Rapid Non-Contact Tension Force Measurements on Stay Cables
∗
†
Marcus Schmieder, Alex Taylor-Noonan, Roland Heere
Abstract
‡
may exceed the replacement cost, especially when
the repairs involve full or partial closure. Therefore,
Vibration to Tension Force calculations are known in
maintaining the structure and xing the issues when
the engineering community through the Taut String
they arise will be most cost ecient in the long run.
Theory. Precise calculations require accurate identication of the natural frequencies and incorporation
Periodic
of correction factors.
Additionally, the duration of
ten include a measurement of the cable tension
individual measurements must be optimized to allow
forces. Traditional methods to measure cable tension
for sucient data aquisition. The work presented in
forces in stay cables can be classied into 2 main
this paper describes a cost eective and optimized
categories:
approach to determine the tension force in stay ca-
Lift-o
bles. This optimization is necessary to accommodate
inspection
plans
for
these
bridges
of-
direct measurement by the lift-o method,
using hydraulic jacks.
any funding restrictions.
Frequency-based
1 Introduction
indirect
measurement
via
the
natural frequency of the cable.
The natural frequency-based method centers on the
Cable Stay bridges gain more and more popularity
Taut String Theory, where the tension force
in the structural engineering world. The reasons for
cable of known length
T in a
L and unit weight w can be cal-
the popularity of these structures include the cost ef-
culated from the
n-th
natural frequency of vibration
ciency of construction and the pleasing appearance
fn
by
of cable stay bridges. However, the main structural
elements holding the deck in place are the stay cables
T =
which are in most cases not thoroughly inspected
over the lifetime of the bridge.
The life span of
where
the structure may be shortened due to corrosion,
slippage, settlement of the structure or part of the
An
structure, resulting in load imbalances in the cables.
Eng.
(1)
is the gravitational constant.
experimental
validation
of
the
determina-
tion of tension force through natural frequencies was
The costs of repairing signicant structural damages
∗ M. Ind.
g
4wL2 fn2
n2 g
published in Russell and Lardner [1998].
& Mgmt., NDT Specialist, Metro Test-
ing Laboratories Ltd, Burnaby, British Columbia.
As
Email:
[email protected]
the
size
of
bridges
presents
challenges
for
accelerometer installation, remote sensing methods
† B.A.Sc. E.I.T., Materials Engineer, Metro Testing Labo-
have been explored.
ratories Ltd. Email: [email protected]
A Laser Vibrometer is one
such device, in which a laser beam is directed at
‡ M.A.Sc. P.Eng., Senior Materials Engineer, Metro Testing
the
Laboratories Ltd. Email: [email protected]
1
location
of
interest
and
the
velocity
of
the
2
COMPARISON OF METHODS
1
vibration is measured .
2
Systems of this kind are
measured directly.
However, this method requires
capable of conducting measurements from distances
heavy jacking equipment which is costly to mobilize,
of up to 200 m. In 1999, Cunha and Caetano used
install and operate.
a
to
accessible and clear of lling materials. Furthermore,
those from cable-mounted accelerometers. Excellent
a suitable bearing location for the jacking mechanism
agreement was found.
is required.
laser
vibrometer
and
compared
the
results
A laser vibrometer was also
The anchorage points must be
used successfully by Miyashita and Nagai [2008] and
Chen and Petro [2006] to determine the frequency of
Using
bridge cables.
measurements requires no access to the anchorage
accelerometers
to
conduct
frequency-based
points but does require access to the cable surface or
This
paper
ca-
sleeve surface. A suciently sensitive accelerometer
laser-based
needs to be physically mounted onto the cable at
vibration measurements. First, the main methods of
a position where vibrations of multiple modes can
bridge cable tension assessment are compared. Next,
be
the authors present the results of original research
for instance).
on the importance of measurement duration. Three
lightweight (usually between 50 and 500 grams), this
methods for identifying the natural frequency from
task typically requires a boom lift for a technician.
the frequency plot are compared for varying mea-
Then, cabling must be run back to the frequency
surement durations.
analyzer.
ble
tension
describes
an
application
determination
through
of
stay
Then, a literature review on
detected
and
measured
(near
the
third-point
While the sensors themselves are
Trac control, or a bridge closure, may
the eects of sag-extensibility and bending stiness
be required if this equipment or cabling interferes
is conducted, and a system of correction factors is
with normal operation. Each stay cable may require
presented.
between 20 to 60 minutes for sensor installation and
cabling.
2 Comparison of Methods
As outlined previously, the primary methods to measure bridge stay cable tension are:
•
Lift-o method
•
Frequency-based method
By
using
non-contact
laser
based
sensors
like
a
Laser Vibrometer, no physical access to the cable
is required.
This allows the location of the mea-
surement point to be at third or midpoint of the
cable,
not limited by the reach of the available
boom lifts.
per location.
Set-up time is only a few minutes
At a rate of $100-150 per hour per
technician on site, reducing the set up time per caUsing accelerometers mounted on the cable
Conducting remote measurements using a
laser vibrometer.
ble will generate signicant cost savings for the client.
A
full
investigation
of
a
cable
stay
structure
may include more than 100 cables. To evaluate these
The lift-o method has the advantage that each
cables a simultaneous or sequential measurement
strand can be jacked and its tension load may be
can be selected. The simultaneous approach usually
1 The
involves using many accelerometers at once.
principle behind this device is that a laser beam re-
ected o a moving surface will experience a phase change
Given
that enough accelerometers and signal cables are
and will also have its frequency shifted through the Doppler
available, and the frequency analyzer has enough
eect.
A Laser Vibrometer is in eect an interferometer; in
input channels (these devices can often support 8,
which a single laser beam is split into a measurement beam
16 or 32 seperate channels), many measurements
and reference beam. After reecting o the vibrating surface,
the measurement beam is recombined with the reference beam
can be conducted simultaneously.
However,
this
(which never left the instrument) and the interference pattern
requires the installation of many accelerometers and
is measured.
signal cables while ensuring that no cable confusion
3
IMPORTANCE OF MEASUREMENT DURATION
occurs.
Conversely,
a laser vibrometer can only
measure one point at a time,
and therefore the
3
As the duration of the measurement increases, the
frequency resolution
fs
improves, and the natural
data collection phase becomes a large part of the
frequency can be determined to within a smaller
overall project.
interval.
Section 3 outlines the implications
of choosing an appropriate measurement duration
Insucient
and an experimental verication is performed.
measurement
durations
will
result
in inadequate frequency resolution of the FFT analThe cost & time expenditures for each method
ysis. However, measurements beyond the minimum
are compared in Table 1. Because the time savings
required time will extend the time on site and may
during setup of each measurement location greatly
lead to higher costs for the client. The objective is to
outweigh the extra time spent during measurement,
minimize data acquisition time whilst still identifying
conducting
pro-
the natural frequency with adequate precision. This
gramme using a laser vibrometer is much quicker
section presents three analytical methods to increase
than with accelerometers.
the accuracy in identifying the natural frequency
a
cable
vibration
measurement
from
measurements
of
shorter
duration.
These
Overall, contactless laser vibrometer systems require
methods are then compared using experimental data
less equipment and less time and are therefore a cost
in order to determine an appropriate measurement
ecient
length.
based
alternative
and
lift-o
to
traditional
methods.
accelerometer-
Furthermore,
laser
vibrometers are generally quite transportable (an
The
entire system can be wheeled by one technician
Peak Method, where the rst natural frequency is
and checked as airline luggage) and provide similar
read from the FFT. Given a frequency resolution
measurement precision as accelerometer systems.
of
simplest
fs ,
identication
method
is
the
First
identifying the natural frequency (f1 ) via the
First Peak Method only gives us the certainty that
fs
fs
< f1T rue < f1F P +
2
2
3 Importance of measurement
duration
3.1
The number of samples collected during data acqui-
Methods that consider multiple use harmonics use
sition is critical for the precise determination of the
the relation of each
natural frequency.
quency
The natural frequency and har-
f1F P −
Considering multiple harmonics
frequency-domain representation of a time-domain
function. For a discrete time series sampled at rate
fa
for duration
T,
tains a series of discrete frequency bins uniformly distributed between 0 Hz and
fa .
bins is the frequency resolution
n-th
harmonic, we have the certainty that
fnM easured −
fs
fs
< fnT rue < fnM easured +
2
2
(5)
The width of these
fs ,
which is indepen-
dent of the frequency setting of the analyzer, and is
simply related to the measurement time
fs =
(4)
If we consider our certainty of identication of the
the FFT result is also a discrete
sequence of the same length (fa T ). This result con-
n-th harmonic to the natural frefn = nf1
monics are determined by nding the peaks of the
Fast Fourier Transform (FFT), which provides the
(3)
fa
1
=
fa T
T
T
fnM easured −
by
(2)
fnM easured −
n
fs
fs
< nf1T rue < fnM easured +
2
2
fs
2
< f1T rue <
fnM easured +
n
fs
2
(6)
3
IMPORTANCE OF MEASUREMENT DURATION
4
Table 1: Comparison of cost and time expenditures for measurement methods
Task
Lift-o Method
Frequency-based Methods
Accelerometers
Laser Vibrometer
Mobilization &
demobilization
Extensive
Moderate
Quick
Extensive
Quick
Extensive
(20-60 minutes per
(1-3 minutes per
location)
location)
to/from site
Set up/teardown for
each cable
Simultaenous at
Each point
multiple locations;
Data acquisition
Rapid
seperately; choice of
little penalty for
measurement
long measurement
duration important
duration
Conversion of gauge
Data processing and
interpretation
pressure to force,
Natural frequency identication,
rapid.
sag-extensibility and bending stiness
corrections, moderate
Futhermore, each measurement can be considered to
be uniformly distributed between these two bounds,
so
(
f1T rue ∼ UNI
fnM easured −
n
with the variance
VAR (f1T rue )
fs
2
(
def
=
σn2
=
fn
+
, M easured
n
fs
n
fs
2
)
12
(7)
smaller variance.
many
measurements,
each
with
variance
monics in a simple average:
N
1 ∑ fn
N n=1 n
In
the
Variance-Weighted
VAR (f1M P )
=
f1V W =
wn fn
∑N
n=1
wn =
√
∑N
By this, the authors propose two methods of peak
variance.
N
∑
(10)
n=1
(8)
identication that use multiple peaks to minimize the
each
tion (square root of variance).
where
∑N
Method,
that is inversely proportional to its standard devia-
σn2 ,
average will be
Average
identied peak is taken into account with a weighting
are combined and averaged, then the variance of this
2
n=1 σn
n
(9)
3.1.2 Variance-Weighted Average Method
It is seen that the variance of each measurement
is proportional to 1/n2 , so higher peaks will have a
If
The Simple Average Method utilizes multiple har-
f1SA =
)2
f2
= s2
12n
3.1.1 Simple Average Method
wn =
n=1
√
σn
σn
fs2
12n2
fs2
12n2
∑N
=
1
n=1 n
1
n
4
SAG-EXTENSIBILITY AND BENDING STIFFNESS CORRECTIONS
n
wn = ∑N
n=1
(11)
n
5
In this gure, the results of the First Peak method
are presented as an error bar (in red),
thereby
showing the certainty range dened by Equation 3.
3.2
Experimental Validation
The authors conducted three long-duration measurements (102.4 seconds) on each of two side-by-side
(and therefore identical) cables.
The parameters
of these cables are presented in Table 2.
The following remarks are made:
•
The best indicator we have of the true natural
frequency is the rst peak of the longest duration
This
measurement. Therefore, the latter two methods
parameter table includes the results of a correction
will be considered eective if they can identify
for sag-extensibility and bending stiness, using the
the natural frequency as close to this value, using
formulas presented in Section 4.
a shorter duration time series.
As an example,
Figure 1 depicts the time series of the rst measurement of the East Cable.
•
The rst peak identication method is very erratic for shorter measurement durations (less
than
In order to validate and compare the performance
of the three identication methods on timeseries of
dierent durations, each 102.4 second time series
•
2
commercial FFT analyzers .
Once
again
using
the
rst
The simple average method provides a marked
improvement over the rst peak method.
was also truncated to four shorter durations (Table
3) . These durations match those available in many
51.2 s).
•
The variance-weighted average method provides
an improvement still over the simple average
measurement
of
method especially for measurement durations
the
less than
East Cable as an example, a plot comparing the
FFT for each duration is presented in Figure 2.
From this plot we can see the peaks are sharper
for the longer-duration measurements.
Conversely,
•
25.6 s.
It is seen that the latter two methods are very
stable
for
measurement
durations
25.6 s and
25.6 s
longer, and therefore it is conrmed that
the peaks are much less dened for shorter-duration
is an appropriate choice of measurement dura-
measurements, increasing the possibility of misiden-
tion that provides a good balance of accuracy
tication.
and eciency.
3.2.1 Conclusions of Experimental Study
After using the three identication methods on the
six time series, the results were plotted in Figure 3.
2 An
ecient, and widely used, method for calculating the
FFT is the radix-2 Cooley-Tukey algorithm, which works with
N samples, where
a time series of 2
N
is a postive integer.
4 Sag-extensibility and bending
stiness corrections
In the preceeding sections, the relation of natural
frequency to tension force was based on the Taut
Consequently, many commercial FFT analyzers oer the user
String Theory, which assumes the cable experiences
a choice of time series lengths that comply with this require-
no sag and has no bending stiness.
ment. Also, to remove the requirement that the user be aware
of the eects of aliasing and properly choose a sampling fre-
In actuality,
neither of these conditions hold, and for cable stayed
quency higher than the Nyquist rate, many commercial FFT
bridges, the cable is also inclined.
analyzers have a sampling frequency that is 2.56 times the dis-
these eects are detailed well in Ni et al. [2002].
played frequency. Therefore, the user selects a measurement
frequency based on the highest expected frequency, the analyzer runs at a higher frequency to avoid aliasing, and the user
is only returned the portion of the FFT below the measurement frequency selected.
The causes of
4
SAG-EXTENSIBILITY AND BENDING STIFFNESS CORRECTIONS
6
0.004
Velocity
0.002
0
-0.002
-0.004
0
20
40
60
80
100
Time [s]
Figure 1: Time-series of East cable, Measurement 1
3
6.4 s
12.8 s
2.5
25.6 s
51.2 s
102.4 s
2
1.5
1
0.5
0
0
1
2
3
4
5
Frequency [Hz]
Figure 2: Comparison of 0-5 Hz region of FFT for varying measurement lengths (East Cable, Measurement
1)
Natural Frequency [Hz]
2.05
2.05
Natural Frequency [Hz]
2.05
Natural Frequency [Hz]
SAG-EXTENSIBILITY AND BENDING STIFFNESS CORRECTIONS
Natural Frequency [Hz]
4
7
East 1 First Peak
2
East 1 Simple Average
East 1 Variance Weighted
1.95
1.9
1.85
1.8
1.75
1.7
2
East 2 First Peak
1.95
East 2 Simple Average
1.9
East 2 Variance Weighted
1.85
1.8
1.75
1.7
1.65
1.6
1.55
1.5
East 3 First Peak
2
East 3 Simple Average
East 3 Variance Weighted
1.95
1.9
1.85
1.8
1.75
1.7
1.9
West 1 First Peak
1.85
West 1 Simple Average
1.8
West 1 Variance Weighted
1.75
1.7
1.65
1.6
1.55
1.5
1.45
1.4
Natural Frequency [Hz] Natural Frequency [Hz]
1.35
1.95
West 2 First Peak
1.9
West 2 Simple Average
1.85
West 2 Variance Weighted
1.8
1.75
1.7
1.65
1.6
1.55
1.5
2.05
West 3 First Peak
2
West 3 Simple Average
1.95
West 3 Variance Weighted
1.9
1.85
1.8
1.75
1.7
0
10
20
30
40
50
60
70
80
90
100
Measurement Duration [s]
Figure 3: Comparison of identication methods for varying measurement durations
110
4
SAG-EXTENSIBILITY AND BENDING STIFFNESS CORRECTIONS
8
Table 2: Summary of parameters for considered cable
Parameter
Symbol
Value
Cable Length
L
w
H
EA
EI
Le
λ2
ξ
α
βn
69.233 m
0.236 kN/m
1620 kN*
236 × 103 kN†
94 kN/m2 †
69.233 m*
Cable Unit Weight
Tension Force (via Taut String Method)
Extensibility
Bending Stiness
Length after sag (See Eq. 18)
Sag-extensibility parameter (See Eq. 12)
Bending stiness parameter (See Eq. 16)
Correction factor for sag-extensibility (See Eq. 14)
Correction factor for bending stiness (See Eq. 15)
0.0075*
276
1.00028*
1.007 (1
st
th
mode) to 1.011 (8
mode)
* frequency dependent, presented parameter was calculated using the average frequency observed.
†
Estimated from tabulated values of bridge stay cables by interpolating diameter.
Table 3: Considered measurement durations
Frequency
Samples
Measurement
Resolution
after
Duration
fs
Tabatabai et al. [1998] present a correction fac-
γ
tor
that
is
[Hz]
[s]
Samples
FFT
6.4
1024
400
0.156
12.8
2048
800
0.078
25.6
4096
1600
0.039
51.2
8192
3200
0.020
102.4
16384
6400
0.010
applied
to
the
measured
natural
frequency.
γ=
α
of
µ
ωn
= αβn − 0.24
ωns
ξ
(13)
is the correction factor accounting for the eect
sag-extensibility
on
the
rst
mode
(in-plane)
frequency determined by the taut string equation.
Correction
factor
βn
accounts
for
the
eect
of
bending stiness on the natural frequency.
Table 4: Modulus of Elasticity for steel cable
E,
Cable
Helical strand
Both
GPa
Structural strand
170
Parallel-wire strand
190 to 210
α
and
βn
are
dependent
They introduced a
fundamental cable parameter of
2
λ =
H
HLe
EA
√
with
ξ=L
L
2
cos θ
inclina-
(14)
2 4 + n 2π
βn = 1 + +
ξ
ξ2
In 1974, Irvine and Caughey considered the problem
( wL )2
the
α = 1 + 0.039µ
2
of sag in a horizontal cable.
on
tion angle of the cable and materials properties:
140
(12)
The modulus of elasticity for steel cables depends on
the rope conguration. Approximate values for each
type, compiled by Irvine [1981], are given in Table 4.

2

λ
µ= 0


0
2
H
EI
(15)
(16)
for n=1 (in-plane)
for n > 1 (in-plane)
(17)
for all n (out-of-plane)
[
Le = L 1 +
( wL cos θ )2 ]
H
8
(18)
5
CONCLUSION
9
of the Royal Society of London. A. Mathematical
where
H
is the initial estimate of tension force along the
chord, determined by Equation 1.
λ2 is as dened in Equation 12.
For higher modes of vibration (n
tion 13 becomes
and Physical Sciences, 341(1626):299, 1974.
T. Miyashita and M. Nagai. Vibration-based Structural Health Monitoring for Bridges using Laser
> 1),
ωn
= βn
ωn s
Equa(19)
The bending stiness and sag-extensibility correction
factors in this section should be calculated and applied to each measured peak before utilizing any of
the three methods presented previously in Section 3.
Doppler Vibrometers and MEMS-based Technologies.
International Journal of Steel Structures, 8
(12):325331, 2008.
YQ Ni, JM Ko, and G. Zheng. Dynamic analysis of
large-diameter sagged cables taking into account
exural rigidity. Journal of Sound and Vibration,
257(2):301319, 2002.
J. C. Russell and T. J. Lardner.
5 Conclusion
Experimental de-
termination of frequencies and tension for elastic
cables.
Stay bridge cable tension measurements are sometimes required to conrm the cables are indeed carrying the designed loads, calculated by the bridge de-
Journal of Engineering Mechanics, 124
(10):10671072, 1998. doi: 10.1061/(ASCE)07339399(1998)124:10(1067).
URL
http://link.aip.org/link/?QEM/124/1067/1.
signer. Laser-based vibration measurement systems
H. Tabatabai, A.B. Mehrabi, and P.Y. Wen-huei.
provide a fast and ecient solution to measure vi-
Bridge stay cable condition assessment using vi-
bration frequencies on stay cables without the need
bration measurement techniques.
for physical access to the cables. Identifying multiple
of SPIE, volume 3400, page 194, 1998.
harmonics allows the natural frequency to be identied to a greater precision by using methods such as a
Variance-Weighted Average. Due to the non-contact
approach and the long measurement range, these systems allow for fast set up and data acquisition. A fast
set up and quick data acquisition results in signicant
cost savings for the client.
References
S.E. Chen and S. Petro. Nondestructive bridge cable
tension assessment using laser vibrometry. Exper-
imental Techniques, 29(2):2932, 2006.
A. Cunha and E. Caetano. Dynamic measurements
on stay cables of cable-stayed bridges using an
interferometry laser system.
Experimental Tech-
niques, 23(3):3843, 1999.
H.M. Irvine. Cable Structures. MIT Press, 1981.
H.M. Irvine and T.K. Caughey. The linear theory of
free vibrations of a suspended cable.
Proceedings
In Proceedings