Dynamic Behavior of Cable Supported Bridges … are developed for cable supported bridges based on both suspension and cable-stayed configurations, ...
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COMSOL CONFERENCE EUROPE Milan, 10-12 October
Department of Structural Engineering University of Calabria, Via P. Bucci, Cubo39-B, 87030, Rende, Cosenza, Italy
Lonetti P., Pascuzzo A., Sarubbo R.
Dynamic Behavior of Cable Supported Bridges Affected by Corrosion Mechanisms under
Moving Loads
Excerpt from the Proceedings of the 2012 COMSOL Conference in Milan
INTRODUCTION TO LONG SPAN CABLE SUPPORTED BRIDGE
Suspended Bridge Cable-Stayed Bridge
Long slender structures
High dynamic amplification effects on the structure are expected
TYPES OF BRIDGES
KEY FEATURES AND STRUCTURAL PROBLEMS
Initial Configuration: Specific initial stresses in the suspension system to ensure that the deck stays in the undeformed configuration during the application of the dead loads
Several damage phenomena, which produce a reduction of the mechanical properties of the bridge constituents
Live loads are comparable with the dead load
MOTIVATION AND SUMMARY OF THE WORK
AIM OF THE WORK
I n v e s t i g a t e t h e i n f l u e n c e o n c a b l e s u p p o r t e d b r i d g e s t r u c t u r e s o f c o r r o s i o n m e c h a n i s m s i n t h e c a b l e - s t a y e d
a n d s u s p e n s i o n s y s t e m s
SUMMARY
Review the main equations of the bridge in a dynamic framework
Develop the finite element implementation and a parametric study to quantify numerically the dynamic amplification effects produced by the moving loads for the cases of damaged and undamaged structures
Analyze the structural behavior of cable system reproducing local vibration effects, by means of a geometric non-linear approach and an explicit damage law for the corrosion mechanisms.
Reproduce accurately the inertial description of the moving loads including non-standard forces produced relative motion with the girder
BRIDGE FORMULATION AND ASSUMPTIONS
OBJECTIVES AND ASSUMPTIONS OF THE MODEL
Dynamic behavior and local vibration effects of the cable system
Moving loads and girder deformation
Simulation of the damage mechanisms in the cable system
INITIAL CONFIGURATION OF THE BRIDGE: “OPTIMIZATION PROBLEM”
1 3 2, ,..., , ,− − = T P G G G
L n nU U U U U
Vector obiective function:
1 2 1, ,..., ,c
T C C C Cn nS S S S S− =
Vector control variable
General optimization problem
( )min
0S
i
U S
S
>
l L/2
UL
P
S1
C
S2
C
S3
C
S 4
C
S j
C Sj+1
C
Sn/2C
Uj
G
Uj+1
G Un/2-1
G
H
p
( ) ( )( )
2
,
, , 0k
k k k kS
dUU S S p U S p S o SdS λ
+ ∆ = + ⋅∆ + ∆ ≅
( )( )
1
,
,k
k kS p k
dUS U SdS
λ−
∆ = −
1k k kS S S+ = + ∆
Iterative method
FORMULATION OF THE CABLE SYSTEM
G
3UG
linear configuration
static profiledynamic configuration
3X
1X2X
ij
ij
1U2U
3U
P
P
P
1UG
2U
Initial deformed configuration
Geometric nonlinearity based on the Green-Lagrange strain measure
Dynamic equations of the i-th stay
2
2
d z dsH mgdx dx
= −
12
ji k ki jT
j i i jT TT T
uu u ux x x x
ε ∂∂ ∂ ∂ = + + ⋅ ∂ ∂ ∂ ∂
Tn gTt tε ε=
Localized elastic damage based on the CDM approach
1 2 31 1 1 1 1 2 1 2 3
1 1 1 1 1 1
0, 0, 0c c cd dU d dU d dUN N b U N U N b U
dX dX dX dX dX dXµ µ µ
+ − − = − = − − =
*0effA A A= − [ ]
*0
0 0
with 0,1effA A AD D
A A−
= = ∈ effeff
TA
=σ1eff Dσσ =−
(1 )eff
effE E D Eσσ σε = = =
− 0
effeff
AE E
A=
Effective Area Damage definition: Corrosion ratio Effective Stress
Lemaitre’s equivalent strain principle Effective modulus of elasticity
FORMULATION OF THE MOVING SYSTEM
UX3
m
R(s) R(s+ds)
dss
3
X3
X1
λRX
c
λ
RX3
X3
X2
( )( )3
1
31 λ λ
=
= +
m
X
s X
dUddR dX g s tdt dt
Moving load description
Balance of linear momentum
Selfweigth loads Transient loads (mass and path time dependent)
Governing equations of the girder
( )( ) ( )( )3
1
23 3
1 2
λλ λ=
= + +
m m
X
s X
dU d UddR dX g s t s tdt dt dt
( )2 2 2 23 3 3 3 3 3
2 2 22∂ ∂ ∂ ∂ ∂ ∂
= + = + + ∂ ∂ ∂ ∂ ∂ ∂ ∂
m m m m m ms td U U U U U Ud c cdt dt t t t t t s s
Time dependent derivative rule
( ) ( ) ( )3 2 3 3 1 1 1, , , ,= +Φm
U X X t U X t X t e
Bridge kinematic
GIRDER-MOVING SYSTEM EQUATIONS (PDE)
3
3
2 2 2 2 2 23 1 3 1 3 3 3 1 1 1
2 2 2 21 1 1 1 1 1 1
2 2λλ λ λ ∂ ∂Φ ∂ ∂Φ ∂ ∂ ∂ ∂ Φ ∂ Φ ∂ Φ = = + + + + + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
XX
dR d U U U U Up g e c e c c e c cdX dt t t X X t t X X t t X X
( )11 1 1 1
1 31 1 1 1 1
1 2
2 2 211, 1, 2, 3,
1
42 2 23
3 2, 024 1, 1, 2, 3, 3,1 1
42 2
1 2 3, 0341 1 1
1 0, 2
1 02
0,
G XX X X X
G X XX X X X X
X X
dEA U U U U U pdX
d U dEI EA U U U U U U I pdX dX
d U d dUEI N AU I pdX dX dX
µ
µ
ρ
+ + + − + =
− + + + + + −Φ + =
+ + −Φ + =
1 1
2 2 20 1 1 1 1 1
01 1 2 21,1 1 1
2 0t X X
dGJ I e g e c e c cdt t X t t X X
λ ρρ ρλ
∂Φ ∂Φ ∂ Φ ∂ Φ ∂ Φ Φ − Φ − + − + − + + = ∂ ∂ ∂ ∂ ∂ ∂
Moving loads equations
R R 1v
λ
λ
v
1
1
21 1 1
21 1
λ λ ∂ ∂ ∂
= = + + ∂ ∂ ∂
XX
dR d U U Up cdX dt t X t
2
2
22 2 2
22 1
λ λ ∂ ∂ ∂
= = + + ∂ ∂ ∂
XX
dR d U U Up cdX dt t X t
Girder Equilibrium equations
VARIATIONAL FORMULATION AND FE IMPLEMENTATION
( )
( ){ } ( )
( )
1 1
1 1 1 11
1 1 1
2
1 1, 1, 1 1 1 1 1 1 1 1 11
1 22 2, 1 3 2, 1 3 2 1 3 3, 2 1,
21 2 3 3, 3, 2 1
1 0,
2
i i ie e e
i i ie e e
ie
G G G G GX X c j j
jl l l
G G G G G GX X X g XX
l l l
G G GX X X
l
N U w dX U w dX b w dX N U
M w N U w dX U w dX U cU w dX
H H U cU c U g w dX
µ
µ λ δ δ
λ
=
+ − − − =
− − − − + + +
− + + + −
∑∫ ∫ ∫
∫ ∫ ∫
∫
( ){ }( )
1 1 11
1 1 1 1
2 2
3 3 2 31 1
2 2
3 3, 1 2 3, 1 2 3 1 2 2 3 2,1 1
201 21 4, 1 01 1 4 1 1 11, 1, 4 1
0,
0,
2
i ie e
i i ie e e
G G G Gj j j j
j j
G G G G G G G GX X X g j j j jX
j jl l
G G G G GX X X X
l l l
T U M
M w N U w dX U w dX T U M
M w dX I w dX e g H H c c w dX
µ
λλλ
λ
= =
= =
− Φ =
− − − − Φ =
− Φ − + Φ + Φ + Φ +
+
∑ ∑
∑ ∑∫ ∫
∫ ∫ ∫
( )1
2
1 2 1 1, 4 1 1 11
0,ie
G G G GX j j
jl
c w dX Mδ δ=
− + Φ + Φ − Φ = ∑∫
PDE VARIATIONAL FORMULATION
FE IMPLEMENTATION
1X
2Xe
i
3X
j
Non standard forces produced by the inertial description of the moving loads
Girder Variational Equations Girder element i-j
VARIATIONAL FORMULATION AND F.E. IMPLEMENTATION
Cable variational equations
( )( )
( )
( )
1
1
1
2
1 0 1 1, 1 1 1 1 1 1 1 1 11
2
1 0 2, 1 2 2 1 1 21
2
1 0 3, 1 3 3 1 3 3 1 1 31
1 0,
0,
0,
i i ie e e
i ie e
i i ie e e
C C C C CX c j j
jl l l
C C C C CX c j j
jl l
C C C C CX c j j
jl l l
N N U w dX U w dX b w dX N U
N N w dX U w dX N U
N N w dX U w dX b w dX N U
µ
µ
µ
=
=
=
+ + − − − =
+ − − =
+ − − − =
∑∫ ∫ ∫
∑∫ ∫
∑∫ ∫ ∫
G
3UG
linear configuration
static profiledynamic configuration
3X
1X2X
ij
ij
1U2U
3U
P
P
P
1UG
2U
Constraint equations: Girder-Pylons /Cable System
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )
3 1 3
1 3 1
1 1 2 2 3 3
, , ,
, , ,
, , , , , , , ,
i i i
i i i
G G CC C C
G G CC C C
P C P C P CP P P P P P
U X t X t b U X t
U X t X t b U X t
U X t U X t U X t U X t U X t U X t
−Φ =
+Φ =
= = =
Girder/Cable System
RESULTS – SUSPENDED BRIDGE
With respect to the undamaged bridge configuration a maximum percentage increment of the maximum displacement equal to 26.66
Amplification slightly variable with the speed
RESULTS – CABLE-STAYED BRIDGE
UD D1 D2 % Amp. D1
% Amp. D2
Hshaped 5.13 5.67 6.92 10.45 34.70 Ashaped 5.16 5.72 7.05 10.91 36.66
UD D1 D2 % Amp. D1
% Amp. D2
Hshaped 6.63 8.09 11.1 22.15 67.51
Ashaped 6.82 8.16 9.64 19.80 41.43
A partial damage in the anchor cable is able to produce high amplifications of the bridge displacements with respect to the undamaged configuration
Speed-dependent amplification
CONCLUDING REMARKS
A general model to predict the dynamic response of long span bridges is proposed including the effects of the local vibration of the stays, the damage mechanisms due to corrosion phenomena and moving loads/girder interaction
Cable-stayed bridges are much more affected by the presence of the damage and the transit speed of the moving loads, since larger values of the bridge displacements with respect to the undamaged configuration are observed
In the framework of cable-stayed bridges, the analyses, denote that the presence of a partial damage in the anchor cable is able to produce high amplifications of the bridge displacements with respect to the undamaged configuration
Analysis are developed for cable supported bridges based on both suspension and cable-stayed configurations, adopting similar properties for the main constituents of the bridge structures, i.e. girder, cable system and pylons
The bridge deformations are quite dependent for the assumed damage scenario
The presence of corrosion in the main cable suspension bridges significantly increases displacements already low-speed
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