Analysis of the Displacements of the Pierre

Transcription

ANALYSIS OF THE DISPLACEMENTS
OF THE PIERRE-LAPORTE SUSPENSION BRIDGE
AS MEASURED BY PRECISE GPS SURVEYS
ROCK SANTERRE, LUC LAMOUREUX AND STÉPHANIE MICHAUD
DÉPARTEMENT DES SCIENCES GÉOMATIQUES
UNIVERSITÉ LAVAL
QUÉBEC, CANADA
ABSTRACT
Algorithms for On-The-Fly ambiguity resolution have been modified for the
deformation monitoring of a suspension bridge. Instantaneous relative
positioning of a deformation network at a precision of about 5 mm horizontally
and 10 mm vertically has been achieved using GPS L1 phase observations.
Particular attention has been paid to the modelling of relative tropospheric
delay and to the phase centre calibration between antennae of different
makes. Moving averages on station coordinates have also been applied to
reduce multipath effects.
The Pierre-Laporte bridge is a 6 lane, 1040 m length suspension bridge which
crosses the St. Lawrence river in Québec City. Three 48 hour GPS sessions
have been conducted during the months of July and October 1996 and during
February 1997. For each session, 5 GPS receivers were observing at a data
rate of 2 seconds.
Daily variation in the vertical position of the antenna located in the middle of
the deck shows clear correlation with respect to temperature and vehicle
loading. Transverse movement of the deck has been monitored and correlated
with (transverse) wind speed. Seasonal variation in temperature caused a
vertical displacement of the middle of the deck, a contraction of the towers as
well as a displacement of the towers towards the river banks.
Santerre, Lamoureux, Michaud, Département des sciences géomatiques
RÉSUMÉ
Des algorithmes “On-The-Fly” pour la résolution des ambiguïtés de phase GPS
ont été modifiés pour la mesure des déformations d’un pont suspendu. Le
positionnement relatif instantané d’un réseau de déformation à une précision
de 5 mm horizontalement et de 10 mm verticalement a été atteint en utilisant
les mesures de phase GPS L1. Une attention particulière a été portée à la
modélisation du délai troposphérique relatif et au calibrage des centres de
phase entre antennes de différents manufacturiers. Des moyennes mobiles au
niveau des coordonnées des stations GPS ont été calculées dans le but de
réduire l’effet des multitrajets.
Le pont Pierre-Laporte est un pont suspendu d’une longueur totale de 1040 m
et compte 6 voies de circulation. Il traverse le fleuve St-Laurent en amont de la
ville de Québec. Trois sessions d’observations de 48 heures chacune ont été
menées durant les mois de juillet et octobre 1996 et février 1997. Lors de ces 3
sessions, 5 récepteurs étaient en opération avec un taux d’enregistrement de 2
secondes.
La variation journalière de la position verticale de l’antenne située au milieu du
tablier montre une forte corrélation avec la variation de température et le flot de
circulation automobile. Le mouvement transversal du tablier a été mesuré et
corrélé avec la vitesse (transversale) du vent. La variation saisonnière de la
température a causé un déplacement vertical du milieu du tablier, une
contraction des tours et un déplacement des tours vers les rives.
OUTLINE OF THE PRESENTATION

Objectives of the research

Modified OTF algorithms

GPS error sources and modelling

Deformation monitoring network

Bridge deck movements, as functions of:
1
- temperature
- wind speed
- traffic flow

Distance between the 2 towers (comparison with EDM)

Seasonal displacements (towers and deck)
- thermal dilatation of the 2 towers
- sag effect at the middle of the deck

Conclusions and further research
2
OBJECTIVES OF THE RESEARCH

To adapt GPS-OTF (On-The-Fly) algorithms for deformation
monitoring of “kinematic” engineering structures such as a
suspension bridge

To develop software for the post-processing of GPS L1 phase
observations with the modified OTF algorithms

To deal with the most important errors, namely: relative
tropospheric delay, antenna phase center variation and
multipath

To apply the methodology in real cases on the Pierre-Laporte
suspension bridge and at different seasons

To correlate the bridge displacements with temperature
variation (thermal dilatation of the bridge structure), wind
speed and traffic flow

To compare GPS results with EDM measurements
MODIFIED OTF ALGORITHMS (1/2)
3
Y
Start
A: A priori
coordinates
B: A priori
ambiguity
Test 1:
on ambiguity
search range
Next
satellite
?
N
Test 5:
on cycle
slips
C: L.S.A. for each
ambiguity set
Y
End
N
Next
epoch
?
Y
Next
ambiguity
set ?
Test 4:
on ratio of
variance factors
Test 2:
on L.S.A.
correction terms
Y
N
N
E: Keep and
display the L.S.A.
solution
N
Test 3:
on individual
residuals
Y
D: Store L.S.A.
solution(s)
MODIFIED OTF ALGORITHMS (2/2)
4
B:
   L 1        d tro p 
N  

 L1

A:
Initialisation from static
processing of the first 30 minutes
of data; and for the other epochs,
coordinates from the
previous epoch
1: Check if the decimal part of eqn. (B)
2: |Coordinate corrections| < 10 cm ?
3: |Residuals| < 10 cm ?
C:
X  ( A T A ) 1 A T  W
 V  A X   W
 =  ( A T A )
2
X
departs by more than 0.25 cycle
from the closest interger value
D:
1
Store all solutions which
pass tests 2 and 3
o
4: Ratio of the second smallest and the
smallest variance factors > 2 ?
E:
If one L.S.A. solution
passes test 4
5: If phase variation between 2 epochs ~
prediction from Doppler frequency
L.S.A.: Least Squares Adjustment
5
GPS ERROR SOURCES AND MODELLING
Type of error
Solution
Troposphere
Meteorological values extrapolated in height from values
recorded at the reference station (Rothacher et al., 1986)
=> Inputs for Hopfield model
Multipath
Antenna siting
and moving average on adjusted coordinates
Phase center
variation
Relative calibration between antennae on a known
and short baseline
(calibration beam)
Ionosphere,
Orbit and S/A
Errors assumed negligible for short baselines
(D < 1 km)
6
DEFORMATION MONITORING NETWORK
a
RIN1
e
d
a
b
f
UL2005
d
e
f
D (km) 0.3 0.7 1.0 0.1 3.5 3.5
h (m)
RIN2
c
59 -4
59 -2 17 19
b
TON
c
N
TACE
TOS
100 m
GPS Station Identification
UL2005: Laval University Campus
RIN1 : Primary Reference Station
RIN2 : Secondary Reference Station
TON
: North Tower (Tour nord)
TOS
: South Tower (Tour sud)
TACE : Deck Center (Centre du tablier)
PIERRE-LAPORTE SUSPENSION BRIDGE:
CHARACTERISTICS AND LOCATION OF GPS STATIONS
TOS
TON
RIN1
70 m
110 m
TACE
670 m
1040 m
TON
TON
TOS
Anemometer
TOS
TACE
T
A
C
E
TOS
TACE
7
Vertical displacement as a function of Temperature
Station TACE (07/96)
8
Eastern Daylight Time
22:00
4:00
10:00
16:00
22:00
4:00
30
Vertical displacement (m)
25
0.25
20
Temperature (oC)
16:00
0.5
15
0
-0.25
-0.5
0
6
12
18
24
30
36
42
48
Elapsed Time (hours)
Vertical displacement due to a car accident
9
0.5
13:15
13:30
13:45
14:15
14:00
0.25
30 min.
0
40 cm
-0.25
-0.5
3:00
3:15
3:30
3:45
4:00
4:15
4:30
Transverse displacement as a function of wind speed
17:00
0.2
23:00
5:00
11:00
17:00
23:00
5:00
25
East
Transverse displacement (m)
50
0.1
0
-25
0
Transverse wind speed (km/h)
10
-0.1
West
-0.2
0
6
12
18
24
30
36
42
48
Vertical displacement and Traffic flow (1/2)
0.5
1:00
7:00
13:00
19:00
1:00
7:00
13:00
3000
2500
2000
1500
0.25
1000
500
0
0
Number of vehicles /15 min.
11
-0.25
-0.5
0
6
12
18
24
30
36
42
48
Vertical displacement and Traffic flow (2/2)
0.1
High traffic flow: 3514 vehicles
Low traffic flow: 257 vehicles
0
-0.1
-0.2
3:30
3:40
4:00 8:00
3:50
8:10
8:30
8:20
Amplitude (m)
12
0.015
FFT
FFT
0.010
0.005
0
0
3
6
12
9
Period (minutes)
15
0
3
6
9
12
15
Period (minutes)
13
DISTANCE BETWEEN THE TWO TOWERS
Distance pillar to pillar
DI2000 : 663.548 m ±3 mm
at a temperature of -4.5oC
Comparison with GPS
(for a mean temperature of -4oC)
663.57
1:00
7:00
13:00
19:00
1:00
7:00
13:00 (EDT)
DI2000 : 663.548 m
663.55
663.53
663.51
0
-5
T(DI2000) : -4.5oC
0
6
12
18
24
30
36
42
-10
48
Temperature (oC)
Distance (m)
mean: 663.542 m ±10 mm
14
DEFORMATION OF THE PIERRE-LAPORTE BRIDGE
(BETWEEN SUMMER AND WINTER SEASONS)
TOS
3.8
3.8
45.5
West
4.5
TON
TACE
4.0
3.4
East
displacements in cm
mean temperature variation: -25oC
mean difference of transverse wind speed: 21 km/h
15
THERMAL DILATATION OF THE TOWERS
TON
TOS
110 m
3.8 cm
Vertical displacement between
summer and winter seasons
TOS
using the Differential Essen&Froome
tropo. model (Rothacher at al., 1986)
TON: 3.4 cm
TOS: 3.8 cm
for a temperature variation of 25.3oC
Height of the towers : 110 m
Dilatation coeff. (TON): 0.034 m / 110 m / 25.3oC = 12.2 ppm / oC
Dilatation coeff. (TOS): 0.038 m / 110 m / 25.3oC = 13.7 ppm / oC
Thermal dilatation coefficient of steel: 11.7 ppm / oC
vertical displacement of 3.3 cm
SAG EFFECT AT THE MIDDLE OF THE DECK
16
TOS
TON
67 m
110 m
TACE
664 m
Vertical variation of station TACE (between summer and winter seasons)
1) With the average coordinates of stations TON, TOS and TACE, for the
summer session, obtain the constants of the sag equation.
cable length between the 2 towers (L): 681.38 m.
2) Calculate the theoretical cable length, for the winter session, with the
thermal dilatation coefficient of steel (11.7 ppm/ oC) for a change of
temperature of -25oC.
L: 681.18 m.
3) With this theoretical cable length, the coordinates of stations TON and TOS
(for the winter session) and the sag equation, calculate the vertical
displacement of the middle of the deck:
51 cm.
4) The vertical displacement of station TACE (from GPS) is 46 cm.
CONCLUSIONS
17
A methodology, algorithms and software have been developed for
bridge deformation monitoring
The precision of instantaneous relative positioning is about ±5 mm
horizontally and ±10 mm vertically (PDOP < 6)
The analysis of the displacements shows:
i) a transverse movement of the deck center due to transverse wind speed
of about: 2 mm / 1 km/h
ii) a vertical displacement of the deck center with respect to temperature
variation of about: -2 cm / 1OC (explainable with sag theory and thermal
dilatation of the cables)
iii) a vertical variation of the deck correlated with traffic flow
iv) an elongation of the tower with respect to temperature variation of about
1.5 mm / 1OC (compatible with the thermal dilatation coefficient of the
towers)
v) a displacement of the towers towards the river banks of about 1.5 mm/1OC
vi) an agreement of a few mm in the comparison of distance between the 2
towers from GPS and EDM measurements
18
FURTHER RESEARCH

Detailed study of the high frequency movements of the bridge
(sessions with a higher GPS data rate)

Addition of more GPS receivers with chokering antennae

Real-time implementation of deformation monitoring

Application to other engineering structures: towers,
skyscrapers, dams, etc.
ACKNOWLEDGEMENTS



Ministère des Transports du Québec
Natural Sciences and Engineering Research Council of
Canada (NSERC)
Fonds d’enseignement et de recherche - Laval University,
Faculty of Forestry and Geomatics

Analysis of the Displacements of the Pierre

Transcription

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