Plates to Moving Loads

plantcitybusinessUrban and Civil

Nov 26, 2013 (3 years and 6 months ago)


Theoretical Response of Floating
Plates to Moving Loads

Roger J Hosking

School of Mathematical Sciences

University of Adelaide



In cold regions, floating ice sheets are often
convenient for operating land
based vehicles
or landing aircraft. However, a moving load
may produce a response in the ice sheet
significantly different from when the load is

and in particular, much larger

The floating ladder rail track has significant
advantages over conventional cross
tie tracks,
including a superior bearing capacity and
reduced vibration and noise. The floating
ladder track has been installed in the suburban
rail system in Japan

but can it be safe at high

“Moving Loads on Ice Plates”

In cold regions, floating ice sheets are often
convenient for operating land
based vehicles
and landing aircraft.

However, a moving load may produce a
response in the ice sheet significantly different
from when the load is stationary

and in
particular, much larger deflexions!

The monograph on “Moving Loads on Ice
Plates” by V.A. Squire et al. (Kluwer 1996),
intended to be widely accessible for field
scientists and engineers, discusses quite
extensive observations consistent with
theoretical results obtained from mathematical
modelling, and with some more recent results.

Ice Road Truckers

(picture courtesy of Vernon Squire)

Ice drilling check after the accident

(picture courtesy of Vernon Squire)

The time
dependent response in a

plate model, due to
point load moving at various speeds is discussed in K.Wang et al.,
J. Fluid Mechanics 521, 295
317 (2004).

The visco
elastic plate model predicts a finite response at all load
speeds, with a most pronounced deflexion at ONE critical speed,
corresponding to the minimum wave phase speed: the evolution at
critical, then critical and super
critical, and then at the water
wave speed (not critical) and even higher load speed as follows….


The traditional cross
tie rail track has transverse
sleepers at intervals along its length, in a familiar
system usually supported by substantial gravel
ballast and extensive earthwork en route.

However, there are now quite different rail track
systems in place or under development.

Many modern rail systems use longitudinal rather
than transverse cross
tie sleepers. Longitudinal
sleepers are constructed from reinforced steel or

stressed concrete beams, and provide superior
rail support.

Ladder Tracks

Engineers at the RTRI of Japan Railways appreciated that

a rail track with two parallel
sleepers should

maintain the transverse distance between the sleepers and

compare with rail tracks using traditional cross
tie sleepers

in weight per unit track length. This led them to construct

and extensively test “ladder tracks”.

As shown in the following Figure, a ladder sleeper onto which the

rails are fastened is a well integrated structure consisting of twin

longitudinal concrete beams and transverse steel connectors, that

maintain a uniform gauge (i.e. the transverse distance between the

twin beams).

Cross tie rail tracks suffer from relatively poor

load distribution and therefore subside much

more under loads than a ladder track.

Figure 2 contrasts the subsidence as a function

of total load carried for ladder and cross


Floating Ladder Track

design shown below,
where the “combined rail” is mounted on
periodic railpads, produces much less vibration
& noise…

Ballasted and floating ladder tracks now

installed in the greater Tokyo region are

shown in the following two photographs.

Ballasted Ladder Track

Floating Ladder Track

Fast Rail on FLT??

A major issue is whether the Floating Ladder
Track may be implemented in fast rail systems,
such as the Shinkansen.

Over 80 years ago, Timoshenko used the model
of a beam on a continuous elastic support, to
predict a significantly enhanced response at
the critical (resonance) speed that sets an
upper limit on the safe speed of rail travel …..
and whilst extremely high on quite solid
supports, there are inherently softer periodic
supports for the FLT.

The simplest mathematical model involving

a Bernoulli
Euler beam on periodic elastic

supports successfully predicted frequency
(Hosking et al. 2004 ) and the critical speed
(Hosking &

2007) for the design.

The anticipated maximum

and the
wave pattern variation with load speed have
also been calculated.

FLT response at various load speeds

Although this safety issue is not readily open to experiment(!),

the graph below from Hosking & Milinazzo (MMAS, 2007) has

been confirmed in careful simulations at the JR RTRI in Tokyo.

Gamma denotes the relative stiffness of
the resilient supports (the railpads),
alpha is the dimensionless wavenumber
using the distance between the railpads
as characteristic length, and the critical
speed in metres per second is shown on
the right
hand side. Fortunately, RTRI
has Gamma approximately one in its
current design, and Gamma should
increase with time

i.e. railpads get
stiffer with age.


Mathematical modelling and computation
define the variable response of an ice sheet or
rail track, that depends upon the speed of the
moving load.

Extensive field observations have confirmed
theoretical predictions in the ice sheet context,
using a simple floating flexible plate model.

The dynamical response of the FLT to a moving
load has been predicted remarkably well, by a
very simple beam and periodic support model.

Although the investigation of the safety of the
FLT for high speed rail systems is encouraging,
load inertia may prove an important factor.