesSingle-Sliding-Surface Body Inclination AnglesAxial Offset AnglesStructural Surface Friction Coefficients
Parameters20°20°5°tan5°
30°30°10°tan10°
45°45°15°tan15°
50°50°20°tan20°
55°55°25°tan25°
Table 3. Stress distributions of the wedge-shaped body inclination angles.
Table 3. Stress distributions of the wedge-shaped body inclination angles.
ParametersMaximum Stress (MPa)Stress Distribution
20°54.1 MPaFront end of the slide body
30°52.4 MPaFront end of the slide body
45°67.8 MPaFront end of the slide body
50°43.1 MPaFront end and both sides of the slide body
55°68.4 MPaTop left of the slide body
Table 4. Stress distributions of the single-sliding-surface body inclination angles.
Table 4. Stress distributions of the single-sliding-surface body inclination angles.
ParametersMaximum Stress (MPa)Stress Distribution
20°44.8 MPaMidline at the top of the slide body
30°55.9 MPaMidline at the front end of the slide body
45°75.3 MPaMidline at the front end of the slide body
50°68.3 MPaMidline at the front end of the slide body
55°47.6 MPaThe right side of the front end of the slide body
Table 5. Stress distributions of axial offset angles.
Table 5. Stress distributions of axial offset angles.
ParametersMaximum Stress (MPa)Stress Distribution
5°114.2 MPaFront end of the slide body
10°64.6 MPaThe left side of the front end of the slide body
15°82.5 MPaFront end of the slide body
20°63.4 MPaMidline at the front end of the slide body
25°64.9 MPaThe left side of the front end of the slide body
Table 6. Stress distributions of friction coefficients.
Table 6. Stress distributions of friction coefficients.
ParametersMaximum Stress (MPa)Stress Distribution
tan5°59.9 MPaFront end of the slide body
tan10°76.4 MPaMidline at the front end of the slide body
tan15°67.5 MPaMidline at the front end of the slide body
tan20°69.4 MPaMidline at the front end of the slide body
tan25°69.5 MPaMidline at the front end of the slide body
Table 7. Frequency and spectrum values for the wedge-shaped body inclination angles.
Table 7. Frequency and spectrum values for the wedge-shaped body inclination angles.
ParametersExcellent Frequency BandMaximum-Amplitude Frequency
30°26 Hz, 67 Hz, 130 Hz23 m/s2 at 67 Hz
45°32 Hz, 120 Hz, 178 Hz29 m/s2 at 120 Hz
50°32 Hz, 160 Hz, 210 Hz26 m/s2 at 160 Hz
55°43 Hz, 122 Hz, 220 Hz23 m/s2 at 220 Hz
Table 8. Frequency and spectrum values for single-sliding-surface body inclination angles.
Table 8. Frequency and spectrum values for single-sliding-surface body inclination angles.
ParametersExcellent Frequency BandMaximum-Amplitude Frequency
20°37 Hz, 64 Hz, 101 Hz12 m/s2 at 101 Hz
30°45 Hz, 93 Hz, 123 Hz40 m/s2 at 123 Hz
45°33 Hz, 120 Hz, 183 Hz23 m/s2 at 120 Hz
50°40 Hz, 104 Hz, 160 Hz20 m/s2 at 104 Hz
Table 9. Frequency and spectrum values of axial offset angles.
Table 9. Frequency and spectrum values of axial offset angles.
ParametersExcellent Frequency BandMaximum-Amplitude Frequency
5°24 Hz, 96 Hz, 157 Hz26 m/s2 at 157 Hz
10°24 Hz, 87 Hz, 169 Hz20 m/s2 at 169 Hz
15°20 Hz, 123 Hz, 159 Hz24 m/s2 at 123 Hz
25°22 Hz, 114 Hz, 157 Hz18 m/s2 at 114 Hz
Table 10. Frequency and spectrum value friction coefficients.
Table 10. Frequency and spectrum value friction coefficients.
ParametersExcellent Frequency BandMaximum-Amplitude Frequency
tan5°89 Hz, 143 Hz, 197 Hz21 m/s2 at 143 Hz
tan10°87 Hz, 130 Hz, 197 Hz26 m/s2 at 130 Hz
tan20°32 Hz, 93 Hz, 128 Hz24 m/s2 at 128 Hz
tan25°16 Hz, 85 Hz, 117 Hz25 m/s2 at 117 Hz
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Du, Y.; Xu, W.; Han, W.; Huang, B.; Liu, H.; Du, X. Dynamic Response Analysis of Wedge-Shaped Rock Slopes under Harmonic Wave Action. Buildings 2023, 13, 2623. https://doi.org/10.3390/buildings13102623

AMA Style

Du Y, Xu W, Han W, Huang B, Liu H, Du X. Dynamic Response Analysis of Wedge-Shaped Rock Slopes under Harmonic Wave Action. Buildings. 2023; 13(10):2623. https://doi.org/10.3390/buildings13102623

Chicago/Turabian Style

Du, Yihan, Wenzhi Xu, Wei Han, Bo Huang, Hui Liu, and Xuze Du. 2023. "Dynamic Response Analysis of Wedge-Shaped Rock Slopes under Harmonic Wave Action" Buildings 13, no. 10: 2623. https://doi.org/10.3390/buildings13102623

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