An analysis of the influence of residual stresses on material fatigue is presented in this paper. Residual stress distribution in railroad rails subjected to simulated service loads is considered. A mechanical model based on the plastic shakedown theory was used to determine residual stresses and the Dang Van fatigue criterion was applied

Pazdanowski, Michał

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* Ph.D. Michał Pazdanowski, Institute for Computational Civil Engineering, Faculty of Civil 
Engineering, Cracow University of Technology.
TECHNICAL TRANSACTIONS
CIVIL ENGINEERING
4-B/2014
CZASOPISMO TECHNICZNE
BUDOWNICTWO
MICHAŁ PAZDANOWSKI*
RESIDUAL STRESSES AS A FACTOR OF RAILROAD 
RAIL FATIGUE
NAPRĘŻENIA RESZTKOWE JAKO CZYNNIK ZMĘCZENIA 
SZYN KOLEJOWYCH
A b s t r a c t
An analysis of the influence of residual stresses on material fatigue is presented in this paper. 
Residual stress distribution in railroad rails subjected to simulated service loads is considered. 
A mechanical model based on the plastic shakedown theory was used to determine residual 
stresses and the Dang Van fatigue criterion was applied.
Keywords: residual stresses, material fatigue, Dang Van criterion
S t r e s z c z e n i e
W artykule przedstawiono analizę wpływu naprężeń resztkowych wywołanych symulowa-
nym obciążeniem szyn kolejowych kołami taboru kolejowego na zmęczenie materiału szy-
ny. Do wyznaczenia rozkładu naprężeń resztkowych zastosowano model mechaniczny opar-
ty na teorii plastycznego przystosowania, a jako kryterium zmęczeniowe przyjęto kryterium 
Dang Vana.
Słowa kluczowe: naprężenia resztkowe, zmęczenie materiału, kryterium Dang Vana
40
1. Introduction
Railroad rails may catastrophically fail during service resulting in loss of human life and 
significant damage to rolling stock [1, 2]. The experimental investigations [3] revealed that 
residual stresses induced in a rail during manufacturing (roller straightening) and service 
(contact loads at the rail/wheel interface, exceeding the elastic bearing capacity of the rail 
material) may constitute an important factor affecting crack nucleation and growth.
In the current paper, the influence of residual stresses induced in a railroad rail during 
simulated service on the fatigue life of such a rail is given consideration.
Residual stresses in rails subjected to service conditions may be found through either 
numerical analysis [21], or through experimental investigations of specimens taken out of 
standard revenue tracks [22] or test tracks [23]. Experimental analysis is time consuming, 
costly and in the case of specimens taken out of revenue tracks, the loading history 
of a specimen may be impossible to ascertain. On the other hand, numerical analysis may 
be prohibitively time consuming when an exact elasto-plastic incremental analysis would 
have to be performed in order to find the final residual stress state in a rail after a significant 
number of loading cycles [21]. This obstacle may be avoided when one settles for an estimate 
of residual stress distribution in a rail subjected to simulated service loads computed using 
the mechanical model based on the elasto-plastic shakedown theorem initially proposed 
in [4, 5]. Although the application of this mechanical model yields only an estimate 
of residual stresses induced by simulated service loads, numerous tests have shown that this 
estimate is of reasonably good quality [6].
Multiaxial high cycle fatigue criteria for metals may be divided into three main 
groups [7]: critical plane approaches, such as Findley [10], Matake [11], McDiarmid [12], 
Dietmann [13]; approaches based on stress invariants, such as Marin [14], Crossland [15], 
Kakuno-Kawada [16], Deperrois [17]; approaches based on stress averages within the 
elementary volume, such as Grubisic and Simburger [18], Liu and Zenner [19], or Dang Van 
[20]. According to [7], criteria belonging to the group of approaches based on the stress 
averages within material volume yield results closest to experiments for so called ‘hard’ 
metals (which include steels). Therefore, the simplest of these, the Dang Van [20] criterion, 
was used in further analysis.
Fig. 1. Initiation and growth of fatigue crack in the head of rail subjected to service loads
41
2. Residual stress evaluation method
The residual stress calculation method based on shakedown theorems [9], in its simplest 
form may be stated as the following minimization problem [5]:
find:
 min ( ) ( )
σ
σ σ σ σ
ij
r kl
r
kl
V
T
ijkl ij
r
ijC dV− ⋅ ⋅ − ⋅∫ 0 0  (1)
subject to:
 σij j
r
, = 0  – for each point in V  (2.1)
 σij
r
jn⋅ = 0  – for each point on dV (2.2)
 Φ( ( ))σ σ σij
r
ij
E
yt+ £  – for each point in V (2.3)
where:
σ σij
r
kl
r,  – time independent residual stresses induced in the considered body by 
the actual loading program,
σ σij kl
0 0,  – initial residual stresses existing in the considered body prior to the 
application of current loading program,
σij
E  –  time dependent elastic stresses induced in the body by current loading 
program changing in time,
sy – material yield limit,
Cijkl – elastic compliance matrix,
nj – vector perpendicular to the body boundary.
Formula (1) denotes the total complementary energy of residual stresses while formulas 
(2.1) and (2.2) denote the internal equilibrium conditions and zero static boundary conditions 
of those stresses. Formula (2.3) denotes the yield condition, which has to be satisfied in every 
moment of time. Of course, all the constraints (2) have to be satisfied in every point of the 
considered body.
3. The Dang Van fatigue criterion
The Dang Van fatigue criterion may be counted among the fatigue criteria based on 
the mesoscopic scale approach, i.e. the scale of metal grains of a metallic aggregate [7]. 
Thisfatigue criterion is based on an average measure of the plastic strain accumulated in all 
the flowing crystals within an elementary volume of the material.
This criterion condenses the history of six stress tensor components into the load path 
defined by two components, and thus simplifies fatigue damage calculations [8]. The criterion 
combines hydrostatic pressure σH and momentary maximum shearing stress τa calculated 
according to the Tresca criterion:
42
 τa
v vt s t s t( ) ( ) ( )= ⋅ − 
1
2 I III
 (3)
evaluated for the part of stress deviator tensor which varies in time; this part is defined as:
 s t s t s t t sij
v
ij ij
c
ij ij H ij
c( ) ( ) [ ( ) ( )]= − = − ⋅ −σ δ σ  (4)
where sij
c  constitutes the solution of the following minimax problem:
 s s t s s t sij
c
s t
ij ij ij ij
ij
= − ⋅ −
∗
∗ ∗minmax( ( ) ) ( ( ) )  (5)
i.e. such value of the time independent stress deviator sij
∗  for which the maximum of the 
norm (5) reaches the lowest value. Shear stress and hydrostatic pressure are then combined 
linearly to yield an equivalent scalar:
 τ τ α σeq DV t a H
t t, max( ( ) ( ))= + ⋅  (6)
which in turn may be used to estimate the fatigue damage [8].
The constant  is determined as follows:
 α
τ
σ
= ⋅ −





3
1
2
D
D
 (7)
where τD and σD represent fatigue limits in torsion and tension-compression, respectively. 
According to the Dang Van criterion, time independent residual stresses do not affect 
the momentary maximum shear stress τa (3), but affect only the hydrostatic stress term σH [8].
4. Service load simulation program
The contact load acting on the rail crown has been simulated by biparabolic pressure 
distribution spanned over the rectangular contact area applied at seven evenly distributed 
discrete contact locations shown in Fig. 2, where location 1 is centered on the rail’s 
longitudinal axis of symmetry while location 7 is offset by 25 mm to the left of this axis. 
Peak pressure p0 and patch dimensions a × b have been determined using elastic Hertz 
contact formulae to compute the contact ellipse area for the given contact load, and later 
on, to determine a rectangle with equivalent area and biparabolic pressure distribution 
balancing this contact load. For the purpose of current calculations, 132RE rail (US type) 
made of steel exhibiting the following material data have been assumed: 206 GPa Young 
Modulus, 483 MPa yield limit, 0.3 Poisson’s ratio. Three values of wheel load have been 
considered, namely: 147 kN, 160 kN, 173 kN. These values correspond to standard wheel 
loads on heavy haul rail lines in North American practice.
A 3D Finite Element Method computational model was used to find the necessary 
momentary elastic stress distributions σij
E  (2.3), while a 2D Meshless Finite Difference 
Method computational model was applied to determine the rail longitudinal axis independent 
distributions of residual stresses σij
r .
43
T a b l e  1
Peak residual and elastic hydrostatic stress levels introduced in rail
Location
(Fig. 2)
Load
[kN]
Compression
[MPa]
Tension
[MPa]
residual σH
r
 elastic σH
r residual σH
r elastic σH
r
2
147
–107.900 –496.460 78.733 –197.260
4 –130.622 –438.299 80.716 –197.779
6 –140.146 –432.423 97.209 –188.910
2
160
–120.254 –407.357 79.491 –209.653
4 –126.147 –423.772 81.502 –211.301
6 –140.023 –459.753 108.147 –192.702
2
173
–142.589 –446.350 107.429 –187.212
4 –147.962 –371.866 116.255 –187.289
6 –171.047 –369.222 153.002 –165.418
The distribution of hydrostatic residual stress σH
r  (directly affecting the Dang Van 
fatigue criterion) in the railhead for all three considered values of wheel load applied at the 
rightmost load application point depicted in Fig. 2 is presented in Fig. 3. For better readability, 
tensile and compressive parts of this stress are depicted separately on the right and left, 
respectively. Peak residual hydrostatic pressure levels introduced by each of the considered 
Fig. 2. Load application points on the rail crown and mesh used during numerical analysis
44
loading scenarios at selected loading locations are presented in Table 1. and compared to 
the highest elastic (momentary) hydrostatic pressure levels introduced by simulated wheel 
load introduced at the same locations in the railhead.
Locations of peak hydrostatic residual stresses σH
r  (compressive component denoted 
by ○, and tensile component denoted by ●) as well as von Mises equivalent residual stresses 
σ0
r  (denoted by □) corresponding to load application points indicated in Fig. 2 are depicted 
in Fig. 4.
Fig. 3. Hydrostatic residual stress σH
r  decomposed into positive (compression – at left) 
and negative (tension – at right) parts. Contour interval 14 MPa
Fig. 4. Location of extreme values of hydrostatic residual stress and von Mises equivalent 
residual stress induced by contact loads applied at locations indicated in Fig. 2
45
5. Conclusions
Initial calculations performed so far indicate that the residual stress levels due to 
simulated contact loads may reach levels on a par with actual (momentary) elastic stresses 
created by these loads, and thus should be included in fatigue calculations. Values of these 
stresses tend to increase substantially (by up to 23.5% in compression and up to 30.1% 
in tension for the results presented in Table 1) as the load application area shifts away 
from the center of the railhead. When the load application area gets very close to the gauge 
side of the rail an additional stress concentration zone occurs at the gauge side of railhead. 
This phenomenon may additionally be aggravated should two point contact load occur 
(for instance on a curved track). Thus, two point contact loads will be subjected to analysis 
in further work.
The results presented in the 5th column of Table 1 are of special interest from the practical 
point of view, as the tensile stresses have an adverse influence on the fatigue life of the body 
subject to cyclic loads. The residual hydrostatic tension for loading scenarios considered so 
far reaches a level of almost 32% of the material yield limit, thus indicating that these stresses 
significantly affect rail fatigue life, and the safety of railroad operation.
The location of extreme residual hydrostatic stresses below the running surface of the 
rail remains fairly stable, regardless of the load application point location, though the depth 
of extreme stresses seems to be affected by the load magnitude for tensile stresses only.
At the locations indicated in Fig. 4 by ●, the residual hydrostatic pressure for higher 
wheel loads is on par with the elastic hydrostatic pressure. The positive sign of this pressure 
(tension) indicates substantially increased risk of rail failure fatigue in this zone.
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Residual stresses as a factor of railroad rail fatigue

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Residual stresses as a factor of railroad rail fatigue

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