Preventing wheel-climb derailments

Written by Dr. Leonid Katz, vice president of engineering, I. G. Technologies, research consulting, Harsco/Zeta-Tech

This study focused on applying new research results to develop preventing and practical understanding of wheel climb derailments.

 

Wheel climbing over the rail accidents represents one of the most frequent and dangerous kinds of derailments and attract continuous attention from railway engineers and scientists. This derailment mode constitutes a complicated phenomenon, especially in curve negotiation, due to the non-linearity of the creep forces involved and the wheel-rail interface. Several useful practical approaches have been developed, based on analytical simplifications, linearization and confined in a set of major parameters. As mentioned by Liu and Magel1, despite a large body of research “wheel climb continues to be a persistent problem and pose serious challenges for commuter and freight railroads alike.”1 Recent derailments dramatically stressed the issue.

After the 1960s, railway scientists started to recognize the problems associated with the previously omitted climb-up parameters, specifically forces due to the spin between the wheel and rail in their area of contact. Modern studies4, 6 show that the angle of attack, or yaw angle, as well as contact creep forces must be used as prim parameters of incipient derailment investigation.

Initial and still commonly used practical criterion (limit) of the wheel-climb derailment was introduced by French scientist M.J. Nadal in his article “Theorie de la stabilite des Locomotives” in 1896, based on simplified equilibrium assumption. Considering typical climb-up conditions, as associated with large lateral L and reduced vertical V forces, Nadal established the following well-known formulation for the critical L/V ratio:

L TAN(β) – μ
–– = –––––––––––
V 1 + μTAN (β)

The formula shows that derailment possibility depends only on the wheel flange/rail angle of attack β and the amount of friction µ. Notwithstanding the several essential weaknesses4,6,

Nadal’s criterion above is believed to represent a conservative estimate of a possible derailment in steady state conditions, therefore included in many railroad standards, international research and measurement models. According to the Federal Railroad Administration’s point of view “current use of Nadal’s criterion in combination with other safety standards, including vehicle track interaction standards, has been demonstrated to be an efficient means to determine the root cause of wheel-climb derailments and to accurately assess the safety of vehicle/truck dynamics.”

However, a lot of railroad scientists follow the viewpoint, that Nadal limit and approach, in spite of simplicity and effectiveness, does not provide the necessary level of accuracy in many practical applications.4, 7

As it was admitted by Dr. Blader, certain weaknesses of the Nadal’s approach could be eliminated by a comprehensive space (x, y, z) study of the wheel/rail contact geometry, allowing for true space angle of the wheel-climb derailment definition. Magel, Liu and Weinstock admitted limitations of the Nadal formula, noting that longitudinal forces are usually also present. The following three-dimensional space study of the climb-up derailment was undertaken in order to remove Nadal’s approach limitations by including missing longitudinal information and determine a real level of safety, associated with the common practical and international use of Nadal’s criteria.

Three-dimensional space geometry of the wheel/rail contact zone (Figures 1, 2) can be described by the geometrical set of equations, using both track and angle of attack oriented coordinate systems as shown in Figures 3 and 4.

The space distribution of contact forces can be represented, taking into consideration that a vehicle, traversing a curve, develops lateral and longitudinal forces generated at the wheel-rail contacts. These forces lead to an angle of attack at the leading wheelset, as it is forced around the curve with significant creepage. Note, that Nadal’s approach was criticized3,7 for omitting the influence of substantial longitudinal contact forces, associated with rolling contact spin and relative rotation of the contact bodies.

The physical nature of the longitudinal force components represents a rather complicated mixture of parameters of influence, including yaw angle of the wheelset, creep level at contact area, variable resistance of the climbing wheel, free and spring slack shock as result of the train cars interaction, etc.

To find amplitude and direction of the aggregate climb-up force P, three-dimensional force system was rearranged into two separate groups, acting in- and out-of the climbing plane (u, s). At this point, aggregate matrix of the kinematical and dynamic equations can be used to identify real three-dimensional angle of climb, determined by the wheel-climb direction (z). Taking this into consideration, common condition of steady-state equilibrium for space distribution of forces after substituting kinematical and dynamic parameters, rearranging and solving for the lateral to vertical forces ratio (L/V), yields to the following final equation:

where f=nonlinear function of parameters[] above
and prim parameters are as follows:

α Three-dimensional real angle of climb
β1 Lateral under angle of attack climb angle
γ Wheel angle of attack
L Lateral wheel rail force
component
D Tangential wheel rail force component
V Vertical wheel rail force
μ Coefficient of friction

This combining equation represents nothing but generalized three-dimensional Nadal climb-up derailment criterion. It is important to stress, that for the special two-dimensional case of tangential wheel rail force D=0, this three-dimensional-climb limit above yields to the well known derailment condition, developed by Nadal:

L TAN(β1) – μ
–– = –––––––––––
V 1 + μTAN (β1)

It should be pointed out that, unlike Nadal’s derailment conditions, three-dimensional climb-up criterion (3D-climb limit) incorporates not only a level of longitudinal dynamics, but is also important for practical derailment investigation to determine real aggregate angle of the climb α. Another way of putting it is, that angle α represents nothing but derailed wheel trajectory angle, marked on the rail surface.

3D-climb limit implementation is presented in Figure 3, showing critical (L/V) ratio as a function of longitudinal to lateral force ratio (D/L) for new rail conditions. This Figure is displaying fundamental trend, according to which Nadal’s criterion is conservative and safe only for the relatively small (D/L) ratios, up to 0.4, where both 3D and Nadal limits practically coincide. It can be seen however, that already for moderate level of longitudinal force D, the range of the critical space (L/V) ratios is located substantially lower than Nadal’s, which should be considered in future climb-in derailment investigation and research.

Figure 4 defined 3D-climb limits for (L/V), showing critical (L/V) ratio as a function of longitudinal to lateral force ratio (D/L) for worn rail conditions. Again, both Nadal’s and 3D limits show close results only for the relatively small (D/L) ratios, up to 0.4, with the 3D criterion curves for moderate and large (D/L) diminishing steeply than for new rail conditions.

It is essential that, in the matrix of parameters under consideration, the three-dimensional model above does not contain any additional approximations or assumptions, comparing to the Nadal’s widely-used approach. Consequently, with the same levels of analytical accuracy and missing out-of-lateral plane forces for Nadal limit, 3D study shows, that popular Nadal criterium has substantial limitation, specifically, it is reliable and conservative only for the small (D/L) ratios and longitudinal forces. It should be particularly underscored for the practical applications and derailment preventions, that the space between Nadal limit line and corresponding 3D limit curve (see Figures 3 and 4) represents a hidden area of possible climb-up derailment for relatively low (L/V) ratios.

For those derailments to occur, reasons for high longitudinal forces D are many and varied, often associated from railroad engineering point of view with train handling, dynamic overloading in turnouts, train acceleration and deceleration, specifically sudden emergency breaking, as in the May 2015 Amtrak derailment. Including 3D-limit and approach into the engineering set of safety tools allows operators to control and reduce the risk of climb-up derailments.

Acknowledgments
The author expresses his appreciation to Dr. Allan M. Zarembski, PE, FASME, Hon. Mbr. AREMA, for discussion and formulation of Generalized Three-Dimensional Nadal Criterion and would like to acknowledge as support research, performed by railway scientists over years, including H. Weinstock, Magdy El-Sibaie, A. Zarembski, E. Magel, J. Kalker, J. A. Elkins and many others.

References
1. Liu, Y. and Magel, E. December 2007. “Understanding wheel-climb derailments.” Railway Track & Structures, pp.37-41. Chicago, Ill.
2. Zarembski A. M. 2007. “What is an acceptable level of risk.” Railway Age/Zeta-Tech/Railway Track & Structures Risk Management & Safety for Railroad Infrastructure & Equipment Conference.
3. Nadal M.J. “Theory de la stabilite des Locomotives, Annales des Mines, Vol.10, 1896.
4. Weinstock H., “Wheel climb derailment criteria for evaluation of rail vehicle safety”, US DOT, Cambridge, 02142, 1985.
5. Kalker J.J. “Survey of Wheel-Rail Rolling Contact Theory”, Vehicle Systems Dynamics, Vol.5, 1979.
6. Johnson K.L. “The effect of tangential contact force upon the rolling motion of an elastic sphere on a plain,” Journal of Applied Mechanics. 1958, pp.25.
7. Gilchrist A.O. and Brickle B.V. “A re-examination of the proneness to derailment of a railway wheelset. Journal of Mechanical Engineering Science, Vol.18, no.3, 1976.

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