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71 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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72 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.