# Seismic coefficient

## Methodology

In a pseudo-static slope stability analysis, a factor of safety (FS) is computed using a static limit equilibrium method in which a horizontal earthquake-induced inertial force $$F_{h.max}$$ is applied to the potential sliding mass. This horizontal inertial force represents the destabilising effects of the earthquake shaking and can be expressed as the product of a seismic coefficient $$k_{max}$$ and the weight $$W$$ of the potential sliding mass $$F_{h.max}=k_{max}\,W$$. The limit equilibrium method should satisfy all three conditions of equilibrium to ensure that a reliable FS is calculated. In pseudo-static slope stability analysis, the use of $$k=k_{max}$$ and FS>1.0 with conservative strengths is equivalent to zero displacements (in other words, the maximum driving force never exceeds the resisting force). Adoption of $$k<k_{max}$$ and FS <1.0 indicates that some displacement level will be expected.51

In order to perform pseudo-static stability analyses, an estimate of the horizontal seismic coefficient is carried out. The seismic coefficient represents the maximum horizontal equivalent acceleration ($$\bar{a}_{h.max}$$) expected in a failure wedge in slip potential and is obtained by averaging the horizontal (instantaneous) forces in the deformable sliding mass, according to the equation: $k_{max}=\bar{a}_{h.max}/\mathrm{g}=\mathrm{max}\,{\bar{a}_{h}(t)}=\mathrm{max}\left(\sum_{z=0}^H\,m(z)a_h(z,t)\right)$ In preliminary design stages, a simplified procedure based on an empirical model from Bray et al52 can be adopted, which although having a much larger error term, does not require prior selection of site-specific seismic records. From dhis approach, the seismic coefficient $$k_{max}$$ for a given target displacement $$D_a$$ [cm] can be estimated from the following expression: $\mathrm{ln}\,k_{max}\simeq\dfrac{-a+\sqrt{b}}{c}$ where $$(a,b)=f(S_a(T_d),D_a,M_w)$$ are two parameters from the model that depends on the pseudo-accelerations evaluated in the “shifted” period of the embankment $$T_d$$ and on the magnitude of the earthquake $$M_w$$, and $$c$$ is a constant. The “shifted” period of the embankment $$T_d$$ is obtained from the fundamental period of the embankment $$T_n$$ as $$T_d \approx \alpha T_n$$ , where $$\alpha$$ is a factor between 1.3 and 1.5.

The fundamental period of slopes and embankments can be estimated from a shear-beam model,53 assuming a truncated-wedge geometry with a shear module varying as $$G_m\simeq G_0(z/H_{max})^m$$ where $$z$$ is the distance from the top, $$G_0$$ is the shear module at the surface, and $$H_{max}$$ is the truncated-wedge height. The equation is: $T_n\simeq\dfrac{4\,\pi\,H_{max}}{a_n\,(2-m)\,V_s}$ where $$a_n=f(m,\lambda)$$ is the $$n$$th-root of the characteristic equation (eigenvalues) $$\lambda$$ is the truncation ratio defined as $$\lambda=h/H_{max}=b/B_{max}$$. The parameter $$a_n$$ can be obtained from the model in Dakoulas and Gazetas.54

When waste rock dumps or tailings deposits are founded on soils with shear wave velocities less than 500 m/s, the fundamental period of the embankment may be longer than estimated, depending on the degree of interaction of the foundation-embankment system. For the preliminary design stages (PFS,FS) it is possible to obtain an estimate of the fundamental period of the embankment using the Rayleigh method, assuming a certain stratification of the terrain down to the bedrock. The fundamental period of the foundation-embankment system may be taken as some value between the maximum value between the two periods and the sum of the two periods. In summary, estimation of the seismic coefficient according to Bray methodology, requires the following steps:

Step 1: Define the target level of allowable displacements $$D_a$$ of the TSF according to the performance objectives during the operation, closure and post-closure periods.

Step 2: Estimate the shear-wave velocity of the top 30 m ($$V_{s30}$$) from the shear wave profiles and estimate the amplification factor $$AF^*$$ according to.55 If the stratigraphy of the foundation soil is not yet known, as is often the case in pre-feasibility stages, assume a NEHRP site-class and estimate the site amplification factor $$AF^*$$ according to56

Step 3: Estimate the fundamental period of the embankment $$T_s$$

Step 4: Get spectral ordinates at degraded period $$T_d \approx 1.3 T_s - 1.5 T_s )$$ for target site class $$Sa(T_d)^* \approx AF^* S_a(T_d)^{rock}$$.

Step 5: For a given target magnitude $$M_w$$, obtain an estimate of the seismic coefficient $$k_{max}^* = f\left(M_w,T_d,Sa(T_d)^*\right)$$ from Bray’s model.57 Get the horizontal seismic coefficient defined as $$k_h^* = k_{max}^*/PGA^*$$, where $$PGA^*\approx AF^* PGA^{rock}$$

The hazard level for this model should be fixed at 50% exceedance level (median values).

## Results

The Newmark displacements and the seismic coefficient for a target displacement under two service scenarios for representative embankments, cut slopes and waste rock dumps are reported in this section.

The proposed methodology to calculate the Newmark displacements requires as an additional parameter the fundamental period of the embankment $$T_s$$. For operational scenarios, rigid structures with fundamental periods between 15 and 30 Hz, representing natural soil cuts and low-to-medium height compact embankments will be assumed. For closure and post-closure scenarios, more flexible structures will be assumed, with natural periods between 1.5 - 3.5 Hz, representing medium to high waste rock dumps with relatively loose, uncompacted materials. The structural response is controlled by the “shifted” period for which the pseudo-spectral accelerations on site class C are calculated $$S_a(T_d=\alpha\,T_s)$$. The Newmark displacements for different slip surfaces in terms of their yield coefficient $$k_y$$ [g] are summarized in Table 7 for natural soil cuts and low-to-medium embankments and Table 8 for high waste rock dumps.

 Newmark displacement (+84%) [cm] NEHRP Ts ky TR=500 TR=1000 TR=2500 TR=5000 TR=10000 C 0.03 0.01 1.6 4.0 12.4 26.2 51.2 C 0.03 0.05 0.1 0.3 1.1 3.1 7.8 C 0.03 0.10 0.0 0.1 0.3 0.8 2.4 C 0.03 0.15 0.0 0.0 0.1 0.4 1.1 C 0.03 0.20 0.0 0.0 0.1 0.2 0.6 C 0.03 0.30 0.0 0.0 0.0 0.1 0.2
 Newmark displacement (+84%) [cm] NEHRP Ts ky TR=500 TR=1000 TR=2500 TR=5000 TR=10000 C 0.3 0.01 1.8 3.7 8.9 16.4 29.4 C 0.3 0.05 0.1 0.2 0.8 1.7 3.8 C 0.3 0.10 0.0 0.0 0.2 0.4 1.1 C 0.3 0.15 0.0 0.0 0.1 0.2 0.5 C 0.3 0.20 0.0 0.0 0.0 0.1 0.2 C 0.3 0.30 0.0 0.0 0.0 0.0 0.1

From the Newmark displacements obtained in the tables above it is possible to define two performance objectives. For slopes and natural soil cuts to be designed for a service life equal to the mine operation (O), a design earthquake with an AEP of 1:2500 years and a maximum permanent deformation close to 1” (2.5 cm) is adopted.

For waste rock dumps and critical structures that must remain stable after closure and post-closure stages, a ground-motion with 1:10.000 years AEP on class “C” site conditions is adopted as the target service level and a maximum permanent deformation of the order of 10” (25 cm) has been adopted as a performance objective. Thus, the levels of service for closure and post-closure are consistent with the prescriptions of the ANCOLD 2019, the GISTM and the D5 design guidelines. The recommended seismic coefficient values correspond to the mean value.

For the design of natural soil slopes and low compacted embankments during the construction and operation stages (O), the seismic coefficient associated with the target performance previously stated results $$k_h \approx$$ 0.34 relative to a $$PGA=$$ 0.058 $$\mathrm{g}$$, equivalent to a pseudo-static coefficient $$k_{max} \approx$$ 0.0197 $$\mathrm{g}$$. The pseudo-spectral acceleration for the “shifted” period of the slope is $$Sa(T_d)\approx$$ 0.096 $$\mathrm{g}$$.

For the design of WRD slopes during closure and post-closure stages (C), the seismic coefficient associated with the target performance previously stated (25”) results $$k_h \approx$$ 0.053 relative to a $$PGA=$$ 0.146 $$\mathrm{g}$$, equivalent to a pseudo-static coefficient $$k_{max} \approx$$ 0.0077 $$\mathrm{g}$$. The pseudo-spectral acceleration for the “shifted” period of the slope is $$Sa(T_d)\approx$$ 0.12 $$\mathrm{g}$$.

1. J. D. Bray and T. Travasarou, “Simplified Procedure for Estimating Earthquake-Induced Deviatoric Slope Displacements,” Journal of Geotechnical and Geoenvironmental Engineering 133 (2007): 381–92.↩︎

2. “Simplified Procedure for Estimating Seismic Slope Displacements for Subduction Zone Earthquakes,” Journal of Geotechnical and Geoenvironmental Engineering 144 (2018); “Procedure for Estimating Shear-Induced Seismic Slope Displacement for Shallow Crustal Earthquakes,” Journal of Geotechnical and Geoenvironmental Engineering 145 (2019).↩︎

3. Dakoulas G. and P. Gazetas, “A Class of Inhomogeneous Shear Models for Seismic Response of Dams and Embankments,” International Journal of Soil Dynamics and Earthquake Engineering 4 (1985): 166–82.↩︎

4. Stewart et al., “Expert Panel Recommendations for Ergodic Site Amplification in Central and Eastern North America.”↩︎

5. P., H., and Y., “Amplification Factors for Spectral Acceleration in Tectonically Active Regions.”↩︎

6. Bray, Macedo, and Travasarou, “Simplified Procedure for Estimating Seismic Slope Displacements for Subduction Zone Earthquakes”; Bray and Macedo, “Procedure for Estimating Shear-Induced Seismic Slope Displacement for Shallow Crustal Earthquakes.”↩︎