This presentation introduces the concept of "impact reduction factor" and a new method that are both developed by Dr. Niyazi Özgür Bezgin to estimate vertical impact forces on railways due to track stiffness variations.

1. APPLICATION OF A NEW CONCEPT AND A METHOD TO
ESTIMATE THE VERTICAL IMPACT FORCES ON RAILWAY TRACKS
DUE TO TRACK STIFFNESS VARIATIONS
Niyazi Özgür BEZGİN, Ph.D
A s s o c i a t e P r o f e s s o r
o z g u r. b e z g i n @ i s t a n b u l . e d u .t r
I S TA N B U L U N I V E R S I T Y
D e p a r t m e n t o f C i v i l E n g i n e e r i n g
J a n u a r y 8 , 2 0 1 8

2. Introductory statement
1. This presentation will introduce a set of analytical equations that is part
of an on-going research at a preliminary stage.
2. The proposed equations base on an underlying hypothesis, the
estimates of which agree with existing empirical equations.
3. The analytical equations will be experimentally validated.
4. The estimates of these equations will be compared to the estimates of
advanced software.

3. Presentation outline
1. A qualitative account of track imperfections that lead to vertical impact
forces.
2. Introduction of a new concept of impact reduction factor.
3. Introduction of a new method that estimates vertical impact forces .
4. Application of the new method to estimate the vertical impact forces
due to variations in track stiffness.
5. On-going and future work and conclusions.

4. 1. The vertical impact forces, vary with the train speed, track/rolling stock
stiffness/damping characteristics and with imperfections1 along the
track or the wheel.
2. These imperfections generate the potential energy2 to the tributary
mass of the train axle due to the vertical differential height they create.
3. The rate of development of an imperfection and the speed of the train
determine the part of this potential energy3 that releases onto the
track superstructure and hence the impact force and acceleration4.
Hypothesis:

5. Imperfections: Mechanical and/or geometrical
1. Variation of track profile.
2. Variation of vertical stiffness along the track.
3. Structural discontinuities along the track.
4. Compromised circularity of the train wheel (wheel flats).
 The following imperfections generate the potential energy of the
tributary mass of the axle:

6. 1. Variations in track profile
Photo: TCDD
 Construction error or construction tolerance.
 Problems with rail linearity.

7. 2. Variations in track stiffness
 Transition from ballasted track to slab track.
 Passage over a culvert with a shallow embedment.
 Transition into and out of viaducts, bridges and tunnels.
 Passage over locally degraded track.

8. Train motion along a structurally perfect track
 Perfection:
1. Track stiffness is constant along the track.
2. Track profile is level.
 This condition does not generate potential energy and hence does not
generate vertical impact forces as the wheel rolls from position 1 to 2
and 3.

9. Train motion along a structurally imperfect track – 1
 Imperfection – 1:
1. Track stiffness is constant along the track.
2. Track profile is variable.

10. Train motion along a structurally imperfect track – 2/1
 Imperfection – 2/1:
1. Track stiffness is variable along the track and
decreases along the direction of motion.
2. Track profile is constant.

11. Train motion along a structurally imperfect track – 2/2
 Imperfection – 2/2:
1. Track stiffness is variable along the track and
increases along the direction of motion.
2. Track profile is constant.

12. Train motion along a structurally imperfect track – 3
 Imperfection – 3:
1. Track stiffness is variable along the track.
2. Track profile is variable.
OR + OR

13. Advanced methods to estimate vertical forces today
 Multi-body simulation softwares – They are necessary…Complicated...
Contact mechanics...Deeper scientific extent... Estimates with higher
precision... Useful for design finalization and maintenance assesment.
• Simpack®
• Vampire®
• Universal Mechanism®
• Simulia®
• ANSYS®
• Etc...
 However… They are so capable that one may unconditionally yield to their
estimates thereby relinquishing the engineering judgement… The
programs may be not be available…

14. Some empirical methods to estimate vertical forces
Talbot
Eisenmann
Clarke
Deutsche Bahn
Japan Railways

15. Some of the existing methods to estimate vertical forces
 Empirical equations – Simple... Implicit characteristics…Variable estimates

16. 1. What are the limitations of these empirical equations, do we still have
their user’s manuals ?
2. What do poor, normal and good track conditions numerically mean?
3. How does the rate of profile and stiffness variation change the vertical
forces?
4. What should be the stiffness of a track ?
Constructive critique of the existing empirical methods

17. 1. Can we develop a simple analytical tool that can explicitly relate the
particular track profile and stiffness variations to vertical impact forces?
2. Can this tool be simple enough that can enable its beneficiary to conduct
a preliminary manual assesment of a track structure?
Simple questions that initiated the presented work

18. The new concept of impact reduction factor1 supported by kinematics2
and
The development of an energy based3 new method developed by
algebra4

19.  Potential energy of the axle mass releases into the track to be stored as
potential energy of the deformed track.
 Potential energy can develop as a variation in the track profile and/or the
track stiffness.
 The train speed, influences the ‘percentage of potential energy’ of the
tributary mass of the axle released onto the track.
 This rate is quantified by the impact reduction factor (f).
A new concept  Impact reduction factor (f)

20.  Time to traverse the deviated track length:
L  tpass
 Time to hypothetically free-fall or elevate through the deviation height:
h  tfall
Impact reduction factor (f) developed with respect to h due to profile
deviation

21. f = 1 −
tfall
tpass
If
tfall < tpass
tfall = tpass
tfall > tpass
than
0 < f ≤ 1
f = 0
f < 0
→
Partial to full reduction
No reduction
Amplification
Impact reduction factor (f)

22. Development of ‘f’ through kinematics and its evaluation
f = 1 −
tfall
tpass
tfall =
2h
g
tpass =
L
v
f = 1 −
v
L
2h
g
 Important findings:
1. For a given speed (v), the smaller the length (L), the smaller is the f
2. For a given (v) and (L), the higher the (h), the smaller is the f.

23. A mass placed on a linear-elastic spring
W = m. g
Fs = k. 𝑎
𝐚 =
𝐦. 𝐠
𝐤

24.  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 +
0
𝑐
m. g. dx
Ps =
0
c
k. x. dx = 1
2 k. c2

25.  For a given free-fall height ‘h’, the impact displacement ‘c’ and the impact
force ‘Fi’ are directly related to the supporting structure’s stiffness ‘k’.
1. Impact force is directly proportional to stiffness.
2. Impact force is directly proportional to h/a.
 The value of h has a very important implication in terms of track
imperfection.
𝐅𝐢 = 𝐅𝐬. 𝟏 + 𝟐𝐡
𝐚 + 𝟏 where 𝐚 =
𝐦. 𝐠
𝐤
Dynamic impact displacement and impact force
With kind acknowledgement of Hacettepe University Physics Engineering Department
and with kind rememberance of the late Professor Egor Popov

26. Track vertical stiffness
 Track stiffness is the amount of force
required to cause a unit displacement
measured under the vertically imposed
axle force.
 The lateral and vertical extent of the
effective track length and effective
depth under the track, influences the
track stiffness.
 Stiffness is an inherent quality of any structure.
∆
𝐤 =
𝐏
∆

27. Details for the next 4 slides are available in my poster session tomorrow:

28. 𝐊 𝐁 = 𝟏 +
𝟐𝐡
𝐚
(𝟏 − 𝐟)
Dynamic impact factor (KB) due to profile deviation
Fi = Fs. 1 + 2h
a (1 − f)
 Vertical force exerted by the wheels is amplified due to c.
 An important finding is that the effect of h depends on a which
relates to the track stifness.
 Another important finding is that the effect of h depends on its rate
of development, which is summarized by f.

29. Impact force factors (KB) for L = 50 m (164 ft), a=5 mm (0.2 in)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 50 100 150 200 250 300 350 400
Impactforcefactorestimates
Train speed (km/h)
Bezgin, h/a=12, L=50m
Bezgin, h/a=10, L=50m
Bezgin, h/a=8, L=50m
Bezgin, h/a=6, L=50m
Bezgin, h/a=4, L=50m
Bezgin, h/a=2, L=50m
Bezgin, h/a=0
Deutsche Bahn
Eisenmann, n=0.1
Eisenmann, n=0.2
Eisenmann, n=0.3
Japanese Railways
Talbot
, L=50m

30. Impact force factors (KB) for L = 25 m (82 ft), a=5 mm (0.2 in)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 50 100 150 200 250 300 350 400
Impactforcefactorestimates
Train speed (km/h)
Bezgin, h/a=8, L=25m
Bezgin, h/a=6, L=25m
Bezgin, h/a=5, L=25m
Bezgin, h/a=4, L=25m
Bezgin, h/a=2, L=25m
Bezgin, h/a=1, L=25m
Bezgin, h/a=0, L=25m
Deutsche Bahn
Eisenmann, n=0.1
Eisenmann, n=0.2
Eisenmann, n=0.3
Japanese Railways
Talbot

31. An application of the proposed concept and method to estimate the
impact forces that develop as a result of track stiffness variation

32. Development of impact force due to transition from high track
stiffness to low track stiffness
E3 = E2 − E1
. f = m. g. c − a − m. g. b − a . f
 As the wheels roll to a region of lower stiffness, they negotiate an
elevational difference ‘h’, due to varying stiffness.
 Depending on the speed of the train, the length of transition and h, part
of this enery releases into the track thereby causing an impact.

33. m. g. c − a − m. g. b − a . f =
k2.b+k2.c
2
. (c − b)
 Transferred energy, deforms the track and temporarily stores in the track.
Part of the potential energy tranfers into the track depending on ‘f’

34. Fi = k2. c = k2. b. 1 + 1.414 1 − f +
a
b
. (f − 1) = Fs. 1 + 1.414 1 − f +
a
b
. (f − 1)
𝐊 𝐁𝟏 = 𝟏 + 𝟏. 𝟒𝟏𝟒 𝟏 − 𝐟 −
𝐚
𝐛
. (𝟏 − 𝐟)
where a≤b and k1≥k2
After a simple algebraic development… Impact factor proposed by
Bezgin: KB1: Due to transition from HIGH TO LOW stiffness track

35. Development of impact force due to transition from low track
stiffness to high track stiffness
E3 = E2 − E1
= m. g. a − c − m. g. a − b . f
 As the wheels roll to a region of higher stiffness, they again negotiate an
elevational difference (h) due to varying stiffness.
 Depending on the speed of the train, the length of transition and h, part
of this enery releases into the track thereby causing an impact.

36. m. g. a − c − m. g. a − b . f =
k2. b + k2. c
2
. (c − b)
 Transferred energy, deforms the track and temporarily stores in the track.
Part of the potential energy tranfers into the track depending on f

37. Fi = k2. c = k2. b. 1.414 1 + f +
a
b
. (1 − f) − 1 = Fs. 1.414 1 + f +
a
b
. (1 − f) − 1
𝐊 𝐁𝟐 = 𝟏. 𝟒𝟏𝟒 𝟏 + 𝐟 +
𝐚
𝐛
. (𝟏 − 𝐟) − 𝟏
where a≥b and k1≤k2
After a simple algebraic development… Impact factor proposed by
Bezgin: KB2: Due to transition from LOW TO HIGH stiffness track

38. Applications of the proposed equations

39. KB2 = 1.414 1 + f +
a
b
. (1 − f) − 1
KB1 = 1 + 1.414 1 − f −
a
b
. (1 − f)
KB = 1 + 1.414
h
a
(1 − f)
A summary of the proposed equations up to this point
f = 1 −
v
L
2h
g
For a≤b and k1≥k2
For a≥b and k1≤k2
Impact reduction factor
Impact factor proposed by
Bezgin, based on profile
deviation. k1=k2
Poster session 583
Hall E, January 9
@10.15 AM

40. KB1 and KB2 estimates for varying stiffness ratios and f values

41. Stiffness ratios vary from 0.1 to 10 and f values vary from 0 to 1
Threshold for passenger
comfort, a=2 m/s2
Threshold for
derailment

42.  Stiffness of an approach track to a bridge is k1 = 40 kN/mm (228 kip/in)
 Stiffness of the track above the bridge is k2 = 120 kN/mm (684 kip/in)
 Track profile remains constant along the transition.
 Track transitions to the bridge within a length of L=2 m (6.56 ft)
 The static track deflection at the approach track and the bridge under an
axle load of 200 kN (45 Kips) is a= 5 mm (0.2 in) and b=1.7 mm (0.07 in)
respectively.
 h = a – b = 5 mm – 1.7 mm = 3.3 mm (0.0033 m)
An example – 1/4

43.  For a given service speed of v = 150 km/h = 41.7 m/s (93 mph), the
impact reduction factor (f) is:
An example – 2/4
𝐟 = 1 −
v
L
2h
g
= 1 −
41.7
2
2 ∗ 0.0033
9.81
= 𝟎. 𝟒𝟔
𝐊 𝐁𝟐 = 1.414 1 + f +
a
b
. (1 − f) − 1
= 1.414 1 + 0.46 +
5
1.7
1 − 0.46 − 1 = 𝟏. 𝟒𝟕

44.  For a given service speed of v = 250 km/h = 69.4 m/s (155 mph), the
impact reduction factor (f) is:
An example – 3/4
𝐟 = 1 −
v
L
2h
g
= 1 −
69.4
2
2 ∗ 0.0033
9.81
= 𝟎. 𝟏
𝐊 𝐁𝟐 = 1.414 1 + f +
a
b
. (1 − f) − 1
= 1.414 1 + 0.1 +
5
1.7
1 − 0.1 − 1 = 𝟏. 𝟕𝟑

45.  If the allowable impact factor due to transition is KB=1.1, (vertical
acceleration = 1 m/s2), what should the length of the transition (L) be?
An example – 4/4
f = 0.896 = 1 −
v
L
2h
g
= 1 −
69.4
L
2 ∗ 0.0033
9.81
→ 𝐋 = 𝟏𝟕. 𝟖 𝐦 (𝟓𝟗 𝐟𝐭)
KB2 = 1.414 1 + f +
a
b
. (1 − f) − 1
= 1.414 1 + f +
5
1.7
1 − f − 1 = 1.1
→ 𝐟 = 𝟎. 𝟖𝟗𝟔

46. Conclusions
1. This study presented a simple pen and pencil method to estimate
impact forces along any track.
2. The study considered the effect of track roughness on the impact forces
due to a change in track stiffness.
3. In terms of track roughness, the rate of deviation (h and L) from the
normal and the track stiffness (k) are required to estimate the impact.
4. The faster (v) the train travels over a certain ‘h’ and ‘L’, the higher is the
generated impact force.
5. For a given h, L and v, the higher the stiffness (lower ‘a’) the higher is
the generated impact.

47. On-going and proposed future work
1. The concepts presented here are also applied to the impact effects of
wheel flats that produced promising results.
2. Damping is introduced into the equations as a factor that represents
the part of potential energy that is damped out of the system.
3. New developments will incorporate non-linear track stiffness into the
equation.
4. Effects of wheel-rail contact duration with respect to track response
time will be incorporated.
5. Comparisons of the proposed estimates with the estimates of
advanced analytical programs and physical validation are required.

48. Teşekkür ederim
Thank you
o z g u r . b e z g i n @ i s t a n b u l . e d u . t r