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Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments (2008)

Chapter: Chapter 6 - Height-Dependent Seismic Coefficients

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Page 55
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
Page 56
Page 57
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Page 58
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
Page 63
Page 64
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
Page 64
Page 65
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
Page 65
Page 66
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
Page 66
Page 67
Suggested Citation:"Chapter 6 - Height-Dependent Seismic Coefficients." National Academies of Sciences, Engineering, and Medicine. 2008. Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments. Washington, DC: The National Academies Press. doi: 10.17226/14189.
×
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55 This chapter summarizes the results of seismic wave inco- herence or scattering studies. These scattering studies were conducted to evaluate the variation in average ground accel- eration behind retaining walls and within slopes, as a func- tion of height. The primary objectives of these studies were to • Evaluate the changes in ground motion within the soil mass that occur with height and lateral distance from a reference point. The consequence of this variation is that the average ground motion within a soil mass behind a retaining wall or within a slope, which results in the inertial force on the wall or within the slope, is less than the instantaneous peak value within the zone. • Develop a method for determining the average ground motion that could be used in the seismic design of retain- ing structures, embankments and slopes, and buried struc- tures based on the results of the scattering evaluations. The wave scattering analyses resulted in the development of a height-dependent seismic coefficient. These results are described in the following sections of this chapter. The dis- cussions provide background for the scattering studies, the results of the scattering analyses for a slope and for retaining walls, and recommendations on the application of the scat- tering effects. These results also will form the basis of discus- sion in sections proposed for use in the AASHTO LRFD Bridge Design Specifications. 6.1 Wave Scattering Evaluations Current practice in selecting the seismic coefficient for re- taining walls normally assumes rigid body soil response in the backfill behind a retaining wall. In this approach the seismic coefficient is defined by the PGA or some percentage of the PGA. A limit equilibrium concept, such as the M-O equation, is used to determine the force on the retaining wall. A similar approach often is taken when assessing the response of a slope to seismic loading. In this case the soil above the critical fail- ure surface is assumed to be a rigid mass. By assuming a rigid body response, the ground motions within the rigid body are equal throughout. For wall or slope heights in excess of about 20 to 30 feet, this assumption can be questioned. The follow- ing sections of this chapter summarize the results of the wave scattering analyses. This summary starts with a case study for a 30-foot high slope to illustrate the wave scattering process. This is followed by a more detailed evaluation of the scatter- ing effects for retaining walls. 6.1.1 Scattering Analyses for a Slope Wave propagation analyses were conducted for an em- bankment slope that was 30 feet in height and had a 3H:1V (horizontal to vertical) slope face. A slope height of 30 feet was selected as being representative of a case that might be en- countered during a typical design. The objective of the analysis was to determine the equivalent average seismic coefficient that would be used in a limit equilibrium slope stability evaluation, taking into consideration wave scattering. Figure 6-1 depicts the slope model employed in the wave propagation study. 6.1.1.1 Slope Model The wave propagation analysis was carried out for a two-dimensional (2-D) slope using the computer program QUAD-4M (1994). For these analyses the seismic coefficient was integrated over predetermined blocks of soil. The seismic coefficient is essentially the ratio of the seismic force induced by the earthquake in the block of soil divided by the weight of that block. Since the summation of forces acting on the block is computed as a function of time, the seismic coefficient is computed for each time step, yielding a time history of the seismic coefficient for the block. In this study, three soil blocks bounded by potential failure surfaces shown in Figure 6-1 were evaluated. C H A P T E R 6 Height-Dependent Seismic Coefficients

56 The model used for these analyses had the following characteristics: • Soil properties assigned for the finite element mesh are shown in Figure 6-1. These properties reflect typical com- pacted fill properties with a uniform shear wave velocity of 800 feet per second (ft/sec). • Ground motions in the form of acceleration time histories were assigned as outcrop motions at the base of the model where a transmitting boundary was provided. • The half-space property beneath the transmitting boundary was assigned a shear wave velocity of 800 ft/sec, identical to the soil mesh above the transmitting boundary. The velocity of the half-space was assigned the same veloc- ity as the embankment to avoid introduction of an impedance contrast in the finite-element model (hence an artificial natu- ral frequency defined for the system). Assigning a uniform soil property above and below the half-space transmitting bound- ary meant that the resultant ground shaking would implicitly be compatible to the intended free-field ground surface con- dition, as defined by a given design response spectrum. To further explain this aspect, reference is made to the left and the right side boundaries of the finite element mesh shown in Figure 6-1. These boundaries are specifically estab- lished as being sufficiently far from the slope face to avoid boundary effects. With the half-space and soil mesh proper- ties as discussed earlier, it is observed that at the left and right edge soil columns, the response should approach the theo- retical semi-infinite half-space problem of a vertically propa- gating shear wave (as modeled by the one-dimensional com- puter program SHAKE—Schnaebel et al., 1972). Therefore, the overall problem at the free-field ground surface, with the exception of the region locally adjacent to the slope face in the middle, should approach a level ground reference outcrop benchmark condition. Rigorously speaking, free-field response at the left side (top of slope) versus the right side (bottom of slope) will be of lit- tle difference in amplitude of shaking, reflecting a slight time delay due to wave passage over a 30-foot difference in soil col- umn height in the model. Introduction of any impedance contrast in either the soil mesh or what is implied by the transmitting boundary effectively introduces a boundary condition into the problem and results in a natural frequency in the boundary value problem. This will result in a free-field ground surface shaking condition deviating from the in- tended level-ground outcrop response spectrum design basis. Likewise, introduction of an impedance contrast would in- troduce complexities to the ground motion design defini- tions. Solutions involving such impedance contrast will, however, be relevant for site-specific cases, as discussed in Chapters 7 and 8 of this Final Report. 6.1.1.2 Earthquake Records Used In Slope Studies Several earthquake time histories were used for input exci- tation; each one was spectrum matched to lower bound, mid, or upper bound spectra, as discussed in Chapter 5. Further documentation of the input motions used for the analyses can be found in Appendix E. Prior to presenting results of the equivalent seismic coeffi- cient evaluations, Figure 6-2 shows a representative accelera- tion time history extracted from a node on the free-field sur- face at the left side boundary (that is, at the top of the slope). The time history is for the Imperial Valley input motion that was used to match the mid target spectrum. This time history can be compared to the reference outcrop motion shown in the same figure. As can be seen from the comparison, the two Figure 6-1. QUAD-4M model for 30-feet high wall.

motions are rather similar as intended by the use of the trans- mitting boundary and a uniform set of soil properties. The Rayleigh damping parameters are intentionally chosen to be sufficiently low to avoid unintended material damping that would lower the resultant shaking at the free-field surface from wave propagation over the small height in soil column used for analysis. 6.1.1.3 Results of Scattering Analyses for Slopes Figures 6-3 through 6-5 show comparisons of seismic co- efficient time histories (dark lines) against the input outcrop motion (light lines) for three acceleration time histories fitted for the lower bound spectral shape. Figures 6-6 through 6-8 and Figures 6-9 through 6-11 present the corresponding comparisons for the mid and upper bound spectrum, respec- tively. In each figure, three traces of seismic coefficient were computed for the three blocks as compared to the light col- ored reference outcrop motion. 6.1.1.4 Observations from Evaluations It can be observed from Figures 6-2 through 6-11 that the variation in the seismic coefficient amongst the three blocks for a given earthquake motion is rather small. However, there is a clear reduction in seismic coefficient from the integrated seismic coefficient time history (dark lines) as compared to the input outcrop motion (light lines). From the comparison, it is also clear that the reduction in shaking in the seismic co- efficient time history as compared to the reference input de- sign time history is highly frequency dependent. The reduction in shaking is much more apparent for the lower bound spectrum records (see Figures 6-3 through 6-5) relative to the mid and upper bound cases. The reduction in shaking for the analyses associated with the mid and the upper bound spectra indicates that the reduction in shaking is justified for the several relative peaks at the time of strong ground shaking, but the reduction becomes much less ap- parent for other portions of the response time history, espe- cially toward the end of the time history. The scattering phe- nomenon results from the fact that several relative peaks at the time of peak earthquake loading will be chopped off, as opposed to a uniformly scaling down of the overall time his- tory motion record. As observed from time-history comparisons for the aver- age seismic coefficients resulting for the three failure blocks in each of the figures, the high frequency cancellation effect, or variation in seismic coefficient among the three failure blocks, appears to be relatively small in the lateral dimension. As discussed more fully in the summary of wave scattering analyses for retaining walls, it appears that the resultant ratio decreases with increasing lateral dimension in the failure 57 Figure 6-2. Comparison QUAD 4M input outcrop motion (top figure) versus free field ground surface response motion (bottom figure). O ut cr op In pu t M ot io n (A cc ), g -1 -0.5 0 0.5 1 0 5 10 15 20 25 30 35 Fr ee F ie ld M ot io n (A cc ), g -1 -0.5 0 0.5 1 0 5 10 15 Time, s 20 25 30 35

58 Figure 6-3. Scattering results for lower bound spectral shape, Cape Mendocino record. 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 Time, s 25 30 35 Input Outcrop Seismic Coeff 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 Input Outcrop Seismic Coeff 35 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 Time, s 25 30 35 Figure 6-4. Scattering results for lower bound spectral shape, Dayhook record.

59 Figure 6-5. Scattering results for lower bound spectral shape, Landers EQ record. 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 Time, s 25 30 Input Outcrop Seismic Coeff 35 Figure 6-6. Scattering results for mid spectral shape, Imperial Valley EQ record. 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 25 30 35 Time, s Input Outcrop Seismic Coeff

60 Figure 6-7. Scattering results for mid spectral shape, Loma Prieta EQ record. 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 Time, s 25 30 Input Outcrop Seismic Coeff 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 Figure 6-8. Scattering results for mid spectral shape, San Fernando EQ record. Input Outcrop Seismic Coeff 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 20 Time, s 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35

61 Figure 6-9. Scattering results for upper bound spectral shape, Imperial Valley EQ record. Input Outcrop Seismic Coeff 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 Time, s 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35 Figure 6-10. Scattering results for upper bound spectral shape, Turkey EQ record. Input Outcrop Seismic Coeff 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 Time, s 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35

block. However, the change appears to be much smaller (on the order of 10 percent among the three blocks). Such variations seem insignificant compared to scattering analyses involving the vertical dimensions of the soil mass. This observation can be explained by prevalent assumptions in wave propagation phenomena interpreted from strong motion data. For example, data from closely spaced strong motion arrays in- dicate that the wave passage effect in the lateral direction in space tends to be correlated to a very high apparent wave speed (say 2.0 to 3.5 km/sec) range, whereas the apparent wave speed in the vertical direction (for example, from downhole arrays) is related to shear wave velocity at the site. The apparent wave speed in the horizontal direction would typically be 10 to 20 times the apparent wave speed in the vertical direction. This would imply that the wave length in the vertical direction would be much smaller than the horizontal direction. Consis- tent with this observation, the wave scattering analyses used an identical input motion at all the nodes across the base of the finite-element mesh. Given the uniform motion input at the base, along with the side boundary conditions chosen to create a vertically propagating shear wave, a relatively minor variation in the motion in the horizontal direction should be expected. Wave scattering analyses presented in this section for slopes provide a qualitative illustration of the wave scattering phenomena. A more comprehensive set of wave scattering analyses is presented for retaining walls. The retaining wall was used to evaluate wave scattering reduction factors (termed an α factor) which could be applied to a site-adjusted PGA to determine an equivalent maximum average seismic coefficient. This equivalent seismic coefficient was than applied to the soil mass for force-based design. 6.1.1.5 Conclusions from Scattering Analyses for Slopes From these studies using the three sets of time histories for each spectral shape (lower bound, mid, and upper bound), reduction factors that can be applied to the peak ground ac- celeration were estimated. For the 30-foot slope, these scat- tering factors will be on the order of 0.5 for the lower bound spectral shape, 0.6 for the mid spectral shape, and 0.7 for the upper bound spectral shape. For slopes higher than 30 feet, further reductions due to canceling of high frequency mo- tions in the vertical dimension due to incoherency effects from the wave scattering phenomenon could be anticipated, as shown in the wall height study. The primary parameter controlling the scaling factor for a height-dependent seismic coefficient is related to the frequency content of the input motion with a lower seismic coefficient associated with the high, frequency-rich lower bound spectrum 62 Figure 6-11. Scattering results for upper bound spectral shape, El Centro EQ record. Input Outcrop Seismic Coeff 1 0.5 0 Bl oc k 3 k -0.5 -1 0 5 10 15 Time, s 20 25 30 35 1 0.5 0 Bl oc k 2 k -0.5 -1 0 5 10 15 20 25 30 35 1 0.5 0 Bl oc k 1 k -0.5 -1 0 5 10 15 20 25 30 35

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

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

65 Seismic Coefficient Response Input Motion Ratio of Seismic Coefficient Response vs. Input Motion I Model File Name Block PGV (max) PGV (max) Sa1 File Name PGV (max) PGV (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 Table 6-1. Results of scattering analyses. (continued on next page)

66 Seismic Coefficient Response Input Motion Ratio of Seismic Coefficient Response vs. Input Motion I Model File Name Block PGV (max) PGV (max) Sa1 File Name PGV (max) PGV (max) Sa1 PGA PGV Sa1 g in/s g g in/s g g in/s g 59 80 ft wall w80-day-.Q4K 2 0.224 9.800 0.41 DAY-L.acc 0.936 11.684 0.39 0.24 0.84 1.05 60 80 ft wall w80-day-.Q4K 3 0.202 9.545 0.40 DAY-L.acc 0.936 11.684 0.39 0.22 0.82 1.03 61 80 ft wall w80-lan-.Q4K 1 0.257 14.593 0.38 LAN-L.acc 0.771 15.173 0.36 0.33 0.96 1.06 62 80 ft wall w80-lan-.Q4K 2 0.243 14.504 0.38 LAN-L.acc 0.771 15.173 0.36 0.32 0.96 1.06 63 80 ft wall w80-lan-.Q4K 3 0.221 13.858 0.37 LAN-L.acc 0.771 15.173 0.36 0.29 0.91 1.03 Average of Above 9 0.275 12.732 0.403 L.B. Spectrum 0.867 14.076 0.380 0.319 0.899 1.061 64 80 ft wall w80-imp-.Q4K 1 0.607 37.264 1.12 IMP-M.acc 0.812 37.054 1.12 0.75 1.01 1.00 65 80 ft wall w80-imp-.Q4K 2 0.599 37.154 1.13 IMP-M.acc 0.812 37.054 1.12 0.74 1.00 1.01 66 80 ft wall w80-imp-.Q4K 3 0.550 36.002 1.10 IMP-M.acc 0.812 37.054 1.12 0.68 0.97 0.98 67 80 ft wall w80-lom-.Q4K 1 0.672 41.988 1.22 LOM-M.acc 1.026 32.275 1.20 0.65 1.30 1.02 68 80 ft wall w80-lom-.Q4K 2 0.635 41.563 1.22 LOM-M.acc 1.026 32.275 1.20 0.62 1.29 1.02 69 80 ft wall w80-lom-.Q4K 3 0.569 39.643 1.19 LOM-M.acc 1.026 32.275 1.20 0.55 1.23 0.99 70 80 ft wall w80-san-.Q4K 1 0.762 45.732 1.24 SAN-M.acc 0.948 42.312 1.18 0.80 1.08 1.05 71 80 ft wall w80-san-.Q4K 2 0.732 44.796 1.23 SAN-M.acc 0.948 42.312 1.18 0.77 1.06 1.04 72 80 ft wall w80-san-.Q4K 3 0.669 42.321 1.18 SAN-M.acc 0.948 42.312 1.18 0.71 1.00 1.00 Average of Above 9 0.644 40.718 1.181 Mid Spectrum 0.929 37.214 1.167 0.697 1.104 1.012 73 80 ft wall w80-elc-.Q4K 1 0.895 42.781 1.76 ELC-U.acc 1.083 45.320 1.78 0.83 0.94 0.99 74 80 ft wall w80-elc-.Q4K 2 0.878 43.230 1.77 ELC-U.acc 1.083 45.320 1.78 0.81 0.95 0.99 75 80 ft wall w80-elc-.Q4K 3 0.828 42.279 1.73 ELC-U.acc 1.083 45.320 1.78 0.76 0.93 0.97 76 80 ft wall w80-erz-.Q4K 1 1.181 52.435 1.77 ERZ-U.acc 1.089 52.950 1.69 1.08 0.99 1.05 77 80 ft wall w80-erz-.Q4K 2 1.135 52.091 1.77 ERZ-U.acc 1.089 52.950 1.69 1.04 0.98 1.05 78 80 ft wall w80-erz-.Q4K 3 1.055 49.750 1.70 ERZ-U.acc 1.089 52.950 1.69 0.97 0.94 1.01 79 80 ft wall w80-tab-.Q4K 1 1.025 43.980 1.83 TAB-U.acc 1.060 46.922 1.76 0.97 0.94 1.04 80 80 ft wall w80-tab-.Q4K 2 1.011 42.697 1.83 TAB-U.acc 1.060 46.922 1.76 0.95 0.91 1.04 81 80 ft wall w80-tab-.Q4K 3 0.936 40.261 1.78 TAB-U.acc 1.060 46.922 1.76 0.88 0.86 1.01 Average of Above 9 0.994 45.500 1.771 U.B. Spectrum 1.077 48.397 1.743 0.922 0.939 1.016 82 120 ft wall w12-cap-.Q4K 1 0.221 12.815 0.47 CAP-L.acc 0.894 15.370 0.39 0.25 0.83 1.21 83 120 ft wall w12-cap-.Q4K 2 0.202 12.610 0.46 CAP-L.acc 0.894 15.370 0.39 0.23 0.82 1.18 84 120 ft wall w12-cap-.Q4K 3 0.199 12.263 0.43 CAP-L.acc 0.894 15.370 0.39 0.22 0.80 1.10 85 120 ft wall w12-day-.Q4K 1 0.195 10.675 0.45 DAY-L.acc 0.936 11.684 0.39 0.21 0.91 1.15 86 120 ft wall w12-day-.Q4K 2 0.176 10.497 0.44 DAY-L.acc 0.936 11.684 0.39 0.19 0.90 1.13 87 120 ft wall w12-day-.Q4K 3 0.189 10.159 0.42 DAY-L.acc 0.936 11.684 0.39 0.20 0.87 1.08 88 120 ft wall w12-lan-.Q4K 1 0.241 14.801 0.44 LAN-L.acc 0.771 15.173 0.36 0.31 0.98 1.22 89 120 ft wall w12-lan-.Q4K 2 0.224 14.223 0.43 LAN-L.acc 0.771 15.173 0.36 0.29 0.94 1.19 90 120 ft wall w12-lan-.Q4K 3 0.203 13.376 0.41 LAN-L.acc 0.771 15.173 0.36 0.26 0.88 1.14 Average of Above 9 0.206 12.380 0.439 L.B. Spectrum 0.867 14.076 0.380 0.240 0.881 1.156 91 120 ft wall w12-imp-.Q4K 1 0.625 40.256 1.24 IMP-M.acc 0.812 37.054 1.12 0.77 1.09 1.11 92 120 ft wall w12-imp-.Q4K 2 0.574 39.312 1.21 IMP-M.acc 0.812 37.054 1.12 0.71 1.06 1.08 93 120 ft wall w12-imp-.Q4K 3 0.516 37.327 1.16 IMP-M.acc 0.812 37.054 1.12 0.64 1.01 1.04 94 120 ft wall w12-lom-.Q4K 1 0.486 39.153 1.33 LOM-M.acc 1.026 32.275 1.20 0.47 1.21 1.11 95 120 ft wall w12-lom-.Q4K 2 0.435 38.141 1.30 LOM-M.acc 1.026 32.275 1.20 0.42 1.18 1.08 96 120 ft wall w12-lom-.Q4K 3 0.450 36.063 1.26 LOM-M.acc 1.026 32.275 1.20 0.44 1.12 1.05 97 120 ft wall w12-san-.Q4K 1 0.521 40.379 1.45 SAN-M.acc 0.948 42.312 1.18 0.55 0.95 1.23 98 120 ft wall w12-san-.Q4K 2 0.504 38.840 1.41 SAN-M.acc 0.948 42.312 1.18 0.53 0.92 1.19 99 120 ft wall w12-san-.Q4K 3 0.449 37.336 1.33 SAN-M.acc 0.948 42.312 1.18 0.47 0.88 1.13 Average of Above 9 0.507 38.534 1.299 Mid Spectrum 0.929 37.214 1.167 0.556 1.047 1.113 100 120 ft wall w12-elc-.Q4K 1 0.863 55.709 1.93 ELC-U.acc 1.083 45.320 1.78 0.80 1.23 1.08 101 120 ft wall w12-elc-.Q4K 2 0.843 53.682 1.90 ELC-U.acc 1.083 45.320 1.78 0.78 1.18 1.07 102 120 ft wall w12-elc-.Q4K 3 0.774 50.337 1.83 ELC-U.acc 1.083 45.320 1.78 0.71 1.11 1.03 103 120 ft wall w12-erz-.Q4K 1 0.921 55.895 1.81 ERZ-U.acc 1.089 52.950 1.69 0.85 1.06 1.07 104 120 ft wall w12-erz-.Q4K 2 0.860 54.019 1.77 ERZ-U.acc 1.089 52.950 1.69 0.79 1.02 1.05 105 120 ft wall w12-erz-.Q4K 3 0.820 50.339 1.68 ERZ-U.acc 1.089 52.950 1.69 0.75 0.95 0.99 106 120 ft wall w12-tab-.Q4K 1 0.874 43.529 2.09 TAB-U.acc 1.060 46.922 1.76 0.82 0.93 1.19 107 120 ft wall w12-tab-.Q4K 2 0.825 41.913 2.03 TAB-U.acc 1.060 46.922 1.76 0.78 0.89 1.15 108 120 ft wall w12-tab-.Q4K 3 0.738 40.690 1.93 TAB-U.acc 1.060 46.922 1.76 0.70 0.87 1.10 Average of Above 9 0.835 49.568 1.886 U.B. Spectrum 1.077 48.397 1.743 0.775 1.027 1.081 Table 6-1. (Continued). this ratio systematically decreased for increasing wall height and lowering of the spectral shape at long periods. Therefore, averaging the ratios (shown in the right-most column) from the three failure mechanisms evaluated in this study would seem to be reasonable. Cursory review of the data supports to some degree, the presumptive historical practice of adopting about 1⁄2 to 2⁄3 of PGA for pseudo-static design analysis. How- ever, as noted above, rather than the prevalent view that the reduction is to account for the time variation in PGA, the re- duction being introduced in this discussion is for wave scat- tering. Any further reduction for the duration of earthquake loading should be determined by the structural designer. 6.2 Conclusions Figure 6-13 provides a basis for determining a reduction factor (that is, the α factor) to be applied to the peak ground acceleration used when determining the pseudo-static force

in the design of retaining walls and slopes. Further discussion of the use of the α factor is provided in Chapter 7. Results of these height-dependent seismic coefficient studies are general enough that they can be applied to either the seis- mic design of retaining walls, embankments and slopes, or buried structures. The design process involves first determin- ing the response spectra for the site. This determination is made using either guidance in the 2008 AASHTO LRFD Bridge Design Specifications or from site-specific seismic hazard studies. Note that spectra in the 2007 AASHTO LRFD Bridge Design Specifications do not distinguish between CEUS and WUS shapes and are not consistent with this recommended approach; however, the newly adopted AASHTO ground motion maps will account for this difference. The only assumption made is that the ground motion design criteria should be defined by a 5 percent damped design response spec- trum for the referenced free-field ground surface condition at the project site. Once the design ground motion is established for a site, the analyses could proceed following the methodology outlined in this chapter. This methodology involves defining the seis- mic coefficient for the evaluation of retaining walls, slopes and embankments, or buried structures, as follows: • The design ground motion demand is characterized by a design response spectrum that takes into account the seis- mic hazard and site response issues for the site. This re- quirement is rather standard, and should not present undue difficulties for the designer. The selection of the appropri- ate spectra shape should focus on the 1-second ordinate. • Starting from the design response spectrum, the designer would normalize the response spectrum by the peak ground acceleration to develop the normalized spectral shape for the specific project site. This spectrum is then overlaid on the spectral shape shown on Figure 5-4 to determine the most appropriate spectral curve shape for the design condition. • After selecting the appropriate spectral shape (that is, in terms of UB, Mid, and LB spectral shapes), Figure 6-13 is used to select the appropriate reduction factor (the α factor). The approach described above was further simplified for use in the proposed Specifications by relating the α factor to height, PGA, and S1 in a simple equation, as discussed in Chapter 7. Either the approach discussed in this chapter or the equation given in Chapter 7 is an acceptable method of determining the α factor. As discussed earlier, wave scattering theory represents one of the several justifications for selecting a pseudo-static seismic co- efficient lower than the peak ground acceleration. In addition to the wave scattering α factor, additional reduction factors may be applied as appropriate, including that some permanent movement is allowed in the design, as discussed in Chapter 7. Consideration also can be given to the use of a time-averaged seismic coefficient based on the average level of ground shak- ing, rather than the peak, as long as the structural designer con- firms that the average inertial force is permissible for design. 67 Figure 6-13. Resultant wave scattering  coefficients for retaining wall design.

Next: Chapter 7 - Retaining Walls »
Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments Get This Book
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TRB’s National Cooperative Highway Research Program (NCHRP) Report 611: Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments explores analytical and design methods for the seismic design of retaining walls, buried structures, slopes, and embankments. The Final Report is organized into two volumes. NCHRP Report 611 is Volume 1 of this study. Volume 2, which is only available online, presents the proposed specifications, commentaries, and example problems for the retaining walls, slopes and embankments, and buried structures.

The appendices to NCHRP Report 611 are available online and include the following:

A. Working Plan

B. Design Margin—Seismic Loading of Retaining Walls

C. Response Spectra Developed from the USGS Website

D. PGV Equation—Background Paper

E. Earthquake Records Used in Scattering Analyses

F. Generalized Limit Equilibrium Design Method

G. Nonlinear Wall Backfill Response Analyses

H. Segrestin and Bastick Paper

I. MSE Wall Example for AASHTO ASD and LRFD Specifications

J. Slope Stability Example Problem

K. Nongravity Cantilever Walls

View information about the TRB Webinar on Report 611: Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments: Wednesday, February 17, 2010

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