Estimation of cables’ tension of cable-stayed footbridge using measured natural frequencies
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Estimation of cables' tension of cable-stayed footbridge using measured natural frequencies
Przemysław Jakiel 1
Zbigniew Mańko 0
0 International University of Logistic and Transport , Soltysowicka, No. 19B, 51-168 Wrocław , Poland
1 Opole University of Technology , Katowicka Street, No 48, 45-061 Opole , Poland
This paper presents analysis of cables' tension of steel cablestayed footbridge using their field-test natural frequencies. A vibration method is usually used for the measured cable tension during the construction of cable systems stiffened with inclined cables. Practical formulas for the vibration method applied herein, mainly based on cablesag and vertical angle effects (a survey measurement), have been verified on the one-tower steel cable-stayed bridge. The bridge is situated in Sieradz (Poland) and it was the structure with the longest span concerning all the cable-stayed bridges in Poland until 1999. The obtained cable axial forces for estimated natural frequencies of low- and high-order modes are verified using FEM models. The final conclusions drawn on the basis of conducted studies can be useful for technical diagnosis, monitoring programs and repair works of similar class of cable-stayed bridges.
1 Introduction
A knowledge of cable tensions is very important concerning suitable geometry of the
cablestayed bridges and it allows to create their detailed calculation models [
1
]. The cable axial
force can be estimated by the method using measured natural frequencies, which depend
not only on axial force but on flexural stiffnes, cable sag and inclination of the cable chord
as well. According horizontal cable configuration with relatively small sag the cable tension
F we can obtain from well known formula F = A L2f2/g (A - cross section area, - material
density, L - length, f – natural frequency, g - gravitational acceleration) applicable for
firstorder mode only, but the result may be burden with error concerning longer tendons [
2
].
In this paper, verification of the accurate method which allows to estimate the cable
axial force using measured its natural frequencies, taking into consideration aforementioned
material and geometrical parameters is presented.
Considering the very slender tendons, i.e. those that are the structural members in the
cable-stayed bridges for instance is difficult to excite the cable artificially to first or
secondorder mode oscillation. In this case, one should use the results obtained from stationary
vibrations, in which modes of high order are usually dominant.
So called vibration method proposed by Zui, Shinke and Namita [
3
] is herein briefly
described with unification of practical formulas, and applied to the results of natural
frequencies obtained from the measurement conducted on a cable-stayed pedestrian bridge
in Sieradz (Poland) and the accuracy is confirmed using FEM models [
4
].
2 Short description of footbridge structure
The cable-stayed footbridge in Sieradz was built in 1978 for pedestrian traffic and
emergency vehicle, e.g. fire engine. The total length of this bridge is 142 m with effective
spans equal to 9.12+75.88+3x19.00 m, respectively. The steel superstructure consists of
one portal tower, consisted of closed, rectangular cross-section, cable system arranged in
harp layout and an orthotropic plate deck (Fig. 1a). Total height of the tower transversally
braced in the locations of cable joints is 43.48 m. The cables are fixed to the tower and
anchored to the deck. The tower is rigidly fixed to the main girders and supported by the
steel hinge bearings situated on a pier P2 (Fig. 1b).
LC1 - dead load
LC2 - dead and live loads
live load
C1
C2
C3
A1 P1
P2
P3
P4
A2
CL Asphalt 40 mm
Steel deck plate
10 mm
4200/2
b)
c)
a)
The steel deck structure, which is 4.10 m wide, is composed of two I-beams 550 mm
main girders, the openwork and plain girder double-tee cross-beams of 525 mm and 250
mm high respectively (Fig. 1c). The cross-beams are alternately spaced: the major ones
7.40 m and the minor ones 1.90 m. The steel plate orthotropic deck consists of 10 mm thick
plate and the longitudinal stiffening ribs (12×200). The plate deck is directly topped with a
50 mm thick layer of asphalt [
5
].
The cables are made of helical wires closed by the z-profile wires in the cross-section in
diameter of 48 mm. A peculiar feature of this bridge are the rocker bearings at each support
except the pier P2, which enable limited longitudinal and vertical displacements of the
spans at their supports.
3 Vibration method’s formula
A simplified algorithm of presented herein the vibration method is used for free-oscilation
frequencies of the cable with its span length denoted as L, sag d and angle of cable
inclination α (Fig. 2a) [
3
].
Introducing a dimensionless parameter (1):
L
F / EJ ,
(1)
DYN-WIND'2017
where EJ is flexural rigidity of cable, the range of application of the method is specified as
any region (...truncated)