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MIDAS UK Expert Engineer Webinar Series
Dynamic Analysis of
High Speed Two (HS2)
Pere Alfaras,
Principal Bridge Engineer
ARCADIS UK
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1. About
2. Introduction to the problem
3. Eurocode requirements
4. Stiffness & Mass
5. Eigenvalue analysis
6. Time history analysis
7. Results interpretation
8. Conclusion
CONTENTS
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1. About
About myself
About Arcadis
About HS2
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About myself
BLABLABLA
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Pere Alfaras
→ Principal Bridge Engineer
→ Arcadis UK
→ 10+ years of experience
→ Bridge and Structural design
→ Lecturer / Finite Element Method
@ UPC/BarcelonaTech for 5 years
Lower Thames Crossing - UK High Speed Two (HS2) - UK
Jeddah Metro - KSA Abi Bakr Al-Siddiq Highway - KSA
High Speed Line Warsaw-Poznan-
Wroclaw - Poland “Eix Diagonal” Motorway - Spain
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About Arcadis
→ Global Design & Consultancy
→ Natural and Built Assets
→ NCE100 Company of the Year
→ We are 27,000 people
→ Over 300 offices
→ Active in over 70 countries
→ €3.2 billion in revenues
→ Extensive bridge expertise
→ UK based team circa 100 staff
→ www.arcadis.com
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Millau Viaduct - France Tower Bridge, London - UK
Albert Bridge, London - UK Rotherhithe Brige, London - UK
Te Wero Bridge – New Zealand Vasco de Gama Bridge - Portugal
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About HS2
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Overall budget: £55.7bn
→ Design JV: Arcadis, Setec & COWI
→ Consortium with Eiffage Kier JV
→ Scope: packages C2 & C3
→ 80km of railway line
→ 86 Bridges
→ 14 Viaducts (3.6km total)
→ 3 green tunnels (5.0km total)
→ Civils works complete by 2022
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2. Introduction to the problem
Background
Resonance and dynamic magnification
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Background
Classic code requirements
First French HSL: Paris-Lyon
ERRI D214 Committee studies
→ Static Analysis with Dynamic amplification factor
→ Following issues were observed:
• Resonance phenomena
• Ballast degradation
• Rapid track deterioration
• Short-span structures specially affected
→ Concluded that for speeds over 200 km/h:
• Likelihood of resonance effects
• Dynamic amplification factor unable to predict resonance
• Deck acceleration must be assessed
→ Established rules for dynamic assessment - now implemented in
Eurocodes
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Resonance and dynamic magnification
Ballasted Vs Ballastless tracks
Simply supported Vs Continuous
Resonant speed
→ Ballast grains loose its grain interlock when a > 0.7g
→ Ballastless tracks wheel-rail contact is reduced beyond
acceptable limits when a > g
→ Single-span structures specially susceptible to resonance
→ Resonance effects are significantly reduced on continuous
structures
→ Resonance speed usually 200km/h < v < design speed
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3. Eurocode requirements
Dynamic amplification factor
Requirements for a static or dynamic analysis
Acceleration check
Load models, combinations, design speed
Structural damping
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Dynamic amplification factor
If dynamic analysis not required
If dynamic analysis required
Ф x (LM71”+”SW/0)
• Ф depends on track irregularities and determinant length LФ
Most unfavourable value of:
Ф x (LM71”+”SW/0)
or
1 + 𝜑′𝑑𝑦𝑛 + Τ𝜑′′ 2 𝑥
𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇
+ Acceleration check
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Maximum dynamic response
Increase resulting fromtrack defects andvehicle imperfections
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Is a dynamic analysis required? (simple structures)
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Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
NA to BS EN 1991-2:2003 Figure NA.12
Figure NA.14
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Acceleration check
Maximum peak values
[EN 1990-2002 A2.4.4.2.1]
→ To ensure traffic safety, the Eurocodes recommend:
- bt = 3.5 m/s2 for ballasted track (ballast stability)
- df = 5.0 m/s2 for ballast-less track (wheel-rail contact)
→ EN 1990-2002 UK Annex: The maximum peak values of bridge
deck acceleration and the associated frequency limits should be
determined for the individual project.
→ Passenger comfort criteria is covered elsewhere in the code
(EN 1990-2002 A2.4.4.3.1)
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4. Stiffness & Mass
Stiffness
Mass
Mass control parameter
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Stiffness
Bridge stiffness
Young’s modulus
Shear deformation
Cracked stiffness
→ Any overestimation of bridge stiffness will overestimate the
natural frequency of the structure and speed at which resonance
occurs
→ A lower bound estimate of the stiffness throughout the structure
shall be used
→ Short term concrete elastic modulus for concrete elements
→ Should be considered
→ Assessment of cracked stiffness is essential, since a reduced
cracked stiffness lead to lower fundamental frequencies hence
lower resonant speeds
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Mass
Upper and lower bound estimates
of mass
Self-weight
Ballast
Other superimposed loads
→ a lower bound estimate to predict maximum deck accelerations
→ an upper bound estimate of mass to predict the lowest speeds at
which resonant effects are likely to occur
→ According to EN 1991-1-1 (enhanced density values may be
used if confirmed via testing and approved by relevant authority)
→ minimum likely dry clean density and minimum thickness of
ballast
→ maximum saturated density of dirty ballast with allowance for
future track lifts
→ rails, sleepers, parapets, OLE, others
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Case Study – General Arrangement
Span arrangement
Cross-section
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Example – Is a dynamic analysis required?
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Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
Figure NA.14Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
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5. Eigenvalue analysis
Frequencies to be considered
Mass participation factors
Bending and torsional modes
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Eigenvalue analysis
Frequencies to be considered
[BS EN 1990-2002 A2.4.4.2.1]
Bending and torsional modes
Mass participation factors
Up to the greater of:
→ 30 Hz
→ 1,5 times the frequency of the fundamental mode of vibration of
the member being considered
→ The frequency of the third mode of vibration of the member
→ Need to be identified to assess n0 and nT
→ Can be used to identify the relevant modes
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Case Study – Eigenvalue analysis results
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Displacement Rotation
Mode No Frequency X Y Z X Y Z
(Hz) mpm (%) mpm (%) mpm (%) mpm (%) mpm (%) mpm (%)
1 6.14 0 0 82.67 0 0 0
2 16.49 0 82.67 0 0 0 0
3 20.04 0 0 0 82.82 0 0
4 20.56 0 0 0 0 0 0
5 28.69 81.91 0 0 0 0 0
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Case Study – Is a dynamic analysis required?
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Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
30
6.1nT > 1.2n0
Is Skew < 15
May use the eigenforms for bending only
Dynamic analysis required
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6. Time history analysis
Time step
Structural Damping
Train Load Models
Model input
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Setting up the Time History Analysis
Linear or Non-linear?
Modal or Direct Integration?
Transient or Periodic?
→ Generally structural behaviour within linear range
→ Modal integration (modal superposition method) should
generally be used with the first modes of the structure (in
accordance to BS EN 1990-2002 A2.4.4.2.1)
→ This is a transient problem
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Time step
ERRI D214 (e), 1999 → recommends to choose a time step not greater than:
where:
𝑓𝑚𝑎𝑥 : maximum frequency used on the modal analysis;
𝐿𝑚𝑖𝑛: minimum span;
𝑛: number of modes used on the modal analysis;
𝑣: train speed.
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ℎ1 =1
8𝑓𝑚𝑎𝑥ℎ2 =
𝐿𝑚𝑖𝑛
200𝑣ℎ3 =
𝐿𝑚𝑖𝑛
4𝑛𝑣ℎ4 = 0.001𝑠
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
0 ms 50 ms 100 ms 150 ms 200 ms 250 ms 300 ms 350 ms
Am
plit
ude
Δt=5ms
Δt=40ms
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Structural damping
Eurocode recommendations
[BS EN 1991-2:2003 6.4.6.3]
→ Recommended damping values
→ Additional damping: TOTAL= +
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Train Load Models
Load models for the acceleration
check and dynamic factor
[BS EN 1991-2:2003 6.4.6.1]
Fatigue loads
[BS EN 1991-2:2003 Annex D]
Speeds to be considered
[BS EN 1991-2:2003 6.4.6.2]
→ HSLM-A: for spans over 7m or complex structures
10 variations (A1 to A10)
→ HSLM-B: for simple structures with spans less than 7m
→ Real train
→ 12 train types
→ traffic mixes
→ 40 m/s vi 1,2 x Maximum Line Speed
→ Reduced speed steps in the vicinity of resonant speeds
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Dynamic nodal loads
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How to transform a moving load to dynamic loads using time functions:
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0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Load [kN
]
Time [s]
Dynamic nodal loads
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0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Load [kN
]
Time [s]
How to transform a series of moving loads to a time function:
delay
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7. Results interpretation
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Graphical outputs
Time History Graph
Fourier transform
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Case Study - Graphical outputs
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Envelope results - to identify the critical locations on the deck
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Case Study -Acceleration Time History
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Time Domain Response – to ensure that the critical time has been captured
Free vibrationForced excitation
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Case Study -Acceleration Response Spectrum
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Frequency Domain Response - to identify critical modes/frequencies
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Additional Example – Continuous Structure
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Animation may help to spot irregularities
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Case Study -Acceleration check
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Peak values must be plotted against speeds to identify resonant/critical speeds.
0
0.5
1
1.5
2
2.5
3
3.5
140 190 240 290 340 390
Peak
acc
ele
ration [m
/s2]
Train speed [km/h]
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
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Case Study - Dynamic amplification factor
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-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Deflect
ion a
t m
idsp
an [m
m]
0
2000
4000
6000
8000
10000
12000
14000
16000
140 165 190 215 240 265 290 315 340 365 390 415
Bendin
g M
om
ent [k
Nm
]
Train speed
[km/h]
Ф x (LM71”+”SW/0)
1 + 𝜑′𝑑𝑦𝑛 + Τ𝜑′′ 2 𝑥
𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇
→ Dynamic responses of all deck
members must be checked and
compared to the equivalent static
responses
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8. Conclusion
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Conclusion
Resonance and dynamic
magnification
Resonant speed
Model properties
Analysis
Result interpretation
→ It is relevant for speeds over 200 km/h
→ Short span structures are particularly prone to resonance
→ Difficult to anticipate the resonant speeds for most structures
→ A dynamic analysis is required to assess acceleration and
dynamic amplification factor for a range of speeds
→ Bridge stiffness and mass have to be carefully assessed
→ Upper and lower bounds must be considered
→ Requires numerous time history cases, which is time consuming
→ Vital to ensure accurate results
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