NONLINEAR AEROSTATIC ANALYSIS OF SELF- ANCHORED AND BI-STAYED CABLE-STAYED BRIDGES USING SAP: 2000

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JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

CIVIL ENGINEERING

ISSN: 0975


6744|
NOV 09 TO OCT 10

| Volume 1, Issue 1


Page
15


NONLINEAR AEROSTATIC ANALYSIS OF SELF
-
ANCHORED AND BI
-
STAYED CABLE
-
STAYED BRIDGES
USING SAP: 2000


1

N D SHAH
,
2
DR. J A DESAI


1
Head
, Department of
Civil Engineering, Education

Campus
,Chana,

CHARUSAT

University, Ch
anga
-
388 421, Dist: Anand
, India.

2
Prof
essor

& Head
, Department of A
pplied Mechanics
,

SV National Institute of Technology
,

Surat
-
395 007,
India


nirajshah.cv@ecchanga.ac.in
,

jad@amd.svnit.ac.in



AB
STRACT

:
For bridging the long and unsupported spans, the cable
-
supported bridges present the most
elegant and

efficient structural solution. And hence, they are increasingly being constructed all over the world.
As the span of the cable
-
stayed bridge inc
reases, the nonlinearities also go on increasing. These nonlinearities
are due to sag in the cable, axial force
-
bending moment interaction in the girder and tower and due to large
deformations of the overall structure. Further, the nonlinearity magnifies w
ith the influence of wind loading.

The paper presents finite element approach for the geometric nonlinear aerostatic analysis of cable
-
stayed
bridges with vehicular interaction. The concept of longer span is elaborated here with help of parametric study.
A
gain the effect of anchoring top cables of cable stayed bride i.e. bi
-
stayed concept is also carried out. The
results shows that the concept of spread pylon proved useful in reducing the cable tensile forces whereas the bi
-
stayed bridge concept is useful i
n reducing the forces in cable, girder and pylon.



Keywords:
Cable
-
stayed Bridges, Nonlinear, Aerostatic, Bi
-
stayed Bridge, Vehicular Loading


1.
INTRODUCTION

Achieving larger spans by inventing new bridge
systems has always been a fascinating intellect
ual
challenge. To increase the maximum span of cable
-
stayed bridges, Uwe Starossek [6] has developed a
modified statical system. The basic idea of this new
concept is the use of pairs of inclined pylon legs
that spread out longitudinally from the foundatio
n
base or from the girder level. The system geometry
entails steeper and shorter cables. The horizontal
cable force component introduced into the deck is
smaller. Additionally, cable sag is reduced.
However, there is a need to study the behaviour of
this b
ridge system under vehicular and wind
loaning to compare its stiffness with the bridge
system having conventional pylons.

As opposed to the classical suspension bridge, the
cable
-
stays are directly connected to the bridge
deck resulting in a much stiffer s
tructure. A large
number of closely spaced cable
-
stays support the
bridge deck throughout its length, reducing the
required depth and bending stiffness of the
longitudinal girder to a minimum, thereby allowing
the construction of relatively longer spans. T
he
structural action is simple in concept: the cable
carries the deck loads to the towers and from there
to the foundation. The primary forces in the
structure are tension in the cable
-
stays and axial
compression in the towers and deck; the effect of
bendi
ng and shear is considered secondary.

2.

FINITE ELEMENT FORMULATION

Based on the finite element concept, a cable
-
stayed
bridge can be considered as an assembly of a finite
number of cable and beam
-
column (for girder and
tower) elements. In this study some ass
umptions
are made as follows. The material is homogeneous
and isotropic. The stress

strain relationship of all
material remains within the linear elastic range
during the whole nonlinear response. The external
loads are displacement independent. Large
disp
lacements and large rotations are allowed, but
strains are small. All cables are fixed to the tower
and to the girder at their joints of attachment.


2.1 STIFFNESS ELEMENTS

In general, the relationship of element forces and
element displacements of a finit
e element can be
expressed as:
0
j jk k j
S KE u S
 




(1)

Where S
j

is the generalized element forces, S
0
j
,
generalized initial element forces, u
k
, generalized
element coordinates, KE
jk
, element stiffness matrix.
In the following,
the element properties of the cable
and beam
-
column elements used in the study will
be briefly summarized.


2.1.1. Cable element with sag

The elastic cable is assumed to be perfectly flexible
and possesses only tension stiffness; it is incapable
of resisti
ng compression, shear and bending forces.
When the weight of the cable is neglected, the
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

CIVIL ENGINEERING


ISSN: 0975


6744|
NOV 09 TO OCT 10

| Volume 1, Issue 1


Page
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cable element can be considered as a straight
member. But under action of its own dead load and
axial tensile force, a cable supported at its end will
sag into a caten
ary shape, as seen in Fig. 1. The
axial stiffness of a cable will change with changing
sag. When a straight cable element for a whole
inclined cable stay is used in the analysis, the sag
effect has to be taken into account. On the
consideration of the sag
nonlinearity in the inclined
cable stays, it is convenient to use an equivalent
straight cable element with an equivalent modulus
of elasticity, which can well describe the catenary
action of the cable. The concept of a cable
equivalent modulus of elastici
ty was first
introduced by Ernst [1]. If the change in tension in
a cable during a load increment is not large, the
axial stiffness of the cable will not change
significantly and the equivalent modulus of
elasticity of the cable can be considered constant
during the load increment, and is given by

2
3
( )
1
12
eq
E
E
wL AE
T





(2)

in which E
eq
, equivalent cable modulus of
elasticity, E, effective cable material modulus of
elasticity, A, cross sectional area, w, cable weight
per

unit length, L, horizontal projected length of
the cable and T , tensile force in the cable. The
cable equivalent modulus of elasticity combines
both the effects of material and geometric
deformation. The value of the equivalent modulus
is dependent upon
the weight and the tension in
cable. Hence, the axial stiffness of the equivalent
element combing cable sag and cable tension
determined by the above equation is the same as
the axial stiffness of the actual cable.















Fig 1: Plane Cable Element

with Sag

When sag effect exists and the inclined cable stay
is represented by a single equivalent straight cable
element with one coordinate (relative axial
deformation) u
1
= ∆
l
, as seen in Fig. 1, the stiffness
matrix KE
jk

of the cable element has the va
lue as
follows:

1 1
[ ] [/] 0&[0] 0
jk eq
KE KE E A l for u for u
   
(3)

where
l

is chord length of cable element. The cable
stiffness vanishes and no element force exists for u
1

<
0,
i.e., when shortening occurs. If the cable sag
effect is neglected, the stiffness matrix for a linear
cable element becomes KE
jk

= [KE] = [EA/
l
],
where E
eq

is replaced by E.


2.1.2. Beam
-
column element

Since a high pretension force exists in inclined
cable stays, the towers and part of the girders are
subjected to large compression; this means that the
bea
m
-
column effect has to be taken into
consideration for girders and towers of the cable
-
stayed bridge. The beam
-
column element is straight
and it has constant cross
-
section. The cross
-
sectional area of the elements remains unchanged
during deformation. For
the beam
-
column element,
the engineering beam theory is employed and no
shear strain is considered. In a beam
-
column,
lateral deflection and axial force are interrelated
such that its bending stiffness depends on the
element axial forces, and the presence
of bending
moments will affect the axial stiffness. The element
bending stiffness decreases for an axial
compressive force and increases for a tensile force.
The beam
-
column effect can be evaluated by using
the stability functions [2, 3]. The plane beam
-
co
lumn element shown in Fig. 2 is employed in this
study. It has three element coordinates, two for end
rotations, u
1
, u
2

and one for the relative axial
deformation u
3
=∆
l
, where ∆
l

is the element axial
elongation or shortening. The element force
corresponding to u
j

is denoted by S
j
, in which S
1
and S
2

are the end moments and S
3

is the axial
force.
















Fig 2: Three DOF plane beam
-
column element


When the beam
-
column effect has to be taken into
consideration, the beam
-
column element stiffness
matrix has the following form [2, 3].

0
[ ] 0
0 0 (/)
s t
t s
t
C C
EI
KEjk KE C C
l
R A I
 
 
 
 
 
 



(4)

JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

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ISSN: 0975


6744|
NOV 09 TO OCT 10

| Volume 1, Issue 1


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where E is modulus of elasticity, A, cross
-
sectional
area, I, moment of inertia of th
e cross
-
sectional
area and
l
, element length.


3.

GEOMETRIC NONLINEARITY

Normally, an iterative procedure [5] is required to
solve the nonlinear equilibrium problem. In this
paper, the Newton
-
Raphson method is employed
with the linear solution as a first app
roximation.
For successive iterations, the actual strain and
stress are determined by taking into account both
the linear and appropriate nonlinear contributions
of the previous approximation. The tangent
stiffness matrix and the external and internal forc
e
vectors are formed by using the usual assembly
procedure for the current structural configuration.
The improved trial solution for the
(i +1
)
th

iteration
is obtained as:

q
i+1

=
q
i

+ ∆
q

i+1






(5)


4.

AEROSTATIC LOAD

Under the wind effect, the bridge is subjected to,
and acts to resist drag force, lift force and pitching
moment. Consider a section of bridge deck in a
smooth flow, as shown in Fig. 3. Assuming

that
under the effect of the mean wind velocity V with
the angle of incidence
0

, the torsional
displacement of deck is

. Then the effective wind
angle of attack is
0
  
 
. The components of
wind

forces per unit span acting on the deformed
deck can be written in wind axes as:

Drag Force:





2
1
2
y Z Y
F V C D
  



(6)

Lift Force:





2
1
2
z Z z
F V C B
  



(7)

Pitching Moment:




2 2
1
2
Z M
M V C B
  



(8)

Where,







,,
Y Z M
C C C
  
is the coefficient of
drag force, lift force and pitching moment in local
axis respectively.


B:is the deck width; D:is the vertical projected area

:is the air density, V
Z

:is the design wind speed

The wind forces in (6
)

(8
) are t
he function of the
torsional displacement of structure. They vary as
the girder displaces.
The above wind forces can be
transformed to the wind forces in global bridge axis
as:





2
1
2
y Y
F V C D
  







(9)






2
1
2
z z
F V C B
  





(10)





2 2
1
2
M
M V C B
  




(11)

Where,





0
0
tan
( ) sec
L
y D
N
C B
C C
A
 
  
 
 
 
 

,






0
0
tan
sec
D N
z L
C A
C C
B
 
  
 
 
 
 





2
0
sec
M M
C C
  
 

 

,

0
cos
Z
V V





Where,
1 2 3
Z b
V V K K K
   
,



V
b

= Basic Wind Speed

K
1
= Risk co
-
effi
cient [cl. 5.3.1
-
IS: (Part 3) 1987]

K
2

= Terrain height and structure size factor
[cl.5.3.2
-
IS: 875 (Part 3) 1987]

K
3

= Topography factor [cl.5.3.3
-

IS: 875 (Part 3)
1987]

A
N
= Vertical Projected Area of Bridge Deck

C
D
, C
L
and C
M

= Static aerodynamic co
-
efficient in wind axes. (To be obtained from
wind tunnel tests)










Fig. 3:
T
hree Components of Wind loads


5.

STUDY UNDERTAKEN

To increase the maximum span of cable
-
stayed
bridges, the use of pairs of inclined pylon legs that
spread out longitudinally
from the girder level was
considered. To examine the sensitivity of the
behaviour, aerostatic load based on 55m/s wind
speed along with vehicular loading as per IRC
6:2000 are used as input. The analysis is carried out
using the standard software “SAP
-
2000
”. The
models generated for the analysis shown in Fig. 4.
The span of bridges is 200m and 400m.
The
primary load cases and load combinations
considered consist of dead load, aerostatic load and
vehicular load as per IRC6:2000.

In all cases, the ratio of ma
in span to side span is
two. Towers have double plane system. Tower
height is one fourth of the main span lengths. The
deck is fully suspended at towers position and its
lateral displacement with respect to the tower is
restrained. All the cables have the
same section in
double plane system where as the double cross
section area than double plane system in single
plane system as there is only one plane of cables.
JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

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ISSN: 0975


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| Volume 1, Issue 1


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The cables have different pre
-
stressing forces to get
null vertical deflection or weight and us
e loads.
The Fig.
5

to 16

describes analysis results for both
considered span of bridge. The charts will be useful
for the bridge designers regarding the use of spread
pylon system in longitudinal direction to reduce the
cable forces and ultimately to redu
ce the cable cost
at the initial stage of cable stayed bridges.

Again, the responses of self
-
anchored type bridge
and partially earth
-
anchored (bi
-
stayed) type were
evaluated. It is found that the response primarily
consists of forces in cable, girder and
pylon as well
as the moment in girder and pylon. Cable stays
restrain the movement of the top of the tower and
transfer directly to ground central span load.



Self Anchored Spread Pylon System





Bi
-
Stayed Spread Pylon System

Fig. 4: Spread Pylon Mod
els





Fig. 5: Effect on Cable Force (200m)






Fig. 6: Effect on Cable Force (400m)







Fig. 7: Effect on Deck Force (200m)


JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

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ISSN: 0975


6744|
NOV 09 TO OCT 10

| Volume 1, Issue 1


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Fig. 8: Effect on Deck Force (400m)





Fig. 9: Effect on Deck Moment (200m)






Fig. 10: Effect on Deck Moment (400m)



Fig. 11: Effect on Pylon Force (200m)





Fig. 12: Effect on Pylon Force (400m)






Fig. 13: Effect on Pylon Deflection (200m)


JOURNAL OF INFORMATION, KNOWLEDGE AND RESEARCH IN

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ISSN: 0975


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NOV 09 TO OCT 10

| Volume 1, Issue 1


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Fig. 14: Effect on Pylon Deflection (400m)






Fig. 15: Effect on Deck Deflection (200m)






Fig. 16: Effect on Dec
k Deflection (400m)


6.

CONCLUSIONS

The main objective is to investigate the effects of
inclination of pylon. From the result analysis,
following points can be concluded:

1.

The system geometry entails steeper and
shorter cables. The horizontal cable force
compo
nent introduced into the deck is smaller.
Additionally, cable sag is reduced.

2.

The effect of nonlinearity magnifies as the
span of bridge increases because with the
increase in span, the bridge becomes more
flexible and susceptible to wind oscillations.

3.

Spr
ead Pylon Bridge with 30
0

spread angle
shows minimum forces in cables, deck and
pylon.

4.

Again the Spread Pylon system is effective in
reducing the pylon top deflection due to cross
ties provided at the top of spread pylon.

5.

For long spans, bi
-
stayed system s
hows
considerable reduction of axial force and
bending moment in the deck than that of self
-
anchored system. It also imparts better flexural
rigidity, which ultimately results in less
deflection.


7.

REFERENCES

[1].

Ernst HJ.
Der E
-
Modul von Seilen unter
Beruecksichtigung des Durchhanges. Der
Bauingenieur 1965;40(2):52

5.

[2].

Fleming J.F., “Nonlinear Static Analysis of
Cable
-
Stayed Bridges”, Computers &
Structures 1979; 10:621

635.

[3].

Ghali A, Neville A.M., “Structural Analysis: A
Uni
fied Classical and Matrix Approach”,
London: Chapman & Hall; 1978.

[4].

O. C. Zienkiewicz, “The Finite Element
Method”, 3rd Edn, Ch. 19. McGraw
-
Hill, New
York (1979).

[5].

R. Kao, “A Comparison of N.R. Methods and
Incremental Procedures for Geometrically
Nonlinear A
nalysis”, Computers & Structures
1974; 4: 1091
-
1097.

[6].

Uwe S., “Cable Stayed Bridge Concept of
Longer Spans”, Journal of Bridge
Engineering, Aug 1996, Vol
-
1, 99
-
103.