Civil, Structural & Geotechnical Engineering
Newmark Seismic Sliding-Block Formula: Yield Acceleration k_y & Permanent Slope Displacement
Evaluate earthquake slope performance beyond pseudo-static failure using Nathan M. Newmark’s (1965) Critical Yield Acceleration k_y = (FS_static - 1) * SIN(beta) and Ambraseys-Menu permanent co-seismic displacement D_N (cm).
Why brief earthquake spikes exceeding FS = 1.0 cause finite centimeters of slip — NOT instant collapse!
Pseudo-static slope analysis treats earthquake acceleration as a permanent gravity tilt, condemning safe earth dams and embankments whenever brief 0.1-second ground-motion pulses push FS transiently below 1.0! In his 5th Rankine Lecture, Nathan M. Newmark (1965, Géotechnique) modeled the landslide mass as a rigid block on an inclined plane of thrust angle beta: downslope slip initiates ONLY during instants when horizontal ground acceleration k_max = PGA / g exceeds the slope’s Critical Yield Acceleration k_y = (FS_static - 1) * SIN(beta)!
Whenever k_max <= k_y, permanent co-seismic displacement is identically 0.00 cm! When 0 < k_y / k_max < 1, Nicolas Ambraseys and J. M. Menu’s (1988, Earthquake Eng. Struct. Dyn.) widely used median regression LOG10(D_cm) = 0.90 + LOG10(((1 - k_y / k_max)^2.53) * ((k_y / k_max)^(-1.09))) estimates permanent Newmark slip D_N in centimeters directly from the acceleration ratio r = k_y / k_max!
| Newmark Quantity (FS_static = 1.35, Thrust Angle beta = 24 deg, k_max = 0.36g) | Formula | Evaluated Value |
|---|---|---|
| Critical Yield Acceleration k_y (in g) | (FS_static - 1) * SIN(RADIANS(beta)) | 0.1424g |
| Yield-to-Peak Acceleration Ratio k_y / k_max | k_y / k_max | 0.3954 |
| Ambraseys-Menu Median Permanent Slip D_N | 10^0.90 * (1 - r)^2.53 * r^(-1.09) | 6.11 cm |
Newmark Critical Yield Acceleration Coefficient k_y (Fraction of g)
Computes (FS_static - 1) * SIN(RADIANS(beta)) for static factor of safety B2 > 1 and slip-block thrust inclination beta (0 < B3 < 90 deg).
=IF(OR(B2 <= 1, B3 <= 0, B3 >= 90), "Requires Static FS > 1.0 and 0 < beta < 90 deg", ROUND((B2 - 1) * SIN(RADIANS(B3)), 4))Ambraseys-Menu Median Co-Seismic Permanent Displacement D_N (cm)
Returns 0 when k_max <= k_y, or evaluates 10^0.90 * (1 - k_y/k_max)^2.53 * (k_y/k_max)^(-1.09) when k_max > k_y.
=IF(OR(B2 <= 1, B3 <= 0, B3 >= 90, B4 <= 0), "Invalid Slope or Seismic PGA Inputs", IF(B4 <= (B2 - 1) * SIN(RADIANS(B3)), 0, ROUND((10 ^ 0.9) * ((1 - ((B2 - 1) * SIN(RADIANS(B3))) / B4) ^ 2.53) * ((((B2 - 1) * SIN(RADIANS(B3))) / B4) ^ (-1.09)), 2)))Try the example
Use the original values or change them to test your understanding.