A simplified approach to assess seismic stability of tailings dams

Sanjay Nimalkar,V.S.Ramakrishna Annapareddy,Anindya Pain

aSchool of Civil and Environmental Engineering,Faculty of Engineering and Information Technology,University of Technology Sydney,Sydney,NSW 2007,Australia

bGeotechnical Engineering Group,CSIR-Central Building Research Institute(CBRI),Roorkee,247667,India

Keywords:Tailings dam Horizontal slice method(HSM)Limit equilibrium Strain-dependent property Pseudo-static method

A B S T R A C T In the zones of high seismic activity,tailings dam should be assessed for the stability against earthquake forces.In the present paper,a simplified method is proposed to compute the factor of safety of tailings dams.The strain-dependent dynamic properties are used to assess the stability of tailings dams under seismic conditions.The effect of foundation soil properties on the seismic stability of tailings dams is studied using the proposed method.For the given input parameters,the factor of safety for low frequency input motions is nearly 26%lower than that for high-frequency input excitations.The impedance ratio and the depth of foundation have significant effect on the seismic factor of safety of tailings dams.The results from the proposed method are well compared with the existing pseudo-static method of analysis.Tailings dams are vulnerable to damage for low-frequency input motions.

1.Introduction

Tailings dams are one of the largest geo-embankments built by the geotechnical engineers and are often constructed with steep slopes using the coarse fraction of tailings for economical designs.Tailings dam retains the mine’s waste produced from the oredressing process.Generally,the wastes are in slurry form and are pumped into the sedimentation pond.The potential failure of such structures is of major concern because of the issues associated with public health,safety,environmental pollution and infrastructure damage.The propagation of seismic waves(S-and P-waves)through a geo-structure may produce seismic inertial forces which lead to instability of the structure,therefore,these forces must be considered in the analysis (Kouretzis et al.,2013,2014;Suchowerska et al.,2014).Several failure case histories on tailings dams have been reported in the last few decades.According to ICOLD(2001),the failure rate of tailings dams over the last one hundred years is calculated as 1.2%and is much higher than that of the conventional water retaining dams(i.e.0.01%).Rico et al.(2008)systematically presented a comprehensive report on tailings dam failure in Europe.Azam and Li(2010)presented a detailed review report on tailings dam failure over the last hundred years.An extensive study conducted by the International Commission of Large Dams and the United Nations Environmental Program in 2001 found that one major tailings dam incident occurred each year.Some of these failures were caused by slope instability,“unpredictable”climate and geological events,such as floods and earthquakes.Kossoff et al.(2014)summarized the causes of tailings dam failure,environmental and economic impacts due to the failure of tailings dam and the remedial measures for mine tailings dams.

A significant number of analytical and numerical studies were conducted on stability assessment of tailings dams under both static and seismic conditions.Study on the seismic stability of tailings dams has gained importance over the past decades(Okusa and Anma,1980;Penman,2001)as a significant number of such structures have failed to sustain the strong earthquakes mainly in Chile and Japan.Therefore,the seismic slope stability analysis is an important part of earthquake resistance design of tailings dams.A reliable seismic slope stability analysis requires a relatively precise evaluation of seismic acceleration generated at different levels in the dam.Psarropoulos and Tsompanakis(2008)numerically examined the stability of tailings dams subjected to static and seismic loading conditions using PLAXIS and QUAD4M,respectively.However,numerical studies are mainly case-specific and time-consuming(Sun and Shen,2017;Sun et al.,2018).A few studies are available which deal with the seismic stability of tailings dam where the seismic forces are assumed to be pseudo-static in nature.The pseudo-static method is used to calculate the permanent displacement of earth dams and embankments under seismic conditions(Sarma,1975;Makdisi and Seed,1978).Some researchers attempted to use the pseudo-dynamic method to compute the factor of safety under seismic conditions(Nimbalkar et al.,2006;Nimbalkar and Choudhury,2007,2010;Choudhury and Nimbalkar,2009).But the pseudo-dynamic method also suffers from some shortcomings.For example,the pseudo-dynamic method does not satisfy the boundary conditions,and it follows a simple approach to consider the acceleration amplification(Choudhury and Nimbalkar,2005,2006;2007,2008;Bellezza,2014).In that approach,one needs to assume an amplification factor,and a linear variation of the acceleration is considered in the analysis(Pain et al.,2015,2017).Also,the pseudo-dynamic method in any form does not consider the damping ratio of rock fill material.

Horizontal slice method(HSM)is a newly developed formulation to compute internal stability of reinforced soil retaining structure(Shahgholi et al.,2001;Nouri et al.,2006,2008).HSM has an advantage over the conventional method of the vertical slice.The advantage allows HSM to have different seismic acceleration values at different depths.Nouri et al.(2008)considered the amplification of acceleration in the internal stability analysis of reinforced soil retaining structure.The amplification of acceleration considered in that study was in accordance with the one in the centrifuge experiments reported by Nova-Roessig and Sitar(2006).But the amplification of acceleration is not a fixed quantity;it depends on several factors such as shear wave velocity of soil,and frequency of input excitation.In the present study,a plane wave is considered to be propagating vertically in a Kelvin-Voigt homogeneous medium.Bellezza(2014)proposed the solution for determining the seismic active earth pressure for plane SH-wave propagating vertically in a Kelvin-Voigt homogeneous medium.The method satisfied the boundary conditions and considered the nonlinear variation of the seismic acceleration.Pain et al.(2017)compared the acceleration ratio,i.e.the ratio of surface acceleration to the input acceleration,obtained using the proposed method and DEEPSOIL linear frequency domain ground response analysis.Annapareddy et al.(2017)applied the same method for seismic translational stability assessment of municipal solid waste land fills.The same procedure coupled with HSM was extended by Nimbalkar and Pain(2017)to evaluate the seismic stability of tailings dams.However,the above method does not consider a reduction in shear modulus and increase in damping ratio with the increase in shear strain which is well-known facts from earthquake geotechnical engineering(Kramer,1996).As an alternative,the above method used the small strain shear modulus and a fixed damping ratio,but the shear strain generated during a seismic event is quite significant.The existing methods assumed that the dam rests over a rigid stratum.

To the best of authors’knowledge,no analytical method is available that can consider the effect of foundation soil properties on slope stability of tailings dams under seismic conditions.Therefore,a new simplified method is proposed in the present paper to assess the seismic stability of tailings dams using straindependent dynamic properties of tailings dam material.The present method is able to consider the effect of foundation soil properties on the seismic stability of tailings dam.In the present study,seismic forces are computed considering the dam material as viscoelastic and dividing the dam intoNhorizontal slices.Kelvin-Voigt model,one of the most widely used mathematical models for dynamic analysis of soil,is used in the present study to idealize soil.In the present study,two foundation cases are considered for tailings dam:(i)dam resting over a rigid stratum,and(ii)dam resting over a flexible stratum( finite thickness of sand layer is considered)and the bedrock is subjected to harmonic shaking.The seismic factor of safety is computed by considering the force equilibrium of failure wedge.

2.Proposed model

The details of tailings dam model considered in the present studyare shown in Fig.1.The vertical height of the tailings dam and the depth of the foundation are represented byHandHf,respectively.The top width of the dam is indicated withb.The solutions to a plane SH-wave travelling vertically upwards and being reflected back through the dam material and foundation are given by Eqs.(1)and(2),respectively(Kramer,1996):

Each of the above equations has two parts:the first part represents the incident wave travelling in the negativeY-direction,and the second part represents the reflected wave travelling in the positiveY-direction(Fig.1).The coefficients in the above equations(A,Af,BandBf)can be solved using four important boundary conditions,i.e.zero shear stress at the free surface,stress continuity and strain compatibility at the interface between the dam and foundation soil,and strain compatibility at the interface between the foundation soil and bedrock.Assuming a base displacementuh=uhocos(ωt),the final simplified expressions for the horizontal displacement in the dam and foundation can be represented by Eqs.(3)and(4),respectively:

Fig.1.Tailings dam model considered in the present study.

whereωH/VsandωHf/Vsfare the normalized frequencies of the dam and foundation soil deposit,respectively;Vsis the shear wave velocity of the dam material;Vsfis the shear wave velocity of the foundation soil;andλis the impedance ratio,which is defined as

The shear strain generated in the dam and foundation can be obtained using Eqs.(15)and(16),respectively:

The expressions for horizontal accelerations in the dam and foundation soil can be obtained by differentiating Eqs.(3)and(4)twice with respect to time,respectively.The horizontal accelerations in the dam and foundation medium are the function of time and depth,as shown in Eqs.(17)and(18),respectively:

wherekhis the horizontal seismic acceleration coefficient at the bedrock,andkhg=-ω2uho.

In the present study,failure surface is assumed to be an arc of a circle having radiusR,as shown in Fig.1.The whole tailings dam is divided intoNhorizontal slices.Fig.2 shows the for cesacting on thej-th slice.According to Shahgholi et al.(2001),simple formulation of HSM comprises 2N+1 unknowns and 2N+1 equations.

The mass of thej-th slice is given by

Fig.2.Geometry and the forces acting on the j-th slice.

The total horizontal inertial force(qhj)acting on thej-th slice may be expressed as

After computing the seismic inertial forces,force equilibrium is satisfied.The equation for vertical equilibrium for each slice is mentioned below:

whereVjandVj+1are the vertical inter-slice forces and are assumed to be equal to the overburden pressures over the horizontal slices(Shahgholi et al.,2001;Nouri et al.,2006).

Again, τr= （τf/FS）(for each slice),whereFSis the factor of safety,thus we have

whereSjandNjare the shear and normal forces acting on the base of each slice,respectively.

By substitutingSjfrom Eq.(22)into Eq.(21),the following equation can be obtained:

Satisfying the horizontal equilibrium for the whole wedge,one can obtain

In the present analysis,reduction in the shear modulus and increase in damping ratio with the increase in shear strain are considered.For sand,the stiffness degradation and damping increase curves proposed by Seed and Idriss(1970)are adopted.In the present study,dry conditions are assumed;therefore,the stiffness degradation and damping increase curves of solid tailings after Zeng(2003)are used in the analysis.The modified hyperbolic model proposed by Matasovic and Vucetic(1993)is used to obtain the mathematical expressions for stiffness degradation and damping increase curves of both sand and tailings material.The expressions for the normalized modulus reduction and the damping ratio curves are provided in Eqs.(25)and(26),respectively:

where γuis the percentage shear strain;γr,ε andsare the model parameters to obtain the best- fit of normalized modulus reduction curve;anda,bandcare the model parameters to obtain the best- fit of damping ratio curve.The values for the curve fitting parameters are given in Table 1.

Figs.3 and 4 show the target and fitted curves for normalized shear modulus and damping ratio,respectively.In the present analysis,an entry-exit method is used to search for the critical failure surface.The entry and exit points of the failure surface are pointsBandD,respectively,as indicated in Fig.1.To find out the critical failure surface for which the factor of safety is minimum,θo,θhandt/T(tis the time,andTis the period of excitation)are varied independently.To identify all the valid failure surfaces,two criteria are introduced.The two criteria are given below.

The width of the dam can be given as

The radius of the failure slip can be given as

3.Results and discussion

A computer program is developed in MATLAB(The MathWorks Inc.,2015)platform for computations.A trial-and-error method based iterative solution scheme is adopted,in which θoand θhare varied individually between 0°and 90°at an interval of 0.2°.Furthermore,t/Tis varied in the range of 0-1 at an interval of 0.01.The value of factor of safety is minimized with respect to these optimized variables.

3.1.Comparison of results from the present method with that from the pseudo-static method

Fig.5 shows the comparison of the factor of safety values from the present method with that from the pseudo-static method for different values ofkh.The trends of the results from the present study are very similar to that from the pseudo-static method.As can be seen,the factor of safety values from the present analysis(with rigid and flexible foundations)and the pseudo-static method are the same forkh=0(i.e.under static conditions,both methods yield similar results).It is also observed that the factor of safety isdecreased with the increasedkh.The factor of safety obtained from the present analysis with the rigid foundation is a maximum of 27%higher compared to that from the pseudo-static method.Similarly,the factor of safety from the present analysis with the flexible foundation is a maximum of 16%greater than that from the pseudostatic method;however,it is decreased to a maximum of 15%compared to the result from the present analysis with the rigid foundation.The probable reason for the above trends can be explained with the help of Fig.6.It shows the variation of horizontal acceleration coefficient at different levels of a tailings dam for the pseudo-static method and the present analysis with rigid and flexible foundations.The same acceleration profiles are used to compute the seismic inertial forces from the respective methods.In case of the pseudo-static method,a constant acceleration coefficient is assumed at all the levels of the tailings dam,but a nonlinear variation of acceleration coefficient is observed from the present method,as shown in Fig.6.Moreover,for the given input frequency,the tailings dam vibrates in the second mode of vibration where parts of inertial forces act in one direction and the remaining act in the opposite direction(refer to Fig.6).This is the reason that the factor of safety values from the present analysis with rigid and flexible foundations are higher than that from the pseudo-static method.As can be seen,the net acceleration coefficient from the present analysis with the flexible foundation is significantly greater than that with rigid foundation,due to the consideration of a finite thickness of sand layer as a foundation for the analysis.For this reason,the factor of safety from the present analysis with the flexible foundation is lower than that with rigid foundation.

Table 1 Values of fitting parameters for modulus reduction and damping ratio curves used in the present study.

Fig.3.Best- fit of stiffness degradation curves of tailings dam material and sand used in the present study.

Fig.4.Best- fit of damping ratio curves of tailings dam material and sand used in the present study.

Fig.5.Comparison of factor of safety from the present method with that from the pseudo-static method for different values of kh.H=44 m,b=4 m, φ =35°,D=Df=0.1%,f=2.5 Hz,λ=1.92,β =21.8°,and Hf=H/3.

Fig.6.Variation of horizontal acceleration coefficient along the height of the tailings dam.H=44 m,b=4 m,D=Df=0.1%,f=2.5 Hz,andλ=1.92.

Fig.7.Comparison of factor of safety from the present method with that from the pseudo-static method for different values of kh.f=1 Hz and 2.5 Hz,λ=1.92 and 1.78,H=44 m,b=4 m,φ =35°,D=Df=0.1%,β =21.8°,and Hf=H/3.

Fig.7 illustrates the effect of frequency of the input motion on the seismic factor of safety of a tailings dam for different values ofkh.For an input frequencyf=1 Hz,the factor of safety from the present analysis with the flexible foundation is a maximum of 12.5%lower than that from the pseudo-static method.Conversely,for another input frequencyf=2.5 Hz,the factor of safety from the present analysis with the flexible foundation is a maximum of 16%greater than that from the pseudo-static method.This can be attributed to the mode change behavior of the dam and can be described with the help of Fig.8.The variation of the horizontal acceleration coefficients for the input frequencies of 1 Hz and 2.5 Hz is depicted in Fig.8.The tailings dam vibrates in the first mode of vibration for the input frequency of 1 Hz.In this mode,all the inertial forces act in the same direction,as shown in Fig.8.But for the input frequency of 2.5 Hz,the dam enters into the second mode of vibration.This can be the possible reason for a lower factor of safety atf=1 Hz and a higher factor of safety atf=2.5 Hz compared to the pseudo-static method.Therefore,it is indicated that the tailings dam may be prone to damage for low-frequency input excitations.

Fig.8.Variation of horizontal acceleration coefficient along the height of the tailings dam.f=1 Hz and 2.5 Hz,λ=1.92 and 1.78,H=44 m,b=4 m,and D=Df=0.1%.

3.2.Parametric study

Fig.9 shows the variation of the factor of safety from the present study with rigid and flexible foundations for different values of internal friction angle.As can be seen,increase in the internal friction angle results in a higher factor of safety.For the set of input parameters,the factor of safety is increased by 56%when the internal friction angle is varied from 20°to 40°.Fig.9 also shows the variation of factor of safety for two different values ofkh.For the set of input parameters,atkh=0.1,the factor of safety from the present analysis with rigid and flexible foundations is nearly 8%-9%and 14%-15%higher than those atkh=0.2,respectively.The factor of safety from the present analysis with the rigid foundation is greater than that from the analysis with flexible foundation.The reason for this has been already explained in the previous sections.

Fig.9.Variation of factor of safety from the present method with rigid and flexible foundations for different values of φ.kh=0.1 and 0.2,H=44 m,b=4 m,f=3 Hz,λ =1.43,Hf=H/3,β =21.8°,and D=Df=0.1%.

Table 2 Factor of safety values from the present method with flexible foundation for different values of kh.f=1.5 Hz,Hf=H/3,H=44 m,b=4 m,φ =35°,β =21.8°,and D=Df=0.1%.

Fig.10.Variation of horizontal acceleration coefficient along the height of the tailings dam.f=1.5 Hz,λ=2,1.5 and 1,H=44 m,b=4 m,and D=Df=0.1%.

Table 2 lists the factor of safety values from the present analysis with the flexible foundation for different values ofkhandλ(impedance ratio).For the given input parameters,the factor of safety is significantly decreased whenλis changed between2 and 1.The impedance ratioλ＞1 signifies a softer foundation.Whenλis decreased from 2 to 1,the amplification of seismic acceleration occurs,as shown in Fig.10.Consequently,the net amount of seismic acceleration acting on the dam is also increased which in turn causes the decrease in the factor of safety.Furthermore,when the impedance ratio is equal to 1,it indicates that no reflected wave is produced and the impedances on each side of the interface are the same.The least factor of safety is witnessed for the impedance ratio of 1 andkh=0.3.

Table 3 gives the factor of safety from the present analysis with the flexible foundation for different values ofkhand normalizedfoundation depthHf/H.For the given set of input parameters,the factor of safety is decreased bya maximum of 17.5%when the depth of foundation is varied fromH/3 to 2H/3.With the further increase in the depth of foundation to 3H/2,the factor of safety is increased by a maximum of 21%compared to the values atHf=2H/3.Therefore,it is indicated that the thickness of the foundation has a significant effect on the seismic factor of safety of a tailings dam.

Table 3 Factor of safety values from the present method with flexible foundation for different values of Hfand kh.f=2.5 Hz,λ=1,H=44 m,b=4 m,φ =35°,β =21.8°,and D=Df=0.1%.

Analysis of the mode change behavior is an important prerequisite for the safe and economical design of tailings dam.The physical meaning of“mode change”behavior in the dam is elucidated further using Fig.11a and b.In the case of low-frequency or long-wavelength shear waves,the dam vibrates in the first mode in which the total seismic acceleration along the depth of the failure wedge acts in one single direction,as shown in Fig.11a,and therefore,it attracts greater inertial force which in turn leads to lower factor of safety.In contrast,for high-frequency or shortwave length shear waves,the dam enters into a higher mode of vibration(second and greater)in which a portion of seismic acceleration acts in one direction and the remaining acts in the opposite direction,as shown in Fig.11b.Consequently,the net amount of seismic inertial force acting on the dam is reduced,and thus a higher factor of safety is induced.

4.Conclusions

The seismic stability analysis is an essential component of the earthquake resistance design of tailings dams.The existing methods suffer from numerous shortcomings as discussed in the Introduction.Therefore,in the present paper,a new simplified method has been developed and presented.The proposed method is capable of considering the effect of foundation soil properties on the factor of safety of tailings dams.An attempt has been made to use strain-dependent dynamic properties(shear modulus and damping ratio)for the evolution of seismic stability of tailings dams.

Under the static conditions(i.e.kh=0),the factor of safety from the present method is exactly the same as that from the pseudostatic method.Under the given combination of input parameters,the factor of safety for low-frequency in put excitations is nearly 26%lower than that for high-frequency input excitations.This can be attributed to the mode change behavior of the tailings dam.

The mode change behavior of tailings dam should be considered for its safe and economical design.The impedance ratio(at the interface between the dam and foundation)and the depth of foundation have the significant effect on the seismic factor of safety of tailings dams.Tailings dams are vulnerable to damage for low frequency input motions.

Fig.11.Schematic diagram representing mode change behavior in tailings dam.

Conflicts of interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

List of notation

ahjHorizontal seismic acceleration in the dam

ahjfHorizontal seismic acceleration in the foundation

bTop width of the dam

DDamping ratio of the tailings dam material

DfDamping ratio of the foundation soil

fFrequency of base excitation

FSFactor of safety of tailings dam

gAcceleration due to gravity

GmaxSmall strain shear modulus

GsecSecant shear modulus

HHeight of the tailings dam

HfDepth of the foundation

hjThickness of thej-th slice

k*Complex wave number of the tailings dam material

Complex wave number of the foundation soil

khHorizontal seismic acceleration coefficient

ljTop width of thej-th slice

mjMass of thej-th slice

NjNormal force acting on the base of thej-th slice

qhjHorizontal inertial force acting on thej-th slice

RRadius of circular slip circle

SjShear force acting on the base of thej-th slice

TPeriod of lateral shaking

tTime

uhjHorizontal displacement in the dam

uhjfHorizontal displacement in the foundation

VjVertical inter-slice force for thej-th slice

VsShear wave velocity of the tailings dam material

VsfShear wave velocity of the foundation soil

YjDistance from the free surface of the dam to the topj-th slice

WjWeight of thej-th slice

αjAngle of the base of thej-th slice with horizontal

β Slope angle of the dam

γ Unit weight of tailings dam material

γfUnit weight of foundation soil

γsShear strain in the dam

γsfShear strain in the foundation soil

θhExit angle for the slip surface

θoEntry angle for the slip surface

λ Impedance ratio

ρ Density of tailings dam material

ρfDensity of foundation soil

φ Internal friction angle

ω Angular frequency of base shaking