Int. J. Forensic Engineering, Vol. x, No. x, xxxx 1

Copyright 20xx Inderscience Enterprises Ltd.

Stochastic failure analysis of the gusset plates in the Mississippi River Bridge

Mojtaba Mahmoodian, Amir Alani and Kong Fah Tee*Department of Civil Engineering,University of Greenwich,Central Avenue, Chatham Maritime,Kent ME4 4TB, UKE-mail: M.Mahmoodian@gre.ac.ukE-mail: M.Alani@gre.ac.ukE-mail: K.F.Tee@gre.ac.uk*Corresponding author

Abstract: The I-35W Mississippi River Bridge over Minneapolis, Minnesota, collapsed suddenly on August 1, 2007. Previous studies showed that the demand-to-capacity ratio for one of the gusset plates had become extremely high after about 40 years of service of the bridge and therefore, the failure of the gusset plate caused the failure of the whole bridge. A forensic assessment using stochastic reliability analysis is carried out in this research to check whether the collapse of the bridge could have been predicted at the design stage. For this purpose, the probabilities of failure for different types of stresses in the gusset plate are estimated. To consider the uncertainties involved in dead load and live load increments with time, the gamma process concept is employed to model stress increments. It is shown that the probability of failure in the year 2007 was higher than the recommended value. Therefore, it can be concluded that if the results of this study had been available at the design stage, the lack of reliability in 2007 could have been predicted and the collapse of the bridge and its disastrous consequences could have been prevented. It is also concluded that stochastic reliability analysis can be used as a rational tool for failure analysis and reliability assessment of bridges to prevent the risk of collapse.

Keywords: probabilistic failure analysis; Mississippi river bridge; gusset plates; reliability index; bridge collapses.

Reference to this paper should be made as follows: Mahmoodian, M., Alani, A. and Tee, K.F. (xxxx) Stochastic failure analysis of the gusset plates in the Mississippi River Bridge, Int. J. Forensic Engineering, Vol. x, No. x, pp.xxxxxx.

Biographical notes: Mojtaba Mahmoodian is a PhD candidate working on reliability analysis and service life prediction of infrastructures, including pipelines and bridges. His working experience includes working for about eight years for construction companies and with consulting engineers, mainly in quality control and assessment of structures and infrastructure such as bridges, dams, tunnels and high rise buildings. His publications include journal and conference SDSHUV LQ WKH HOG RI VWRFKDVWLF UHOLDELOLW\ DQDO\VLV FRUURVLRQ RI FRQVWUXFWLRQDOmaterials (concrete, steel and cast iron), concrete technology and quality control and optimisation methods in civil engineering.

2 M. Mahmoodian et al.

Amir Alani is Head of the Department of Civil Engineering at the University of Greenwich and the Bridge Wardens Chair in Tunnel and Bridge Engineering, both since 2009. He previously was Head of the Department of Mechanical and Design Engineering at the University of Portsmouth. He completed his MSc and PhD in Saturated Soil Media with applications in Petroleum Engineering at the University of Science and Technology of Montpellier in France. His research interests are bridge and tunnel engineering (assessment and monitoring), applications of non-GHVWUXFWLYHWHVWLQJPHWKRGVLQWKHKHDOWKPRQLWRULQJRIVWUXFWXUHVEUHUHLQIRUFHGconcrete, the reliability of cementitious concrete sewer pipes and repair and maintenance of buildings and structures.

Kong Fah Tee is a Senior Lecturer in infrastructure engineering at the University of Greenwich. He did his BEng (Hons) in Civil Engineering from the University Putra Malaysia and his PhD from the National University of Singapore. He also FRPSOHWHG D SRVWJUDGXDWH FHUWLFDWH LQ +LJKHU (GXFDWLRQ DW WKH 8QLYHUVLW\ RIGreenwich and is a Fellow of the Higher Education Academy. His research effort KDVEHHQIRFXVHGRQVWUXFWXUDOV\VWHPLGHQWLFDWLRQVWUXFWXUDOKHDOWKPRQLWRULQJexperimental stress analysis, fatigue, fracture mechanics, structural dynamics and control and reliability. He was an invited visiting foreign expert at the Nanjing University of Aeronautics and Astronautics.

1 Introduction

The I-35W bridge over the Mississippi River in Minneapolis, Minnesota, collapsed suddenly on 1 August, 2007. Approximately 300 m of the 570 m long bridge collapsed onto the water and ground, resulting in 13 fatalities and 145 injuries, with 111 vehicles involved in the collapse. The collapse of the bridge in a major US downtown area was unprecedented. This event was peculiar because the bridge was a typical steel truss structure and the collapse occurred under normal operating conditions. In general, steel truss bridges such as the I-35W Mississippi River bridge are a very common form for long-span bridges in the world. Before this event occurred, steel truss bridges had earned the reputation of being economical and reliable. Although the small redundancy of the trusses may be of concern, it is believed that mandated maintenance procedures assure that this structural system is as safe and reliable as any other (Liao et al., 2011).

A failure analysis of the I-35W bridge (Astaneh-Asl, 2008; Hill et al., 2008) showed that the bridge failure was initiated at the gusset plates that connected the top chord members to a compression diagonal and tension diagonal. In a comprehensive structural analysis, Holt and Hartmann (2008) concluded that the demand-to-capacity ratio of one gusset plate (U10) was high enough to cause the failure of the plate and eventually, the failure of the whole bridge. Gusset plates are designed to bear up stresses at the Whitmore section (Whitmore, 1952), to block shear from tension and buckling under compression; the stresses are calculated based on simple beam equations. The stress distribution in gusset plates for bridge trusses differs substantially from that in simple tension or compression connections and varies with member arrangements (Liao et al., 2011).

To resolve uncertainties about loads and stresses at the design stage, the use of stochastic models for reliability analyses of bridges can be considered. In a stochastic model, temporal

Stochastic failure analysis of the gusset plates in the Mississippi 3

variability associated with the stresses can be taken into account. In this study, a stochastic reliability analysis is conducted on the critical gusset plate (U10) in the collapsed I-35W bridge to check the predictability of the collapse based on the probability of failure. To model the monotonic progression of stresses, this paper proposes a gamma process model for stress variations due to loads increments.

A gamma process is a stochastic process with independent, non-negative increments having a gamma distribution with an identical scale parameter and a time-dependent shape parameter. A random variable X that is gamma-distributed with shape and scale is denoted: X ~ (, *DPPD, ). A stochastic process model, such as the gamma process, incorporates the temporal uncertainty associated with the evolution of stress. The gamma process allows modelling of gradual stress increment monotonically accumulating over time. The mathematical aspects of gamma processes can be found in Dufresne et al.(1991), Ferguson and Klass (1972), Singpurwalla (1997) and van der Weide (1997).

In this study, based on the available information, the loading condition of the bridge at the design stage (year 1967) is determined and stresses are calculated. For two consequential 20-year periods (ending in the years 1987 and 2007), stresses are then predicted. The gamma model parameters (scale and shape parameters) are estimated using the maximum likelihood method or the method of moments. The probability of failure of the bridge due to different stresses (shear, tension and compression) is calculated and the probabilities of failure for the gusset plate are checked with the maximum acceptable probability of failure suggested by design codes such as LRFD AASHTO (1998).

2 Description of I-35W Mississippi River bridge failure

According to the reports from the Minnesota Department of Transportation (Mn/DOT, 2008) and the National Transportation Safety Board (NTSB, 2008b and 2008d), construction of the I-35W Mississippi bridge started in 1964 and after four years, in 1967, the bridge was opened to the public. The Mississippi River bridge comprised 14 spans and 13 piers consisting of the south approach spans, the north approach spans and the central deck-truss spans. The length between the two end supports was 581 m. The spans from the south and the north approaches were rested respectively at the main trusses at panel points U0 and U7KHPDLQWUXVVZDVRQroller supports at Piers 5, 6 and 8 and a hinge support at Pier 7. The main truss was 11.6 m long DQGLQWRWDOSDQHOVDQGPDLQWUXVVHVZHUHMRLQHGE\RRUWUXVVHVWKDWVSDQQHGWUDQVYHUVHO\between the panels (Figure 1). The bottom trusses were supported on longitudinal stringers, ZKLFKVXSSRUWHGWKHUHLQIRUFHGFRQFUHWHGHFNDQGWUDIFORDG%HORZWKHRRUWUXVVHVVZD\IUDPHVLQDFKHYURQFRQJXUDWLRQZHUHMRLQHGWRWKHWZRPDLQWUXVVHV7KHPDLQWUXVVHVZHUHalso connected by lateral bracing spanning among the upper chords and lower chords.7KHGHFNZDVVHSDUDWHGIRUVRXWKERXQGDQGQRUWKERXQGWUDIF(DFKKDOIRIWKHGHFNZDV

meant to carry four 3.7PWUDIFODQHVDQGWZRm shoulders. The total concrete thickness was 215 mm, which, according to the original design, should have been 165 mm thick, but the thickness was increased by 50 mm because of concrete overlay in later years. External barriers and middle railings were also added to address maintenance and operational issues. At the time of the collapse, a bridge patching and overlay project had been underway since June, 2007. Two northbound outside lanes and two southbound inside lanes were closed to KHDY\WUDIF(TXLSPHQWDQGPDWHULDOZHUHVHWXSRQWKHVHFXUHVRXWKERXQGWUDIFODQHVQHDUthe location of the failure.

4 M. Mahmoodian et al.

2QFHWKHFROODSVHRIWKHEULGJHWRRNSODFHHOGLQYHVWLJDWLRQVZHUHSHUIRUPHGE\WKH176%and the Mn/DOT. Observations (Astaneh-Asl, 2008; Hill et al., 2008; NTSB, 2008d) showed that the collapse was initiated at panel point U10. Figure 2(a) is a picture taken before the FROODSVH176%FVKRZLQJWZRJXVVHWSODWHVOLQNLQJYHWUXVVPHPEHUVDWSDQHOSRLQW8RIWKHZHVWPDLQWUXVV8:)LJXUHEVKRZVWKHYHWUXVVPHPEHUVXSSHUFKRUGVU9/U10 and U10/U11, diagonals members L9/U10 and U10/L11 and a vertical member 8/7KHYH WUXVVPHPEHUVZHUH FRQQHFWHG WKURXJKDSDLU RIPP WKLFNJXVVHWplates of ASTM A441 grade 50 steel, using river bolts of 25 mm diameter. Upper chords U9/U10 and U10/U11 and compression diagonal member L9/U10 were precast box sections, while tension diagonal member U10/L11 and vertical member U10/L10 were W sections. All the truss members were welded prefabricated sections. Because U10 was located near an LQHFWLRQSRLQWRIWKHFRQWLQXRXVWUXVVWKHIRUFHLQWKHXSSHUFKRUGXFWXDWHGIURPWHQVLRQon one side to compression on the other side, as can be seen in Figure 2(b); the diagonal members exerted considerable compression and tension on the joining plate. Consequently, DKXJHVKHDUIRUFHZDVIRUPHGRQDFULWLFDOODWHUDOVHFWLRQDVVKRZQLQWKHJXUH

Figure 2 Panel point U10 before collapse: (a) Photo U10W (NTSB, 2008c); (b) member descrip-tion and truss forces (units in kN)

Source: Liao et al., 2011

Based on commonly used design checks, Holt and Hartmann (2008) suggested that if the thickness of the gusset plate was twice that of the original one at panel point U10, it would

Figure 1 Plan and elevation of I-35 Mississippi River Bridge

Source: Liao et al., 2011

Stochastic failure analysis of the gusset plates in the Mississippi 5

have been much safer to transfer the shear forces and design loads. Figure 3(a) shows a picture of panel point U10W taken after the collapse (NTSB, 2008c). Figure 3(b) shows the three primary cracks seen in Figure 3(a); these were commonly observed in all four U10W and U10E gusset plates. Unbelievably, fracture of gusset plates was detected in the U10 panel points. This observation, in addition to the study by Astaneh-Asl (2008) and by Holt DQG+DUWPDQZKLFKLVEULH\SUHVHQWHGLQWKHQH[WVHFWLRQSRZHUIXOO\VXJJHVWWKDWthe collapse of the I-35W Bridge started in the U10 gusset plate.

Figure 3 Panel point U10 after collapse: (a) Photo U10W (NTSB, 2008c); (b) Reported locations of fracture (see online version for colours)

Source: Liao et al., 2011

3 Previous results of the structural analysis of the gusset plates

The study by Holt and Hartman (2008) on the truss gusset plates indicates that the Demand-to-Capacity (D/C) ratios of some of the gusset plates (especially gusset plate U10) were more than the allowable values. The D/C ratio is a comparative measure of WKHHIFLHQF\RIWKHGHVLJQ$'&YDOXHRIOHVVWKDQLQGLFDWHVDFRQVHUYDWLYHGHVLJQD'&UDWLRRILQGLFDWHVDQHIFLHQWGHVLJQDQGD'&UDWLRJUHDWHUWKDQLQGLFDWHVDliberal design with a reduction in the intended factor of safety. They found out that in designing the gusset plates, as usually is the case, the width and length of many of them were dictated by the number of connectors (rivets) needed to fasten the truss members to the gusset plate. The study indicated that the thicknesses of the gusset plates supplied at L5, L7, L9, U12 and L13 were the result of the demands of the applied loading. All of these gusset plates have one or more D/C ratios equal to or greater than 0.80. They showed that all the gusset plates at U2, L3, U4, U10 and L11 failed the D/C ratio check for shear along Section A-A and the gusset plates at U10 and L11 on Section B-B also IDLOHG WKH VKHDU'& UDWLR FKHFNE\D VLJQLFDQWPDUJLQ ,Q DGGLWLRQ WKHVH WZRSODWHValso had principal tension and compression overstresses that were again a result of the dominating shear on the section.

Holt and Hartman (2008) also concluded that only the U10 gusset plate violated the unsupported edge limitations. Reviewing the entire set of D/C ratios in total reveals that the

6 M. Mahmoodian et al.

gusset plates at U10 were the most under-designed plates. Coupling the lack of capacity of these gusset plates with a violation of the detailing requirements for the unsupported edge, makes the U10 gusset plates the most vulnerable plates in the truss. Figure 4 illustrates the demand to capacity ratios for gusset plate U10 obtained by Holt and Hartman (2008). In almost all cases, the ratio is more than one.

Figure 4 Demand to capacity ratio, results from Holt and Hartman, 2008 (see online version for colours)

$QRWKHUVWXG\E\+DRZKLFKXVHGDQRQOLQHDU'QLWHHOHPHQWFRPSXWDWLRQEDVHGload rating, also indicated that some of the gusset plates had almost reached their yield limit when the bridge experienced the design load condition. Both previous studies (Holt and Hartman, 2008; Hao, 2010) showed that the capacity inadequacies for the U10 gussets were considerable for all conditions investigated with the plate, providing approximately 50% of the resistance required by the design loadings. In the next section, a reliability-based methodology is introduced to analyse the reliability of the gusset plate.

4 Problem formulation for reliability analysis

In the theory of structural reliability, the criterion that should be checked for reliability analysis can be expressed in the form of a limit state function (or safety margin) as follows (Melchers 1999):

Z R S t R t S t, , ,( ) = ( ) ( ) (1)where:

R(t): structural resistance or capacity at time tS(t): load effect or stress at time t.

For an in-service bridge, the stresses SLQFUHDVHZLWKWLPHGXHWRWKHLQFUHDVHLQWUDIFQHZpavement overlays and so on. The capacity of the structure or the resistance of the material also may change due to deterioration processes and aging. In practical examples, there are

Stochastic failure analysis of the gusset plates in the Mississippi 7

always uncertainties when predicting stresses in the future because the prediction of changes LQ WUDIF DQG UHSDLU SODQV LV XQFHUWDLQ7KLV LVVXH OHDGV LQIUDVWUXFWXUHPDQDJHUV WR DSSO\stochastic analyses for the reliability assessment of bridges.

Because no corrosion in gusset plates had been reported in the Mississippi River bridge, in this study the resistance (R) is assumed to be constant with time (i.e., there is no material deterioration and resistance reduction within the bridges lifetime). However, because of the addition of more dead load from overlaying new pavements and remarkable changes LQ WUDIF YROXPHV RQ WKH EULGJH WKH VWUHVVHV RQ WKH EULGJH DUH FRQVLGHUHG DV VWRFKDVWLFprocesses of the loads effect.

5 Stochastic process of stresses

For an in-service bridge, dead loads increase due to repair and maintenance actions and OLYHORDGVLQFUHDVHGXHWRKLJKHUWUDIFGHPDQG:LWKWKHVHDVVXPSWLRQVDVWUXFWXUDOORDGeffect (stress) that varies in time can be modelled as a gamma process. The occurrence of a VWUHVVPD\EHGHVFULEHGE\DJDPPDGLVWULEXWLRQIXQFWLRQ7KHPDWKHPDWLFDOGHQLWLRQRIthe gamma process is given in Equation (2). Recall that a random quantity S has a gamma distribution with shape parameter > 0 and scale parameter > 0 if its probability density function is as given by Papoulis (1965):

Ga S S e S( , )( )

.|

=

1 (2)

Let (t) be a non-decreasing, right continuous, real-valued function for t 0. The gamma process is a continuous-time stochastic process {S(t), t 0} with S(t) denoting the stress at time t, t 0; the probability density function is S(tLQDFFRUGDQFHZLWKWKHGHQLWLRQRIWKHgamma process, given by

f S Ga S tS t( ) ( ) ( | ( ), ),= (3)

with mean and variance as follows:

E S t t( ( )) ( ) ,=

(4)

Var S t t( ( )) ( ) .= 2

(5)

A structure is considered to fail when its stress, denoted by S(t), becomes more than its resistance (which is to say that the limit state function, Z, in Equation (1), is negative). Assuming that resistance R is deterministic, the time at which failure occurs is denoted by the lifetime T. By considering the probability density function (Equation (3)), the probability of failure can then be written as:

Pr( ) Pr( ( ) ) ( ) ( ( ), )( ( ))

,( )T t S t R f S dt RtS t SR

= = =

(6)

where v x t e dtv tt x

, ( ) = =

1 is the incomplete gamma function for x 0 and v > 0.To model stress change due to load increment, in terms of a gamma process, the question which remains to be answered is: how does the expected stress increase over time (i.e., the

8 M. Mahmoodian et al.

shape function for the gamma process model)? It is very common to model progressive processes such as structural deterioration by using a power law formulation (Ellingwood and Mori, 1993; van Noortwijk and Klatter, 1999). In this study, the same formulation is used for modelling stress increments within the service life of the structure:

( ) ,t ctb= (7)

where c and b are constants. Most of the time, by considering a few predicted values for stress, there is engineering knowledge available about the shape of the expected stress and therefore, parameter b in Equation (7) can be assumed.

To model the stress as a gamma process with shape function (t) = ctb and scale parameter , the parameters c and should also be estimated. For this purpose, statistical methods are suggested. The two most common methods that can be used for parameter estimation are maximum likelihood and the method of moments. Both methods for deriving the estimators of c and were initially presented by Cinlar et al. (1977) and were developed by van Noortwijk and Pandey (2003).

5.1 Maximum Likelihood EstimationIn statistics, Maximum Likelihood Estimation (MLE) is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, 0/(SURYLGHVHVWLPDWHVIRUWKHPRGHOVSDUDPHWHUV,QJHQHUDOIRUD[HGVHWRIGDWDDQGDQunderlying statistical model, the method of maximum likelihood selects values of the model parameters that produce a distribution giving the predicted data the greatest probability (i.e., parameters that maximise the likelihood function). Let us assume n predicted stresses are denoted by S1, S2, , Sn. The principle of maximum likelihood assumes that the sample data set is representative of the population, with a probability density function of fS (S1, S2, ,Sn DQGFKRRVHV WKDWYDOXH IRU WKH XQNQRZQSDUDPHWHU WKDWPRVW OLNHO\ FDXVHG WKHpredicted data to occur, i.e., once predictions S1, S2, , Sn are given. fS (S1, S2, , SnLVDIXQFWLRQRIDORQHDQGWKHYDOXHRIWKDWPD[LPLVHVWKHDERYHSUREDELOLW\GHQVLW\IXQFWLRQLVWKHPRVWOLNHO\YDOXHIRU

In the current study a typical data set consists of prediction times ti, i = 1, , n where 0 = t0 < t1 < t2 < ... < tn and the corresponding predictions of the cumulative amounts of stresses Si, i = 1, , n are assumed to be given as inputs of the model. The maximum likelihood estimators of c and can be determined by maximising the logarithm of the likelihood function of the increments. The likelihood function of the predicted stress increments i i iS S i n= =1 1, , ,! is a product of independent gamma densities:

11

1

[ ][ ] 1

1 ( ) ( )1 1 1

(( , , | , ) ( ) .( [ ])

b bi i b b

i i

i i

c t tn n c t tn i i ib bS t S ti i i i

l c f ec t t

= =

= =

!

(8)

To maximise the logarithm of the likelihood function, its derivatives are set to zero.

c n c

ib

ib n

b

nl cc t t

ctS

c tlog ( , , | )

[ ]log log ( [

1

1

! =

iib

ib

ib

ib

inb

ni

i

t

c t tctS

+ ( )

=

1

1 1

])

[ ] log 11

11

1

n

nb n

b

nib

ib

i

n

i ib

ibt

ctS

t t c t t

=

+ =

log [ ] log [ ]]( ){ } = 0

(9)

Stochastic failure analysis of the gusset plates in the Mississippi 9

where the function (x) is the derivative of the logarithm of the gamma function:

( ) ( )( )

log ( ) .'

x xx

xx

= =

(10)

It follows that the maximum likelihood estimator of is (van Noortwijk and Pandey, 2003):

bn

n

ctS

= (11)

where c must be computed iteratively from the following equation:

( ){ }1 11 [ ] [ ] log log .b

n b b b b b ni i i i i ni

n

ctt t c t t t

S =

= (12)

5.2 Method of momentsIn statistics, the method of moments is a method of estimation of population parameters such as mean and variance by equating sample moments with unpredictable population moments and then solving those equations for the quantities to be estimated. The method of moments estimates c and can be found from van Noortwijk and Pandey (2003):

1

11

c

n

ii n

n bb b ni i

i

Stt t

=

=

= = =

(13)

( ) ( )( )2

21112

11 1

1 .

n b b ni i b bn i

i i in b b ii i i

t tSt t

t t

=

==

=

(14)

7KHUVWHTXDWLRQIURPERWKPHWKRGVPD[LPXPOLNHOLKRRGDQGPHWKRGRIPRPHQWVLVWKHsame and the second equation in the method of moments is simpler, because this method PDNHVLWXQQHFHVVDU\WRGRLWHUDWLRQIRUQGLQJWKHXQNQRZQSDUDPHWHU ( )c .

6 Resistance capacity

$FFRUGLQJWRRIFLDOGRFXPHQWV176%DQGDWKHEULGJHVVWHHOPHPEHUVZHUHPDGHRIJUDGHPLOGVWHHOZLWK

10 M. Mahmoodian et al.

LHWKH\HDU7KHORDGVIRUWKHWZRVXEVHTXHQW\HDUSHULRGVHQGLQJLQWKH\HDUDQGLQWKH\HDUFDQEHSUHGLFWHGEDVHGRQHVWLPDWLRQRIWKHLQFUHDVHVLQGHDGORDG DQG OLYH ORDG GXH WR UHSDLU DQGPDLQWHQDQFH DFWLRQV DQG JURZLQJ WUDIF GHPDQGUHVSHFWLYHO\6KHDUSULQFLSDOWHQVLRQDQGSULQFLSDOFRPSUHVVLRQVWUHVVHVDWWKHJXVVHWSODWHsections then can be determined accordingly. The calculated and predicted stress values are presented in Table 1.

Table 1 6WUHVVHVDWJXVVHWSODWH8IRU6HFWLRQ$$DQG6HFWLRQ%%LQWKH\HDUVDQG

Stress type(S)

Calculated stresses at year

t0 = 0 03D

Predicted stresses at year 1987,

t1 = 20 yr 03D

Predicted stresses at year 2007,

t2 = 40 yr 03D

6HFWLRQ$$ 6KHDU 103 155 212

PrincipalTension

212

PrincipalCompression

151 212

6HFWLRQ%%

6KHDU 103 PrincipalTension

PrincipalCompression

151

To model the stress increments as a gamma process, the scale and shape parameters (i.e., (t)and ) should be estimated. For this purpose, the two statistical methods mentioned above PD[LPXPOLNHOLKRRGDQGPHWKRGRIPRPHQWVDUHXVHG7KHUHVXOWVIRUVHFWLRQV$$DQG%%RIWKHJXVVHWSODWH8)LJXUHDUHSUHVHQWHGLQ7DEOH

Figure 5 )UHHGLDJUDPRIJXVVHWSODWH8VHHRQOLQHYHUVLRQIRUFRORXUV

Source +ROWDQG+DUWPDQ

Stochastic failure analysis of the gusset plates in the Mississippi 11

Table 2 *DPPDPRGHOSDUDPHWHUVHVWLPDWHGE\PD[LPXPOLNHOLKRRGPHWKRGDQGPHWKRGRImoments

Stress type(S) Parameter

Maximum likelihood method Method of moments

6HFWLRQ$$

6KHDU (t) 0.064t 0.105t 0.012 0.019

PrincipalTension

(t) 0.039t 0.032t 0.007 0.006

PrincipalCompression

(t) 0.052t 0.049t 0.009 0.009

6HFWLRQ%%

6KHDU (t) 0.059t 0.073t 0.013 0.016

PrincipalTension

(t) 0.022t 0.025t 0.005 0.005

PrincipalCompression

(t) 0.027t 0.030t 0.006 0.007

7.2 Calculation of the probability of failureThe estimated gamma parameters are used in Equation (6) for calculation of the SUREDELOLW\RIIDLOXUH7KHUHVXOWVDUHLOOXVWUDWHGLQ)LJXUHVDQG7KHUHVXOWVVKRZWKDWin all cases, there is a good agreement between the results obtained from the maximum likelihood method and the method of moments. For instance, in Figure 6a, considering the acceptable probability of failure of 0.1, the service life of the gusset plate from the PD[LPXPOLNHOLKRRGPHWKRGZRXOGEH\HDUVZKLOHIRUWKHPHWKRGRIPRPHQWVWKHVHUYLFHOLIHLV\HDUV'HVLJQFRGHVVXJJHVWDUHOLDELOLW\LQGH[RIIRUEULGJHGHVLJQ/5)'$$6+72

This value of reliability index corresponds to a probability of failure Pf = 0 232 10 3. . An

RYHUDOO UHYLHZRI WKH UHVXOWVSUHVHQWHG LQ)LJXUHDVKRZV WKDW WKHJXVVHWSODWH8ZLOOUHDFKWKLVSUREDELOLW\RIIDLOXUHDIWHU\HDUVIRUFURVVVHFWLRQ$$7KHOLIH WLPHRI WKHJXVVHWSODWHFRQFOXGHGIURPRWKHUJUDSKVIRUFURVVVHFWLRQ$$)LJXUHVEDQGFDUHHYHQshorter. This means that if the current study had been carried out at the design stage or before WKHEULGJHFROODSVHGLWZDVFOHDUO\SUHGLFWDEOHWKDWDIWHU\HDUVRIVHUYLFH WKHEULGJHVstability would be at high risk.7KHREWDLQHGUHVXOWVSUHVHQWHG LQ)LJXUHVDQGDUHDOVR LQDJRRGDJUHHPHQWZLWK

WKHUHVXOWVRIWKHSUHYLRXVVWXG\E\+ROWDQG+DUWPDQSUHVHQWHGLQ)LJXUHRIWKHFXUUHQWVWXG\7KLVKDVEHHQPRUHFOHDUO\LOOXVWUDWHGLQ)LJXUH7KHSUHVHQWHGUHVXOWV LQ)LJXUHVKRZWKDWWKHSUREDELOLW\RIIDLOXUHRIWKHJXVVHWSODWH8LQLVPXFKKLJKHUthan the permissible value ( Pf =

0 232 10 3. ).Although it is expected to have a deterministic conclusion in a forensic analysis, the

QDWXUHRIDVWRFKDVWLFVWXG\ZLOOHQGXSWRDSUREDELOLVWLFIRUPRIWKHUHVXOWV+RZHYHULQthe current stochastic study, the criterion of Pf <

0 232 10 3. is checked to reach certain conclusions about the failure or safety of the structure and therefore, the consequence of the study is certain and reliable.

12 M. Mahmoodian et al.

Figure 6 3UREDELOLW\RIIDLOXUHRIWKHEULGJHGXHWRGLIIHUHQWW\SHRIVWUHVVHVRQJXVVHWSODWH8VHFWLRQ$$

Figure 7 3UREDELOLW\RIIDLOXUHRIWKHEULGJHGXHWRGLIIHUHQWW\SHRIVWUHVVHVRQJXVVHWSODWH8VHFWLRQ%%

Stochastic failure analysis of the gusset plates in the Mississippi 13

8 Conclusion

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References$$6+72 /5)' %ULGJH 'HVLJQ 6SHFLFDWLRQV QG HG $PHULFDQ $VVRFLDWLRQ RI 6WDWH+LJKZD\DQG7UDQVSRUWDWLRQ2IFLDOV:DVKLQJWRQ'&

$VWDQHK$VO$3URJUHVVLYHFROODSVHRIVWHHOWUXVVEULGJHVWKHFDVHRI,ZFROODSVH7thInternational Conference on Steel Bridges*XLPDUHV3RUWXJDO

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14 M. Mahmoodian et al.

)HUJXVRQ76DQG.ODVV0-$UHSUHVHQWDWLRQRILQGHSHQGHQWLQFUHPHQWSURFHVVHVZLWKRXW*DXVVLDQFRPSRQHQWVAnn. Math. Stat.9RO1RSS

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