Seismic Earth Pressure (Mononobe–Okabe)

Method summary

Mononobe–Okabe: \( P_{AE} = \tfrac{1}{2} K_{AE}\,\gamma H^2 (1 - k_v) \), \( \psi = \tan^{-1}\frac{k_h}{1 - k_v} \),
\( K_{AE} = \dfrac{\cos^2(\phi - \theta - \psi)}{\cos\psi \cos^2\theta \cos(\delta + \theta + \psi)\left[1 + \sqrt{\dfrac{\sin(\phi + \delta)\sin(\phi - \beta - \psi)}{\cos(\delta + \theta + \psi)\cos(\beta - \theta)}}\right]^2} \), valid while \( \phi - \beta - \psi \ge 0 \).
Passive: \( K_{PE} \) uses \( \cos^2(\phi + \theta - \psi) \), \( \cos(\delta - \theta + \psi) \) and \( \left[1 - \sqrt{\cdot}\right]^2 \) with \( \sin(\phi + \beta - \psi) \); planar wedges overestimate resistance when \( \delta > \phi/2 \).
Dynamic increment (Seed & Whitman 1970): \( \Delta P_{AE} = P_{AE} - P_A \) acting at \( 0.6H \); for a vertical wall with level backfill \( \Delta K_{AE} \approx \tfrac{3}{4} k_h \).
Submerged, restrained water (Matsuzawa et al. 1985): \( \psi_e = \tan^{-1}\!\left(\frac{\gamma_{sat}}{\gamma'}\frac{k_h}{1 - k_v}\right) \), \( P_{AE} = \tfrac{1}{2} K_{AE}(\psi_e)\,\gamma' H^2 (1 - k_v) + \tfrac{1}{2}\gamma_w H^2 \).
Surcharge: \( P_q = K\,(1 - k_v)\,q H \frac{\cos\beta\cos\theta}{\cos(\beta - \theta)} \). The critical wedge angle comes from a trial wedge search.