Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Temperature effects on tension forces and frequencies of suspended cables
Yaobing Zhao1, Yueyu Zhao1,2, Ceshi Sun1, Zhiqian Wang2
College of Civil Engineering, Hunan University, Changsha, Hunan 410082, PR China
2
College of Mechanical and Vehicle Engineering, Hunan University, Changsha, Hunan 410082, PR China
email:[email protected]
1
ABSTRACT: In order to study the effects of temperature changes on cable tension forces and mode frequencies, the thermal
stressed configuration is introduced and an analytical model of the suspended cable is proposed. Numerical results show that the
temperature effects do not change the natural relationships between tension forces and frequencies. Moreover, it is shown that
the temperature changes and tension forces have the negative correlations, and it is the same for the relationships between the
frequencies of the anti-symmetric (higher order symmetric) modes and temperature changes. Nevertheless, as to the frequency
of the first symmetric mode, both the negative and positive correlations which are largely dependent on the initial horizontal
tension force are found. Furthermore, the sensitivity analysis show that the sensitivities of tension forces and frequencies to
warming and cooling are not symmetric, but with the growth of the initial tension forces, the differences become smaller and
smaller.
KEY WORDS: Suspended cable; Temperature effects; Tension force; Frequency; Sensitivity analysis.
1
INTRODUCTION
Cables have been widely applied in many mechanical systems
and engineering structures, and these structures are always
subjected to time-variant environment conditions, such as the
temperature, humidity, traffic load and wind. Generally
speaking, the temperature effects on the material properties
and vibration characteristics of engineering structures are
obvious, according to the literature review by Xia [1].
Nevertheless, the investigation of cable dynamics when
thermal effects are taken into account has not been received
too much attention, although the thermal effects on these
typical structures were considered by Irvine as early as 1981
[2]. Recently, with the development of temperature sensitive
materials and the practical need in engineering, numerical
studies addressed the thermal effects on suspended and
inclined cables [3][4][5], cable-stayed bridge[6] and the longspan suspension bridge[7]. Actually, some existed experiment
results also reveal that the temperature changes could affect
the nonlinear dynamic phenomena of suspended cables
apparently [8].
Although many analysis methods are proposed to study the
temperature effects in cable structures, such as the regression
model [9] and the neural network technique [10]. However,
both in the estimation of cable tension force and structural
health monitoring, it is still a challenge task to separate the
effects of temperature changes on vibration properties of
structures. Therefore, it is necessary to reveal the relationships
between the temperature changes and tension forces and
frequencies of cables based on a mathematical model,
therefore, such an investigation is attempted in this study.
The structural of the paper is organized as follows: firstly,
the thermoelastic equilibrium equations of suspended cables
including the static and dynamic state are derived in section 2.
In section 3, the governing equations are solved by numerical
method (eg. Newton-Raphson method), and the effects of
temperature changes on cable horizontal tension forces, the
symmetric and anti-symmetric mode frequencies are
illustrated and analyzed. Moreover, the sensitivity analysis of
tension forces and frequencies to temperature changes are
given at the end of this section. Finally, some concluding
remarks are drawn in section 4.
2
GOVERNING EQUATION
Figure 1 Three configurations of the suspended cable
Fig. 1 shows the coordinate system and three configurations
of the suspended cable. The left support O is the origin of the
coordinate and the direction OB is taken as the x-coordinate,
and the direction perpendicular to OB is the y-coordinate of
which the descending direction is taken as positive. L , b and
T are the span, sag and initial tension force of the suspended
cable, respectively; H and V are the horizontal and vertical
support reaction; mg is the self-weight per unit length; Lx is
the arc-length of the cable; the displacement of the point are
denoted by u(x, t) and v(x, t) along the x and y directions,
respectively.
For the sake of brevity, the following assumptions are
proposed in this study:
(1) The sag-to-span ratio is sufficient small (b/L < 1/8) and
the profile of the suspended cable could be expressed by a
parabola;
(2) Only a perfectly-flexible transversal behavior is
considered, and any flexural, torsion and shear effects are
negligible;
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
(3) The cable is very long, and the temperature changes are
uniform along the length and cross-section;
(4) The Young modulus and boundary conditions are
independent on the temperature changes.
Firstly, as to the state of initial static under self-weight, the
static equilibrium of the x and y directions of the cable located
as (x,y) requires that
dx
T
=H,
ds
(1)
2
d2 y
⎛ dy ⎞
H 2 = − mg 1 + ⎜ ⎟ ,
dx
⎝ dx ⎠
where the approximate profile of the suspended cable could be
expressed as
mgL2
4b
y = 2 ( L − x ) x, b =
(2)
8H
L
Secondly, assuming that the temperature of environment
condition changes, the equations of static equilibrium
configuration under temperature effects are as follows
∂ ⎡
(T + T ′)⎛⎜ dx + ∂u ⎞⎟⎤⎥ = 0,
⎢
∂s ⎣
⎝ ds ∂s ⎠⎦
(3)
∂ ⎡
⎛ dy ∂v ⎞⎤
(T + T ′)⎜ + ⎟⎥ = −mg ,
∂s ⎢⎣
⎝ ds ∂s ⎠⎦
where u and v are the additional cable displacement owing to
temperature effects and T ′ is the additional tension force
( T ′dx/ds = H ′ ).
Moreover, adopting an exact kinematic formulation and
linear elasticity of the material, the following dimensionless
cubic equation for the additional tension h is expressed as
follows [3]
⎛
λ2 ⎞⎟ 2 ⎛⎜
λ2 ⎞⎟
h 3 + ⎜⎜ 2 + β +
h
+
1
+
2
β
+
h+ β = 0,
(4)
⎜
24 ⎟⎠
12 ⎟⎠
⎝
⎝
where
2
H ′ 2 ⎛ mgL ⎞ EA
EA
,λ =⎜
, β = αΔtLt
,
⎟
H
HLe
⎝ H ⎠ HLe
(5)
2
⎡ 1 ⎛ mgL ⎞ 2 ⎤
⎡
1 ⎛ mgL ⎞ ⎤
Le = L ⎢1 + ⎜
L
L
+
,
=
1
,
⎥
⎢
⎥
⎟
⎜
⎟
t
⎢⎣ 8 ⎝ H ⎠ ⎥⎦
⎢⎣ 12 ⎝ H ⎠ ⎥⎦
where α is the thermal expansion coefficient, Δt is the values
h=
∂ 2v
mg
H,
(8)
H + H′
∂t
where H is the external horizontal dynamic tension and ω is
the circular frequency.
Furthermore, the boundary conditions are as follows
v (0) = 0 , v ( L) = 0.
(9)
Therefore, it is convenient to obtain the solutions of Eq. (7),
and the symmetric and anti-symmetric modes are
distinguished in our study.
On the one hand, in the case of the anti-symmetric mode,
no overall additional tension would be generated ( T = 0 ), and
the expression for the circular frequencies is given by
ω~ = 2nπ (n = 1,2,3...) ,
(10)
where
( H + H ′)
2
+ mω 2 v =
m
(11)
.
H (1 + h )
On the other hand, in the case of the symmetric mode, the
additional tension is induced. Therefore, the frequencies of
symmetric modes are obtained by solving the following
transcendental equation
3
ω~ ω~ 4 ⎛ ω~ ⎞
tan = − ~2 ⎜ ⎟ ,
(12)
2 2 λ ⎝2⎠
where
~
λ2
λ2 =
.
(13)
(1 + h )3
ω~ = ωL
3
NUMERICAL RESULTS AND DISCUSSIONS
In the following numerical analysis, the dimensional
parameters and material properties of the suspended cable are
selected [3]: the cable span L = 200 m, the area of cross
the Young modulus
section A = 7.069 ×10 −2 m2,
11
E = 2.0 ×10 Pa, the density ρ = 7800 kg/m3 and the thermal
expansion coefficient α = 1.2 ×10 −5 / o C.
of the temperature changes and λ2 is the Irvine parameter
which collects the geometrical and mechanical properties of
suspended cables.
Finally, giving a slightly disturbance to the suspended cable,
and the dynamic equilibrium equations are expressed as
∂ ⎡
∂ 2u
⎛ dx ∂u ∂u ⎞⎤
(6)
+
⎟⎥ = m 2
⎢ T +T′+T ⎜ +
∂s ⎣
∂t
⎝ ds ∂s ∂s ⎠⎦
(
)
∂ ⎡
∂ 2v
⎛ dy ∂v ∂v ⎞⎤
(7)
+
⎟⎥ = m 2 − mg
⎢ T +T′+T ⎜ +
∂s ⎣
∂t
⎝ ds ∂s ∂s ⎠⎦
where u and v are the dynamic component of the x and y
directions and T is the additional dynamic tension.
Neglecting the longitudinal component displacement and
substituting the afore-mentioned static equilibrium equations,
therefore, the equation of motion is finally reduced to
(
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)
Figure 2: Comparison of the results obtained by Zhao et al.
and Treyssede for the first two symmetric/anti-symmetric
natural frequencies versus temperature changes( f s1 /f a1 : the
first symmetric/anti-symmetric mode natural frequency,
f s 2 /f a 2 : the second symmetric/anti-symmetric mode natural
frequency)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
First of all, in order to guarantee the validity of the
following numerical results, a comparison of the solutions
obtained in literature [3] is performed. Fig. 2 shows the first
two symmetric and anti-symmetric mode frequencies when
the temperature effects are taken into account. In this case, the
initial horizontal tension force H = 9.38 ×106 N. As could be
clearly observed in this figure, a very good agreement is
presented.
3.1
effect on the first symmetric mode frequency is significant, in
a particular range of tension force, with the decrease of the
tension force, the frequency is increasing. Referring to Zui et
al. [11] and Fang and Wang [12], the same conclusion based
on the experiment results was proposed. For this reason, in the
estimation of cable tension force, provided that the tension
force in the cable is not large, higher order modes would be
adopted [12].
Effects of temperature changes
In this section, the equation which involves the relationships
between the frequencies and cable tension forces when the
temperature changes are taken into consideration is solved by
the iterative method (eg. Newton-Raphson method). Moreover,
as a variable parameter, the initial horizontal tension H ranges
from 7.0×10 6 N to 2.5×10 7 N and the corresponding sag-tospan ratio ranges from 5.5 ×10 −4 to 2.0 × 10 −3 .
Fig. 3 shows the three dimensional graphs of temperature
changes with the first two symmetric and anti-symmetric
mode frequencies and the horizontal tension. As a matter of
fact, a comparison of the anti-symmetric modes (Fig. 3(a) and
3(b)) and symmetric modes (Fig. 3(c) and 3(d)) evidences that
the temperature effects on the first symmetric modes are much
more complicated than those on the anti-symmetric ones and
the higher order symmetric ones. Nevertheless, with the
increase of the order of symmetric mode, the temperature
effects on the symmetric modes tend to be more identical to
that on anti-symmetric modes, as shown in Fig. 3.
(a)
(b)
(c)
(d)
Figure 4. Relationships between the horizontal tension
H + H ′ and the first symmetric/anti-symmetric mode
frequencies ( ω s1 /ω a1 ) .
From a physical viewpoint, most of the cables in
engineering are made of steel. Provided that the environment
temperature changes, the length of the cable varies,
correspondingly. Fig. 5 shows the effects of temperature
changes on horizontal tension forces of suspended cables. As
could be observed in Fig. 5, no matter the initial tension force
is, the temperature changes have a global negative effect on
the tension forces. Therefore, in practical designing, the
effects of temperature changes on the pre-stressed losses
should be taken into consideration seriously. More
importantly, it is noted that the slope of every curve in Fig. 5
is different. In other words, as to different initial horizontal
tension, the sensitivity of tension force to temperature changes
is different.
Figure 3. Three dimensional graphs of temperature changes
with the first two symmetric/anti-symmetric mode frequencies
and the horizontal tension: (a) the first anti-symmetric mode,
(b) the second anti-symmetric mode, (c) the first symmetric
mode, (d) the second symmetric mode.
In Fig. 4, the relationships between the first symmetric
(anti-symmetric) mode frequency ω s1 / ω a1 and the horizontal
tension under three different temperature conditions are
illustrated. It should be pointed out that the temperature
effects do not change the natural relationships between the
frequencies and tension force. Furthermore, because the sag
Figure 5. Effects of temperature changes on horizontal tension
forces of the suspended cable.
Fig. 6 exhibits the relationships between the temperature
changes and the first symmetric (anti-symmetric) mode
frequency ω s1 / ω a1 respectively. In the case of anti-symmetric
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
mode (Fig. 6(a)), with the increase of temperature changes,
the frequency ω a1 decreases. In contrast, in the case of
symmetric mode (Fig. 6(b)), provided that the temperature
changes are rising, whether the frequency ω s1 is ascending or
descending is largely dependent on the initial horizontal
tension force, as shown in Fig. 6(b). Specifically, when the
initial horizontal tension is small (eg. H = 7.0 × 10 6 N), the
frequency has a positive correlation with the temperature
changes. The conclusion was confirmed by the finite element
results obtained by Treyssede [3]. On the other hand, provided
that the initial tension is large (eg. H = 2.5 × 10 7 N), the sag
effect is negative. Therefore, with the increase of the
temperature changes, the frequency is decreasing. It should be
pointed out that, as to the moderate initial horizontal tension
(eg. H = 1.5 × 10 7 N), the frequency is not a monotonic
function with the temperature changes. More importantly, it is
interesting to find out that there are some cross-points in Fig.
6(b). Hence, assuming that the temperature changes are
properly chosen, the suspended cables which have different
initial tension forces may have the same first symmetric mode
frequency, it is because of that the temperature changes may
lead to the variation of the Irvine parameter. By comparison,
no cross-points are found in the case of the anti-symmetric
mode (Fig. 6(a)).
Figure 6. Temperature changes versus the frequencies: (a) the first symmetric mode, (b) the first anti-symmetric mode.
3.2 Sensitivity analysis
In practical, the researchers could not estimate the tension
forces and mode frequencies of suspended cables at the same
environment conditions, especially the temperature. Moreover,
different cables, even the same cable, have different values of
tension forces during different stages. Therefore, it is very
important and necessary to investigate the sensitivity of
tension forces and frequencies to different values of
temperature changes. For the sake of brevity and simplicity,
only the first symmetric and anti-symmetric mode frequencies
are considered in this study. Hence, three following nondimensional parameters are defined:
ω Δt
ω Δt
H Δt
χ1 =| 0 − 1 | , χ 2 =| s01 − 1 | , χ 3 =| a01 − 1 |, (14)
H
ω s1
ω a1
where H Δt , ω sΔ1t and ω aΔ1t are the values of tension forces, the
first symmetric and anti-symmetric mode frequencies when
the value of temperature changes is Δt . Correspondingly, H 0 ,
ω s01 and ω a01 are the relative ones when Δt = 0 .
Generally speaking, Figs. 7 and 8 show the sensitivity of
tension forces and frequencies ( ω s1 and ω a1 ) to four different
values of temperature changes. It manifests from these figures
that the effects of warming and cooling are not symmetric, but
with the increase of the initial horizontal tension forces, the
differences become smaller and smaller.
Figs. 7(a) and 8(a) exhibit the sensitivity of tension forces to
temperature changes. In Fig. 7(a), there are critical values of
initial horizontal tension forces, no matter the initial tension
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force is larger or smaller than the critical one, the sensitivity
of tension forces is decreasing. In other words, as to the nonprestressed and high-prestressed cables, the temperature
effects could be neglected. Furthermore, the critical value is
largely dependent on the value of the temperature changes. In
Fig. 8(a), it is noted that the effect of cooling is more apparent
than that of warming in a large range of initial horizontal
tension force. Figs. 7(b) and 8(b) report the sensitivity of
frequency ω s1 to four different values of temperature changes.
As described in Fig. 7(b), with the increase of the initial
tension force, the sensitivity behavior of the frequency is very
complex and oscillatory. Moreover, in a particular range of
initial tension forces, it is interesting to figure out that even
the absolute value of temperature changes is smaller, but the
sensitivity of this frequency may be larger. In Fig. 8(b), it is
recognized that the sensitivity of frequency to cooling is not
always larger than that to warming. In other words, different
ranges of initial forces would draw different conclusions on
the sensitivity of frequency ω s1 . In addition, the sensitivity of
frequency ω a1 to different temperature changes is shown in
Figs. 7(c) and 8(c). By comparing Figs. 7(a) and 7(c), not too
many differences could be observed. Nevertheless, by
comparing Figs. 8(a) and 8(c), it is easily to realize that when
the initial tension force is small, the sensitivity of frequency to
cooling is a little higher that to warming. However, provided
that the initial tension force is large, the sag effect is small, in
this case, slightly higher sensitivity behavior of frequency to
warming than that to cooling could be observed.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Figure 7. Initial horizontal tension forces versus three parameters of the cable under different values of temperature changes: (a)
tension force; (b) the first symmetric mode; (c) the first anti-symmetric mode
Figure 8. Initial horizontal tension forces versus the sensitivity difference under different values of temperature changes: (a)
tension force; (b) the first symmetric mode; (c) the first anti-symmetric mode
4
CONCLUSIONS
In this study, the thermal stress equilibrium configuration of
suspended cable is introduced, and then an analytical model is
proposed to study the influences of temperature changes on
tension forces and frequencies.
Generally speaking, the temperature changes have a global
negative effect on the tension forces, the anti-symmetric mode
and higher order symmetric mode frequencies. Whereas for
the first symmetric mode frequency, both negative and
positive effects are found, and they are largely dependent on
the initial tension forces of the suspended cable. Furthermore,
there are some peak values in the sensitivity analysis of
tension forces and frequencies, and these values are largely
dependent on the temperature changes. On the other hand, it is
important to emphasize that the sensitivity of tension forces
and frequencies to warming and cooling is not symmetric,
although with the increase of the initial tension force, the
difference becomes smaller and smaller. Finally, it should be
pointed out that the temperature effects do change the values
of tension forces and frequencies, but it does not change the
natural relationships between frequencies and tension forces.
ACKNOWLEDGEMENT
The work was supported by the National Natural Science
Foundation of China (No.11032004) and the Funding Method
for Hunan University Graduate Students to Participate in
High-level International Academic Conference (HNU[2009].
No. 13).
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