A comparative study of pseudo-static slope stability analysis using different design codes

Xin-guang Yang *,En-i Zhai,Yuan Wang ,Zhong-o Hu

a Central Research Institute of Building and Construction Co.,Ltd.,MCC Group,Beijing 100088,China

b Goldwind Science&Technology Co.,Ltd.,Beijing 100176,China

c College of Mechanics and Materials,Hohai University,Nanjing 210098,China

d China Three Gorges Corporation,Beijing 100038,China

Abstract Many researchers have developed new calculation methods to analyze seismic slope stability problems,but the conventional pseudo-static method is still widely used in engineering design due to its simplicity.Based on the Technical Code for Building Slope Engineering(GB 50330-2013)of China and the Guidelines for Evaluating and Mitigating Seismic Hazards in California(SP117),a comparative study on the pseudo-static method was performed.The results indicate that the largest difference between these two design codes lies in determination of the seismic equivalence reduction factor(f eq).The GB 50330-2013 code specifies a single value for f eq of 0.25.In SP117,numerous factors,such as magnitude and distance,are considered in determining f eq.Two case studies show that the types of slope stability status evaluated by SP117 are in agreement with those evaluated by the seismic time-history stability analysis and Newmark displacement analysis.The factors of safety evaluated by SP117 can be used in practice for safe design.However,the factors of safety evaluated by GB 50330-2013 are risky for slope seismic design.

Keywords:Earthquake;Slope stability;Pseudo-static method;Design code

1.Introduction

Landslides constitute a common type of secondary geological disaster that occurs during an earthquake and results in widespread damage and large economic losses(Kou et al.,2018;Han et al.,2018).It is estimated that around 20%of recorded landslides have been triggered by earthquakes(Wen et al.,2004).For example,the 1994 Northridge earthquake,which registered a magnitude of 6.7 on the Richter scale,triggered more than 11000 landslides over an area of approximately 10000 km2.Some of these landslides damaged and destroyed homes and other structures,blocked roads,disrupted pipelines,and caused other serious damage(Parise and Jibson,2000).On May 12,2008,a magnitude-8.0 earthquake struck Wenchuan County in Sichuan Province,China.Due to the notably high magnitude and long duration of vibration of the earthquake and the complicated geoenvironment of the disaster area,the Wenchuan earthquake caused a significant number of landslides.Dozens of landslides with volumes greater than 107m3occurred(Huang,2009).According to damage statistics(Huang and Li,2009),landslides and other geohazards,including the Wangjiayan landslide,accounted for one third of all deaths caused by the earthquake.Consequently,assessment of the seismic stability of slopes has attracted considerable attention in geotechnical engineering and earthquake engineering.

The state-of-the-art seismic stability analysis method for slopes is time-domain analysis,which provides a powerful tool for seismic design of a geotechnical structure.However,this technique requires reliable constitutive models and appropriate dynamic boundary conditions,which are not usually available in practice(Kontoe et al.,2013).These difficulties limit the application of the dynamic approach.Consequently,such advanced techniques are justified only in major projects or for extremely large earthquakes,and the conventional pseudo-static method is still widely used in engineering design due to its simplicity(Baker et al.,2006;Loaiciga,2015;Loukidis et al.,2003;Shinoda,2015;Yang and Chi,2014;Zhao et al.,2016).

In the pseudo-static method,the seismic loading is modeled as a statically applied inertial force,the magnitude of which is a product of a seismic coefficientkand the weight of the potential sliding mass.Generally,thekvalues are much smaller thanamh/g,whereamhis the maximum horizontal acceleration expected at the site,andgis the acceleration of gravity.For comparison with the displacement analyses,Seed(1979)calibrated the pseudo-static method for seismic stability analysis of earth dams with soils that do not suffer significant strength loss during earthquakes.It was concluded that for earthquake magnitudes of 6.5 and 8.5,k=0.10 andk=0.15 are recommended,respectively,together with are quired factor of safety of1.15.The work of Seed(1979)is regarded as an important milestone because of its wide use and development.Similarly,Hynes-Griffin and Franklin(1984)recommended that thekvalue should be half of the peak bedrock acceleration and a design factor of safety of 1.0 should be required,though this method is not recommended for areas that are subject to large earthquakes or that have embankments with liquefiable soil.Bray et al.(1998)also developed a pseudo-static method for analyzing the stability of solid-waste land fills.The procedure calls for akvalue that is 0.75 times the maximum bedrock acceleration.By reviewing recommendations of different codes,Baker et al.(2006)found that,although the pseudo-static method is recommended for conventional projects by most design guidelines and codes(e.g.,Eurocode 8(ECS,2004),SP117(CGS,2008),GB 50330-2013,and IITK-GSDMA(2005)),the magnitudes of the recommendedkvalues and the corresponding stability safety criteria are different.TheTechnical Code for Building Slope Engineering(GB 50330-2013)of China and theGuidelines for Evaluating and Mitigating Seismic HazardsinCalifornia(SP117)are the most representative design codes.Therefore,a comparative study on the pseudo-static methods based on GB 50330-2013 and SP117 was performed.This paper first summarizes the pseudo-static slope stability analysis method based on the two design codes and subsequently examines the factors influencing the seismic coefficientk.Two example analyses were comparatively performed to verify the reliability and rationality of the pseudo-static method.

2.Comparison of GB 50330-2013 and SP117

2.1.Introduction of GB 50330-2013

GB 50330-2013 recommends seismic stability analysis for permanent slopes in earthquake zones where the basic intensity is 7 or greater.If no important buildings are present in the sliding areas,a pseudo-static method combined with the limit equilibrium method(LEM)or finite element method(FEM)can be used to assess the seismic slope stability.The horizontal seismic loadsQeacting on each slip mass or element can be calculated as follows:

whereGis the gravity of the slip mass or element;andkhis the horizontal seismic coefficient,which is determined by the basic earthquake intensity and maximum horizontal acceleration(amh)expected at the site,as shown in Table 1.

Generally,the seismic coefficientkhcan be calculated as follows:

wherefeqis the seismic equivalence reduction factor used to narrow the difference between the calculation results and the actual seismic behavior of slopes during earthquakes.According to Table 1 and Eq.(2),thefeqvalues are equal to 0.25 for any maximum horizontal acceleration.In other words,the pseudo-static method recommended by GB 50330-2013 specifies a single value forfeq.

According to GB 50330-2013,the stability of slopes can be divided into four types of status:unstable,stable-,stable,and stable+according to the factor of safety,as shown in Table 2.

For permanent slope engineering under earthquake conditions,Fstvalues for different safety levels of slope engineering are shown in Table 3.

The safety level of slope engineering can be determined by the severity of the slope failure,slope types,and slope heightH.For soil slopes,the safety level can be determined as shown in Table 4.

Table 1 Horizontal seismic coefficient recommended by GB 50330-2013.

Table 2 Status of slope stability.

Table 3 Critical factor of safety F st.

Table 4 Safety level for soil slope.

2.2.Introduction of SP117

If the sites are located in the Seismic Hazard Zone maps published by the California Division of Mines and Geology on the Newmark displacement analysis in SP117.Bray and Rathje(1998)found that the Newmark displacement(u)is a function ofky,kmax,andD5-95,and presented a relationship to predict the median value of the slope displacement.The yield seismic coefficientkyis the ratio of the seismic accelerationay,yielding a factor of safety equal to unity,to the acceleration of gravityg;kmaxis the ratio of maximum horizontal equivalent acceleration(amhe)over the duration of earthquake shaking to the acceleration of gravityg;andD5-95is the significant duration of shaking measured as the time between 5%and 95%of the normalized Arias intensity.The median values ofD5-95（（D5-95）med）on rock can be estimated as follows(Abrahamson and Silva,1996):

(CDMG),the pseudo-static method is recommended by SP117(Blake et al.,2002)for screen analysis.The purpose of the screen analysis is to filter out sites that have no potential or low potential for landslide development.IfFs＞1,the screen analysis will satisfy the requirement for the stability of seismic slopes.IfFs≤1,a quantitative evaluation such as the Newmark displacement analysis(Newmark,1965)will be performed to assess the seismically induced landslide hazards.

A de-stabilizing horizontal seismic coefficient is utilized with a conventional LEM.The seismic coefficient represents the fraction of the weight of the sliding mass that is applied as an equivalent horizontal force acting through the centroid of the sliding mass.The seismic coefficient to be used in the analyses is as follows:

whereamhris the maximum horizontal acceleration at the site for a soft rock site condition.Unlike GB 50330-2013,thefeqvalues specified by the SP117 guidelines are not constant,but a function of magnitude and site-source distance(Stewart et al.,2003).This approach has a rational basis.In addition to the slope height,slope angle,and unit weight and shear strength of the geomaterial,the main factors influencing seismic slope stability areamhr,magnitude,and site-source distance.It is easy to understand that earthquakes with greateramhrand magnitude tend to result in poorer slope performance than earthquakes with smalleramhrand magnitude.The reasons for this outcome are that greateramhrmeans stronger shaking,and earthquakes with greater magnitude have longer durations of shaking.

Depending on the magnitude and distance,thefeqvalues are identified using a model for seismic slope displacements based whereMdenotes magnitude,andrdenotes site-source distance.

Bray and Rathje(1998)presented a relationship to predict the median value of slope displacement,which is described as follows:

The seismic equivalence reduction factorfeqcan be related to magnitude,distance,andamhbased on the following assumptions and observation:

(1)feqis related toky/kmax.The rationale for this relationship is described in detail by Stewart et al.(2003).

(2)Two values of the threshold Newmark displacement are used in SP117:5 cm and 15 cm.It should be noted that the Newmark displacements provide only an index of slope performance.The 5-cm threshold value distinguishes conditions in which very little displacement is likely from conditions in which moderate or higher displacements are likely.The 15-cm threshold value distinguishes conditions in which small to moderate displacement are likely from conditions in which large displacements are likely(Stewart et al.,2003).The use of these two threshold displacements is intended to enable engineers and regulatory agencies to exercise judgment over the level of performance that they wish to enforce.

(3)kmaxis related to the product ofamhr/gandFnr,whereFnris the factor that accounts for the nonlinear response of the materials above the slide plane and can be approximated as follows:

for 0.1＜amhr/g＜0.8(Bray et al.,1998).

Thus,feqcan be obtained as follows:

2.3.Comparison of GB 50330-2013 and SP117

Both SP117 and GB 50330-2013 recommend the use of the pseudo-static method for seismic stability analysis of slopes.As mentioned previously,the pseudo-static method is used for a screen analysis in SP117.If the site fails the screen,a Newmark analysis will be performed(Blake et al.,2002).According to GB 50330-2013,the pseudo-static method is recommended to divide the seismic slopes into four types of stability status:unstable,stable-,stable,and stable+.However,the largest difference between SP117 and GB 50330-2013 in terms of their utilization of the pseudo-static method is the calculation procedure for the horizontal seismic coefficientkh.The factors ofkhconsidered in SP117 and GB 50330-2013 are shown in Table 5.

According to GB 50330-2013,khis determined based on the basic earthquake intensity andamh.The problems posed by basic earthquake intensity are significant to seismic structure design because of its subjectivity.Moreover,thefeqvalues are equal to 0.25 for anyamhvalues and are independent of magnitude or distance.The duration of strong shaking has a significant influence on the seismic stability of slopes,but it is not considered in the calculation offeqin GB 50330-2013.

Numerous factors,including magnitude,distance,and threshold displacement,are considered in the determination ofkhin SP117(Blake et al.,2002).In addition to the duration of the earthquake,feqis also related to the slope threshold displacement becausefeqis introduced to narrow the difference between the calculation results and actual seismic performance,and the slope displacement is an effective calibration for measuring the actual seismic behavior.Consequently,in the pseudo-static method,it is necessary to consider the different levels of slope performance as indexed by displacement.

3.Examples

3.1.Example 1

A seismic stability problem of a simple homogeneous soil slope with a slope inclination angle of α =36.87°and a height ofH=15 m was investigated,as shown in Fig.1.The soil unit weight γ was 20 kN/m3,the cohesive strengthcwas 35 kPa,and the internal friction φ was 15°.The seismic equivalence reduction factors and factors of safety for different ranges ofamhr,M,r,anduwere calculated using the Morgenstern-Price method(Morgenstern and Price,1965)with the pseudo-static method recommended by GB 50330-2013 and SP117.The results are shown in Figs.2 and 3.

Table 5 Factors of k h considered in SP117 and GB 50330-2013.

Fig.1.Graph for calculation of homogeneous slopes.

The following conclusions can be drawn from Figs.2 and 3:

(1)The values offeqspecified by GB 50330-2013 are equal to 0.25 and are much smaller than those of SP117 except forM=6.0,r=10 km,amhr=0.1g,andu=15 cm.The seismic inertial forces applied on the sliding mass calculated by GB 50330-2013 are much smaller than the inertial forces calculated by SP117 for most cases.Thus,under the same conditions,the factors of safety of the seismic slopes evaluated by GB 50330-2013 are greater than those evaluated by SP117,as shown in Fig.3.It is also shown that with increasingamhr,the difference in the factors of safety obtained by two design codes also increases.

(2)When the threshold displacement is equal to 5 cm,the averages offeqfrom SP117 are 0.479,0.591,and 0.712 forM=6.0,7.0,and 8.0,respectively.When the threshold displacement is 15 cm,the averages offeqfrom SP117 are 0.316,0.428,and 0.549 forM=6.0,7.0,and 8.0,respectively,because with the increase of the threshold displacement,the slope performance inevitably worsens,and thefeqvalues decrease.Thus,the factors of safety obtained by SP117 increase with the threshold displacement,as shown in Fig.3.Because thefeqvalues are independent of the threshold displacement in GB 50330-2013,the factors of safety are invariable with the threshold displacement.

(3)The results also show that for a givenamhr,thefeqvalues obtained by SP117 increase with the magnitude,and the factors of safety decrease,i.e.,it is thought that the duration of the earthquake is one of the most important factors.

(4)With the increase in the distancer,thefeqvalues increase,and the factors of safety obtained by SP117 decrease.However,with increasing magnitude,the influence of distance onfeqand the factors of safety both decrease.This result is also due to the effect of the duration on the seismic stability of slopes.

(5)With the increase ofamhr,thefeqvalues from SP117 first increase and subsequently decrease,and the factors of safety decrease throughout.

(6)The slip surfaces predicted by SP117 and GB 50330-2013 are nearly the same.The slip surfaces forM=7.0,r=20 km,amhr=0.2g,andu=5 cm are shown in Fig.4.

Fig.2.Relationship between f eq and a mhr.

Fig.3.Relationship between F s and a mhr.

Fig.4.Slip surfaces from SP117 and GB 50330-2013(M=7.0,r=20 km,a mhr=0.2g,and u=5 cm).

3.2.Example 2

For further comparison of the pseudo-static methods in SP117 and GB 50330-2013,the seismic stability of two homogeneous soil slopes was evaluated using the time-history analysis method and the Newmark displacement analysis method.

The dynamic response of a slope subjected to a given time history of an earthquake is governed by the following equation:

where M,C,and K denote the mass,damping,and stiffness matrices,respectively;¨u,˙u,and u denote the vectors of slope acceleration,velocity,and displacement,respectively;and¨ugdenotes the vector of earthquake acceleration.Rayleigh damping was used in the analysis,i.e.,damping in the form of C= ηM+ ξK,where η = λω,ξ= λ/ω,λ denotes the damping ratio,and ω denotes the natural angular frequency.Eq.(8)can be solved using the Wilson-θ method.

According to the static and dynamic stresses obtained by the FEM,the factor of safety of the seismic time-history stability analysis can be calculated as follows:

whereciand φiare the cohesive strength and internal friction angle at elementifrom undrained triaxial shear tests,respectively;liis the length of slip surface at elementi;and σniand τiare the normal and shear stresses on the slip surface at elementi,respectively:

where σx=are the static and dynamic horizontal stresses of the element,respectively;andare the static and dynamic vertical stresses of the element,respectively;andandare the static and dynamic shear stresses of the element,respectively.The seismic stability of the slopes in the time history analysis is evaluated using the minimum mean factor of safetyproposed by Liu et al.(2003)as follows:

whereFs0denotes the factor of safety in the static status,andFsmindenotes the minimum factor of safety in the seismic time history.

The Newmark displacement analysis(Newmark,1965)is also applied to assessment of the seismic stability of slopes.Based on the LEM,the concept of yield acceleration is used in Newmark displacement analysis.The permanent displacement of the slopes can be determined by two integrations of the average acceleration,which exceeds the yield acceleration.

The seismic stability of two simple homogeneous soil slopes(shown in Fig.1)was evaluated using the Morgenstern-Price method(Morgenstern and Price,1965)with the pseudostatic method(recommended by GB 50330-2013 and SP117),the time-history analysis method,and the Newmark analysis method.The slope information and soil shear strength parameters are shown in Table 6.The threshold displacement of the slopes was assumed to be 5 cm.

For the time-history analysis and Newmark analysis,eight sets of observed earthquake waves(PEER,2010)were input at the bottom of the slopes(as shown in Table 7 and Fig.5).

If the minimum mean factor of safety＜1 and the Newmark displacementuis larger than the threshold displacement(5 cm for this example),the slope is identified as unstable.Otherwise,the slope is identified as stable.According to the magnitude and distance of the input earthquake waves,thefeqvalues of the pseudo-static method can be calculated,and the factor of safety can be determined.The slope status can also be divided into stable(Fs＞1)and unstable(Fs＜1)categories using the pseudo-static method recommended by SP117 and GB 50330-2013.

Compared with the time-history analysis,the results of the pseudo-static method recommended by SP117 or GB 50330-2013 can be deemed coincident,conservative,or risky.If the slope status evaluated by the pseudo-static method is the same as that by the time-history analysis,the results of the pseudostatic method can be regarded as coincident.If the slope is unstable as evaluated by the pseudo-static method,while the slope is stable according to the time-history analysis,the results of the pseudo-static method are said to be conservative.Otherwise,the results are determined to be risky.The results are shown in Tables 8 and 9.

The results indicate that the types of slope stability status evaluated by SP117 are in agreement with those of the seismic time-history stability analysis and Newmark displacement analysis.Note that although the seismic factor of safetyof Slope A is 0.958 when the 2000 Tottori earthquake wave is input,the Newmark displacement is only 2 cm.Hence,the stability status is identified as stable by the time-history analysis.However,the pseudo-static factor of safety evaluated by SP117 is 0.933 under the same conditions,and thus the stability status is considered to be unstable.Although the types of stability status evaluated by the seismic time-history analysis and SP117 are different,the results from SP117 can be regarded as conservative.Similarly,when the 2000 Tottori and 2014 Jinggu earthquake waves are input to Slope B,the results from SP117 are regarded as conservative as well.Consequently,the factors of safety determined by SP117 can be used in practice to supply a safe design.

Table 6 Slope information and soil parameters.

Table 7 Earthquake wave information.

For slopes A and B,the factors of safety calculated by GB 50330-2013 are greater than 1.0 under all seismic conditions,and even the Newmark displacement far exceeds the threshold value.As a result,the types of stability status evaluated by GB 50330-2013 are not in agreement with those of seismic time history stability analysis and Newmark displacement analysis.The factors of safety calculated by GB 50330-2013 are risky for seismic slope design.

4.Conclusions

(1)Both SP117 and GB 50330-2013 recommend the use of the pseudo-static method for the seismic stability analysis of slopes.Seismic slopes can be divided into four different types of stability status according to the factors of safety from GB 50330-2013,but the pseudo-static method is used for screen analysis in SP117.

(2)According to GB 50330-2013,thefeqvalues are equal to 0.25 for anyamhand are independent of magnitude or distance,which lacks a rational basis.Thefeqvalues specified by SP117 are dependent on magnitude and distance.Additionally,the slope threshold displacement is considered.

(3)The pseudo-static method recommended by SP117 and GB 50330-2013 was applied to the seismic stability problem of a single homogeneous soil slope.The results show that thefeqvalues obtained from GB 50330-2013 are much smaller than those from SP117 for most cases.Consequently,the factors of safety determined by GB 50330-2013 are much greater than those of SP117.With the increase ofamhr,the difference in the factors of safety obtained by two codes increases.The results also indicate that the magnitude,distance,and threshold displacement have a certain influence on thefeqand factors of safety obtained by SP117.

Fig.5.Time histories of earthquake waves.

Table 8 Seismic stability results of Slope A.

(4)Comparison of the time-history stability analysis,the Newmark displacement analysis,and the pseudo-static method recommended by the two codes indicates that the types of slope stability status evaluated by SP117 are in good agreement with the seismic time-history stability analysis and Newmark displacement analysis.The factors of safety determined by SP117 can be used in practice to supply a safe design.

Table 9 Seismic stability results of Slope B.

(5)The types of stability status evaluated by GB 50330-2013 are not in agreement with seismic time-history stability analysis and Newmark displacement analysis.The factors of safety determined by GB 50330-2013 are risky for seismic slope design.

(6)For future revision of GB 50330-2013,it is advised that thefeqvalues and safety standard should be properly improved based on further composite analysis of the seismic behavior and stability of slopes under earthquake conditions.For larger values ofamh,assessment of seismic slope stability should be combined with time-history stability analysis and quantitative evaluation analysis,such as the Newmark displacement method.