95
A
DVANCED
E
NGINEERING
3(2009)1, ISSN 1846-5900
THERMAL AND STRESS ANALYSIS OF BRAKE DISCS
IN RAILWAY VEHICLES
Oder, G.; Reibenschuh, M.; Lerher, T.; Šraml, M.; Šamec, B.; Potrč, I.
Abstract: Present paper shows a thermal and stress analysis of a brake disc for railway vehicles
using the finite element method (FEM). Performed analysis deals with two cases of braking; the
first case considers braking to a standstill; the second case considers braking on a hill and
maintaining a constant speed. In both cases the main boundary condition is the heat flux on the
braking surfaces and the holding force of the brake calipers. In addition the centrifugal load
1
is
considered.
Keywords: railway transport, brake disc, finite element method (FEM), thermal and
stress analysis
1 INTRODUCTION
The main problem of braking and stopping a heavy train system is the great input of
heat flux into the disc in a very short time. Because of high temperature difference the
material is exposed to high stress. The result is a heat shock. The problem can be
solved only by applying a non stationary and numerical calculation. The analysis is
carried out for two models of the disc (Fig. 1) (the disc is supposed to be symmetrical)
and for two modes of load. The material of the brake disc is rounded graphite defined
under the standard SIST EN 1563:1998(en) [2] with a characteristic EN-GJS-500-7
(EN-JS 1050). The disc is screwed on the hub, which is upset upon the axle of the train
wagon. During the analysis only one part of the working cycle is considered, and that
is warming up and cooling down.
Fig. 1. Break disc
1
The paper work Stress analysis of a brake disc considering centrifugal and thermal load
96
2 NUMERICAL MODEL
2.1 Modeling and preparing the 3D model
In this analysis two models of the disc are considered:
• brake disc without wear (Fig. 2);
• worn out brake disc with a 7 mm wear on each side.
Fig. 2. Section of the new disc with basic measurements
2.2 Determination of the load
Two different loads are used:
• braking down from the maximum speed of 250 km/h to a standstill. The initial
temperature of the disc and the surrounding is 50°C.
Fig. 3. Stop braking load
• braking by the maximum speed of 250 km/h on a hill and maintaining
constant speed. Because of previous braking the temperature of the disc is
150°C. The temperature of the surrounding is 50 °C.
Fig. 4. Drag braking load
P(t) [W]
t [s]
t
b
P
max
Q
pr
[J]
P(t) [W]
t [s]
Q
pr
[J]
P=const
.
97
The goal is to find out how the temperature is distributed on the whole construction
of the disc by braking on a flat track to a standstill (Fig. 3). The deceleration factor is
1.4 m/s
2
. The part of the braking energy that transfers on the surrounding air is not
considered. The reason for that is the high braking power which causes the dominant
effect of the heat flux. An assumption is made that the heat flux uses the convection
with a heat transfer coefficient of 10 W/m
2
K.
Because of a constant heat flux into the disc, braking on a hill (Fig. 4) is a bigger
disadvantage, than braking on a flat track. An assumption is made that the heat transfer
coefficient of the forced convection is 100 W/m
2
K. The goal is to determine how the
temperatures and stress rise if the disc reaches the maximum working temperature of
350°C.
In both cases the effect of the humidity in the air and the heat transfer with
radiation is not considered.
2.3 Determination of the physical model
Braking on the flat track derives from the physical model for determination of the heat
transfer in dependency from the braking time. Beside that the weight distribution of the
vehicle is considered. The weight arrangement is 60/40 [3] in the favor of the front part
of the carriage. This means that the front part of the carriage takes 60% of the whole
load. In our case only 10 % of the whole brake force is applied to one disc from the
forward part of the carriage.
Because of the mentioned weight distribution, only the front part of the carriage is
analyzed. Every carriage is consistent of for axles with three brake discs attached to
each axle. The kinetic energy [3] for one wheel considering constant deceleration is:
( )
( )
dt
t
v
F
dt
t
P
v
M
z
z
t
disc
disc
t
∫
∫
⋅
=
=
⋅
⋅
⋅
0
0
2
0
2
2
1
1
,
0
,
(1)
The change of energy is equal to the heat flux on the surface of the disc. This ratio
is used to calculate the thermal load on the brake disc. Other data used for the analysis
are listed in table 1.
Mass of the vehicle – M [kg]
70 000
Start velocity – v
0
[m/s]
70
Deceleration – a [m/s
2
] 1,4
Braking time – t
b
[s]
50
Effective radius of the braking disc – r
d
[m]
0,247
Radius of the wheel – r
w
[m]
0,460
Incline of the track – δ [‰] 11
Friction coefficient disc/pad – μ [/]
0,4
Surface of the braking pad A
c
– [mm
2
] 20000
Tab. 1. Data for calculating the heat flux
Forces which work on the brake disc [3]:
98
]
[
5
,
9125
t
a
2
1
t
v
r
r
2
v
M
2
1
1
,
0
F
2
z
z
0
w
d
2
0
disc
N
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
−
⋅
⋅
⋅
⋅
⋅
⋅
=
.
(2)
Instant heats flux entering one side of the braking disc [3]:
( )
( )
(
)
]
[
t
6860
343000
t
a
v
r
r
F
t
v
F
t
Q
0
w
d
disc
disc
disc
W
⋅
−
=
⋅
−
⋅
⋅
=
⋅
=
. (3)
In the case of braking on a flat track 26 time steps, each step 2 seconds long, are
considered.
In the case of braking on a hill, a physical model is used to determine the heat flux
in dependency of the potential energy. The vehicle is maintaining a constant speed of
250 km/h. Consequently the heat flux is constant. The energy is considered to be
equally divided between the 12 discs of the vehicle. The energy balance is:
b
t
Q
h
g
M
⋅
⋅
=
Δ
⋅
⋅
12
,
(4)
With the consideration of trigonometry and constant speed, the brake power for
one disc is:
]
W
[
43711
12
sin
0
=
⋅
⋅
⋅
=
δ
v
g
M
Q
,
(5)
In this case 52 time steps with a constant heat flux are used.
2.4 Force determination for the brake caliper
The surface pressure between the disc and the pad, on behave of the calculated force
applied to the disc, needs to be determined. In the case of braking on a flat track the
pressure is:
]
MPa
[
14
,
1
=
⋅
=
μ
c
disc
A
F
p
.
(6)
In the case of an inclined track the force is:
]
[
233
r
r
v
2
Q
F
d
w
0
disc
N
=
⋅
⋅
⋅
=
μ
. (7)
Surface pressure is:
]
MPa
[
03
,
0
4
,
0
20000
233
=
⋅
=
p
.
(8)
These additional loads are considered in both cases of the analysis.
2.5 Determination of boundary conditions, mesh properties and loads
The calculated heat flux is considered for both sides of the disc. Because the stress is
also analyzed, the disc needs to be properly fixed. The disc is put together rigid where
the disc is screwed onto the hub (Fig. 5).
The forming of the volume mesh is automatic. The mesh is consistent of 84354
tetrahedral elements (designation of the element is C3D4AT – it allows us to perform a
thermo – deformational analysis) each approximately 6 mm big.
99
Fig. 5. The meshed disc section with the load and fixing spot
To perform the analysis the material properties from the table 2 are used.
Heat conductivity – λ [W/mK]
35,2
Density – ρ [kg/m
3
] 7100
Specific heat – c
p
[J/kgK]
515
Module of elasticity – E [MPa]
169000
Poisson number – ν [/]
0,275
Tab. 2. Material properties
3 THE ANALYSIS OF THE RESULTS
3.1 Thermal analysis
The first results show the case of a flat track and a new disc (the initial temperature of
the disc and the surrounding is 50°C). Considering that the disc during braking phases
does not cool down to 50°C a presumption is made that the new temperature of the disc
is 150°C. The temperature of the surrounding is still 50°C.
In case of braking on the flat track the highest temperatures reach up to 174°C for
the new disc. This temperature is reached after a time period of 30 s. Because the heat
flux is decreasing, the temperature falls after 52 s down to 154°C. The cooling ribs and
the place where the disc is bolted to the hub are almost unaffected by the changing
temperatures (Fig. 6, a).
a)
b)
Fig. 6. Temperature fields for the new disc a) and the worn out disc b) braking on a flat track
Q
fixing spot
[°C]
[°C]
100
For the worn out disc with the same load, the maximum temperatures of 211°C are
achieved after 38 s. They appear on areas where the wreath of the disk and the cooling
ribs are not connected. In this case the ribs are heavier exposed to the heat flux because
the disc wreaths are thinner. The temperatures are from 30 – 40 °C higher. After a time
period of 52 s the temperature of the disc reaches 195°C (Fig. 6, b).
For the worn out disc on an inclined track the temperature after 104 s reach 240°C.
The highest values are on the contact surface between the brake disc and brake pad.
Because of the high traveling speed, the temperature of the cooling ribs first fall from
150°C to 125°C. After a while the temperature rise back to 140°C (Fig 7, a).
a)
b)
Fig. 7. Temperature fields for the new disc a) and the worn out disc
b) braking on an inclined track
The temperature for a worn out disc after 104 s are 258°C. The hottest areas appear
on the same spots as they do in the first test – flat track. Beside that, at first the
temperature of the cooling ribs fall from 150°C to 130°C. After a while they rise back
to 147°C (Fig. 7, b).
The area directly beneath the braking pad carries the main burden. This is also the
place where the highest temperatures are achieved. The figures show how the
temperatures toward the hub fall. This information is needed to determine the influence
of the heat flux on the disc.
3.2 Analysis of the stress
Thermal stresses in the disc appear because the temperatures rise. Beside the thermal
stress, the centrifugal load and the holding force of the brake caliper is also considered.
The goal of this analysis is to determine the influence of the centrifugal load in
comparison with the thermal load. The comparison stress is given on von Mises.
In the case of a flat track and considering the centrifugal load, the stresses are
185 MPa. On spots where the thermal stresses are the highest, the value is 110 MPa
(Fig. 8, a).
[°C]
[°C]
101
a)
b)
Fig. 8. Stress field for the new disc a) and the worn out disc b) braking on a flat track
In case of the worn out disc with the same load, the maximum value is 174 MPa.
The maximum values appear on the passage of the holding teeth (Fig. 8, b).
In the case of braking on an inclined track the stress for the new disc reach up to
162 MPa (Fig.9, a).
a)
b)
Fig. 9. Stress field for the new disc a) and the worn out disc b) braking on an inclined track
In the last case of the analyzed worn out disc the stress are 148 MPa (Fig. 9, b).
The maximum values appear on the passage of the holding teeth.
[MPa]
[MPa]
[MPa]
[MPa]
]
MPa
[
185
max
=
σ
]
MPa
[
174
max
=
σ
]
MPa
[
162
max
=
σ
]
MPa
[
148
max
=
σ
102
4 DISCUSSIONS OF THE RESULTS
Of all the cases, the highest temperatures arise from the worn out disc on an inclined
track. In 104 s the temperatures rise to 258°C. The actual braking time is shorter and
amounts to 65 s. The highest allowed temperature of the disc is 350°C (long-term).
From the results we can see that centrifugal loads contribute 10 – 20 MPa of stress,
depended on the model of the disc and the load. The highest comparison stress on von
Mises is 185 MPa – that value is still smaller than the permitted value of 213 MPa,
which considers a safety factor of 1,5 (table 3).
Numerical analyze
T
[°C] σ
ther
[MPa] σ
ther_cent
[MPa]
New disc
174
170
185
Flat track
Worn out disc
211
158
174
New disc
240
141
162
Inclined track
Worn out disc
258
130
148
Tab. 3. The results of numerical analyze
We achieved the highest difference in values by braking on a flat track. The reason
why - there is a greater temperature difference in the first case as it was in the second.
5 CONCLUSIONS
Temperatures and stress in discs under different loads are very high. Although they are
fulfilling the buyer’s requirements for safety, we did not considered shearing forces,
residual stress and the cyclic loads during brake discs lifespan. The results need to be
compared with experimental results, which is also our suggestion for future work.
References:
[1] ABAQUS 6.7.1 – tutorial. 2008.
[2] SIST EN 1563:1998(en). Founding – Spheroidal graphite cast irons. SIST, Ljubljana
1998.
[3] Mackin, T.J.: Thermal cracking in disc brakes. Engineering Failure Analysis (2002), no.
9, str.63-76.
[4] Oder, G.: Determination of non stationary thermal and stress fields in brake discs.
Maribor: Faculty of Mechanical Engineering, 2008.
[5] Reibenschuh, M.: Stress analysis of a brake disc under centrifugal and thermal load.
Maribor: Faculty of Mechanical Engineering, 2008.