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82 using the Mononobe-Okabe method specified in Appendix A11, Newmark sliding block analyses. In effect, this represents Article A11.1.1.1: an uncoupled analysis of deformations as opposed to a fully The wall system and any structures supported by the wall can coupled dynamic analysis of permanent wall deformations. tolerate lateral movement resulting from sliding of the struc- However, this approach is commonly used for seismic ture. slope stability analyses, as discussed in Chapter 8. The wall base is unrestrained against sliding, other than soil The existing AASHTO LRFD Bridge Design Specifications friction along its base and minimal soil passive resistance. use an empirical equation based on peak ground acceleration If the wall functions as an abutment, the top of the wall must also be restrained, e.g., the superstructure is supported by slid- to compute wall displacements for a given wall yield acceler- ing bearings. ation. This equation was derived from studies of a limited number of earthquake accelerations, and is of the form: The commentary for this Article notes that, d = 0.087 (V 2 kmax g )( k y kmax ) -4 In general, typical practice among states located in seismically (7-3) active areas is to design walls for reduced seismic pressures cor- responding to 2 to 4 inches of displacement. However, the where amount of deformation which is tolerable will depend on the nature of the wall and what it supports, as well as what is in front ky = yield acceleration; of the wall. kmax = peak seismic coefficient at the ground surface; V = maximum ground velocity (inches/sec), which is the Observations of the performance of conventional cantilever same as PGV discussed in this report; and gravity retaining walls in past earthquakes, and in particu- d = wall displacement (inches). lar during the Hyogoken-Nambu (Kobe) earthquake in 1995, have identified significant tilting or rotation of walls in addition Based on a study of the ground motion database described to horizontal deformations, reflecting cyclic bearing capacity in Chapter 5, revised displacement functions are recom- failures of wall foundations during earthquake loading. To mended for determining displacement. accommodate permanent wall deformations involving mixed For WUS sites and CEUS soil sites (Equation 5-8) sliding and rotational modes of failure using Newmark block failure assumptions, it is necessary to formulate more complex log ( d ) = -1.51 - 0.74 log ( k y kmax ) + 3.27 log (1 - k y kmax ) coupled equations of motions. Coupled equations of motion may be required for evaluat- - 0.80 log ( kmax ) + 1.59 log ( PGV ) ing existing retaining walls. However, from the standpoint of performance criteria for the seismic design of new conven- For CEUS rock sites (Equation 5-6) tional retaining walls, the preferred design approach is to limit log ( d ) = -1.31 - 0.93 log ( k y kmax ) + 4.52 log (1 - k y kmax ) tilting or a rotational failure mode, to the extent possible, by ensuring adequate ratios of capacity to earthquake demand - 0.46 log ( kmax ) + 1.12 log ( PGV ) (that is, high C/D ratios) for foundation bearing capacity fail- ures and to place the design focus on performance criteria where that ensure acceptable sliding displacements (that is lower kmax = peak seismic coefficient at the ground surface; and C/D ratios relative to bearing or overturning). For weaker PGV = peak ground velocity obtained from the design foundation materials, this rotational failure requirement may spectral acceleration at 1 second and adjusted result in the use pile or pier foundations, where lateral seis- for local site class (that is, Fv S1) as described in mic loads would be larger than those for a sliding wall. Chapter 5. Much of the recent literature on conventional retaining wall The above displacement equations represent mean values seismic analysis, including the European codes of practice, and can be multiplied by 2 to obtain an 84 percent confidence focus on the use of Newmark sliding block analysis methods. level. A comparison with the present AASHTO equation is For short walls (less than 20-feet high), the concept of a back- shown in Figure 7-18. fill active failure zone deforming as a rigid block is reasonable, as discussed in the previous paragraph. However, for higher 7.7 Conventional Gravity walls, the dynamic response of the soil in the failure zone leads and Semi-Gravity Walls-- to non-uniform accelerations with height and negates the Recommended Design Method rigid-block assumption. for External Stability For wall heights greater than 20 feet, the use of height- dependent seismic coefficients is recommended to deter- Based on material presented in the previous paragraphs, the mine maximum average seismic coefficients for active fail- recommended design methodology for conventional gravity ure zones, and may be used to determine kmax for use in and semi-gravity walls is summarized by the following steps:

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83 Figure 7-18. Comparison between all except CEUS-Rock and AASHTO correlations for PGV 30 kmax. 1. Establish an initial wall design using the AASHTO LRFD 11. Determine horizontal driving and resisting forces as a Bridge Design Specifications for static loading, using appro- function of k (using spreadsheet calculations) and plot priate load and resistance factors. This establishes wall as a function of k as shown in Figure 7-20b. The values dimensions and weights. of ky correspond to the point where the two forces are 2. Estimate the site peak ground acceleration coefficient equal, that is, the capacity to demand ratio against slid- (kmax) and spectral acceleration at 1 second (S1) from the ing equals 1.0. 1,000-year seismic hazards maps adopted by AASHTO 12. Determine the wall sliding displacement (d) based on the (including appropriate site soil modification factors). relationship between d, ky /kmax, kmax, and PGV described 3. Determine the corresponding PGV from the correlation in Section 7.6. equation between S1 and PGV (Equation 5-11, Chapter 5). 13. Check bearing pressures and overturning criteria to con- 4. Modify kmax to account for wall height effects as described firm that the seismic loads meet performance criteria in Figure 7-16 of Section 7.5. for seismic loading (possibly maximum vertical bear- 5. Evaluate the potential use of the M-O equation to deter- ing pressure less than ultimate and overturning factor mine PAE (Figure 7-10) as discussed in Section 7.2, taking of safety greater than 1.0). into account cut slope properties and geometry and the 14. If step 13 criteria are not met, adjust footing dimensions value of kmax from step 3. and repeat steps 6-12 as needed. 6. If PAE cannot be determined using the M-O equation, use 15. If step 13 criteria are satisfied, assess acceptability of slid- a limit-equilibrium slope stability analysis (as described ing displacement (d). in Section 7.4) to establish PAE. 7. Check that wall bearing pressures and overturning criteria for the maximum seismic load demand required to meet performance criteria. If criteria are met, check for sliding potential. If all criteria are met, the static design is satisfac- tory. If not, go to Step 8. 8. Determine the wall yield seismic coefficient (ky) where wall sliding is initiated. 9. With reference to Figure 7-19, as both the driving forces [PAE(k), kWs, kWw] and resisting forces [Sr(k) and PPE(k)] are a function of the seismic coefficient, the determination of ky for limiting equilibrium (capacity to demand = factor of safety = 1.0) requires an interactive procedure, using the following steps: 10. Determine values of PAE as a function of the seismic co- efficient k (