Slope Parameters
Earthquake Parameters
Risk: Low
Permanent Displacement D vs Peak Acceleration amax/g (current point ●)
Safety Factor FS vs Slope Angle β (current point ●)
Theory & Key Formulas
Yield accel.: $k_y = (FS-1)\sin\beta$
Permanent displacement (Ambraseys-Menu):
$$D = 0.087\frac{v_{max}^2}{a_{max}}\left(\frac{k_y}{a_{max}/g}\right)^{-2.53}$$
FS via infinite slope limit equilibrium
What is the Newmark Method for Slope Stability?
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What exactly is the Newmark method? I've heard of static slope stability, but what makes this "dynamic"?
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Basically, it's a way to estimate how much an earthquake can make a slope slide. In static analysis, you just check if the slope is stable under gravity. The Newmark method adds the shaking from an earthquake, treating the slope like a rigid block that only slides when the shaking force exceeds a critical threshold. Try moving the "Peak Acceleration" slider in the simulator to see how stronger shaking affects the results.
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Wait, really? So the slope isn't failing the whole time during the quake? What's this "yield acceleration" I see in the output?
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Exactly! That's the key insight. The yield acceleration ($k_y$) is the seismic acceleration level needed to just overcome the slope's strength and start it sliding. Below that, the slope holds. Above it, it accumulates permanent displacement. In practice, for a dam or embankment, you'd calculate this to see if shaking from a design earthquake would cause dangerous movement. Change the soil's friction angle ($\phi$) and cohesion ($c$) above to see how they directly influence $k_y$.
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So the final "Permanent Displacement" number is the total slide distance? How do you get that from just the acceleration and duration?
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Great question. The simulator uses an empirical formula (like Ambraseys-Menu) that correlates displacement with the excess shaking. It integrates the effect of every moment the earthquake acceleration exceeds the yield acceleration. A common case is a highway cut in a seismic zone; engineers use this displacement value to judge risk. Try reducing the slope angle ($\beta$) and notice how both the yield acceleration and the predicted displacement change.
Physical Model & Key Equations
The analysis starts with a static Factor of Safety (FS) against sliding. The dynamic tipping point is the yield acceleration, which scales with how much stronger the slope is than the minimally stable condition.
$$k_y = (FS - 1) \sin \beta$$
Here, $k_y$ is the yield acceleration (in $g$), $FS$ is the static factor of safety, and $\beta$ is the slope angle. A lower $k_y$ means the slope is closer to failure and will start sliding under weaker shaking.
The permanent displacement is estimated using a regression model derived from analyzing records of real earthquake-induced slides. The Ambraseys-Menu equation is one common model.
$$D = 0.087 \frac{v_{max}^2}{a_{max}}\left( \frac{k_y}{a_{max}/g}\right)^{-2.53}$$
Where $D$ is the permanent displacement (cm), $v_{max}$ is the peak ground velocity, $a_{max}$ is the peak ground acceleration, $k_y$ is the yield acceleration, and $g$ is gravity. This shows displacement is highly sensitive to the ratio of yield to peak acceleration.
Real-World Applications
Dam Safety Assessment: Engineers use this method to evaluate the seismic stability of earthfill and tailings dams. For instance, if a predicted displacement exceeds 15 cm for a critical dam, it might trigger a redesign with stronger materials or flatter slopes to reduce risk.
Highway & Railway Embankments: When building transportation corridors in earthquake-prone areas like California or Japan, this analysis ensures that embankments won't suffer excessive settlement or collapse during a quake, blocking vital evacuation and supply routes.
Landslide Hazard Zoning: Geologists and planners apply the Newmark method regionally using GIS to map areas with high displacement potential. A common case is assessing landslide risk for neighborhoods built on steep hillsides after a major fault rupture.
Mine Slope Design: In open-pit mining, the stability of high, steep slopes is critical for worker safety and economic operation. Dynamic analysis helps plan mining sequences and evaluate the need for reinforcement if blasting or regional seismicity could trigger a slide.
Common Misunderstandings and Points to Note
When you start using this tool, there are several pitfalls that engineers, especially those with less field experience, often fall into. A major misunderstanding is thinking that the calculated permanent displacement D directly equals the collapse distance. For example, even if D=0.5m, it does not mean the entire slope will slide 0.5m all at once. The Newmark method provides an estimate of the "average" displacement due to the accumulation of shear strain. Actual failure involves this displacement concentrating locally or developing into a surface slide, so you should treat the D value as a relative indicator for risk comparison.
Next is the setting of input parameters. You must not use the "cohesion c" and "internal friction angle φ" directly from geotechnical investigation reports. During an earthquake, strength degrades (dynamic strength reduction) due to cyclic loading, so it's common to set them at about 70-80% of the static strength. For instance, if static tests yield c=30 kN/m² and φ=30°, for dynamic analysis you would typically use c=24 kN/m² and φ=24°. Be careful not to forget this correction in the tool, as it will lead to an overestimated factor of safety and an underestimated permanent displacement.
Finally, regarding seismic motion input. The tool requires you to input $v_{max}$ and $a_{max}$ as single values, but the period characteristics of the actual seismic wave are crucial. For example, long-period seismic motions affect deeper parts of the slope, increasing the risk of deep-seated slides, not just surface slides. In practice, the basic approach is a "multiple-case analysis," where you input several seismic waves considering the expected earthquake's source characteristics and ground amplification factors, and adopt the most critical result. Remember, this tool is intended for initial screening.
Worked Example
Sandy slope: Beta=32°, C=8 kPa, Phi=32°. Using standard unit weight γ=18 kN/m³, the static FS calculates as 1.28. For a magnitude 7.0 earthquake with peak ground acceleration (PGA)=0.35g, yield acceleration ay≈0.18g. Integrating double-integration of the acceleration pulse duration (typical 15 seconds) with Newmark's sliding-block model yields permanent displacement approximately 120 mm. If FS drops below 1.0 during shaking, slope failure initiates; displacement accumulates throughout strong-motion duration.