Seismic response analysis of jointed rock slope under pulse-like ground motions

Because of the directional effect, energy concentration and hanging wall effect of pulse-like ground motions, it is easy to cause rock fall, collapse, and landslide of jointed rock slope. To reveal the deformation law and failure model of jointed rock slope under pulse-like ground motions, a discrete element numerical model of jointed rock slope in the northern Yunnan province was established to study the acceleration, dynamic displacement, and slope failure modes of jointed rock slope under bi-directional pulse-like ground motions. The results show that the period of the pulse signal extracted from the RSN147 seismic wave is 2.03s, and the energy ratio is 70.3%. Under the action of RSN147, the time of peak acceleration and peak displacement of the slope are consistent with the peak pulse velocity, and the predominant frequency is also the same. The low-frequency component of ground motion causes the global displacement of the slope. The peak dynamic displacement of the slope is about 5~6cm, but the maximum residual displacement is only 11.3% of the peak dynamic displacement, so the slope is stable. The strength reduction method was used to analyze the overall stability of the slope. The slope displacement �rst appears at the slope toe. With the continuous development of deformation, the slope is in a critical state, and the strong weathering rock mass slides along the fractures. The research results can provide support for the construction of jointed rock slopes in strong weathered areas.


Introduction
With the development of the strategy for developing transportation, the strategy for western expansion, and the Belt and Road strategy, the infrastructure construction program in the western region is gradually increasing (Wei et al., 2023).The western part of China is located on the Eurasian plate.Under internal and external actions for a long time, a large number of jointed rock slopes have developed in the nearfault region.Near-fault pulse-like ground motions with short duration, long period, and large peak velocity have signi cant slip effects, hanging wall effects, and directional effects (Song et al., 2014).Under the action of pulse-like ground motions, the joint rock slope has caused rock falls, collapse landslides, and other geological disasters, which brings great hidden dangers to the construction and operation of infrastructure along the line.As a kind of natural geological material, the rock mass is usually cut by discontinuous fractures, and its mechanical properties are characterized by discontinuity, heterogeneity, and anisotropy.The fractures of rock mass slope often play a controlling role in the stability of rock mass and its deformation and failure mode (Li et al., 2022).
The research on jointed rock slopes mainly focuses on the mechanical response, failure mode, and disaster mechanism under internal and external actions.Wang et al. (2006) carried out an analysis of the propagation effect of seismic waves in the jointed rock mass, and the transmission coe cient of waves is mainly related to joint spacing and wavelength.The work mentioned above mainly focuses on the dynamic characteristics and disaster mechanism of jointed rock slopes and reveals the deformation and instability mechanism of rock slopes.However, the established laboratory test model and numerical model of jointed rock slope contain a limited number of structural planes, which cannot accurately re ect the fracture characteristics of jointed rock slope under internal and external dynamic actions, and then cannot reveal the instability evolution process of jointed rock slope.To research the in uence of random fractures formed by internal and external actions on the stability of jointed rock slope, a three-dimensional numerical model of jointed rock slope was established by using 3DEC discrete element software, and the dynamic response rule and failure model of jointed rock slope under the action of pulse ground motion were investigated by various methods, to further reveal the deformation and instability mechanism of jointed rock slope.

Geological conditions of the study area
The research site is a jointed high steep rock slope in northwest Yunnan province, with an elevation of 1500m ~ 1650m.The site survey results show that the slope is located in a wing of the synclinal geological structure.In the tectonic process, the strati ed structural plane is developed in the strata, the structural plane spacing is 5m, and the slope angle is 47°.The lithology is dominated by diorite, and the main minerals are plagioclase, hornblende, quartz, and biotite.The fracture of the slope is developed, the fracture surface is rough, and the thickness of the strong weathering layer is about 20m.The site is close to a fault, and under the action of pulse ground motion, the jointed rock mass is easy to slide along the structural plane, resulting in rockfall and local landslide.After eld investigation, the occurrence of the layered structural plane is 153°∠85°, and the strong weathering layer of slope mainly consists of 3 groups of fracture, which are listed in Table 1. 3 Numerical calculation

Analysis of pulse-like ground motions
The typical characteristics of near-fault pulse-like ground motions include strong energy concentration, permanent ground displacement, directional effect, large velocity pulse, and hanging wall effect.According to the earthquake intensity(eight degree) and fault type of the site (strike-slip fault), RSN147 ground motion was selected for analysis.Since the ground motion is mainly a low-frequency component (0 ~ 5Hz) that accounts for most of the ground motion energy, the Butterworth low-pass lter was used for processing, and the maximum frequency is 20Hz.To avoid distortion of calculation results, linear baseline correction was adopted.The acceleration time history, corresponding Fast Fourier Transform (FFT) spectrum curve, and velocity time history of the processed RSN147 are shown in Fig. 1.
The velocity time history of RSN147 ground motion has an obvious bi-directional pulse effect.The maximum positive velocity is 0.32m/s, the corresponding time is 3.57s, and the maximum negative velocity is 0.29m/s, the corresponding time is 3.02s.Due to the wide frequency band of ground motion, the amplitude of pulse velocity is identical to the peak value of original signal (Alavi et al., 2004).A simpli ed pulse mathematical model was adopted to t the velocity data, and a pulse time history that can epitomize the pulse performance of the original signal was obtained, and the peak point method was used to determine the pulse signal period.The mathematical model of the pulse is as follows ( Where V p is the peak velocity of the pulse-like ground motion, T p is the period, N c is the number of pulse signal, T pk is the corresponding time of peak velocity, is the phase, with the range of -60°~105°.The number of pulses, phase, and period were gradually changed, and the least square method (Eq.2) was used to determine the pulse function.When α is the smallest, the corresponding function is the pulse signal of the original signal.The pulse signal of RSN147 ground motion was extracted, as shown in Fig. 2 . The period of the pulse signal is 2.03s.

Numerical model of jointed rock slope
The research model is an anti-dip bedded jointed rock slope, and the surface 20m range is a strongly weathered jointed rock mass, with three groups of fractures.The slope was formed by 3DEC software (Fig. 3).The slope width is 210m.The length of the slope top is 1.5 times the length of the slope surface, which is 84m.The occurrence of the layered structural plane is 153°∠85°.To simulate the three groups of fracture on the surface of the slope, a three-dimensional discrete fracture network is established according to the on-site statistical data of the fractures, and the discrete fracture network is embedded into the slope to form the strongly weathered rock mass.To monitor the evolution law of deformation under the action of pulse-like ground motion, three survey lines (A, B, C) were arranged in the slope, with a total of 27 monitoring points to monitor the dynamic response of rock mass such as acceleration, displacement, and velocity.The monitoring points and detailed dimensions of the slope are described in Fig. 4.
The slope is mainly diorite, according to the engineering experience and the laboratory test results, the basical parameters of rock are listed in Table 2, and the main parameters of the structural plane and fracture are shown in Table 3.In the numerical calculation, the mohr-coulomb constitutive model was selected for diorite rock, and the coulomb friction constitutive model is used for layered structure planes and fractures.Firstly, the static calculation was carried out to obtain the initial stress under gravity.In the process of static calculation, the normal displacement constraints were applied on the four sides and the bottom gridpoints of the slope.In the process of dynamic calculation, RSN147 stress time history was applied on the bottom gridpoints of the model, and the conversion relationship between velocity and stress is shown in Equations 3 and 4. A free eld was set around the model to absorb seismic waves to avoid the re ection of seismic waves affecting the numerical calculation results, and a viscous damping boundary was set on the bottom gridpoints of the model.The model adopts local damping, the critical damping ratio is 5%, and the damping value is 0.157.

4
Where, and are the normal and tangential stresses applied at the bottom of the model respectively; is the density of rock mass.and are the speeds of the P wave and S wave respectively.and are the normal and tangential velocity components of the input ground motion respectively.

Analysis of slope acceleration
To analyze the acceleration response of the slope under pulse-like ground motions, the acceleration time histories of A4, A8, B4, B8, C4, C8 were measured, FFT is applied to the acceleration time history of monitoring points to analyze the acceleration spectrum characteristics of rock blocks in the frequency domain, as shown in Fig. 5.The peak values of acceleration and velocity of RSN147 are mainly concentrated in the 2s ~ 4s, and the main frequency is 1.1Hz.The acceleration time history of the slope monitoring points shows that the peak acceleration of the 6 monitoring points also appears in this time.
When the velocity of the pulse-like ground motion reaches the peak, the seismic energy is concentrated, the acceleration response of the slope is consistent in time.
As can be seen from the acceleration spectrum curve in Fig. 5, the predominant frequency of the 6 monitoring points is consistent with the predominant frequency of ground motion, which is 1.1Hz, indicating that the energy of pulse-like ground motion is mainly concentrated at the time of 2 ~ 4s, this is veri ed by the energy ratio of pulse signal.The acceleration energy of the six monitoring points is mainly concentrated in 0 ~ 5Hz, there is only one peak value at the three monitoring points of A4, B4, C4.A8, B8, C8 have a large elevation, and with the superposition of ground motion energy, the acceleration spectrum curve is similar to the original ground motion, with the characteristics of double peaks.
A ampli cation factor is de ned as the ratio of peak ground acceleration(PGA) of the monitoring point to point A1.The PGA ampli cation coe cients of 27 monitoring points are described in Fig. 6.The 9 points of line A are arranged on a vertical line.With the increase of elevation, the ground motion energy has a cumulative effect, and the PGA ampli cation coe cient increases gradually.In lines B and C, the magni cation coe cient of PGA generally increases with the increase of elevation.However, some monitoring points are located at the interface of the rock mass.Under the earthquake, rock blocks collide with each other, resulting in abrupt changes in PGA, and the local ampli cation effect is greater than the elevation effect.

Analysis of slope acceleration
To analyze the dynamic displacement under RSN147, the dynamic displacement time histories of A4, A8, B4, B8, C4, C8 were measured, as shown in Fig. 7.The energy of the pulse component of RSN147 can reach 70.3%, which can be released within 2s.The peak ground displacement(PGD) at the 6 monitoring points is 5 ~ 6cm, the corresponding time is the same as the time of peak velocity of the pulse signal, the slope shows a global displacement during this period.Combined with the FFT spectrum of acceleration in Fig. 5, the global displacement is mainly the results of the low-frequency component of ground motion (0 ~ 5Hz).RSN147 is a bi-directional ground motion with a short interval between a positive peak and a negative peak, and the two peaks are similar.After the peak displacement, the slope moves to the initial position, and the displacement increases in reverse.The maximum positive displacement is about 2cm.
Thereafter, the amplitude of ground motion is small and the displacement of slope uctuates near zero.
There was no signi cant difference in PGD at the 27 monitoring points, with the range of 4.5 ~ 6cm, and the PGD occurred at the time of peak pulse velocity, as shown in Fig. 8.Because of the cumulative phenomenon of seismic action and the spur effect of slope, the PGD of the three measuring lines increased gradually with the increase of elevation, and the variation of measuring line C is the largest, 21.2%.After the earthquake, the maximum residual dis-placement of the 27 monitoring points is only 11.3% of PGD, so the slope is stable.

Analysis of failure mode
The SRM was used to calculate the safety factor and deformation law of the slope in the unstable state, as shown in Fig. 9.The principle of the strength reduction method is to reduce the cohesion and internal friction angle to calculate the stability of the slope.When the deformation is in a critical state, the reduction coe cient is the safety factor of the slope.The safety factor of the model is 1.95.In the process of model calculation, slope displacement rst appeared at the foot of the slope.Due to the mutual cutting of fractures, the jointed rock mass slipped downward, resulting in local instability at the foot of the slope, as shown in Fig. 9(a).With the continuous development of deformation, the jointed rock mass slides along the structure plane, while the deformation is small in the deep zone.However, in the anti-dip strati ed rock slope, the strati ed structural plane and wind-induced fractures cut each other, and the interlocking effect between the jointed rock mass is better, which increases the stability of the slope.

Conclusion
In the paper, 3DEC discrete element software is used to build a re ned three-dimensional discrete element model of a jointed rock slope in northern Yunnan Province.The dynamic characteristics of the jointed rock slope under pulse-like ground action are studied, and the failure mode of the jointed rock slope is revealed.The main conclusions are as follows:      Slope failure mode deformation and instability mechanism of slope(Song et al., 2020;Song et al., 2021).Zhou et al. (2022) conducted a series of foundation friction tests for rock slopes with intermittent joints in open-pit mines, studied the ground failure process and displacement evolution law of rock slopes with intermittent joints, and put forward a slope stability evaluation model from the perspective of energy.Feng et al. (2022) studied the cumulative effect of rock slope under multiple strong earthquakes through a large-scale shaking table test and analyzed the cumulative damage degree of slope by using residual deformation rate and Arias strength magni cation.Arnold (2016) established a discrete element model of jointed rock slope to research the fracture development and expansion evolution process under earthquake and analyzed the in uence of ground motion duration and geological structure on the stability of jointed rock slope.Ma et al. (2013) conducted numerical centrifuge tests on rock slopes to study the in uence of slope inclination on slope failure mode, 30° is the critical value for slope failure mode transformation.Wang et al. (2019) constructed a geo-mechanical model of an anti-tipping rock slope and proposed an analytical method of secondary tipping failure for a jointed rock slope.Zhou et al. (2019).established a cusp abrupt change theoretical model for the instability of jointed rock slopes and discussed the in uence of factors such as connectivity rate, dip angle, friction angle, and cohesion on slope stability and failure mode.Zhang et al. (2018) modi ed the kinematic element method and determined that the main controlling factors of jointed rock slope are the location of the rock bridge, the penetration degree of fracture, and the dip angle.

Figure 4 Monitoring
Figure 4

Figure 7 Dynamic
Figure 7 Song et al. carried out a shaking table test and a large number of numerical calculations for jointed rock slope, analyzed the dynamic response law of jointed rock slope from the time domain, frequency domain, and time-frequency domain, and better revealed the