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Pedro Alves Costa
Geotechnical Challenges in High Speed Railway Lines:The critical speed issue
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1. Introduction
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1. Introduction
NEW PARADIGM – THE CRITICAL SPEED OF THE TRACK-GROUND SYSTEMS CAN EASILLY BE ACHIEVED!!
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LOAD
Soil-
structure
Quasi-static mechanism
t
t
F
uPuw
F
1. Introduction
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LOAD
Soil-
structure
Quasi-static mechanism
Dynamic mechanism
F
uP
t
t
1. Introduction
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When the train speed reaches the "critical barrier", a radiation of energy
occurs, similar to...
…but it can be even more complex when dealing with geotechnical materials!
1. Introduction
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2. Basics of Critical Speed
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2. Basics of Critical speed
( ) ( )
<<−
×=0
||,||22
1
,0,,byaxctx
batyxpz
δ
( ) ( ) ( )111 ,,0,, ckykfykpz −= ωδω
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2. Basics of Critical speed
Usually the ground is not homogeneous over depth…
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2. Basics of Critical speed
Usually the ground is not homogeneous over depth…
0 0.2 0.4 0.6 0.8 1 1.2 1.40.5
1
1.5
2
M=c/Cs
Co
eficie
nte
s d
e a
mp
lific
ação
din
âm
ica
H=6 mH=3 m
y
x
z
2.0 m
2.0 mc
H=3.0m
E=200 MPa
ν=0.35
ρ=2000 kg/m
ξ=0.03
3
E=50 MPaν=0.35
ρ=2000 kg/m
ξ=0.03
3
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
f (Hz)
k1 (
/m)
M=0.5
M=0.7
M=0.934M=1.0
M=1.5
M=1.9
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3. Track-ground critical speed
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3. Track-ground critical speed
Pzc
xRail: EI , m
Railpad: k , c
Sleepers: mz
Ballast: E , ρ ,h hb b
s
p p
r r
Ground layer 1
Ground layer 2
Ground layer n
Simplified semi-analytical modelling
Sheng, X., C. Jones, and D. Thompson, A theoretical study on the influence of the track on train-induced ground vibration. Journal of Sound and Vibration, 2004. 272: p. 909-936.Costa, P. A., A. Colaço, R. Calçada and A. S. Cardoso 2015. Critical speed of railway tracks. Detailed and simplified approaches. Transportation Geotechnics 2: 30-46.
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3. Track-ground critical speed
Simplified semi-analytical modelling
V1
(ballasted track)
V2
(slab track)
V3
(slab track)
Rail UIC60 UIC60 UIC60
msleeper (kg/m) 490 - -
hballast/slab (m) 0.35 0.35 0.44
Ebal/lastlslab (Pa) 130e6 30e9 30e9
ρρρρ ballast/lslab (kg/m3) 1700 2500 1990
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3. Track-ground critical speed
Homogeneous ground
0.2 0.4 0.6 0.8 1 1.21
1.5
2
2.5
3
3.5
4
4.5
5
M=c/Cs
FD
A-v
ert
ica
l d
isp
lace
me
nt
Cr
Ballasted track
Slab track - V1
Slab track
(V2)
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
f (Hz)
k1 (
/m)
P-SV mode
(GROUND)
Ballasted track
Slab track - V2
Slab track
(V3)
The critical speed of the system is fully dominated by the properties of the
ground.
Costa, P. A., A. Colaço, R. Calçada and A. S. Cardoso 2015. Critical speed of railway tracks. Detailed and simplified approaches. Transportation Geotechnics 2: 30-46.Mezher, S. B., D. P. Connolly, P. K. Woodward, O. Laghrouche, J. Pombo and P. A. Costa. 2016. Railway critical velocity - Analytical prediction and analysis. Transportation Geotechnics 6: 84-96.
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3. Track-ground critical speed
Non-homogeneous ground
The critical speed of the system is affected by the stiffness of the track.
The analysis of the critical speed can be performed by simplified methods.
0.5 1 1.5 20.8
1
1.2
1.4
1.6
1.8
2
M=c/Cs
FDA
- v
ert
ica
l dis
pla
cem
en
t
Ballasted trackSlab track Slab track
(V3)
0 10 20 300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
f (Hz)
k1 (
/m) Ballasted track
P-SV mode
(ground)
Slab track
Slab track
(V3)
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4. Track-embankment-ground critical speed
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4. Track-embankment-ground critical speed
Numerical modelling
CL2.5D FEM
ITM
Ground
2.5 BEM
( ) ( )ω=ωω+ω−+++ ,kp~,ku~)),k(KMKkKkKikK( 1n1n1global5
global2global4
41
global3
21
global21
global1
Regular geometry – 2.5D BEM
Complex geometry – 2.5D FEMAlves Costa, P., R. Calçada, and A. Cardoso, Track–ground vibrations induced by railway traffic: In-situ measurements and validation of a 2.5D FEM-BEM model. Soil Dynamics and Earthquake Engineering, 2012. 32: p. 111-128.
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4. Track-embankment-ground critical speed
Numerical modelling – the role of embankment properties
Continuos slabtrack
Vs={220;300} m/s
ρ=2000 kg/m3ν=0,35
ξ=0,03
Vs={185;220} m/s
ρ=2000 kg/m3
ν=0,35
ξ=0,03
Embankment 2
Embankment 1 h={1;2;3}m
h=2 m
Vs=variable
ρ=1900 kg/m3ν=0.45ξ=0.03In
-situ
so
il
-20 m
150 m/s
480 m/s
-3 m
Shear wave
velocity
z
AnalysisVS,Embankment1
(m/s)
VS,Embankment2
(m/s)
hEmbankment1
(m)
A1 220 185 1
A2 220 185 2
A3 220 185 3
A4 300 220 1
A5 300 220 2
A6 300 220 3
Does embankmentproperties affect the criticalspeed value?
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4. Track-embankment-ground critical speed
Numerical modelling – the role of embankment properties
AnalysisVS,Embankment1
(m/s)
VS,Embankment2
(m/s)
hEmbankment1
(m)
A1 220 185 1
10 m/s160 m/s176 m/s195 m/s
Due to the track-embankment properties the critical speed can be larger
than the shear wave velocity in the shallow layer of the ground
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4. Track-embankment-ground critical speed
Numerical modelling – the role of embankment properties
AnalysisVS,Embankment1
(m/s)
VS,Embankment2
(m/s)
hEmbankment1
(m)
A1 220 185 1
A2 220 185 2
A3 220 185 3
Analysis
Critical speed (m/s)
Error (%)Direct
approach
Simplified
approach
A1 176 164 -6.0
A2 181 173 -4.4
A3 182 181 -0.5
http://vcritical.simpleaxis.com/
Preliminiary assessement of
critical speed in few seconds
Colaço, A.; Alves Costa, P.; Geotechnical challenges in very high speed railway tracks. The
numerical modelling of critical speed issues. NUMGE 2018
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4. Track-embankment-ground critical speed
Numerical modelling – the role of embankment properties
AnalysisVS,Embankment1
(m/s)
VS,Embankment2
(m/s)
hEmbankment1
(m)
A4 300 220 1
A5 300 220 2
A6 300 220 3
Analysis
Critical speed (m/s)
Error (%)Direct
approach
Simplified
approach
A4 195 188 -3.6
A5 204 199 -2.4
A6 206 201 -2.4
Embankment
height
Embankment
stiffness
Critical
Speed
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5. The critical speed effects
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5. The effects
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5. The effects
Large rail displacements
Risk of derailment
Large strains and stresses in the
subgrade
Colaço, A., P. Alves Costa and P. Lopes, Dynamicground stress‐path evolution due to the railwaytraffic: A parametric study. Revista Internacional de Métodos Numéricos para Cálculo y Diseño enIngeniería, 2015. 31(2): p. 120-131.
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Large rail displacements
Risk of derailment
Large strains and stresses in the
subgrade
Colaço, A., P. Alves Costa and P. Lopes, Dynamic ground stress‐path evolution due to therailway traffic: A parametric study. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 2015. 31(2): p. 120-131.
Shakedown limit can be achieved
Accumulation of plastic strains
Track deterioration
5. The effects
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5. The effects
The critical speed and the shakedown limit
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5. The effects
The critical speed and the shakedown limit
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5. The effects
The critical speed and the shakedown limit
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5. The effects
The critical speed and the shakedown limit
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5. The effects
0.25
0.30
Slab: E= 30 GPa
ρ=2500 kg/m
ν=0.2; ξ=0.001
3
PCL: E= 5 GPa ρ=2500 kg/m
ν=0.2; ξ=0.001
3
Rail: UIC60
Railpad: k=40 kN/mm
c=8 kN.s/m
Ground: Cs=120 m/s
ρ=1900 kg/m ν=0.3; ξ=0.03
K =0.5
3
0
2.60
0 50 100 1500.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Velocity (m/s)
DA
F (
-)
Original scenario
Without poor
concrete layer
Critical
speed
20 40 60 80 100 1200
500
1000
1500
2000
2500
Velocity (m/s)
����
�
Original scenario
without poor concrete layer
Service speed should be around
0.6-0.7 of the critical speed to avoid
strong reduction of shakedown limit
VcrThe critical speed and the shakedown limit
Alves Costa, P.; Lopes, P. Silva Cardoso, A. Soil shakedown analysis of slab railway tracks:
numerical approach and parametric study. (2018) Computers and Geotechnics (submitted)
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5. Case study - Ledsgard
Alves Costa, P., et al., Influence of soil non-linearity on the dynamic response of high-speed railway tracks.Soil Dynamics and Earthquake Engineering, 2010. 30(4): p. 221-235.
Alves Costa, P., R. Calçada, and A. Silva Cardoso, Track-ground vibrations induced by railway traffic, in
Applications of Computational Mechanics in Geotechnical Engineering,, L. Sousa, et al., Editors. 2012,
Balkema.
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5. The effectsLedsgard - General description
2. Numerical Model
3,352,502,702,502,702,503,35
1,35
Via oriental
Travessase=0,67 m
Carris UIC60
Via ocidental
Sentido Norte -Sul
Camadas Granulares
Maciço de Fundação
10-4
10-3
10-2
10-1
100
101
0
0.2
0.4
0.6
0.8
1
Distorção (%)G
sec/
Gm
ax
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Experimental results
0 5 10 15 20 25 30 35 40-15
-10
-5
0
5
10
15
Tempo (s)
De
slo
cam
en
to v
ert
ica
l (m
m)
0 1 2 3 4-15
-10
-5
0
5
10
15
Tempo (s)
De
slo
cam
en
to v
ert
ica
l (m
m)
0 50 100 150 200-15
-10
-5
0
5
10
15
Velocidade de circulação (km/h)
De
slo
cam
en
to d
e p
ico
(m
m)
Train speed (km/h)
Pe
ak v
ert
ica
l dis
pla
ce
me
nt (m
m)
Ve
rtic
al d
isp
lace
me
nt (m
m)
Ve
rtic
al d
isp
lace
me
nt (m
m)
5. The effectsLedsgard - Measurements
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Ledsgard – Elastic numerical modelling
5. The effects
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A
B
C
G Gm ax s ec
1 1
Shear strain
Shear stress
D amping
Strains 10-6 10-5 10-4 10-3 10-2 10-1
Small Medium High Failure
Elastic
Elastoplatic
Effect of
cyclic loading
Analysis
approach
Linear
elastic
Equivalent
linear analysis
Non-linear analysis
Soil behaviour is clearly non-linear;
Demand of new prediction
approaches
The critical speed is dependent on
train properties;
5. The effectsLedsgard – Non linear behaviour
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5. Case study - Ledsgard
0 1 2 3 4-15
-10
-5
0
5
10
15
Tempo (s)
De
slo
ca
me
nto
ve
rtic
al (
mm
)
0 50 100 150 200 250-20
-15
-10
-5
0
5
10
15
Train speed (km/h)
Pe
ak
dis
pla
cem
en
t (m
m)
Ledsgard – Linear equivalent analysis
Time (s)
Time (s)
Vert
ical d
ispla
cem
ent (m
m)
Vert
ical d
ispla
cem
ent (m
m)
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6. Some mitigation solutions
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Wittenberge (High speed railway line Hamburg – Berlin, Germany):
6. Some mitigation solutionsStone columns
• Stone columns
• Diameter: 0.6m to 0.8m
• Spacing: 2.0m x 1.25m
[1] V.R. Raju – “Ground improvement techniques for railway embankments”,
Keller Grundbau GmbH, Technical Paper 10-59E
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6. Some mitigation solutionsStone columns
Aterro
Sub-balastroBalastro
TravessaCarris
[m]
1,6 1,6 1,6 1,6 1,4 1,4 1,6 1,61,61,61,61,41,41,6
21,6
Colunas de brita
(Diâmetro = 0,8 m)
E [MPa] ρ [kg/m3] ν [-] D [-] Cs [m/s]
160 2000 0,3 0,03 175
Stone columns
Embankment
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Cs = 100 m/s 12 m
Cs = 200 m/s
2 mCs = 178 m/s
5. Some mitigation solutionsStone columns
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Velocidade de circulação [m/s]
Fa
cto
r d
e A
mp
lific
açã
o D
inâ
mic
a [
-]
sem colunas
com colunas (E=160MPa)com colunas (E=240MPa)
com colunas (E=320MPa)
Load speed (m/s)
Dyn
am
ica
mp
lifica
tio
nfa
cto
r (-
)
Reference
scenario
Stone columns
E=160 MPa
Stone columns
E=240 MPa
Stone columns
E=320 MPa
In layered ground, installation of stone columns allows to increase the critical
speed
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6. Some mitigation solutions
Cutter-Soil-Mixing (CSM)
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6. Some mitigation solutions
Embankment reinforced with geogrids and deep soil mixing columns
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1,6 1,6 1,6 1,6 1,4 1,4 1,6 1,61,61,61,61,41,41,6
21,6
Aterro
Sub-balastroBalastro
TravessaCarris
Barretas de DeepSoil Mixing
(Largura = 0,8 m)
[m]
6. Some mitigation solutions
Cutter-Soil-Mixing (CSM)
E [MPa] ρ [kg/m3] ν [-] D [-] Cs [m/s]
650 2000 0,3 0,03 354
1000 2000 0,3 0,03 439
CSM panels
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Cs = 100 m/s 12 m
Cs = 200 m/s
2 mCs = 178 m/s
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Velocidade de circulação [m/s]
Fa
cto
r d
e A
mp
lific
aç
ão
Din
âm
ica
[-]
sem barretas
com barretas (E = 650 MPa)
com barretas (E = 1000 MPa)
Dyn
am
ica
mp
lifica
tio
nfa
cto
r (-
)
Load speed (m/s)
6. Some mitigation solutions
Cutter-Soil-Mixing (CSM)
Large increase of the critical speed
Critical speed is governed by the embankment properties
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7. Conclusions
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7. Conclusions
The problem of “critical speed” due to moving loads was revisited.
It was shown that the critical speed is equal to the lower phase velocity of the
system track-embankment-ground.
A simplified approach for the assessment of the critical speed of elastic linear
systems is proposed. This simplified approach can very useful for scoping
analysis, in order to identify the railway stretches that should be submitted to
detailed analysis.
It was shown that the critical speed should be assessed taking into account
the particular site conditions and not based in prescriptive rules. Critical
speed is governed by the ground stiffness and layering, embankment
properties and track type.
.
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7. Conclusions
It was shown that when the train speed becomes close to the critical speed,
the strains experienced in the ground are not compatible with a linear
approach and that a strong reduction of the elastic shakedown limit load
occurs.
A numerical approach based on the 2.5D FEM, where the ground is
simulated by an equivalent linear model, is proposed as a suitable method
for the assessment of vibrations induced by very high-speed traffic.
Some measures to increase critical speed were presented and discussed. It
was shown that 2.5D approach is a suitable method for the support on the
design of new high-speed railway lines.
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Thank you for your attention Pedro Alves Costa____________________Assistant ProfessorFaculty of EngineeringUniversity of PortoPortugal
Email: [email protected]: +351 962934245