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

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

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

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

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

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

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

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