Ichinose, Luiza Hiroko

Kitada, Toshiyuki

Shirooka, Masakazu

Yamaguchi, Takashi

Osaka City University Repository

Mern. Fac. Eng., Osaka City Univ., Vol. 40, pp. 53 - 57 (1999)
Development of a Program for Analyzing
Visco-Elastic Behavior of Cable Supported Steel Bridges
Takashi YAMAGUCHI *, Toshiyuki KITADA**,
Masakazu SHIROOKA *** and Luiza Hiroko ICHINOSE****
(Received September 30, 1999)
Synopsis
Developed in the present study is an analytical program for cable supported steel bridges, in which the viso-
elastic behavior of bridges can be considered. The analytical program consists of two parts, one is for the analysis
of the visco-elastic behavior, and another part is for the FEM analysis of the overall structures. The visco-elastic
behavior is evaluated by using a three-element model. Then the applicability of the program is discussed using
simplified models. Finally, the long-term behavior of cable-stayed bridges with steel cables or carbon fiber
reinforced plastics (CFRP) cables are investigated by using this program.
KEYWORDS: visco-elasticity, cable-stayed bridges, long-term behavior, steel cable, CFRP cable
1. Introduction
In Japan, bridges with cables such as cable-stayed bridges have and Nielsen bridges been frequently
constructed recently due to their rational mechanical properties. However, since the cables, which are very
important structural elements, show visco-elastic behavior, their stress and strain vary with time. As a result, due
to such a visco-elastic behavior significantly large stress or large displacement may occur during the bridge
service period. Therefore, in order to maintain the performance of such bridges, long-term behavior should be
precisely predicted in advance.
In the present study, the analytical program for two dimensional framed models considering the visco-elastic
behavior of cables was developed at first. In this program, finite deference method is used for the analysis on the
visco-elastic behavior of cables, and FEM is utilized for the structural analysis at each time step. Secondly, in
order to investigate long-term behavior of cable-stayed bridges, the program was applied to two cable-stayed
bridge models, one with steel cables, another with carbon fiber reinforced plastics cables (CFRP).
2. Development of Analytical Program
2.1 Outline of the developed analytical program
A flow-chart of the program is shown in Fig. 1. First, the FEM analysis!) is carried out at time to. Then, in
order to obtain the increment of tensile force in each cable, ~hc, the time dependent behavior analysis of cables is
carried out by using the finite deference method at t = to + Ll t. Using the obtained incremental forces in each cable,
Llhc, the FEM analysis the at time to +~ t is carried out. As a result, the bridge behavior can be obtained at t = to +
L1 t. This process shall be repeated until the target time.
*
**
***
****
Lecturer, Department of Civil Engineering
Professor, Department of Civil Engineering
Graduate student, Department of Civil Engineering
Japan Industrial Testing Corp.
53
54
where
T:filal time
ill : total rumer ofcables
i : mrmer ofcables
t:tinJ:
e e: creep strait it cables
e :cables strain
a :cables stress
.1 E ic :mcmemale e
.1fic : increm:mal tensen ofcables
Ej: Young's rrodub ofcables
A; :sectim area ofcables
Fig. 1 Flow-chart of the developed program
(2)
a
--+
Voigt rmdel
E2
n
a
+-
Constants, such as £}) £2 and T} are determined based on
the experimental measurements.
E c : creep strain
a : stress
T} : coefficient of viscosity
E e : elastic strain
2.2 Modeling of Visco-Elastic Behavior on Steel Cables
The three-element model as shown in Fig. 2 is applied
for modeling the visco-elastic behavior of steel cables. This
model is well known as one of the simplest models for
representing such a behavior. It 'consists of both a spring
element and the Voigt (Kelvin) element. In this figure, £1
and £2 denotes Young's modulus and n, the coefficient of
viscosity. Constitutive equation of this model is as follows:
d Cc £2 a
-=--c +- (1)dt n C n
2.0
3
• Experimental results
-- Evaluated curve
0.0 +-------.,...-----,....------.,
o 1 2
loglot (hour)
Fig. 3 Relaxation rate vs. time
Material properties of CFRP cableTable 1
~ nominal diameter norminal cross characteristic elastic modulus unit weightsectional area streo2th
unit 2 kN N/mm 2 N/mmmm mm
CFRP Cable 35 698.8 980 1.18xl0s 1.10 X 10.2
Fig. 2 Three-element model
a CFRP cable obtained
from experiments is shown
in Fig. 3, and the material
properties, in Table 12).
Based on these experimental results, the relationship between relaxation rate vs. time is evaluated by non-linear
regression analysis as follows:
2.3 Modeling of behavior on Carbon Fiber Reinforced
Plastics Cables (CFRP)
CFRP cables have better characteristics, such as higher
strength, resistance to erosion, and lighter weight, compared
with steel cables. Therefore, significant decrease in dead
load can be expected by using these cables.
The relaxation curve of
55
L1a 2
- = 0.0675(loglot) +0.18851og10t +0.1685ao
(3)
where
t : time (hour)
a 0 : initial stress
L1 a : increment stress
From this figure, it is found that the regression results agree well with the experimental ones.
10mCable
10m
Node 1
Node 1
••••••0 Node
Beam
Table 2 Material properties
Node 2 98kN
,,~--------,-..
Cable --. L
.,'MI-------.... 0 x
10m
(a) model A
MenDer E1 (N/nm) E2(N/rrm1 TJ(year· N/nm1
Cable l.%Xlrf 1.%Xlrf l.%Xlrf
Beam 2<riXlrf - -
i.8kN oLx
(b) model B
Fig. 4 Outline of the simplified models
':' Analytical Result
- Theory
______....--.,-0.955 )(10-5
0.4 ~--.---.-----r---r----.-.......,
° 2 4 6 8 10 12
• Iime (year)
FIg. 6 ReSUlts of model B
X10·S
1.1
_1.0
SO.9
c
eO.8
~0.7 ,
~.6 .
a0.5 -0.499)( 10-5
" Analytical Result
- Theory
X10·S
1.1
_1.0
!.0.9
c
eO.8
~0.7
~0.6Q
0.5 J 0.5 )( 10-5
0.4 >---~---.-~-..------.-----,
° 2 4 6 8 10 12
Time (yearl
Fig. 5 Results ot" model A
2.4 Verification of the Developed Analytical Program
z;
In order to check the applicability of the developed program, the visco-
elastic behavior of simplified models, as shown in Fig. 4, have been
investigated using this program. In case of the model A, the stress is kept
constant, and on the other hand, the stress is variable in case of the model B.
In this analysis, the time step of 0.1 year is adopted. Material properties
applied in the analysis are summarized in Table 2.
The analytical results are shown in Figs. 5 and 6. At the same time,
theoretical solutions are also shown in these figures. From these figures, it is
observed that the analytical results of these two cases are almost identical to
the theoretical results. Accordingly, the developed program can be applied
to the analysis of the visco-elastic behavior of cable bridges.
Overall view of the cable-stayed bridgeFig. 7
( ) : Cable No.
(unit: m)
Fig. 8 Dimensions and boundary conditions of the
half-model
3. Application to long-term behavior analysis of cable-stayed bridges
3.1 Analytical model
The analytical model of an adopted cable-
stayed bridge is shown in Fig. 7. This model is
based on the Ikuchi bridge which is one of the
Simanami bridges connecting Onomichi City
and Imabari City in Japan. Considering the
symmetry of the bridge, the half model as
shown in Fig. 8 was used. In particular, two
types of cables were applied to this bridge, one
is steel cables, another is CFRP cables.
The section modulus and material
properties used in the analysis are summarized
in Table 33) 4}. The applied load is shown in
Table 43}.
56
Table 3 Section modulus and material properties
~ Qoss E(N/mmz) A (mz) I (m4) If:z(N/mmz '1 (year· N/mmz,Section No.
Girder <D 2.06x1lt 1.302 2.258
~ 2.06x1cr 1.179 1.996
@ 2.06x1cr 1.449 2.365
@ 2.06x1lt 1.317 2.269
@ 2.06x1cr 1.045 1.767
Tower @ 2.06x1cr 0.640 4.538
(j) 2.06x1cr 0.781 5.583
(ID 2.06x1cr 0.800 5.818
@ 2.06x1lt 0.847 6.167
@ 2.06x1lt 0.996 6.754
CablcNo.
Steel Cable (1) 1.92x1cr 1.74xlO·z 2.40xIO·' 8.87x106 4.38xlo'
(2)-(11) 1.92x1o' 1.20x10·z 1.15xIO·' 8.87x106 4.38xIO'
(12)-(14) 1.92xlcr 5.81x10·3 2.69xlO·6 8.87xI06 4.38xIO'
CFRP Cable (1) 1.18xlcr 8.39xlO·3 S.60x10·6
(2)-(11) 1.18x1cr S.S9xlO·' 2.49xlO·6
(12)-(14) 1.18xlo' 2.80xlO·3 6.22xIO·7
Table 4 Applied load (dead load)
In the previous analysis in Ref. 3), the cable pre-
stresses has been determined in such a way .that
deflections do not occur at the anchorages of the cables
in the main girder, when the dead load is applied.to the
analytical model.
-----
Girder Tower Cable No. Steel Cable CFRP Cable
(1) 2.8655 0.1317
Dead Load 182.3 185.2 (2)-(11) 1.9737 0.0878
(N/mm) (12)-(14) 0.9604 0.0439
3.2 Results and Comments
Analytical results of the two types of cable-stayed
bridges are shown in Figs. 9--11. As for the bridge with CFRP cables, the analysis was carried out until only
1,000 hours, because the experimental data over 1,000 hour are not yet available.
Fig. 9 shows the displacement of the center of the bridge. The horizontal axis is the time, and the vertical axis
is the normalized displacements divided by the initial displacements. It is found that in case of the bridge with
steel cables, the final displacement is about 2.5 times the initial displacement, and after 0.3 year, the displacement
stabilizes. On the other hand; in case of the bridge with CFRP cables, it is found that the final displacement is
more than 4.0 times the initial displacement. Therefore, from these results, steel cables can be considered better
than the CFRP cables from the viewpoint of relaxation.
Secondly, the changes of the tensile force in the cable (1) of the side span as shown in Fig. 8 is shown in Fig.
10. The horizontal axis is the time, and the vertical axis, the normalized cable axial force divided by the initial
cable axial force. It can be seen that in case of the bridge with steel cables the final normalized axial force will be
almost 0.9957 after 0.3 year. On the other hand, in case of the bridge with CFRP cables the predicted value for the
final normalized axial force ratio is about 0.994.
Fig. 11 illustrates the change of bending moment with the elapsed time. Vertical axis shows the normalized
bending moments divided by the initial ones. It is understood from this figure that M/MtO of the steel cable model
converges into 1.40 during about 0.3 year. On the other hand, that of the CFRP cable model can be predicted to
converge to 1.43.
.Predicted
---------~--------------
---
O.9 +----r----.----.-~___,.-.___- --r---r----.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (year)
1.0
1.5
1.4
1.3
~
~ 1.2
::E
1.1
--- CFRP Cable
-+- Steel Cable
'------------ z~~~~ _----.- CFRP Cable
-+- Steel Cable
1.001
1.000
:0.999
o
'Vi 0.998
____.......~...............~........_+_e ~ 0.997
c
.~ 0.996
c
~ 0.995
0.994
0.993 +---'---~----.----,--~___,.-r---T--~---'
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (year)
4 _-----------"'\-------------
/- predicted
O+----r---r-----.-----..----,.-r--.----.----r----.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Time (year)
Fig. 9 Displacements at the center Fig. 10
of the bridge
Cable forces at the upper Fig. 11
of the side span
Bending moments at the
base of the tower
57
4. Concluding Remarks
In the present study, in order to investigate the long-term behavior of bridges considering visco-elastic
behavior, the analytical program was developed. Analyzing the long-term behavior of two types of cable-stayed
bridges, one with steel cables, another with CFRP cables by using this analytical program, its applicability to
bridge structures was discussed. Concluding remarks and comments for future research needs obtained from this
study are as follows:
(1) The long-term behavior analytical program for 2D framed structures considering visco-elastic behavior was
developed. It was confirmed that the developed program is efficient.
(2) In the present study, the long-term behavior of a bridge with CFRP cables was investigated. However, due to
the limitation of the experimental data on the relaxation rate, the bridge behavior at the final stage could not
be analyzed. Therefore, in order to investigate the ultimate state of the bridge, long-term tension tests should
be carried out.
(3) The visco-elastic behavior of cables can be considered to be very small. However, considering that the
influence of concrete elements on the long-term behavior of the bridge structure cannot be neglected.
Improvements in the program considering other materials such as concrete are, therefore, necessary.
References
1) Fujitani, Y. : Mechanics of Frame Structures, Maruzen Book Company, pp. 1-107, March 1993 (in Japanese).
2) Kajima Corp., Nippon Steel Corp., Nippon Steel Chemical Co., Ltd. and Suzuki Metal Industry Co., Ltd. :
Fatigue Tests on 9 CFRP Strands (200tf class) and Application Test for External Cables of PC Members, Report,
April 1994 (in Japanese).
3) Uchida, S. : Application of Hybrid Cables Fabricated by New Materials and Steel Wires to Bridge Structures,
Master Thesis, Osaka City University, pp. 36-39, March 1998 (in Japanese).
4) Watanabe, E., Kamei, M., Inoshita, Y. and Nakade, O. : Creep and Relaxation Tests of Full Size Cables, Journal
of Structural Engineering, Vol. 36 A, JSCE, pp. 51-61, March 1990 (in Japanese).


Development of a Program for Analyzing Visco-Elastic Behavior of Cable Supported Steel Bridges

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Development of a Program for Analyzing Visco-Elastic Behavior of Cable Supported Steel Bridges

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