Ground Motions and Liquefaction – The Loading Part of the Equation
Steve Kramer*, Roy Mayfield, and Robert Mitchell
Department of Civil and Environmental Engineering
University of Washington
Seattle, WA 98195-2700, USA
Ph: (206) 685-2642; Fax: (206) 543-1543
Emails: kramer@u.washington.edu (Kramer); roym@ce.washington.edu (Mayfield);
ram@shanwil.com (Mitchell)
*Corresponding author
ABSTRACT
While the evaluation of soil liquefaction hazards has advanced tremendously over the
past 40 years, the development of performance-based earthquake engineering will place
additional demands on geotechnical engineers charged with evaluating liquefaction hazards.
Since performance-based earthquake engineering is generally formulated in a probabilistic
manner, uncertainties associated with the prediction of liquefaction must be quantified; this will
require characterization of uncertainties in both loading and resistance. This paper discusses
issues associated with characterization of earthquake loading from the standpoint of liquefaction
hazard evaluation, and suggests a new approach to characterization of earthquake loading that
may have significant advantages for performance-based liquefaction hazard evaluation.
Examples based on recorded ground motions from the Chi-Chi earthquake are also presented and
discussed.
Key Words: Liquefaction, intensity, performance-based earthquake engineering
Introduction
Evaluations of liquefaction potential generally involve comparison of loading and
resistance (or demand and capacity). The loading is a function of the level of ground shaking
produced by the earthquake and the resistance is a function of the characteristics of the soil.
Whether expressed in terms of cyclic shear stresses, cyclic shear strains, or some other metric
such as energy, liquefaction potential depends on both loading and resistance.
The development of performance-based earthquake engineering [1-3] will place
additional demands on engineers responsible for liquefaction hazard evaluations. While
conventional liquefaction hazard evaluations are typically performed for one, or perhaps two,
levels of ground motion, most implementations of PBEE require integration of hazard over a
wide range of ground shaking levels. Furthermore, they require estimation of the probability of
liquefaction over that wide range. In recent years, a number of investigators using a variety of
different approaches have proposed probabilistic liquefaction models [4-7], i.e. models that allow
estimation of the probability of liquefaction given some measures of loading and resistance. A
recent set of studies supported by the Pacific Earthquake Engineering Research (PEER) Center
[8], for example, took a very careful look at the case history data from which empirical
liquefaction potential evaluations are developed. By systematically and consistently evaluating
and considering uncertainties in the various factors that influence both loading and resistance,
these researchers were able to reduce a good deal of the uncertainty inherent in previous
liquefaction evaluation procedures.
The overwhelming majority of liquefaction potential evaluations are performed using the
cyclic stress method [9] in which both loading and resistance are measured in terms of cyclic
shear stress amplitudes, commonly normalized by initial vertical effective stress to produce a
cyclic stress (or resistance) ratio. Loading is actually characterized by means of two variables in
the cyclic stress approach – one that describes the amplitude of the shear stress and the other that
serves as a proxy for duration and frequency content. In the commonly used “simplified
method,” peak ground acceleration, PGA, serves as the amplitude parameter and magnitude, M,
as the duration/frequency content parameter (through the so-called Magnitude Scaling Factor,
MSF). The necessity of specifying two parameters to describe earthquake loading for
liquefaction potential evaluation purposes presents no difficulty when ground motion hazards are
specified by deterministic seismic hazard analysis (DSHA) since such procedures produce
scenario events with easily identified PGA and magnitude values. The procedure is less
apparent, however, when hazards are specified by probabilistic seismic hazard analysis (PSHA)
because PGA becomes a weighted average of many different scenarios that span a wide range of
magnitudes (and, in many cases, multiple sources). De-aggregation analyses can be used to
select a magnitude that is consistent with the PGA value, but this procedure works better for
some sites than others. Vector-based PSHAs, which would produce multiple ground motion
parameters with proper consideration of correlation, would allow a more rigorous definition of
PGA and magnitude, but such procedures are only now beginning to be developed.
Performance-Based Earthquake Engineering
PBEE is generally formulated in a probabilistic framework to evaluate the risk associated
with earthquake shaking at a particular site. The risk can be expressed in terms of cost, fatalities,
or other measures. Minimizing the uncertainty in these risk estimates requires minimizing the
uncertainties in the variables and relationships between the variables that go into the calculation
of risk. If we limit ourselves to the risk associated with liquefaction, we can consider
uncertainties in the level of shaking and uncertainties in the response of the soil deposit given
some level of shaking. Obviously, we would like to be able to accurately predict (a) the level of
shaking accurately and (b) the response of a liquefiable soil deposit, given that level of shaking.
The PBEE framework being developed by PEER computes risk as a function of ground
shaking through the use of several intermediate variables. The ground motion is characterized
by an Intensity Measure, IM, which could be any one of a number of ground motion parameters
(e.g. PGA, Arias intensity, Ia, etc.). The effects of the IM on a system of interest are expressed in
terms that make sense to engineers in the form of Engineering Demand Parameters, or EDPs
(e.g. excess pore pressure, settlement, etc.). The physical effects associated with the EDPs are
expressed in terms of Damage Measures, or DMs (e.g. settlement-induced crack width). Finally,
the risk associated with the DM is expressed in a form that is useful to decision-makers by means
of Decision Variables, DV (e.g. repair cost). The mean annual rate of exceedance of various DV
levels, DV, can be expressed in terms of the other variables as
 DV 
N DM N EDP N IM
  
k 1
j 1 i 1
P[ DV | DM k ]P[ DM k | EDP j ]P[ EDP j | IM i ]   IMi
where P[a|b] describes the probability of exceeding a given b, and where NDM, NEDP, and NIM are
the number of increments of DM, EDP, and IM, respectively; accuracy increases with increasing
number of increments. The geotechnical engineer’s contribution to this process for liquefaction
hazards comes primarily in the evaluation of P[EDP|IM].
Optimum Intensity Measure
A good intensity measure for use in the PEER PBEE framework would be one that is
efficient, in the sense of being closely related to the EDP of interest, and sufficient, in the sense
of not needing to be accompanied by additional information to improve its relationship to the
EDP [10]. Additionally, it should be accurately predictable, i.e. have relatively low dispersion
in its attenuation relationship.
The relative quality of various IMs can be evaluated by comparing their performance
with respect to these three criteria. Three potential IMs are considered here – PGA, Ia, and a
modified version of cumulative absolute velocity that will be referred to as CAV5. CAV5 is
defined in the same manner as cumulative absolute velocity, but with a threshold acceleration of
5 cm/sec2.
Efficiency
The efficiency of these three parameters was evaluated through a series of onedimensional, nonlinear site response analyses described by [11]. A series of nine hypothetical
soil deposits spanning a range of liquefiable layer thicknesses and densities was subjected to over
450 input motions from 22 different earthquakes of magnitudes ranging from 5.2 to 7.4. For
each soil profile, the depth-averaged (over the thickness of the liquefiable layer) excess pore
pressure ratio was computed for each input motion. The depth-averaged pore pressure ratios
were then plotted against each of the three ground motion parameters and a curve fit to each plot;
examples of these plots for one of the nine cases are shown in Figure 1. The sum of squared
residuals (SSR) was computed for each soil profile as was the mean SSR for all nine profiles. As
indicated in Figure 1, both visually and in terms of the computed SSR, the uncertainty in depthaveraged pore pressure ratio is significantly lower for Arias intensity than for PGA, and
significantly lower yet for CAV5 compared to Arias intensity. In fact, CAV5 proved to have the
lowest variability in depth-averaged pore pressure ratio (conditional upon the IM) in a group of
some 300 candidate IMs.
Sufficiency
The sufficiency of the parameters can be illustrated by comparing the variation of their
computed residuals with magnitude and distance (Figure 2). Linear regression on the residuals
shows the relative sufficiency of the three parameters. Note the significant slope of the PGA
residuals plotted against magnitude; this magnitude dependence explains why a magnitude
scaling factor is needed in addition to PGA in order to make good estimates of liquefaction
potential. CAV5 is clearly the most sufficient of these parameters with respect to magnitude, and
is of about equal sufficiency to the other parameters with respect to distance.
Predictability
The predictability of the various parameters can be evaluated by comparing the standard
errors from attenuation relationships for each. Obviously, many attenuation relationships for
peak ground acceleration are available and relationships have recently been developed for Arias
intensity; these relationships are generally of somewhat different form and based on different
data sets. To eliminate differences associated with the form of the relationships and with the
data on which the attenuation relationships are based, attenuation relationships for peak
acceleration, Arias intensity, and CAV5 were developed using the same basic equation and the
same set of ground motions. The form of the equation selected was equivalent to that of
Traversarou et al. [12], i.e.
ln( IM )  c1  c2 (M  6)  c3 ln( M / 6)  c4 ln( R 2  h 2 )  f1FN  f 2 FR
(1)
where M is moment magnitude, R is closest distance to the rupture surface, h is a fictitious
hypocentral depth determined by regression and FN and FR are style-of-faulting variables (FN =
FR = 0 for strike-slip, FN = 1 and FR = 0 for normal, and FN = 0 and FR = 1 for reverse and
reverse-oblique faulting); the data set was the one used for the previously described site response
analyses. The results of those analyses are shown graphically for CAV5 in Figure 3.
The uncertainty involved in IM predictability is usually expressed in terms of a standard
error term, ln IM. This term represents the standard deviation of the residuals from the regression
analysis on which the attenuation relationship is based; since IMs are usually lognormally
distributed, the standard deviation is of the residuals of the logarithm of the IM. The standard
error terms obtained using a mixed effects regression model are presented in Table 1. The values
for PGA and Ia are consistent with standard error values in other attenuation relationships for
those parameters; since CAV5 is a “new” IM, there are no other attenuation relationships to
compare it to. It is notable, however, that PGA exhibits the best predictability (lowest ln IM), Ia
exhibits the worst predictability (highest ln IM), and that the predictability of CAV5 is much
closer to that of PGA than Ia.
Summary
In terms of the criteria of efficiency, sufficiency, and predictability, the historical
liquefaction IMs of peak acceleration and Arias intensity have relative strengths and weaknesses.
Peak acceleration is relatively inefficient and distinctly insufficient with respect to magnitude,
but has good predictability. Arias intensity is relatively efficient and sufficient, but has poor
predictability. The new parameter, CAV5, on the other hand, has distinctly better efficiency and
sufficiency than the others, and has predictability that is nearly as good as that of peak
acceleration. Based on these criteria, it would appear that CAV5 could be a superior IM for
performance-based liquefaction hazard evaluations.
Some Characteristics of CAV5
Because CAV5 is a new parameter whose utility for liquefaction hazard evaluation is just
now being explored, it is useful to consider some of its basic characteristics relative to other
intensity measures that have been used for liquefaction hazard evaluation.
Amplitude Characteristics
The fact that CAV5 makes use of an acceleration threshold means that low-level
acceleration pulses do not contribute to the parameter, which implies that they do not contribute
to excess pore pressure generation. This ins consistent with the theoretically and experimentally
observed existence of a threshold shear strain below which no volume change of an assemblage
of soil particles will occur under drained conditions. Under undrained conditions, the threshold
shear strain represents a limit below which no excess pore pressure is generated.
Temporal Characteristics
CAV5 is an integral intensity measure, i.e. one whose value builds up with time. The
calculation of CAV5 can be illustrated in a series of steps, as illustrated in Figure 4. Figure 4(a)
shows the CHY044-W ground motion record measured in the Chi-Chi earthquake. Figure 4(b)
shows the absolute value of the recorded accelerogram and Figure 4(c) shows the absolute value
of acceleration after application of the 5 cm/sec2 threshold acceleration. Finally, Figure 4(d)
shows the integrals of absolute acceleration (i.e. of the curve plotted in Figure 4(b)) and absolute
acceleration after application of the threshold (curve in Figure 4(c)). The final values of the
curves in Figure 4(d) are CAV (upper curve) and CAV5 (lower curve). Application of the
threshold can be seen to influence the function that is being integrated at the beginning and end
of the record, but it also has some influence in the central portion of the record. Low-level
oscillations, which contribute to CAV but don’t lead to the generation of significant pore
pressure, do not contribute to CAV5. For this particular record, CAV5 is about 10% smaller than
CAV; that statistic, however, varies from record to record.
Frequency Domain Characteristics
Peak acceleration is well recognized as a “high frequency” ground motion parameter, i.e.
one that reflects the higher frequency components of a ground motion record. Arias intensity is
also considered to be a relatively high frequency parameter. To determine the frequency domain
characteristics of CAV5 relative to peak acceleration and Arias intensity, the characteristics of the
three parameters were evaluated for a set of 20 ground motion records from the Chi-Chi
earthquake. The motions, obtained from the PEER strong motion database and listed in Table 2,
are from distances of 11 – 26 km and have moderate amplitudes (0.1  PGA  0.3 g);
obviously, the magnitude of the earthquake that produced them was the same so no magnitude
scaling factor would be needed to account for differences in frequency content and duration in a
PGA-based liquefaction hazard evaluation.
Response spectra were computed for each of the motions and the degree of correlation
between the three parameters and the spectral acceleration at different periods were computed.
The results, expressed in terms of R2 for the best fit obtained from a series of functions (linear,
logarithmic, power, exponential), are shown in Figure 5. These results clearly show that peak
acceleration and Arias intensity are most closely correlated to lower period (higher frequency)
components of the motions than CAV5. Thus, CAV5 appears to better reflect the longer period
(lower frequency) components of the motions. Since pore pressure generation is known to be
closely related to strain amplitude, and strain amplitude is proportional to particle velocity, and
particle velocity reflects longer period components of a ground motion than PGA, the fact that
CAV5 is related to longer periods suggests that it might have a closer relationship to pore pressure
generation than PGA and Arias intensity.
Efficiency With Respect to Chi-Chi Surface Motions
The parameter, CAV5, was originally identified by correlations between depth-averaged
pore pressure ratios and input motion (rock outcrop) intensity measures. Liquefaction hazard
evaluations, however, are customarily performed using intensity measures from ground surface
motions. The efficiency of CAV5 relative to peak acceleration and Arias intensity at the surface
level was evaluated by the following process.
1. A soil profile consistent with the conditions at Berth 4 of the Port of Taichung was
developed for equivalent linear site response analysis.
2. To ensure that computed pore pressure ratios would not be extremely high (in which case
dispersion in pore pressure generation would be minimal), a simplified analysis was
performed to identify the PGA level for which a 5% probability of liquefaction would be
predicted by the procedure of Seed et al. (2003). This PGA was approximately 0.1g.
3. The 20 motions listed in Table 2 were individually scaled so that they would produce a
peak ground surface acceleration of 0.1g in equivalent linear analyses of the Port of
Taichung soil profile. The average Arias intensity and CAV5 values for the 20 scaled
motions were 0.265 m/sec and 5.39 m/sec, respectively.
4. The suite of motions was scaled twice more – first to obtain a set of rock motions that
produced ground surface motions with Ia = 0.265 m/sec, then again to obtain a set of rock
motions that produced ground surface motions with CAV5 = 5.39 m/sec.
5. All three sets of motions, which produced constant ground surface PGA, Ia, and CAV5
values, were used as input motions to a nonlinear, effective stress model of the Port of
Taichung profile.
Excess pore pressures computed from the analyses described in Step 5 are presented in
Figure 6. A perfect intensity measure would produce no dispersion in computed pore pressure
ratios, i.e. motions with equal IM would produce equal pore pressures. Obviously, no IM can be
perfect. The coefficients of variation associated with each set of analyses are shown in Figure 7.
These results show that the uncertainty in excess pore pressure ratio, conditional upon IM, is
lowest when CAV5 is used as the IM and highest when PGA is the IM, even when the magnitude
dependence problem of PGA is removed by consideration of motions from a single earthquake.
Liquefaction Intensity Measures in the Chi-Chi Earthquake
Because CAV5-based liquefaction evaluation procedures have not yet been developed,
direct evaluations of the usefulness of CAV5 for characterization of loading cannot be made.
However, the values of CAV5 in the Chi-Chi earthquake can be compared with expected values
of CAV5 from California earthquakes using the previously described attenuation relationships.
The fact that peak accelerations in the Chi-Chi earthquake were lower than predicted by
previously existing attenuation relationships has been widely reported. Figure 8 shows the CAV5
values from N recording stations in the Chi-Chi earthquake, along with the CAV5 values
predicted by the CAV5 attenuation relationship (based on California data) described previously.
The recorded CAV5 values are generally lower than the values predicted by the attenuation
relationship, particularly at relatively short distances where they are substantially lower. Within
about 10 km of the fault, it appears that similar soil deposits with equal resistance to liquefaction
would be about equally likely to liquefy.
Summary and Conclusions
Over the past 40 years, significant advances have been made in the development of
procedures for evaluation of liquefaction potential. However, most of the advances, and most
recent research, have been associated with improved evaluation of liquefaction resistance. It is
important to recognize, particularly as the earthquake engineering profession moves toward
performance-based concepts for evaluation and design, that advances can also be made with
respect to the loading part of the liquefaction hazard evaluation equation.
This short paper was intended to explore some of the demands that emerging
performance-based earthquake engineering procedures will place on engineers involved with
liquefaction hazard evaluations, and to discuss their implications for characterization of
earthquake loading in such evaluations. The concepts of efficiency, sufficiency, and
predictability can be used to evaluate the relative suitability of different intensity measures for
performance-based liquefaction hazard evaluations. Intensity measures that are more efficient,
sufficient, and predictable will lead to more accurate (less uncertain) estimates of liquefaction
potential; in the probabilistic framework of PBEE, such parameters will also lead to less
conservative estimates of liquefaction hazard.
A new intensity measure for performance-based liquefaction hazard evaluation, CAV5,
was introduced and shown to have a combination of efficiency, sufficiency, and predictability
that exceeds those of intensity measures historically used for liquefaction hazard evaluations. It
appears that CAV5 is a promising parameter for use in performance-based liquefaction hazard
evaluations.
Acknowledgments
The work described in this paper was supported by the Washington State Department of
Transportation and the John R. Kiely Professorship at the University of Washington. This
support is gratefully acknowledged.
References
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http://pitch.stanford.edu/rmsweb/RMS_Papers/pdf/nico/EQ_Spectra01.pdf
11. Mitchell, R.A. and Kramer, S.L. (2003). “Ground motion intensity measures for liquefaction
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Table 1. Standard errors for IM attenuation relationships
IM
ln IM
PGA
0.620
Ia
1.070
CAV5
0.708
Table 2. Characteristics of ground motions recorded in Chi-Chi earthquake.
Motion
als-e
als-n
chy010-e
chy010-n
chy034-e
chy035-n
chy035-w
chy041-w
tcu029-n
tcu029-w
tcu036-n
tcu036-w
tcu039-n
tcu039-w
tcu046-n
tcu046-w
tcu048-n
tcu048-w
tcu056-n
tcu056-w
R (km)
15.29
15.29
25.39
25.39
20.23
18.12
18.12
25.96
24.71
24.71
13.69
13.69
16.70
16.70
14.34
14.34
14.38
14.38
11.11
11.11
Lat.
23.465
23.465
23.521
23.520
23.520
23.439
24.559
24.559
24.449
24.449
24.492
24.492
24.469
24.469
24.181
24.181
24.159
24.159
Long.
120.544
120.544
120.545
120.584
120.584
120.596
120.749
120.749
120.696
120.696
120.784
120.784
120.854
120.854
120.593
120.593
120.624
120.624
PGA (g)
0.183
0.163
0.225
0.173
0.248
0.246
0.251
0.301
0.200
0.165
0.131
0.139
0.145
0.205
0.115
0.133
0.184
0.123
0.134
0.134
Ia (m/sec)
0.962
0.907
0.722
0.671
1.444
1.300
1.561
1.548
1.962
0.578
0.739
0.734
0.843
0.943
0.348
0.435
0.702
0.554
0.842
0.877
CAV5
11.556
11.486
8.934
9.275
14.13
12.588
14.117
15.1
10.237
8.763
10.629
10.142
10.531
11.88
5.923
6.695
11.475
9.402
11.956
11.804
Figure 1. Efficiency plots for (a) peak ground acceleration, (b) Arias Intensity, and (c)
cumulative absolute velocity.
Figure 2. Magnitude sufficiency for (a) PGA, (b) Ia, and(c) CAV5. Distance sufficiency for
(d) PGA, (e) Ia, and (f) CAV5.
Figure 3. Attenuation relationship for CAV5.
Figure 4. Illustration of CAV5: (a) original accelerogram, (b) squared accelerogram, (c)
squared accelerogram after application of 5 cm/sec2 threshold acceleration, (d)
buildup of CAV and CAV5 as integrals of squared acceleration and squared
acceleration after threshold application.
Figure 5. Variation of coefficient of determination with period for relationship between
spectral acceleration and three intensity measures.
Figure 6. Excess pore pressure profiles for scaled motions: (a) motions scaled to
produce constant ground surface PGA value, (b) motions scaled to produce
constant ground surface Ia value, and (c) motions scaled to produce constant
ground surface CAV5 value.
Figure 7. Variation of coefficient of variation of excess pore pressure with depth for
motions scaled to produce constant ground surface PGA, Ia, and CAV5 values.
Figure 8. Variation of CAV5 with distance as recorded in Chi-Chi earthquake and predicted by
attenuation relationship based on California data.