Dynamic Assessment of PSC bridge structures under moving vehicular

loads

H. Hao and X.Q. Zhu

School of Civil and Resource Engineering

The University of Western Australia

Background

Vehicle-bridge interaction analysis

Bridge Load Carrying Capacity

OutlineOutline

Background

Can the existing bridges carry the modern traffic loads?

capacity of a bridge decrease

vehicle weight increase

a2S a1S

Iv ,θv

Mv ,yv

v

y2 y1

y3

cs1ks1

m1y4

cs2ks2

m2

ct1kt1kt2 ct2

Quarter-car model Half-car model

Vehicle-bridge interaction

P2 P1

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0.01 0.1 1 10

Spatial frequency, n (cycle/m)

Displacement power

spectral density

(m2/cycle/m)

A

B

C

D

E

F

G

H

Formulation of the vehicle and bridge model

a2S a1S

Iv ,θv

v int

vvvvFYKYCYM −=++ &&&

The vehicle and bridge modelVehicle model

Mv ,yv y2 y1

y3

cs1 ks1

m1 y4

cs2 ks2

m2

ct1 kt1 kt2 ct2

)( PFΦΩqqχq +=++g

T

nM&&&

)()()(1311311

zyczyktPtt

&& −+−=

)()()(2422422

zyczyktPtt

&& −+−=

Bridge model

Vehicle-bridge interaction forces

Two-dimensional vehicle model (4-DOF)

fKuuCuM =++ &&&

Combining the vehicle and bridge model, the coupled equation of motion is as follows

Formulation of the vehicle and bridge model

fKuuCuM =++ &&&

rC

I

0

Φ

rK

I

0

Φ

f

0

0

Φ

f ′

−

+

−

+

=t

T

n

t

T

n

g

T

n M

v

MM

=fd+fr

where

Deterministic part from the vehicle weight

Random part from road surface roughness

)()()()()( ttttt rid bbzAz ++=& i=1,2

Formulation of the vehicle and bridge model

By introducing the state-vector, the equation can be written as follows

where

Note:i=1, rampi=2, road surface roughness

rirri

t fMb =)(

[ ]TTT uuz &=

−−=

−− CMKM

I0A

11)(t

drd

t fMb =)(

=

−1M

0M

r

0ff ≤

0ff > cycle/m2/1

0π=f

05.2

00 )/()()( −⋅= fffSfS dc

44.1

00 )/()()( −⋅= fffSfS dc

Road surface roughness (Dodds and Robson, 1973)

Analysis of the response due to the vehicle weight

)()()()()( ttttt rid bbzAz ++=&

Computational algorithm

Analysis of the response due to the ramps

Analysis of the response due to the road surface roughness

)()()()( tttt ddd bzAz +=&

)()()()( 11 tttt rrr bzAz +=&

)()()()( 222 tttt rrr bzAz +=&

Covariance matrix of the zr2(t)

∫∫

∫∫∞+∞+

+∞

∞−

+∞

∞−

+−

⋅+=

ωωωωωωωωωω

ωωωωωωωωω

dttSvdttSiv

dttSivdttSttE

TT

TTT

rr

),(),()(),(),()(

),(),()(),(),()(])()([

22

22

12

211122

QQQQ

QQQQzz

∫∫∞−∞−

+− ωωωωωωωωωω dttSvdttSiv ),(),()(),(),()( 2212 QQQQ

∫−=

t

xi

k dtt0

)(ˆ

11e)(),(),( τττω τω

SHQ

∫=t

xi

kdtt

0

)(ˆ

11e)(),(),( τττω τω

SHQ

∫−=

t

xi

cdtt

0

)(ˆ

21e)(),(),( τττω τω

SHQ

∫=

t

xi

cdtt

0

)(ˆ

21e)(),(),( τττω τω

SHQ

Computational algorithm

Covariance matrix of the zr2(t) and br2(t)

ωωωωω

ωωωω ω

dttvttjv

ttjvtteSttE

TT

T

c

T

k

vtiT

rr

))(),()(),(

)(),()(),(()()]()([

22

1122

SQSQ

SQSQbz

+−

+= ∫+∞

∞−

Covariance matrix of the

ωωωωω dttvttjvT

c

T

k))(),()(),(

2

22

2SQSQ +−

)]()([)()]()([

)]()([)()()]()([)()]()([

2222

222222

ttEtttE

ttEttttEtttE

T

rr

TT

rr

T

rr

TT

rr

T

rr

BBAzB

BzAAzzAzz

++

+=&&

)(2 trz&

-1

0

1

2

3

4

5

0 0.3 0.6 0.9 1.2

Vehicle position (vt/L)

Dis

pla

cem

ent (m

m)

v=15 m/sv=25 m/sv=35 m/sv=25 m/s (Green&Cebon)

389

390

391

392

393

394

395

0 0.3 0.6 0.9 1.2

Vehicle position (vt/L)

Fo

rce

(kN

)

v=15 m/s

v=25 m/s

v=35 m/s

ApplicationsApplications

2

2.5

512×10-6

(m3/cycle)Damaged

0

0.5

1

1.5

2

2.5

10 15 20 25 30 35 40

Vehicle speed

DLC

0

32×10-6

128×10-6

512×10-6

(m3/cycle)

Very good

Good

Average

Damaged

stat

total

δ

δ=DAF

Dynamic Amplification factor (DAF)

Front axle

0

0.5

1

1.5

10 15 20 25 30 35 40

Vehicle speed

DA

F

0

32×10-6

128×10-6

Very good

Good

Average

Vehicle speed

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

10 15 20 25 30 35 40

Vehicle speed

DLC 0

32×10-6

128×10-6

512×10-6

(m3/cycle)

Very goodGood

Average

Damaged

Dynamic Load Coefficient (DLC)

stat

dyn

P

P=DLC

Rear a

xle

Load Carrying capacity evaluation

Description of the example bridge

Finite element model (FEM)

Evaluation of the load carrying capacity of Evaluation of the load carrying capacity of the bridge with FEM

Update FEM based on the dynamic test data

Evaluation of the load carrying capacity of the bridge with updated model

Bridge description

A

17.8 18.3 17.8

It’s a three span slab on girder bridge.

The bridge decks are composed of 21 precast

prestressed concrete girders and a cast-in-situ

slab.

The girders are connected to the slab with shear

links.

A

Abutment 1 Abutment 2Pier 1 Pier 2

610

1422.4

1980

762

28

40

11

80

Elevation of Bridge

Section A-A

Bridge description

Centre of

Bridge

14

08

64

.8

2% Fall

37

4.7

Detail ADetail B

Shear connector φ16

330.2

89

10

21

02

4-φ12.7

14

0

1422

S2 φ10

The slab is average 140mm thick and 9.2 m width

Spacing of girders is 1.4 m

There are 22 strands and 4 reinforcements in each girder.

The design prestressing force is 84.8 kN per strand.

Shear link is deformed rebar 12 mm in diameter

304

4570

1422 1422 1422

431.8

152

43

1.8

14

01

02

86

4.8

20-φ12.7strands

4-φ12.7

S1 φ10

S3 φ16

Section of deck

Section of girder

Model of concrete members

Abutment diaphragm

Girder

Slab

See Detail A

See Detail B

A

A

A-A Section

Detail A

Cap beam

Detail B

SlabPier diaphragm

Cap beam

GirderGirder

Bearing

Finite element model of the bridge

Model of reinforcement

Center line of

girder

Layer 1

Layer 2

Layer 3Layer 4Shear connector φ16

330.2

10

21

02

14

0

1422

S2 φ10

Rebar layers in one girder

Rebar layers in the girders

LayerNumber

Area per bar (mm2)

Space (mm)

1 4 126.68 82.55

2 8 94 19.05

3 8 94 53.98

4 6 94 71.69Rebar layers in the girders

431.8

152

43

1.8

14

01

02

89

86

4.8

20-φ12.7strands

4-φ12.7

S1 φ10

S2 φ10

S3 φ16

Model of shear connectors and stirrups

The shear connectors transfer

the longitudinal shear flow

between the slab and the girders,

and provide constraint against

vertical separation between the

slab and the girders.

There are 81 shear connectors

35 76.2×5 152×11 229×9 270 305×10 381×4

Center line

of girder

Shear connector

There are 81 shear connectors

for each girder and more than

1000 in whole bridge.

To reduce the finite element

model size, a cluster of the shear

connectors are lumped together

based on the area equivalent.

The shear connectors are

modelled by a CARTESIAN

connector element

Center of abutment

17832GirderDiaphragm

1219 1353 2438 1353 12193658 3658 2438

Location of the shear connectors in half girder

Location of shear connectors in the FE model

Boundary conditions

The columns of the abutments are buried and unlikely to move due to the resistance of the soil and rocks around the abutment.

The cap beams of the abutments are fixed in the model.

The columns in the piers are fixed in the model.

Load action for ultimate load carrying capacity evaluation

Traffic load M1600 is used in this study based on the influence line analysis

Load factors considered in the analysis

Number of lanes

Elevation

360 kN 360 kN6 kN/m

360 kN360 kN

0.4 3.2m standard design 0.6

Number of lanes

Accompanying lane factors

Dynamic load allowance

Ultimate load factor

17.838 18.288 17.838

5.42 3.75 4.0 5.0

Section A

Load position for the most adverse positive moment

Plan

0.6

0.2

1.251.25 3.75 1.25 varies 6.25m min. 5.01.251.25 1.251.25

2.0m

1.25

M1600 moving traffic load

Load action for ultimate load carrying capacity evaluation

The transverse position is decided to achieve the most adverse moment in girder.

First row of the wheel is located on the second girder

Two traffic lines are applied. Two traffic lines are applied.

Longitudinal position for the two lanes are same

The spacing is 1.2 m

Magnitude of the Traffic Loads for Ultimate Load

Carrying Capacity Evaluation

No. Wheel load Distributed component

Magnitude Magnitude width

Lane 1 140.4 KN 4.39 KN/m2 3.2 m

Lane 2 112.3 KN 3.51 KN/m2 3.2 m

Load position on the slab

Material parameters

Concrete

Girder:

fcm= 48.3 Mpa

E0 = 38.0 GPa

f = 2.38 MPa

Prestress strands:

Es= 195 GPa

fy =1487.5 MPa

f = 1750 MPa

Steel

cmfE ××= 043.05.1

0ρ

ff ×= 4.0

Concrete

fcr = 2.38 MPa

r = 2500 kg/m3

u = 0.2

fu = 1750 MPa

r = 7800 kg/m3

εu= 0.35

Cap beam and slab:

fcm= 31.0 Mpa

E0 = 30.5 GPa

fcr = 2.2 MPa

r = 2500 kg/m3

u = 0.2

Reinforcement:

Es= 200 GPa

fy = 500 MPa

fu = 550 MPa

r = 7800 kg/m3

εu= 0.16

cmcrff ×= 4.0

Prestress strands

uyff ×= 85.0

35.0≥u

ε

Reinforcement

yuff ×= 10.1

16.0≥u

ε

Material Model

0E

Concrete in compression

is the Young’s modulus of concrete

Concrete in tensionStrai

Stress (MPa)

fcm

0.9fcm

ecr

0E

1E2E

01 05.0 EE =

02 018.0 EE −=

(Balakrishnan and Murray 1988 )

(Wang and Hsu 2001)

Concrete in tension

Reinforcement and prestress strandsA plasticity model with strain hardening in both tension and compressionfy yield stressfu ultimate stressεu ultimate strain

Strai

n0

fc

r

ecr

50ecr

A

BC

Uniaxial stress-strain relationship for concrete

4.0

1

1

=

ε

εσ cr

crf crcr f5.0, 11 ≥> σεε

Strain

Stress

fy

fu

0

Es

Stress-strain relationship for reinforcement

εu

40

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60

Test results Mesh 1 Mesh 2

Mesh 3 Mesh 4

Test done by Rabczuk and Eibl (2004) were used to calibrate the model

FE Model of the Bridge from Design Drawings

Summary of the Element Information

Component Element type Element number

Girder Solid element 7056

Slab Shell element 384

Reinforcement

Surface element 1197

Diaphragm Solid element 128

Cap beam Solid element 128

Shear linkCartesian connector

168

BearingCartesian connector

28

Pier column Beam element 12

Meshed finite element model

Mesh of the

rebar layers in

half girder

Mesh of the

girder

section

Results:

Displacement pattern of the slab

Results:

1

1.2

1.4

1.6

1.8

The flexure cracks first occurred at 0.57 times of the ultimate design load in girder B-1. The stress of the bottom layer strands is 1042 MPa in the cracking section.

Load

rat

io t

o t

he

nom

inat

ed l

oad

s A2

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

Then the stress of strands increased to 1487.5 MPa at 1.34 times of the ultimate design loads with 110 mm displacement

The maximum load the structure can carry before softening is 1.67 times the ultimate design load.

Displacement (mm)

Load

rat

io t

o t

he

nom

inat

ed l

oad

s

Load-displacement relationship

A1

Finite element analysis on the original model

Effect of the shear connectors

Rigid connection

assumption of the empirical formula

1.2

1.4

1.6

1.8

Design

condition

Perfect connection

Load

rat

io t

o t

he

nom

inat

ed l

oad

s A1

B1

formula

Design parameters

the original condition

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

No shear connectors

Displacement (mm)

Load

rat

io t

o t

he

nom

inat

ed l

oad

s

Load-displacement curves

Completely damaged shear connectors results in the initial stiffness of the bridge decrease by 25.3%. And the ultimate capacity decrease by 21%.

Minimum principal stress of

girder at failure stage

Results

Perfect connection Designed shear connectors

No shear connectors

Model updating

The objective of model updating is to obtain values of the chosen parameters so that the difference between the dynamic properties of the numerical model and the measurements is minimal.

The model updating procedure used is the one implemented by Xia et al. The model updating procedure used is the one implemented by Xia et al. (2006) utilizing the optimisation algorithm.

In the current model, the stiffness of the girders, the slabs and the shear connectors are updated.

An updated finite element model of the bridge is derived based on the data from the dynamic field tests to simulate the actual condition of the bridge after 30 years of usage.

Generally speaking, the updated finite element model will provide more reliable and accurate results than the initial model established according to the design drawings.

Model updating results

Frequency comparison before

and after updating Mode shape comparison before

and after updating

Model updating results:

The red color bar indicate the increase in stiffness. The blue color bar indicate the decrease in stiffness.

Stiffness of most girder Stiffness of most girder elements increase about 40%, implying that the real modulus of the girders are higher than the initial ones.

The stiffness decrease in the middle of the first and second span indicate the damage in this area.

Change ratio of girder stiffness

Model updating results:

The deck’s stiffness is lower than its designed value by about 20% to 40%, showing some degree of deterioration

Change ratio of slab stiffness

some degree of deterioration after serving 30 years, especially in the central part.

The increase of stiffness in the kerb areas may be due to the strengthening effect from the guardrail.

Model updating results

Girder

A

Girder

G

In first span, most of the shear connectors are in good condition.

In the central span, Girder B F and G

2nd

span

1st span

3rd span

Change ratio of shear connectors stiffness

Girder B F and G demonstrate relatively large damage.

For the 3rd span, Girders A, B and F have some damage, and Girders D, E and G contain a fair amount damage at the supports.

Condition Assessment of Slab-girder Bridges

MRWA Bridge By model updating

By local damage indexBridge Layout

The finite element analysis with updated model

The initial stiffness of the structure increases by 12%.

The flexure crack appears in the middle of the first girder in first span at 0.80 of the rating loads.

The strands start to yield at 1.39

Original model

1.2

1.4

1.6

1.8

Original model

A2B2

The strands start to yield at 1.39 times the nominated loads in middle of the first span

Comparing this with the original model, more significant shear cracks develop in the updated model.

The calculation stops at 1.49 times of the nominated load as the flexure mode of failure happens.

Updated model

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300

Load

rati

o

Updated model

Displacement (mm)

A1

B1

Load-displacement relationship - comparison

between original and updated model

Conclusions

Nonlinear finite element analysis and model updating method are applied

to calculate the load carrying capacity of a practical bridge.

The finite element analysis carried out on the original model shows the

load carrying capacity of the bridge is higher than that estimated from load carrying capacity of the bridge is higher than that estimated from

code or empirical method.

The bridge carries 1.67 times of the ultimate load specified in the design

code, and 20% higher than the capacity calculated from the empirical

formula.

The finite element analysis of the updated model indicates that the

bridge ultimate load carrying capacity is about 1.49 of the code value.

All the results indicates the bridge still safe for the modern traffic loads.