Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 695-705, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.695

A FINITE ELEMENT APPROACH TO DEVELOP TRACK GEOMETRICAL

IRREGULARITY THRESHOLDS FROM THE SAFETY ASPECT

Amin Miri

School of Civil Engineering, Sharif University of Technology, Tehran, Iran

e-mail: amin.miri.iust@gmail.com

Saeed Mohammadzadeh, Hoda Salek

School of Railway Engineering, Iran University of Science and Technology, Tehran

Riding quality and safety of rail tracks are directly influenced by track geometry; hence,
their degradation along time could reduce safety and cause serious accidents. Standards
propose thresholds for track geometrical parameters to keep track safety and riding comfort
at an acceptable level. In this study, amethod is proposed to select or define a set of proper
thresholds for geometrical parameter irregularities according to desirable to safety level.
The impact of track geometry irregularities on the derailment index has been investigated
through the finite elementmodel. The results suggest that twist and gauge shortage have a
greatereffecton thederailment indexcomparedto theverticalprofileof the track.Having the
critical values of geometrical irregularities that result in derailment, safety factors of Iranian
and Euro code standards in determining their thresholds are calculated and compared. It
is shown that each standard has a unique set of safety factors that depend on speed and
geometrical parameters.

Keywords: maintenance and inspection, track geometry parameters, finite element,
Adams/rail software

Notations

Q,Y – vertical and lateral forces between rail and wheel flange, respectively
β – contact angle of rail and wheel flange
µ – coefficient of friction between rail and wheel flange
Z – alignment of modeled track
X – longitudinal coordinate of modeled tracks
I – alignment irregularity value
mo,mb,mw – mass of car body, bolster and wheel set, respectively
Iox,Ioy,Ioz – inertia of car body around x, y and z axis, respectively
Iwx,Iwy,Iwz – inertia of wheel set around x, y and z axis, respectively
KPV ,KSV – primary and secondary vertical dampers stiffness, respectively
KLS – lateral dampers stiffness

1. Introduction

Safety and ride comfort are two important characteristics of any railway network. These cha-
racteristics are closely related to the geometrical condition of track superstructure, meaning
whenever the track superstructure is at standard condition, so are the safety and ride comfort
of passing trains. Tomonitor geometrical conditions of railway tracks, track quality indexes are
proposed that are based on track geometrical irregularities.



696 A.Miri et al.

Whether a geometrical parameter is considered irregular or normal, it is assessed through
comparing themeasuredgeometrical parameterwith thepre-determined set of thresholdspropo-
sed in standard guidelines and standards. Table 1 presents the geometrical parameter threshold
as stated in Iranian guideline and Euro code standard (Alert limit state) (Iran Ministry of Ro-
ads and Transportation, 2005; EN-13848-5, 2008). It is clear from Table 1 that for a certain
geometrical parameter at a certain speed, each guideline proposes different threshold values due
to different operational regime and characteristics of the networks in which they are applied.

Table 1. Iran and Euro-code thresholds (all values in mm)

V>230 200<V<230 160<V<200 120<V<160 80<V<120 V<80
Speed Track
[km/h] parameter

– – 12 16 16 18 Iran Vertical
16 20 20 23 26 28 Euro code profile

– – 6 8 10 12 Iran
Gauge +

28 28 28 35 35 35 Euro code

– – 2 2 2 2 Iran
Gauge -

5 7 7 10 11 11 Euro code

– – 5 5 5 5 Iran
Alignment

10 12 12 14 17 22 Euro code

– – 1.5 1.5 1.5 1.5 Iran
Twist

5 7 7 7 7 7 Euro code

The point to keep in mind regarding the thresholds is that maintenance operations are
dependent upon them. So proposing restricted values for thresholds would result in frequent
maintenance operations and higher maintenance expenditures, while loose thresholds may in-
crease the possibility of accidents due to excessive irregularities. In this regard, developing an
optimized set of thresholds would not only reduce expenses, but also guarantee safety of the
operation.

Although defining thresholds of track geometrical irregularities seems to be of great impor-
tance, the focus of studies is rather on track geometrical parameters. Some papers have propo-
sed novel track quality indexes based on track geometrical parameters (Madejski andGrabczyk,
2002;ORE, 1981;Miri andMohammadzadeh, 2014). Others studied the effect of track geometri-
cal irregularities on the performance and conditional assessment of the track, such as initiation
and evolution of rail corrugation (Jin et al., 2005), effects of maintenance operations on track
quality (Ataei et al., 2014), proposing an inspection interval for track geometrical parameters
(ArastehKhouy et al., 2013), and evaluating the probability of derailment (Mohammadzadeh et
al., 2011). Also, there are emerging technologies such as the performance-based track geometry
inspection system that relate geometrical parameters of track to vehicle performance in real time
(Li et al., 2006). In a recent study by Arasteh Khouy et al. (2014) a cost model is proposed to
specify the cost-effective maintenance limits for track geometry maintenance. The model con-
siders degradation rates of different track sections and takes into account the costs associated
with inspection, tamping, delay time penalties and risk of accidents due to poor track quality
to come up with the cost-effective intervention limit of the longitudinal level for tamping.

The aim of this paper is to propose a method that could be used to develop a set of track
geometrical irregularity thresholds from the safety point of view. Track geometrical parameters
considered in this paper include vertical profile, alignment, track gauge, cross level (also known
as “cant”) and twist, which are schematically presented in Fig. 1. Tomeasure safety of proposed
thresholds for each track geometrical parameter, the derailment index is considered as the safety
indicator.



A finite element approach to develop track geometrical irregularity thresholds... 697

Fig. 1. (a) Vertical profile, (b) alignment, (c) gauge, (d) cant and twist irregularities

Derailment occurswhen carwheels run off the rail that supplies support and guidance of the
vehicle. Many papers have been published concerning this subject since it is crucial to railway
safety (Yadav, 2007;Wu andWilson, 2006; Elkins andWu, 2000; Nadal, 1908;Weinstock, 1984;
Karmel and Sweet, 1981). Derailment is divided into two main categories: sudden derailment
and flange climbing derailment, of which the second category is taken into account in this study.

Flange climb derailment is the case in which wheels climb to the top of the rail individually
and then run over it. In this condition, forces causing derailment are higher than resisting forces,
but not powerful enough to cause the first case (Yadav, 2007). Suchderailment occurs as vertical
forces tend to decrease or as a result of increasing lateral forces which are combined with the
forwarding movement of the vehicle. In the case of the vertical profile, the vertical forces Q
would decrease and as a result, Y/Q increases andmay reach the critical limit. This case is also
known as “unloading”.

There are different theories in the field of the derailment phenomenon such as Nadal theory
(Nadal, 1908), Weinstocks (Weinstock, 1984) and Kereszty (Yadav, 2007). According to Nadal
(1908), if the resisting forces are higher than the derailing forces, no derailment will occur.
Therefore, by analyzing forces applied to the flange – according to the condition stated above –
it could be concluded that

µ(Y sinβ+Qcosβ)+Y cosβ¬Qsinβ (1.1)

which could be restated as

Y

Q
¬
tanβ−µ

1+µtanβ
(1.2)

in which Y andQ are the lateral and vertical forces between the rail andwheel flange, respecti-
vely.β is the contact angle of the rail andwheel flange andµ is the coefficient of friction between
the flange and rail. For most wheel types, β is equal to 68◦ (Yadav, 2007). Also depending on



698 A.Miri et al.

the geometry and roughness of contact surface, µ varies between 0.25 and 0.27 (Yadav, 2007).
Using these values, equation (1.2) simplifies as

Y

Q
¬ 1.4 (1.3)

Fig. 2. Free-body diagram of wheel-rail contact forces

Considering the simplicity and comprehensiveness of Nadal theory, it is used as derailment
theory in this study. Different values have been stated as the critical limit of the Y/Q ratio.
According to UIC 518 and EN14363, in tracks with curve radii of more than 250 meters, the
Y/Q ratio shall not be greater than 0.8 according to the sliding mean over 2m of the track.
Also, 0.8 has been stated as the critical value for Y/Q ratio in various works (Elkins and Wu,
2000; Karmel and Sweet, 1981; Kik et al., 2002). In this regard, the critical value of the Y/Q
ratio is considered to be 0.8 throughout the paper.
Another important factor in developing thresholds of geometrical irregularities is the chord

length. Most guidelines and standards consider chord lengths of 10, 19, and 37 meters. Thro-
ughout this paper, a 10 meter chord is selected to control short wavelength defects that can
result in high wheel forces over a short portion of the track. These forces may not produce
excessive car bodymotion yet their action on thewheels and trackmay cause derailment, which
is in line with the purpose of this paper.

2. Methodology of research

The aim of this paper is to find the amplitude of track geometrical irregularities that result
in derailment. To do so, tracks with a length of 100 meters are modeled in Adams/Rail. Solid
elements are used to model the track, and the space between each sleeper is divided into two
distinct elements. Geometrical defects are simulated based on the definition of track geometrical
parameters usingmathematical functions which are characterized bywavelength and amplitude
and applied to a 10 meter chord of the modeled tracks.
Next, a freight wagon (characteristics of the wagon are presented in Table 2) passes over the

modeled track with a geometrical defect, and dynamic analyses are carried out to determine
lateral and vertical forces. Hence, the derailment index is determined for a certain amplitude
of geometrical irregularity. Next, the amplitude of geometrical irregularity is increased and the
whole process repeats until the derailment index reaches the critical threshold of 0.8; so the
corresponding amplitude of geometrical irregularity that results in derailment is determined.
Since the speed is an effective parameter in the derailment phenomenon, the whole process is
carried out for speeds of 40, 80, and 120km/h.



A finite element approach to develop track geometrical irregularity thresholds... 699

Table 2.Characteristics of freight wagon used in simulations

Symbols Values Unit Name

mo 32000 kg mass of car body

mb 1503 kg mass of a bolster

mw 1503 kg mass of wheel set

Iox 5.68 ·10
4 kgm2 inertia of car body around x axis

Ioy 1.97 ·10
6 kgm2 inertia of car body around y axis

Ioz 1.97 ·10
6 kgm2 inertia of car body around z axis

Iwx 810 kgm
2 inertia of wheel set around x axis

Iwy 810 kgm
2 inertia of wheel set around y axis

Iwz 112 kgm
2 inertia of wheel set around z axis

KPV 6 MN/m primary vertical dampers stiffness

KSV 6 MN/m secondary vertical dampers stiffness

KLS 6 MN/m lateral dampers stiffness

3. Investigating the effects of alignment

The alignment of the track increases lateral forces on the rail which, according to Nadal theory,
could result in derailment if the irregularity is high enough. A polynomial function of the 4th
order is chosen to model the alignment, which is as follows

z=x4−2lx3+ l2x2 (3.1)

inwhich z is the alignment of themodeled track, x is the longitudinal coordinate of themodeled
track, and l is the alignment irregularity value. Figure 3 shows a 12mm left rail alignment in
a 100 meters track that starts from the point at 50 meters and continues up to the point at
60 meters. Dynamic analysis for this track is carried out and Fig. 4 demonstrates the results of
the front axle of front and rear bogies, at a speed of 80km/h.

Fig. 3. Model of the alignment defect



700 A.Miri et al.

Fig. 4. Result of dynamical analysis of the modeled track with the alignment defect in Adams rail

Fig. 5. Derailment index (Y/Q) versus alignment irregularity for speeds of 50, 80, and 140km/h

Figure 5 presents the results of dynamic analyses of tracks with the alignment defect for
speeds of 50, 80, and 140km/h. It is evident that as train speed increases, derailment occurs
at lower wavelengths of the alignment defect. For speeds of 50, 80, and 140km/h, derailment
occurs at alignment irregularities of 27, 21, and 18, respectively. These values are close to the
thresholds stated in Euro code standard, but considerably higher than those stated in Iranian
standard.

4. Investigating the effects of the vertical profile

Vertical forces of the wheel on the rail could decrease in the presence of the vertical profile
irregularity which increases the Y/Q value and could result in derailment. The vertical profile
is modeled using the same geometrical function used tomodel the alignment defect. The results
of dynamic analysis are presented in Fig. 6. In a lower speed of 50km/h, the derailment ratio
remains below the critical value of 0.8 for vertical profiles of up to 350mm. But at 140km/h,
derailment occurs at a vertical profile irregularity of 40mm.Generally, it could be seen that the
derailment ratio is less sensitive to the vertical profile than to the alignment, since Y/Q remains
fairly below the critical value for higher vertical profile irregularities.



A finite element approach to develop track geometrical irregularity thresholds... 701

Fig. 6. Derailment index versus vertical profile irregularity for speeds of 50, 80, and 140km/h

5. Investigating the effects of the track gauge

Track gauge variations can lead to large lateral wheel forces resulting in derailment. Both extra
and shortage of the gauge are undesirable, since an extra gauge may end up in wheels falling
off the rail, while shortage of gauge could result in wheels climbing on the top of the rail. A
trapezoidal function is used tomodel extra and shortage of the track gauge, as shown in Fig. 7.
Figures 8a and 8b present the results of dynamical analysis for the modeled tracks with extra
and shortage of the gauge, respectively.

Fig. 7. Geometrical function used to model track gauge irregularities

Fig. 8. Derailment index versus (a) extra gauge irregularity and (b) gauge shortage irregularity for
speeds of 50, 80, and 140km/h



702 A.Miri et al.

6. Investigating the effects of twist

To model the twist irregularity, sinusoidal waves with amplitudes of 1, 2 and 3mm and wave-
lengths of 2-6mm are used. The results of dynamical analysis of the modeled tracks with twist
defects are presented in Figs. 9a to 9c. As these figures suggest, derailment is sensitive to both
wavelength and amplitude of the twist. Keeping the amplitude constant, derailment occurs at
shorter wavelengths. On the other hand, an increasing amplitude results in higher Y/Q ratios.
The twist with an amplitude of 3mm results in derailment in speeds of 80 and 140km/h regar-
dless of thewavelength, as shown inFig. 9c. These results suggest that the twist is an influential
parameter affecting derailment.

Fig. 9. Derailment index versus twist with an amplitude of (a) 1mm, (b) 2mm and (c) 3mm for speeds
of 50, 80, and 140km/h

7. Comments on the results

Critical amplitudes of track geometrical irregularities are concluded in Table 3 for speeds of
50, 80, and 140km/h. Having the amplitude of track geometrical irregularities that result in
derailment, thresholds could be developed by applying the safety factor to the values inTable 3.
Moreover, by dividing the critical values in Table 3 to the thresholds stated in the guidelines,
it is possible to estimate the safety factors considered in the guidelines from the safety point
of view. Figure 10 presents the safety factors determined for Iranian and Euro code standards.
Alert limit state of Euro code is considered, since it is recommended based on safety issues
related to derailment (EN-13848-5, 2008).
According to Fig. 10, Iranian standard is very conservative in defining thresholds since all

safety factors arebelow0.5,whileEuro code considers logical safety factors. For example, Iranian
standard considers a safety factor of 0.1 for the gauge shortage for a speed of 50km/h,while this
value is 0.8 inEuroCode standard.This shows that each standardconsiders auniqueset of safety
factors that dependonmanyparameters, such as thenetwork inwhich it is applied,maintenance
operations frequency and quality, rolling stock characteristics and operational regimes. Also, the



A finite element approach to develop track geometrical irregularity thresholds... 703

Table 3.Critical values of geometrical irregularities that result in derailment

Parameter
Speed [km/h]
50 80 140

Alignment 27 21 18

Profile – 150 40

Gauge + 49.5 41 37

Gauge - −13.5 −13 −11.5

Twist (1mm) – 2.5 4.5

Twist (2mm) 3 5 –

Twist (3mm) 4.5 – –

Fig. 10. Safety factors of Iranian and Euro code standard thresholds from the safety point of view

safety aspect is not the sole determinant of the thresholds, and restricted thresholds could be
result from considering a plethora of factors for determining the thresholds.

According to Fig. 10, a single value is not considered as the safety factor for all geometrical
parameters.For a speedof 140km/h, safety factors for thealignment, vertical profile, extragauge
and shortage of gauge are 0.8, 0.6 and 0.9, respectively, according to Euro code. As expected,
the safety factor of a geometrical parameter varies with speed as well. For example, Euro code
takes safety factors of 0.8, 0.6, and 0.9 for the shortage of gauge irregularity at speeds of 50, 80,
and 140, respectively.

8. Conclusion

Track performance is highly dependent on the condition of geometrical parameters. Using a
predefined set of thresholds, it is possible to monitor geometrical irregularities of the track and
determine the quality index for a portion of the track. In this study, a method is proposed
which could be used to select or define a new set of thresholds from the point of view of safety.
To do so, geometrical irregularities are modeled in Adams/Rail finite element software and
dynamical analyses are carried out to determine the derailment index. Varying the amplitude
of geometrical irregularities, it is possible to determine the critical geometrical irregularity that
results in derailment of the train.

The results suggest that derailment is highly correlated to twist and gauge shortage. Ac-
cording to the results, the twist with an amplitude of 3mm results in derailment at speeds of
80 and 140km/h regardless of the wavelength. On the other hand, derailment is less sensitive
to variations in the vertical profile irregularity, since no derailment occurs the vertical profile
irregularity of 350mm at a speed of 50km/h.

Multiplying the critical values of track geometrical irregularities by safety factors, it is po-
ssible to develop a new set of thresholds from the safety aspect. The reverse could be done to



704 A.Miri et al.

determine safety factors of guidelines for developing thresholds, which is calculated in this paper
for Euro code and Iranian standards. It is observed that each standard considers a unique set of
safety factors to determine thresholds of geometrical irregularities. Iranian standard considers
very restrictive safety factors, while Euro code choses a logical approach. This is mainly due to
the fact that characteristics of the railway networks in which they are applied are different.

Considering high costs of maintenance operations and the importance of geometrical para-
meters to the safety of tracks, the selection of an optimized set of thresholds for a railway track
is crucial. Using the proposedmethod of this paper could lead to an optimum set of thresholds
that satisfies both financial and safety aspects at the same time. The same finite element model
could also be used to determine the thresholds from any other aspect, such as ride comfort.

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Manuscript received October 16, 2016; accepted for print January 5, 2017