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Figure 7-5. Cut slope construction.
4. MSE walls (Article 11.10.7). Seismic design provisions are 7.2 The M-O Method and Limitations
very explicit and are defined for both external and internal
stability. For external stability the dynamic component of The analytical basis for the M-O solution for calculating
the active earth pressure is computed using the M-O equa- seismic active earth pressure is shown in Figure 7-6 (taken
tion. Reductions due to lateral wall movement are per- from Appendix A11.1.1.1 of the AASHTO LRFD Bridge
mitted for gravity walls. Fifty percent of the dynamic earth Design Specifications). This figure identifies the equations for
seismic active earth pressures (PAE), the seismic active earth
pressure is combined with a wall inertial load to evaluate
pressure coefficient (KAE), the seismic passive earth pres-
stability, with the acceleration coefficient modified to ac-
sure (PPE), and the seismic active pressure coefficient (KPE).
count for potential amplification of ground accelerations.
Implicit to these equations is that the soil within the soil is
In the case of internal stability, reinforcement elements are
a homogeneous, cohesionless material within the active or
designed for horizontal internal inertial forces acting on
passive pressure wedges.
the static active pressure zone.
5. Prefabricated modular walls (Article 11.11.6). Seismic de-
7.2.1 Seismic Active Earth Pressures
sign provisions are similar to those for gravity walls.
6. Soil-nail walls. No static or seismic provisions are currently In effect, the solution for seismic active earth pressures is
provided in AASHTO LRFD Bridge Design Specifications. analogous to that for the conventional Coulomb active pres-
However, an FHWA manual for the design of nail walls sure solution for cohesionless backfill, with the addition of a
(FHWA, 2003) suggests following the same general pro- horizontal seismic load. Representative graphs showing the
cedures as used for the design of MSE walls, which involves effect of seismic loading on the active pressure coefficient KAE
the use of the M-O equation with modifications for iner- are shown in Figure 7-7. The effect of vertical seismic loading
tial effects. is traditionally neglected. The rationale for neglecting verti-
cal loading is generally attributed to the fact that the higher
The use of the M-O equations to compute seismic active frequency vertical accelerations will be out of phase with the
and passive earth pressures is a dominant factor in wall design. horizontal accelerations and will have positive and negative
Limitations and design issues are summarized in the follow- contributions to wall pressures, which on average can rea-
ing sections. sonably be neglected for design.
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Seismic Active Earth Pressure
PAE = 0.5 H 2 (1 - kv ) K AE
Figure 7-7. Effect of seismic coefficient and soil
friction angle on active pressure coefficient.
Seismic Passive Earth Pressure
PPE = 0.5 H 2 (1 - kv ) K PE
of 38° in a = 35° material. The M-O solution increases sig-
where nificantly if the seismic coefficient increases to 0.25 for the
same case, as the failure plane angle decreases to 31°. In prac-
cos 2 ( - - )
K AE = tice, however, as shown in Figures 7-3 to 7-5, the failure plane
cos cos 2 coos ( + + ) would usually intersect firm soils or rock in the cut slope
-2 behind the backfill rather than the slope angle defined by a
sin ( + ) sin ( - - i )
× 1 - purely cohesionless soil, as normally assumed during the
cos ( + + ) cos ( i - ) M-O analyses. Consequently, in this situation the M-O solu-
cos 2 ( - + ) tion is not valid.
K PE = A designer could utilize an M-O approach for simple non-
cos cos 2 cos ( - + )
homogeneous cases such as shown in Figure 7-10 using the
sin ( + ) sin ( - + i )
-2
following procedure, assuming 1 < 2:
× 1 -
cos ( - + ) cos ( i - )
= unit weight of soil (ksf)
H = height of wall (ft)
= friction angle of soil (°)
= arc tan (kh/(1 - kv))(°)
= angle of friction between soil and wall (°)
kh = horizontal acceleration coefficient (dim.)
kv = vertical acceleration coefficient (dim.)
i = backfill slope angle (°)
= slope of wall to the vertical, negative as shown (°)
Figure 7-6. M-O solution.
Figure 7-8 shows the effect of backfill slope angle on KAE
as a function of seismic coefficient, and illustrates the design
dilemma commonly encountered of rapidly increasing earth
pressure values with modest increases in slope angles. Fig-
ure 7-9 indicates the underlying reason, namely the fact that
the failure plane angle approaches that of the backfill slope
angle , resulting in an infinite mass of the active failure Figure 7-8. Effect of backfill slope on the seismic
wedge. For example, for a slope angle of 18.43° (3H:1V slope) active earth pressure coefficient using M-O
and a seismic coefficient of 0.2, the failure plane is at an angle equation, where CF = seismic coefficient.
