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63 input motion. Hence a smaller scaling factor (less than unity) spectral shape. Ratios of peak average seismic coefficient re- should be expected for a CEUS seismological condition and sponse versus input motion (as measured by PGA) tabulated in for rock sites. In typical design applications, both the seismo- the third column from right in the table were used to develop logical and geotechnical conditions should be implicit in the the scaling factors (defined as ) applied to the PGA to deter- adopted reference ground surface outcrop design response mine the peak average seismic coefficients acting on a block of spectrum following seismic loading criteria defined by the soil for pseudo-static seismic analyses of the retaining walls. NCHRP 20-07 Project. Results in Table 6-1 are based on the average ground motions The second parameter controlling the scaling factor for within each set of analyses. The time-dependent change in seismic coefficients is related to the height of the soil mass PGA, PGV, and S1 is not considered. Use of the scaling factor (that is, slope height in the context of the presented slope re- does not, therefore, account for changes in inertial loading sponse analysis) or the height of a retaining wall as discussed with time. In other words the scaled PGA is the peak loading below. In general, the scattering analyses show that the effect and will be less for most of the earthquake duration. The of height on PGV (a parameter of interest for Newmark slid- average inertial force over the duration of shaking can vary ing block analyses) is relatively small. from less than one-third to two-thirds of the peak value, depending on the magnitude, location, and other characteris- tics of the earthquake. 6.1.2 Scattering Analyses Similar to the observation made earlier from the slope scat- for Retaining Walls tering analyses, the variation in the coefficient was not very The wave scattering analyses discussed in the previous sec- significant among the three failure blocks evaluated, and tion have been extended from a slope configuration to con- therefore, results from the three failure blocks were averaged. figurations commonly encountered for retaining wall de- Also, results from the three time histories each matched to the signs. Wave scattering analyses were conducted to establish same response spectrum were averaged. The resultant solu- the relationship between peak ground acceleration at a given tions for the coefficients categorized by wall height and the point in the ground to the equivalent seismic coefficient. In spectral shapes (that is, upper bound, mid and lower bound this context the equivalent seismic coefficient was the coeffi- spectral shapes) are summarized in Figure 6-13. cient that should be applied to a soil mass to determine the The reduction in PGA shown in the above figure arises peak force amplitude used in pseudo-static, force-based de- from a wave scattering reduction in the peak PGA for design sign of a retaining wall. The product of the equivalent seismic analyses. There are other factors that provide further justifi- coefficient and soil mass defined the inertial load that would cation for reducing the PGA value, as discussed here: be applied to wall surface from the retained backfill. 1. Average versus peak response. As noted previously, a pseudo-static analysis treats the seismic coefficient as a con- 6.1.2.1 Retaining Wall Model stant horizontal static force applied to the soil mass. How- Figure 6-12 provides a schematic description of the wave ever, the peak earthquake load from a dynamic response scattering analyses performed for the retaining wall problem. analysis occurs for a very short time--with the average seis- Similar to the slope scattering study described in the previous mic force typically ranging from 30 to 70 percent of the peak subsection, the QUAD-4M program was used during these depending on the characteristics of the specific earthquake analyses. event. Hence further reduction in the force demand reflect- Nine input motions were used for the analyses. Features of ing the overall average cyclic loading condition might be these records are described in Appendix E. These records justified, where a structural system is designed for some de- were used as input motion at the base of the finite-element gree of ductile yielding. The acceptability of an additional mesh. The analyses included use of a transmitting boundary time-related reduction should be decided by the structural element available within the QUAD-4M program. A free design, since it will depend on the method of analysis and boundary at the wall face was assumed. the design philosophy. The Project Team decided that a-priori reduction in the PGA after adjustment for wave scattering by time-related factor was not appropriate, and 6.1.2.2 Results of Wave Scattering Analyses therefore this additional reduction has not been introduced for Retaining Walls into the design approach. This decision also means that it is Table 6-1 summarizes the results from the wave scatter- very important for the geotechnical engineer to very clearly ing analyses for the retaining structure. Data presented in define whether the resulting seismic coefficient is the in- Table 6-1 are from 36 QUAD-4M runs covering four wall stantaneous peak or an average peak corrected for the du- heights, three spectral shapes, and three time histories for each ration of ground shaking.

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64 PLANE 3 PLANE 2 PLANE 1 20' 20-FEET WALL MESH PLANE 3 PLANE 2 PLANE 1 40-FEET WALL MESH 40' PLANE 3 PLANE 2 PLANE 1 80' 80-FEET WALL MESH PLANE 3 PLANE 2 PLANE 1 0 100' 150' 150-FEET WALL MESH Figure 6-12. Models used in scattering analyses. 2. Load fuse from wall movements. Another justification for retaining wall is designed to slide at a specific threshold designing to a value less than the PGA arises from the fact load level as discussed in Chapter 7. that many retaining walls are implicitly designed for wall movements when the wall is designed for an active earth From Table 6-1 it can be observed that the ratio of equiva- pressure condition. The wave scattering analyses in this lent seismic coefficient (for a block of soil) to the PGA (at a evaluation were based on linear elastic analyses and fur- single point on the ground surface) did not change drastically ther reduction in the force demand is justified when the for the three failure planes studied in the analyses. However,

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65 Table 6-1. Results of scattering analyses. Ratio of Seismic Coefficient Response Seismic Coefficient Response Input Motion vs. Input Motion PGV PGV PGV PGV I Model File Name Block (max) (max) Sa1 File Name (max) (max) Sa1 PGA PGV Sa1 g in/s g g in/s g g in/s g 1 20 ft wall w20-cap-.Q4K 1 0.579 13.506 0.33 CAP-L.acc 0.894 15.370 0.39 0.65 0.88 0.85 2 20 ft wall w20-cap-.Q4K 2 0.590 13.862 0.34 CAP-L.acc 0.894 15.370 0.39 0.66 0.90 0.87 3 20 ft wall w20-cap-.Q4K 3 0.518 12.344 0.30 CAP-L.acc 0.894 15.370 0.39 0.58 0.80 0.77 4 20 ft wall w20-day-.Q4K 1 0.740 10.486 0.34 DAY-L.acc 0.936 11.684 0.39 0.79 0.90 0.87 5 20 ft wall w20-day-.Q4K 2 0.730 10.705 0.35 DAY-L.acc 0.936 11.684 0.39 0.78 0.92 0.90 6 20 ft wall w20-day-.Q4K 3 0.670 9.286 0.31 DAY-L.acc 0.936 11.684 0.39 0.72 0.79 0.79 7 20 ft wall w20-lan-.Q4K 1 0.759 12.033 0.31 LAN-L.acc 0.771 15.173 0.36 0.98 0.79 0.86 8 20 ft wall w20-lan-.Q4K 2 0.761 12.297 0.32 LAN-L.acc 0.771 15.173 0.36 0.99 0.81 0.89 9 20 ft wall w20-lan-.Q4K 3 0.699 10.173 0.28 LAN-L.acc 0.771 15.173 0.36 0.91 0.67 0.78 Average of Above 9 0.672 11.632 0.320 L.B. Spectrum 0.867 14.076 0.380 0.783 0.830 0.842 10 20 ft wall w20-imp-.Q4K 1 0.670 31.047 0.97 IMP-M.acc 0.812 37.054 1.12 0.83 0.84 0.87 11 20 ft wall w20-imp-.Q4K 2 0.685 31.884 1.00 IMP-M.acc 0.812 37.054 1.12 0.84 0.86 0.89 12 20 ft wall w20-imp-.Q4K 3 0.602 28.298 0.87 IMP-M.acc 0.812 37.054 1.12 0.74 0.76 0.78 13 20 ft wall w20-lom-.Q4K 1 0.992 31.034 1.06 LOM-M.acc 1.026 32.275 1.20 0.97 0.96 0.88 14 20 ft wall w20-lom-.Q4K 2 1.010 31.719 1.08 LOM-M.acc 1.026 32.275 1.20 0.98 0.98 0.90 15 20 ft wall w20-lom-.Q4K 3 0.855 27.454 0.94 LOM-M.acc 1.026 32.275 1.20 0.83 0.85 0.78 16 20 ft wall w20-san-.Q4K 1 0.742 40.453 1.03 SAN-M.acc 0.948 42.312 1.18 0.78 0.96 0.87 17 20 ft wall w20-san-.Q4K 2 0.758 41.297 1.06 SAN-M.acc 0.948 42.312 1.18 0.80 0.98 0.90 18 20 ft wall w20-san-.Q4K 3 0.655 33.340 0.92 SAN-M.acc 0.948 42.312 1.18 0.69 0.79 0.78 Average of above 9 0.774 32.947 0.992 Mid Spectrum 0.929 37.214 1.167 0.830 0.886 0.850 19 20 ft wall w20-elc-.Q4K 1 0.986 40.725 1.56 ELC-U.acc 1.083 45.320 1.78 0.91 0.90 0.88 20 20 ft wall w20-elc-.Q4K 2 0.981 41.631 1.60 ELC-U.acc 1.083 45.320 1.78 0.91 0.92 0.90 21 20 ft wall w20-elc-.Q4K 3 0.890 35.655 1.37 ELC-U.acc 1.083 45.320 1.78 0.82 0.79 0.77 22 20 ft wall w20-erz-.Q4K 1 1.068 43.290 1.43 ERZ-U.acc 1.089 52.950 1.69 0.98 0.82 0.85 23 20 ft wall w20-erz-.Q4K 2 1.094 44.468 1.47 ERZ-U.acc 1.089 52.950 1.69 1.00 0.84 0.87 24 20 ft wall w20-erz-.Q4K 3 0.978 39.040 1.26 ERZ-U.acc 1.089 52.950 1.69 0.90 0.74 0.75 25 20 ft wall w20-tab-.Q4K 1 1.091 41.827 1.54 TAB-U.acc 1.060 46.922 1.76 1.03 0.89 0.88 26 20 ft wall w20-tab-.Q4K 2 1.103 42.756 1.58 TAB-U.acc 1.060 46.922 1.76 1.04 0.91 0.90 27 20 ft wall w20-tab-.Q4K 3 0.938 37.597 1.38 TAB-U.acc 1.060 46.922 1.76 0.88 0.80 0.78 Average of Above 9 1.014 40.777 1.466 U.B. Spectrum 1.077 48.397 1.743 0.942 0.845 0.840 28 40 ft wall w40-cap-.Q4K 1 0.543 14.021 0.32 CAP-L.acc 0.894 15.370 0.39 0.61 0.91 0.82 29 40 ft wall w40-cap-.Q4K 2 0.530 14.543 0.34 CAP-L.acc 0.894 15.370 0.39 0.59 0.95 0.87 30 40 ft wall w40-cap-.Q4K 3 0.470 13.677 0.33 CAP-L.acc 0.894 15.370 0.39 0.53 0.89 0.85 31 40 ft wall w40-day-.Q4K 1 0.441 12.190 0.36 DAY-L.acc 0.936 11.684 0.39 0.47 1.04 0.92 32 40 ft wall w40-day-.Q4K 2 0.410 12.414 0.38 DAY-L.acc 0.936 11.684 0.39 0.44 1.06 0.97 33 40 ft wall w40-day-.Q4K 3 0.385 11.284 0.36 DAY-L.acc 0.936 11.684 0.39 0.41 0.97 0.92 34 40 ft wall w40-lan-.Q4K 1 0.449 11.961 0.33 LAN-L.acc 0.771 15.173 0.36 0.58 0.79 0.92 35 40 ft wall w40-lan-.Q4K 2 0.427 12.771 0.34 LAN-L.acc 0.771 15.173 0.36 0.55 0.84 0.94 36 40 ft wall w40-lan-.Q4K 3 0.411 12.045 0.33 LAN-L.acc 0.771 15.173 0.36 0.53 0.79 0.92 Average of Above 9 0.452 12.767 0.343 L.B. Spectrum 0.867 14.076 0.380 0.524 0.916 0.904 37 40 ft wall w40-imp-.Q4K 1 0.734 31.666 0.99 IMP-M.acc 0.812 37.054 1.12 0.90 0.85 0.88 38 40 ft wall w40-imp-.Q4K 2 0.745 33.017 1.05 IMP-M.acc 0.812 37.054 1.12 0.92 0.89 0.94 39 40 ft wall w40-imp-.Q4K 3 0.696 31.165 0.99 IMP-M.acc 0.812 37.054 1.12 0.86 0.84 0.88 40 40 ft wall w40-lom-.Q4K 1 0.968 35.371 1.09 LOM-M.acc 1.026 32.275 1.20 0.94 1.10 0.91 41 40 ft wall w40-lom-.Q4K 2 0.993 37.374 1.15 LOM-M.acc 1.026 32.275 1.20 0.97 1.16 0.96 42 40 ft wall w40-lom-.Q4K 3 0.903 35.285 1.09 LOM-M.acc 1.026 32.275 1.20 0.88 1.09 0.91 43 40 ft wall w40-san-.Q4K 1 0.804 40.479 1.07 SAN-M.acc 0.948 42.312 1.18 0.85 0.96 0.91 44 40 ft wall w40-san-.Q4K 2 0.839 42.883 1.13 SAN-M.acc 0.948 42.312 1.18 0.89 1.01 0.96 45 40 ft wall w40-san-.Q4K 3 0.772 39.551 1.06 SAN-M.acc 0.948 42.312 1.18 0.81 0.93 0.90 Average of Above 9 0.828 36.310 1.069 Mid Spectrum 0.929 37.214 1.167 0.891 0.982 0.916 46 40 ft wall w40-elc-.Q4K 1 0.785 43.411 1.60 ELC-U.acc 1.083 45.320 1.78 0.72 0.96 0.90 47 40 ft wall w40-elc-.Q4K 2 0.814 45.795 1.69 ELC-U.acc 1.083 45.320 1.78 0.75 1.01 0.95 48 40 ft wall w40-elc-.Q4K 3 0.766 43.155 1.60 ELC-U.acc 1.083 45.320 1.78 0.71 0.95 0.90 49 40 ft wall w40-erz-.Q4K 1 1.229 45.744 1.48 ERZ-U.acc 1.089 52.950 1.69 1.13 0.86 0.88 50 40 ft wall w40-erz-.Q4K 2 1.267 48.699 1.56 ERZ-U.acc 1.089 52.950 1.69 1.16 0.92 0.92 51 40 ft wall w40-erz-.Q4K 3 1.179 45.240 1.47 ERZ-U.acc 1.089 52.950 1.69 1.08 0.85 0.87 52 40 ft wall w40-tab-.Q4K 1 1.017 44.276 1.55 TAB-U.acc 1.060 46.922 1.76 0.96 0.94 0.88 53 40 ft wall w40-tab-.Q4K 2 1.020 46.188 1.63 TAB-U.acc 1.060 46.922 1.76 0.96 0.98 0.93 54 40 ft wall w40-tab-.Q4K 3 0.913 43.438 1.55 TAB-U.acc 1.060 46.922 1.76 0.86 0.93 0.88 Average of Above 9 0.999 45.105 1.570 U.B. Spectrum 1.077 48.397 1.743 0.927 0.935 0.900 55 80 ft wall w80-cap-.Q4K 1 0.380 14.464 0.43 CAP-L.acc 0.894 15.370 0.39 0.43 0.94 1.10 56 80 ft wall w80-cap-.Q4K 2 0.371 14.270 0.43 CAP-L.acc 0.894 15.370 0.39 0.41 0.93 1.10 57 80 ft wall w80-cap-.Q4K 3 0.340 13.829 0.42 CAP-L.acc 0.894 15.370 0.39 0.38 0.90 1.08 58 80 ft wall w80-day-.Q4K 1 0.240 9.725 0.41 DAY-L.acc 0.936 11.684 0.39 0.26 0.83 1.05 (continued on next page)