# 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.88

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: $k_{max}=\bar{a}_{h.max}/\mathrm{g}=\mathrm{max}\,{\bar{a}_{h}(t)}=\mathrm{max}\left(\sum_{z=0}^H\,m(z)a_h(z,t)\right)$ The empirical model from Bray et al89 allows to estimate Newmark displacements $$D_n$$ for different yield coefficients $$k_y$$. This procedure, although having a much larger error term, does not require prior selection of site-specific seismic records. The expression is:

$\mathrm{ln}\,(D_n)=A\,f(k_y,\,M_w,\,T_s)+B(k_y)\,\mathrm{ln}\,[S_a(T_d)]+C\,(\mathrm{ln}\,[S_a(T_d)])^2+\varepsilon$ where $$S_a(T_d)$$ is the pseudo-acceleration evaluated in the “shifted” period of the embankment $$T_d$$. This period is obtained from the fundamental period of the embankment $$T_s$$ as $$T_d \approx \alpha T_s$$, where $$\alpha$$ is a factor between 1.3 and 1.5. The error term $$\varepsilon$$ is a random variable with a normal distribution with mean $$\mathrm{E}[\varepsilon]=0$$ and variance $$\mathrm{Var}[\varepsilon] \approx \sigma_{BM}^2$$.

By considering the epistemic uncertainties of the seismic hazard and the site response the mean and the error term of $$D_n$$ must be corrected. The mean renders:

$\mathrm{ln}\,(D_n)=A\,f(k_y,\,M_w,\,T_s)+B(k_y)\,\mu_{\mathrm{ln}\,S_a}+C\,\left(\,\mu_{\mathrm{ln}\,S_a}^2+\sigma_{\mathrm{ln}\,S_a}^2\right)+\sigma_{BM}^2$ And the error term: $\sigma_{\mathrm{ln}\,D_n}^2=\sigma_{BM}^2+B^2\,\sigma_{\mathrm{ln\,S_a}}^2+4\,B\,C\,\mu_{\mathrm{ln\,S_a}}\,\sigma_{\mathrm{ln\,S_a}}^2+4\,C^2\,\mu_{\mathrm{ln\,S_a}}^2\,\sigma_{\mathrm{ln\,S_a}}^2+2\,C^2\,\sigma_{\mathrm{ln\,S_a}}^4$ For a target displacement $$D_a$$ [cm] the seismic coefficient $$k_{max}$$ 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 and $$c$$ is a constant.

The fundamental period of slopes and embankments can be estimated from a shear-beam model,90 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: $T_s\simeq\dfrac{4\,\pi\,H_{max}}{a_n\,(2-m)\,V_s}$ 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.91

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. An estimate of the fundamental period of the embankment is possible to obtain 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.92 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 to93

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

Step 4: Get spectral ordinates at the 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.94 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 for a characteristic slope that represents the GGM tailings storage dam and the seismic coefficient for a target displacement under two service scenarios are reported in this section.

The tailings storage storage facility will have a maximum height $$H_{max}=40\,\mathrm{m}$$ (at Stage 6), a base $$b_0=28\,\mathrm{m}$$ and a slope of $$\beta≈24\,^\circ$$ (Figure 30). It is mainly composed of two materials: general mine waste material of varying composition, ranging from fine-grained saprolite to fresh rock (identified as zone B) and selected waste material inferred as sandstone/sand dune cover (zone C) and extremely weathered/upper saprolite (zone A), with an estimated main friction angle of $$\phi=30^{\circ}$$. The estimated shear modulus value for the dam base, calculated with the procedure described above, resulted in $$G_0=190 \pm 25\,\mathrm{MPa}$$, equivalent to a shear-wave velocity $$V_s=310\pm 30\,\mathrm{m/s}$$.

The fundamental period of the tailings storage facility for the given conditions is $$T_{n,1}≈0.423\pm 0.031\,\mathrm{s}$$. The main parameter that controls the dam response is 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 16.

 Newmark displacement (+84%) [cm] NEHRP Ts ky TR=500 TR=1000 TR=2500 TR=5000 TR=10000 C 0.423 0.01 6.1 13.0 30.0 51.8 85.4 C 0.423 0.05 0.4 1.2 3.8 8.0 16.2 C 0.423 0.10 0.1 0.3 1.1 2.5 5.4 C 0.423 0.15 0.0 0.1 0.4 1.1 2.5 C 0.423 0.20 0.0 0.1 0.2 0.6 1.4 C 0.423 0.30 0.0 0.0 0.1 0.2 0.6

The pseudo-static coefficient $$k_{max}$$ are estimated for the GGM for two service scenarios. For a service life equal to the mine operation, a design earthquake with an AEP of 1:2475 years is adopted. For critical slopes that must remain stable during closure and post-closure stages, a ground-motion with 1:10,000 years AEP is adopted as the target service level. These levels of service for closure and post-closure are consistent with the prescriptions of the ANCOLD 2019 and the GISTM design guidelines. The fundamental period for the projected geometry of the tailings storage is $$T_{n,1}≈0.423\pm 0.031\,\mathrm{s}$$. The pseudo-static coefficient for different target displacements are presented in Table 17 for the ground motions under certain exceedance probabilities, and in Table 18 for the scenario-based ground motion.

 AEP-based pseudo-static coefficients [g] NEHRP Ts Da TR=500 TR=1000 TR=2500 TR=5000 TR=10000 C 0.423 0.25 0.0429 0.0701 0.1247 0.1857 0.2728 C 0.423 1.00 0.0213 0.0358 0.0658 0.0999 0.1494 C 0.423 2.50 0.0123 0.0214 0.0405 0.0627 0.0955 C 0.423 5.00 0.0076 0.0136 0.0268 0.0423 0.0655 C 0.423 10.00 0.0042 0.0080 0.0166 0.0271 0.0430
 Scenario-based pseudo-static coefficients [g] Ts Repi Mw SID Da Kmax 0.423 5 5 C 0.25 0.1921 0.423 5 5 C 1.00 0.0994 0.423 5 5 C 2.50 0.0600 0.423 5 5 C 5.00 0.0388 0.423 5 5 C 10.00 0.0233

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.”↩︎