New Procedure for Estimating Cable Force in Cable

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New Procedure for Estimating Cable Force in Cable-stayed Bridge
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Dong-Ho Choi, Ph.D., Professor
Department of Civil and Environmental Engineering, Hanyang University
222 Wangsimni-ro, Seongdong-gu
Seoul 133-791, Korea
+82-2-2220-4155 (Phone)
+82-2-2220-4322 (Fax)
[email protected]
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Wan-Soon Park *, Ph.D., Post-Doctoral Fellow
Department of Civil and Environmental Engineering, Hanyang University
222 Wangsimni-ro, Seongdong-gu
Seoul 133-791, Korea
+82-2-2220-4155 (Phone)
+82-2-2220-4322 (Fax)
[email protected]
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Hani Nassif, Ph.D., P.E., Professor
Department of Civil and Environmental Engineering
Rutgers, The State University of New Jersey
623 Bowser Road, Piscataway, NJ 08854
(848)-445-4414 (Phone)
(732)-445-0577 (Fax)
[email protected]
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* corresponding author
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Submission date : July 30, 2012
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Word count :7,142 words (4,892 words for text + 2,250 words for five figures and four tables)
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TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
1
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ABSTRACT
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In structural health monitoring of the cable-stayed bridges, estimation procedure for the cable
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tension influences the accuracy of monitoring system. Although some formulas are used in many
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monitoring systems, the results from these formulas are not accurate because they neglect the
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initial deflection, the natural frequency changing, and the initial curvature shortening of an
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oscillating cable. This paper presents a new estimation procedure that considers these neglected
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effects in the existing formulas and achieves the modified accuracy in estimating the tension
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force in the cable. Considering the initial deflection of the cable at the static status derives a
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clear explanation about the natural frequency changing for the change of the slenderness ratio of
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the cable. And by considering the stretching force induced by the initial curvature shortening, the
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additional forces acting on the supports of the cable conservatively are also investigated. In this
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paper, the concept of total tension force is proposed to increase the accuracy in tension force
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estimation for the cable. The re-estimation for the cable-stayed Alamillo Bridge, Spain shows
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that the new procedure estimates tension force of the cables more accurately and that this can be
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used as an alternative procedure in the structural health monitoring system.
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Key Words:
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Cable force
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Cable-stayed bridges
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Structural health monitoring
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Tension force
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TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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INTRODUCTION
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Over the last decade, the number of structural health monitoring (SHM) systems for cable-stayed
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bridges has increased. These SHM systems have become more complicated (1, 2, 3) due to their
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demand on understanding the behaviour of cable-stayed bridges. Until now, the exact estimation
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of the cable tension force has been the main objective of these SHM systems. For this purpose,
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some formulas were developed to incorporate the raw data collected from the SHM systems. But
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the accuracy of the estimation procedure has remained as the unsolved problem. The simple
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estimation formulas used in the vibration method for the bridge cable can be derived from the
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transverse vibrations of a taut string with the assumption of no sag. However, bridge cables do
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not behave as taut strings because of their flexural rigidities. The flexural rigidity is extremely
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important in calculating the tension force of bridge cables just as it was investigated by many
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Japanese researchers.
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Shinke et al. (4) proposed their first estimation formula that can account for the flexural
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rigidity effects in tension force calculations; this formula is very practical, but the flexural
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rigidity of the deflected cable is needed in the calculation. For more practical use, Zui et al. (5)
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proposed a modified formula in which the first, second, and high natural frequencies of deflected
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cables can be used. Utsuno et al. (6) of Kobe Steel, Ltd. also proposed an estimation method that
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determines the tension force and the flexural rigidity of deflected cables simultaneously. This
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method is based on the same partial differential equations used by Zui et al. but it simplifies the
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solution by assuming the boundary conditions. Since publication, this method has been widely
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used owing to its convenience and practicality. Outside of Japan, Robert et al. (7) proposed a
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simple estimation formula that considers the flexural rigidity in calculating cable tension, and
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Mehrabi and Tabatabai (8) modified the aforementioned formula to consider both flexural
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rigidity and sag effects in bridge cables. Ren et al. (9) proposed a practical formula for rapid
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calculation considering flexural rigidity and the sag effect. In their formula, the estimation
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procedure offers one of two cases: considering only the cable sag effect and considering only the
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cable bending stiffness effect. To estimate the cable tension only using the first natural frequency,
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Kim et al. (10) derived a new formula by using the variational method.
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To correct the difference between the calculated results by the existing formulas and the
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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1
measured data from field tests, Ahn (11) proposed compensating formulas. To increase the
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accuracy of the cable tension estimates, Park et al. (12) proposed the system identification theory
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using the finite element method. For the thermal axial force or thermal bending moment applied
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to the cable, Treyssède (13) proposed a new formula considering thermal stress effects.
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Installation of a damper or tube at the support of a cable changes the characteristics of the
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vibration. However, the existing formulas do not consider this effect. To solve this problem, Yan
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et al. (14) proposed a simplified new formula that considers the support stiffness. This formula
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has a similar form to Utsuno et al.'s formula.
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The boundary condition of the cable is the factor that affects the natural frequency of the
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cable. The existing formulas for estimating the tension force assume hinge support conditions. In
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fact, the supports of a cable in a bridge are partially fixed. If the slenderness ratio of the cable
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decreases, the support stiffness of the cable increases and the support conditions are closer to
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that of the fixed support condition. However, if the slenderness ratio of the cable increases, the
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support stiffness of the cable decreases and the support conditions are close to that of the hinge
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support condition. Because the slenderness ratio of the cables in a bridge is greater than 200 in
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most cases, the support stiffness of the cable is very small and the support condition of the cable
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can be assumed to match the hinge support condition.
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The free vibrations of axially loaded beams under gravity effects were first studied by
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Shih et al. (15). They explained the vibration of axially loaded beam as the sum of static and
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dynamic deflections. An approximate solution for the dynamic response was derived by
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perturbation methods based on nonlinear partial differential equations, and the governing
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equation for dynamic response in their study was very similar to the existing equation for cable
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tension estimation. Hughes and Bert (16) found an error in the governing equation of Shih et al.
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and proposed a new governing equation for the dynamic response, which has the same form as
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the existing equation for cable tension estimation. Although Hughes and Bert proposed an exact
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governing equation, they did not clearly explain the dynamic response in symmetric and anti-
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symmetric modes.
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To consider the gravity effect, the nonlinear solution for the static deflection is needed to
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transform the beam from the static status to the dynamic status. This problem is related to the
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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curvature shortening effect. If immovable ends are assumed at the supports, the curvature
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shortening effect is restrained. Zaslavsky (17) investigated the curvature shortening behaviour
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and the reaction at immovable ends for changes in the support position.
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In this paper, a new practical procedure is developed to increase the accuracy in
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estimation of the cable tension in the cable-stayed bridge. In the new, practical procedure, the
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initial deflection of the cable by the gravity effect, which is not considered in the existing
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formulas, is taken into account. By considering this effect, the dynamic characteristics of
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nonlinear cable oscillations can be understood in depth. The problem of underestimating the
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value of the cable tension when using the existing formulas results from ignoring a self-
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stretching force, which is induced from the boundary condition of the immovable ends is also
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cleared. To verify the effectiveness of the new procedure, this paper included the re-estimation
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of the tension forces for the cable-stayed Alamillo Bridge, Spain. For the re-estimation, the
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research paper about the tension force estimation of the Alamillo Bridge's cable by Casas (18, 19)
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was referenced.
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CORRELATED NOLINEAR INFLUENCES
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Initial Deflection
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The differential equation for the static deflection of the bridge cable to account for the curvature
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shortening effect can be described as follows (20):
EI
d 4 ws
dx 4
L
2
ª
º d 2w
EA § dws ·
s
»
d
x
« Na 2
2 L ¨© dx ¸¹
«
»
dx
0
¬
¼
³
q
q
U gA (1)
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where EI is the flexural rigidity of the undeflected cable, EA is the axial rigidity of the cable, L is
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the undeflected length of the cable, Na is the pure applied axial load, q is the uniform weight per
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unit length, is the cable density, g is the gravity acceleration, and s is the static deflection
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when tension force is applied. The relation between the resultant axial load N and the pure
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applied axial load Na is defined as follows:
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
L
N
Na 5
2
EA § dws ·
dx
2 L ¨© dx ¸¹
0
³
(2)
L
1
If Na becomes zero, then ws
EI
2
d 4 wo
dx 4
ª EA L § dw · 2 º d 2 w
o
o
«
¨
¸ dx »
« 2 L 0 © dx ¹
» dx 2
¬
¼
³
2
EA § dwo ·
dx.
2 L ¨© dx ¸¹
0
³
L
q
(3)
2
EA § dwo ·
dx
2 L ¨© dx ¸¹
0
³
(4)
Equation (3) can be rewritten as follows:
EI
d 4 wo
dx 4
N conserv
d 2 wo
dx 2
(5)
q
L
4
If Na becomes very large, then
EI
5
and Equation (1) becomes
where o is the static deflection when tension force is not applied, and we denote
N conserv
3
wo , N
d 4 ws
dx 4
Na
d 2 ws
dx 2
q (N
2
EA § dws ·
ws | 0,
dx | 0.
2 L ¨© dx ¸¹
0
³
and Equation (1) becomes
(6)
Na )
Under the immovable ends condition, the linear static deflection s() becomes
ws x § § L x · Na ·
§x
sinh ¨ ¨ ¸
sinh ¨
¸
¨ © 2 2 ¹ EI ¸
¨2
q x L x 2 EI q
©
¹
©
2
2Na
§
·
Na
L Na
cosh ¨
¨ 2 EI ¸¸
©
¹
Na
EI
·
¸¸
¹
(7)
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Changing of Nonlinear Natural Frequency
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The total deflection of vibration () consists of the initial static deflection s() in Equation (7)
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and the continuously changing dynamic deflection d():
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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f
ws ( x) wd ( x)
w( x)
¦
wi sin(
i 1
f
Sx
Sx
i) an (t ) sin( n)
L
L
n 1
¦
(8)
1
wherei is the coefficient of the Fourier sine series for initial static deflection, and n is the
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coefficient of the Fourier series for dynamic deflections. To build the differential equation for the
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vibration of the bridge cable, the inertia and dynamic stretching forces are added to Equation (6).
EI
4
2
L
ª
º 2
w 4 w( x, t ) «
w w( x, t )
w 2 w( x, t )
EA § ww x, t ·
»
N
d
x
A
U
¨
¸
« a 2 L ¨© wx ¸¹
» wx 2
wx 4
wt 2
0
¬
¼
³
q
(9)
For i=n, substituting Equation (8) in Equation (9) gives
f
¦a
n
''(t ) g n1an (t ) g n 2 an 2 (t ) g n3 an3 (t )
0
(10)
n 1
5
where
g n1
4
§ EI · § n S · §
3 Awi 2
N L2
¨¨1 2 a2 ¨
¸¨
¸
4I
© U A ¹ © L ¹ © n S EI
4
§ EI · § n S · §
N L2
¨¨1 2 a2
¨
¸¨
¸
© U A ¹ © L ¹ © n S EI
·
¸¸
¹
·
¸¸
¹
(for n 1,3,5...)
(for n
(11)
2, 4, 6...)
4
gn2
§ EI · § n S · § 3 Awi ·
¨
¸¨
¸
¸ ¨
© U A ¹ © L ¹ © 4I ¹
(for n 1,3,5...)
0
(for n 2, 4,6...)
(12)
4
g n3
§ EI · § n S · § A ·
¨
¸¨
¸ ¨ ¸
© U A ¹ © L ¹ © 4I ¹
(for n 1, 2,3...)
(13)
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By using the multiple scale method, the approximate solution of Equation (10) can be expressed
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as follows (21):
an t 8
a H cos(Zn t E 0 ) a 2H 2 g n 2 a 2H 2 g n 2 cos(2Zn t 2 E 0 )
O (H 3 )
2 g n1
6 g n1
(14)
where n is the angular nonlinear natural frequency of the system and is calculated as follows:
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
Zn
7
ª a 2H 2 9 g g 10 g 2
n 3 n1
n2
g n1 «1 2
«
24 g n1
¬«
º» O(H
»
¼»
3
)
(15)
1
where a is the parametric amplitude, is the perturbation parameter, and O(3) represents terms
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equal to or higher than 3. By eliminating nonlinear terms from Equation (15), the angular linear
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natural frequency n0 is determined as follows:
Zn0
(16)
g n1
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If EI is zero, Equation (15) results in the natural frequency of the taut string. For the nonlinear
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natural frequency n, the parameter a H is needed and dependent on the initial displacement and
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velocity at the initial excitation. To determine a H , the initial displacement n0 and initial velocity
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n0 of n(t) are assumed as follows:
sn0
an0 cos En0 , vn0
an0Zn0 sin En0 (17)
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where an0 and n0 are the amplitude and the phase angle of the initial excitation, respectively.
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Because of the disturbance due to nonlinear oscillation, the natural frequency of the system
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changes from n0 at the initial excitation to n. Substituting the initial conditions of Equation (17)
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in Equation (14) gives
aH
aH
§
g n1sn 0 3 g n1sn 0 8an20 g n 2 4 sn20 g n 2
¨
an 0 ¨ 3 g n1sn 0 3an 0
an20
¨
©
(4an20 2 sn20 ) g n 2
·¸
¸
¸
¹
(for n 1,3,5...)
(18)
an0 (for n 2,4,6...)
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In the nonlinear vibration of the cable, the amplitude an0 of the initial excitation may be several
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times the initial midpoint deflection parameter i. In addition, the phase angle n0 may be
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considered as zero for convenient calculation.
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Restraining of Initial Curvature Shortening
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As the cable becomes heavier or longer, the stretching force due to initial curvature shortening
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by self-weight increases. If the cables in cable-stayed bridges have small diameters and are light
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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1
(slenderness ratio < 200), this stretching force is much smaller than the applied axial load Na;
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therefore, the tension force of the cable can be estimated with only a small error margin using
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the existing formulas while neglecting this stretching force. If the slenderness ratio of the cable
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is greater than 200, or if the weight of cable is further increased by grouting, this stretching force
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should be considered in the tension force calculation.
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If the support conditions of the cable is simply supported, this force moves the end side
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of cable on roller support to the opposite side (large displacement occurs). In this case, the cable
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cannot act as the structural member. To sustain the applied tension force and vibrate normally,
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the cable should have the immovable-end conditions. And under this support conditions, the
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stretching force due to initial curvature shortening acts as the conservative force restraining the
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large displacement of the cable. This force can be calculated using a second order elastic
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analysis or via the relation between the elongation of stretching force Nconserv and the static
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deflection o () by self-weight in Equation (4). However, to use Equation (4), an iteration
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calculation is needed to find the stretching force Nconserv. This iterative procedure may be too
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much trouble. The stretching force Nconserv can be also determined by the relation between the
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nonlinear natural frequency and the applied axial load Na. As shown in Equation (18), there is
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the border line to divide the imaginary natural frequency and the real natural frequency. If the
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radicand in Equation (18) is positive, the nonlinear natural frequency n of the cable becomes
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the real number. This means that the cable will be vibrating under the given condition. However,
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if it is negative, the nonlinear natural frequency n becomes the imaginary number. In this case,
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the cable would sway but the vibration of the cable would not occur. By equating the radicand in
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Equation (18) with zero and solving for Na, the axial load to divide the imaginary natural
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frequency region and the real natural frequency region can be decided. This axial load becomes
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the stretching force Nconserv which can be described as follows:
N conserv
EI S 2
L2
§ 22 EA L2 q 2
¨
¨
S4
©
1/ 3
·
¸¸
¹
(19)
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TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
9
1
NEW TENSION ESTIMATION PROCEDURE
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Comparison with Existing Formulas
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Figure 1 shows the difference between the linear natural frequency f10 and nonlinear natural
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frequency f1 with the natural frequencies calculated by the existing formulas. In addition, the
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sag-to-span ratio , defined as 1/L, is plotted in Figure 1. For the nonlinear natural frequency f1 ,
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an amplitude an 0 equal to the initial midpoint deflection parameter 1 is assigned ( = 1.0). The
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frequency curves of the taut string and derived from the work of Zui et al. (5) and Mehrabi and
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Tabatabai (8) are also plotted.
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As the applied axial load Na increases, the nonlinear natural frequency f1 increases and
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the sag-to-span ratio decreases. If the applied axial load Na is slightly greater than the
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stretching force Nconserv , the increasing rate of the nonlinear natural frequency f1 becomes large,
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and the decreasing rate of the span ratio becomes slow. If the applied axial load Na is much
13
greater than the stretching force Nconserv , the increasing rate of the nonlinear natural frequency f1
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and decreasing rate of the span ratio are constant.
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As the applied axial load Na increases, all frequency curves approach the taut string’s
16
frequency curve. The curve obtained from the formula of Zui et al. (5) is slightly larger than the
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taut string’s natural frequency curve. This is because in the formula of Zui et al. (5), some
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constant frequency (related to the flexural rigidity of the cable) is reflected. However, this
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formula does not catch the nonlinear phenomenon when the applied axial load Na is small. The
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natural frequency curve from the formula of Mehrabi and Tabatabai (8) is similar to that of the
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linear natural frequency f10 . Considering the fact that Mehrabi and Tabatabai (8) used parameter
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Ɖ2 as proposed by Irvine (22) in their formula, this similarity may be expected. However, the
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formula of Mehrabi and Tabatabai (8) gives incorrect results when small axial loads are applied
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to the cable.
TRB 2013 Annual Meeting
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Choi, Park, and Nassif
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1
(a) Overall comparison (Na = 0 ~ 3000kN)
2
3
4
(b) High tension region comparison (Na=2500~3000 kN)
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FIGURE 1 Comparison between new and existing formulas for tension force calculations (continued).
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
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1
2
(c) Comparison for short span beam (L=10m)
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FIGURE 1 Comparison between new and existing formulas for tension force calculations.
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5
Total Tension Force and Estimation Procedure
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The behaviour of an axial loaded cable with immovable ends is different from that of an axial
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loaded and simply supported cable. If the applied axial load Na becomes zero, there is no tension
8
force in the cable, but the reaction forces at the immovable ends still exist. The value of the
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reaction force is the same as that of the stretching force Nconserv. The stretching force Nconserv acts
10
as the force sustaining force on the cable, and induces the reaction forces at the immovable ends.
11
The stretching force Nconserv can be considered a conservative force for the change of the applied
12
axial load Na. If the applied axial load Na becomes greater than zero, the cable is tensioned and
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the total reaction force becomes Na + Nconserv . If an axial loaded beam is placed on immovable
14
ends, the total tension force Ntotal of the sagged cable consists of an applied axial load Na and
15
stretching force Nconserv :
Ntotal
N a Nconserv
TRB 2013 Annual Meeting
(20)
Paper revised from original submittal.
Choi, Park, and Nassif
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1
Table 1 shows the new estimation procedure for the cable tension using the first mode
2
nonlinear natural frequency. To estimate cable tension, only four constants ( , EI, EA, L) are
3
needed. In calculating the nonlinear natural frequency of the cable, the assumption about the
4
initial condition of the excitation using Equation (17) is needed. Also, to calculate the stretching
5
force Nconserv , the iterative calculation is needed. If the iterative calculation is cumbersome, the
6
approximate method proposed by Zaslavsky (17) or in Equation (19) can also be used.
7
8
TABLE 1 Estimation procedure for the cable tension using the first mode nonlinear natural frequency
Step
1
Procedure for each step
Check the material characteristics of the cable
Parameters to
calculate
U , EI , EA, L
Calculate the initial midpoint deflection parameter
2
w1
4qL4
S 3 L2 N a EI S 2
w1
Calculate the coefficients of nonlinear oscillations
4
§
N L2 3 Awi 2
¨¨ 1 2a 4I
© S EI
g n1
§ EI · § S ·
¨
¸¨ ¸
© U A ¹© L ¹
gn2
§ EI · § S · § 3 Awi ·
¨
¸¨ ¸ ¨
¸
© U A ¹ © L ¹ © 4I ¹
g n3
§ EI · § S · § A ·
¨
¸¨ ¸ ¨ ¸
© U A ¹ © L ¹ © 4I ¹
3
·
¸¸
¹
g n1 , g n 2 , gn3
4
4
Calculate the linear natural frequency
4
5
Z10
Z10
g11
Assign the initial conditions (recommend: a10 w1 , E10 0 )
a10 , E10
Calculate the nonlinear natural frequency
6
Z1
ª a 2H 2 9 g g 10 g 2
13 11
12
g11 «1 2
«
24 g11
¬«
TRB 2013 Annual Meeting
º» O(H
»
¼»
3
)
Z1
Paper revised from original submittal.
Choi, Park, and Nassif
7
13
Calculate the axial load from the measured first natural frequency
Na
Calculate the stretching force due to self-weight
8
N conserv
EI S 2
L2
§ 22 EA L2 q 2
¨
¨
S4
©
1/ 3
·
¸¸
¹
Calculate the total tension force
9
Ntotal
N a Nconserv
Nconserv
Ntotal
1
2
3
VERIFICATION EXAMPLE
4
The Alamillo Bridge, Spain
5
6
FIGURE 2 Layout of cables from the inclined pylon to the deck (by Casas (19)).
7
8
To verify the new formula, the tension force of the Alamillo Bridge cables was re-estimated
9
using the reported data. As shown in Figure 2, the Alamillo Bridge is the cable-stayed bridge
10
with just one inclined pylon. The parallel cable system consists of 26 cables. Each parallel cable
11
line has two cables that connect on each side of the pedestrian passage. Figure 2 also shows the
12
layout of the 13 parallel cables from the inclined pylon to the deck. For the anchorage of the
TRB 2013 Annual Meeting
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Choi, Park, and Nassif
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1
cable, DYWIDAG DB-E61 is used. For the C1~C12 cables, the cable consists of 60 epoxy
2
coated strands. The C13 cable consists of 45 epoxy coated strands. The diameter of the epoxy
3
coated strands that satisfies ASTM A416 Grade 270 is 15.24 mm.
4
5
For the Alamillo Bridge, Casas (18) tested the natural frequencies of the cables and
6
reported the estimated cable tensions. The estimated results by Casas (18) show a clear
7
contradiction between the calculation results using the existing formulas and the direct
8
measurement by jack pressure and strain gauges. In this paper, for the length and the natural
9
frequency of the cables, the work of Casas (18) is referenced, and for the material properties of
10
the strands, the data from DYWIDAG is used.
11
12
Table 2, Table 3, and Table 4 show the measured first mode natural frequencies and the
13
estimated tension forces for 26 cables using the formulas of Zui et al. (5), Mehrabi and
14
Tabatabai (8), and the new proposed formula. The values used in tables are as follows: for
15
C1~C12 cables, density of 8,811 kg/m
, axial rigidity of 1.64718u109 N, and flexural
16
rigidity of 1.14326u106 N˜m , and for C13 cables, density of 8,853 kg/m
, axial rigidity
17
of 1.23585u109 N, and flexural rigidity of 646,744 N˜m . The letters L and R attached to
18
the cable numbers in Tables refer to the left and right cables, respectively, in the parallel cable
19
position. In the Alamillo Bridge, two parallel cables are installed on the same transverse line and
20
act as a one-cable element in the global structural system because they are very near to each
21
other. However, the measured frequencies for the two parallel cables were different.
22
23
The average tension forces Nz_avg estimated by Zui et al.’s formula (5) show the greatest
24
values. The average tension forces Nmt_avg estimated by Mehrabi and Tabatabai’s formula (8)
25
show the second greatest values. Finally, Na_avg estimated by the new formula shows the
26
minimum values. The estimated tension forces showed considerable differences compared with
27
the average design values Td.. As mentioned before, this difference can be complemented by
28
adding the stretching force Nconserv due to initial curvature shortening to the estimated tension
29
forces. The total tension forces Ntotal_avg which include the stretching force Nconserv estimated by
30
Equation (19), agree well with the average design values Td given in Table 4. are the
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
15
1
calculated values in the design analysis of the Alamillo Bridge. This values are checked by the
2
jack pressure. In design, the parallel cables are assumed to be equally tensioned (non-eccentric
3
tensioning) and have the same tension force in the analysis.
4
5
TABLE 2 Tension force estimates for the Alamillo Bridge (Zui et al. (5))
Cable
No.
Length
[m]
Meas.
first
freq.
[Hz] (18)
8L
8R
9L
9R
10L
10R
11L
11R
12L
12R
13L
13R
169.2
169.2
186.0
186.0
202.6
202.6
219.5
219.5
236.1
236.1
253.0
253.0
0.69
0.69
0.63
0.64
0.56
0.55
0.52
0.52
0.46
0.47
0.50
0.50
Stretching
force
[ kN ]
Nconserv
Total Tension
force
[ kN ]
Nz_avg + Nconserv
Design
value
[ kN ] (18)
Td
4133
1036
5169
5121
4236
1104
5340
5013
3841
1169
5010
4640
3962
1233
5195
4562
3667
1295
4962
4199
3700
1024
4723
4365
Estimated tension by
Zui et al.'s formula
[ kN ]
Nz
Nz_avg
4133
4133
4169
4303
3911
3772
3962
3962
3588
3746
3700
3700
6
7
TABLE 3 Tension force estimates for the Alamillo Bridge (Mehrabi and Tabatabai (8))
Cable
No.
Length
[m]
8L
8R
9L
9R
10L
10R
11L
169.2
169.2
186.0
186.0
202.6
202.6
219.5
TRB 2013 Annual Meeting
Estimated tension by Stretching
Meas.
force
first Mehrabi and Tabatabai's
formula
[
kN
]
[
kN ]
freq.
[Hz] (18)
Nmt
Nmt_avg
Nconserv
0.69
0.69
0.63
0.64
0.56
0.55
0.52
4009
4009
4018
4163
3695
3535
3709
Total Tension
force
[ kN ]
Nmt_avg + Nconserv
Design
value
[ kN ] (18)
Td
4009
1036
5045
5121
4090
1104
5194
5013
3615
1169
4784
4640
3709
1233
4942
4562
Paper revised from original submittal.
Choi, Park, and Nassif
11R
12L
12R
13L
13R
16
219.5
236.1
236.1
253.0
253.0
0.52
0.46
0.47
0.50
0.50
3709
3182
3391
3541
3541
3287
1295
4581
4199
3541
1024
4565
4365
1
2
TABLE 4 Tension force estimates for the Alamillo Bridge (New estimation procedure)
Cable
No.
8L
8R
9L
9R
10L
10R
11L
11R
12L
12R
13L
13R
Length
[m]
Meas.
first
freq.
[Hz] (18)
169.2
169.2
186.0
186.0
202.6
202.6
219.5
219.5
236.1
236.1
253.0
253.0
0.69
0.69
0.63
0.64
0.56
0.55
0.52
0.52
0.46
0.47
0.50
0.50
Esitmated Tension
by New procedure
[ kN ]
Na
3930
3930
3906
4064
3502
3314
3467
3467
2695
2996
3397
3397
Stretching
force
[ kN ]
Total Tension
Design
force
value
[ kN ]
[ kN ] (18)
Ntotal_avg
Td
(= Na_avg+ Nconserv)
Na_avg
Nconserv
3930
1036
4967
5121
3985
1104
5089
5013
3408
1169
4577
4640
3467
1233
4700
4562
2846
1295
4140
4199
3397
1024
4421
4365
3
4
Figure 3 compares the total tension forces in the cables, in which the stretching forces
5
Nconserv due to initial curvature shortening given in Table 2 and Table 3 were added to the
6
estimated average axial loads Nz_avg , Nmt_avg.. The new procedure was more accurate, and the
7
stretching force Nconserv should be included for tension force estimation of the cable-stayed
8
bridge cables. Therefore, the application of other formulas to the cable-stayed bridges should be
9
considered carefully.
10
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
17
1
2
FIGURE 3 Estimated results for the total tension forces of the Alamillo Bridge cables.
3
4
CONCULUSION
5
In this study, a new estimation procedure for the total tension force of bridge cables is proposed.
6
The existing methods based on the taut string theory consider additionally the flexural rigidity of
7
the cables. However, they do not consider the initial deflection, the natural frequency changing,
8
and the initial curvature shortening of the cables. To incorporate these correlated nonlinear
9
influences in the tension force estimations, the new estimation procedure considers the total
10
tension force consisting of the applied axial load and the stretching force due to the initial
11
curvature shortening for the cable. A new approximation method for this stretching force was
12
also derived from the nonlinear natural frequency formula for the cable. Each step of the new
13
estimation procedure was clearly proposed in this paper. The reliability of the new method was
14
verified by re-estimating the tension forces of the Alamillo Bridge cables.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
18
1
2
ACKNOWLEDGEMENTS
3
This work is a part of a research project supported by Korea Ministry of Land, Transportation
4
Maritime Affairs (MLTM) through Core Research Project 1 of Super Long Span Bridge R&D
5
Center. The authors wish to express their gratitude for the financial support. This research was
6
supported by Basic Science Research Program through the National Research Foundation of
7
Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A
8
2007054).
9
10
REFERENCES
11
1.
Gil, H. B., J. C. Park, J. S. Cho, and G. J. Jung. Renovation of Structural Health Monitoring
12
System for Seohae Bridge, Seoul, South Korea. In Transportation Research Record:
13
Journal of the Transportation Research Board, No. 2201, Transportation Research Board of
14
the National Academies, Washington, D.C., 2010, pp. 131-138.
15
2.
Rohleder Jr., W. J., B. Tang, T. A. Doe, N. F. Grace, and C. J. Burgess. Carbon Fiber-
16
Reinforced Polymer Strand Application on Cable-Stayed Bridge, Penobscot Narrows, Maine.
17
In Transportation Research Record: Journal of the Transportation Research Board, No.
18
2050, Transportation Research Board of the National Academies, Washington, D.C., 2008,
19
pp. 169-176.
20
3.
Berube, K. A., R. A. Lopez-Anido, and V. Caccese. Integrated Monitoring System for
21
Carbon Composite Strands in Cable-Stayed Bridge, Penobscot Narrows, Maine. In
22
Transportation Research Record: Journal of the Transportation Research Board, No. 2050,
23
Transportation Research Board of the National Academies, Washington, D.C., 2008, pp.
24
177-186.
25
4.
Shinke, T., K. Hirokana, H. Zui, and H. Nishimura. Practical Formulas for Estimation of
26
Cable Tension by Vibration Method. Journal of the Japan Society of Civil Engineers, Vol.
27
294, 1980, pp. 25-34 (in Japanese).
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
1
5.
19
Zui, H., T. Shinke, and Y. Namita. Practical Formulas for Estimation of Cable Tension by
2
Vibration Method. Journal of Structural Engineering, ASCE, Vol. 122, No. 6, 1996, pp.
3
651-656.
4
6.
Utsuno, H., I. Yamagiwa, K. Endo, and K. Suguii. Identification of Flexural Rigidity and
5
Tension of One-Dimensional Structure Part 4: Transfer Function Method Applied to
6
Rotational Stiffness Boundary. Proceedings of the Japan Society of Mechanical Engineers,
7
No. 98-9, 1998, pp. 308-311 (in Japanese).
8
7.
by the Vibratory Method. Bulletin de liaison des laboratoires des ponts et Chaussees, Vol.
9
173, 1991, pp.109-114 (in French).
10
11
8.
Mehrabi, A. B. and H. Tabatabai. Unified Finite Difference Formulation for Free Vibration
of Cables. Journal of Structural Engineering, Vol. 124, No. 11, ASCE, 1998, pp. 1313-1322.
12
13
Robert, J. L., D. Bruhat, J. P. Gervais, and J. Chatelain. The Measurement of Cable Tension
9.
Ren, W. X., G. Chen, and W. H. Hu. Empirical Formulas to Estimate Cable Tension by
14
Cable Fundamental Frequency. Structural Engineering and Mechanics, Vol. 20, No. 3, 2005,
15
pp. 363-380.
16
10. Kim, H. K., J. W. Hwang, and M. J. Lee. Fundamental Frequency Extraction of Stay Cable
17
Based on Energy Equation. Journal of the Korean Society of Civil Engineers, Vol. 28, No.
18
1A, 2008, pp. 125-133 (in Korean).
19
20
11. Ahn, S. S. Vibration Investigation and Countermeasures for Stay Cables. Journal of the
Korean Society of Civil Engineers, Vol. 22, No. 3-A, 2002, pp. 663-678 (in Korean).
21
12. Park, T. H., S. Y. Moon, H. J. Joo, and B. H. Kim. Estimating Tensile Force of Hangers in
22
Suspension Bridges using Frequency Based SI Technique: I. Theory. Journal of the Korean
23
Society of Civil Engineers, Vol. 27, No. 2A, 2007, pp. 165-172 (in Korean).
24
13. Treyssède, F. Vibration Analysis of Horizontal Self-Weighted Beams and Cables with
25
Bending Stiffness Subjected to Thermal Loads. Journal of Sound and Vibration, Vol. 329,
26
2010, pp. 1536-1552.
TRB 2013 Annual Meeting
Paper revised from original submittal.
Choi, Park, and Nassif
20
1
14. Yan, B., G. Wang, and Z. Wang. Tension Force Measurement of Sagged Cable with An
2
Intermediate Support. Proceedings of the IABSE-IASS Symposium, London, England, No.
3
0760, 2011.
4
5
6
7
8
9
10
11
12
13
14
15. Shih, C. F., J. C. Chen, and J. Garba. Vibration of A Large Space Beam under Gravity
Effect. AIAA Journal, Vol. 24, 1986, pp. 1213-1216.
16. Hughes, G. C. and C. W. Bert. Effect of Gravity on Nonlinear Oscillations of A Horizontal,
Immovable-End Beam. Nonlinear Dynamics, Vol. 3, 1992, pp. 365-373.
17. Zaslavsky, A. Beams on Immovable Supports. Publications of the International Association
for Bridge and Structural Engineering, Vol. 25, 1965, pp. 353-362.
18. Casas, J. R. A Combined Method for Measuring Cable Forces: The Cable-Stayed Alamillo
Bridge, Spain. Structural Engineering International, Vol. 4, 1994, pp. 235-240.
19. Casas, J. R. Full-Scale Dynamic Testing of the Alamillo Cable-Stayed Bridge in Sevilla
(Spain). Earthquake Engineering and Structural Dynamics, Vol. 24, 1995, pp. 35-51.
20. Choi, D.-H. and W.-S. Park. Tension Force Estimation of Extradosed Bridge Cables
15
Oscillating Nonlinearly under Gravity Effects. International Journal of Steel Structures, Vol.
16
11, No. 3, 2011, pp. 383-394.
17
21. Nayfeh, A. H. and D. T. Mook. Nonlinear Oscillations. Wiley, New York, 1979.
18
22. Irvine, H. M. and T. K. Caughey. The Linear Theory of Free Vibration of A Suspended
19
Cable. Proceedings of the Royal Society of London, Series A, Mathematical and Physical
20
Sciences, Vol. 341, No. 1626, 1974, pp. 299-315.
21
22
23. Timoshenko, S. Vibration Problems in Engineering. D. Van Nostrand Company, Inc., New
York, 1937.
TRB 2013 Annual Meeting
Paper revised from original submittal.