This presentation introduces the concept of impact reduction factor and a method both developed by Dr. Niyazi Özgür Bezgin that can estimate vertical impact forces on railways due to changes in track profile. The Bezgin Impact Factors KB1 and KB2 are introduced.

1. 1. Introduction
• Vertical railway track irregularities may increase the vertical forces acting on the railway track.
• Track construction and maintenance faults as well as allowable track construction tolerances may generate
irregularities in the form of vertical deviations in the track profile along a length of track and can result in increased
vertical wheel forces.
• Aside from computerized techniques, there are also well-established semi-empirical equations such as those proposed
by Talbot, Clarke and Eisenmann among many others that allow the designers to estimate the impact forces.
• The applicability of the existing empirical methods is limited in scope due to particular track and railway vehicle
conditions that generated them. One cannot claim that an empirically developed equation applies to every other track
condition.
• The author attempts to overcome this shortcoming by developing an analytical method and an explicit analytical
equation to estimate the impact forces, based on the principle of energy conservation, rules of kinematics and a
new concept of impact reduction factor developed by the author earlier.
• The proposed method concentrates on the track conditions, track stiffness, and the rate of change of track profile
irregularity.
I S T A N B U L U N I V E R S I T Y
Department of Civil Engineering
APPLICATION OF A NEW CONCEPT AND A METHOD
TO ESTIMATE THE VERTICAL IMPACT FORCES ON RAILWAY TRACKS
DUE TO TRACK PROFILE IRREGULARITIES
Niyazi Özgür BEZGİN, Ph.D
Associate Professor, ozgur.bezgin@istanbul.edu.tr
4. Development of a new concept and a method to estimate the dynamic impact forces
• The level of impact as the wheels roll over the defective track length, relates to the time it takes to hypothetically
free-fall from the height of vertical deviation h, referred to as tfall and the time to traverse the defective track length
L, referred to as tpass.
• Part of the potential energy of the tributary axle mass that transfers into the track as it rolls down or climbs up the
variable profile is quantified by a parameter proposed as the “IMPACT REDUCTION FACTOR (f)”
𝐟 = 𝟏 −
𝐭 𝐟𝐚𝐥𝐥
𝐭 𝐩𝐚𝐬𝐬
If
𝐭 𝐟𝐚𝐥𝐥 < 𝐭 𝐩𝐚𝐬𝐬
𝐭 𝐟𝐚𝐥𝐥 = 𝐭 𝐩𝐚𝐬𝐬
𝐭 𝐟𝐚𝐥𝐥 > 𝐭 𝐩𝐚𝐬𝐬
than
𝟎 < 𝐟 ≤ 𝟏
𝐟 = 𝟎
𝐟 < 𝟎
→
Partial to full reduction
No reduction
Amplification
𝑡𝑓𝑎𝑙𝑙 =
2ℎ
𝑔
, 𝑡 𝑝𝑎𝑠𝑠 =
𝐿
𝑣
𝐟 = 1 −
tfall
tpass
= 1 −
2. h
g
L
v
= 𝟏 −
𝐕
𝐋
.
𝟐𝐡
𝐠
E1 = m. g. h + c − a
E2 = m. g. h + c − a − 𝐦. 𝐠. 𝐡. 𝐟
E3 =
1
2
k a + c c − a
h + c − a − f. h =
1
2
a+c c−a
a
2. a. h + 2. a. c − 2. a2
− 2. a. f. h = c2
− a2
a2
+ 2. a. h. 1 − f − 2. a2
= c2
− 2. a. c
2. a. h. 1 − f − 2. a2
= c − a 2
− a2
2. a. h. (1 − f) = c − a
𝐜 = 𝐚 𝟏 + 𝟐𝐡
𝐚 (𝟏 − 𝐟)
𝐅𝐢 = 𝐤. 𝐜 = k. a. 1 + 2h
a (1 − f)
= 𝐅𝐬. 𝟏 + 𝟐𝐡
𝐚 (𝟏 − 𝐟)
Impact force factor proposed by Bezgin (KB)
𝐊 𝐁 = 𝟏 + 𝟐𝐡
𝐚 (𝟏 − 𝐟)
5. Variable profile and constant stiffness along the track
• The variation in the track profile generates a potential energy to the
tributary mass of the wheel.
a: Maximum static deflection of the track.
c: Maximum dynamic deflection of the track.
h: Difference in vertical track profile along a track length of L.
k: Track stiffness.
E: Potential energy.
• Depending on the impact reduction factor, this potential energy generates
the impact force of wheels on the track.
𝐦. 𝐠. 𝐡 + 𝐦. 𝐠. 𝐝𝐱
𝐜
𝟎
= 𝐤. 𝐱. 𝐝 𝐱
𝐜
𝟎
m. g. h + c =
1
2
. k. c2
h + c =
1
2.a
. c2
2. a. h + 2. c. a = c2
2. a. h = c − a 2
− a2
a2
. 1 + 2.h
a = c − a 2
c = a. 1 + 1 + 2.h
a
Fi = k. c = k. a. 1 + 1 + 2.h
a
𝐅𝐢 = 𝐅𝐬. 𝟏 + 𝟏 + 𝟐. 𝐡
𝐚 = 𝐅𝐬. 𝐊
2. Constant profile and stiffness along the track
• Given that the train wheels are perfectly circular, a perfectly
level track with a constant stiffness, would not induce
dynamic impact forces.
• The potential energy of the tributary mass of the wheels
would not change with respect to the deformed or
undeformed states of the track between 1 and 2.
3. Impact force on the track induced by the rate of change of potential energy of the axle mass
• Let us first consider the development of an impact force induced by a free-falling mass on an undamped linear –
elastic spring through the principle of energy conservation.

2. Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
tfall/tpass, L=30.5 m (100 ft)
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.01 0.02 0.04 0.06 0.08 0.10 0.12
0.00 0.01 0.03 0.06 0.09 0.12 0.15 0.17
0.00 0.02 0.04 0.07 0.11 0.14 0.18 0.21
0.00 0.02 0.04 0.08 0.12 0.16 0.21 0.25
0.00 0.02 0.05 0.09 0.14 0.18 0.23 0.28
Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
f = 1- tfall/tpass
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 0.99 0.98 0.96 0.94 0.92 0.90 0.88
1.00 0.99 0.97 0.94 0.91 0.88 0.85 0.83
1.00 0.98 0.96 0.93 0.89 0.86 0.82 0.79
1.00 0.98 0.96 0.92 0.88 0.84 0.79 0.75
1.00 0.98 0.95 0.91 0.86 0.82 0.77 0.72
Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
KB
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.14 1.20 1.29 1.35 1.41 1.45 1.50
1.00 1.24 1.34 1.48 1.59 1.68 1.76 1.84
1.00 1.33 1.46 1.65 1.80 1.92 2.03 2.13
1.00 1.41 1.57 1.81 1.99 2.15 2.28 2.41
1.00 1.48 1.68 1.96 2.17 2.36 2.52 2.66
h
(mm)
(in)
h/a for
a=10
mm
(0.4 in)
tfall
(s)
0 0.0 0.000
10 (0.4) 1.0 0.045
20 (0.8) 2.0 0.064
30 (1.2) 3.0 0.078
40 (1.6) 4.0 0.090
50 (2.0) 5.0 0.101
h
(mm) (in)
h/a for
a=10
mm
(0.4 in)
0 0.0
10 (0.4) 1.0
20 (0.8) 2.0
30 (1.2) 3.0
40 (1.6) 4.0
50 (2.0) 5.0
h
(mm) (in)
h/a for
a=10
mm
(0.4 in)
0 0.0
10 (0.4) 1.0
20 (0.8) 2.0
30 (1.2) 3.0
40 (1.6) 4.0
50 (2.0) 5.0
Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
tfall/tpass, L=15.2 m (50 ft)
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.02 0.04 0.08 0.12 0.16 0.21 0.25
0.00 0.03 0.06 0.12 0.17 0.23 0.29 0.35
0.00 0.04 0.07 0.14 0.21 0.29 0.36 0.43
0.00 0.04 0.08 0.16 0.25 0.33 0.41 0.49
0.00 0.05 0.09 0.18 0.28 0.37 0.46 0.55
Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
f = 1-tfall/tpass
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 0.98 0.96 0.92 0.88 0.84 0.79 0.75
1.00 0.97 0.94 0.88 0.83 0.77 0.71 0.65
1.00 0.96 0.93 0.86 0.79 0.71 0.64 0.57
1.00 0.96 0.92 0.84 0.75 0.67 0.59 0.51
1.00 0.95 0.91 0.82 0.72 0.63 0.54 0.45
Train speed (km/hour) (mph)
0 25 50 100 150 200 250 300
0 16 31 62 93 124 155 186
KB
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.20 1.29 1.41 1.50 1.57 1.64 1.70
1.00 1.34 1.48 1.68 1.84 1.96 2.08 2.18
1.00 1.46 1.65 1.92 2.13 2.31 2.46 2.60
1.00 1.57 1.81 2.15 2.41 2.62 2.81 2.99
1.00 1.68 1.96 2.36 2.66 2.92 3.14 3.35
Train speeds (v) Pass-time, tp=L/v (s)
km/h mph m/s ft/s
For L=15.2
m (50 ft)
For L=30.5
m (100 ft)
For L=45.7
m (150 ft)
For L=61 m
(200 ft)
0 0 0.0 0.0 Infinite Infinite Infinite Infinite
25 16 6.9 22.8 2.19 4.39 6.58 8.78
50 31 13.9 45.6 1.10 2.19 3.29 4.39
100 62 27.8 91.1 0.55 1.10 1.65 2.19
150 93 41.7 136.7 0.37 0.73 1.10 1.46
200 124 55.6 182.3 0.27 0.55 0.82 1.10
250 155 69.4 227.8 0.22 0.44 0.66 0.88
300 186 83.3 273.4 0.18 0.37 0.55 0.73
𝐟 = 𝟏 −
𝐭 𝐟𝐚𝐥𝐥
𝐭 𝐩𝐚𝐬𝐬
𝐊 𝐁 = 𝟏 + 𝟐𝐡
𝐚 (𝟏 − 𝐟)
If
𝟎 < 𝐟 ≤ 𝟏
𝐟 = 𝟎
𝐟 < 𝟎
→
Partial to full reduction
No reduction
Amplification
1. Occurrence of impact forces relates to combined influence of track irregularities and track stiffness.
2. Along a track with a very high stiffness, even a minute vertical deviation from the track profile may produce a high
impact force on the track.
3. Whether a certain level of vertical deviation along a misaligned track length makes the track, a poor track or a good
track is also a matter of the track stiffness and the resulting static deflection.
4. The base limit condition for h/a=0 represents perfect track alignment and therefore the track does not induce
impact forces on rolling wheels.
5. Note that the estimated impact factors by the proposed method decrease as the defective track length along which
a given value of vertical deviation develops increases.
6. The effect of a vertical deviation (h) on the developing impact also relates to its ratio to the static track deflection.
7. The proposed method by Bezgin responds to the particular conditions of track (h, a and L) and provide different
estimates for differing conditions.
8. The proposed equation suggests that if the track stiffness were so high to produce so small static deflections, the
high track stiffness would amplify the impact effect of a very small vertical irregularity along the track.
Impact reduction factor (f)
Impact factor proposed by Bezgin (KB)
6. Application of the proposed method to estimate impact forces
8. Conclusions
7. Impact factor estimates of a selection of semi-empirical methods