912
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2 IMPLEMENTATION OF NEWMARK-TYPE
PREDICTIVE MODELS
Τhe Newmark conventional analytical rigid block method is
used to predict cumulative slope displacements obtained by
integrating twice with respect to time the parts of an earthquake
acceleration-time history that exceed the critical or yield
acceleration, a
c
(k
y
·g) (e.g. threshold acceleration required to
overcome shear soil resistance and initiate sliding). The second
approach is a two-parameter vector (PGA, PGV) model
proposed by Rathje and Antonakos (2011) applied herein to
evaluate co-seismic slope displacements. This model is
recommended for use in practice due to its ability to
significantly reduce the variability in the displacement
prediction. For flexible sliding, k
max
(e.g. peak value of the
average acceleration time history within the sliding mass) is
used in lieu of PGA and k–vel
max
(e.g. peak value of the k-vel
time history provided by numerical integration of the k-time
history) is used to replace PGV. The third one is the Bray and
Travasarou (2007) model. In this model cumulative
displacements are calculated using the nonlinear fully coupled
stick-slip deformable sliding block model proposed by Rathje
and Bray (2000) to capture the dynamic response of the sliding
mass. They use a single intensity parameter to characterize the
equivalent seismic loading on the sliding mass, i.e. the ground
motion’s spectral acceleration S
a
at a degraded period equal to
1.5T
s
, which was found to be the optimal one in terms of
efficiency and sufficiency (Bray 2007).
The first goal is to study the influence of the earthquake
characteristics and the dynamic response of the slope on the
magnitude of the residual slope displacements using the
aforementioned three predictive models. In this respect,
permanent displacements as a function of the critical
acceleration ratio (e.g. k
y
/k
max
or k
y
/PGA) are computed using
the three approaches considering different earthquake input
motions and compliance of the sliding surface. Comparisons
between the models allowed evaluating their reliability. Mean
displacements were calculated using the Newmark rigid block
model, as reference, whereas median values ±1 standard
deviation and median and 16
th
- 84
th
percentiles were derived
for the decoupled and coupled approximations respectively.
Table 1. Parameters describing the characteristics of the ground motions
nd the dynamic response of the sliding mass
a
Earthquake
record name
Valnerina
1979- Cascia_L
Northridge
1994- Pacoima
Dam_L
Earthquake code
Cascia
Pacoima
Moment
magnitude (M
w
)
5.9
6.7
PGA (g)
0.15
0.41
Fundamental
period T
p
(sec)
0.23
0.48
Mean Period T
m
(sec)
0.295
0.507
Scaled PGA (g)
0.3
0.7
0.3
0.7
PGV (cm/sec)
10.3
30.9
14.6
43.9
Natural period of the
sliding mass T
s
(sec)
0.16
0.032
0.16
0.032
S
a
(1.5T
s
)/PGA
scaled
2.93
1.07
2.26
1.03
T
s
/T
m
0.54
0.11
0.32
0.06
The seismic input consists of two real acceleration time
histories recorded at rock outcropping conditions and scaled at
two levels of PGA, i.e. 0.3 and 0.7g. Table 1 presents the
parameters describing some basic characteristics of the ground
motions and the flexibility of the potential sliding surface. The
displacements were computed for nearly rigid (T
s
=0.032sec)
and relatively flexible (T
s
=0.16 sec) sliding masses. The derived
(mean or median) permanent displacements for the three
different predictive models and for the different considered
earthquake scenarios plotted as a function of the critical
acceleration ratio, k
y
/k
max
or k
y
/PGA, are illustrated in Figures
2a, 2b and 2c when considering the nearly rigid sliding surface.
Moreover in Figures 3a and 3b we compared between them the
three analytical models for the Pacoima 0.7g input motion for
the nearly rigid and the relatively flexible sliding mass
respectively.
(a)
(b)
(c)
Figure 2. Newmark (a), Rathje and Antonakos (b) and Bray and
Travasarou (c) displacement versus k
y
/k
max
considering a nearly rigid
sliding mass for different acceleration time histories (cascia, pacoima)
scaled at different levels of PGA (0.3g, 0.7g)
The results prove the important role of the amplitude and
frequency content of the earthquake as well as the compliance
of the sliding surface on the magnitude of the computed
displacements. As it should be expected, time histories scaled at
0.7g produce larger displacements compared to those scaled at
0.3g for the same critical acceleration ratios. For the Newmark
and Rathje and Antonakos models the lower frequency input
motion (Pacoima- f
p
=2.1Hz) generally yields larger
displacements in relation to the higher frequency input motion
(Cascia- f
p
=4.4Hz). For the Newmark model (see Fig. 2a) this
trend becomes more pronounced with the increase of the critical
