EURODYN 2008 – Southampton, July 7-9

Dynamic analysis for fatigue safety examination of existing

short span concrete railway bridges

Mário Pimentel
Joaquim Figueiras
Eugen Brühwiler

University of Porto, Faculty of Engineering

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

Index

1. Introduction
2. Mathematical model
3. Validation
4. Parametric study
5. Conclusions

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

1. Introduction
• Modern railway systems sustainability demands for larger traffic volumes, higher
traffic loads and higher trains speeds;
• These issues are already reflected in recent studies of the ERRI regarding the
formulation of a new load model, the LM2000;
• For the design of new railway bridges in European international lines, the Eurocode 1
already recommends the adoption of a classification factor of a=1.33 for the LM71 and
SW/0 load models;
• The existing bridge stock also needs to be checked against these demanding loading
conditions and decisions must be taken concerning their upgrade to withstand the new
loading scenario (axle loads up to 30ton);
• However, the economical impact of strengthening measures is much larger than the
additional cost due to a conservative design;
• It can be then understood that, in the case of existing bridges, besides new traffic
models, also the development of methodologies enabling the exploitation of
hidden reserves is mandatory.

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

1. Introduction
• In a scenario of increasing maximum axle loads the fatigue life of short span
reinforced concrete bridges is drastically reduced:

500 25
Fatigue life, FL (years)

FL(33t) / FL(25t) (%)

Heavy traffic mix
400 20

10%
33t traffic mix
300 15

200 10

100 5

0 0
4 5 6 7 8 9 10 11 4 5 6 7 8 9 10 11
Span (m) Span (m)

Fatigue life of simply supported railway RC Ratio between the fatigue lives with the
short span bridges under different traffic 25ton and 33ton traffic scenarios
scenarios

• Reinforcement fatigue may become a conditioning limit state.

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

1. Introduction
• Fatigue life is very sensitive to some underlying assumptions usually adopted in the
fatigue verification procedure:
250

% of reference fatigue life

L=10.0m
200
L=7.5m
Precision factor: 150
L=5.0m

x= Dss,real/Dss,calc 100

50

0
0.85 0.9 0.95 1 1.05 1.1 1.15
Precision factor, x
• Looking only to the “action side” of the problem, several aspects contribute to the
differences between the real and calculated stress amplitude spectra:
1. Continuous rails over the supports may provide significant partial fixity;
2. Load distribution through the rail/sleepers/ballast is relevant when analysing
short span bridges;
3. Real axle loads and axle spacing can provide significantly different stress
amplitude spectra than obtained with the load models;
4. Dynamic amplification of traffic effects can be different from the code DAF’s.

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

2. Mathematical model
• Bridge

Equations of motion: Pl(t)

 2 w x, t  w x, t 
Lw x, t   m  x   Cx   pt  vl*tl
Pl+1(t)
t 2
t dx,l
dy,l vl+1*tl+1

P t   x  vx tl   y  v y tl 
Nl dx,l+1
pt    l rect  dx dy,l+1

l 1 d x ,l d y ,l
R  d x ,l
rect
  dy 
   ,l 

Uncoupled equations of motion – Modal decomposition technique:


w x, t    Y t   x 
n n
n 1

Yk  2 x k k Yk  k2 Yk  Pb,k  t  , k  1,..., 

Nl
Pb,k  t     v t  P  t 
med
k l l
l 1

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges
M=293ton
2. Mathematical model M(t)
L=26m
f1=5.9Hz
• Bridge V(t) w(t) v=144Km/h
1.2E-03

Quasi-static correction procedure: 1.0E-03 Static

1 Mode
8.0E-04
3 Modes

P  t 
6.0E-04 1 Mode + Correct.
N mod e
 Nl

w (m)
 x, t    A ,k x  Yk  t   b,k 2    x, v t  P  t  w(t)
4.0E-04 FEM - DIANA

k 
l l

2.0E-04
k 1 l 1
0.0E+00
0 0.2 0.4 0.6 0.8 1
-2.0E-04

  x, t  - dynamic effect under consideration

1600
-4.0E-04
1400
Static
t (s)
(shear force, bending moment, 1200
1000
1 Mode
3 Modes
displacement, strain component, etc.) 800
21 Modes

M (kN.m)
1 Mode + Correct.
600 FEM - DIANA
M(t)
A ,k  x  - contribution to   x, t  due to a unit
400
200
displacement in the kth mode 0
-200 0 0.2 0.4 0.6 0.8 1
250
-400
V(t)
  x, v tl  - static influence field (or influence
t (s)
200 Static
1 Mode

line) of the effect   x, t  150 3 Modes

30 Modes

V (kN)
100 3 Modes + Correct.
FEM - DIANA
50

0
0 0.2 0.4 0.6 0.8 1
-50
t (s)

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

2. Mathematical model
• Vehicles – Freight wagons

Running gear Linearized models

Vehicle equations of motion

derived using the Lagrange’s
formulation

d  T  T V WD
   0
dt  uv ,i  uv ,i uv ,i uv ,i

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

2. Mathematical model
• Track

Simplified formulation (acceptable if the track response as well as the high

frequency content of the wheel/rail contact force is not of interest):

Contribution to the partial fixity at

the supports can be taken into
account, if appropriate, in the
bridge finite element model, and
consequently in the bridge modal
shapes and influence fields

• Time dependent contact force exerted by the lth axle:

P l t   kt ,l uv,l t   r v t l   wv t l  ct ,i u v,l t   rv t l   w

 v t l   0, l  1,..., N l

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

2. Mathematical model
Bridge equations of motion (NmodexNmode) Vehicle equations of motion (Nv,dofxNv,dof)
  C Y
Mb Y   K Y  P t   v  Cv u v  K v u v  Pv t 
Mv u
b b b

Can be expressed in terms of the system unknowns Yk, uv,l and its derivates

Coupled equations of motion [(Nmode+Nv,dof)x(Nmode+Nv,dof)]

Mb 0   Y  Cb  C bb Cbv   Y   K b  K bb K bv   Y  Fb 

 0        
 M v  u
v   C vb C v  u v   K vb K v  uv  Fv 

Time dependent coupling matrices

No iterative procedure required;

Small size of the system of equations;
 Efficient calculation method

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

2. Mathematical model
• Computational implementation
 Currently the model is implemented in Matlab;

 The coupled system of differential equations is solved with the fourth order
Runge-Kutta method and an automatic adaptative time stepping scheme;

 An event driven integration scheme is adopted to deal with situations where loss
of contact between the wheels and the bridge is detected. If contact is lost, the
integration is stopped, the contact parameters adjusted, the integration
restarted and vice-versa (automatic procedure).

 The model needs the input of the bridge modal parameters and influence fields
(or influence lines). In simple cases these can be analytically derived. They can
also be most generally obtained from detailed (and eventually updated) finite
element models.

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

3. Validation
• Simple bridge travelled by a two axle system with a large wheel flat in the
rear axle
3 modes included in the response;
Rounded wheel flat modelled as a analytical sinusoidal function;
Axles excited before entering the bridge by imposing initial
conditions in the vehicle degrees of freedom;
Bridge, vehicle and wheel flat properties according to Fryba (1999)
The wheel flat “hits” the deck
The wheel flat in approximately at midspan
The first axle the rear axle “hits”
enters the bridge the deck

Proposed model
Fryba (1999)

Midspan bending moment

Contact force in the rear axle

Proposed model
Fryba (1999)

Midspan displacement

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

4. Preliminary parametric study

• Model parameters

Simply supported RC bridges Freight train

L=10m L=5m Freight train composed
by 3 carriages with two
f1=12Hz f1=22Hz axles each

mb=8.3t/m mb=7.0t/m
20t axle loads: mc=38t, mw=1t, Ic=7mc
x=2.2% x=2.55% 30t axle loads: mc=58t, mw=1t, Ic=7mc

Suspension properties Track

Equivalent viscous damping ratio: 2.8 •No damping;
2.6
x = 5% – 10% 2.4 •Linearized spring:
Frequency fv1 [Hz]

2.2Hz
2.2
2.0Hz
2
1.8Hz kv = 80MN/m; 160MN/m.
Linearized spring stiffness: 1.8
1.6
fv1 = 1.8Hz-2.0Hz-2.2Hz 1.4
1.2
Body bounce frequency 1
100 150 200 250 300
Max. Axle Load [kN]
Body bounce frequencies for loaded
wagons

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

4. Preliminary parametric study

• Irregularity types
1  2 
Type 1 - Large track depression: Lrr = 3.0m; Arr = 6mm
r x   Arr 1  cos x 
2  Lrr 
Lrr
Type 2 - Medium track depression: Lrr = 1.0m; Arr = 2mm
Arr
Type 3 – Worn rail head identation: Lrr = 0.1m; Arr = 1.3mm

Type 4 - 10 random profiles generated according to

the FRA PSD function for class 4 tracks:

Grr  f rr ,i Df rr cos2 f rr ,i x   i 

N harm
r x   2
i 1

• Dynamic amplification

Max  dyn 
DAF  1   
Max  sta 

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

4. Preliminary parametric study

• Varying axle load and irregularity type

1.4 1.4
20t - Irreg. Type 1 20t - Irreg. Type 3
1.3 30t 1.3 30t

Amplification (1 + )
Amplification (1 + )

20t - Irreg. Type 2 20t - Irreg. Type 4

30t EC1 fatigue DAF •10m span bridge;
1.2 1.2 30t
• xv=10%;
1.1 1.1
• fv1=2Hz;
1 1
Moving Loads Moving Loads • kv = 80MN/m.
0.9 0.9
40 60 80 100 120 140 40 60 80 100 120 140
Speed [km/h] Speed [km/h]

1.4 1.4
20t - Irreg. Type 1 20t - Irreg. Type 3
30t 30t EC1 fatigue DAF
1.3 1.3
Amplification (1 + )

Amplification (1 + )

20t - Irreg. Type 2 20t - Irreg. Type 4

30t 30t
•5m span bridge;
1.2 1.2
• xv=10%;
1.1 1.1
• fv1=2Hz;
1 Moving Loads 1
Moving Loads • kv = 80MN/m.
0.9 0.9
40 60 80 100 120 140 40 60 80 100 120 140
Speed [km/h] Speed [km/h]

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

4. Preliminary parametric study

• Varying track stiffness
1.4
kt=80MN/m - Irreg. Type 3
kt=160MN/m EC1 fatigue DAF
1.3
Amplification (1 + )

kt=80MN/m - Irreg. Type 4 •5m span bridge;

kt=160MN/m
1.2
• xv=10%;
1.1 • fv1=2Hz;
1 • 30t axles
0.9
40 60 80 100 120 140
Speed [km/h]

• Dispersion of the DAF values

1.4
kt=160MN/m - Irreg.Type 4
EC1 fatigue DAF
1.3 •5m span bridge;
Amplification (1 + )

Max
1.2 • xv=10%;
1.1 Mean • fv1=2Hz;

1 • 30t axles
Min
0.9
40 60 80 100 120 140
Speed [km/h]

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

5. Conclusions

• The proposed algorithm proved highly efficient, requiring very low calculation times;
• The model is appropriate for performing statistical analyses and parametric studies
requiring a large number of calculations as well as for a detailed 3D dynamic analysis
of a specific bridge;
• The undertaken parametric study, although of limited extension, revealed that the
EC1 DAF for fatigue verifications might be conservative and revealed a trend for the
DAF to decrease with increasing axle loads;

Ongoing work/Research needs

• Field tests are underway to validate some modelling assumptions;
• A consistent definition of a DAF for fatigue verifications is required;
• Definition of mean and standard deviation values of wheel defects;
• Extend the parametric study.

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering
Dynamic analysis for fatigue safety examination of existing short span concrete railway bridges

END

Laboratory for the Concrete Technology and Structural Behaviour

University of Porto, Faculty of Engineering