This presentation sums up the proposed method to estimate the dynamic impacts on railway tracks. This work progressing by the inclusion of wheel stiffness and damping as well as the contact duration between the wheel and the rail.

1. 5th International Conference
on Road and Rail Infrastructure
CETRA 2018
Niyazi Özgür BEZGIN
Associate Professor
I s t a n b u l U n i v e r s i t y
Proposal of a new analytical method to estimate the
vertical impact forces on railway tracks due to
changes in track profile and track stiffness

2. Content
 Introduction to dynamic impact forces on railway tracks
 Proposal of a new analytical method that estimates impact forces
and a brief review of previously presented work
 Impact forces due to descending and ascending track profiles.
 Impact forces due to increasing and decreasing track stiffness.
 Conclusions

3. Introduction
 Perfectly level running track profile:
 Running track profile with irregularities are:
1. Variation of track profile along a length of track.
2. Variation of track stiffness along a length of track.
3. Variation of wheel circularity (wheel flats).

4. Variation in track profile
 Variation in track profile generates the difference in the potential
energy of the tributary wheel mass.

5. Variation in track stiffness
 Variation in static track deflection generates the difference in the
potential energy of the tributary wheel mass.

6. Variation in wheel circularity
 Each of these variations, generate a change in the potential
energy of the tributary mass of the wheel, as the wheel rolls
along the track.
 Variation in the abrupt change in wheel diameter generates the
difference in the potential energy of the tributary wheel mass.

7.  Empirical equations – Simple with limited precision.
Some empirical estimations

8. Some empirical estimation

9.  Multi-body simulation software : They provide engineering
estimates with higher precision and they are useful for design
finalization and assesment of track maintenance.
• Simpack®, Vampire®, Universal Mechanism®, Simulia® and
more…
 One may unconditionally yield to their estimates thereby
relinquishing the engineering judgement.
 One may lack, the time, tool or the budget for such an analysis.
 Can we develop a simple pencil and paper method?
Some advanced analytical estimation methods

10.  Potential energy for the tributary wheel mass can develop as a
variation in the track profile, track stiffness or wheel circularity.
 Part of this potential energy of the tributary mass releases into
the track to be stored as potential energy of the deformed track.
 The stiffness of the track and the wheel, roughness of the track
and/or the wheel and the train speed, influence the generated
impact on the track.
Potential energy of the tributary wheel mass

11. A new concept: Impact reduction factor
 This rate of change is embodied in a new concept named the
impact reduction factor (f) (Bezgin, 2017).

12. Use of kinematics
If
tfall < tpass
tfall = tpass
tfall > tpass
then
0 < f ≤ 1
f = 0
f < 0

13. Time variation of descending track profile for wheel
tributary mass

14. a
 Static wheel force P due to the tributary wheel mass, causes
vertical rail deformation a.
 Track stiffness per rail k relates to the force and the
deformation.
𝐤 =
𝐏
𝐚
=
𝐦𝐠
𝐚
Linear-elastic idealization of track stiffness

15. Conservation of energy of the wheel tributary mass
for descending track profile

16. Impact factor for the descending track profile (KB,d)

17. The concept of f and KB,d first presented and published
at:

18. Comparisons of KB,d with some of the existing empirical
equations, presented at TCG 2017 in St. Petersburg

19. KB,d revised and presented and published at 97th. TRB
Meeting in Washington, DC:
 Where the theory underlying the equation KB,d was explained
in detail and also notational presentation improved.

20. Further comparisons for KB,d at 97th. TRB

21. Time variation of track profile due to decreasing track
stiffness for wheel tributary mass

22. Storage of potential energy of the wheel tributary mass
in the track for decreasing track stiffness
𝐦. 𝐠. 𝐜 − 𝐚 − 𝐦. 𝐠. 𝐛 − 𝐚 . 𝐟 =
𝐤 𝟐. 𝐛 + 𝐤 𝟐. 𝐜
𝟐
. (𝐜 − 𝐛)
Fi = k2. c = k2. b. 1 + 1.414 1 − f +
a
b
. (f − 1)
𝐊 𝐁𝟏 = 𝟏 + 𝟏. 𝟒𝟏𝟒 𝟏 − 𝐟 −
𝐚
𝐛
. (𝟏 − 𝐟) , 𝐰𝐡𝐞𝐫𝐞 𝐚 ≤ 𝐛

23. Time variation of track profile due to increasing track
stiffness for wheel tributary mass

24. Storage of potential energy of the wheel tributary mass
in the track for increasing track stiffness
𝐦. 𝐠. 𝐚 − 𝐜 − 𝐦. 𝐠. 𝐚 − 𝐛 . 𝐟 =
𝐤 𝟐. 𝐛 + 𝐤 𝟐. 𝐜
𝟐
. (𝐜 − 𝐛)
Fi = k2. c = k2. b. 1.414 1 + f +
a
b
. (1 − f) − 1
𝐊 𝐁𝟐 = 𝟏. 𝟒𝟏𝟒 𝟏 + 𝐟 +
𝐚
𝐛
. (𝟏 − 𝐟) − 𝟏 , where a≥b

25. KB1 and KB2 estimates for 0 ≤ f ≤ 1 and various a/b ratios

26. Impact equations due to track stiffness transitions (KB1
and KB2) were presented at the 97th TRB in January 2018.

27. Time variation of ascending track profile for wheel
tributary mass

28. Conservation of energy of the wheel tributary mass
for ascending track profile

29. Impact factor for the ascending track profile (KB,a)

30. KB,d and KB,a estimations: An example
 Track stiffness per rail is k=43 kN/mm.
 The static axle force is Fs=170 kN
 The static rail deflection is a= 85 kN÷ 43 kN/mm = 2 mm.
 The vertical variation of track profiles are: h=4 mm, h= 8 mm
and h=12 mm.
 Therefore h/a values are: 4/2= 2, 8/2= 4 and 12/2= 6.
 Rough track lengths are: L=10m, L=25 m and L=70 m.

31. The durations required to traverse (tp) the rough track
lengths (L)
10 25 70
km/h m/s
50 13.9 0.72 1.80 5.04
100 27.8 0.36 0.90 2.52
200 55.6 0.18 0.45 1.26
300 83.3 0.12 0.30 0.84
Train speed
L (m)
tp (s)

32. 50 100 200 300 50 100 200 300
0 0.00 0.000 0.000 0.000 0.000 0.000 0 0.00 0.000 0.000 0.000 0.000 0.000
0.004 2.00 0.029 0.016 0.032 0.063 0.095 0.004 2.00 0.029 0.006 0.011 0.023 0.034
0.008 4.00 0.040 0.022 0.045 0.090 0.135 0.008 4.00 0.040 0.008 0.016 0.032 0.048
0.012 6.00 0.049 0.027 0.055 0.110 0.165 0.012 6.00 0.049 0.010 0.020 0.039 0.059
50 100 200 300 50 100 200 300
0 0.00 1.000 1.000 1.000 1.000 0 0.00 1.000 1.000 1.000 1.000
0.004 2.00 0.984 0.968 0.937 0.905 0.004 2.00 0.994 0.989 0.977 0.966
0.008 4.00 0.978 0.955 0.910 0.865 0.008 4.00 0.992 0.984 0.968 0.952
0.012 6.00 0.973 0.945 0.890 0.835 0.012 6.00 0.990 0.980 0.961 0.941
50 100 200 300 50 100 200 300
0 0.00 1.00 1.00 1.00 1.00 0 0.00 1.00 1.00 1.00 1.00
0.004 2.00 1.25 1.36 1.50 1.62 0.004 2.00 1.15 1.21 1.30 1.37
0.008 4.00 1.42 1.60 1.85 2.04 0.008 4.00 1.25 1.36 1.51 1.62
0.012 6.00 1.57 1.81 2.15 2.41 0.012 6.00 1.34 1.49 1.69 1.84
h
(m)
h/a
tf
(s)
Train speed (km/hour)
tf/tp for L=25 m
f = 1-tf/tp for L=25 m
h
(m)
h/a
Train speed (km/hour)
KB,d for L=25 m
h
(m)
h/a
Train speed (km/hour)
h
(m)
h/a
tf
(s)
Train speed (km/hour)
tf/tp for L=70 m
h
(m)
h/a
Train speed (km/hour)
f = 1-tf/tp for L=70 m
h
(m)
h/a
Train speed (km/hour)
KB,d for L=70 m
Estimations for KB,d

33. Estimations for KB,a
50 100 200 300 50 100 200 300
0 0.00 0.000 0.000 0.000 0.000 0.000 0 0.00 0.000 0.000 0.000 0.000 0.000
0.004 2.00 0.029 0.016 0.032 0.063 0.095 0.004 2.00 0.029 0.006 0.011 0.023 0.034
0.008 4.00 0.040 0.022 0.045 0.090 0.135 0.008 4.00 0.040 0.008 0.016 0.032 0.048
0.012 6.00 0.049 0.027 0.055 0.110 0.165 0.012 6.00 0.049 0.010 0.020 0.039 0.059
50 100 200 300 50 100 200 300
0 0.00 1.000 1.000 1.000 1.000 0 0.00 1.000 1.000 1.000 1.000
0.004 2.00 0.984 0.968 0.937 0.905 0.004 2.00 0.994 0.989 0.977 0.966
0.008 4.00 0.978 0.955 0.910 0.865 0.008 4.00 0.992 0.984 0.968 0.952
0.012 6.00 0.973 0.945 0.890 0.835 0.012 6.00 0.990 0.980 0.961 0.941
50 100 200 300 50 100 200 300
0 0.00 1.00 1.00 1.00 1.00 0 0.00 1.00 1.00 1.00 1.00
0.004 2.00 1.02 1.03 1.06 1.09 0.004 2.00 1.01 1.01 1.02 1.03
0.008 4.00 1.04 1.09 1.17 1.25 0.008 4.00 1.02 1.03 1.06 1.09
0.012 6.00 1.08 1.16 1.31 1.45 0.012 6.00 1.03 1.06 1.11 1.17
h
(m)
tf/tp for L=25 m tf/tp for L=70 m
h/a
tf
(s)
Train speed (km/hour)
h
(m)
h/a
tf
(s)
Train speed (km/hour)
h
(m)
h/a
h
(m)
h/a
Train speed (km/hour)
f = 1-tf/tp for L=25 m f = 1-tf/tp for L=70 m
Train speed (km/hour)
h
(m)
h/a
h
(m)
h/a
Train speed (km/hour)
KB,a for L=25 m KB,a for L=70 m
Train speed (km/hour)

34. Variations of KB,d and KB,a for descending and ascending
track conditions
Let us not forget: a = static rail deflection = 2 mm

35. Discussion of results
 Track with a high stiffness, amplifies the effect of a profile
variation by increasing h/a, whereas increased static wheel
forces reduce the effect of profile variation by decreasing h/a.
 For a track with track stiffness per rail of k, the dynamic impact
force of the wheel at a speed of v relates to h/L.
 Estimated impact force values due to ascending and descending
track profiles for a given h/L are different.
 For a given track stiffness, train speed and |h/L|, the
descending track profile produces higher impacts compared to
ascending track profiles.

36. Discussion of results
 Estimated impact force values due to increasing and decreasing
track profiles are different.
 n-fold variation and 1/nth variation in track stiffness produces
different impact values on the track.
 Proposed method yielded clear and explicit analytical
equations that relate the dynamic impact forces on railway
tracks to track stiffness, track roughness and train speed.
 The first four equations: KB,d, KB,a, KB1, KB2 are proposed by
Bezgin (2017 and 2018) and the fifth equation KB3 is proposed
by Kolukırık and Bezgin (2017)….Session 5.A on May 18th.

37. 𝐊 𝐁,𝐝 = 𝟏 +
𝟐𝐡
𝐚
(𝟏 − 𝐟) , for descending track profile
𝐊 𝐁,𝐚 = 𝟐.
𝐡
𝟐𝐚
. 𝟏 − 𝐟 + 𝟏 − 𝟏 , for ascending track profile
𝐊 𝐁𝟏 = 𝟏 + 𝟏. 𝟒𝟏𝟒 𝟏 − 𝐟 +
𝐚
𝐛
. (𝐟 − 𝟏) , for k1≥k2 where a≤b
𝐊 𝐁𝟐 = 𝟏. 𝟒𝟏𝟒 𝟏 + 𝐟 +
𝐚
𝐛
. (𝟏 − 𝐟) − 𝟏 , for k1≤k2 where a≥b
where 𝒇 = 𝟏 −
𝒕 𝒇𝒂𝒍𝒍
𝒕 𝒑𝒂𝒔𝒔
= 𝟏 −
𝑽
𝑳
.
𝟐𝒉
𝒈
𝐊 𝐁𝟑 = 𝟏 +
𝟐𝐡
𝐚
(𝟏 − 𝐟) , for wheel flats
where 𝒇 = 𝟏 −
𝒕 𝒇𝒂𝒍𝒍
𝒕 𝒔𝒑𝒊𝒏
= 𝟏 −
𝟒.𝑽.𝒔𝒊𝒏
𝜽
𝟐
𝒓
𝒈
𝜽.𝒓

38. Thank You
Hvala vam
Teşekkür ederim