Dynamic Assessment of PSC bridge structures under …old.enea.it/eventi/eventi2008/WIM280908/23Hao_...
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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.