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Importance of vertical rail track stiffness on dynamic overloading: Limitations of the Eisenmann
formulation

Eisenmann’s formula considered the quality of the rail track and the confidence interval. This
formula has been widely proven for maximum speeds of 200 km/h.
Since the introduction of the high-speed rail in Europe in the 1980s, rail vehicles have been
designed to ensure that their loads, which are transmitted to the rail track, are substantially
less than the loads transmitted by conventional vehicles. The quality of the newer rail tracks is
significantly higher than the quality of existing rail tracks. Consequently, Eisenmann’s formula
from 1969, which was developed using data from conventional vehicles on conventional lines,
was not valid for determining the stresses caused by high-speed trains, such as France’s
highspeed trains (Train à Grande Vitesse, TGV). In 1993, Eisenmann proposed a modification
of his previous formula to adapt it to the case of high-speed vehicles and lines. The following
expression was defined for the parameter for speeds from 201 km/h to 300 km/h:

In the 1970s, the National Society of French railways (SNCF) developed important theoretical and
experimental research to analyze the effect of rail tracks and vehicles on dynamic overloads.
This research was performed by Prud’Homme, who used a classic model for modeling a rail track-
vehicle system and its behavior and analyzed the excitations produced by the irregularities of
a rail track. Prud’Homme applied the theory of random vibrations to develop the well-known
formula for calculating the dynamic overloads produced by unsprung masses of a vehicle:

where

: standard deviation of the dynamic overloads due to unsprung masses,
∆
: running speed of the vehicle, km/h,
: variable related to rail track defects and vehicle defects,

: unsprung mass of the vehicle,

: vertical rail track stiffness, t/mm,
(): rail track damping

The significance of Prud’Homme’s formula is that it introduces new criteria and reveals how the
vertical rail track stiffness, the unsprung mass of the vehicle, and the quality of the rail track,
in addition to the speed of the vehicle, affect the dynamic overloads.

As expressed by this formula, for a given speed and rail track quality, different geotechnical
and geometric compositions of infrastructures, which determines the value of K, produces
different dynamic overloads. This fact was not considered by Eisenmann’s formula and exposes
its limitations. The objective of this article is to analyze these limitations and the relationship

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