Simplified method for analyzing soil slope deformation under cyclic loading

Ga Zhang,Yaliang Wang,Fangyue Luo

State Key Laboratory of Hydroscience and Engineering,Tsinghua University,Beijing,100084,China

Keywords:Soil slope Cyclic load Deformation Centrifuge model test Slice method

ABSTRACT Reasonable assessment of slope deformation under cyclic loading is of great significance for securing the safety of slopes.The observations of centrifuge model tests are analyzed on the slope deformation behavior under cyclic loading conditions.The potential slip surface is the key for slope failure and follows two rules: (i) the relative horizontal displacement along the potential slip surface is invariable at an elevation,and (ii) the soil along the slip surface exhibits the same degradation pattern.These rules are effective regardless of the location of the potential slip surface throughout the entire deformation process of a homogeneous slope,ranging from the initial deformation stage to the failure process and to the post-failure stage.A new,simplified method is proposed by deriving the displacement compatibility equation and unified degradation equation according to the fundamental rules.The method has few parameters that can be determined through traditional element tests.The predictions from the proposed method agree with the centrifuge test results with vertical loading and shaking table loading.This result confirms that the proposed method is effective in predicting the full deformation process of slopes under different cyclic loading conditions.

1.Introduction

Soil slopes experience various cyclic loads,such as earthquakes,traffic loads,and wind generator applications from its foundation.Such cyclic loads exhibit significant differences in magnitudes and frequency.The stability level and deformation magnitude of a slope under cyclic loading conditions need an effective evaluation in many geotechnical examples.Quasi-static methods are widely used for the stability analysis of slopes under cyclic loading.Terzaghi(1950) proposed a seismic coefficient to determine the equivalent inertia force and analyzed the stability level of a slope during an earthquake.The seismic coefficient plays an essential role in an accurate quasi-static analysis; thus,a comprehensive chart for the seismic coefficient was advised by considering the dynamic behavior of the soil and the time-dependency of the earthquake(Seed and Martin,1966).Leshchinsky and San (1994) derived an analytical solution for the normal stress distribution along the potential slip surface based on the quasi-static method and accordingly evaluated the dynamic stability of the slope.The quasistatic method was combined with the strength reduction method to derive a design chart for the slope stability (Baker et al.,2006).

Newmark (1965) pointed out that the dynamic stability of a slope is not accurately analyzed by quasi-static methods and should be evaluated through the accumulated slope displacement.He proposed a rigid sliding body method to predict the permanent slope displacement due to dynamic loading(Newmark,1965).Such a method has been widely adopted and modified for a more precise prediction.The yield acceleration for the Newmark method was investigated using the limit equilibrium scheme by considering planar,circular and logarithmic spiral slip surfaces (Seed and Goodman,1964; Prater,1979; Sarma,1981),and a linear decrease with increasing displacement was assumed by considering the soil strength reduction during vibration (Matasovic et al.,1997).Houston et al.(1987)replaced the static strength parameters by the cyclic strength to determine the yield acceleration of a slope.Tika-Vassilikos et al.(1993) proposed a formulation to consider the effect of the shear rate on the shear strength according to direct shear tests.A binary mass model was used to compare the difference in permanent displacement predicted by the traditional and modified Newmark rigid sliding body methods (Kramer and Smith,1997).

The Newmark methods could be used to predict the post-yield deformation of a slope with the application of cyclic loading;however,slopes have been observed to exhibit significant deformation before failure.More rigorous numerical analysis methods,e.g.the finite element method,could provide the deformation process of a slope during cyclic loading.With such methods,the irreversible cyclic deformation of the soil was described using various types of elastoplasticity models (Tian and Yao,2018; Ye et al.,2018) or empirical relationships between the irreversible strain and cyclic number (Vucetic and Dobry,1988; Subramaniam and Banerjee,2013; Camargo et al.,2016; Cai et al.,2018).Accordingly,the densification of the soil under cyclic loading,which has a significant influence on the soil deformation,could be considered in numerical analysis.The prediction accuracy of the numerical methods is suspicious for the slope deformation near failure due to the numerical challenge of strain localization.In addition,the complexities of the algorithms and parameter determination make such methods difficult in the practical design.A simplified analysis method is necessary to predict the full deformation process of soil slopes under cyclic loading conditions,from the initial to the post-failure stage.

Various centrifuge model tests have been performed to analyze the cyclic slope deformation behavior,as they are an effective way to produce the equivalent prototype gravity stress field in a small model (Wang et al.,2010,2018a; Enomoto and Sasaki,2015).The authors conducted a series of centrifuge model tests and observed the deformation and failure behaviors of slopes under cyclic loading conditions (Wang et al.,2017; 2018b).In this study,fundamental rules of slope deformation due to cyclic loading are revealed through a thorough analysis of centrifuge model tests.A new,simplified method is proposed within the framework of the slice method to analyze the full deformation process of soil slopes during cyclic loading,including the initial deformation stage,prefailure stage,failure process and post-failure stage.Finally,the effectiveness of this method is examined by comparing the predictions with various types of centrifuge model test results.

2.Fundamental rules

The observations of centrifuge model tests (Wang et al.,2017;2018b) were analyzed for the fundamental rules of slope deformation under cyclic loading conditions.According to the scaling law,the slope height,loading plate width and measured displacement can be transformed to the prototype dimension by multiplying the g level,i.e.centrifugal acceleration,in the centrifuge model tests.The loading pressure is identical for the model and prototype scales.In those centrifuge model tests,sine cyclic loads with different amplitudes (40 kPa,30 kPa,20 kPa and 10 kPa)and mean pressures(60 kPa,50 kPa and 40 kPa)are applied at the slope top via a loading plate with a width of 50 mm (2.5 m in the prototype) at 50g of centrifugal acceleration (Fig.1).Here the amplitude refers to half of the difference between the peak and valley of the sine cyclic load pressure.In a test,the load increases monotonically to the mean pressure and then the load vibrates around the mean pressure in a sine shape with a frequency of 1 Hz.The corresponding frequency in the prototype is 0.02 Hz and thus the inertia forces could be ignored.Various magnitudes and distributions of the deformation and slip surfaces with different failure processes are yielded and measured in the tests.

Fig.1 shows the size of the model slope with a height of 250 mm and a gradient of 1.5:1.The corresponding slope height is 12.5 m in the prototype.Two types of cohesive soils are used for the centrifuge model tests.One is silty soil with a plastic limit of 18.5%and a density of 1.5 g/cm3,termed S1 soil.The other is clay with a plastic limit of 15.5% and a density of 1.6 g/cm3,termed S2 soil.

Fig.1.Schematic view of a test model of the slope (unit: mm).

2.1.Displacement compatibility rule

According to the cyclic loading-induced progressive failure mechanism of a slope(Wang et al.,2018b),the slip surface occurs in the deformation localization zone.This result indicates that the deformation concentrates around the position where the slip surface is located prior to slope failure.To examine the deformation behavior around the potential slip surface,a couple of points are selected at a horizontal line and located on the two sides of the slip surface.The distance of the point couples is selected as 8 mm according to the observations of local failure along the slip surface of the slope model in centrifuge model tests.The relative displacement of the point couple is measured via a correlation-based analysis on the image series of the slope during cyclic loading(Zhang et al.,2009).In the correlation-based analysis,the measurement point is indicated using a square area with it as the center in an image.The maximum correlation factor of the distribution of grey level of the area is searched to determine the position of the measurement point on the image series.As a result,the displacement of the measurement point is obtained with an accuracy of 0.03 mm in the centrifuge model tests.

Fig.2 shows the histories of the relative horizontal displacement of the point couples at different elevations under cyclic loads.It should be noted that the cyclic loading includes the process in which the load increases to the mean pressure of the cyclic load,which cannot be described using the cyclic number.Thus,the settlement at the loading plate is used as an internal variable to indicate the loading process.It can be seen from Fig.2 that the relative horizontal displacements at all elevations increase during cyclic loading.It is worth noting that the relative horizontal displacements along the potential slip surface at different elevations are fairly close for the same loading cycle during the entire loading,from the beginning of the loading to the post-failure of the slope(Fig.2).It can be concluded that the relative horizontal displacement along the potential slip surface is identical for various elevations of the slope at a load.In other words,the relative horizontal displacement at opposite sides of the potential slip surface is dependent only on the features of the slope and loading.This finding shows that the potential slip surface could be regarded as the key portion of the slope with the displacement compatibility rule,i.e.an identical relative horizontal displacement along the potential slip surface for various locations.The displacement compatibility rule describes the essential deformation features of the slope and is valid throughout the entire deformation process of the slope.

Fig.3 shows the relative horizontal displacement along the slip surface of a slope after an earthquake loading is applied on the slope model base in a centrifuge model test(Wang et al.,2010).In the test,the peak acceleration is observed to gradually increase with increasing altitude.The relative horizontal displacement is nearly equal at various elevations.Thus,the displacement compatibility rule of the slope is preliminarily confirmed to be effective under earthquake conditions.

Fig.2.Development process of relative horizontal displacement of point couples along the slip surface in different tests:(a)μ=60 kPa,A=40 kPa;(b)μ=60 kPa,A=20 kPa;(c)μ=60 kPa,A=10 kPa;and(d)μ=50 kPa,A=20 kPa.μ-mean pressure of the sine wave;A-amplitude of the sine wave;Δu-relative horizontal displacement of point couples;ssettlement at the loading plate; y-vertical coordinate.

Fig.3.Relative horizontal displacement of point couples along the slip surface at the end of seismic loading.x-horizontal coordinate; magnitude in the bracket-relative horizontal displacement of point couple.

2.2.Unified degradation rule

In this study,an index,termed the diversity factor of displacement(DFD),is used to quantify the slope deformation as follows:

where n is the number of measurement points in a square element and set to 25,u is the horizontal displacement,and L is the element side length and set to 8 mm.Compared with the strain that is usually used to quantify the deformation,DFD can be easily calculated with more image-based gauging points to increase the analysis accuracy and has been successfully used for slope deformation analysis (Wang et al.,2018b).

Fig.4 shows the DFD histories of the slope under sine cyclic loading.It can be seen that the deformation of the element close to the slip surface (elements #1 and #3) is remarkably larger than those of the elements in the base body (element #4) and in the sliding body (element #2).

During cyclic loading,the peak load is maintained;however,the deformation of all the elements increases with increasing loading cycles.This result demonstrates that the soil element exhibits significant degradation due to an increase in the number of loading cycles.The degradation of the soil is described using the degradation index,δ,which was proposed by Idriss et al.(1978).The degradation index is determined using the following equation:

Fig.4.Histories of the DFD of elements of the slope with application of cyclic load with a mean pressure of 60 kPa and an amplitude of 30 kPa.N-number of cycles.

where GS1and GSNare the secant moduli of the soil at the first and Nth loading cycles,respectively.

For stress-controlled cyclic triaxial tests,the increment of axial strain during a loading cycle could reflect the degradation of moduli of the soil.Thus,a method is proposed to determine the degradation index based on such type of cyclic loading test(Zhou and Gong,2000),as follows:

where εC1and εCNare the axial strains of the soil at the first and Nth loading cycles,respectively.Since the amplitude of the cyclic load is maintained during the tests,the degradation index of the soil element could be determined with a similar pattern to Eq.(3),as follows:

where ΔDFD1and ΔDFDNare the DFD increments at the first and Nth loading cycles,respectively.

Fig.5 shows the histories of the degradation index of several elements of the slope.It can be observed that the degradation index of all the elements decreases from 1 during cyclic loading.The decrease rate of the degradation index decreases as the number of loading cycle increases.This result demonstrates that the degradation of the soil modulus exhibits a tendency to stabilize during cyclic loading.The degradation of the soil near the slip surface developed rapidly compared to those of the soil in the base body and in the sliding body.In particular,close examination shows that the elements close to the slip surface(elements#1 and#3)exhibit a close relationship with the increasing number of loading cycles.

Fig.6 shows histories of degradation index of the soil elements around the slip surface under cyclic loads with different amplitudes and mean pressures.The history curves of the degradation index of the elements at different elevations nearly coincide.This result indicates that the soil along the slip surface exhibits a unified degradation rule that is independent of the location.In other words,the degradation process for all the soil elements along the slip surface could be formulated with one equation.In addition,the degradation process is found to accelerate with increasing amplitude or peak of the cyclic load; this demonstrates that the amplitude of cyclic load is an indicator to the degradation behavior of the slope.

Fig.5.Histories of the degradation index of elements for the slope with application of cyclic load with a mean pressure of 60 kPa and an amplitude of 30 kPa.

3.Framework

According to the new findings on the fundamental rules,the potential slip surface plays the key role in describing the slope deformation behavior under cyclic loading.Thus,the simplified method for cyclic deformation analysis of slopes could be set up based on the framework of the traditional slice method (Fig.7).In this study,the simplified Bishop slice method is selected,and the following assumptions are adopted:(1)the slip surface is circular;and(2)the slice is rigid,and the deformation of the slope occurs in the slip surface only,which comprehensively considers the slope deformation due to various types of cyclic loading.In addition to those assumptions,two essential equations are proposed based on the fundamental rules,as follows.

3.1.Displacement compatibility equation

According to the assumption that the displacement appears only along the slip surface,the horizontal displacement of the slices is equal to satisfy the displacement compatibility rule,as follows:

where uiis the horizontal displacement at the bottom of the ith slice.In this study,the shear band is assumed to have a constant thickness that is specified as 5%of the slope height.The shear strain at the bottom of the slice is derived using the following equation:

where γiand αiare the shear strain and inclination at the bottom side of the ith slice,respectively.According to Eq.(6),the distribution of the shear strain throughout the slip surface could be calculated with the shear strain of the initial slice.

3.2.Unified degradation equation

According to the unified degradation rule,the soil along the slip surface exhibits an identical degradation process independent of the location.Thus,such degradation could be captured using only one equation.In this study,a power function is proposed to describe the relationship between the degradation index and number of loading cycles,as follows:

Fig.6.Histories of degradation index of elements along slip surfaces in different tests:(a)μ=60 kPa,A=40 kPa;(b)μ=60 kPa,A=30 kPa;(c)μ=60 kPa,A=10 kPa;and (d)μ=50 kPa,A=20 kPa.

Fig.7.Schematic view of the slice method.

where t is the degradation parameter,which is unique for a slope under cyclic loading.Fig.8 shows that the fitting lines using Eq.(7)agree well with the observations from centrifuge model tests.The fitting results also demonstrate that the degradation parameter t increases when the amplitude of the cyclic load increases.

The degradation parameter is dependent on the slope features,such as the slope height and gradient.The monotonic capacity of the slope,Fs,i.e.the limit load on a slope under monotonic loading conditions,is introduced to describe the slope features in a comprehensive way.The monotonic capacity could be determined by replacing cyclic loading with monotonic loading while other conditions remain the same(e.g.loading position and loading area)using the simplified Bishop method.Consequently,the mobilized level,fs,is defined by

Another index,termed the cyclic loading level,fc,is introduced to quantify the magnitude of the cyclic load with reference to the capacity of the slope,which is determined by

The following equation is proposed to determine the degradation index t:

Fig.8.Equation fitting between the degradation index and number of loading cycles: (a)μ=60 kPa,A=30 kPa; (b)μ=60 kPa,A=20 kPa; (c)μ=60 kPa,A=10 kPa;and (d)μ=50 kPa,A=20 kPa.

where k is the loading factor.It should be noted that k could be set to a constant of 0.72 for various cyclic loading conditions(Fig.9).In other words,k is unnecessary to determine for practical cases.

Fig.9.Equation fitting for the degradation parameter under different cyclic loading conditions.

4.Formulations

4.1.Regular cyclic loading condition

The regular cyclic load (e.g.sine wave and rectangular wave) is divided into two components,i.e.a monotonic load and a constant amplitude cyclic load.The slope deformation due to the two components is analyzed in turn using the proposed method.Before the cyclic load-induced deformation analysis,the stress field of the slope under self-weight conditions should be obtained.Thus,the slope deformation analysis under regular cyclic loading conditions includes following three stages.

(1) Self-weight loading stage

The strength behavior of the soil is captured using the Mohr-Coulomb criterion.For the ith slice,the limit tangential resistance,Ti,max,is calculated as follows:

where Niis the normal force at the bottom of the ith slice,liis the bottom length of the ith slice,and ciand φiare the strength parameters of the soil at the bottom of the ith slice.When the soil does not reach the shear strength,the tangential force at the bottom of the ith slice,T0,i,is calculated by

where γ0,iis the shear strain at the bottom of the ith slice at the selfweight loading stage,and Giis the static shear modulus of the soil at the bottom of the ith slice.When the soil reaches failure,the tangential force is maintained as Ti,maxeven if the shear strain continues to increase.

Through analyzing the force equilibrium of the ith slice in the vertical direction,the following equation is obtained:

where miis the mass of the ith slice,αiis the inclination at the bottom side of the ith slice,and N0,iis the normal force at the bottom of the ith slice at the self-weight loading stage.The overall torque balance of the sliding body is as follows:

where diis the distance from the ith slice to the center of the slip surface (Fig.7) and r is the radius of the slip surface.

The tangential force at the bottom of each slice,T0,i,could be obtained through an iteration solution for the simultaneous equations of Eqs.(6)and (12)-(14).

(2) Monotonic loading stage

The monotonic load could be divided into a few subloads to assure the iteration accuracy.For a subload,the induced shear strain increment of the slice,Δγi,can be calculated as follows.

When the shear stress of the soil does not reach the shear strength,the subload-induced increment of the tangential force at the bottom of the ith slice,ΔTi,is calculated by

If the shear stress of the soil reaches failure,the tangential force is maintained as Ti,max.In the vertical direction,the equilibrium equation is

where ΔFi,yis the vertical component of the subload of the ith slice,and ΔNiis the subload-induced increment of tangential and normal forces at the bottom of the ith slice.The overall torque balance of the sliding body is as follows:

where ΔFiis the subload of the ith slice,and hiis the distance from the subload of the ith slice to the center of the slip surface.

The subload induced increments of the tangential force and shear strain at the bottom of each slice,ΔTiand Δγi,can be obtained through an iteration solution for the simultaneous equations of Eqs.(6)and(15)-(17).Thus,the tangential force and shear strain at the bottom of each slice,Tiand γi,can be obtained as follows:

where Ti,pand γi,pare the previous tangential force and shear strain at the bottom of each slice before the application of the loading increment,respectively.It should be noted that the shear strain is zero at the beginning of monotonic loading.

(3) Constant amplitude cyclic loading stage

An equivalent viscoelastic model is used to capture the dynamic stress-strain relationship of the soil (Shen and Wang,1980).According to the model,the dynamic shear modulus,Gd,and damping ratio,λ,are obtained as follows:

where k1,k2,n and λmaxare the model parameters;σ′mis the mean effective stress; pais the atmospheric pressure; and γcis the reference shear strain determined by

where γd,maxis the maximum shear strain in the analysis period.

For the Nth loading cycle,the dynamic shear strain,γd,i,satisfies the following equation:

where GdN,iand λiare the dynamic shear modulus and damping ratio of the soil for the Nth loading cycle,respectively; Fd,iis the component of the amplitude of the cyclic loading in tangential direction to the slice bottom;and Hiand Hi+1are the horizontal forces of the neighboring slices.Thus,the equation for the whole sliding body is derived as follows:

According to the unified degradation equation,the dynamic shear modulus of the soil for the Nth loading cycle is

The calculation procedure can be described as follows:

(1) According to Eq.(6),the cyclic loading-induced increment of the shear strain at the bottom of each slice at the Nth cycle,γdN,i,is expressed as a function of the shear strain at the bottom of the first slice,γdN,1.Thus,Eq.(23) becomes an equation with an unknown variable,γdN,1.

(2) An iteration solving method,the Wilson-θ scheme,is used to solve the equation for γdN,1.

(3) γdN,iis obtained with γdN,1using Eq.(6).The shear strain at the bottom of each slice,γi,can be obtained as follows:

where γi,pis the previous shear strain at the bottom of each slice before the application of Nth cyclic loading.

4.2.Irregular cyclic loading condition

Fig.10.Schematic view of equivalent cyclic number.

For irregular cyclic loading(e.g.earthquake),the mean pressure and amplitude of the load exhibit a significant variation during loading.For the Nth loading cycle,the following analysis scheme is used to consider the mean pressure and amplitude of the cyclic load:

(1) The variation in the mean pressure of the cyclic load in the Nth cycle and the (N-1)th cycle is considered using the computation category of the monotonic loading stage under regular cyclic loading conditions.

(2) The variation of the amplitude of cyclic load in the Nth cycle and the (N-1)th cycle is considered using the equivalent cyclic number on the basis of the criterion that the combination of amplitude and cyclic number induces equal shear strain.For example,the amplitudes of the(N-1)th and Nth cycles are A1and A2,respectively.For a slice,the shear strain at the bottom after the(N-1)th cycle is γ2.As shown in Fig.10,the cyclic number is altered from N1to N2since the amplitude is changed from A1in the (N-1)th cycle to A2in the Nth cycle.It should be noted that the degradation index should be involved in the equivalent cyclic number analysis using Eq.(7).

4.3.Parameters

The parameters in this method could be divided into the static group and dynamic group.The static parameter group includes the strength parameters of the soil,c and φ,and the shear modulus,G.These parameters could be determined by triaxial or direct shear tests.The dynamic parameter group includes four parameters: k1,k2,n and λmax.These parameters can be determined by dynamic triaxial or resonant column tests.

5.Verifications

The predictions from the proposed method are compared with the centrifuge model test results to validate the proposed method.The slip surfaces of the slopes used for calculation are determined according to the test data.The parameters of the soil were determined with resonance column tests and are listed in Table 1.According to the assumption of the proposed method that the slice is rigid and the slope deformation occurs along the slip surface only,the settlement at the slope top could be derived using the calculated shear strain of the corresponding slice and is compared with the measurement results in the centrifuge model tests.

Table 1 Parameters of the soil for the centrifuge model tests.

5.1.Vertical loading centrifuge model tests

Figs.11 and 12 compare the histories of settlement at the slope top that are obtained using predictions from the proposed method and centrifuge model test results.The settlement increases during loading at a decreasing rate.The predicted settlement exhibits a close match with the test measurement.This result indicates that the proposed method can analyze the deformation of slopes under vertical cyclic loading conditions with accuracy.

Fig.11.Comparison of predictions from the proposed method and centrifuge model test results for the top settlement of the slope under vertical loading condition (soil S1): (a)μ=60 kPa,A=30 kPa; and (b) μ=60 kPa,A=20 kPa.Δs-settlement at slope top due to cyclic loading.

5.2.Centrifuge shaking model tests

Fig.12.Comparison of predictions from the proposed method and centrifuge model test results for the top settlement of the slope under vertical loading condition (soil S2): (a)μ=60 kPa,A=10 kPa; and (b) μ=40 kPa,A=30 kPa.

A shaking table test at a centrifugal acceleration of 40g on the soil slope is used to confirm the effectiveness of the proposed method (Wang,2013).The slip surface for calculation is shown in Fig.3.Fig.13a shows the history of the input seismic wave.Fig.13b compares the histories of settlement at the slope top that are obtained using predictions from the proposed method and test results.The settlement increases during the earthquake,and the rate of increase decreases after strong shaking.The predicted postearthquake settlement at the slope top is 5.3 mm at the model dimension,which is close to the test measurement of 5.7 mm.

Fig.13.Comparison of predictions from the proposed method and centrifuge model test results for the slope under seismic condition at prototype dimension (soil S3):(a)History of input seismic wave,and(b)Predictions from the proposed method and test results.ss-settlement at slope top; a-seismic acceleration; ts-time.

Another shaking table test on a rockfill embankment is used to validate the proposed method (Wang et al.,2018a).Fig.14a shows the schematic view of the embankment model and the slip surface for calculation.The static safety factor of the embankment is fairly large,indicating that the slope is sufficiently stable before the shaking application.A sine wave with an amplitude of 0.21g,a frequency of 50 Hz and a cyclic number of 40 at the prototype dimension is input in the tests.Fig.14b compares the histories of the top settlement of the embankment that are obtained using predictions from the proposed method and test results at the prototype dimension.The settlement increases at a decreasing rate during the vibration input.The predicted post-vibration settlement at the slope top is 346 mm at the prototype dimension,which is close to the test measurement of 330 mm.

Fig.14.Comparison of predictions from the proposed method and centrifuge model test results for the embankment under seismic condition(soil S4):(a)Schematic view of the dam model (unit:mm),and(b)Predictions from the proposed method and test results.

The above two comparison results indicate that the proposed method can predict the deformation of slopes under earthquake conditions with accuracy.

6.Conclusions

Based on the slope deformation analysis in centrifuge model tests under cyclic loads with different amplitudes and mean pressures,it is revealed that the potential slip surface can be regarded as the key portion of the slope involving two fundamental rules of the cyclic loading-induced deformation of slopes throughout the entire deformation process of a homogeneous slope,ranging from the initial deformation stage to the failure process and to the postfailure stage,including: (1) the displacement compatibility rule: the relative horizontal displacement at the opposite sides of the potential slip surface is independent of the location; and (2) the unified degradation rule: the soil along the slip surface exhibits an identical degradation pattern regardless of the location in the slope.

A new,simplified analysis method is proposed within the framework of the slice method to predict the deformation of homogeneous soil slopes under cyclic loading conditions.The displacement compatibility equation and unified degradation equation are assumed on the basis of new findings on the fundamental rules on the slope deformation.The formulation of the method is described by considering regular and irregular cyclic loads.The method has few parameters that could be determined through traditional element tests.

The predictions of the proposed method agree with the measurement results of the centrifuge model test with vertical cyclic loading and shaking table loading.The proposed method is confirmed to effectively predict the full deformation process of the slope under various cyclic loading conditions,including the initial deformation stage,pre-failure stage,failure process and postfailure stage.

It should be noted that the rule assumptions of the proposed method need to be verified with more slope data from field observations,model tests and numerical analysis.Thus,the proposed method could be improved and confirmed in further studies.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The study is funded by Tsinghua University Initiative Scientific Research Program,State key Laboratory of Hydroscience and Engineering (Grant No.2020-KY-04) and National Natural Science Foundation of China (Grant No.52039005).