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