Seismic Reliability Analysis of Cable Stayed Bridges Against First Passage Failure

study the of This book deals with earthquake-resistant structures, such as, buildings, bridges and liquid storage tanks. It contains twenty chapters covering several interesting research topics written by researchers and experts in the field of earthquake engineering. The book covers seismic-resistance design of masonry and reinforced concrete structures to be constructed as well as safety assessment, strengthening and rehabilitation of existing structures against earthquake loads. It also includes three chapters on electromagnetic sensing techniques for health assessment of structures, post earthquake assessment of steel buildings in fire environment and response of underground pipes to blast loads. The book provides the state-of-the-art on recent progress in earthquake-resistant structures. It should be useful to graduate students, researchers and practicing structural engineers.


Introduction
In many structural applications, the ultimate purpose of using stochastic analysis is to determine the reliability of a structure, which has been designed to withstand the random excitations. The present study is concerned with one of the failures, which are the results of the dynamic response of the stable cable stayed bridges. If Y(t) is the dynamic response (either deflection, strain or stress) of the bridge at a critical point, the bridge may fail upon the occurrence of one of the following cases (Y.K. Lin, 1967): i. Y(t) reaches, for the first time, either an upper level A or lower level -B, where A and B are large positive numbers ( Fig. 1(a)). ii. Damage to the structure accumulated as Y(t) fluctuates at small or moderate excursion which are not large enough to cause a failure of the first type, and failure occurs when the accumulated damage reaches a fixed total. iii. Failure may occur if a response spends too much of its time off range, i.e., it is. outside the set limits for more than a minimum fraction of its lifetime ( Fig. 1(b)) The first case, which is called first passage failure (also called first excursion failure), is the objective of this chapter. According to definition, the first passage failure analysis means determining the probability that a prescribed threshold level (displacement, stress or other response level) will be exceeded, for the first time, during a fixed period of time. The first passage failure does not lead to the catastrophic failure of the bridge, but in view of serviceability consideration it is important. Therefore, the purpose of designing the structures against first passage failure is to reduce the probability of such failure, over the expected lifetime of the structure, to an acceptable level. In most random vibration problems, there is a probability close to unity that any given high response threshold level will be exceeded if the structure is excited for a long enough period of time. Seismic response of cable stayed bridge due to the random ground motion is obtained in this chapter using frequency domain spectral analysis. The ground motion is assumed to be a partially correlated stationary random process. In the response analysis, the quasi-static response of the bridge deck produced due to the support motion is dully considered. The probability of first passage failure for the possibility of future earthquake is then presented for the analysis of various threshold levels (taken as fraction of yield stress) is considered. A parametric study is performed to show the effect of some important parameters such as threshold level, soil conditions, degree of correlation, angle of incidence of earthquake etc. on the reliability of cable stayed bridge against the first passage failure.
(a) Failure occurs when Y(t) first reaches the level Y(t) = A .
(b) Failure occurs if Y(t)  A for more than an acceptable fraction of the total elapse time.

Brief review of earlier works
A brief review of the earlier works on different aspects of reliability analysis of structures especially cable supported bridges subjected to seismic forces / dynamic excitations is presented below in order to highlight the need for the present work.

Seismic response of cable supported bridges
The vibration of cable stayed bridges under the earthquake excitation has been a topic of considerable research for many years. The dynamic behaviour of cable stayed bridges, subjected to earthquake ground motion, has been studied by several researchers. Morris (1974) utilized the lumped mass approach for the linear and non-linear dynamic responses of two dimensional cable stayed bridges due to sinusoidal load applied at a node. It was concluded that four mode solutions were sufficient for both linear and nonlinear solutions. However, a time increment of 10% of the period corresponding to the highest mode gave satisfactory numerical results for all the dynamic analyses carried out. Abdel Ghaffer and Nazmy (1987) investigated the effects of three dimensionality, multisupport excitation, travelling waves and non-linearity on the seismic behaviuor of cable stayed bridges. Their studies indicated that non-linearities become more important as the span of the bridge increases, and the effect of multiple support seismic excitation is more pronounced for a higher structural redundancy. Nazmy and Abdel Ghaffar (1991), (1992) studied the effects of non-dispersive travelling seismic wave and ground motion spatial variability on the response of cable stayed bridges considering the cases of synchronous and non-synchronous support motions due to seismic excitations. The responses were obtained by time history analysis. They concluded that depending on the dynamic properties of the local soils at the supports, as well as the soils at the surrounding bridge site, the travelling seismic wave effect should be considered in the seismic analysis of these bridges. Also, there is multi-modal contribution from several modes of vibration to the total response of the bridge, for both displacement and member forces. Hong Hao (1998) analyzed the effects of various bridge and ground motion parameters on the required seating lengths for bridge deck to prevent the pull-off and -drop collapse using random vibration method. He analyzed two span bridge model with different span lengths and vibration frequencies and subjected to various spatially varying ground excitations. Ground motions with different intensities, different cross correlations and different site conditions were considered in the analysis. Soyluk and Dumanoglu (2000) carried out asynchronous and stochastic dynamic analysis of a cable stayed bridge (Jindo Bridge) using finite element method. In the asynchronous dynamic analysis, various wave velocities were used for the traveling ground motion. They found that response in the deck obtained from asynchronous dynamic analysis are much higher than the response obtained by the stochastic analysis. Further, shear wave velocity of ground motion greatly influences the response of the Jindo cable stayed bridge. Lin (1967), Bolotin (1965) and Crandall, Mark (1963), Abbas and Manohar (2005a, b;2007) discussed various models of structural failure under random dynamic loading and classified them into the following types: (i) failure due to the first excursion of the response beyond a safe level; (ii) failure due to the response remaining above a safe level for too long a duration and (iii) failure due to the accumulation of damage. All these types of failure are associated with the reliability estimate of Structures subjected to dynamic loading like earthquake. A number of investigations are reported in the literature on the reliability analysis of Structures against the aforementioned types of failure. Relatively recent ones and a few important old ones are reviewed here. Konishi (1969) studied the safety and reliability of suspension bridges under wind and earthquake actions. He treated the suspension safety of the structure under random ground motion from the standpoint of threshold crossing of a specified barrier. Vanmarcke (1975) dealt with the problem of the probability of first-passage beyond a threshold value by a time dependent random process. Barrier was classified into three distinct categories: a single barrier, a double barrier, and a barrier defined for the envelope of the random process. The assumption that barrier crossings are independent, so that they constitute a Poisson process, is nearly true for high barrier levels. But for relatively low barrier levels, the Poisson process assumption is in error, as it does not account for the clustering effect in case of a narrow-band process while for a wide-band process, the actual time spent in the unsafe domain is not considered. Vanmarcke suggested improvements in the Poisson process assumption to allow for the above effects. Chern (1976) dealt with the reliability of a bilinear hysteretic system, subjected to a random earthquake motion, considering first excursion beyond a specified barrier and low-cycle fatigue. Solomon and Spanos (1982) studied the structural reliability under a non-stationary seismic excitation, based on the first excursion beyond a specified barrier by the absolute value and envelope of the response process. Schueller and stir (1987) reviewed various methods to calculate the failure probabilities of structural component or system in the light of their accuracy and computational efficiency.

Reliability analsysis of structures
They analyzed that for problems of higher dimensions, approximation techniques utilizing linearization like FOSM introduced considerable error. In view of these difficulties, they provided an alternate method to calculate the failure probabilities, which combines the advantages of both the importance sampling technique and the design point calculation. Mebarki et al. (1990) presented a new method for reliability assessment, called Hypercone method based on the principles of level-2 methods of reliability analysis. The main aim of the method was to evaluate reliability index 'beta' and to deduce the values of the probability of failure by considering the whole geometry of the failure domain. The restrictions and the practical application of the method were discussed. The results reported showed that the value of probability of failure deduced from the Hasofer-Lind index beta, assuming linearity of the limit state surface, and from Monte Carlo Simulations are in accordance with those deduce from the Hypercone method. Zhu (1993) reviewed several methods for computing structural system reliability. A discretization or cell technique for determining the failure probabilities of structural system is proposed. The gaussian numerical integration method is introduced to improve its computational accuracy and can be applied to gaussian or non-gaussian variables with linear or non-linear safety margin. Harichandran et al. (1996) studied the stationary and transient response analyses of the Golden Gate suspension bridge, and the New River Gorge and Cold Spring Canyon deck arch bridges subjected to spatially varying earthquake ground motion (SVEGM). They found that transient lateral displacements of the suspension bridge center span significantly overshoot the corresponding stationary displacements for the filtered Kanai-Tajami excitation power spectrum; this spectrum may therefore be unsuitable for analyzing very flexible structures.

Assumptions
The following assumptions are made for the First Passage Failure analysis: i. It is assumed that the response process is stationary Gaussian with zero mean value. ii. As in spectral analysis method, the evaluated response is always positive, therefore the single barrier level (called type B barrier, according to Crandall et al.1966) is used. iii. It is assumed that the threshold level crossing occur independently according to a Poisson process. iv. The structure is assumed to be linear and lightly damped. v. The bridge deck (girder) and the towers are assumed to be axially rigid. vi. The bridge deck, assumed as continues beam, does not transmit any moment to the towers through the girder-tower connection. vii. Cables are assumed to be straight under high initial tensions due to the dead load and well suited to support negative force increment during vibration without losing its straight configuration. viii. An appropriate portion of the cable is included (in addition to deck mass) in the dynamic analysis of the bridge deck and is assumed to be uniformly distributed over the idealized deck. ix. Beam-column effect, in the stiffness formulation of the beam is considered for the constant axial force in the beam and its fluctuating tension in the cable is ignored. Further, cable dynamics is ignored for the bridge deck vibration, i.e., the tension fluctuations in the cables are assumed to be quasi-static, and do not introduce any nonlinearity in the system.

Seismic excitation
The seismic excitation is considered as a three component stationary random process. The earthquake ground motion is assumed as stationary random although in many cases it is assumed as a uniformly modulated non-stationary process. The response analysis remains the same for both cases. The response derived by assuming the process to be stationary can be multiplied by an envelope function to take care of the non-stationary. The components of the ground motion along an arbitrary set of orthogonal directions will be usually statistically correlated. However, as observed by Penzien and Watable (1975), the three components of ground motion along a set of principal axes are uncorrelated. These components, directed along the principal axes, are usually such that the major principal axis (u) is directed towards the expected epicenter, the moderate principal axis (w) is directed perpendicular to it (horizontally) and the minor principal axis (v) is directed vertically as shown in Fig.2. Nigam and Naranayan (1995) highlight the critical orientation of the alpha angle between the two sets of axes. Der Kiureghian (1996) developed the model for the coherency function describing spatial variability of earthquake ground motions. The model consists of three components characterizing three distinct effects of spatial variability, namely, the incoherence effect that arises from scattering of waves in the heterogeneous medium of the ground and their differential super positioning when arriving from an extended source, the wave-passage effect that arises from difference in the arrival times of waves at different stations, and the site-response effect that arises from difference in the local soil conditions at different stations. Abbas and Manohar (2002) developed a critical earthquake excitation models with emphasis on spatial variability characteristics of ground motion. In this study, the three components of the ground motion are assumed to be directed along the principal axes of the bridge x, y, z or shifted with an angle . Each component is assumed to be a stationary random and partially correlated process with zero mean characterized by a psdf. The psdf of the ground acceleration is defined by Clough and Penzien (1975) (1) in which S o is the spectrum of the white noise bedrock acceleration; Hi  a n d  2 2 Hi  are the transfer functions of the first and the second filters representing the dynamic characteristic of the soil layers above the bedrock, where in which g  , g  are the resonant frequency and damping ratio of the first filter, and f  , f  are those of the second filter. The clough and Penzien is a double filter for spectral density of www.intechopen.com the ground acceleration for which the corresponding displacement psdf does not become unrealistic. In case of Kanai-Tajimi spectrum, although the psdf of acceleration is simpler but it has problem that the corresponding displacement psdf become undefined at zero frequency.
in which ij r is the separation distance between stations i and j measured in the direction of wave propagation; c is a constant depending upon the distance from the epicenter and the inhomogeneity of the medium; V s is the shear wave velocity of the soil; and is the frequency (rad/sec) of the ground motion.

www.intechopen.com
For one sided spectrum it is well known that   is the variance of ground acceleration. The Modified Mercalli intensity of an earthquake Is is a measures of the strength of the shaking at a particular location and will vary with distance, substrate conditions and other factors. The empirical relation between the standard deviation of peak ground acceleration and earthquake intensity Is is given as where K * is a peak factor given as The empirical relation between the magnitude of earthquake and intensity of earthquake is given as In Eqn. (9), N 0 is the mean rate of zero crossing and is given by By defining the filter characteristics g ,ξ g , f , ξ f and specifying a standard deviation of the ground acceleration g f   , the psdf of the ground acceleration can be completely defined.
The ground motion is represented along the three principal directions (u,v,w) by defining ratio R u , R v and R w along them such that (12) and the psdfs of the ground acceleration in the principal directions of the ground motion (u,v,w) can be defined as where R x , R y ,and R z are the ratios of the ground motion along the principal axes of the bridge and given as and R u , R v , and R w are ratios of the ground motion along the principal directions of the ground motion (u,v,w) as shown in Fig.2.

Distribution function of magnitude of earthquake
Two types of distribution functions of the magnitude of earthquake are considered in the study.

Exponential distribution
This type of probability distribution function of the magnitude of earthquake is based on the Gutenberg -Ritcher Recurrence law (Kagan, 2002a) where  m is the mean annual rate of exceedence of magnitude m ; 10 a is the mean yearly number of earthquakes greater than or equal to zero,; and b describes the relative where,  = 2.303b, and m 0 is the lower threshold magnitude of earthquake, earthquakes smaller than which are eliminated, and m is the magnitude of earthquake. The cumulative distribution function (CDF) of magnitude of earthquake for exponential distribution is given by the following expression

Gumbel type-I distribution
The cumulative distribution function of magnitude of earthquake for Gumbel type-I distribution is given by the following expression where  and u are the parameters for Gumbel Type-1 distribution given by in which M and  m are the mean and standard deviation of the magnitudes of earthquake respectively.

Free vibration analysis of cable stayed bridge deck (girder)
The bridge deck, as shown in Figs.3(a) is idealized as a continuous beam over the outer abutments and the interior towers, and the effect of cable is taken as vertical springs at the points of intersections between the cables and the bridge deck shown in Fig.3(b). Further, the effect of the spring stiffness is taken as an additional vertical stiffness to the flexural stiffness of the bridge. Referring to Fig.3(c) and Fig.3(d), the fluctuations of tension in the cable at any instant of time (t) can be written as where, / The deflections of the tower at the cable joints can be obtained by assuming that the tower behaves like a vertical beam fixed at the bottom end and constrained horizontally at the level of the bridge deck and subjected to the transverse forces h i (t) cos i (i=1, Nc) at the cable tower joints as shown in Fig.3(b) and are given by where the elements of the matrix (C) can be easily obtained from the deflection equation of the tower (vertical beam) subjected to concentrated load, as mentioned above, using standard structural analysis procedure. Eliminating {Δ} from Eqns. (24) and (25) ,, , , in which E d and I r are the modulus of elasticity of the bridge deck and vertical moment of inertia of the beam segment r of the deck respectively. P(x r ,t) is defined as the load induced due to seismic excitations at different support degree of freedom and is given by j=1,2, --------, 8 are the accelerations at the different support degrees of freedom and g jr (x r ) is the vertical displacement of the r th segment of the bridge deck due to unit displacement at the j th degree of the supports. In Eqns. (28) and (29), it is assumed that for lightly damped system, the effect of damping term associated with quasi-static movement of the supports is negligible (Clough and Penzien, 1993 where {C} r is the vector of integration constants containing A nr etc., and (T) r is the matrix integration constants. The sign conventions used in the dynamic stiffness formulation are shown in Fig.4. The explicit expressions for the elements of (K) r and (T) r are given by Chatterjee (1992). Assembling the stiffness (K) r for each element (r) and adding the vertical stiffness due to cables (K C,V ), the overall stiffness of the bridge (K) is obtained. The condition for the free vibration of the bridge deck may then be written as

Modal transformation
The modal transformation of the relative vertical displacement y(x r ,t) for any point in the r th deck segment is given as where R jn is the modal participation factor given by in which g jr (x r ), 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 j th direction of support movement.
where [G n ] is the generalized force coefficients for the n th mode and can be obtained by Eqns. (36 & 37) i.e. G 1n =R 1n , -----, G 8n = R 8n. The psdf of the response y(x r ,t) is given by

Evaluation of the quasi-static displacement
The is the psdf matrix for the ground displacements at the support d.o.f.s and can be obtained in terms of the psdf of ground acceleration g g ff S   with the help of both the coherence function, the ratio between the three components of ground motion and Eqns. (11, 16 & 17).

Evaluation of the total displacement
The total displacement at any point of the r th beam segment of the bridge deck at any time (t) can be written as Y(x r ,t) = y (x r , t)+ g (x r , t) The psdf of the total vertical displacement can be expressed as   EId g x d x is obtained from the quasi-static analysis of the entire bridge using the stiffness approach as mentioned before.

Statistical parameters of response
For studying the statistical properties of the response process, the first few moments of the response power spectral density function are needed. The j th moment of the PSDF function may be defined as follows: The zeroth and second moments may be recognized as the variances of the response and the first time derivative of the response respectively. The mean rate of zero crossing at positive slopes is by Another quantity of interest is the dispersion parameter q given by The value of q lies in the range [0,1]. It can shown that q is small for a narrow band process and relatively large for a wide band process. The mean rate of crossing a specified level A at a positive slope by a stationary zero mean Gaussian random process z(t) can be expressed by Lin (1967) Where, L 0 is the probability of starting below the threshold, is the decay rate, and T is the duration of the response process. At high barrier levels, L o is practically equal to one, and the decay rate is given by the following expressions for processes with double barrier and one sided barrier respectively (Vanmarcke, 1975;Lin, 1967) 2 In case of relatively low threshold levels, an improved value can be obtained by using the expressions for the probability of starting below the threshold and the decay rate (Vanmarcke, 1975)

Reliability estimation against first passage failure
For an earthquake with given magnitude M, the probability of First Passage Failure, i.e., the probability that the response is larger than a threshold level A, can be determined from the following relationship   If the rate of earthquake occurrence for the seismo -tectonic region considered in the study is a constant and n is the average number of earthquakes per year in the magnitude range of interest for the source region, the probability of atleast one failure due to earthquake in "m" years can be expressed as   .

Numerical study
A double plane symmetrical harp type cable stayed bridges, (Morris, 1974 ) used as illustrative example is shown in Fig.3.3(a). The structural data of the bridge is shown in Table-1.
In addition, the following data are assumed for the analysis of the problem, E c = E d ;  = 0.02 for all modes; and the tower -deck inertia ratio, the ratio between three components of the ground motion ( R u :R v :R w ), , duration of earthquake, values for exponential distribution of magnitude of earthquake are taken 4, (1.0:1.0:1.0) , 0.0 0 , 15 sec and (1.5, 2.303, 2.703) respectively unless mentioned otherwise. Also, the ground motion is assumed to be partially correlated in firm soil unless mentioned otherwise. The random ground motion is assumed to be homogeneous stochastic process which is represented by Clough and Penzien double filter psdf given by two sets of filter coefficients representing the soft and firm soils respectively. For the soft soil , the coefficients are  g = 6.2832 rad/sec;  f = 0.62832 rad/sec ;  g =  f = 0.4, while those for the firm soils are  g = 15.708 rad/sec;  f = 1.5708 rad/sec;  g =  f = 0.6. The two psdfs corresponding to the two sets of filter coefficients are shown in Fig.5. The spatial correlation function used in the parametric study is given by Eqn.5 in which the value of c =2.0, Vs = 70 m/sec and Vs = 330 m/sec for the first and second psdfs respectively. The r.m.s (root mean square) ground acceleration is related to intensity of earthquake by empirical equation given by Eqn.7.
Intensity of earthquake I s in turn is related to magnitude of earthquake given by Eqn. 9 Table 2. Fundamental frequencies of the bridge deck for different tower -deck inertia ratio.

Effect of the tower -Deck inertia ratio (I t /I d )
The first 5fundamental frequencies are obtained for different ratios of the tower -deck inertias i.e. 2, 4 and 6 respectively as shown in Tables-2. It is seen from the tables that with the increase of the tower -deck inertia ratio, the frequencies of the bridge deck increase. This is due to the fact that the increase in the tower stiffness increases the component of the vertical stiffness of the bridge provided by the cables.

Effect of the quasi-static component on the response
Total and relative displacement of the bridge decks obtained for the firm soil is shown in Tables-3. It is seen from the Tables that the contribution of the quasi-static component to the total response is significant for displacement where as it is small for bending moment.  Table 3. Effect of the quasi -static part of the response on the r.m.s responses

Effect of barrier level on the reliability
Effect of the barrier level on the reliability of the bridge is shown in Figs. 7. The barrier level are taken as 15%, 20%, 25%, 30%, 33%, 40%, 50% and 70% of the yield stress assuming that the barrier level is the difference between yield stress and the pre-stress in the girder ( deck). It is seen from the figures that the reliability increases as the barrier levels increases as it would be expected. However, the variation is not linear; it tends to follow an S shaped curve. For certain condition, the variation of reliability with barrier level may be very steep in the lower range of barrier levels. The same figures also compare between the reliabilities for firm and soft soils conditions. It is seen from the Fig.7 that the reliability for a particular barrier level is higher for firm soil. The difference between the two is considerably more at the lower end of the barrier level.

Effect of magnitude of earthquake on the reliability
The effect of the distribution of magnitude of earthquake on the reliability is shown in Fig.8. The figure shows the variation of reliability with barrier level for different distributions of the magnitude of earthquake obtained by exponential and gumbel distribution. It is seen from the figures that the reliability increases with the increase in beta values for exponential distribution. The difference between reliabilities obtained for two beta values considerably is more for lower values of barrier level. Above a certain value of beta, the reliability nearly approaches unity for all barrier levels. Further, Gumbel distribution provides much higher value of reliability as compared to Exponential distribution (for beta = 1.5). For soft soil condition (Fig.9), the effect of the distribution of the magnitude of earthquake is more produced.     Fig.12 show the effect of angle of incidence on the variation of reliability with barrier level. Three values of angle of incidence are considered namely, 0 0 i.e. major direction of earthquake is along the longitudinal axis of the bridge and the other two cases are having 30 0 and 70 0 angle of incidence with the longitudinal axis of the bridge. It is seen from the figures that 0 0 angle of incidence provides minimum reliability while 70 0 angle of incidence provides maximum reliability. This is expected because 0 0 angle of incidence produces maximum stress in the bridge.  Fig.13 shows the variation of reliability with barrier level for three cases of ground motion that is fully correlated, partially correlated and uncorrelated. It is seen from the figures that fully correlated ground motion provides the highest reliability while uncorrelated ground motion gives the lowest value. The partially correlated ground motion gives reliability in between the two. This is the case because uncorrelated / partially correlated ground motion induces additional bending moment in the deck due to the phase lag of ground motion between different supports. The difference between the reliabilities for the three cases is nor very significant for the hard soil. However, the difference between them is significant for the soft soil (Fig.14). Further, the difference between the reliabilities are considerably reduced at the higher end of the barrier level. www.intechopen.com Fig.15 shows the variation of reliability with duration of earthquake for a barrier level of 33% . It is seen from the figures that reliability decreases mildly with the increase of duration of earthquake for both soft and firm soil. Thus, the duration of earthquake does not have significant influence on the reliability.

Conclusions
Reliability against first passage failure of cable stayed bridges under earthquake excitation is presented. The responses of cable stayed bridges are obtained for random ground motion which is modeled as stationary random process represented by double filter power spectral density function (psdf) and a correlation function. The responses are obtained by frequency domain spectral analysis. Conditional probability of first crossing the threshold level for a given RMS ground acceleration is obtained by using the moments of the psdf of the response. The RMS value of the ground acceleration is related to the magnitude of earthquake by an empirical equation, and the probability density function of the earthquake is integrated with the conditional probability of failure to find the probability of first passage failure. Using the above method of analysis, two cable stayed bridges are analyzed and probabilities of first passage failure are obtained for a number of parametric variations. The results of the numerical study lead to the following conclusions: i. The reliability against first passage failure increases sharply with the increase in barrier level in the lower range of its values. ii. For the soft soil condition, the reliability is considerably less as compared to the firm soil condition. iii. Gumbel distribution of the magnitude of the earthquake provides a very high estimate of reliability and gives values close to those obtained by exponential distribution with high values of the parameter . iv. Uncorrelated ground motion provides lower estimates of the reliability as compared to the fully correlated ground motion. The difference significantly more for the soft soil.
v. The ratios between the components of ground motion have considerable influence on the reliability estimates. For soft soil condition, the difference between the reliability estimates, especially in the lower range of barrier level. vi. In the lower range of barrier level, considerable difference between reliabilities is observed for 0 0 and 70 0 angle of incidence with respect to the longitudinal axis of the bridge. For higher value of the barrier level, there is practically no difference between the two reliabilities. vii. The duration of ground motion does not have significant influence on the reliability estimates. viii. It is found that reliability decreases with the increase of average number of earthquake per year. The variation is nonlinear and more steep for the soft soil condition. For an average number of earthquake 0.5 per year , the reliability could be as 0.3.