This presentation presents the concept of "Impact Reduction Factor" and the "Bezgin Impact Factor" to estimate vertical impact forces on railways due to track profile variation.

1. Development of a new and an explicit analytical
equation that estimates the vertical dynamic impact
loads of a moving train
Dr. Niyazi Özgür BEZGIN
Ist anbul U niversit y, C ivil Engineering D epart ment
o z g u r. b e z g i n @ i s t a n b u l . e d u . t r
1 8 . 5 . 2 0 1 7

2. Estimation of the vertical loads of a moving train
Photo: TCDD
 Structural design of a railway track requires an estimation of the
imposed forces on the track due to the train motion.

3.  The vertical gravitational forces vary with the train speed, track –
rolling stock stiffness – damping characteristics and with the faults
along the track and/or the train rolling mechanisms.
Structural design of railway track superstructure
Photo: Wikipedia

4. Faults: Mechanic and geometric
 Variation of vertical stiffness along the track.
 Variation of vertical alignment along the track.
 Mechanical or geometric discontinuities along the track.
 Compromised circularity of the train wheel (wheel flats).

5. Track stiffness and track alignment
Photo: TCDD

6. Existing methods to estimate vertical forces – 1
 Empirical equations – Simple...Estimates with a level of precision.

7.  What are the limitations of these empirical equations?
 What do poor, normal and good track conditions mean?
 How do the geometric faults correlate with track length?
 How does the track stiffness influence the vertical forces?
 What should be the stiffness of a track ?
Constructive critique of the existing empirical methods

8.  Would a simple analytical tool that can explicitly relate the
particular track deformities and track stiffness values to impact
loads be useful?
Simple question that initiated the presented work

9. Existing methods to estimate vertical forces – 2
 Multi-body simulation softwares – Complicated...Contact
mechanics...Deeper scientific extent...Engineering estimates with
higher precision... Useful for design and track maintenance
assesment.
1. Simpack
2. Vampire
3. Universal Mechanism

10. Development of an analytic equation
 The development of the equation relies on the rules of
kinematics1 and the principle of conservation of energy2.

11. A mass placed on a linear-elastic spring
W = m. g
Fs = k. xs
𝐱 𝐬 = 𝐦. 𝐠/𝐤

12.  Potential energy of the mass (Pm) releases into the spring to be
stored as potential energy of the deformed spring (Ps).
A mass released on a linear-elastic spring
Pm = m. g. h + m. g. dx
xi
0
Ps = k. x. 𝑑𝑥
xi
0
= 1
2 k. xi
2

13.  For a given free-fall heigt of ‘h’, the impact displacement and the
impact force are directly related to the supporting structure’s
stiffness ‘k’.
 Fi ∝ k
 Fi ∝ h/xs
𝐅𝐢 = 𝐅𝐬. 𝟏 + 𝟐𝐡
𝐱 𝐬
+ 𝟏 where 𝐱 𝐬 = 𝐦. 𝐠/𝐤
Dynamic impact displacement and impact force

14. Track vertical stiffness
 Track support stiffness is the
amount of force required to cause
a unit track displacement
measured under the vertically
imposed axle load.
 The lateral and vertical extent of
the effective track length and
effective depth under the track
influences the track stiffness.

15. Train motion along a structurally perfect track
 Perfection:
1. Track stiffness is constant along the track.
2. Track horizontal alignment is level.
Maximum track displacements per wheel location.

16. Train motion along a structurally imperfect track
 Imperfection:
1. Track stiffness is constant along the track.
2. Track horizontal alignment is not level.
Envelope for maximum track displacements along the track.

17.  Potential energy of the mass releases into the track to be stored as
potential energy of the deformed track.
 The lateral train speed, influences the ‘percentage of potential
energy’ of the tributary train mass of the axle released into the
track.
 This rate is quantified by the impact reduction factor (f).
A new concept  Impact reduction factor (f)

18.  Time to traverse and pass along the horizontal track distance:
‘L’  (tpass)
 Time to hypothetically free-fall through a vertical height:
‘h’  (tfall)
Impact reduction factor (f)

19. 𝐟 = 𝟏 −
𝐭 𝐟𝐚𝐥𝐥
𝐭 𝐩𝐚𝐬𝐬
Impact reduction factor (f)
If
tfall < tpass
tfall = tpass
tfall > tpass
then
0 < f ≤ 1
f = 0
f < 0
→
Partial reduction
No reduction
Amplification

20. Development of ‘f’ through kinematics
f = 1 −
tfall
tpass
𝑡𝑓𝑎𝑙𝑙 =
2ℎ
𝑔
𝑡 𝑝𝑎𝑠𝑠 =
𝐿
𝑣
𝐟 = 𝟏 −
𝐯
𝐋
𝟐𝒉
𝒈

21. Transfer of energy as the wheel descends
m. g. h + xi − xs − m. g. f. h =
1
2
k. xs + xi xi − xs
 Part of the potential energy of
the tributary mass of moving
train axle releases into the track
to be stored as potential energy
of the deformed track.

22. h + xi − xs − 𝐟. h =
1
2
xs+xi xi−xs
xs
• 2. 𝑥 𝑠. ℎ + 2. 𝑥 𝑠. 𝑥𝑖 − 2. 𝑥 𝑠
2
− 2. 𝑥 𝑠. 𝑓. ℎ = 𝑥𝑖
2
− 𝑥 𝑠
2
• 2. 𝑥 𝑠. ℎ. 1 − 𝑓 − 𝑥 𝑠
2
= 𝑥𝑖
2
− 2. 𝑥 𝑠 . 𝑥𝑖
• 2. 𝑥 𝑠. ℎ. 1 − 𝑓 − 𝑥 𝑠
2 = 𝑥𝑖 − 𝑥 𝑠
2 − 𝑥 𝑠
2
• 2. 𝑥 𝑠. ℎ. 1 − 𝑓 = 𝑥𝑖 − 𝑥 𝑠
2
• xi = xs + 2. xs. h. (1 − f)
𝐱 𝐢 = 𝐱 𝐬 𝟏 + 𝟐𝐡
𝐱 𝐬
(𝟏 − 𝐟)
Development of ‘xi’ through conservation of energy

23. 𝑲 = 𝟏 + 𝟐𝐡
𝐱 𝐬
(𝟏 − 𝐟)
Dynamic impact factor (K)
𝑓 = 1 −
𝑣
𝐿
2ℎ
𝑔

24. An application of the proposed equation

25. tp values for various train speeds and L values

26. Track static displacement: xs=0.005 m
∆
𝐏
𝐤
= ∆
 h will be compared with ∆.
 Linear and elastic domain.

27. Ratios of time values up to h/xs=20 for L=100m
 Static track displacement taken as xs = 5 mm = 0.005 m.
h
(m)
h/xs
tf
(s)
Train speed
(km/hour)
50 100 150 200 250 300 350
tf/tp (Ratio of the fall-time to pass-time) for L=100 m
0 0.00 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.01 2.00 0.045 0.006 0.013 0.019 0.025 0.031 0.038 0.044
0.02 4.00 0.064 0.009 0.018 0.027 0.035 0.044 0.053 0.062
0.03 6.00 0.078 0.011 0.022 0.033 0.043 0.054 0.065 0.076
0.04 8.00 0.090 0.013 0.025 0.038 0.050 0.063 0.075 0.088
0.05 10.00 0.101 0.014 0.028 0.042 0.056 0.070 0.084 0.098
0.06 12.00 0.111 0.015 0.031 0.046 0.061 0.077 0.092 0.108
0.07 14.00 0.119 0.017 0.033 0.050 0.066 0.083 0.100 0.116
0.08 16.00 0.128 0.018 0.035 0.053 0.071 0.089 0.106 0.124
0.09 18.00 0.135 0.019 0.038 0.056 0.075 0.094 0.113 0.132
0.1 20.00 0.143 0.020 0.040 0.059 0.079 0.099 0.119 0.139

28. Impact reduction factors (f) for L=100 m
h
(m)
h/xs
Train speed
(km/hour)
50 100 150 200 250 300 350
f = 1-tf/tp (Impact reduction factor) for L=100 m
0 0.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.01 2.00 0.994 0.987 0.981 0.975 0.969 0.962 0.956
0.02 4.00 0.991 0.982 0.973 0.965 0.956 0.947 0.938
0.03 6.00 0.989 0.978 0.967 0.957 0.946 0.935 0.924
0.04 8.00 0.987 0.975 0.962 0.950 0.937 0.925 0.912
0.05 10.00 0.986 0.972 0.958 0.944 0.930 0.916 0.902
0.06 12.00 0.985 0.969 0.954 0.939 0.923 0.908 0.892
0.07 14.00 0.983 0.967 0.950 0.934 0.917 0.900 0.884
0.08 16.00 0.982 0.965 0.947 0.929 0.911 0.894 0.876
0.09 18.00 0.981 0.962 0.944 0.925 0.906 0.887 0.868
0.1 20.00 0.980 0.960 0.941 0.921 0.901 0.881 0.861

29. 𝐊 = 𝟏 + 𝟐𝐡
𝐱 𝐬
(𝟏 − 𝐟)
Impact load factors (K) for L=100 m
h
(m)
h/xs
Train speed
(km/hour)
50 100 150 200 250 300 350
K (Impact factor) for L=100 m
0 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.01 2.00 1.16 1.22 1.27 1.32 1.35 1.39 1.42
0.02 4.00 1.27 1.38 1.46 1.53 1.60 1.65 1.70
0.03 6.00 1.36 1.51 1.63 1.72 1.81 1.88 1.96
0.04 8.00 1.45 1.63 1.78 1.90 2.00 2.10 2.19
0.05 10.00 1.53 1.75 1.92 2.06 2.18 2.30 2.40
0.06 12.00 1.61 1.86 2.05 2.21 2.36 2.49 2.61
0.07 14.00 1.68 1.96 2.18 2.36 2.52 2.67 2.80
0.08 16.00 1.75 2.07 2.30 2.51 2.68 2.85 2.99
0.09 18.00 1.82 2.16 2.43 2.65 2.84 3.02 3.18
0.1 20.00 1.89 2.26 2.54 2.78 2.99 3.18 3.36

30. 𝐊 = 𝟏 + 𝟐𝐡
𝐱 𝐬
(𝟏 − 𝐟)
h
(m)
h/xs
Train speed
(km/hour)
50 100 150 200 250 300 350
K (Impact factor) for L=50 m
0 0.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.01 2.00 1.22 1.32 1.39 1.45 1.50 1.55 1.59
0.02 4.00 1.38 1.53 1.65 1.75 1.84 1.92 2.00
0.03 6.00 1.51 1.72 1.88 2.02 2.14 2.25 2.35
0.04 8.00 1.63 1.90 2.10 2.27 2.42 2.55 2.68
0.05 10.00 1.75 2.06 2.30 2.50 2.67 2.83 2.98
0.06 12.00 1.86 2.21 2.49 2.72 2.92 3.10 3.27
Impact load factors (K) for L=50 m

31. Impact load factors (K) for L = 100 m

32. Impact load factors (K) for L = 50 m

33. Impact load factors (K) for L = 25 m

34. h = 40 mm,
L = 25 m,
h/L = 625
h = 15 mm,
L = 25 m,
h/L = 1670
Rate of development of track vertical irregularity is
critical

35. Discussion – 1
 EN 13848-5 - Railway applications -Track - Track Geometric Quality
refers to the limitations in L and in h.

36. Discussion – 2
 If L and in h are limited, than one must determine the limit for the
highest track stiffness (k) for a given axle load since the stiffness
influences (xs) which determines the critical ratio (h/xs).
Maximum h values for L = 25 m at particular speed intervals
(km/h)
v ≤ 80 80 < v ≤120 120 < v ≤160 160 < v ≤ 230 230 < v ≤300
h (AL - max) (mm) 18 16 15 12 10
h/xs up. bound 5 5 5 5 5
xs (mm) 3.6 3.2 3 2.4 2
k170 (kN/mm) 47.2 53.1 56.7 70.8 85.0
k250 (kN/mm) 69.4 78.1 NA NA NA

37. Conclusion – 1
 An analytic equation is established that explicitly relates the
dynamic impact load factor to:
1. Vertical track degradation (h)
2. Vertically degraded track length (L)
3. Track stiffness (k, xs)
4. Train speed (v)
𝐅𝐢 = 𝐤. 𝒙 𝒔 𝟏 + 𝟐𝐡
𝐱 𝐬
(𝟏 − 𝐟) = 𝑭 𝒔. 𝐊

38.  The established equation can also estimate the impact loads
generated along a perfectly level track with variable track
stiffness.
Conclusion – 2

39.  The established equation has the potential to investigate the
development of vertical dynamic impact loads along tracks where
both the horizontal track alignment and the track stiffness varies.
 Such a case occurs especially along track-bridge transition zones.
Proposal for future work – 1
http://engineering.illinois.edu.
Photo courtesy of Donald Uzarsk.

40.  Studies are under way to upgrade the developed equation to
include the combined track – rolling mechanism stiffness and
damping.
Proposal for future work – 2

41. Спаси́ бо большо́е
Çok teşekkür ederim
Thank you very much
oz gur.bezgin@is tanbul.edu.tr