Theoretical Response of Floating Plates to Moving Loads

1. Theoretical Response of Floating Plates to Moving Loads Roger J Hosking School of Mathematical Sciences University of Adelaide Email: marjhg@gmail.com

2. BACKGROUND 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 stationary – and in particular, much larger deflexions! 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 speed?

3. “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.

4. Ice Road Truckers (picture courtesy of Vernon Squire)

5. Ice drilling check after the accident(picture courtesy of Vernon Squire)

6. The time-dependent response in a visco-elastic 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 sub-critical, then critical and super-critical, and then at the water wave speed (not critical) and even higher load speed as follows….

10. RAIL TRACKS 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 pre-stressed concrete beams, and provide superior rail support.

11. Ladder Tracks Engineers at the RTRI of Japan Railways appreciated that a rail track with two parallel longitudinal 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).

13. 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-tie sleepers.

15. The Floating Ladder Track design shown below, where the “combined rail” is mounted on periodic railpads, produces much less vibration & noise…

17. Ballasted and floating ladder tracks now installed in the greater Tokyo region are shown in the following two photographs.

18. Ballasted Ladder Track

19. Floating Ladder Track

20. 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.

21. 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 & Milinazzo 2007) for the design. The anticipated maximum deflexion and the wave pattern variation with load speed have also been calculated.

22. FLT response at various load speeds

23. 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.

24. 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.

26. CONCLUSIONS • 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.