Short Note
Inertial Forces from Earthquakes on a Hyperloop Pod
by Thomas H. Heaton
Abstract
High-speed transit (
1300
km
=
hr) using pods traveling in evacuated tubes
has been proposed. This Short Note addresses how earthquake ground shaking is
changed when it is experienced by a high-speed pod that is confined to a track. In
particular, earthquake motions can cause lateral deformations of the tube that cause
centripetal forces in the pod. I discuss the nature of these forces for the cases of (1) a
tube that crosses a fault offset, (2) a tube that is deformed by traveling waves in the
Earth, and (3) a tube that resonates between fixed points (e.g., a simple bridge). I
suggest several schemes to control the peak centripetal accelerations of the pod.
Introduction
Elon Musk has suggested that very high-speed transpor-
tation (
350
m
=
sor
1300
km
=
hr) could be achievable using
sealed pods that move within evacuated tubes (3.5 m diameter;
Garber, 2012
). Several companies are currently developing de-
signs for such a system and it has been suggested that a hyper-
loop system may be appropriate for California. What are the
dynamics of a high-speed vehicle moving in a fixed tube that
is subject to earthquake motion? Obviously, bends in the tube
will cause centripetal accelerations that are perpendicular to
the tube. A pod traveling at velocity
V
P
in a tube of radius of
curvature
R
experiences a centripetal acceleration
a
centripetal
of
EQ-TARGET;temp:intralink-;df1;55;354
a
centripetal
V
2
P
R
:
1
If
V
P
350
m
=
s, then the radius of curvature that corre-
sponds to
1
g
is about 12 km. What are the inertial accelera-
tions of a high-speed pod traveling in a fixed tube that is
elastically deforming in an earthquake? It seems that there
are several different possibilities: (1) the tube is deformed
in the immediate vicinity of a static fault offset, (2) the tube
is deformed by the transient motion of traveling seismic
waves, and (3) the tube is deformed by lateral resonances
of the tube (e.g., the modes of a bridge). I discuss each of these
issues separately.
Accelerations Due to a Fault Offset
Assume that there is a fault offset
D
and that the static
motion of the ground in the direction perpendicular to the
tube is described by
EQ-TARGET;temp:intralink-;df2;55;129
u
g
y
D
π
arctan
x
L
;
2
in which
y
is perpendicular to the tube;
y
can be either hori-
zontal or vertical.
x
is the distance along the tube such that
x
0
at the fault trace, and
L
is a scaling factor that deter-
mines the width of the deformation zone. The total width of
the zone is on the order of
10
L
. If time
t
is measured with
respect to the time at which the pod intersects the fault, then
the position of the pod is
EQ-TARGET;temp:intralink-;df3;313;426
u
P
y
t
D
π
arctan
V
P
t
L
D
π
arctan
τ
;
3
in which
τ
≡
t=T
L
and
T
L
≡
L=V
P
, which is the time it takes
for the pod to traverse the deformation zone. If the position of
the pod is
x
P
V
P
t;
−
∞
<t<
∞
, then the pod
’
s velocity
perpendicular to the track is given by
EQ-TARGET;temp:intralink-;df4;313;342
_
u
P
y
D
π
1
τ
2
−
1
d
τ
dt
D
π
1
tV
P
L
2
−
1
V
P
L
D
T
L
π
1
t
T
L
2
−
1
:
4
Then, the acceleration is
EQ-TARGET;temp:intralink-;df5;313;260
u
P
y
−
2
tD
T
3
L
π
1
t
T
L
2
−
2
−
2
D
T
2
L
π
τ
1
τ
2
−
2
:
5
The acceleration directly above the fault crossing,
t
0
,is
zero, and the maximum accelerations occur when
EQ-TARGET;temp:intralink-;df6;313;197
d
3
u
P
y
d
τ
3
0
−
2
D
T
2
L
π
1
1
τ
2
2
−
2
τ
2
1
τ
2
3
;
6
or when
EQ-TARGET;temp:intralink-;df7;313;141
1
τ
2
−
2
τ
2
0
:
7
Therefore, the maximum accelerations occur when
τ
1
,or
EQ-TARGET;temp:intralink-;df8;313;99
u
P
y
j
max
D
2
π
T
2
L
DV
2
P
2
π
L
2
:
8
2521
Bulletin of the Seismological Society of America, Vol. 107, No. 5, pp. 2521
–
2524, October 2017, doi: 10.1785/0120170054
The deformation length required to limit the acceleration to less
than
u
P
y
j
max
is then
EQ-TARGET;temp:intralink-;df9;55;709
L
V
P
D
2
π
u
P
y
j
max
s
:
9
If the speed of the pod is
350
m
=
s, then a maximum acceler-
ation of
1
g
occurs when
L
≈
44
D
p
. For example, if the fault
offset is 9 m, then the maximum accelerations can be kept be-
low
1
g
by smoothly deforming the tube over a 1320-m-wide
stretchspanningthefault.Thismaybefeasiblebyemployinga
flexurally stiff tube that is mounted to stiff pylons with fric-
tional bearings that can slide laterally. In this case, the mount-
ing system would need to be able to slide up to 4.5 m in the
vicinity of the fault crossing. Construction of a system to dis-
tribute fault offset over a length of pipeline was employed by
the Alyeska Pipeline Service Company to distribute strain in
the TransAlaska pipeline at the point where it intersects the
Denali fault. The support system worked as intended during
the 2002
M
7.9 Denali earthquake and, despite a 6 m offset
of the pipeline, the pipeline def
ormation was sufficiently dis-
tributed to avoid rupture of the pipe (
Honegger
et al.
, 2004
). In
the case of the TransAlaska pipeline, the designers included an
S
curve in the pipe that helped distribute strain during the off-
set. High centripetal forces would preclude the use of
S
curves
in a hyperloop system.
Although distributing the pipe deformation over the
largest possible distances is dynamically desirable, cost is-
sues may favor designs that minimize the width of the de-
formation zone. This suggests another strategy to deform
the pipe at a fault crossing. That is, to force the pipe to bend
in constant curvature sections that induce constant pod accel-
erations
a
0
that are in opposite directions on either side of
the fault crossing. In this case, the acceleration of the pod can
be described as
EQ-TARGET;temp:intralink-;df10;55;316
u
P
y
a
0
Y
t
1
2
T
0
−
Y
t
−
1
2
T
0
;
10
in which
T
0
is the time it takes the pod to traverse each of the
curved sections and
EQ-TARGET;temp:intralink-;df11;55;250
Y
t
≡
1
j
t
j
<
1
2
0
j
t
j
>
1
2
:
11
I can doubly integrate equation
(10)
to obtain
EQ-TARGET;temp:intralink-;df12;55;195
u
P
y
a
0
8
>
>
>
<
>
>
>
:
0
t
T
0
<
−
1
1
2
t
2
−
1
<
t
T
0
<
0
1
2
t
2
T
2
0
0
<
t
T
0
<
1
T
2
0
t
T
0
>
1
:
12
That is, the total offset of the pipe across the deformation
zone is
a
0
T
2
0
. The total width of the deformation zone is
W
2
V
P
T
0
, so the total offset can be written as
D
a
0
W=
2
V
P
2
. Finally, the required width of the defor-
mation zone is then obtained as a function of the fault offset
and the pod acceleration, or
W
2
V
P
D=a
0
p
. If the fault
slip is 9 m and the acceleration is
1
g
, then a
350
m
=
s pod
would require a total deformation width of 738 m, which
is substantially smaller than the 1320 m required by a pipe
with an arctan deformation shape.
The analyses above can be used for any component of
deformation perpendicular to the track. However, deforming
the track into a smooth shape seems easier to accomplish if
the motion is horizontal. That is, if there is significant vertical
offset of the track at the Earth
’
s surface, it may be difficult to
design a system that minimizes tube curvature; if
L
is small,
the accelerations become extreme because they increase
as
1
=L
2
.
Tube Deformed by a Traveling Seismic Wave
Assume that a pod is traveling in the
x
direction at veloc-
ity
V
P
and that the surface of the Earth is deformed by a wave
for which the
y
component of motion is given by a nondis-
persive wave that is also traveling in the
x
direction with
apparent velocity
c
. Then, the
y
component of the ground
displacement is
u
g
y
t
−
x=c
(the
x
component is irrelevant
because the pod is not constrained in the
x
direction). Now
the
x
position of the pod is just
x
P
tV
P
, so the
y
compo-
nent of the pod motion
u
P
y
t
is just
EQ-TARGET;temp:intralink-;df13;313;437
u
P
y
t
u
g
y
t
−
x
P
c
u
g
y
t
1
−
V
P
c
:
13
Now, I can determine the
y
component of the pod
’
s velocity as
EQ-TARGET;temp:intralink-;df14;313;379
_
u
P
y
t
d
dt
u
g
y
t
1
−
V
P
c
1
−
V
P
c
_
u
g
y
t
1
−
V
P
c
:
14
Differentiating a second time, I obtain the acceleration
EQ-TARGET;temp:intralink-;df15;313;305
u
P
y
t
1
−
V
P
c
2
u
g
y
t
1
−
V
P
c
:
15
Notice that
EQ-TARGET;temp:intralink-;df16;313;246
u
P
y
t
j
V
P
→
c
0
:
16
That is, if the pod is traveling at the apparent wavespeed, it
experiences no acceleration. Of course, this solution is not
physical because a pod that is traveling at the wavespeed
would not experience the wave, because the wave would never
catch up to the pod. If, on the other hand, the pod is traveling
at the wavespeed but in the opposite direction from the wave
propagation, then the acceleration on the pod would be four
times larger than the ground acceleration. Notice that the tim-
ing of the acceleration experienced by the pod is compressed
by a factor of
1
−
V
P
=C
.
Although the pod can experience large accelerations if it
travels into the propagating wavefield, the amplitudes of the
2522
Short Note
transient displacements are unchanged by the pod
’
s motion;
it is just that these motions happen faster in the pod
’
s refer-
ence frame.
Estimating the apparent velocity of the wave along the
surface of the Earth is a common problem in seismology and
it depends on the seismic velocities of the crust and the
geometry of the earthquake relative to the station. An appar-
ent horizontal wavespeed of
5000
m
=
s would be typical for a
shear wave located in the near-source region of damaging
earthquakes. In this case, the velocity of a hyperloop pod is
not large enough to cause significant amplification of the ac-
celerations (factor of 0.5%). However, short-period surface
waves in basins can have group velocities of less than
300
m
=
s, and in this case the accelerations would be strongly
affected (the particulars depend on a number of factors that
can be calculated from standard seismological analysis).
Accelerations from Standing Waves
I can also calculate the accelerations of a pod traveling
through a tube that is experiencing a standing harmonic wave
(e.g., lateral modal resonance of a bridge, similar to the fun-
damental modal vibration of a tensioned string). Assume that
the inertial motion on the bridge is the sum of the motion of
the ground (the bridge pier) plus the deformation of the
bridge. I assume that the bridge consists of the hyperloop
pipe that is supported at both ends of the bridge. If the lateral
motion of the pipe is harmonic, then I assume that
EQ-TARGET;temp:intralink-;df17;55;401
u
P
y
x; t
u
g
y
x; t
A
cos
ω
0
t
cos
k
0
x
;
17
in which
A
is the amplitude of the standing wave, and
ω
0
and
k
0
are the natural frequency and wavenumbers of the funda-
mental-mode resonance, respectively. Assuming that the po-
sition of the pod is a linear function of time and that the inertial
motion of the piers is approximately equal to the ground mo-
tion at the middle of the bridge, then equation
(17)
becomes
EQ-TARGET;temp:intralink-;df18;55;299
u
P
y
t
u
g
y
x
0
;t
A
cos
ω
0
t
cos
k
0
V
P
t
:
18
The corresponding velocity and acceleration is obtained by
differentiating equation
(18)
with respect to time:
EQ-TARGET;temp:intralink-;df19;55;245
_
u
P
y
x
0
;t
_
u
g
y
x
0
;t
A
−
ω
0
sin
ω
0
t
cos
k
0
V
P
t
−
cos
ω
0
t
k
0
V
P
sin
k
0
V
P
t
19
EQ-TARGET;temp:intralink-;df20;55;198
u
P
y
x
0
;t
u
g
y
x
0
;t
A
−
ω
2
0
cos
ω
0
t
cos
k
0
V
P
t
ω
0
sin
ω
0
t
k
0
V
P
sin
k
0
V
P
t
A
ω
0
sin
ω
0
t
k
0
V
P
t
sin
k
0
V
P
t
−
cos
ω
0
t
k
0
V
P
2
cos
k
0
V
P
t
u
g
y
x
0
;t
−
A
ω
2
0
k
2
0
V
2
P
cos
ω
0
t
cos
k
0
V
P
t
2
ω
0
k
0
V
P
sin
ω
0
t
sin
k
0
V
P
t
:
20
The peak acceleration of the pod at the midpoint of the span
occurs when
x
0
and
t
0
,inwhichcase
(20)
simplifies to
EQ-TARGET;temp:intralink-;df21;313;709
u
P
y
x
0
;t
0
u
g
y
x
0
;t
−
A
ω
2
0
k
2
0
V
2
P
≈
SA
ω
0
1
V
2
P
c
2
;
21
in which
c
ω
0
=k
0
is the wavespeed for a flexural wave in a
pipe, and SA
ω
0
≡
max
d
2
=dt
2
position
of
bridge
center
in
an
inertial
frame
is the response spectral acceleration at
ω
0
. The acceleration response of an oscillator vibrating at
ω
0
with a displacement amplitude of
A
is approximately
ω
2
A
.
The ratio of the acceleration of the moving pod to the station-
ary pod is then approximated as
EQ-TARGET;temp:intralink-;df22;313;561
SA
pod
SA
tube
1
k
2
0
V
2
P
ω
2
0
1
V
2
P
c
2
:
22
In the case of a resonant beam of length
L
that is fixed at its
ends, the flexural wavespeed is related to the fundamental
resonant period
T
0
by
c
2
L=T
0
,soSA
pod
=
SA
tube
1
V
2
P
T
2
0
=
4
L
2
.Bernoulli
–
Euler beam theory can be used
to determine the wavespeed of a harmonic bending wave
u
y
cos
kx
−
ω
t
, that is traveling along a cylindrical tube
of inner and outer diameters of
d
i
and
d
o
,respectively,
EQ-TARGET;temp:intralink-;df23;313;437
c
2
π
λ
EI
ρ
S
s
23
(
Housner and Vreeland, 1965
, p. 331), in which
λ
2
L
2
π
=k
0
is the wavelength of the standing wave,
E
is Young
’
s
modulus,
ρ
is material density,
S
π
=
4
d
2
o
−
d
2
i
,and
I
π
=
64
d
4
o
−
d
4
i
.Equation
(23)
can be simplified to
EQ-TARGET;temp:intralink-;df24;313;344
c
π
L
c
L
I
S
r
π
L
c
L
d
4
0
−
d
4
i
16
d
2
o
−
d
2
i
s
π
L
c
L
1
16
d
2
o
d
2
i
r
≈
π
d
o
2
2
p
L
c
L
;
24
in which I assume that the tube wall thickness is small com-
pared with its diameter. I also identified that
c
L
E=
ρ
p
,
which is the wavespeed of a longitudinal wave in a slender
bar (about
5000
m
=
s for steel). I can now estimate the period
of the fundamental mode as
EQ-TARGET;temp:intralink-;df25;313;203
T
0
2
L
c
4
2
p
π
d
o
c
L
L
2
≈
1
:
8
d
o
c
L
L
2
:
25
Therefore, I can estimate the period of the 3.5 m steel tube to
be
T
0
≈
1
:
03
×
10
−
4
L
2
. That is, a 100 m span would have a
period of about 1 s, whereas a 200 m span would have a period
of about 4 s. The flexural wavespeed can now be written as a
function of the span length, or
EQ-TARGET;temp:intralink-;df26;313;104
c
2
L
T
0
π
d
o
c
L
2
2
p
L
:
26
Short Note
2523
I am now able to rewrite the ratio of the high-speed pod spec-
tral acceleration relative to the stationary pod using equa-
tion
(22)
,
EQ-TARGET;temp:intralink-;df27;55;697
SA
pod
SA
tube
1
V
2
P
c
2
≈
1
V
P
2
2
p
L
π
d
o
c
L
2
≈
1
0
:
81
L
d
o
2
V
P
c
L
2
:
27
That is, a 100 m span would amplify the 1 s spectral accel-
eration of a
350
m
=
s pod by a factor of about 4.24. Notice that
this amplification grows rapidly with the length of the span; a
200 m span would be a natural resonance at about 4 s and the
corresponding 4-s spectral acceleration would be amplified by
a factor of about 11.5. It appears that long unsupported spans
of the pipe may have the greatest potential to produce large
pod accelerations. As an example of a worst-case scenario,
consider a 230-m-long bridge that consists of a pipe that is
supported at its ends and that is subjected to the motions
recorded at station KATNP during the 25 April 2015
M
7.8
Gorkha, Nepal, earthquake (
Galetzka
et al.
,2015
). A 230-m
span length would produce a 5-s natural period with a corre-
sponding amplification of spectral acceleration of about 17.
The KATNP site recorded a 5-s spectral acceleration of about
0
:
5
g
, so the expected acceleration of a
350
m
=
spodthat
crosses the bridge when it was experiencing maximum defor-
mation is about
8
:
5
g
, provided the system remains linear.
Control Strategy to Minimize Inertial Forces on a
Bridge
Notice that the pod could be so lucky as to cross the
bridge just as the tube is approximately straight. A
350
m
=
s
pod would require only 0.66 s to cross a 230-m-long bridge
that vibrates with an approximate period of 5 s. The bridge is
approximately straight when
ω
t
n
1
=
2
π
in equa-
tion
(18)
, and then there is no lateral acceleration from the
fundamental mode. This suggests a scheme that adjusts
the speed of the pod such that the pod reaches the center of
the bridge when the tube is temporarily straight. If the pod
speed is adjusted so that it enters the bridge at the appropriate
time, there should be no lateral acceleration due to resonance
of the bridge. If there is an accelerometer at the center of the
bridge that records the acceleration in the transverse directions
to the bridge, the record could be integrated with respect to
time to obtain the bridge particle velocity with time. If the
maximum bridge velocities occur at times
t
n
t
0
n=
2
T
0
,
in which
T
0
is the bridge fundamental period, and if the pod is
initially traveling at velocity
V
P
, and if a braking deceleration
A
is applied at time
t
B
and distance
x
B
from the center of the
bridge, the position of the pod will be
EQ-TARGET;temp:intralink-;df28;55;120
x
t
x
B
−
V
P
t
−
t
B
−
1
2
A
t
−
t
B
2
:
28
I can determine the deceleration
A
such that the pod is at the
bridge center when it is straight by solving
EQ-TARGET;temp:intralink-;df29;313;733
x
t
t
n
0
x
B
−
V
P
t
0
−
t
B
n
2
T
0
−
1
2
A
t
0
−
t
B
n
2
T
0
2
29
or
EQ-TARGET;temp:intralink-;df30;313;671
A
2
x
B
−
V
P
t
0
−
t
B
n
2
T
0
t
0
−
t
B
n
2
T
0
2
:
30
Obviously, it is advantageous to send a message to the pod
earlier, rather than later. The required accelerations decrease
with the distance that the pod is from the bridge.
Discussion
I will admit that I was skeptical when Brogan BamBro-
gan came to Caltech to argue for the viability of a hyperloop
system in California. The thought of high-speed motion
through a pipe deforming in earthquakes seemed alarming.
However, the simple analysis of this Short Note seems to
indicate that the dynamic challenges are manageable. That
said, I do not recommend using the analysis in this note for
engineering design decisions; finite-element simulations of
more realistic descriptions of the Earth are undoubtedly bet-
ter tools for these problems. Nevertheless, this simple analy-
sis can provide an insight into this problem.
Data and Resources
No data were used in this Short Note.
Acknowledgments
The author is grateful to an anonymous reviewer for helpful comments
and for careful review of the equations, including several important correc-
tions.
References
Galetzka, J., J.-P. Avouac, J. Genrich, and K. Hudnut (2015). Slip pulse and
resonance of Kathmandu basin during the 2015
M
w
7.8 Gorkha earth-
quake, Nepal imaged with geodesy,
Science
doi:
10.1126/science.
aac6383
.
Garber, M. (2012). The real iPod: Elon Musk
’
s wild idea for a
“
Jetson
Tunnel
”
from S.F. to L.A.,
The Atlantic
, July 13.
Honegger, D., D. Nyman, E. Johnson, L. Cluff, and S. Sorensen (2004).
Trans-Alaska pipeline system performance in the 2002 Denali fault,
Alaska, earthquake,
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20,
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–
738.
Housner, G. W, and T. Vreeland Jr. (1965).
The Analysis of Stress and
Deformation
, California Institute of Technology, Pasadena, California,
440 p.
Department of Mechanical and Civil Engineering
California Institute of Technology
MC 104-44
Pasadena, California 91125
heaton@caltech.edu
Manuscript received 17 February 2017;
Published Online 25 September 2017
2524
Short Note