lower bounds on ground motion at point reyes during the...
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Lower Bounds on Ground Motion at Point Reyes1
During the 1906 San Francisco Earthquake from Train2
Toppling Analysis3
by Swetha Veeraraghavan, Thomas H. Heaton, and Swaminathan Krishnan4
Address of authors:5
Swetha Veeraraghavan6
Idaho National Laboratory (formerly at Caltech)7
2525 North Fremont Avenue, Idaho Falls, ID 83415.8
10
Thomas H. Heaton11
Caltech12
1200 East California Blvd, MC 104-44, Pasadena, CA 91125.13
15
Swaminathan Krishnan16
Arup (formerly at Caltech)17
12777 W Jefferson Blvd #100, Los Angeles, CA 90066.18
20
21
1
Abstract22
Independent constraints on the ground motions experienced at Point Reyes station during23
the 1906 San Francisco earthquake are obtained by analyzing the dynamic response of a train24
that overturned during the earthquake. The train is modeled as a rigid rectangular block for25
this study. From this analysis, we conclude that the PGA and PGV at Point Reyes station26
would have been at least 4 m/s2 and 0.5 m/s, respectively. This lower bound is then used to27
perform simple checks on the synthetic ground motion simulations of the 1906 San Francisco28
earthquake. It is also shown that the hypocenter of the earthquake should be located to the south29
of Point Reyes station for the overturning of the train to match an eye-witness description of30
the event.31
Introduction32
The 18th April, 1906 San Francisco earthquake (Mw 7.8) and the subsequent investigation of33
this earthquake (Lawson and Reid, 1908) marked the birth of modern earthquake science in the34
United States. Mount Hamilton, at a distance of approximately 130 km from the San Andreas35
fault, was the location closest to the source where ground motion was recorded (by a three-36
component pendulum). Using this record as a constraint, Boore (1977) and Lomax (2005)37
located the earthquake hypocenter offshore (off the coast of San Francisco). Recently, efforts38
have also been made to recreate strong ground motion from this earthquake using the limited39
observations and inferences about the distribution of fault slip in that event (Song et al., 2008;40
Aagaard et al., 2008). Given the sparsity of recorded data, there is still significant uncertainty41
in the epicentral (and hypocentral) location(s) and the intensity of near source ground motions42
during this earthquake.43
A train, pulled by a narrow gauge locomotive [engine number 14, built in 1891 by Brooks44
and scrapped in 1935 (Dickinson et al., 1967)], overturned near Point Reyes station during this45
earthquake. The train headed south towards San Francisco was stopped on a siding near the46
station for refueling at the time of the earthquake. A photo of the overturned train, taken after47
the incident, is shown in Fig. 1 and a map with the train location (black circle) in relation to48
the San Andreas fault (red line) is shown in Fig. 2a. The overturning of the train during the49
2
earthquake was witnessed by a conductor and documented by Jordan (1907). The following50
passage is from Jordon’s description of the event.51
At Point Reyes Station at the head of Tomales Bay the 5:15 train for San Francisco52
was just ready. The conductor had just swung himself on when the train gave a great53
lurch to the east, followed by another to the west, which threw the whole train on its54
side. The astonished conductor dropped off as it went over, and at the sight of the55
falling chimneys and breaking windows of the station, he understood that it was the56
Temblor. The fireman turned to jump from the engine to the west when the return57
shock came. He then leaped to the east and borrowing a Kodak he took the picture58
of the train here presented.59
Estimating the ground shaking intensity (at least in the direction of toppling) needed to60
overturn the train in the manner documented by Jordan can supplement the limited available61
data from this earthquake. To this end, Anooshehpoor et al. (1999) idealized the rocking62
behavior of the train in 2-D to that of a rigid rectangular block of height 3.76 m and width63
0.91 m supported on a rigid horizontal surface. The block was assumed to rock about two64
corners [O orO′ on Fig. 2b] that correspond to the two sets of train wheels seated on the tracks.65
Collisions between the wheel and the track during rocking were assumed to be inelastic, i.e.,66
it was assumed that bouncing did not occur upon impact. The fault-normal (orthogonal to the67
N-S alignment of the train and the line) ground motion was idealized to a full sinusoidal pulse.68
Their goal was to analytically determine a lower bound on the sine wave amplitude (and the69
corresponding pulse period) required to overturn the train model in the manner documented by70
Jordan. From moment balance about the contact point O (and O′), the equation(s) of motion71
for the rocking response of a rectangular block subjected to horizontal base excitation [ug(t)]72
is (are):73
θ + p2sin[−α− θ(t)] = −p2 ug(t)g
cos[−α− θ(t)] θ(t) < 0 (1)
θ + p2sin[α− θ(t)] = −p2 ug(t)g
cos[α− θ(t)] θ(t) > 0 (2)
where m is the mass of the block, g is the acceleration due to gravity, R is the distance of74
the contact point O or O′ from the center of mass (c.g.), I is the moment of inertia of the block75
3
about either contact point, and p2 = mgR/I .76
This equation of motion is nonlinear and discontinuous. The trigonometric terms causing77
the nonlinearity are often linearized so that an approximate analytical solution may be obtained78
(Housner, 1963; Yim et al., 1980; Spanos and Koh, 1984; Shi et al., 1996; Anooshehpoor et al.,79
1999). Anooshehpoor et al. used the linearized form of Eq. 1 with ug(t) = Asin(ωt+ψ) and80
estimated the minimum amplitude A of the sinusoidal ground acceleration required to topple81
the train to be 0.35 g [3.4 m/s2], 0.5 g [4.9 m/s2], and 1.05 g [10.3 m/s2] at frequencies of 1 Hz,82
1.5 Hz and 2 Hz, respectively. While the response of the train model under idealized pulses83
did provide interesting insights into the dynamics of the problem, Anooshehpoor et al. cor-84
rectly recognized that the model’s response under an earthquake excitation can be significantly85
different from that under an equivalent idealized pulse (Makris and Rousson, 1998; Voyagaki86
and Vamvatsikos, 2014). So they sought to analyze the train model under two scaled accelero-87
grams, the Lucerne record from the 1992 Landers earthquake and a synthetic accelerogram88
at Point Reyes from a hypothetical Mw 8 earthquake rupture propagating northwest on the89
San Andreas fault with epicenter near the Golden Gate bridge (obtained from John Anderson90
through personal communication).91
While the case of the simple full sine pulse excitation is amenable to solving analytically,92
this approach becomes intractable for complex excitation histories such as earthquake ground93
motion. The discontinuity in the equations (Eqs. 1 and 2), which arises due to a change in the94
point of rotation fromO toO′ (or vice-versa) upon impact of the block with the ground, is han-95
dled analytically by determining the times at which θ(t) goes to zero, and switching between96
the solutions of the two equations at these times. In doing this computation, the discontinuity97
arising out of the velocity reduction applied at impact to simulate perfectly inelastic collisions98
(Housner, 1963; Shi et al., 1996) must be honored as well. Another source of discontinuity99
in the equation of motion is the change in the equation of motion with the form of the ground100
excitation. Even in the case of a block subjected to a full-sine wave, once this excitation ends,101
two different equations of motion arise and the time of impact in relation with the excitation102
period determines which equation of motion needs to be solved next. The solution for θ(t)103
in all equations contains cosh and sinh terms and the time at which θ(t) goes to zero, i.e.104
time of impact, cannot be solved analytically. Therefore, different cases have to be formulated105
4
based on whether impact occurs before or after excitation ends. If this approach were to be106
employed for earthquake excitation discretized in a piece-wise linear fashion, there would ex-107
ist a separate equation of motion for each piecewise part of the excitation, and this coupled108
with the discontinuity in equation of motion arising from impact, would result in analytically109
intractable number of cases.110
The nonlinearity and discontinuity in the rocking dynamics of the rectangular block also111
make it difficult to superpose analytical results obtained for simple ground excitations such as112
sinusoidal pulses and impulses to estimate its response under complex excitation histories such113
as earthquake ground motion (using Fourier series or other techniques). Recognizing these114
difficulties, Anooshehpoor et al. (1999) numerically solved the equations of rocking motion115
[Eqs. 1 and 2] for the rectangular train model under the two seismograms, scaled to different116
levels. The Lucerne record scaled to a PGA level of 0.76 g [7.5 m/s2] topples the train model,117
whereas the synthetic seismogram from the hypothetical San Andreas fault earthquake scaled118
to a PGA level of 1.1 g [10.8 m/s2] topples the model. They also used lowpass-filtered (with119
a corner frequency at 3 Hz) versions of these accelerograms and found that high-frequency120
ground motion plays a significant role in initiating the rocking motion which can then be121
sustained with low-amplitude, low-frequency ground shaking.122
It has been shown that the solution to the equation of motion is sensitive to small variations123
in excitation frequency, phase and amplitude of excitation (Yim and Lin, 1991; Iyengar and124
Manohar, 1991). Therefore, there is a need to expand the types and number of earthquake125
records used in establishing the toppling fragility characteristics of the train model in order126
to estimate the ground motions experienced at Point Reyes station during the 1906 San Fran-127
cisco earthquake. Furthermore, the Anooshehpoor et al. (1999) analysis was conducted using128
a single horizontal component of shaking. The effects of vertical ground motion were not con-129
sidered. Yim and Lin (1991) showed that vertical ground motion does not systematically affect130
the rocking response of a rectangular block. However, this result was based on the linearized131
rocking equation of motion. Here, we approximate the train using a rectangular block and132
analyze the response of this rectangular train model under both vertical as well as horizontal133
ground motions from 140 worldwide earthquake records to obtain the overturning fragility of134
the train as a function of ground motion parameters. We also analyze the train model under135
5
the ground motions at Point Reyes station from 1906-like earthquake simulations by Aagaard136
et al. (2008) with ruptures initiating at three different locations [indicated by the blue stars in137
Fig. 2a] to arrive at independent constraints on the possible hypocenter location of the 1906138
San Francisco earthquake.139
Overturning fragility of the train140
There have been numerous analytical, numerical and experimental studies on the rocking re-141
sponse of a rectangular block under ground excitation (Housner, 1963; Yim et al., 1980; Zhang142
and Makris, 2001; Purvance et al., 2008; Hinzen, 2009, 2010). Here, we use a rigid body143
dynamics algorithm presented in Chapter 2 of Veeraraghavan (2015) to analyze the rocking144
response of the train model under earthquake excitation. While this algorithm is capable of145
simulating three-dimensional response under 3-component ground motion, we limit ourselves146
to 2-D analysis here. We concur with Anooshehpoor et al. that the train resting on rails may147
be viewed as a very long rectangular block that will predominantly rock in its shorter direc-148
tion, i.e., perpendicular to the tracks. Some rolling may have occurred along the tracks, but149
the response in the two directions may, for all practical purposes, be considered to be uncou-150
pled (given the far greater length of the locomotive compared to its width). We maintain the151
assumptions of rocking only about the two points where the wheels come in contact with the152
tracks (denoted by O and O′ in Fig. 2b) and perfectly inelastic collisions upon impact between153
the wheels and the rails (when the point of rotation switches from O to O′ or vice versa). The154
latter assumption is realized by setting the coefficient of restitution to zero, which causes the155
vertical velocity of the impacting contact point to be reduced to zero. The minimum distance156
between O and O′ is limited by the track width, which is the minimum distance between the157
rail tracks (excluding the width of the rail head). In reality, the point about which rocking158
occurs may shift closer to the outer edge of the rail head as the rocking angle increases. So,159
the distance between O and O′ could be anywhere between the track width (0.91 m) and the160
track width with the inclusion of the rail heads at either end (1.04 m).161
To estimate the height of the train model, the height of the center of gravity (c.g.) of the162
engine is required. Assuming the weight of the engine to be uniformly distributed across its163
6
height (3.76 m - including the chimney) results in the height of the train model being the same164
as that of the engine with the c.g. located at half the height of the train engine. However,165
in reality, the c.g. of the engine might have been a little lower than mid height of the engine166
as the density is probably higher in the lower half of the engine compared to the top. Booth167
(1908) states that locomotives built around 1908 have their c.g. over 1.52 m above the rails.168
Since there may be small differences with the different engine types, a likely lower limit on the169
height of c.g. is 1/3rd height of the engine, which places the c.g. just below the train’s driver170
wheels that have a diameter of 1.27 m. Therefore, the c.g. of the train model could have been171
anywhere between 1.25 and 1.88 m. To address these uncertainties in the height of the train’s172
c.g. and rocking point locations along the rail head, we consider two different rectangular train173
models for this study. Model 1 is a slender train model where the c.g. of the train is located at174
half the train’s height (including the chimney) and the width of the train model is assumed to175
be the track width excluding the width of the rail heads resulting in b = 0.45 m, h = 1.88 m and176
b/h = 0.24 as in the Anooshehpoor et al. study. In contrast, model 2 is a stouter train model177
where the c.g. is located at 1/3rd of the train’s height and the width of the train model is taken178
to be the track width including the width of the rail heads, i.e., b = 0.52 m, h =1.25 m and b/h179
= 0.42.180
The rails prevent the train from sliding in the direction perpendicular to the rails. So, a181
high value of 1.2 is used for static and kinetic coefficients of friction between the train model182
and the ground to prevent the model from sliding. The suspension system installed in the train183
is assumed to be sufficiently stiff for the train to behave as a rigid body. This assumption may184
not be valid for locomotives that have been designed and manufactured in the recent years. It185
is also important to note here that sloshing effects of the liquid within the boiler is not modeled186
this study. Abramson (1966) reported that sloshing forces are usually of little consequence for187
rail vehicles because the weight and volume of the liquid contained is usually sufficiently small188
compared to the vehicle weight and that any resulting sloshing motion have been sufficiently189
suppressed in the past with the use of simple baffles. While it is not clear whether baffles were190
used on this train engine designed in 1891, we assume that the sloshing forces are negligible for191
the purposes of this study due to the smaller weight of the fluid compared to that of the engine.192
It is also assumed that the rocking response of the train engine is independent of the rocking193
7
responses of the tender and other carriages connected to the engine. When these additional194
carriages of differing weight and geometry are included, it is more likely that a higher ground195
motion would be required to overturn the assembly than that required to overturn just the196
engine. So the estimate obtained in this study would most likely still be the lower bound on the197
ground motions experienced at Point Reyes. Accurately taking into account the details of the198
tender and carriages would require much more information about the event than is currently199
available.200
The 140 earthquake records used to analyze the train models come from worldwide earth-201
quakes with magnitudes greater than 6 and source-to-site distances less than 10 km. The list202
of earthquakes considered is a subset of the 154 earthquakes considered by Purvance et al.203
(2012). The records are first normalized such that the peak ground acceleration (PGA) of the204
strong ground motion component is 1 m/s2. The normalized records are then scaled to yield205
records with PGA from 1 m/s2 to 19 m/s2 in steps of 1 m/s2. Because differentiation is a lin-206
ear operation, the peak ground velocities (PGV) also scale by the same scaling factors as the207
corresponding PGAs resulting in PGVs ranging from 0.025 m/s to 7.912 m/s.208
A total of 2660 PGA-scaled time-history analyses of each train model are performed. In209
each instance, the strong component of the horizontal ground motion is applied along the width210
of the train model and the vertical ground motion is applied along its height. The overturning211
probability of both the train models on the PGA-PGV plane are shown in Fig. 3. These plots212
are developed by binning the 140 scaled records at each PGA level into PGV bins of 0.25 m/s213
width. The overturning probability in each bin at each PGA level is the fraction of records214
(in that bin and that PGA level) that overturn the model. The sampling of different regions of215
the PGA-PGV domain can be gauged by the varying thickness of each column (at each PGA216
level). The thickness of the column at each PGA level in a given PGV bin is proportional to217
the fraction of points (out of 140) being sampled in that bin. For example, 16 of the 140 records218
at a PGA level of 4 m/s2 have a PGV between 0 m/s and 0.25 m/s. So, the thickness of the lower219
bin between 0 m/s and 0.25 m/s is proportional to 16/140. Regions with thinning columns are220
regions that are sparsely sampled; obviously, the results there may not be as reliable as the221
densely sampled regions.222
The colored contour lines correspond to overturning probabilities of 0.1, 0.3, 0.5, 0.7 and223
8
0.9. It can be seen from these figures that a minimum PGA of 4 m/s2 is required for the224
slender train model to overturn while that required for the stouter train model is 7 m/s2 (for225
realistic PGV values of less than 2 m/s). For comparison, the quasi-static acceleration needed226
to uplift one corner of the slender and stouter train models and get it to start rocking is 2.5227
m/s2 and 4.1 m/s2, respectively. Beyond these PGA thresholds of 4 m/s2 and 7 m/s2, the228
overturning probability is independent of the PGA as indicated by the horizontally aligned229
contours. It appears that the probability of overturning is quite low when the PGV is below230
1 m/s and 1.5 m/s for the slender and stouter train models, respectively, whereas the overturning231
probability goes up quite rapidly when the PGV exceeds this value [Fig. 3]. A small fraction232
of the records with PGV in the range of 0.5-1.0 m/s and a PGA in the range of 5-10 m/s2 is233
able to overturn the slender train model. Similarly, small fraction of the records with PGV in234
the range of 1.0 - 1.5 m/s and a PGA in the range of 8-15 m/s2 is able to overturn the stouter235
train model. This study shows that due to uncertainties in the train model, the minimum PGA236
and PGV required to overturn the train model lie in the range of 4-7 m/s2 and 0.5-1.0 m/s. This237
range can be further tightened with the availability of more data regarding the distribution of238
weight across the height of the train and details on the rocking points. But in the absence of239
the required data, the minimum PGA of 4 m/s2 and PGV of 0.5 m/s required to overturn the240
slender train model serve as conservative lower bounds on the ground motion experienced at241
the Point Reyes location during the 1906 earthquake.242
The near-horizontal contours on Fig. 3a suggest that the slender train model is not very243
sensitive to high-frequency parts of the ground motion spectrum beyond the PGA threshold244
of 4 m/s2. This can also be observed in the response of the slender train model (Fig. 4) to245
two different earthquake records: (i) the horizontal component (230) of the El Centro Station246
#6 record from the 1979 Imperial Valley earthquake, and (ii) the horizontal component (64)247
of the LA dam record from the 1994 Northridge earthquake. The ElCentro record has PGA248
and PGV of 4.38 m/s2 and 1.13 m/s, respectively. The high PGA pulse at approximately 2.5 s249
along the record causes tiny rotations in the model [noticeable in the angular velocity (θ) time250
history] but it is actually the high PGV pulse around 5.6 s that imparts sufficient momentum to251
rock the train model substantially and to eventually overturn it. From Fig. 3a, the overturning252
probability of the slender train model for PGA of 4.38 m/s2 and PGV of 1.13 m/s is 0.5. For253
9
the 1994 Northridge earthquake recorded at LA dam, the PGA and PGV are 4.17 m/s2 and254
0.75 m/s, respectively, and the overturning probability of the slender train model under this255
ground motion from Fig. 3a is 0.2. It can be seen from the figures to the right in Fig. 4 that256
the slender train model does not overturn under this excitation. Though both ground motions257
have similar PGA, the Imperial Valley ground motion has a higher PGV than the Northridge258
earthquake ground motion which causes the slender train model to overturn. It has to be259
noted that in some ground motion records such as the 1979 Imperial Valley earthquake, the260
highest velocity pulse does not coincide with the highest acceleration pulse and can in fact be261
the result of a low magnitude long-duration acceleration pulse (Hall et al., 1995; Makris and262
Black, 2004). In these cases, even if the PGA is high enough to initiate rocking, if the train263
model comes to rest before the PGV pulse arrives and if the acceleration magnitude at that time264
instant is not sufficient to initiate rocking again, then the block might not rock/overturn even265
though another record with the same PGA and PGV overturns the train model in a scenario266
where both the PGA and PGV are resulting from the same pulse. The 140 earthquake records267
considered for this study include both types of ground motion: PGA and PGV correspond268
to the (i) same coherent pulse and (ii) different pulses. Since we are interested in a blind269
prediction of the lower bounds on ground motion at Point Reyes location during the 1906270
earthquake, no distinction is made on this basis within the considered set of 140 earthquake271
records.272
To further explore the sensitivity of the overturning probability to the frequency content of273
the ground motion, we consider two measures of ground motion time period: (i) PGV/PGA,274
and (ii) the time period (T) that maximizes the pseudovelocity response spectrum. Note that275
these time period measures do not change when the records are scaled. Also, there appears to276
be a linear correlation between PGV/PGA and the time period T that maximizes the pseudove-277
locity response spectrum [see Figure 4.4 in Veeraraghavan (2015)] .278
Using the results from the records scaled to achieve various PGA levels, the overturning279
probability as a function of PGA and PGV/PGA for the slender train model is shown in Fig. 5a.280
As before, the 140 records at each PGA level are divided into PGV/PGA bins of width 0.05 s.281
The thicknesses of the columns along the PGV/PGA axis are proportional to the fraction of282
earthquake records (out of 140) that are sampled in a given PGV/PGA bin. Similarly, the283
10
overturning probability in the PGV-T planes is shown in Figs. 5b. This is developed using the284
normalized records scaled to yield records with PGV ranging from 0.25 m/s to 5 m/s in steps285
of 0.25 m/s. For these figures, the 140 earthquake records at each PGV level are divided into T286
bins of width 0.5 s. The PGA and PGV required to overturn the slender train model decrease287
more or less monotonically with increasing PGV/PGA [Figs. 5a]. However, the PGV required288
to overturn the slender train model appears to share a parabolic relationship with T [Fig. 5b]289
with the records that have periods near 1.6 s requiring the smallest PGVs to overturn the train290
model.291
To better understand this sensitivity of the slender train model to time period, let us go292
back to the equation of motion for the rocking response of the rectangular block [Eqn. 2].293
Linearizing the equation of motion gives:294
θ − p2θ(t) = −p2[ ug(t)g
− α] θ(t) > 0 (3)
The left hand side of the equation of motion is different from that of a spring mass sys-295
tem undergoing simple harmonic motion (SHM) due to the negative sign accompanying θ. As296
mentioned previously, the free vibration solution for θ is a linear combination of non-periodic297
cosh and sinh functions. So rigid bodies do not have a natural propensity to rock at a “natural298
frequency or period” unlike spring-mass oscillators. The time taken to complete one cycle of299
rocking is dependent upon the amplitude of rocking (Housner, 1963), unlike SHM where the300
period of oscillation is a function of the physical properties of the system alone. Therefore,301
resonance cannot occur in rigid-body rocking driven by external excitation and it is not ex-302
pected that the rocking response of a rectangular block will be sensitive to excitations with a303
particular time period.304
The dependence of overturning probability of the slender train model on ground motion305
period (PGV/PGA) was presented in Fig. 5. The ground motion duration, which specifies306
the duration of the earthquake record that contains 90% of the energy, could also affect the307
overturning probability. Here, we calculate the duration of the 140 earthquake records using308
the energy integral formulation developed by Anderson (2004), where the square of the ground309
velocity multiplied by the ground density is used as a measure of the energy density of the310
11
wavefield. Fig. 6a is developed by dividing the PGV scaled earthquake records into bins of311
width 0.25 m/s in PGV and 3 s in duration. Fig. 6b is developed similar to Fig. 5a by dividing312
the 140 earthquake records at each PGA level into bins of width 3 s in duration. Fig. 6a shows313
that the PGV required to overturn the slender train model is more or less independent of the314
duration of the earthquake. However, the PGA required to overturn the slender train model315
decreases more or less uniformly with duration of the earthquake. In other words, a short316
duration of strong acceleration is unlikely to result in a large enough velocity pulse to overturn317
the train model. The overturning probabilities of the slender train model in the PGD-PGA,318
PGD-PGV and PGV-PGD/PGV planes can be found in Chapter 4 of Veeraraghavan (2015).319
The overturning fragility maps obtained for the slender train model may be used to check320
whether the synthetic ground motions from the 1906 San Francisco earthquake simulations321
by Aagaard et al. (2008) are realistic. They simulated several rupture scenarios by modify-322
ing a source model developed by Song et al. (2008) and a recently constructed 3-D seismic323
wave-speed model of northern California. Three of these scenarios involved the rupture of the324
same extent of the northern San Andreas fault, but with rupture initiating at Bodega Bay (to325
the north of San Francisco), offshore from San Francisco in the middle, and San Juan Bautista326
at the southern end [Fig. 2a]. The ruptures nucleating at Bodega Bay and San Francisco are327
bilaterally propagating ruptures, whereas the rupture originating at San Juan Bautista propa-328
gates predominantly in a south-to-north direction. The three scenarios predict PGA between329
4.5 m/s2 and 6.0 m/s2, PGV between 0.8 m/s and 1.6 m/s at Point Reyes. For these ranges330
of ground motion intensities, the overturning probability of the slender train model from the331
fragility maps ranges between 0.4 and 0.8. Thus, Aagaard et al.’s simulations do not over-332
estimate the ground motion intensities at Point Reyes. They may, in fact, be quite realistic.333
For the same ranges of ground motion intensities, the overturning probability of the stouter334
train model from the PGV-PGA fragility map in Fig. 3b is less than 0.2. However, the ac-335
tual train geometry is expected to be more slender than the stouter train model considered in336
this study with the c.g. being approximately 1.5 m above the rails (Booth, 1908) instead of337
the 1.25 m lower bound on c.g. height considered for the stouter train model. So, the low338
overturning probability for the stouter train model does not negate the realistic nature of the339
synthetic ground motion simulations.340
12
Hypocenter location of the 1906 San Francisco earth-341
quake342
The ground motion histories at Point Reyes (station SF432) from the three rupture scenarios343
(with three hypocenter locations) may also be used to determine the most plausible of the three344
hypocenter locations and could independently verify the estimates by Boore (1977) and Lomax345
(2005). Ground velocity time histories at the Point Reyes Station (station SF432) are retrieved346
from a USGS repository of these simulations (Aagaard et al., 2009). These time histories are347
provided in the east-west, north-south and vertical directions.348
To analyze the response of the 2D train model to these time histories, the orientation of349
the train engine just before the earthquake is required so that the horizontal ground motion350
perpendicular to the length of the engine can be used as input. From Jordon’s description351
of the train conductor’s experience, it appears that the head of train engine was pointing to352
the south allowing the engine to rock along the east-west direction. However, a map of the353
railway track near the Point Reyes station (Figs. 7a and 7b) suggests that the head of the train354
engine would have been pointing to the southeast if the train had been stationed on the rail355
track at Point Reyes station before the earthquake. From the images of the toppled train, it356
appears that the train was stationed on a siding, that is located to the left of the actual rail357
track in Fig. 7c. The presence of the train on the siding instead of the actual track is also358
mentioned in Dickinson et al. (1967). The orientation of this siding track is not known but359
the position of this track to the left of the actual track suggests that the train orientation was360
probably closer to south. To avoid relying entirely on the conductor’s account, 10 different361
orientations of the engine are considered, ranging from 0◦ to 45◦ from the south towards east362
in steps of 5◦. The ground motions from the three different earthquake scenarios are rotated to363
obtain the horizontal ground motion component perpendicular to the train for each of these 10364
orientations.365
The slender train model is analyzed under these 30 ground motion histories. The horizontal366
displacement response histories of the c.g. of the slender train model with respect to the ground367
are given in Figs. 8a (hypocenter at Bodega Bay), 8b (hypocenter at San Francisco), and 8c368
(hypocenter at San Juan Bautista) for 0◦ orientation of the train from the south. Displacements369
13
to the right are positive, which corresponds to east for the southern train orientation. All 10370
train orientation scenarios with earthquake hypocenter at either San Francisco or San Juan371
Bautista resulted in train c.g. displacements similar to 8b and 8c, respectively. In the scenarios372
with hypocenter at Bodega Bay, the train model first lurched to the left in all 10 orientation373
scenarios but (i) lurched to the right and overturned in the right as in 8a for train orientations374
between 0◦ and 15◦, (ii) lurched to the right and then lurched again towards left and overturned375
to the left for 20◦, (iii) continued rocking without overturning for orientation of 25◦ and (iv)376
overturned in the left during the first lurch for orientations between 30◦ to 45◦ from the south.377
The initial movement of the slender train model towards left (or west from the conductor’s378
viewpoint) in all scenarios with hypocenter at Bodega Bay varies from the conductor’s account379
of the event. Only the scenarios with the hypocenter located south of Point Reyes (i.e., offshore380
from San Francisco and San Juan Bautista) produce ground motions that overturn the train in381
the manner documented by Jordan (the train first lurching to the east and then overturning in382
the west). Therefore, our analysis places the hypocenter of the 1906 earthquake to the south383
of Point Reyes station. This inference does not conflict with the currently accepted hypocenter384
location near San Francisco (Lomax, 2005).385
Conclusion386
A train overturned at Point Reyes station during the 1906 San Francisco earthquake. In this387
paper, lower bounds on the ground motion experienced at the Point Reyes station during this388
earthquake are obtained by estimating the ground motion parameters required to overturn a389
rectangular block model of this train. The minimum PGA and PGV required to overturn the390
train are 4 m/s2 (compared to a PGA range of 3.4 m/s2 to 10.8 m/s2 estimated by Anooshehpoor391
et al.) and 0.5 m/s, respectively using a slender model of the train with height equal to the actual392
height of the train engine (including the chimney) and width equal to the track width. When393
the uncertainties in the location of the train engine’s c.g. and the location of rocking points are394
considered, the minimum PGA and PGV required to overturn the train model varies between395
4-7 m/s2 and 0.5-1.0 m/s, respectively. These results show that the slender train model provides396
a conservative lower bound on the ground motions at the Point Reyes during this earthquake.397
14
The probability of overturning is quite low when the PGV is below 1 m/s but goes up quite398
rapidly when the PGV exceeds this value. Although it is not expected that the rocking response399
of a rectangular block will be sensitive to excitations with a particular time period, the slender400
train model is marginally more sensitive to earthquake records with predominant time periods401
in the vicinity of 1.6s. It was also observed that the PGV required to overturn the slender402
train model is more or less independent of the duration of the earthquake. However, the PGA403
required to overturn this train model decreases more or less uniformly with duration.404
The fragility (overturning probability) maps for the slender train model are used to perform405
a reality check on the synthetic ground motion at Point Reyes from the 1906-like San Fran-406
cisco earthquake simulations by Aagaard et al. The maps indicate overturning probabilities of407
0.4-0.8 for this train model under the predicted ground motion intensities from three rupture408
scenarios (with hypocenters in Bodega Bay to the north of Point Reyes and offshore from San409
Francisco and San Juan Bautista, both to the south of Point Reyes) indicating that the predic-410
tions by the Aagaard et al. simulations are quite realistic. Time history analysis of the slender411
train model under synthetic ground motion histories at Point Reyes from the three scenarios412
for 10 different train orientations shows this train model overturning in 29 out of the 30 cases.413
However, only the ground motions from the scenarios with the hypocenter to the south of Point414
Reyes reproduce the eye-witness account of the train lurching to the east, then to the west be-415
fore toppling. We conclude that the hypocenter for the 1906 San Francisco earthquake must416
lie to the south of Point Reyes, perhaps offshore from San Francisco as widely believed.417
Data and Resources418
The time histories from the ground motion simulations of the 1906 San Francisco Earthquake419
were downloaded from https://pubs.usgs.gov/ds/413/ and was last accessed in420
August 2014. The earthquake records used in this study were downloaded from the PEER421
strong ground motion database (last accessed in August 2014). The rigid body dynamics algo-422
rithm used for analyzing the train model is from Chapter 2 of Veeraraghavan (2015).423
15
Acknowledgments424
We thank Brad Aagaard of U.S. Geological Survey (USGS) for pointing us towards the data425
from the ground motion simulations. We would also like to thank Dr. Klaus-G. Hinzen, Dr.426
Rasool Anooshehpoor and an anonymous reviewer for their thoughtful review which signifi-427
cantly improved the manuscript. This research project has been supported by National Science428
Foundation (NSF Award EAR-1247029), USGS and Southern California Earthquake Center429
(SCEC).430
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Email Address500
[email protected], [email protected], [email protected]
19
List of Figures502
1 San Francisco- (south-) bound train which overturned at Point Reyes503
Station during the 1906 San Francisco earthquake [reprinted from Anoosheh-504
poor et al. (1999)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22505
2 (a) A map showing the San Andreas fault (red line) and Point Reyes506
Station (black circle). The blue stars indicate the hypocenter locations507
for three 1906-like San Francisco earthquake simulations by Aagaard508
et al. (2008). The star near San Francisco corresponds to the widely509
accepted hypocenter location. (b) 2-D rectangular block model of the510
train. The color version of this figure is available only in the electronic511
edition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22512
3 Overturning probability of the (a) slender and (b) stouter train models513
as a function of PGV and PGA. Each column in this figure contains514
140 earthquake records scaled to a specific PGA level. The varying515
thicknesses of the column are proportional to the fraction of earth-516
quakes (out of 140) being sampled in a given PGV bin. The color517
version of this figure is available only in the electronic edition. . . . . 23518
4 Horizontal ground acceleration (top figure) and velocity time histories519
(second from top) as well as angular displacement normalized with520
respect to π/2 − α (third from top) and angular velocity time history521
(fourth from top) of the slender train model (count-clockwise rota-522
tion of the block about O’ is positive) under the 1979 Imperial valley523
earthquake recorded at ElCentro station (left) and the 1994 Northridge524
earthquake recorded at LA dam (right). . . . . . . . . . . . . . . . . 23525
5 Overturning probability of the slender train model as a function of (a)526
PGA and PGV/PGA, and (b) PGV and T, the period at which the peak527
of the pseudovelocity response spectrum occurs. The PGA required528
to overturn this train model monotonically decreases with increasing529
PGV/PGA but the overturning PGV decreases first and then increases530
with increasing T, attaining a minimum at a T of about 1.6 s. Each row531
in these figures contain 140 earthquake records scaled to a given PGA532
[(a)] or PGV [(b)] level. The column thicknesses are proportional to533
the fraction of earthquakes (out of 140) that are sampled in a given534
PGV/PGA or T bin. The color version of this figure is available only535
in the electronic edition. . . . . . . . . . . . . . . . . . . . . . . . . 24536
6 Overturning probability of the slender train model as a function of (a)537
PGV and duration, and (b) PGA and duration. Each row in these fig-538
ures contain 140 earthquake records scaled to a given PGA [(a)] or539
PGV [(b)] level. The varying thicknesses of the column are propor-540
tional to the fraction of earthquakes (out of 140) being sampled in a541
given duration bin. The color version of this figure is available only in542
the electronic edition. . . . . . . . . . . . . . . . . . . . . . . . . . . 24543
20
7 (a) Pacific railroad map showing the railway route (reprinted from544
Northwestern Pacific Railroad Historical Society’s website - http:545
//www.nwprrhs.org/history.html), (b) expanded version of546
the map showing the track near Point Reyes station, and (c) toppled547
train near a siding to the left of the actual track. . . . . . . . . . . . . 25548
8 Horizontal displacement time histories of the c.g. of the slender train549
with respect to the ground when subjected to the synthetic ground mo-550
tion histories at Point Reyes station from the three 1906-like scenario551
earthquake simulations by Aagaard et al. (2008) with hypocenter lo-552
cated at (a) Bodega Bay (north of Point Reyes), (b) offshore from San553
Francisco (south of Point Reyes) and (c) San Juan Bautista (further554
south of Point Reyes) for 0o from south orientation of the train. Dis-555
placements to the east are positive. . . . . . . . . . . . . . . . . . . . 25556
21
Figure 1: San Francisco- (south-) bound train which overturned at Point Reyes Stationduring the 1906 San Francisco earthquake [reprinted from Anooshehpoor et al. (1999)].
(a)
−124˚
−124˚
−123˚
−123˚
−122˚
−122˚
−121˚
−121˚
36˚ 36˚
37˚ 37˚
38˚ 38˚
39˚ 39˚
0 100
km
SacramentoSanta Rosa
San Francisco
San Jose
1906
Bodega Bay
San Juan Bautista
Point Reyes
(b)
2h
2b
α
θ
O’
c.g.
O
R
Figure 2: (a) A map showing the San Andreas fault (red line) and Point Reyes Station (blackcircle). The blue stars indicate the hypocenter locations for three 1906-like San Franciscoearthquake simulations by Aagaard et al. (2008). The star near San Francisco correspondsto the widely accepted hypocenter location. (b) 2-D rectangular block model of the train.The color version of this figure is available only in the electronic edition.
557
558
559
560
22
(a)
0 5 10 15 20PGA (m/ s2 )
0
1
2
3
4
5
PGV(m
/s)
0.10.3
0.50.7
0.9
0.0
0.2
0.4
0.6
0.8
1.0
OverturningProbability
16/140
(b)
0 5 10 15 20PGA (m/ s2 )
0
1
2
3
4
5
PGV(m
/s)
0.10.30.50.70.9
0.0
0.2
0.4
0.6
0.8
1.0
Overturning
Probability
16/140
Figure 3: Overturning probability of the (a) slender and (b) stouter train models as a functionof PGV and PGA. Each column in this figure contains 140 earthquake records scaled to aspecific PGA level. The varying thicknesses of the column are proportional to the fractionof earthquakes (out of 140) being sampled in a given PGV bin. The color version of thisfigure is available only in the electronic edition.
5
0
5
ug(m/s
2)
1
0
1
ug(m/s
)
1
0
1θ/(π/2−α)
0 2 4 6 8 10Time(s)
1
0
1
θ(ra
d/s
)
5
0
5
ug(m/s
2)
1
0
1
ug(m/s
)
1
0
1θ/(π/2−α)
0 5 10 15 20Time(s)
1
0
1
θ(ra
d/s
)
Figure 4: Horizontal ground acceleration (top figure) and velocity time histories (secondfrom top) as well as angular displacement normalized with respect to π/2 − α (third fromtop) and angular velocity time history (fourth from top) of the slender train model (count-clockwise rotation of the block about O’ is positive) under the 1979 Imperial valley earth-quake recorded at ElCentro station (left) and the 1994 Northridge earthquake recorded atLA dam (right).
23
(a)
0 0.1 0.2 0.3 0.40
5
10
15
20
PGV/PGA (s)
PG
A (
m/s
2)
0.1
0.1
0.5
0.9 0.9
Ove
rtu
rnin
g P
rob
ab
ility
0
0.2
0.4
0.6
0.8
1
10/140
(b)
0 1 2 3 40
1
2
3
4
5
T (s)
PG
V (
m/s
)
0.1
0.3
0.50.7
0.9
Ove
rtu
rnin
g P
rob
ab
ility
0
0.2
0.4
0.6
0.8
1
16/140
Figure 5: Overturning probability of the slender train model as a function of (a) PGA andPGV/PGA, and (b) PGV and T, the period at which the peak of the pseudovelocity responsespectrum occurs. The PGA required to overturn this train model monotonically decreaseswith increasing PGV/PGA but the overturning PGV decreases first and then increases withincreasing T, attaining a minimum at a T of about 1.6 s. Each row in these figures con-tain 140 earthquake records scaled to a given PGA [(a)] or PGV [(b)] level. The columnthicknesses are proportional to the fraction of earthquakes (out of 140) that are sampled in agiven PGV/PGA or T bin. The color version of this figure is available only in the electronicedition.
(a)
1.5 7.5 13.5 19.5 25.50
1
2
3
4
5
Duration (s)
PG
V (
m/s
)
Ove
rtu
rnin
g P
rob
ab
ility
0
0.2
0.4
0.6
0.8
1
15/140
(b)
1.5 7.5 13.5 19.5 25.50
5
10
15
20
Duration (s)
PG
A (
m/s
2)
Ove
rtu
rnin
g P
rob
ab
ility
0
0.2
0.4
0.6
0.8
1
15/140
Figure 6: Overturning probability of the slender train model as a function of (a) PGV andduration, and (b) PGA and duration. Each row in these figures contain 140 earthquakerecords scaled to a given PGV [(a)] or PGA [(b)] level. The varying thicknesses of thecolumn are proportional to the fraction of earthquakes (out of 140) being sampled in a givenduration bin. The color version of this figure is available only in the electronic edition.
561
562
563
564
24
(a)
(b)
(c)
Figure 7: (a) Pacific railroad map showing the railway route (reprinted from North-western Pacific Railroad Historical Society’s website - http://www.nwprrhs.org/history.html), (b) expanded version of the map showing the track near Point Reyesstation, and (c) toppled train near a siding to the left of the actual track.
(a)
0 1 2 3 4 5 6−0.5
0
0.5
Time (s)
Dis
pla
cem
ent (m
) East
West
(b)
0 1 2 3 4 5 6 7−0.5
0
0.5
Time (s)
Dis
pla
cem
ent (m
) East
West
(c)
0 2 4 6 8 10−0.5
0
0.5
Time (s)
Dis
pla
cem
ent (m
) East
West
Figure 8: Horizontal displacement time histories of the c.g. of the slender train with re-spect to the ground when subjected to the synthetic ground motion histories at Point Reyesstation from the three 1906-like scenario earthquake simulations by Aagaard et al. (2008)with hypocenter located at (a) Bodega Bay (north of Point Reyes), (b) offshore from SanFrancisco (south of Point Reyes) and (c) San Juan Bautista (further south of Point Reyes)for 0o from south orientation of the train. Displacements to the east are positive.
25