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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.

ABSTRACT The paper discusses the dynamic analysis and design of the pylon of a cable stayed bridge for earthquake 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”.

INTRODUCTION Rameswaram is one of the places of tourism importance in Tamilnadu. It is an island situated between Tamilnadu and Srilanka. There is a meter gauge railway line 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. When this meter gauge line has to be widened, a new bridge to carry the heavier loads that are anticipated for a broad gauge line should replace this existing bridge. So it is aimed to provide a fruitful solution for the problem by providing a cable stayed Bridge (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.

++

Further the reason for the proposal of cable stayed bridge for this reach may be justified as this type of bridges are proved to be economical and are broadly adopted in recent days for medium and long span. This paper mainly discusses the procedure adopted for the dynamic analysis and design of pylon for earthquake forces (Fig 2.). DYNAMIC ANALYSIS Dynamic 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 of masses for each mode. The second stage is the calculation of spatial distribution of effective earthquake forces based on the concept of Response Spectrum. And the third, the check for stability of pylon for the evaluated earthquake forces. Dynamic analysis of pylon has been carried out for two different directions namely Lateral and Longitudinal directions (degree of freedom in vertical direction is neglected) separately and the forces are superimposed in the third stage. MODAL ANALYSIS The pylon is isolated from the whole structure and the modal analysis has been performed based on the following assumptions; 1.

2. 3. 4.

5.

Axial Stiffness of the pylon is very high and so the natural frequencies in the vertical direction will be very high and the earthquake forces will be minimal. In fact, already we have given sufficient concentration for the forces in vertical direction; hence the degrees of freedom in the vertical direction are neglected. Rotational moment of inertia would practically be negligible and so the rotational degrees of freedom are also condensed. The structure is symmetric about the lateral and longitudinal axes and so there will not be any torsional response of the structure. 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. The damping is assumed as viscous damping with damping ratio ζ = 5%.

2

The first three assumptions are very common and the fourth assumption has been specially adopted for this particular case. The natural frequency of the deck will be low when compared to the pylon and so the response of the deck will be comparatively be higher than that of the pylon and in the same way, the inertia forces 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 cable stays and not by rigid members. Thus the approximation is justified, but this approximation would be slightly on the conservative side. In fact, isolation of pylon is possible only if such approximation is adopted (refer Fig. 3). Thus the complexity of the problem is reduced. Also this approximation is adopted for the modal analysis in lateral direction only. In the case of longitudinal direction the inertia forces will

not be transferred to pylon as the cable can freely roll over the pulley arrangement made in the pylon. Idealized System The 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 lateral direction, 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 the longitudinal direction, there would not be any transfer of inertia force from deck to pylon and so the mass matrix includes only the mass of pylon.

L

[M*] = ∫ m s ( x ) [ϕ ( x )] 2 dx + [m] 0

[C*] = 2 ζ ω n [M*] L

L

0

0

[K*] = ∫ E I ( x ) [ϕ" ( x )] 2 dx − ∫ N [ϕ' ( x )] 2 dx

26

Generalized System The modal analysis based on idealized system approach has some limitations viz; • The inertia forces due to the mass of spring may either be neglected or over estimated as in the Eq.2. • The variation in the natural frequencies due to the presence of any axial or lateral loads can not be represented. The above limitations may not be interfering much in most of the problems and 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 the generalized system approach and the results are compared. The free vibration equation for generalized MDF system (refer Fig 5. for Generalized SDF system) is, [M*] {U”} + [C*] {U’} + [K*] {U} = 0 (3) where,

The mass matrix has been arrived from the axial force on the pylon. For the lateral direction, 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. In the longitudinal direction, it includes only the mass of pylon. But in the calculation of geometric stiffness, the axial force includes the effect of Prestress also. In the Generalized MDF approach, the choice of the shape function φ(x) plays a vital role. For our problem, the Hermitian shape function (refer Eq. 4 and Fig. 6) is used. Although the section is not truly prismatic, the chosen shape function is expected to give results with good degree of accuracy. 1 H (ξ) = < H1 H 2 > = [ 2 − 3ξ + ξ3 2 + 3ξ − ξ3 ] 4 3 2 (4) H' (ξ) = [ − (1 − ξ ) (1 − ξ 2 )] 4 3 H" (ξ) = [ − ξ ξ] 2 The elements of pylon are not truly prismatic so the mass, stiffness, geometric stiffness matrices for the uniformly varying sections are arrived by using a generalized expression for breadth of individual element b(ξ) as shown in Eq.5 (also refer Fig. 7). From which the generalized expression for area and moment of inertia are obtained. The element mass, element stiffness and element geometric stiffness matrices are then reduced to the form as shown below. b(ξ) = α – β ξ (5) where, α = breadth at mid section and β = double thechange of breadth per unit length of the element. Area A(ξ) = b(ξ)2 –( b(ξ) -0.2)2 = (0.4α – 0.04) – (0.4β) ξ A(ξ) =A– B ξ Moment of Inertia, 1 I (ξ) = b(ξ) 4 − ( b(ξ) − 0.2 ) 4 12 3 3 2 2 2 2 2 1 (−0.8 β ) ξ + (2.4 α β − 0.24 αβ ) ξ + (0.48 αβ − 2.4 α β − 0.24αβ ) ξ =   12 + (0.8 α 3 − 0.24 α 2 − 0.032 α − 0.0016) 

[

I (ξ) =

[

]

]

1 C ξ3 + Dξ 2 + Eξ + F 12 Thus, the element mass matrix 72 A 208 A + 16 B  35 5 35 ρL   e [ms ] =  16  72 208 16 A A− B  35 35 5  Element stiffness matrix

(

)

(

)

 3 D+B − 3 D+B 5 E  5   3  L − 3 D + B 3 D+B  5 5   Element geometric stiffness matrix  6 −6  5 N 5 e [K G ] =   L −6  5 6 5  [K ] e =

(

) (

)

Transformation matrix 0 C  [T ] e =    C 0 EVALUATION OF EARTHQUAKE FORCES The next step is to assess the peak acceleration for each uncoupled SDF system. For this purpose the concept of response spectrum is employed. Time History Analysis for various uncoupled SDF systems has been performed and their peak responses are plotted against their natural frequencies, which is referred as Response Spectrum here, unlike the Response Spectrum in the true sense. It has been decided to use the N-S component of horizontal ground acceleration recorded at ELCENTRO - substation, California during Imperial Valley earthquake of May 18, 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 based on numerical integration – linear interpolation method. Then from the response spectrum (Fig. 8), the pseudo-acceleration spectrum (Fig. 9) and the equivalent static forces due to earthquake have been evaluated. For the evaluation of equivalent static forces, 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 = 1.5 Response reduction factor R = 5.0 ANALYSIS FOR EARTHQUAKE LOAD AND DESIGN In this stage the earthquake forces evaluated are given to the pylon, modeled in STAADPro and P-Δ analysis has been performed. Then the stress resultants calculated by the P-Δ analysis are checked against the capacities of the sections provided. And the stability of the pylon may be reported as ‘safe’ or ‘unsafe’ against ELCENTRO earthquake based on the results. SUMMARY OF RESULTS In lateral direction, • The fundamental frequency obtained for idealized system & generalized system are 5.421 rad/sec and 5.407 rad/sec respectively.

• •

•

For idealized system, the participation factors of first and first three modes are 85.67% 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 mode evaluated from idealized system method is 228.03 mm and from generalized system method is 242.3 mm. And for the damped systems, the values are 85.55 mm and 85.22 mm. By idealized system approach, the base shear obtained is 1885 KN and 711.6KN for undamped and damped systems respectively. Whereas by generalized system approach the base shear obtained are 2005.7 KN and 686.9 KN for undamped and damped systems respectively (Table 1).

In longitudinal direction, • The fundamental frequency obtained for idealized system & generalized system are 13.56 rad/sec and 16.755 rad/sec respectively. • For idealized system, the participation factors of first and first three modes are 99.53% 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 mode evaluated from idealized system method is 204.79 mm and from generalized system method is 56.5 mm. And for the damped systems, the values are 44.803 mm and 25.52 mm. • By idealized system approach, the base shear obtained is 1852 KN and 404.42 KN for undamped and damped systems respectively. Whereas by generalized system approach the base shear obtained are 498.69 KN and 224.68 KN for undamped and damped systems (Table 2). • The stress resultants on the pylon as a result of P-Δ analyses are found to be within the section capacities provided. DISCUSSION OF RESULTS AND CONCLUSION In lateral direction, the fundamental frequency obtained by Idealized system approach is slightly higher than the same obtained by generalized system approach. Whereas in the longitudinal direction it is vice versa and also the variation is much higher. This is because the major contribution for mass matrix in lateral direction is the mass from the deck and so the variation of mass matrix between the Idealized system and Generalized system methods has not showed much influence on the natural frequencies but the slight reduction in the values obtained by Generalized system method is due to the reduction in stiffness in the form of Geometric stiffness. But in the longitudinal direction the contribution to mass matrix is 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 the natural frequencies.

The same trend has been observed as far as the other natural frequencies are concerned. In the lateral direction, the values of the natural frequencies for idealized and generalized system approaches are slightly varying and those of generalized system are on the lower side. And in the longitudinal direction, the natural frequencies obtained for generalized system are comparatively higher than those obtained for idealized system. In both the directions, the spatial distribution of earthquake forces showed much variation. In longitudinal direction by idealized system approach, the earthquake force obtained for the node ‘1’ is a high value, even more than the value obtained in lateral direction. This is because the whole mass of the legs of the pylon, assumed to be lumped at that node is very high when comparing to the lumped masses at the other nodes. Such anomalous values are not obtained by the generalized system method as it includes the mass of the pylon that is actually contributing inertia forces. However the exactness of the solution relies on the shape function chosen. In comparing to undamped system, the damped system with ζ = 0.05 showed much reduction in the earthquake forces. This is mainly because the maximum responses have been reduced much for the damped systems and so the pseudo spectral accelerations and hence the forces too (refer Fig no 3 & 4). In fact, undamped system is an ideal case and does not exist and so evaluation of response, forces etc., neglecting the damping seems to be an over estimation. Even though in this project contribution from all the 11 modes of vibration are considered, the participation of first three modes have reached 0.97 in lateral direction and in longitudinal direction, the first mode itself have reached about 0.99. From the project work, 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 lesser when comparing to the longitudinal direction. Further, the designed pylon is reported as “safe against ELCENTRO earthquake”. SUGGESTIONS FOR FURTHER STUDY For 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. It may be replaced by some other which would suit for the uniformly varying elements. • The stability check may be extended to few other, strong motion earthquakes or design 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 Element Analysis’ – 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, New Delhi. [5] Ram Chandra (1971) ‘Design of Steel Structures’ – Standard Book House, New Delhi Vol. I & II. [6] Walther R., Houriet B., Isler W. and MoÏa P. (1985) ‘Cable Stayed Bridges’ – Thomas Telford Ltd, London.

Table 1 EFFECTIVE EARTHQUAKE FORCES IN LATERAL DIRECTION Node No. 1 2 3 4 5 6 7 8 9 10 11

IDEALIZED MDF SYSTEM UNDAMPED 5% DAMPED 536.39 210.19 161.37 63.03 158.00 61.52 165.40 64.18 154.01 59.47 142.16 54.46 149.36 56.32 135.77 49.50 148.45 49.76 96.96 28.66 37.57 14.51 1885.44 711.6

GENERALIZED MDF SYSTEM UNDAMPED 5% DAMPED 503.55 162.25 210.74 67.63 205.60 65.74 212.75 67.74 197.13 62.43 180.38 56.83 183.90 58.09 155.75 50.81 144.29 50.78 52.57 29.47 -40.96 15.19 2005.7 686.96

Table 2 EFFECTIVE EARTHQUAKE FORCES IN LONGITUDINAL DIRECTION Node IDEALIZED MDF SYSTEM GENERALIZED MDF SYSTEM No. 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 3.25 11 14.89 3.14 2.82 1.24 1852.96 404.42 498.69 224.68

Fig. 1. ARIEL VIEW OF THE PROPOSED CABLE STAYED BRIDGE

Fig. 2. QUADRUPED PYLON

Fig. 3. ISOLATED PYLON SHOWING THE DEGREES OF FREEDOM IN ONE DIRECTION

Fig 4. EQUIVALENT MASSSPRING-DASHPOT DIAGRAM

Fig. 5. GENERALIZED SDF SYSTEM

Fig. 6. VARIATION OF DISPLACEMENT AS PER HERMITIAN SHAPE FUNCTION Fig. 7. VARIATION OF PROPERTIES WITH THE LENGTH OF THE ELEMENT

Max. Response (in mm)

300 250 200

ζ=0

150 100 50

ζ = 5%

NATURAL FREQUENCIES (in rad/sed)

UNDAMPED

990.4

790.0

633.4

506.8

406.2

323.0

271.5

251.4

204.7

187.6

160.3

127.0

124.2

94.4

72.6

71.5

51.2

37.8

24.4

13.6

5.4

13.4

0

5% DAMPED

Fig. 8. RESPONSE SPECTRUM 35 30

ζ=0

25 20 15 10

ζ = 5%

5

NATURAL FREQUENCIES (in rad/sed)

UNDAMPED

Fig. 9. PSEUDO ACCELERATION SPECTRUM

990.4

790.0

633.4

506.8

406.2

323.0

271.5

251.4

204.7

187.6

160.3

127.0

124.2

94.4

72.6

71.5

51.2

37.8

24.4

13.6

13.4

0 5.4

Pseudo-Acceleration (in m/sec^2)

40

5% DAMPED