Seismic Stability Analysis of a Cable Stayed Bridge Pylon

8/14/2019 Seismic Stability Analysis of a Cable Stayed Bridge Pylon

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Cite the article: Balasubramanian S R and Ganesan K. (2005). Seismic stability analysis of a

cable stayed bridge pylon. Proc. of Intl. Conf. on Recent Advances in Concrete andConstruction Technology, 7-9 Dec, Chennai, TN, India. pp. 809-821.

809

SEISMIC STABILITY ANALYSIS OF A CABLE

STAYED BRIDGE PYLON

BALASUBRAMANIAN.S.R.+

, K,GANESAN.++

SRM Institute of Science and Technology, Kattankulathur, Kancheepuram (dist).

Alagappa Chettiar College of Engineering and Technology, Karaikudi.

ABSTRACTThe paper discusses the dynamic analysis and design of the pylon of a cable stayed

bridge forearthquake forces. A project has been programmed to check the adequacy of the

solution a system with lumped mass and elasticity approach (referred to as Idealized system)

for the problem. Because this method has some disadvantages that the contribution of the

mass of pylon to the inertia forces cannot be accessed accurately and the effect of

compressive forces, lateral forces to the vibration parameters cannot be included. Therefore a

comparative study between the solutions of system with lumped mass and elasticity (Idealized

system) and system with distributed mass and elasticity (referred to as Generalized system)

has been carried out and the results are compared. For the evaluation of earthquake forces,

response spectrum for the ground acceleration measured at ELCENTRO substation during

Imperial Valley earthquake of May 18, 1940 has been prepared, from which the pseudo

spectral accelerations are arrived. From the report, it is observed that the idealized system

method have resulted in variations in comparing to the Generalized system method. In lateral

direction the influence of idealization is much less when compared to the longitudinal

direction. Further, the designed pylon is reported as safe against ELCENTRO earthquake.

INTRODUCTIONRameswaram is one of the places of tourism importance in Tamilnadu. It is

an island situated between Tamilnadu and Srilanka. There is a meter gauge railwayline and a plate girder bridge for a length of 2200m between Mandapam and

Pamban, which connects Rameswaram island to the other parts of the country. Whenthis meter gauge line has to be widened, a new bridge to carry the heavier loads thatare anticipated for a broad gauge line should replace this existing bridge. So it isaimed to provide a fruitful solution for the problem by providing a cable stayedBridge (refer Fig 1.).

+ Lecturer, School of Architecture, SRM Institute of Science and Technology.++ Assistant Professor, Dept. of Civil Engineering, A C College of Engg. & Tech.


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Further the reason for the proposal of cable stayed bridge for this reach may bejustified as this type of bridges are proved to be economical and are broadly adoptedin recent days for medium and long span. This paper mainly discusses the procedureadopted for the dynamic analysis and design of pylon for earthquake forces (Fig 2.).

DYNAMIC ANALYSISDynamic analysis of Pylon has been performed under three different stages.

The first being Modal analysis, which includes determination of natural frequencies,mode shapes and hence the participation factors and the spatial distribution ofmasses for each mode. The second stage is the calculation of spatial distribution ofeffective earthquake forces based on the concept of Response Spectrum. And the

third, the check for stability of pylon for the evaluated earthquake forces. Dynamicanalysis of pylon has been carried out for two different directions namely Lateraland Longitudinal directions (degree of freedom in vertical direction is neglected)separately and the forces are superimposed in the third stage.

MODAL ANALYSISThe pylon is isolated from the whole structure and the modal analysis has

been performed based on the following assumptions;

1. Axial Stiffness of the pylon is very high and so the natural frequencies inthe vertical direction will be very high and the earthquake forces will be minimal. Infact, already we have given sufficient concentration for the forces in verticaldirection; hence the degrees of freedom in the vertical direction are neglected.

2. Rotational moment of inertia would practically be negligible and so therotational degrees of freedom are also condensed.3. The structure is symmetric about the lateral and longitudinal axes and so

there will not be any torsional response of the structure.4. Inertia forces due to the masses from deck are evaluated based on the

response of pylon itself and not on the actual response of deck.5. The damping is assumed as viscous damping with damping ratio = 5%.

The first three assumptions are very common and the fourth assumption has been specially adopted for this particular case. The natural frequency of the deckwill be low when compared to the pylon and so the response of the deck will becomparatively be higher than that of the pylon and in the same way, the inertiaforces too. However, only a fraction of the actual inertial force generated in from the

deck will be transferred to the pylon as the forces are to be transferred through cablestays and not by rigid members. Thus the approximation is justified, but thisapproximation would be slightly on the conservative side. In fact, isolation of pylonis possible only if such approximation is adopted (refer Fig. 3). Thus the complexityof the problem is reduced. Also this approximation is adopted for the modal analysisin lateral direction only. In the case of longitudinal direction the inertia forces willnot be transferred to pylon as the cable can freely roll over the pulley arrangementmade in the pylon.


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Idealized SystemThe free vibration equation for Idealized MDF system (refer Fig. 4.) is[M] {U} + [C] {U} + [K] {U} = 0 (1)where, [M] = [m] + [ms] (2)

[m] = lumped mass from deck[ms] = mass of pylon[C] = Damping matrix[K] = Stiffness matrix

The mass matrix has been arrived from the axial force on the pylon. For the lateraldirection, it includes the mass of pylon, mass of deck and mass of train (of the

heaviest combination) but not the axial force due to the initial prestress. But in thelongitudinal direction, there would not be any transfer of inertia force from deck topylon and so the mass matrix includes only the mass of pylon.

Generalized System

The modal analysis based on idealized system approach has somelimitations viz; The inertia forces due to the mass of spring may either be neglected or overestimated as in the Eq.2. The variation in the natural frequencies due to the presence of any axial orlateral loads can not be represented.

The above limitations may not be interfering much in most of the problemsand so the popular equation of motion, given in Eq.1 is commonly used. But in the

case of the pylon of a cable stayed bridge it may not be true because,o The mass of the pylon is very high.o All the loads from deck are transferred to the pylon and so the axial force will

be concentrated much on the pylon.Thus it is required to check the adequacy of Eq.1 and to quantify the

variation with the actual values; the same problem has been carried out based on thegeneralized system approach and the results are compared. The free vibrationequation for generalized MDF system (refer Fig 5. for Generalized SDF system) is,

[M*] {U} + [C*] {U} + [K*] {U} = 0 (3)where,

=

=

+=

L

0

L

0

22

n

2L

0

s

dx)]x('[Ndx)]x("[)x(IE*]K[

*]M[2*]C[

]m[dx)]x([)x(m*]M[

The mass matrix has been arrived from the axial force on the pylon. For thelateral direction, it includes the mass of pylon, mass of deck and mass of train (of theheaviest combination) but not the axial force due to the initial prestress. In the


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longitudinal direction, it includes only the mass of pylon. But in the calculation ofgeometric stiffness, the axial force includes the effect of Prestress also.

In the Generalized MDF approach, the choice of the shape function (x)plays a vital role. For our problem, the Hermitian shape function (refer Eq. 4 andFig. 6) is used. Although the section is not truly prismatic, the chosen shape functionis expected to give results with good degree of accuracy.

][2

3)("H

)]1()1([4

3)('H

]3232[4

1HH)(H

22

3321

=

=

++=>


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Element geometric stiffness matrix

=

56

56

56

56

L

N]K[ eG

Transformation matrix

=

C0

0C]T[ e

EVALUATION OF EARTHQUAKE FORCES

The next step is to assess the peak acceleration for each uncoupled SDFsystem. For this purpose the concept of response spectrum is employed. TimeHistory Analysis for various uncoupled SDF systems has been performed and their

peak responses are plotted against their natural frequencies, which is referred asResponse Spectrum here, unlike the Response Spectrum in the true sense. It has

been decided to use the N-S component of horizontal ground acceleration recordedat ELCENTRO - substation, California during Imperial Valley earthquake of May18, 1940, since it has a strong ground motion a maximum acceleration of 0.319g,and have lasted for about 32 sec. The Time history analyses have been done basedon numerical integration linear interpolation method. Then from the responsespectrum (Fig. 8), the pseudo-acceleration spectrum (Fig. 9) and the equivalent staticforces due to earthquake have been evaluated. For the evaluation of equivalent staticforces, the constants adopted are taken from IS:1893-2002 and are as listed below,

Zoning factor Z = 0.36 (for zone III)Importance factor I = 1.5Response reduction factor R = 5.0

ANALYSIS FOR EARTHQUAKE LOAD AND DESIGNIn this stage the earthquake forces evaluated are given to the pylon,

modeled in STAADPro and P- analysis has been performed. Then the stressresultants calculated by the P- analysis are checked against the capacities of thesections provided. And the stability of the pylon may be reported as safe orunsafe against ELCENTRO earthquake based on the results.

SUMMARY OF RESULTSIn lateral direction, The fundamental frequency obtained for idealized system & generalized systemare 5.421 rad/sec and 5.407 rad/sec respectively. For idealized system, the participation factors of first and first three modes are85.67% and 97.98% respectively. And the same for generalized system are 85.09%and 97.52% respectively. The undamped maximum response corresponding to the fundamental modeevaluated from idealized system method is 228.03 mm and from generalized system


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method is 242.3 mm. And for the damped systems, the values are 85.55 mm and85.22 mm. By idealized system approach, the base shear obtained is 1885 KN and711.6KN for undamped and damped systems respectively. Whereas by generalizedsystem approach the base shear obtained are 2005.7 KN and 686.9 KN forundamped and damped systems respectively (Table 1).

In longitudinal direction, The fundamental frequency obtained for idealized system & generalized systemare 13.56 rad/sec and 16.755 rad/sec respectively. For idealized system, the participation factors of first and first three modes are99.53% and 99.97% respectively. And the same for generalized system are 99.17%and 99.95% respectively. The undamped maximum response corresponding to the fundamental modeevaluated from idealized system method is 204.79 mm and from generalized systemmethod is 56.5 mm. And for the damped systems, the values are 44.803 mm and25.52 mm. By idealized system approach, the base shear obtained is 1852 KN and 404.42KN for undamped and damped systems respectively. Whereas by generalized systemapproach the base shear obtained are 498.69 KN and 224.68 KN for undamped anddamped systems (Table 2). The stress resultants on the pylon as a result of P- analyses are found to bewithin the section capacities provided.

DISCUSSION OF RESULTS AND CONCLUSIONIn lateral direction, the fundamental frequency obtained by Idealized

system approach is slightly higher than the same obtained by generalized systemapproach. Whereas in the longitudinal direction it is vice versa and also the variationis much higher. This is because the major contribution for mass matrix in lateraldirection is the mass from the deck and so the variation of mass matrix between theIdealized system and Generalized system methods has not showed much influenceon the natural frequencies but the slight reduction in the values obtained byGeneralized system method is due to the reduction in stiffness in the form ofGeometric stiffness. But in the longitudinal direction the contribution to mass matrixis entirely due to the mass of the pylon only and so the variation of mass matrix

between Idealized system and Generalized system methods influenced much on thenatural frequencies.

The same trend has been observed as far as the other natural frequencies areconcerned. In the lateral direction, the values of the natural frequencies for idealizedand generalized system approaches are slightly varying and those of generalizedsystem are on the lower side. And in the longitudinal direction, the naturalfrequencies obtained for generalized system are comparatively higher than thoseobtained for idealized system. In both the directions, the spatial distribution ofearthquake forces showed much variation. In longitudinal direction by idealizedsystem approach, the earthquake force obtained for the node 1 is a high value, even


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more than the value obtained in lateral direction. This is because the whole mass ofthe legs of the pylon, assumed to be lumped at that node is very high whencomparing to the lumped masses at the other nodes. Such anomalous values are notobtained by the generalized system method as it includes the mass of the pylon thatis actually contributing inertia forces. However the exactness of the solution relieson the shape function chosen. In comparing to undamped system, the dampedsystem with = 0.05 showed much reduction in the earthquake forces. This ismainly because the maximum responses have been reduced much for the dampedsystems and so the pseudo spectral accelerations and hence the forces too (refer Figno 3 & 4). In fact, undamped system is an ideal case and does not exist and soevaluation of response, forces etc., neglecting the damping seems to be an over

estimation. Even though in this project contribution from all the 11 modes ofvibration are considered, the participation of first three modes have reached 0.97 inlateral direction and in longitudinal direction, the first mode itself have reachedabout 0.99.

From the project work, it is observed that the idealized system method haveresulted in variations in comparing to the generalized system method. In lateraldirection the influence of Idealization is much lesser when comparing to thelongitudinal direction. Further, the designed pylon is reported as safe againstELCENTRO earthquake.

SUGGESTIONS FOR FURTHER STUDYFor the actual implementation of the prescribed design, the work done in

this project is not sufficient and the analysis has to be extended. The following are

few studies to be included in further extension, Analytical solutions obtained in this project may be experimentally verified. In this project, the Hermitian shape function for prismatic elements is used. Itmay be replaced by some other which would suit for the uniformly varyingelements. The stability check may be extended to few other, strong motion earthquakes ordesign spectrums may directly be employed.

REFERENCES[1] Arora S.P., Saxena S.C., (1973) Railway Engineering Dhanpat Rai & Sons,

New Delhi.[2] Chandrupatla R.T., Belegundu A.D., (1997) Introduction to Finite ElementAnalysis Prentice hall of India Pvt. Ltd., New Delhi.

[3] Chopra A.K., (2001) Dynamics of Structures Pearson Education (Singapore)Pte Ltd., New Delhi.[4] Mario Paz (1985) Structural Dynamics CBS Publishers & Distributors, NewDelhi.[5] Ram Chandra (1971) Design of Steel Structures Standard Book House, NewDelhi Vol. I & II.[6] Walther R., Houriet B., Isler W. and Moa P. (1985) Cable Stayed Bridges Thomas Telford Ltd, London.


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Table 1 EFFECTIVE EARTHQUAKE FORCES IN LATERAL DIRECTION

Node

No.IDEALIZED MDF SYSTEM GENERALIZED MDF SYSTEM

UNDAMPED 5% DAMPED UNDAMPED 5% DAMPED

1 536.39 210.19 503.55 162.25

2 161.37 63.03 210.74 67.63

3 158.00 61.52 205.60 65.74

4 165.40 64.18 212.75 67.74

5 154.01 59.47 197.13 62.43

6 142.16 54.46 180.38 56.83

7 149.36 56.32 183.90 58.09

8 135.77 49.50 155.75 50.81

9 148.45 49.76 144.29 50.78

10 96.96 28.66 52.57 29.47

11 37.57 14.51 -40.96 15.19

1885.44 711.6 2005.7 686.96

Table 2 EFFECTIVE EARTHQUAKE FORCES IN LONGITUDINALDIRECTION

Node

No.IDEALIZED MDF SYSTEM GENERALIZED MDF SYSTEM

UNDAMPED 5% AMPED UNDAMPED 5% DAMPED

1 1450.08 317.17 343.87 155.27

2 65.26 14.25 26.27 11.85

3 59.78 13.04 23.96 10.79

4 54.28 11.82 21.63 9.73

5 48.73 10.59 19.30 8.67

6 43.22 9.36 16.95 7.60

7 37.64 8.13 14.58 6.52

8 32.03 6.89 12.16 5.42

9 26.38 5.64 9.79 4.34

10 20.67 4.39 7.36 3.2511 14.89 3.14 2.82 1.24

1852.96 404.42 498.69 224.68


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Fig.1.ARIELVIEWO

FTH

EPROPOSEDCABLESTAYEDBRIDGE


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Fig. 2. QUADRUPED PYLON


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Fig. 3. ISOLATED PYLON SHOWING

THE DEGREES OF FREEDOM IN

ONE DIRECTION

Fig 4. EQUIVALENT MASS-

SPRING-DASHPOT DIAGRAM

Fig. 5. GENERALIZED SDF SYSTEM


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0

50

100

150

200

250

300

5.4

13.4

13.6

24.4

37.8

51.2

71.5

72.6

94.4

124.2

127.0

160.3

187.6

204.7

251.4

271.5

323.0

406.2

506.8

633.4

790.0

990.4

NATURAL FREQUENCIES (in rad/sed)

M

ax.

Response(inmm)

UNDAMPED 5% DAMPED

0

5

10

15

20

25

30

35

40

5.4

13.4

13.6

24.4

37.8

51.2

71.5

72.6

94.4

124.2

127.0

160.3

187.6

204.7

251.4

271.5

323.0

406.2

506.8

633.4

790.0

990.4

NATURAL FREQUENCIES (in rad/sed)

Pseudo-Acceleration(inm/sec^2)

UNDAMPED 5% DAMPED

Fig. 6. VARIATION OF

DISPLACEMENT AS PERHERMITIAN SHAPE FUNCTION

Fig. 7. VARIATION OF

PROPERTIES WITH THE LENGTH

OF THE ELEMENT

Fig. 8. RESPONSE SPECTRUM

Fig. 9. PSEUDO ACCELERATION SPECTRUM

= 0

= 5%

= 0

= 5%

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Seismic Stability Analysis of a Cable Stayed Bridge Pylon