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EFFECTS ON NUMBER OF CABLES FOR MODAL ANALYSIS OF CABLE-STAYED BRIDGES

Yang-Cheng Wang Associate Professor & Chairman Department of Civil Engineering

Chinese Military Academy Feng-Shan 83000,Taiwan

Republic of China

ABSTRACT. Due to the use of computer tech- nique and the use of high strength material. The span lengths of cable-stayed bridges have been in- creased. Compared with continuous and suspen- sion bridges, cable-stayed bridge deck has sub- jected to much stronger axial force caused by the horizontal component of cable reactions. The ax- ial forces make the geometric nonlinearity for the bridges. Cable-stayed bridge is supported by ca- bles instead of internal piers. Therefore, the pre- stress of the cable, inclined angle between the cable and the bridge's deck as well as the cross section areas of the cables are the most impor- tant features for this type of structure.

With various number of cables, the cable- stayed bridges may have different prestress, in- clined angles and the cable cross section ar- eas. The stiffness of the bridge deck may be changed due to the axial forces. In this paper, the three dimensional finite element model of the bridge having the similar geometry with Quincy Bayview Bridge has been built. The modal anal- ysis has been carried out by using different num- ber of cables. The natural frequencies and their corresponding mode shapes are found and com- pared with those obtained from ambient test.

The important effects on the number of cables have been drawn. The numerical results have been presented in tabular and graphical forms.

Chun-Ho Hua Lecturer

Department of Civil Engineering Chinese Military Academy

Feng-Shan 83000,Taiwan Republic of China

Keywords: Cable-Stayed Bridge, Cable, Modal Analysis

NOMENCLATURE M Mass Matrix C Damping Matrix K Stiffness Matrix

U Nodal Displacement Vector U Nodal Velocity Vector U Nodal Acceleration Vector F External Force Vector 1. INTRODUCTION

Cable-stayed bridges have been more interest- ing in the recent years due to increasing span length and their aesthetics. Cable-stayed bridges have subjected to strong axial force due to theca- ble reactions compared with those of suspensions and conventional continuous bridges. In order to reduce the cable tension and the axial force act- ing on the bridge deck, Agrawal 1 proposed that the cable tension can be reduced rapidly by using more number of cables.

In the field of dynamic analysis such as aero- dynamic analysis due to wind loading, transient dynamic analysis due to traffic loading and the bridge subjected to seismic loading, these analy- ses are getting important for this type of bridge.

No matter what kind of dynamic analysis is ap-

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plied, the natural frequency of the bridge is nec- essary to be studied. In this paper, cable-stayed bridges having the geometry similar to a realistic cable-stayed bridge supported by different num- ber of cables are analyzed. This type of cable- stayed bridges has been investigated 2 ' 3 ' 4

' 5 by fi- nite element method and ambient test 6 . The nat- ural frequencies and their corresponding mode shapes are found and presented in tabular and graphical forms.

2. PROBLEM IDEALIZATION

In order to effectively model and solve the problem, the following idealizations are made:

• Members were initially straight and piece- wise prismatic.

• The material behavior was linearly elastic and the moduli of elasticity E in tension and compression are equal.

• The effects of residual stresses was negligibly small.

Bayview Bridge crossing Mississippi River is studied in this paper. Quincy Bayview Bridge is designed 7 in 1983 and completed in 1989. The bridge has three spans. It consists of a main span of 900 ft and two equal side spans of 440 ft for total span length of 1780 ft. The structure is described as the following sections;

3.1 Cables

The Quincy Bayview Bridge has 25 different cross section areas of 56 cables ranging from 4.074 in 2 to 13.892 in 2 constructed of 0.25 in di- ameter wires with an ultimate strength of 240 psi.

The inclined angles between the deck and the ca- bles are different ranging from 38° to 69.3°. The cables are 7-wire cables. Twenty-eight of them support the main span and 14 of them support each side span. The cables are connected at the bottom flange of the main girder. The first in- terval from the supports is 62 ft. The other in- tervals on the side span are 63 ft and the interval on main span is 60 ft. The cables are attached to the pylons at 9 ft intervals beginning at 6 ft from the top of the pylon. The cables have the

• The cable element is a straight, tension-only same cross section areas which are symmetric in element with uniform properties from end to the longitudinal direction.

end.

• The modus of elasticity of the steel and cable is 30 x 10 6 psi, and the poisson's ratio, v, is 0.3 and the unit weight is 490 lb/ft 3 .

• The modulus of elasticity of concrete is equal to 4.47 x 10 6 psi, Poisson's ratio, v, is 0.25 and the unite weight is 150 lb/ft 3 .

• The end supports are attached to the ground with a pinned connection.

• The pylon and the deck are also connected with a pinned connection and the bridge deck remains continuous.

3. STRUCTURAL DESCRIPTION

In order to investigate the realistic bridge, a bridge having similar geometry to that of Quincy

3.2 Bridge Deck

Figure 1 represents the typical cross section of the deck. The deck consists of a nine inch precast slab with two precast traffic barriers, five longitudinal steel stringers with equal spacing of 7.25 ft and floor beams transverse the main girder with spacing from 17 ft to 24 ft which transfers stringer loads to the main girder.

Figure 1 The Typical Cross Section of the

Bridge Deck

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The deck is supported by anchor piers at each end nite element model used in this paper is referred with pinned connections. The piers are concrete to Hua's model. The number of supporting of columns whose bases are fixed at the bed rock cables are modified based on the model.

under the water level.

3.3 Pylons

There are two pylons of Quincy Bayview Bridge. Each of the two pylons consists of two concrete columns and two struts. The upper strut connects with the two columns at the level of 78 ft from the top of the pylon and the lower strut supports the bridge deck.

3.4 Boundary Conditions

The boundary of the bases of the pylons are constrained in all of the translation and rotation directions. The both end of the side spans are con::;trained in the vertical direction. It is con- sidered by pinned connection. The bridge deck and the lower strut::; of the pylons are coupling to- gether in longitudinal and vertical direction. All of the boundary condition::; considered are made in the finite element model of the bridge.

3.5 Finite Element Model

The numerical analysis i::; feasible to solve this type of structure. The structure is modeled as a three dimensional finite element model. The cables are discretized a::; a three dimensional tension-only truss element. Each node of the el- ement has three degree of freedom, i.e. transla- tion in x-, y- and z-direction. As the truss ele- ment subjected to compre::>sive forces, its modu- lus of elasticity, E, will be considered as it is ap- proaching to zero. The pylons and the stringers are considered as three dimensional beam ele- ments. Each node of the beam element has six degrees of freedom, i.e., translation in x-, y- and z-direction as well as the rotation about x-, y- and z-direction. The bridge deck and the com- posite girder were modeled as a plate element.

Each node of the plate element also has six de- grees of freedom which are the same as those of beam element. Hua 2 et. al. provided the global model of this cable-stayed bridge. This fi-

4. SOLUTION PROCEDURES

The modal analysis is used to extract the natu- ral frequencies and mode shapes of a linear elastic structure. In the modal analysis, free, undamped vibrations are assumed, i.e. F=O and C=O. The governing equation is expressed as;

MU+GU+KU=F(x,t) (1)

Substituting F=O and C=O, the Equation (1) be- comes;

MU +KU = 0 (2)

For linear system, free vibrations will be har- monic of the form,

U = U

0

coswt (3)

Substituting U and U in the governing equation gives,

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For non-trial solution, the determinant of [K - w 2 M] must be zero;

IK- .\MI = 0 (5)

where >. = w 2 . If n is the order of the matrices, then the equation results in a polynomial of or- der n, which should have n roots; wi, w~, · · · , w~.

This is an eigenvalue problem, whose solution are the eigenvalues, >.i, and the corresponding eigen- vectors Ui. The eigenvalues represent the natu- ral frequencies of the system ( wi = v:\) and the

eigenvectors represent their corresponding mode shapes.

5. NUMERICAL RESULTS

Based on the finite element model and the solution procedure, the numerical results were found. Figure 2 represents the axial force due to the cable reactions acting on the bridge deck.

The horizontal axis of the figure represents the lo-

cation along the bridge deck. The vertical axis of

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the figure represents the normalized axial force.

The unity is considered as the horizontal com- ponent of the anchor cable reaction of the bridge supported by 56 cables (the original bridge). The center-dash line represents the axial forces acting on the original bridge supported by 56 cables.

The solid line represents the axial force acting on the bridge deck supported by 168 cables. The dash line represents the axial forces acting on the bridge supported by 280 cables. Figure 2 in- dicates that the original bridge has the strongest axial forces acting on the main span bridge deck.

-

.~ 7

"

< 6

-

~ 5

"

0 ~

.

"'

"

•• 3

E

"'2

"

c

z '

Cable Nurnb<"r

- - · · - - - b B - , - - . - - -

Bridge Deck

Figure 2 The Axial force Acting on the Bridge Deck for Various Number of Cables

In side spans, as the number of cable increased, the axial force acting on the bridge deck is in- creasing. As the cable number of 280, the axial force in side span is greater than that of main span created by the original bridge's support- ing cables. The bridge deck around the pylon is always subjected to stronger compressive axial forces. On the other hand the midpoint of main span is always subjected to tensile axial forces. It is hard to conclude that the bridge is dominated by the compressive axial force of by the tensile axial forces.

Even though the axial force changes the nat- ural frequency of the structural system, it is not easy to find the location dominating the bridge behavior. Table 1 represents the frequencies of the bridges supported by various number of ca- bles. As the number of cables increased the natu- ral frequency is also increased. It means that the structural stiffness is enhanced. Based on Figure 2, as the number of cables increased, the bridge deck around the pylon is subjected to stronger axial force than that of the original bridge. On the other hand, the bridge deck at midpoint of the main span is subjected to stronger tensile ax- ial force than those of the original bridge.

Table 1 The Frequencies of the Cable-Stayed Bridges Having Various Numbers of Supporting Cables

Number of Cables

Mode 56 168 280

Number Freq. Mode Freq. Mode Freq. Mode (Hz) Shape (Hz) Shape (Hz) Shape

1 0.45383 F 0.55201 L 0.64102 F

2 0.55230 L 0.57196 F 0.70137 L

3 0.61483 F 0.74006 F 0.81327 F

4 0.96661 F 1.2878 T 1.2584 p

5 1.0194 T 1.2959 T 1.2601 p

6 1.1368 F 1.3155 T 1.3774 T

F: Flexural Mode L: Lateral Mode

T: Torsional Mode P: Pylon Buckling Mode

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Therefore, the natural frequency of this type of bridge will be increased as the number of ca- ble increased. Regarding the mode shape corre- sponding to the natural frequency, the first flex- ural mode of the bridge supporting by 168 ca- bles is missing. The first mode of the original bridge is flexural but this mode is missing and the lateral mode is instead for the bridge sup- ported by 168 cables. The corresponding fre- quencies of these two modes have an excellent agreement. Compared with the second flexural mode of the original bridge and the first flexu- ral mode of the bridge supporting by 280 cables, the frequencies have also a good agreement. The difference between these two modes is less than 5%.

If the number of cable increases, the cables take more load. Not only the bridge deck takes care of more loads but also the pylons take care of more loads. The pylons are only subjected to compression. The stiffness of the pylon will be reduced by compressive cable reactions. Table 1 indicates that the fifth and sixth modes are py- lon flexural even though the moment of inertia of the pylon is as 94 times as much as that of the bridge deck. In order to present the mode shape of the natural vibration, Figure 3 shows the lat- eral mode of the bridge supported by 168 cables.

The free vibration modes of the original bridge are referred to Hua's et. al. paper.

Over View

Top View

Figure 3. The First Lateral Mode of the Bridge Supported by 168 Cables.

The second and third modes, i.e. the first and second flexural modes, are presented in Figures 4

and 5, respectively. The fourth and sixth modes, i.e. the first and the second torsional modes, are presented in Figures 6, 8, respectively. The fifth mode, i.e. the first pylon buckling mode, is rep- resented in Figure 7.

Over View

Side View

Figure 4. The First Flexural Mode of the Bridge Supported by 168 Cables.

Over View

Side View

Figure 5. The Second Flexural Mode of the Bridge Supported by 168 Cables.

If it is disregarding the agreement of mode shape, the natural frequencies is only considered.

Figure 9 represents the comparison of the first three frequencies for the bridges supported by 56, 168 and 280 cables. The horizontal axis of Fig- ure 9 represents the mode number of the bridge.

The vertical axis represents the corresponding

frequencies of the first three modes in Hz.

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Over View

Side View

Figure 6. The First Torsional Mode of the Bridge Supported by 168 Cables.

Over View

Side View

Figure 7. The First Pylon Flexural Mode of the Bridge Supported by 168 Cables.

Over View

Side View

Figure 8. The Second Torsional Mode of the Bridge Supported by 168 Cables.

,.-...

~

N '-"

u ~

z ~

~ a

~

~

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

r-.-56~a~l~~- .1~~-C~~e-S 0 280 cab~~: I L _____ - ·-·--- -- __ , _________ I

first mode second mode third mode MODE NUMBER

Figure 9. The Comparison of Natural Frequencies of the First Three Modes of the

Bridge Having Various ~umber of Cables.

As discussed in the previous, if the number of cables is increased. the frequencies are also m- creased but the increments are not linear.

6. CONCLUSIONS

Based on the previous work, the important conclusions can be drawn as the following:

• For realistic bridges. as the supporting ca- bles increased. the bridge deck around the pylon subjected to stronger compressive ax- ial force than that of the bridge supported by less number of cables. The axial forces will reduce the stiffness of the bridge deck.

On the other hand, the midpoint of the main span subjected to stronger tensile force which enhances the stiffness of bridge deck.

In global analysis, it is hard to determine

that the location subjected to compressive

axial force or the location subjected to ten-

sile forces will dominate the structural be-

havior.

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• As tht> number of cables increased, the pylon will subjected to stronger compressive axial force. It will also reduce the stiffness of the pylon.

• For the global analysis, as the number of ca- bles increased, the free vibration frequency will increase for this type of bridge.

8. REFERENCES

[1] T.P. Agrawal," Cable-Stayed Bridges- Para- metric Study", Journal of Bridge Engineer- ing, Vol.2, No.2, 1997, pp.61-67.

[2] Chun-Ho Hua and Yang-Cheng Wang,

"Modeling of a Cable-Stayed Bridge for Dy- namic Analysis", Proceeding of 12nd Inter- national Modal Analysis Conference, Vol.II, pp.1885-1899.

[3] John C. Wilson and Wayne Gravelle," Mod- elling of a Cable-Stayed Bridge for Dynamic Analysis", Earthquake Engineering and

Structural Dynamics, Vol. 20, pp.707-721, 1991.

[4] Ermopoulos, J. CH., Vlahinos, A.S., and Wang, Yang-Cheng "Stability Analysis of Cable-Stayed Bridges" , International Jour- nal of Computers & Structures, Vol. 22 No.

12, pp.1083-1089 June 1993.

[5] Vlahinos, A.S. and Wang, Yang-Cheng

"Nonlinear Dynamic Behavior of Cable- Stayed Bridges" , in the Proceedings of the 12th International Modal Analysis Confer- ence, Vol.II, pp.l335-1341, 1994.

[6] John C. Wilson and Tao Lin," Ambient. Vi- bration Measurements on a Cable-St.ayeG.

Bridge", Earthquake Engineering and Struc- tural Dynamics, Vol. 20, pp.723-747, 1991.

[7] Mojeski and Master, "Structural Drawings

of Quincy Bayview Bridge", Mojeski and

Master Consulting Engineering, Marrisburg,

Pennsylvania, 1983.

References

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