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NONLINEAR FREE VIBRATIONS ANALYSIS OF A CABLE
-

SUPPORTED BRIDGE DUE TO CHANGE OF
THE MAIN BEAM’S RIGIDITY


J. HENDEL
1
, M. KICZKA
2


1

PhD.,
Częstochowa University of Technology, ul.
Akademicka 3, 42
-
200 Częstochowa, POLAND

2
MK Dach System, Gliwice, P
OLAND


ABSTRACT:

N
onlinear free vibrations analysis of a cable
-
stayed bridge with radius system of stay
-
cables are presented
in this article. Relationship between frequency of free vibrations and the main beam’s rigidity is analyzed. Computer
program
Robo
t Millenium
, which bases on the finite element method (FEM), was used for numerical calculations. The
difference between results of nonlinear construction model (geometrical nonlinearity) and linear model is shown.


1. INTRODUCTION

Cable
-
supported bridge
s may be considered as the
transition between the traditional constructed bridges
(beam
-
, frame
-
, arch bridges) and the suspension bridges.
Considering their advantages, among them: clear static
model, possibility of changing stay
-
cable’s tension, large
co
nstruction’s work reliability and a small height of a
stiffening beam (bridge deck) there has been made a huge
progress lately in that type of bridges. Steel bridge decks
constructions advantage is observed, especially
considering long
-
span bridges. Most p
opular systems of
stay
-
cables are fan, harp and radius. Known disadvantage
of cable
-
supported bridges is pretty high strain and
sensitivity of stay
-
cables and stiffening beams for
dynamic behavior. Effect of dynamic behavior (according
to PN
-
82/B
-
2000) is
taken into consideration by executing
a dynamic calculations or by multiplication a
characteristic values by dynamic coefficient. Therefore a
dynamic analysis of construction, especially free vibration
analysis, is important for the cable
-
supported bridges
.


2. CABLE
-
STAYED BRIDGE'S DESCRIPTION

There was taken a cable
-
stayed bridge shown on figure 1
to dynamic analysis as an example.



Fig. 1.

Static model of analyzed bridge.


Construction’s data is following:



Entire length of a bridge 500m (120+260+120)



B
ridge deck made as a complex steel
-
concrete plate,
30m width and 30cm thick. Grid with the main
beams and cross
-
bars is a static model of a bridge
deck.



Heavy pylons made of B75 concrete and squared
cross
-
section 5x5m, made as a hybrid portal frame,
relate
d to the letter “A”.



Round cables for stay
-
cables made of wires with the
Z
-
bar cross
-
section and preliminary tension from
1000kN to 12000kN


The grid of a bridge deck is a good calculation’s model,
furthermore it suites for fast change of the main beam’s
c
ross
-
section parameters very good. Bridge deck’s model
is following: complex steel
-
concrete plate 30cm thick,
cross
-
bars made of HEA 400 profile, fixed in 2m spaces.
Main beams are plate girders, fixed in 10m spaces.

New type of welded plate girder IMK has

been created to
examine the effect of change the main beam’s rigidity for
entire bridge’s free vibrations. Based on existing cross
-
section’s tables for steel constructions (IKS, PRS, etc.)
relationship between the height of a beam’s web and other
beam’s d
imensions was found. Based on that twenty new
cross
-
sections was created. The smallest of them has a
web height 250mm and the largest 4000mm. Gradation of
cross
-
section’s height is 250mm. Due to research
character of this paper, four last cross
-
sections: I
MK
4100, IMK 4200, IMK 4300 and IMK 4400 were designed
with maximum flange’s thickness
cm
15
8

. It is only a
mathematical model of increasing cross
-
section’s rigidity
and does not exist as a real girder plates.

Steel used to design above cr
oss
-
sections characterizes the
density
3
83
,
7852
m
kg



and Young elasticity modulus
GPa
E
205

. Main cross
-
section’s moments and radius
of inertia was shown in the table bellow.








Tab. 1.

Main girder plate’s IMK cross
-
section’s rig
idity data.


Area
Moments of inertia
Radius of inertia
[cm2]
[cm4]
[cm]
IMK 250
163
21 514
11,5
IMK 500
225
119 063
23,0
IMK 750
293
326 649
33,4
IMK 1000
380
644 207
41,2
IMK 1250
460
1 185 838
50,8
IMK 1500
545
2 005 938
60,7
IMK 1750
630
3 123 873
70,4
IMK 2000
680
4 232 477
78,9
IMK 2250
878
6 485 083
86,0
IMK 2500
1 025
9 707 342
97,3
IMK 2750
1 128
12 895 779
106,9
IMK 3000
1 150
14 867 133
113,7
IMK 3250
1 213
17 976 326
121,8
IMK 3500
1 325
23 070 954
132,0
IMK 3750
1 620
31 006 935
138,3
IMK 4000
1 800
40 605 000
150,2
IMK 4100
2 800
82 594 133
171,5
IMK 4200
3 600
116 880 000
180,2
IMK 4300
4 800
168 812 800
187,5
IMK 4400
7 200
274 450 000
195,2


Round cables Ø 30mm, Ø 40mm and Ø 50mm were used
in the discussed static model of cable
-
supported bridge.

Bridge deck’s static model is eight bearing points,
supported at the end of the main girders. These points a
re
pivot bearings of examined bridge. Preliminary tensioned
cables carry all the bridge deck’s weight and the moving
load. Shape of the two pylons is a hybrid of a portal frame
and a letter “A” They are fully fixed in the ground. Cross
-
section is constant
and its dimension is 5x5m. Pylon’s
height is 150m (50+100) on the static model, and a portal
frame’s inclination angle inside to the bridge deck is
7,125°.

Bridge was described using finite element method (FEM)
in such way: pylons were divided for 8 finite

elements,
deck for 1500 finite elements, homogenous cables for 1
finite element, main girders for 48 finite elements, and
cross
-
bars were divided for 3 finite elements. Analyzed
bridge contains globally 1560 nodes and 2560 finite
elements.

Cable elements
taken into consideration were
geometrically nonlinear, because of large displacements
gained.


3. NUMERICAL ANALYSIS

Robot Millenium

program was used for numerical
calculations of described cable
-
supported bridge.

Modal analysis is used to calculate self v
alues, their
derivatives (self frequencies, self periods) and self vectors,
mass share coefficients for self vibrations problem.

Self values and forms are estimated from the following
equation:


0
)
(
2




i
i
U
M
K



where:

K

-

construction’s rigidity
matrix,

M

-

construction’s mass matrix,

i


-

proper frequency (free wheel frequency) of the
‘i’ form,

i
U

-

vector of the self form ‘i’.


The number of self forms solved by the program can be
estimated by the upper
boundary of frequency or period,
or by the lower boundary of the included mass sum
(shown in percents), or directly as the required number of
forms.

Block under spaced iteration method is used to calculate
the general self problem.


0






B
K



which can we meet during the modal analysis or buckling
analysis of construction. There are used following
significations in the above equation:


K


-

rigidity matrix



B=M


-

mass matrix in case of dynamic
analysis, other cases:

B=K
S

-

where
K
S

is the ten
sion’s rigidity
matrix (stress
-
stiffness matrix) cause by
linearization of a proper nonlinear
operator.




-

self value





-

self vector


Block under spaced iteration method consists in
simultaneous vector’s iter
ations in settled dimension (size)
under spaced. Each vector, for which convergence process
is done, is deleted from the work under space and then
new starting vector is added instead of it. Perpendicularity
of these vectors is assured in each iteration’s
step.
Following criteria of convergence is used:


tol
k
i
k
i
k
i






1


where:

i, k



number of form and number of iteration’s step.

tol


tolerance.


Block under space iteration method is recommended in
cases, where it is oblique to find many self pairs

(self
values and vectors)


mostly more than 10.

Results of modal analysis are determined with a specific
precision ε, that is defined by the following equation:


i
i
i
i
B
K









1




Numerical of free vibration’s frequencies, for described
before cab
le
-
supported bridge calculations were done, in
dependence of the main beam’s rigidity, in the range
shown in table 1 and the cross
-
section’s area. Ten
fundamental forms of free vibrations were calculated and
shown on the following pictures.


Form 1.



Who
le oblong symmetric form. It shows half of the wave
length on the bridge span. Pylons moved outside, to the
end of the bridge deck. Boundary frequencies of this form
are
Hz
26
,
0
10
,
0

.



Form 2.



Oblong, anti symmetric form. Two wave lengths on

the
bridge span. Clear transverse deformation (connected with
a cross
-
beam’s rigidity). Pylons moved the same way.
frequencies of this form are
Hz
38
,
0
11
,
0

.


Form 3.



Torsional symmetric form. Linear torsional deformation
of the whole bridge

deck. Half a wave length on the
bridge span. Pylons gently moved outside. Boundary
frequencies of this form are
Hz
43
,
0
28
,
0

.


Form 4.



Whole transverse, symmetric form. Transverse deflection
of a bridge deck corresponds with half wave length

on the
bridge span, and torsional in due shape of a deflected
curve (effect of the cables tension). Minimal rotational
deformation of pylons, according to the transverse
deflection of a bridge deck. Boundary frequencies of this
form are
Hz
46
,
0
34
,
0

.








Form 5.



Hybrid oblong
-
torsional anti symmetric form. Whole
bridge deck deformation with clear torsional transverse
way accented. Half wave length with phase displacement
φ=90°. Lack of clear vertical pylons movement. Boundary
frequencies of this form are
Hz
45
,
0
32
,
0

.


Form 6.



Oblong symmetric form. Bridge deck torsion on the
bearings is none, maximum in the mid
-
span, corresponds
with 0,25 transverse wave len
gth. Clear transverse
deflection of pylons the same way. Boundary frequencies
of this form are
Hz
52
,
0
37
,
0

.


Form 7.



Oblong anti symmetric form. The bridge deck’s torsion is
largest next to the pylons, where the phase is opposite,
correspond
s with a wave length in transverse plain. Large
pylons deflection opposite sides. Boundary frequencies of
this form are
Hz
52
,
0
39
,
0

.




Form 8.



Oblong, symmetric form. Whole deck deformation
corresponding with 1,5 wave length with a phase
di
splacement φ=180°. Clear cross
-
beams torsional form’s
share. Pylons moved outside. Boundary frequencies of this
form are
Hz
49
,
0
36
,
0

.


Form 9.



Oblong anti symmetric form. The bridge deck
deformation corresponds with a two waves length with a

phase displacement φ=270°. The largest share of pylon’s
tower deformation. Boundary frequencies of this form are
Hz
54
,
0
40
,
0

.


Form 10.


Oblong symmetric form. The bridge deck deformation
corresponds with a 1,5 wave length. Large share of the

torsional forms. Clear pylon’s deformation especially in
the mid
-
height, opposite sides. Boundary frequencies of
this form are
Hz
62
,
0
40
,
0

.

Relationship free vibration’s frequencies and the main
beam’s rigidity for the cables diameter 3cm, 4cm

and 5cm
are shown in the graphs bellow:



Relationship between frequency and the cross-section's rigidity
Radius system [stay-cables nonlinear d=3,0cm]
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Frequenciesi [Hz]
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10


Relationship between frequency and the cross-section's rigidity
Radius system [stay-cables nonlinear d=4,0cm]
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Frequencies [Hz]
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10




Relationship between frequency and the cross-section's rigidity
Radius system [stay-cables nonlinear d=5,0cm]
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Frequencies [Hz]
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10

Moreover, the bridge construction’s free vibrations,
without geometrical nonlinearity taken into
consideration (stay
-
cables
as bar’s elements) has been
calculated. Therefore it can estimate what is the effect
of this nonlinearity for model’s free vibration’s
frequencies. Ten fundamental values of the self values
difference are shown below:


Effect of nonlinearity -difference of self values
Radius system; round cross-section of cables (d=3,0cm)
0
1
2
3
4
5
6
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Difference of self values
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10





Effect of nonlinearity -difference of self values
Radius system; round cross-section of cables (d=4,0cm)
0
1
2
3
4
5
6
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Difference of self values
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10



Effect of nonlinearity -difference of self values
Radius system; round cross-section of cables (d=5,0cm)
0
1
2
3
4
5
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
{log} Main beam's rigidity [cm4]
Difference of self values
w1
w2
w3
w4
w5
w6
w7
w8
w9
w10

4. CONCLUSIONS

First and second form of analyzed bridge are oblong. Two
curves corresponding with frequencies
ω
1

and
ω
2

got
almost identical course. These graph’s changeability
corresponds with the square functions. Deflection’s angle
of tangent to the
ω
2

curve is about twice greater then to
the
ω
1

curve. Character of graph’s changeability, showing
relationship be
tween self values, free vibration’s
frequencies and main beam’s rigidity, is similar for first
two self frequencies. For examine bridge of specified
properties none local extremes, for the specified range of
main beams cross
-
section’s rigidity (from 21 tho
usands
cm
4

to 275 millions cm
4
), were stated.

Third form is symmetric, oblong, in which cross
-
bars are
deformed transverse. Graph’s changeability, showing
frequency
ω
3

is much more different, considering values.
Character of graph’s changeability corresponds with
which the
ω
2

graph, excepted the difference of values, that
changes as the rigidity increases. Graphs showing
frequencies
ω
1

and

ω
2

do not have any extremes
for the
specified (by IMK cross
-
sections) rigidity’s range. The
ω
3

graphs includes the local extremes for linear stay
-
cables.
For stay
-
cables nonlinear there are no extremes and
graph’s values has an increasing trend.

Fourth form is fully transverse. Chang
eability of a self
value
ω
4

is synonymous at all. Small increment, gradually
increasing about the cross
-
section of IMK 3500 (23 mln.
cm
4
) and then loss of value. For cable elements that
increment decreases as the cable diameter gets higher.
After the value

reaches maximum, there is a large
decrease.

Fifth form is oblong. For cable elements the graph’s
changeability is almost flat till a cross
-
section IMK 3500
(23 mln. cm
4
), where its maximum is. The other part is
decreasing of logarithmic kind course.

Sixth

form is torsional, with transverse part of pylons
vibration. Graph’s changeability of
ω
6

is similar to
ω
4

and
differs nothing but values.

Seventh form is torsional too with transverse part of
pylons vibration. Graph’s changeability of
ω
7

increase up
to a cross
-
section IMK 2250 (6,5 mln. cm
4
) slowly, then
increases higher, of power kind cour
se. About the cross
-
section of IMK 4100 (82 mln. cm
4
) a graph reaches its
maximum and decreases little.

Eight form is now oblong. Its changeability is almost
constant, from a cross
-
section of IMK 1500 (2 mln. cm
4
)
it increases kind as a logarithmic functio
n to IMK 3750
(31 mln. cm
4
). However, at the end there is a decrease of
same logarithmic kind.

Ninth form is oblong too. The graph’s changeability is a
kind constant, linear. It shows a small, decreasing trend
from a cross
-
section of IMK 3500 (23 mln. cm
4
)
. The
decrease, however insignificant, increases for cables of
higher diameter.

Tenth form, as two above, is oblong. Its graph’s
changeability differs from the others, representing smaller


frequencies. For cable elements the graph is similar
concerning dif
ferent cable diameters. First constant, about
a cross
-
section of IMK 2500 (10 millions of cm
4
) its
character is increasing kind as a power function to IMK
4000 (41 millions of cm
4
), but finally it is kind a constant
graph.


Based on the analysis of numeric
al data and graphs (also
others, not included in this paper) following conclusions
had been made:

1.

Basically, for the first three self values of analyzed
bridges, if the main girders rigidity increases,
frequency of their free vibrations increases too,

2.

The
effect of geometrical nonlinearity increases for
stay
-
cables of higher diameter.

3.

First forms of analyzed bridges were always oblong,
symmetric,

4.

Use of nonlinear cable elements (for description of
the stay
-
cables in FEM) in a visible way appeases
the change
ability of examines graphs. (for examined
events),

5.

Local extremes (visible) for relationship in the linear
truss stay
-
cables moved in higher frequency area for
nonlinear cable stay
-
cables,

6.

For the examined events, change of cable’s tension
has omitted effe
ct on self values of free vibrations


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[1]

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-
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Jolanta HE
NDEL , DSc.

Częstochowa University of Technology

ul. Akademicka 3, 42
-
200 Częstochowa

e
-
mail: jhen@mib.pcz.czest.pl

WWW: http://mib.pcz.czest.pl


Mariusz KICZKA, MSc.

MK Dach System, Gliwice

e
-
mail: khmer@wp.pl