A simplified fragility analysis of fan type cable stayed bridges

Vol.4, No. 1 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION June, 2005

Article ID: 1671-3664(2005)01-0083-12

A simplified fragility analysis of fan type cable stayed bridges

R. A. Khan ~*, T. K. Datta 2~ and S. Ahmad 3~

1. Civil Engineering Department, Jamia Millia lslamia, New Delhi-25, India

2. Civil Engineering Department, Indian Institute of Technology, New Delhi-16, India

3. Department of Applied Mechanics, Indian Institute of Technology, New Delhi-16, India

Abstract: A simplified fragility analysis of fan type cable stayed bridges using Probabilistic Risk Analysis (PRA) procedure is presented for determining their failure probability under random ground motion. Seismic input to the bridge support is considered to be a risk consistent response spectrum which is obtained from a separate analysis. For the response analysis, the bridge deck is modeled as a beam supported on springs at different points. The stiffnesses of the springs are determined by a separate 2D static analysis of cable-tower-deck system. The analysis provides a coupled stiffness matrix for the spring system. A continuum method of analysis using dynamic stiffness is used to determine the dynamic properties of the bridges .The response of the bridge deck is obtained by the response spectrum method of analysis as applied to multi- degree of freedom system which duly takes into account the quasi - static component of bridge deck vibration. The fragility analysis includes uncertainties arising due to the variation in ground motion, material property, modeling, method of analysis, ductility factor and damage concentration effect. Probability of failure of the bridge deck is determined by the First Order Second Moment (FOSM) method of reliability. A three span double plane symmetrical fan type cable stayed bridge of total span 689 m, is used as an illustrative example. The fragility curves for the bridge deck failure are obtained under a number of parametric variations. Some of the important conclusions of the study indicate that (i) not only vertical component but also the horizontal component of ground motion has considerable effect on the probability of failure; (ii) ground motion with no time lag between support excitations provides a smaller probability of failure as compared to ground motion with very large time lag between support excitation; and (iii) probability of failure may considerably increase for soft soil condition.

Keywords: probabilistic risk analysis; fan type cable stayed bridge; fragility analysis; first order second moment

1 Introduction

For seismic probabilistic risk assessment of structures, it is necessary to assess the vulnerability o f structures due to earthquakes forces. The likelihood of structural damage due to earthquakes is usually expressed by a fragility curve, which describes the probability of damage at various levels of ground shaking. Fragility analysis is the key element o f seismic probabilistic risk analysis (PRA) which duly considers the seismicity of the region and local soil condition for obtaining the probability of failure o f the structure. A rigorous probabilistic risk analysis of structures is highly complex. It requires the consideration of many factors such as mode of collapse, non-linear analysis, ductility demand, soil - structure interaction and soil effect in modifying the ground motion, local

Correspondence to: R.A.Khan, Civil Engineering Department, Jamia Millia Islamia, New Delhi-25, India Fax: 0091-61-26981261 E-mail: [email protected]

*Lecturer; ~Professor Received 2004-11-23; Accepted 2005-02-15

seismicity and seismic risk of the region, uncertainties in modelling, analysis procedure, material property etc. It is not possible to consider all factors in one analysis and perform a rigorous PRA of any structure. Therefore, various levels of simplifications have been adopted by different researchers. In particular, simplification regarding the failure mode is invariably made for complex structures, since it is practically impossible to consider probability o f failure for all collapse modes by performing elaborate inelastic analysis. Another simplification which is often made is in the treatment o f different uncertainties like those of material property, ductility evaluation, modelling etc, since data on the probabilistie variations o f these quantities are difficult to ascertain.

There have been several studies on the PRA procedure applied to different kinds o f structures. Ravindra (1989) provided an overview of the seismic PRA methodology and describes some of the recent PRA applications. Shinozuka et aL (1989); Matsubara et al. (1993) presented applications of seismic PRA procedure to buildings. Dubord et al. (1996) used PRA procedure for nuclear industry. Some studies are also reported on PRA of dams (Costantino and Gu, 1991; Lee

84 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.4

et al., 1998). Application of PRA for determining the probability of failure of bridges for earthquake forces is not widely reported in the literature. In particular, there is a lack of study on the use of PRA for complex bridge structures like cable stayed bridges. There are a few studies on the reliability analysis of cable supported bridges (Michel, 1992; Madsen and Rosenthal, 1992; Kiyohiro and Frangopol, 2002; Pourzeynali and Datta, 2002). But they do not consider the reliability analysis for seismic forces.

The difficulty of obtaining the probability of failure of cable supported bridges for any type of loading including the seismic forces is to consider the probability of failure of the complex system of cables, tower and deck in resisting the load action. One of the ways to treat the problem is to determine the probability of failure of each component of the system and component failures may be combined to obtain the failure of the total system. The other way is to identify (or assume) the failure of the most important component the system and determine the probability of failure based on that. Further, assumptions regarding other issues are needed to evolve a simplified and practical procedure for determining the probability of failure of such complex systems.

In this paper, a simplified PRA procedure is presented for obtaining an initial estimate of the probability of failure of fan type cable stayed bridges for seismic forces at a site for which not much earthquake data is available. For this purpose, the failure of the deck is assumed for the failure of the bridge. The uncertainties included in the structure are those due to uncertainties of seismic inputs, material property, capacity of the bridge cross section, modeling, analysis procedure, ductility effect and damage concentration effect. For the response analysis, a risk consistent response spectrum is used since response spectrum is a more popular input for earthquake analysis. The risk consistent response spectrum is obtained from a separate analysis. Effects of a number of important parameters such as the components of ground motion, spatial correlation, angle of incidence of earthquake etc. on the probability of failure are investigated for a cable stayed bridge.

2 Modeling of the bridge deck

The bridge deck is idealized as a continuous beam over the outer abutments and the interior towers as shown in Figs. 1 and 2. The effect of the cables is taken as vertical springs at the points of interactions between the cables and the bridge deck and is obtained by separate analysis (Allam and Datia, 1999). Further, the effect of the spring stiffness is taken as an additional vertical stiffness to the entire flexural stiffness of the bridge.

3 Assumptions

The following assumptions are made for the study:

(i) The bridge deck (girder) and the towers are assumed to be axially rigid.

(ii) The bridge deck, assumed as continuous beam, does not transmit any moment to the towers through the girder-tower connection.

(iii) Cables are assumed to be straight under high initial tension due to the dead load and well suited to support negative force increment during vibration without losing its straight configuration.

(iv) Beam-column effect, in the stiffness formulation of the beam, is considered for the constant axial force in the beam and its fluctuating component in the cable is ignored.

(v) Little earthquake data is available for the region under study, except for the recurrence interval of earthquakes and some indirectly evaluated magnitudes of earthquakes.

(vi) The region is surrounded by multiple sources (point sources) of earthquake.

(vii) Any of the attenuation laws (reported in the literature) could be valid for the region.

(viii) It is assumed that both normalized response spectrum ordinates normalized with respect to maximum acceleration value (SN(T)) is empirically determined function of magnitude M and epicentral distance R, and they are lognormally distributed.

4 Equation of motion

The equation of motion for the relative vertical vibration Y(xJ) of the beam segment r of the idealized deck with constant axial force N , neglecting the shear deformation and rotary moment of inertia is given by

O4Y oey(x~,t) Edlr ~-q-TfT~ 4 + N r + C r

3 X r 3Xr 2 2y < a

g at e

aY(x~,t) ~t

forr = 1 ,2 ,3- - - .N b (1)

in which El and / are the modulus of elasticity of the bridge deck and vertical moment of inertia of the beam segment r of the deck respectively; I~ r ,g, N , N b , C are the dead load per unit length of the beam segment, the acceleration due to gravity, the axial force given to the beam segment r due to cables, number of beam segments and damping of the r ~h segment of the bridge deck respectively.

P(x r, t) is defined as the load induced by seismic excitations at different supports and is given by

8 < P ( X r ' g ) : - - g - /= 1 (2)

No. 1 R.A. Khan et al.: A simplified fragility analysis of fan type cable stayed bridges 85

where j2. ( t ) , j = 1,2,. ..... ,8 are the ground acceleration inputs a(the supports as shown in Fig. 1. Note that there is no excitation applied to the spring support which represents the the cable-deck interaction, gr(x) is the vertical displacement of the r ~h segment of'the bridge deck due to unit displacement given in the direction of f

�9 J

only (one at a time) at the supports. ~ ( x ) is obtained by solving the entire bridge (i.e. deck, towers and cables), taking into account the transfer of moment between the deck and the tower, by a separate analysis using the standard structural analysis procedure. The mode shapes and frequencies of the bridge are obtained using the dynamic stiffness formulation as given by Chatterjee et al. (1993). In the dynamic stiffness formulation, beam segments are considered between the supports 1, 2, 4, etc. as shown in Fig.2. The dynamic stiffness matrices for a beam segment between two support is given in Appendix-I and details of the procedure are available in Chatterjee et al. (1993). Note that at the support nodes, both rotational and vertical degrees of freedom are considered for obtaining the dynamic stiffness of the beam segment between two supports. They are assembled to form the stiffness matrix of the full structure and then the rotational degrees of freedom are

condensed out. Dynamic response at any point within the r th segment of the beam is obtained by solving Eq. (1) using normal mode theory. Bending moment at any section like 3 or 9 can be determined using the dynamic and quasi - static components of displacement response. For section 3, beam segment 2-4 is considered, while for section 9, beam segment between support 8 and the corresponding support on the other symmetrical half is considered.

5 Modal spectral analysis

The modal analysis for the relative vertical displacement y (xr, t) for any point in the r th deck segment is given as

y ( x ~ , t ) : ~ . O . ( x r ) q . ( t ) r = 1, 2, ----, N,o (3) n = l

in which r (Xr) is the n th mode shape of the /h beam segment of the bridge deck and qn (t) is the T/th generalized coordinate. Substituting Eq.(3) into Eq.(1), multiplying by ~b m (xr), integrating w.r.t. L r and using the orthogonality of the mode shapes leads to

3x53.0m I-

t i

6 7 8 9[

i _l

i - 7x53.0m

~ 7.15rn 7.15m

56.25m

33.75m

_i 3x53.0m l "

L ( t ) ~ A ( t )

Fig. 1 Fan type cable stayed bridge considered for parametric study

rth beam segment e-

l 2 3 4 5 6 l 7 8 -

I_ L, ._1 ['- -t

X r

i

I I

Fig. 2 Idealization of the bridge deck


G (t)+ 2~(o G(t)+co2q.(t)= P.(t)

n = 1,'", M (4)

in which (. and co are the damping ratio and the natural frequency of the n" vertical mode; M is the number of modes considered and fi~(t) is the generalized force given as

8

(t)= ZR,o (xr)?, (t) (5) j = l

where Rj. is the modal participation factor given by

Rj n m

g J o ' " (6)

in which &r (Xr)' the quasi static function, is the vertical displacement of the r th beam segment of the bridge deck due to unit displacement given in the fh direction of support movement. Following the principles of modal spectral analysis, the expressions for the PSDF of responses are obtained as:

8 8 M M

j= l k=l n=l m : l

RjnR~mHn*((O)Hm(O))Syfi. k (0)) (7)

8 8

Sgg(X~,O))=EEgj~(xr)gkr(x ) S f j k (o.)) .]=t k=l (8)

8 8 M

EEE Cn(Xr)Rjngkr(Xr)Hn*((1))S)}.[ k (0.)) j=l k=l n=l

(9)

in which Syy(Xr,(O), Sgg(Xr,O) ) and Syg(x, co) are the PSDFs of the relative displacement, the quasi-static displacement and the cross PSDF between them, respectively.

The expression for the PSDF of absolute (total) displacement is given by using Eqs. (7), (8) and (9) as

8 8 M M

Syy (Xr'('O) = E E E E Cn (Xr ~)m (Xr)RjnRkmHn* ((A))~ ./=1 k : l n=l m=l

8 8 M

//,. ((o)Sy)?~ ( ( o ) + 2 Z Z Z <(x~)R,~g~(x~)[~ j = l k=l n=l

8 8

j= l k=l

(lo)

where $2 r (c~ Sy A (co) and S[ zk (c~ are expressed in terms of the PSDF of ground acceleration SMg (co) using the coherence function given by Eq. (16) and the ratio between the three components of the ground motion along the global axes of the bridge (Rx, Ry, Rz) as given by Eqs. (26), (27) and (28).

Similar expressions can be obtained for the PSDF of the bending moment at any point in the r 'h beam segment of the bridge deck as those derived for the total displacement by replacing r r ) and Nr(Xr) by Effrd2r 2 and Ee]rd2gjr(x)/dx 2 respectively. Effrd2&r(xA/dx: is obtained from the quasi-static analysis of the entire bridge using the stiffness approach as mentioned before.

6 Risk consistent response spectrum and coherence function

For obtaining the risk consistent response spectrum at the free field, a risk consistent response spectrum at the bedrock level is first derived using a procedure outlined by Shinozuka et al. (1989). The procedure assumes that the occurrence of earthquake is a poisson process and considers the effect of several earthquake sources which surround a given region. The methodology requires the construction of a conditional probability matrix of the form P(A>alM)k in which A denotes PGA; a is specified value of the PGA; M is the i ~h value of the magnitudes of earthquake for which the conditional probability of the exceedance of PGA is obtained; and k is the/d h source of earthquake surrounding the region. Since it is assumed that not much earthquake data is available, it is very difficult to construct the conditional probability for the region given by [P(A>a)IM]k Therefore, help from available attenuation laws is used to construct this conditional probability. Since it is also not possible to verify the applicability of any attenuation law for the region, it is assumed that any attenuation law out of a collected set of attenuation laws could be valid for the region. Thus, for a given magnitude of earthquake, the PGA value at any epicentral distance defining the site, becomes a random variable. The set of values (say N values), the random variable can assume, is obtained from the set of attenuation laws considered in the study. For example, if N is the number of attenuation laws selected, then N values of PGA are obtained for a given magnitude of earthquake at the site. The probability of occurrence of a PGA value at the site can be then obtained from the N values of the PGA for a given magnitude of earthquake. An empirical relationship is used to obtain the logarithmic spectral acceleration ordinates for different period as function of magnitude of earthquake and epicentral distance given by Takemura et al. (1989). Response spectrum at the free field is obtained by one dimensional wave propagation analysis through the soil with the input spectrum at the bedrock. The free field risk consistent response spectrum is used as input to the


bridge. Apart from the response spectrum, the response analysis of multi-supported structures like bridge requires the specification of a coherence function. This function takes into consideration the lack of correlation between ground motions at different supports in the seismic analysis of the bridge. In the present analysis, the coherence function used is given in the next section.

7 Response analysis of fan type cable stayed bridge

Using modal spectral analysis, the variance of the total vertical displacement of the bridge deck can be obtained by integrating the psdf of the total vertical displacement over the frequency range of interest and is given by

8 8 M M

O ' y 2 ( X r ) = s 1 6 3 2 2r162 m j=l k=I n=l m=l

8 8 M

+ 2 ZZy__,~b,(x~)7j, gk~(x~)ps).s;cri, o~rS~ j=l k=l n=l

8 8

+ 22gir(Xr)ga, . (x , . )Pf: f~cr fo . f~ (1]) j=l k=l

In which 2

o'2 : i H (o)) (o))do) l,,, SD D -o;

~ : i S.6.6 (o))do) (12)

a

1 ! H * (o))do) Pfjnfkm -- - - n(o))Hm(o))S])]'k �9 O'j},, O'fk m

a

1 f H* P i s . A - - - ,, (o ) )S) )y~ (o))do) o.fj Cr,r~ _

a

P6r , - f SM~ (o))do) O'f , O'fk

(13)

(14)

(15)

In obtaining the cross spectrums Sink(co ) etc., the coherence function pu(co) is given by (Loh and Yeh, 1988)

pzj (o)) = exp [ - c ( ~ ) ] " k kz~zv~ )J (16)

stations q~ and qj measured in the direction of wave propagation; V is the shear wave velocity of soil; m is the frequency of ground motion (rad/sec); and c is a constant depending on the distance from epicenter and inhomogeneity of the soil medium. Note that the coherence function p~(co) is used here for homogeneous random field and denotes only the time lag effect between two support excitations for the same seismic waves propagating with a velocity V.

The ground motion is represented along the three principal directions (u,v,w) (Fig.3) by defining ratio R , R and R w along them such that

i i g ( t ) : R j g ( t ) ; V g ( t ) : & f g ( t )

~ g ( t ) = R w L ( t ) (17)

and the psdfs of the ground acceleration in the principal directions of the ground motion (u,v,w) can be defined a s

(o)) = (o));

(o)): R?Sjjg (co);

Sf0g/0g ( C O ) = Rw2S~g~g (0)) (18)

2 R 2 2. 2 = R 2 2 . or_ 2 = RwZo.?z

(19)

j 7

y (Vertical)

/ /

/ /

I I

I Vlinoy,) /

/

/ ' ,----Lii

J / X i -

/(Longitudinal)

~ pZ'Op~ vo.

Fig. 3 Principal directions of the bridge(x,y,z) and the ground motion(u,v,w)

in which ru=]qi- @ is the separation distance between When the ground motion with respect to the


principal direction of the bridge is defined as the angle of inclination (a) between the direction of major component of ground motion with the major direction of the bridge(x) (Fig. 3), the ground motions along the principal directions of the bridge (x, y, z) are defined as

X g ( t ) : ~ g ( t ) C O S { ~ - - "l~g ( t ) s i n a (20)

5g ( t ) : / / e ( t ) s ina - f~g (t) cosa (2l)

j)g(t)=i)g(t) (22)

The psdfs of the ground accelerations along x, y, z can be written as

�9 , , - R 2 (23 ) S ~ = cos 2 a S ~ + sin 2 a S%~ - ~S?~;~

S ~ = sin 2 a S,,,~ + cos 2 a S| = R2~S]~yg (24)

, , 8 2 , = = S i j (25)

where R , R and R are the ratios of the ground motion x y.. . z along the principal axes of the bridge and give as

2 �9 2 Rx 2 = R 2 cos 2 a R w sin a (26)

R 2 = Ru 2 sin 2 a + 8,~ 2 COS 2 C~ (27)

Ry 2 = R~2 (28)

where R , R , and R are ratios of the ground motion along the principal directions of the ground motion (u, v,w).

The development of the response spectrum method of analysis uses the variance of response as given by Eqn. 11, and is briefly presented below.

Let Dj(co,~,) denote the response spectrum for the displacement corresponding to the j 'h support degree of freedom for the frequency co, and damping 4, and let fj ...... be the mean peak ground displacement corresponding to the fh support degree of freedom defined as

.... =E[max~(t) l] ' The relationship between the expected peak and r.m.s values of response can be written as

D . (O,)n, ~n ) : PjnCY]), ; f j . . . . . = P / Y J ) (29)

in which pj, and Pi are the peak factors. Similarly, E[maxlY(x, t)l] can be written as E[maxf~(t)lJ=p~ao in J ,Y. which p~ is the peak factor of the total response. Using Eqs. (13), (14), (15) and (16) and assuming that all peak factors pj, p andp~ are the same, the expected peak value

, J . of the total &splacement can be written as

8 8 M M

E[maxlY(xr,t)l]:-[ZZ . 2 )n(Xr) m(Xr)Rjne]o " j - 1 k 1 n- I m=l

8 8 M

j 1 k=l n=l 8 8

Rj.gk~ (x~ )Pr~,,l~ Dj(c~ fk . . . . -[- Z Z g j r (Xr) " j=lk 1

1 f: ..... L . . . . (30)

Similar expression can be obtained for the expected peak value of total vertical bending moment at any point of the bridge deck by replacing ~b (x r) and &r(Xr) by EJ~d2~/dx 2 and E~.d2gj~(x)/dx 2 respectively. The expected peak value of the total response as given by Eq.(30) can be obtained using the response spectrum as input provided pf,. f~ , pf, o A' Pf, A are known. These quantities are obtained by converting the response spectrum to power spectral density function using the expression given by Der Kiureghian (1981) and the coherence function.

Thus, the expected peak value of the response is obtained with inputs as the risk consistent response spectrum and a coherence function.

8 E v a l u a t i o n o f p r o b a b i l i t y o f f a i l u r e

The load action i.e. moment at any section of the bridge is a random variable due to a number of uncertainties associated with earthquake loading, material properties, method of analysis etc. Similarly, the moment capacity of the section is also a random variable influenced by many uncertainties like uncertainty of material strength, ductility at the joints, damage concentration effect etc.

The moment induced at a section (called Resistance R), a random variable, is considered to be a product of five random variables and is given by

R= MFI F2F3 F4 (31)

where M is a random variable denoting the internal moment at the section produced by the earthquake load. Apart from the random nature of the earthquake loading, the randomness of the moment M arises from a number of uncertainties. Out of those four uncertainties are included in the study namely, (i) uncertainty of input motion; (ii) uncertainty in system parameter; (iii) uncertainty in modeling; and (iv) uncertainty in the analysis procedure such as non-linear analysis being replaced by linear analysis; mean peak response is obtained from response spectrum analysis rather than Monte Carlo simulation technique. These uncertainties are incorporated in the form of Eq. (31) with the help of four independent random variables Ft to F 4 representing deviation of the actual response from M. The random variables M, F1, F 2, F 3 and F 4 are assumed to be log

No.1 R.A. Khan et al.: A simplified fragility analysis of fan type cable stayed bridges 89

normally distributed. As a result,

O"ln R = ~//~12 -1- /~2 2 "t- /~3 2 "~ ~4 2 (32)

in which fla, i = 1 to 4 is the coefficient of variation of random variables F~, i = 1 to 4 and atn e is the logarithmic standard deviation of resistance R.

Note that the median values of F~, F2, F 3 and F 4 are taken as unity and o~1~ = ln(fie +1) for f l < 0.25, al,x--flx in which fl~ is the coefficient of variation ofx.

Similarly, capacity of the section is written as

C = M~ F 5 F 6 (33)

where M e is a random variable denoting the moment capacity and Fs, F 6 are two random variables representing deviation from the actual strength accommodating capacity to resist induced moment at the section. F s is incorporated to account for uncertainties due to energy absorption capacity and ductility effect. F 6 caters to the uncertainty due to damage concentration effect in MDOF system i.e., actual difference between linear and non-linear analysis of MDOF system. All the three random variables are assumed to be independent lognormally distributed variables. Therefore, logarithmic standard deviation of C can be written as

~ : X/(fl~ + f162 + fl2c) (34)

in which fls, f16 and flM~ are coefficients of variation o f f s, F 6 and Mc respectively. The median values ofFs, F 6 and Mc are specified.

First order second moment (FOSM) method is used to calculate the probability of failure by assuming both resistance R and capacity C to be log-normally distributed and by defining the probability of failure as (Ranganathan, 1990).

Pf=q~ 2 2

(O" lnR't-O" lnC) (35)

where, K' and C are the median values and o-1~ and o-~n c are the logarithmic standard deviation of the resistance and capacity of the structure.

9 Evaluation of factors (F 1 to F6)

F~ represents the variability of response caused by input motion's variability. The median value is unity and the logarithmic standard deviation /~1 is evaluated from two response values con'esponding to input motions of the median and 84% non-exceedence spectra i.e fl~ = In ( r84 / rs0 ). F 2 is the factor representing the variability resulting from the system parameter variation. The

logarithmic standard deviation t2 is evaluated in the same manner as fll i. e , as the logarithm of the ratio between the 84 th percentile non-exceedence and the median responses.F 3 accounts for the uncertainty involved in the modeling of the system and in the analytical methods for the evaluation of response. It follows that F~ is to be evaluated as the ratio of observed response to the response calculated with the model. Generally, the median value of F 3 is taken as unity and coefficient of variation ranges between 0.15 to 0.2 (Takeda et al., 1989). F 4 is the factor accounting for the uncertainty resulting from the simplifications in the analysis in the evaluation of the expected response. F 4 is assumed to have a median value of unity with a 0.5 c.o.v (Takeda et al., 1989). The concept of F 5, the energy absorption factor, cater to the energy absorption during nonlinear excursion of a SDOF system. The median value is g e n e r a ~ n to be proportional to the Newmark s formula x/(2/_t - 1) with reduction factor of 0.6, where/~ is the ductility - - tor. F 6 cater to the damage concentration effect of MDOF systems. The median value of F~ is evaluated as an average ratio between the linear and nonlinear analysis (for a number of cases on MDOF systems). Extensive studies indicate that the median value typically lies in between 0.6 to 1.25. The coefficient of variation is assumed to be 0.1 (Takeda et al., 1989). Although the above values have been adopted for the factors F 1 to F 6 for obtaining the probabilities of failure (Pf) for the parametric studies, a separate sensitivity analysis has been conducted to investigate the effect of these factors on the Pc

10 Numerical study and discussion

In order to illustrate the PRA procedure adopted for finding the reliability of fan type cable stayed bridge under seismic excitation, a fan type cable stayed bridge taken by Au et al. (2001) is considered as shown in Fig.1. The bridge is assumed to be located in a site surrounded by three earthquake sources. The attenuation laws used for obtaining the risk consistent spectrum are taken from Gupta et al. (1997). The empirical formula for spectral ordinates is taken from Takemura et al. (1989). The other data for obtaining the spectrum use (i) epicentral distances of the site from three earthquake sources as 153 km, 253 km and 87.5 km respectively; (ii) annual occurrence rate of earthquakes as 4.1,2.3 and 0.8 respectively; (iii) magnitude of earthquake to vary from 5 to 9 with probability density function (pdf) given as PM (m) = /3 exp(-/3(m - m0) ) where, fl = 2.303b, m 0 is the lower threshold magnitude of earthquake, m is the magnitude of earthquake and b is the relative likelihood of large or small earthquakes. The corresponding risk consistent response spectrum at the bed rock level is shown in Fig.4.

Depth of overlying soil on the bedrock is assumed to be 40m. Two types of soil conditions namely soft (V~=80m/sec) and firm (V~=330m/sec) are considered.


O Z

3.0

2.5

2.0

1.5

1.0

0.5

0.0 0.0

~ " ~ ' -Z - - ,.t .~ a. 1 l . . . . . . . . . . . . . . . . . . .

: _-iii Oi06g-oiig] .t 0 16g-O.2gi �9 0.26g-O.3g i

0.8 1.6 2.4 3.2 4.0 4.8 5.6 Time period (s)

Fig. 4 Normalized response spectrum at bedrock (50th percentile)

e~ o

o Z

O

rq

O Z

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0 0

: ...... PGA at bed = 0103g

f 1 2 3 4 5

Time period (s)

(a) V s = 80m/s

0 1 2 3 4 5 Time period (s)

(b) Vs = 330m/s

Fig. 5 Normalized risk consistent spectrum at top

No. 1 R.A. Khan e t al.: A simplified fragility analysis of fan type cable stayed bridges 91

Damping of the soil is taken as 5%. The risk consistent response spectrums for the free field ground motion for Vs = 80m/see and 330 m/sec are shown in Figs. 5 - 6. The mean value of the ductility factor is taken as 4.0. The ratio between the three components of ground motion R u (major), R v (minor) and R w (vertical) is taken as unity; the angle of incidence (i.e., angle of major earthquake direction with the longitudinal axis of the bridge) is taken as zero (shown in Fig. 3); the value of correlation coefficient c is taken as 0.5 as suggested by Hindy and Novak (1980). Since the value of c depends upon the epicentral distance and inhomogeneity of the medium, the value of c may vary. For example, Loh and Yeh (1988), while providing the correlation Eq.(16). suggested the value of c = 0.125. Therefore, another value of c = 0.125 has also been considered for one case in order to study the effect of c on the reliability estimates. The ground motion inputs are provided in the form of response spectrum shown in Figs. 4 to 5. These response spectrums are applied as input at the supports shown in Fig. 1. Note that response spectrums corresponding to both vertical and horizontal ground motions at the supports are having the same normalized shape but the mean square values of the ground motion s in the two directions bear a ratio expressed by Eqs.(18) and (19). The effect of this ratio on the probability of failure is considered as a parameter for study. Further, it is assumed that the horizontal and vertical ground motion at a support is assumed to be fully correlated without any time lag. The spring supports shown in Fig. 2 are the fictitious supports representing the supports provided by the cables to the deck and therefore, no ground motion is applied at the support as stated before. Failure of the bridge is assumed to take place when plastic hinges are formed at section 3 and the corresponding point on the other symmetric half (Fig.l) where the maximum bending moment occurs. This is not an actual failure of the bridge including the failure of towers and stay cables. It denotes a serious impairment in the serviceability condition, and provides a higher level of safety against actual failure. It is further assumed that the cross section of the bridge is subjected to 50% of the yield stress because of the superimposed of dead load and gravity load. Therefore, 50% of o- value is assumed to be threshold limit for seismic effec~

10.1 Effect of components of ground motion on probability of failure

Figure 6 show the effect of the ratio (Rj Rv: R w ) of the component of ground motion. It is seen from the figures that the various combinations of R u , R v and R

w

will lead to different values of Pr When both R and R v components are maximum of all other values used in different combinations, the probability of failure is maximum. For the firm soil, relative increase in probability of failure for the above combination of the earthquake components as compared to other

combinations is more than that for the soft soil. The reason for this is attributed to the difference in shape of the risk consistent response spectrums between the soft and firm soils. Thus, not only the higher value of the vertical component of ground motion gives higher value of Pf but also higher value of Pf is obtained if the longitudinal of ground motion is also more. The reason for this is that longitudinal component of ground motion contribute to the vertical vibration of the deck because of the cables supporting the deck.

10.2 Effect of time lag (correlation) between support excitations on the probability of failure

Figure 7 show the effect of time lag (correlation) between support excitation on the probability of failure. It is seen from the figures that the ground motion without having any time lag between support excitations i.e. c tending to zero (one extreme idealized condition) provides much less value of the probability of failure (Pf) as compared to the ground motion producing very large time lag between support excitation i.e. c tending to infinity (another idealized extreme condition). This is the case because the rms value of the bending moment increases as the time lag increases between the support excitations. Note that the probability of failure is sensitive to the choice of the value of c as shown in Fig.7(a). There is a considerable difference between the probability of failure for c = 0.5 and c = 0.125. Further, it is observed that the difference between probabilities of failure because of time lag, increase with the increase in the value of PGA for both soft and firm soil conditions.

10.3 Effect of angle of incidence of earthquake on probability of failure

Figure 8 shows the effect of angle of incidence on the probability of failure. As it would be expected, the 0 ~ angle of incidence provides higher value of probability of failure compared to an angle of incidence 70 ~ . It is interesting to note that for an angle of incidence of 30 ~ , the reduction in Pf from that of 0 ~ angle of incidence is not very significant. It shows that the variation o fP t with the angle of incidence is not uniform; the rate of decrease of Pf with the increase in angle of incidence significantly increases at higher values of the angle of incidence. This is the case because the angle of incidence affects the correlation length non-uniformly leading to non-uniform changes of stress with the change of angle of incidence. The observations are the same for both soft and firm soil conditions.

t0.4 Effect of nature of input ground motion on the probability of failure

Figure 9 compares between fragility curves obtained with ground motions with white noise and risk consistent response spectrum inputs at the bedrock level. It is seen


0.06 I 0.05

oo4 I oo3 I oo2 I 001 / 0.00 t~

0

1.0:1.0:1.0 1.0:0.4:0.6

+0 .8 :0 .5 :0 .6 ~0 .6 :0 .5 :0 .6

0.1 0.2 0.3 0.4 PGA (g)

(a) G = 80m/s

0.5

0.012

0.010

0.008

0.006

,~ 0.004

0.002

0.000

4 - 1.0:1.0:1.0 1.0:0.4:0.6 /

= ~176176176162

= , , ~ - - - - - ~ , �9 , ~ - - - - ~ , , ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)

0.9

(b) V~ = 330m/s

Fig. 6 Effect of ratio of components of earthquake

0.06 I c=Very large / *

005 c : o 5 / / , "~ 0.04 / - - c=0.125 / / = | c=0

0.03 j

oo2! 0.00 -

0 0.1 0.2 0.3 0.4 0.5 PGA (g)

(a) V~ = 80m/s

0.012

0.010

L~ 0.008

0,006

4 .,= 0.00

0,002

0.000 0

- -~ c=Very large c=0.5 / 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)

(b) Vs = 330m/s

Fig. 7 Effect of time lag between support excitations

0.9

0.06

0.05

0.04

0.03

�9 -~ 0.02

0.01

0.00 0

--*- Angle = 0 ~ Angle = 300

= 70 ~ --4,- Angle

0.1 0.2 0.3 0.4 PGA (g)

(a) G = 80m/s

0.5

0.012

0.010

~_.~ 0.008

0.006

~0.004

0.002

0.00{ 0

Angle = 0 ~ 1 - -~ Angle = 30 ~ 4 - Angle = 7 0 ~

i I , - _ ~ i t , J

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)

(b) V~ = 330m/s

Fig. 8 Effect of angle of incidence of earthquake

0.9

f rom the figure that the white noise input provides m u c h higher probabi l i ty o f failure. This is the case because white noise excitat ion gives more response as it excites more number o f modes o f the structure. Further, the difference be tween the probabil i t ies o f failure for ground motions with c = 0 and c = very large, for white noise ground input, is m u c h more as compared to that for risk consistent spectra and it could be as m u c h as 0.008 for a P G A value o f 0.2g. Therefore , in the absence o f any reliable ear thquake data for free field ground motion,

assumpt ion o f white noise input at bedrock level wi th a reasonable top firm soil layer provides a conserva t ive est imate o f the probabi l i ty o f failure.

10.5 Ef fec t o f sens i t iv i ty s t u d y o f d i f f erent p a r a m e t e r s (ill to f16) on the p r o b a b i l i t y o f fa i lure

In order to pe r fo rm the sensit ivi ty analysis , the logar i thmic standard deviat ions o f the factors are


0.16 - , - c = Very large (White noise) 0.14 --*- c = 0 (White noise) ic Ve large Rskc~ 0.12 - - c = 0 (Ri

.0.10 0.08

�9 "~ 0.06 0.04 0.02 o.oo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 PGA (g)

Fig. 9 Effect of seismic input for V~ = 330 m/s

0.9

Fig. 10

F> rth segment

X3r, F3r

! X~F' 1

Sign conventions used in the dynamic stiffness formulation

Table 1 Sensitivity study of different parameters (ill to f16) for PGA = 0.23g

Probability of failure

fll /?2 /]3 f14 f15 & Vs=80m/s ~=330m/s

0.49 0.46 0.30 0.15 0.14 0.25 0.02275

0.49 0.30 0.30 0.15 0.14 0.25 0.01321

0.49 0.46 0.15 0.15 0.14 0.25 0.01743

0.49 0.46 0.30 0.30 0.14 0.25 0.02872

0.49 0.46 0.30 0.15 0.20 0.25 0.02442

0.49 0.46 0.30 0.15 0.14 0.10 0.01876

0.49 0.46 0.30 0.30 0.20 0.25 0.03005

0.49 0.30 0.15 0.15 0.14 0.10 0.00587

0.00368

0.00149

0.00233

0.00539

0.00415

0.00264

0.00587

0.000376

changed one at a time within a practical range. Note that the logarithmic standard deviation of the factor F l can not be changed since it has a fixed value once risk consistent response spectra for 50 th and 84 th percentile are obtained. The result of the sensitivity analysis is shown in Table 1. It is seen from the table that individual change in the coefficient of variation of the factors does not have significant influence on the probability of failure. Last two rows of Table 1 shows the values of Pf for the combinations of maximum and minimum values of the standard deviations of the factors. It may be seen from the table that the values of Pf for these two combinations are significantly different, especially for the firm soil condition. Thus, when the logarithmic standard deviations of factors (F 2 to F6) are changed together, the change in the probability of failure could be significant.

11 C o n c l u s i o n s

A simplified PRA procedure is presented for the reliability analysis of cable stayed bridges. Reliability analysis is performed by considering the uncertainties inherent in the modeling, analysis procedure, material properties, capacity of the bridge deck, ductility effect and seismic input. The bridge is analyzed by response

spectrum method of analysis which duly takes into account spatial correlation of ground excitations at the support. The probability of failure is obtained by First Order Second Moment method of reliability. Following conclusions can be drawn from the following numerical results:

(i) Probability of failure is considerably more for soft soil condition i.e V~=80m/sec where as the probability o f failure is reduced for stiffer soil i.e Vs=330m/sec.

(ii) The probability of failure increases with the increase in the ratio of vertical to longitudinal components of ground motion and with the increase o f longitudinal component of ground motion itself.

(iii) Ground motion with no time lag between support excitation provides less value of probability of failure as compared to the ground motion with very large time lag.

(iv) The probability of failure decreases with the increase in the angle of incidence of earthquake non - uniformly on the lower end of the values of angle o f incidence, the decrease of Pf with the increase in the angle o f incidence is very small.

(v) When the risk consistent acceleration input spectrum is changed to the response spectrum corresponding to white noise at the bedrock level, the probabilities of failure are considerably increased. Thus, in the absence of a reliable risk consistent input


spectrum, white noise input at the bedrock level is a conservative choice for reliability estimates.

(vi) The standard deviation of uncertainty factors has a moderate influence on the probability of failure when they are varied one at a time. When all the variables are simultaneously increased or decreased, there is a considerable change in the probability of failure.

(vii) The probability of failure is sensitive to the variation of coherence coefficient. The probability of failure increases as the value of coherence coefficient is increases. The probability of failure is increased sharply from c = 0 to c = 0.5 and afterwards its effect on the probability of failure is not significant.

References

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Takeda M, Kai Y and Mizutani M (1989), "Seismic PRA Procedure in Japan and Its Application to a Building Performance Safety Estimation, Part2: Fragility Analysis", Proc. ICOSSAR 89, 5th Int. Conf. Structural Safe~ and Reliability, ASCE, USA, pp.629-636:

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Appendix A

Formulation of dynamic stiffness matrix is provided on Web Site: http://www.iem.cn/eeev or http://mceer.buffalo.edu/eeev

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A simplified fragility analysis of fan type cable stayed bridges

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