Introduction

Rapid urbanization and sustained development in the living standards have rapidly increased municipal solid waste (MSW) generation, all over the world. The three major techniques for MSW treatment and disposal are incineration, composting, and landfills. Landfilling is the most commonly used technique for disposal of waste from different sources, as it provides most convenient and economical way of disposal. Engineered landfills reduce the harmful effects of solid waste on public health and environment. The modern engineered landfill consists of four main components bottom liner system, leachate collection system, gas collection system, and final cover system. Rapid increase in solid waste amounts and the exorbitant rates for acquiring new lands have motivated the engineers to design the landfills with greater heights and steeper slopes in order to increase waste filling capacity per unit area. The stability of these landfills is important for operating landfills and (or) closed landfills under both static and dynamic conditions.

As per the regulations of U.S. Federal Resource Conservation and Recovery Act under subtitle D (40 CFR Part 258, Criteria of Municipal Solid Waste Landfills), the new landfills to be constructed in seismic impact zones should be earthquake resistant. The study on the seismic stability of landfills got importance after Northridge earthquake, 1994 \(\left( M_{w} = 6.7 \right)\). During this earthquake around 22 landfills were subjected to ground accelerations of greater than 0.05 g, and out of these eight were geosynthetic lined landfills. Tearing of geosynthetic liners, cracking of cover soil and some changes in the geometry of landfill were observed [1, 2]. Significant amplification of peak ground accelerations for several earthquake records at OII landfill was observed. The amplification of accelerations was clearly noticed under low-frequency ground motions at OII landfill site [3]. So far, the performance of MSW landfills during earthquakes was good, but very few modern landfills (with geosynthetic liners and cover systems) were exposed to severe ground shaking. The damages to landfill components may cause obstructions for the daily operations of landfill such as landfill gas extraction and leachate collection. The temporary shutdown of gas extraction systems has a significant consideration as it may lead to fire accidents or big explosions.

Seismic stability analysis of landfills is an important part of the seismic design of landfills. The two major failure types identified from landfill failure case histories are the rotational and translational mode of failures. In the case of landfills with geosynthetic base layers, the predetermined critical failure surface is along the liner system which causes translational failure [4,5,6,7,8]. In the present study translational mode of failure is considered because the failure of liner system may cause obstructions for landfill functionality. Shewbridge [9] proposed simplified expression for calculation of yield acceleration of lined landfills of a typical trapezoidal landfill. The researcher assumed the waste mass as a rigid body, but that assumption is only valid for material with infinite shear wave velocity. In reality, the velocity of the shear wave in municipal solid waste is very low. Choudhury and Savoikar [10] proposed a methodology to compute the seismic yield acceleration of landfills with different geometric configurations using a pseudo-dynamic method that considers a finite value of shear wave velocity. The method only considered the vertically propagating shear wave and neglected the interference effect of the shear wave reflected from the top surface of the landfill. And most importantly, the method considered low strain shear wave velocity throughout the calculation whereas 1D ground response analysis reports a considerable reduction of shear wave velocity of MSW material under seismic condition. Qian and Koerner [5, 6] and Choudhury and Savoikar [11] applied the pseudo-static method of analysis to study the seismic translational stability of side hill type MSW landfill, considering the waste shear strength. Savoikar and Choudhury [7, 8] improved the two-part wedge solution by using the pseudo-dynamic method. However, these methods are also having some limitations such as, assuming a linear variation of amplification acceleration and neglecting the damping properties of MSW.

Pain et al. [12,13,14,15] proposed the modified pseudo-dynamic method by addressing the above limitations. The methodology satisfies the boundary conditions. Recently, Annapareddy et al. [16] applied similar method for assessing the translational stability of MSW landfills under seismic conditions. However, the method assumed a linear homogeneous landfill material and also used the constant low strain dynamic properties in their analysis. This linear assumption may hold well for minor earthquakes, but during moderate to severe earthquakes, the behaviour of solid waste is nonlinear and the degree of nonlinearity plays an important role [17]. Moreover, in their analysis the internal cohesion of the solid waste was ignored and which has significant effect on the seismic factor of safety of MSW landfills [5, 7]. Hashash et al. [18] compared the results of 1D equivalent linear and nonlinear seismic response analysis of dry Nevada sand using DEEPSOIL [19] with centrifuge experimental results. The study reported that the results from equivalent linear and nonlinear seismic response analysis are as good as the results from centrifuge experiments.

The parameters those have an influence on the dynamic response of MSW landfills includes geometry and unit weight of landfill, modulus reduction and damping of waste, characteristics of input ground motion etc. Dynamic response of MSW landfills could be studied using the available computer programs such as SHAKE2000 [20], DEEPSOIL [19]. However, these computer programs are not capable of estimating the factor of safety of landfill under seismic condition. Numerical methods based on finite element approach (QUAD4M [21]) are available for computing the factor of safety of landfills. The numerical solution requires advanced constitutive laws, and the main shortcoming of most of these laws is that they involve a great number of parameters some of which even don’t have physical meaning or it is difficult to evaluate from the routine test. Bray et al. [22] carried out 1D site response analysis on model MSW landfills founded on five different subsurface conditions using SHAKE91 [23]. Equivalent linear model is used to model the material non-linearity as a function of cyclic shear strain. Results of this study indicated significant amplification of base acceleration for the entire height of landfill with a maximum amplification near the top of the landfill. Rathje and Bray [24] studied the reliability of using 1D site response analysis to predict dynamic response of landfills. The study compared 1D site response analysis in SHAKE91 with 2D finite element simulation using QUAD4M. The results indicated that 1D analysis could reasonably predict seismic loading and seismically induced permanent displacements for deep sliding surfaces (such as sliding along liners) conservatively. The effects of local site conditions on the dynamic response of MSW landfill was studied by Psarropoulos et al. [17] using 2D finite element program QUAD4M. The results indicated that the dynamic response of landfills depends not only on the sub-surface conditions and input excitations but the geometrical and material properties of landfill also play a crucial role. Choudhury and Savoikar [25] performed 1D equivalent linear site response analysis on five model MSW landfills to study, the effects of landfill height, foundation, material characteristics and base excitations using DEEPSOIL. Results indicated a higher amplification for landfills with smaller heights, and with low seismic base accelerations. The 1D non-linear dynamic response analysis of MSW landfill located in Bangalore, India was done by Anbazhagan et al. [26] using DEEPSOIL. Amplification of base input acceleration was observed at the top surface of the landfill. The high amount of amplification of acceleration may disturb the stability condition or damage the landfill cover system. Recently, Huang and Fan [27] presented a very comprehensive literature review on the external forces (excessive rainfall, earthquake, human engineering activities etc.) affecting the stability of MSW landfills.

To the best of author’s knowledge, there is no analytical method available that may consider the effects of amplification and material non-linearity on the factor of safety of landfills. In the present study, an attempt has been made for the first time to evaluate the seismic factor of safety of MSW landfills using the strain-dependent dynamic properties (shear modulus and damping ratio), an equivalent linear approach. In the present analysis, both internal friction and cohesion of the solid waste is considered. Most importantly, the present method considers the mode change behaviour of the landfill and this must be considered in the evolution of inertial forces for a safe and economical design of landfills. The objective of the present study is to develop an integrated methodology that gives the seismic acceleration profile in the landfill as well as the factor of safety and yield acceleration values for a particular geometric configuration. The amplitude of the stress waves attenuate, as they propagate through a homogeneous elastic medium. The two important sources for the attenuation of stress waves is the material through which the stress wave travels and the other is the geometry of stress wave propagation. The attenuation of wave associated with the material type is named as ‘material damping’ and subsequently, the attenuation of wave associated the geometry is named as ‘radiation damping’. In the present analysis, material damping is addressed. The model MSW landfill is assumed to be rested over a rigid horizontal stratum and thus neglected the effects of radiation damping [17, 24, 28]. Equivalent linear site response analysis is also performed using DEEPSOIL and the acceleration ratios (ratio of surface acceleration of landfill to the input acceleration) from the present analytical method are compared with the DEEPSOIL results.

Present Study

Proposed Methodology

The governing differential equation for a plane SH-wave travelling through a homogeneous Kelvin–Voigt medium of thickness \(H\) is given in Eq. (1) [13, 15, 16].

$$\rho_{sw} \frac{{\partial^{2} u_{h} }}{{\partial t^{2} }} = G\frac{{\partial^{2} u_{h} }}{{\partial z^{2} }} + \eta_{sw} \frac{{\partial^{3} u_{h} }}{{\partial z^{2} \partial t}}$$
(1)

The general solution of Eq. (1) may be represented as follows,

$$u_{h} \left( {z,t} \right) = C_{1} e^{{i\left( {\omega t - k^{*} z} \right)}} + C_{2} e^{{i\left( {\omega t + k^{*} z} \right)}}$$
(2)

where, C1 and C2 are constants. To solve Eq. (2), two boundary conditions are required. As per the boundary conditions, shear stress is zero when \(z = 0\) and the displacement is equal to base displacement when \(z = H\). Assuming base displacement \(u_{b} = u_{ho} \cos (\omega t)\), the equation for horizontal displacement may be represented as given in Eq. (3).

$$u_{h} \left( {z,t} \right) = \frac{{u_{h0} }}{{C^{2} + S^{2} }}\left[ {\left( {CC_{z} + SS_{z} } \right)\cos \left( {\omega t} \right) + \left( {SC_{z} - CS_{z} } \right)\sin \left( {\omega t} \right)} \right]$$
(3)

where

$$C = \cos \left( {y_{1} } \right)\cosh \left( {y_{2} } \right)$$
(4a)
$$S = - sin\left( {y_{1} } \right)sinh\left( {y_{2} } \right)$$
(4b)
$$C_{z} = \cos \left( {\frac{{y_{1} z}}{H}} \right)\cosh \left( {\frac{{y_{2} z}}{H}} \right)$$
(4c)
$$S_{z} = - sin\left( {\frac{{y_{1} z}}{H}} \right)sinh\left( {\frac{{y_{2} z}}{H}} \right)$$
(4d)
$$y_{1} = \frac{\omega H}{{V_{s} }}\sqrt {\frac{{\sqrt {1 + 4\xi^{2} } + 1}}{{2\left( {1 + 4\xi^{2} } \right)}}}$$
(5a)
$$y_{2} = - \frac{\omega H}{{V_{s} }}\sqrt {\frac{{\sqrt {1 + 4\xi^{2} } - 1}}{{2\left( {1 + 4\xi^{2} } \right)}}}$$
(5b)

The partial derivative of Eq. (3) with respect to ‘\(z\)’ gives the expression for shear strain, \(\gamma_{s}\) as a function of depth (z) and time (t), and the expression is as follows;

$$\gamma_{s} (z,t) = \frac{\partial u(z,t)}{\partial z} = \frac{{u_{ho} }}{{C^{2} + S^{2} }}\left[ {A_{h} \cos \left( {\omega t} \right) + B_{h} \sin \left( {\omega t} \right)} \right]$$
(6)

where,

$$A_{h} = y_{2} C_{{c_{z} }} C_{c} - y_{1} S_{{s_{z} }} C_{c} + y_{1} C_{{c_{z} }} S_{s} + y_{2} S_{{s_{z} }} S_{s}$$
(7a)
$$B_{h} = y_{1} S_{{s_{z} }} S_{s} - y_{2} C_{{c_{z} }} S_{s} + y_{1} C_{{c_{z} }} C_{c} + y_{2} S_{{s_{z} }} C_{c}$$
(7b)

where

$$C_{{c_{z} }} = \frac{1}{H}\left[ {\cos \left( {\frac{{y_{1} z}}{H}} \right)\sinh \left( {\frac{{y_{2} z}}{H}} \right)} \right]$$
(8a)
$$S_{{s_{z} }} = \frac{1}{H}\left[ {\sin \left( {\frac{{y_{1} z}}{H}} \right)\cosh \left( {\frac{{y_{2} z}}{H}} \right)} \right]$$
(8b)
$$C_{c} = \cos \left( {y_{1} } \right)\cosh \left( {y_{2} } \right)$$
(8c)
$$S_{s} = \sin \left( {y_{1} } \right)\sinh \left( {y_{2} } \right).$$
(8d)

It is worth mentioning that modulus reduction and damping ratio curves are required in the present solution. These two curves could be developed by performing a series of cyclic triaxial tests on MSW samples from a specific landfill. However, in the absence of specific landfill data, curves available in the literature can be used. In the present analysis, stiffness degradation and damping curves available in the literature are adopted. Zekkos et al. [29] proposed the shear modulus and damping curves of MSW from a landfill in the San Francisco, USA. Similarly, Ramaiah et al. [30, 31] proposed the shear modulus and damping curves of MSW from two waste dumps in Delhi, India. Choudhury and Savoikar [32] proposed the modulus reduction and damping curves of MSW by using worldwide data available in the literature, with the help of best curve fitting techniques. Therefore, in the present study, the curves that are suggested by Choudhury and Savoikar [32] are used. The mathematical expressions for these curves are as follows;

The expression for modulus reduction curve is;

$$\frac{{G_{sec} }}{{G_{max} }} = \frac{1}{{1 + 7.85 \gamma_{u}^{0.95} }}$$
(9a)

The expression for damping ratio curve is;

$$\xi = 30\left( {1 - e^{{ - 1.4 \gamma_{u}^{0.36} }} } \right)$$
(9b)

where, \(\gamma_{u}\) is the percentage shear strain; \(G_{sec}\) is the secant shear modulus; and \(G_{max} = V_{s}^{2} \rho_{sw}\) is the low strain shear modulus.

Differentiate Eq. (3) twice with respect to time to get the expression for horizontal acceleration, and the expression is as follows;

$$a_{h} \left( {z,t} \right) = \frac{{k_{h} g}}{{C^{2} + S^{2} }}\left[ {\left( {CC_{z} + SS_{z} } \right)\cos \left( {\omega t} \right) + \left( {SC_{z} - CS_{z} } \right)\sin \left( {\omega t} \right)} \right]$$
(10)

where, \(k_{h} g = - \omega^{2} u_{h0}\). This expression is used to compute the seismic force acting on the landfill body.

The model MSW landfill used in the present study is shown in Fig. 1, with all the forces acting over the landfill. The entire waste mass is divided into two wedges namely active wedge and passive wedge resting on back slope liner and base liner respectively. These two wedges are separated by an imaginary wall for the mathematical treatment [4]. Following are the assumptions of this method.

Fig. 1
figure 1

Model side-hill type MSW landfill used in the present study

Full size image
  1. 1.

    The FS is assumed to be same along the failure surface.

  2. 2.

    The resultant inter-wedge force is assumed to be inclined at an unknown angle of \(\varepsilon\) with the horizontal and acts at a distance of H/3 above the base of the interface.

  3. 3.

    In order to meet the waste shear failure criteria at the interface, the FS at the interface (FSV) should not be less than unity.

  4. 4.

    The FSV must not be less than the FS.

Inter-wedge forces acting at the interface between active and passive wedges may be given as;

$$\begin{aligned} E_{Vp} & = E_{Hp} \cdot {{\tan \phi_{sw} } \mathord{\left/ {\vphantom {{\tan \phi_{sw} } {FS_{V} }}} \right. \kern-0pt} {FS_{V} }} + {{C_{sw} } \mathord{\left/ {\vphantom {{C_{sw} } {FS_{V} }}} \right. \kern-0pt} {FS_{V} }} \\ & = E_{Hp} \cdot m_{sw} + n_{sw} \quad \quad \left( {\begin{array}{*{20}l} {{\text{say, }}{{{ \tan }\phi_{sw} } \mathord{\left/ {\vphantom {{{ \tan }\phi_{sw} } {FS_{V} = m_{sw} }}} \right. \kern-0pt} {FS_{V} = m_{sw} }}} \hfill \\ {{{C_{sw} } \mathord{\left/ {\vphantom {{C_{sw} } {FS_{V} = n_{sw} }}} \right. \kern-0pt} {FS_{V} = n_{sw} }}} \hfill \\ \end{array} } \right) \\ \end{aligned}$$
(11a)
$$\begin{aligned} E_{Va} & = E_{Ha} \cdot {{\tan \phi_{sw} } \mathord{\left/ {\vphantom {{\tan \phi_{sw} } {FS_{V} + {{C_{sw} } \mathord{\left/ {\vphantom {{C_{sw} } {FS_{V} }}} \right. \kern-0pt} {FS_{V} }}}}} \right. \kern-0pt} {FS_{V} + {{C_{sw} } \mathord{\left/ {\vphantom {{C_{sw} } {FS_{V} }}} \right. \kern-0pt} {FS_{V} }}}} \\ & = E_{Ha} \cdot m_{sw} + n_{sw} \\ \end{aligned}$$
(11b)

Frictional forces acting along the liners of active and passive wedges may be given as;

$$F_{p} = N_{p} {{ \cdot \tan \delta_{p} } \mathord{\left/ {\vphantom {{ \cdot \tan \delta_{p} } {FS_{p} }}} \right. \kern-0pt} {FS_{p} }} + {{C_{p} } \mathord{\left/ {\vphantom {{C_{p} } {FS_{p} }}} \right. \kern-0pt} {FS_{p} }}$$
(11c)
$$F_{a} = N_{a} \cdot {{\tan \delta_{a} } \mathord{\left/ {\vphantom {{\tan \delta_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }} + {{C_{a} } \mathord{\left/ {\vphantom {{C_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}$$
(11d)

Horizontal seismic inertial forces acting on the active and passive wedges may be given as;

$$Q_{Ha} \left( t \right) = \int\limits_{0}^{H} {m_{a} } \left( z \right) \cdot a_{h} \left( {z,t} \right) \cdot dz$$
(12a)
$$Q_{Hp} \left( t \right) = \int\limits_{{H_{1} }}^{H} {m_{p} } \left( z \right) \cdot a_{h} \left( {z,t} \right) \cdot dz$$
(12b)

Consider the force equilibrium of active and passive wedge and then rearrange the terms to obtain EHa and EHp.

$$E_{Ha} = \frac{{\left( {W_{a} + n_{sw} } \right) \cdot \left( {\sin \beta - \cos \beta \cdot {{\tan \delta_{a} } \mathord{\left/ {\vphantom {{\tan \delta_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}} \right) + Q_{Ha} \cdot \left( {\cos \beta + \sin \beta \cdot {{\tan \delta_{a} } \mathord{\left/ {\vphantom {{\tan \delta_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}} \right) - {{C_{a} } \mathord{\left/ {\vphantom {{C_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}}}{{\cos \beta + sin\beta \cdot {{tan\delta_{a} } \mathord{\left/ {\vphantom {{tan\delta_{a} } {FS_{a} + m_{sw} \cdot \left( {\sin \beta - \cos \beta \cdot {{\tan \delta_{a} } \mathord{\left/ {\vphantom {{\tan \delta_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}} \right)}}} \right. \kern-0pt} {FS_{a} + m_{sw} \cdot \left( {\sin \beta - \cos \beta \cdot {{\tan \delta_{a} } \mathord{\left/ {\vphantom {{\tan \delta_{a} } {FS_{a} }}} \right. \kern-0pt} {FS_{a} }}} \right)}}}}$$
(13a)
$$E_{Hp} = \frac{{\left( {W_{p} + n_{sw} } \right) \cdot {{\tan \delta_{p} } \mathord{\left/ {\vphantom {{\tan \delta_{p} } {FS_{p} - Q_{Hp} + {{C_{p} } \mathord{\left/ {\vphantom {{C_{p} } {FS_{p} }}} \right. \kern-0pt} {FS_{p} }}}}} \right. \kern-0pt} {FS_{p} - Q_{Hp} + {{C_{p} } \mathord{\left/ {\vphantom {{C_{p} } {FS_{p} }}} \right. \kern-0pt} {FS_{p} }}}}}}{{1 - m_{sw} \cdot {{\tan \delta_{p} } \mathord{\left/ {\vphantom {{\tan \delta_{p} } {FS_{p} }}} \right. \kern-0pt} {FS_{p} }}}}$$
(13b)

For the equilibrium of entire waste mass, requires \(E_{Ha} = E_{Hp}\). Say, \(FS_{a} = FS_{p} = FS\). After rearranging the terms, the final factor of safety equation may be of the form;

$$a \cdot FS^{2} + b \cdot FS + c = 0$$
(14)

where

$$\begin{aligned} a & = Q_{Hp} \cdot \sin \beta \cdot m_{sw} + W_{a} \cdot \sin \beta + \left( {Q_{Ha} + Q_{Hp} } \right) \cdot \cos \beta \, - n_{sw} \cdot \sin \beta ; \\ b & = - \left[ {\begin{array}{*{20}l} {W_{t} \cdot \sin \beta \cdot \tan \delta_{p} \cdot m_{sw} + W_{p} \cdot \cos \beta \cdot \tan \delta_{p} + W_{a} \cdot \cos \beta \cdot \tan \delta_{a} + Q_{Hp} \cdot \cos \beta \cdot \tan \delta_{a} \cdot m_{sw} } \hfill \\ { + \,Q_{Ha} \cdot \cos \beta \cdot \tan \delta_{p} \cdot m_{sw} - \left( {Q_{Ha} + Q_{Hp} } \right) \cdot \sin \beta \cdot \tan \delta_{a} + n_{sw} \cdot \cos \beta \left( {\tan \delta_{p} - \tan \delta_{a} } \right)} \hfill \\ { + \,C_{p} \cdot \sin \beta \cdot m_{sw} + C_{a} + C_{p} \cdot \cos \beta } \hfill \\ \end{array} } \right]; \\ c & = W_{t} \cdot \cos \beta \cdot \tan \delta_{a} \cdot \tan \delta_{p} \cdot m_{sw} - W_{p} \cdot \sin \beta \cdot \tan \delta_{a} \cdot \tan \delta_{p} - Q_{Ha} \cdot \sin \beta \cdot \tan \delta_{a} \cdot \tan \delta_{p} \cdot m_{sw} \\ & \quad - n_{sw} .\tan \delta_{a} \cdot \tan \delta_{p} \cdot \sin \beta - C_{p} \cdot \sin \beta .\tan \delta_{a} + m_{sw} \left( {C_{p} \cdot \cos \beta .\tan \delta_{a} + C_{a} \cdot \tan \delta_{p} } \right){\kern 1pt} \\ \end{aligned}$$
(15)

Let, \(m_{sw} = 0\,{\text{and}}\,n_{sw} = 0\) (i.e., by neglecting the waste strength) the minimum FS, \(FS_{\rm min }\) can be calculated using Eq. (14). This may be directly used for design as it gives more conservative results. For FS < 1, take FSV = 1 because FSV must not be less than unity based on the previous assumption. For FS ≥ 1, take FSV = FS, which means that maximum values of \(m_{sw} = {{\tan (\phi_{sw} )} \mathord{\left/ {\vphantom {{\tan (\phi_{sw} )} {FS}}} \right. \kern-0pt} {FS}}\,{\text{and}}\,n_{sw} = {{C_{sw} } \mathord{\left/ {\vphantom {{C_{sw} } {FS}}} \right. \kern-0pt} {FS}}\) (i.e., by considering the waste strength fully) the maximum FS, \(FS_{\rm max }\) can be calculated using Eq. (14). After obtaining the lower and upper bound FS, the average FS, \(FS_{avg}\) could be determined using present method, in place of true FS, \(FS_{true}\). Using the present method, the absolute maximum difference between \(FS_{true}\) and \(FS_{avg}\) may also be predicted using, either difference between \(FS_{avg}\) and \(FS_{\rm min }\) or the difference between \(FS_{\rm max }\) and \(FS_{avg}\). Generally, this difference should be within 5% for most of the cases. The procedure for the calculation of factor of safety is briefly explained in the above section. For more detailed explanation readers may refer to [5, 10]. The procedure for the calculation of factor of safety is coded using MATLAB [33] program.

The following are the programming steps:

  1. 1.

    Develop modulus reduction and damping curves of MSW from laboratory tests performed on an MSW from specific landfill. In the absence of those, curves available in the published literature may be used.

  2. 2.

    Assign the small strain values of shear modulus (G) and damping ratio (ξ).

  3. 3.

    At a specific time step use shear modulus (G) and damping ratio (ξ) to compute the maximum shear strain along the depth of the landfill using Eq. (6).

  4. 4.

    Use the maximum shear strain to obtain a new set of values (\(G^{{\left( {i + 1} \right)}}\) and \(\xi^{{\left( {i + 1} \right)}}\)) using the stiffness degradation and damping ratio curves for the successive iteration.

  5. 5.

    Repeat the steps 3–4 until the difference between the calculated shear modulus and damping ratio in two successive iterations is in between than 5–10%.

  6. 6.

    The shear modulus and damping ratio corresponding to the last iteration are the equivalent linear values. Use the same values to obtain the acceleration profile along the depth at that specific time step.

  7. 7.

    Use the acceleration profile to calculate seismic inertial forces at the same time step.

  8. 8.

    Use the same inertial forces to compute the factor of safety values for the same time step using the proposed equations.

  9. 9.

    Repeat the steps 2–8 for the next time step.

The minimum acceleration coefficient required for the yielding of waste mass along the failure plane may be termed as yield acceleration coefficient ky. Yielding of waste mass just starts when the factor of safety is unity. So, the yield acceleration coefficient is the minimum acceleration coefficient corresponding to a factor of safety of unity. Similar to the factor of safety, the average yield acceleration coefficient could be calculated using the present method. Yield acceleration coefficient can be used for the estimation of seismically induced permanent displacements in a landfill.

Linear and Equivalent Linear Ground Response Analysis Using DEEPSOIL

Site-specific ground response analysis is a very handy tool to access the acceleration distribution in an MSW landfill body when it is subjected to base shaking. The ground response analysis may be categorized into three types: linear, equivalent linear and non-linear. Many researchers had employed an equivalent linear approach to quantify the amplification or de-amplification of the input seismic ground motion in the MSW landfill [22, 25, 26]. In the present analysis, a 1-D ground response analysis is carried out using DEEPSOIL linear and equivalent linear analysis. The height of the MSW column is kept equal to the total height of the landfill. A small strain shear wave velocity and a fixed damping ratio are used in the DEEPSOIL linear analysis. The parameters H = 30 m, Vs = 150 m/s, γsw = 10.5 kN/m3 and ξ = 10% are used as input parameters for the linear analysis. The boundary condition in DEEPSOIL and in the proposed method is kept same i.e., the MSW column is resting over rigid bedrock. The base of the MSW column is subjected to a harmonic shaking khgcos(ωt). The duration of excitation, t is kept equal to the period of excitation, T = (2π/ω). The seismic acceleration at the free surface is obtained using the frequency domain DEEPSOIL linear analysis. Acceleration ratio that is defined as the ratio of acceleration at the top of the MSW column to the input acceleration is also computed in the present analysis.

In reality, the shear strain generated during a seismic event is significantly large. It is also well established in geotechnical earthquake engineering that the shear modulus decreases significantly with the increase in the shear strain and at the same time damping ratio value increases. The equivalent linear approach of ground response analysis uses two curves namely modulus reduction curve and damping ratio curve to arrive at a strain-dependent shear modulus and damping ratio value. It is an iterative process. In the present study, the modulus reduction curve and damping ratio curve proposed by Choudhury and Savoikar [32] is used in the DEEPSOIL equivalent linear analysis. Unit weight of the MSW, boundary condition and the base input motion are kept same as the linear analysis. Acceleration ratio is also computed using the DEEPSOIL equivalent linear analysis.

Results and Discussions

Comparison of Acceleration Ratio from a Present Analytical Method with the Results of 1D Site Response Analysis, DEEPSOIL

Distribution of acceleration along the depth of waste mass is very important for the evaluation of seismic inertial forces over the landfill. The acceleration profiles obtained from the linear and equivalent linear analyses using the present analytical method are validated against the results of the corresponding analyses using DEEPSOIL. A non-dimensional parameter named as acceleration ratio is computed. The ratio of surface acceleration to that of input acceleration is known as acceleration ratio or, sometimes called as amplification ratio. The acceleration ratios obtained from the present analytical method are in good agreement with the DEEPSOIL results as shown in Fig. 2. It also shows the comparison of results from linear and equivalent linear analysis using the present method as well as DEEPSOIL.

Fig. 2
figure 2

Comparison of acceleration ratio obtained from the proposed linear and equivalent linear based approach with DEEPSOIL

Full size image

In the case of linear analysis, a constant low strain shear modulus and damping properties are used. In the present analysis low strain Gmax = 24.082 MPa (\(V_{s} = \,150\) m/s), \(\gamma_{sw} = 10.5\) kN/m3 and \(\xi\) = 10% are used as input parameters. For an input frequency of \(f =\) 0.4 Hz (\(\omega = 2\pi f = 2.513\) rad/s), the acceleration ratio from the present linear analysis is 1.127 and from the DEEPSOIL linear analysis is 1.129. Similarly, for another input frequency of \(f = 1\) Hz (\(\omega = 2\pi f = 6.283\) rad/s), the acceleration ratio from the present linear analysis is 3.104 and from DEEPSOIL linear analysis is 3.148. The percentage difference in the case of linear analysis is very less; in fact it is less than 1.5%.

In the case of equivalent linear analysis, the properties such as modulus reduction and damping increase with an increase in cyclic shear strain are considered. The strain-dependent dynamic properties obtained from the present equivalent linear analysis is given in Table 1. For an input frequency of \(f = 0.65\) Hz (\(\omega = 2\pi f = 4.084\) rad/s), the acceleration ratio from the present equivalent linear analysis is 2.61 and from DEEPSOIL equivalent linear analysis is 2.79. Similarly, for another input frequency of \(f = 3\) Hz (\(\omega = 2\pi f\) = 18.849 rad/s), the acceleration ratio from the present equivalent linear analysis is 1.502 and from DEEPSOIL equivalent linear analysis is 1.58. The percentage difference in the equivalent linear analysis is about 5–10%. The equivalent linear analysis graphs are shifted towards the left side of linear analysis graph (see Fig. 2), this might be attributed to the use of strain dependent dynamic properties instead of using low strain dynamic properties.

Table 1 Strain-dependent equivalent linear properties of MSW used in the present study
Full size table

Convergence of the Proposed Iterative Scheme

As explained in the above sections, computation of equivalent linear properties is an iterative procedure. To check the convergence of the proposed method, shear strain at different frequencies is plotted (see Fig. 3). In the present analysis, H = 30 m, \(\gamma_{sw} = 10.5\) kN/m3 and the low strain dynamic properties \(V_{s} = 150\) m/s and \(\xi\) = 10% are used as input parameters. Figure 3 clearly indicates that the results are converging after maximum 15 iterations. The number of iterations needed for convergence of results is changing with input frequency. Furthermore, a change in the input frequency results in the change of maximum shear strain value and its point of occurrence along the depth of the landfill. For an input frequency of f = 0.2 Hz (\(\omega = 2\pi f = 1.256\) rad/s), the maximum shear strain obtained is 0.14% and it occurred in the bottom portion of landfill as shown in Fig. 3a. For an input frequency of f = 0.5 Hz (\(\omega = 2\pi f = 3.142\) rad/s), the maximum shear strain obtained is 0.81% and it occurred just above the base of the landfill (see Fig. 3b). Moreover, the maximum shear strain value decreased significantly with a further increase in the input frequency as the landfill entered into higher mode of vibration. Besides, the point of occurrence of maximum shear strain is shifted towards the top of landfill as in Fig. 3c, d.

Fig. 3
figure 3

Variation of shear strain along the depth of landfill under different input frequencies

Full size image

Comparison of Factor of Safety and Yield Acceleration Coefficient

To access the seismic stability of landfills, the factor of safety values are computed and also compared with the existing pseudo-static and pseudo-dynamic methods. In the present study, a non-dimensional time interval is used in the form of t/T (t is time and T is period of lateral shaking). The factor of safety is minimized by varying t/T at an interval of 0.01 between 0 and 1 in the developed MATLAB [33] program. The input parameters used in the present analysis are H = 30 m, B = 20 m, \(\alpha =\) 14°, \(\beta =\) 18.43°, \(\gamma_{sw} =\) 10.5 kN/m3, \(\delta_{a} = \delta_{p} =\) 10°–40°, \(C_{a} = C_{p} =\) 5–40 kN/m2, \(\phi_{sw} =\) 30°, \(C_{sw} =\) 0–3 kN/m2, \(V_{s} =\) 150 m/s, f = 0–3 Hz and \(\xi\) = 10%. The input parameters used in the present analysis refers to the papers of Savoikar and Choudhury [7, 8].

Effect of interface cohesion of liner materials is shown in Fig. 4 and the comparison of the present study with the pseudo-static and pseudo-dynamic methods is also shown. It may be clearly seen that an increase in interface cohesion increases the factor of safety. The increase in interface cohesion increases the total resistive force, consequently, the factor of safety increases. Results from the present study follow the similar trend as that of pseudo-static and pseudo-dynamic methods. The factor of safety values computed from the present study linear analysis (in which small strain dynamic properties are used) are about 11% higher than the corresponding values from the pseudo-static method of analysis. Moreover, the values are about 7% higher in the case of present study equivalent linear analysis in which equivalent linear dynamic properties are used. For the present set of data, landfill entered into the second mode of vibration which reduces the net seismic force acting on the landfill. Consequently, the values of the factor of safety from the present method are more conservative as compared to the conventional pseudo-static method.

Fig. 4
figure 4

Comparison of the average factor of safety values from the present study with pseudo-static and pseudo-dynamic methods

Full size image

Effect of interface friction angle of liner materials is shown in Fig. 5; the same figure also shows the effect of frequency content. It may be clearly seen that increase in interface friction increases the factor of safety. The increase in interface friction increases the total resisting force and as a result, factor of safety is increased. The factor of safety values is about 25% higher as compare to a pseudo-static method for an input frequency of 2 Hz. But, at the same time factor of safety values are about 4% lesser than the pseudo-static results for an input frequency of 0.2 Hz as shown in Fig. 5. The reason for the change in the factor of safety values by changing input frequency may be drawn with the help of acceleration profiles at same input frequencies (see Fig. 6). When the frequency equals to 0.2 Hz, surface acceleration is amplified by 1.8 times the base input acceleration which increases the magnitude of the seismic inertial force. As a result, the computed factor of safety values are lesser in comparison with the pseudo-static method of analysis. Similarly, when the frequency equals to 2 Hz, landfill mass vibrating in the second mode (i.e., some part of the landfill is moving in one direction and the remaining part is moving in opposite direction as shown in Fig. 6) which reduces the net seismic force acting on the landfill. As a result, the factor of safety values obtained using the proposed method are higher in comparison with the pseudo-static based method. The factor of safety values obtained using present study linear and equivalent linear approach for different interface friction angle values and presented in Table 2. The results are compared with the conventional pseudo-static results. The factor of safety values from the present study are higher than the pseudo-static values and the reason for this stands same as already explained above.

Fig. 5
figure 5

Effect of input frequency on the factor of safety values computed using the proposed equivalent linear based method

Full size image
Fig. 6
figure 6

Acceleration profiles at an input frequency of 0.2 and 2.0 Hz

Full size image
Table 2 Comparison of factor of safety values from the proposed linear and equivalent linear based approach with the conventional pseudo-static analysis for H = 30 m; B = 20 m; kh = 0.05; \(\alpha\) = 14°; \(\phi_{sw}\) = 30°; β = 18.43°; \(\xi =\) 10%; \(\delta_{a}\) = \(\delta_{p}\) = \(\delta\); \(\gamma_{sw}\) = 10.5 kN/m3; \(C_{a} = C_{p} =\) 10 kN/m2; \(C_{sw} =\) 3 kN/m2; \(f =\) 3 Hz; \(V_{s} =\) 150 m/s
Full size table

The yield acceleration coefficients for different combinations of interface shear strength parameters of liner material are computed and presented in Table 3. The interface friction angle is varied from 9° to 21° and the interface cohesion is varied from 0 to 13 kN/m2 and the remaining parameters are kept constant.

Table 3 Average yield acceleration (kyavg) values for different combinations of friction angle (δ) and cohesion (C) of liner materials for H = 30 m; B = 20 m; kh = 0.1; \(\alpha\) = 14°; \(\phi_{sw}\) = 30°; β = 18.43°; \(\delta_{a}\) = \(\delta_{p}\) = δ°; \(\gamma_{sw}\) = 10.5 kN/m3; \(C_{a} = C_{p} =\) C kN/m2; \(C_{sw} =\) 3 kN/m2; \(f =\) 1 Hz
Full size table

Conclusions

Strain-dependent dynamic properties of MSW are computed and used to calculate the acceleration ratio, factor of safety and yield acceleration coefficient for a typical side-hill type model MSW landfill. The following are important conclusions drawn from the present study.

  1. 1.

    Amplification of base acceleration is observed at a low-frequency input motions which is in line with the statements of site response studies of MSW landfills. Since, at low-frequency input motions, the seismic acceleration along the depth of the landfills acts in the same direction.

  2. 2.

    The factor of safety values computed using the proposed equivalent linear based approach are higher compared to the conventional pseudo-static method of analysis for high-frequency input motions. However, the factor of safety values are slightly lower than the pseudo-static values for low frequency input motions. This can be attributed to the mode change behaviour of landfills.

  3. 3.

    The factor of safety values computed using the proposed equivalent linear based approach are lower than the values of linear analysis but, it is still higher than the values of pseudo-static and pseudo-dynamic analysis for a given set of input parameters. This can be attributed to the use of strain-dependent dynamic properties and also the landfill is vibrating at higher mode for the given set of input parameters.

  4. 4.

    The maximum shear strain computed for low-frequency input motion is significantly higher as compared to high-frequency input motion. At low frequency, the seismic force along the landfill depth is acting in the same direction but, at high frequency, some portion of the landfill mass is moving in one direction and the remaining portion is moving in the opposite direction.

  5. 5.

    The Acceleration ratio computed using the proposed linear and equivalent linear based approach is very much comparable with DEEPSOIL linear and equivalent linear results.