Fatigue reliability assessment of retrofitted steel bridges integrating monitored data

Structural Safety 32 (2010) 77–89

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Structural Safety

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Fatigue reliability assessment of retrofitted steel bridges integrating monitored data

Ming Liu, Dan M. Frangopol *, Kihyon KwonDepartment of Civil and Environmental Engineering, Advanced Technology for Large Structural Systems (ATLSS) Center, Lehigh University, Bethlehem, PA 18015-4729, USA

a r t i c l e i n f o

Article history:Received 11 May 2008Received in revised form 13 August 2009Accepted 17 August 2009Available online 17 September 2009

Keywords:Fatigue reliabilityDistortion-induced fatigue crackingFinite element modelingSpatial adjustment factorField monitored dataSteel bridges

0167-4730/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.strusafe.2009.08.003

* Corresponding author. Tel.: +610 758 6103.E-mail address: [email protected] (D.M. Fr

a b s t r a c t

This paper focuses on fatigue reliability assessment of retrofitting distortion-induced cracking in steelbridges integrating monitored data. The fatigue reliability assessment of the connection details is basedon the approach used in the AASHTO standard design specifications with all necessary information fromfinite element modeling (FEM) and structural health monitoring (SHM). Both in-plane traffic loading andout-of-plane relative displacements are considered along with different connection boundary conditions.The primary cause of the observed fatigue cracks before retrofitting is identified as the out-of-plane rel-ative displacements, while the potential fatigue cracking re-initiation after retrofitting depends on theboundary conditions and critical locations that can be identified from the validated FEM. When the iden-tified critical locations are different from the SHM sensor locations, the original monitored data may bemodified by using a spatial adjustment factor (SAF). The proposed approach is illustrated by using anactual bridge monitored by the Advanced Technology for Large Structural Systems (ATLSS) Center, aNational Engineering Research Center at Lehigh University.

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1. Introduction

Distortion-induced fatigue cracking in steel bridges has beenextensively studied [14,22,27]. It has been concluded that distor-tions with the magnitudes even on the order of only 0.5 mm(0.02 in.) may induce high cyclic stress-ranges up to 276 MPa(40 ksi) in small welded gaps [16]. This explains why several hun-dred bridges developed unanticipated fatigue cracks after only afew years of service under normal traffic loadings. Fortunately,these fatigue cracks usually initiate in planes parallel to the tensilestresses under normal traffic loadings, and can be discovered at theearly stages of their propagations. If effective retrofitting methodsare implemented before cracks become perpendicular to the ten-sile stresses under normal traffic loadings, these distortion-in-duced fatigue cracks are not considered to have negative impacton structural safety of steel bridges [11,15].

For a steel tied-arch bridge, the connection between a trans-verse floor-beam and a tie girder often experiences the fatiguecracking induced by both in-plane loading and out-of-plane distor-tion. The in-plane loading from heavy vehicles induces the tensilestresses at the fixed connection details. The out-of-plane distortionis developed because of the relative movement between the tie gir-der and top flange of the floor-beam, when the top flange is notconnected to the tie girder. This relative movement is in the direc-tion of the traffic, and is related to the global deflection of thebridge and local deflection of the floor-beam under heavy vehicle

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angopol).

traffic loading and/or due to wind loading. The floor-beam-to-gir-der connection in a steel tied-arch bridge is typically designedand fabricated as a simple web shear connection, where the con-nection plate or double angles do not cover the full depth of theweb of the floor-beam, resulting in relatively small web gaps be-tween the ends of the connection plate or double angles and theflanges of the floor-beam. In these small web gaps of the floor-beam, the fatigue cracking initiates due to tensile stress concentra-tions and propagates parallel to the flange along the flange–webconnection [10]. The typical retrofit methods include (a) drillinga hole at the end of existing crack to arrest the crack; (b) providinga positive attachment between the tie girder and the top flange ofthe floor-beam to eliminate the relative movements betweenthem; (c) stiffening the entire bridge to prevent the large deforma-tions of the bridge; and (d) softening the connection by cutting offits upper end to allow the relative movements between the topflange and the connection plate or angles of floor-beam [7]. Drillinga crack arrest hole is the most economical retrofit method, but onlyprovides a temporary solution because re-initiation of the distor-tion-induced fatigue cracks often occurs around the drilled hole.Rigidly connecting the top flange of the floor-beam to the tie girderis expensive, and may shift the cracking locations to the floor-beamweb near the stringer-to-floor-beam connections [25]. Obviously,stiffening the entire bridge is a costly retrofit option, and construc-tability on existing steel bridges presents a great challenge for thisretrofit method [19]. On the other hand, softening the connectionis a cost-efficient and effective alternative among all potential ret-rofit methods. Therefore, this paper focuses on evaluating theeffectiveness of the softening connection retrofitting method in

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78 M. Liu et al. / Structural Safety 32 (2010) 77–89

terms of estimating the fatigue reliability of the modified connec-tion details after applications of this retrofit method.

In this study, the fatigue reliability of the connection details isestimated by using the American Association of State Highwayand Transportation Officials (AASHTO) Guide Specifications for Fati-gue Evaluation of Existing Steel Bridges [1] and Standard Specifica-tions for Highway Bridges [2]. All necessary information isobtained from both finite element modeling (FEM) and structuralhealth monitoring (SHM) data. The FEM is validated by comparingthe results from the FEM with the field measurements by the SHM.Both in-plane traffic loading effects and out-of-plane relative dis-placements are considered along with different connection bound-ary conditions between the floor-beams and the bridge girders. Theprimary causes of both observed fatigue cracks before retrofittingand potential fatigue cracking re-initiation after retrofitting aredetermined to be either in-plane traffic loading or out-of-plane rel-atively displacement. The critical locations for potential fatiguecracking after retrofitting are also identified through the validatedFEM models. The fatigue reliabilities at these identified criticallocations are computed by using the field monitored data. Whenthe identified critical locations are different from the sensor loca-tions, the original field monitored data may be modified by usingthe proposed spatial adjustment factors (SAF) from the validatedFEM. The fatigue reliability of the connection details after retrofit-ting is defined as the minimum of the computed fatigue reliabili-ties at these identified critical locations. The proposed approachis illustrated by using an existing bridge, which was monitoredfrom October to December 2003, by the Advanced Technology forLarge Structural Systems (ATLSS) Center, a National EngineeringResearch Center at Lehigh University.

2. Fatigue reliability assessment using AASHTO approach

The AASHTO approach to fatigue assessment is based on the S–N curves in the AASHTO design specifications [2], and the Miner’srule [23]. The corresponding limit state equation for the connec-tion details in consideration can be simply expressed as [24]

Nc � Nt ¼ 0 ð1Þ

where Nc = total number of stress cycles to fatigue failure under var-iable–stress-range S, and Nt = accumulated number of stress cyclesapplied to the connection details during the period from the startof fatigue damages to the time t under consideration. Nc is depen-dent on the variable–stress-range S, and can be expressed as [28]

Nc ¼A � DEðSBÞ

ð2Þ

where A = fatigue–strength coefficient which is based on the cate-gory of the connection details considered and the corresponding

Table 1CAFL and coefficient A based on AASHTO Fatigue Categories (2002).

Category CAFLMPa(ksi)

Design value ofcoefficient AMPa3

(ksi3)

*MeancoefficMPa3

(ksi3)

A 165.0(24.0)

82.0 � 1011

(250.0 � 108)82.0 �(250.0

B 110.0(16.0)

39.3 � 1011

(120.0 � 108)39.3 �(120.0

C 69.0(10.0)

14.4 � 1011

(44.0 � 108)14.4 �(44.0 �

D 48.3(7.0)

7.21 � 1011

(22.0 � 108)7.21 �(22.0 �

E 31.0(4.5)

3.61 � 1011

(11.0 � 108)3.61 �(11.0 �

* See Zhao et al. [28] for computation procedures; since log10 A is assumed to follow a

S–N curve as shown in Table 1; B = fatigue–strength exponent,which can be assigned as the constant of 3.0 representing the slopeof the S–N curves; E(SB) = mean value of SB indicating the Bth mo-ment of S with probability density function (PDF), fS(s), and D = alognormal distributed random variable with the mean value of 1.0and coefficient of variation (COV) of 0.3 for metallic materials[26], which is related to the Miner’s damage-accumulation indexD [23]

D ¼Xk

i¼1

ni

Nci¼ Nc

A� EðSBÞ ¼ Nc

A�Z 1

0SB � fSðsÞh i

ds ð3Þ

where k = maximum number of stress-range levels Si (i = 1, 2 . . . k)under consideration; ni = number of cycles under constant stress-range level Si; and Nci = total number of cycles to fatigue failure un-der constant stress-range level Si.

Similarly, when the accumulated number of stress cycles Nt isrepresented by the time-variant PDF gN (n,t), Nt can be calculatedfor the entire period of time T. A typical S–N curve is extendedfor a detail expected to have finite fatigue life (i.e. linear in loga-rithmic form), whereas it stays constant in the constant-amplitudefatigue limit (CAFL) for theoretically infinite fatigue life (i.e. bi-lin-ear). After reaching the specified number of cycles, NS = A/CAFLB, atthe specified time, TS, the S-N curve can continue to decrease (i.e.finite life) or remain constant (i.e. infinite life). Accordingly, Nt

can be expressed as

Nt ¼Z TS

0

Z 1

0½n � gNðn; tÞ�dn � dt þ

Z T

TS

Z 1

0½n � gNðn; tÞ�dn � dt ð4Þ

It is noted that the second term of Eq. (4) can be ignored in thecalculation of Nt for infinite life (i.e. Nt = NS when t > TS). Using Eq.(4), the limit state Eq. (1) can be re-written in general form as

A � DR10 SB � fSðsÞh i

ds

�Z TS

0

Z 1

0n � gNðn; tÞ½ �dn � dt þ

Z T

TS

Z 1

0n � gNðn; tÞ½ �dn � dt

� �¼ 0

ð5Þ

3. Integration of SHM data into fatigue reliability assessment

Structural health monitoring (SHM) can provide actual informa-tion on fS(s) and gN (n,t). The PDF, fS(s), can be obtained from thestress-range histograms collected by using the rain-flow cyclecounting method or other approaches [12,6] during the monitoringperiod. According to the Miner’s rule [23,1], the equivalent con-stant-amplitude stress-range, Sreff, is equal to the cubic root ofthe mean cube (rmc) of all stress-ranges, Srj (i.e. B = 3), that is

value ofient A

*Standard deviationof coefficient AMPa3

(ksi3)

Standard deviationof log10 A

1012

� 109)36.9 � 1012

(112.5 � 109)0.1865

1012

� 109)17.7 � 1012

(54.0 � 109)0.1865

1012

109)6.48 � 1012

(19.8 � 109)0.1865

1012

109)3.24 � 1012

(9.90 � 109)0.1865

1012

109)1.62 � 1012

(4.95 � 109)0.1865

normal distribution, r(log10 A) = r(ln A)/ln 10.

M. Liu et al. / Structural Safety 32 (2010) 77–89 79

Sreff ¼X nj

Ntotal� S3

rj

� �13

ffiZ 1

0½S3 � fSðsÞ�ds

� �13

ð6Þ

where nj = number of observations in the predefined stress-rangebin Srj; and Ntotal = total number of observations during the monitor-ing period. It should be noted that, for various reasons, the SHM sen-sors may not be always placed at the critical locations. Among suchreasons are the difficulties in installing sensors at certain criticallocations, uncertainties in determining critical locations from struc-tural modeling and FEM stress analysis, and time-variant effects oflocalized corrosions and damages. Therefore, the original field mon-itored data may need to be extrapolated in order to find Sreff at thecritical locations under consideration. For example, Fisher et al.[16] suggested a simple linear gradient approach to extrapolatethe stress-ranges at the weld toe of a gusset plate from the originalreadings of the strain gages that were installed nearby. In this study,FEM models are created to explore the relationships of the stress-ranges between the critical locations and the sensor locations,resulting in the newly developed spatial adjustment factors (SAF).The procedure for obtaining the stress-range histograms at the crit-ical locations by using the SAF and random number generation tech-niques will be discussed in the example presented in this paper.

It should be also noted that the stress-range histograms gener-ated from the original field monitored data typically contain a largenumber of low magnitude stress cycles due to light vehicles, lowwinds or other secondary vibrations, and/or even electrostatic orelectromagnetic noises if the strain gage signals are not filtered[29]. These low magnitude stress cycles make no contributions tocumulative fatigue damages but, when included in the computa-tion of Sreff in Eq. (6), yield smaller values of Sreff. Therefore, the ac-tual fatigue resistances may be overestimated from the S–N curves.In other words, Sreff in Eq. (6) should be computed from the moni-tored stress ranges larger than a predefined threshold only.

From a large number of laboratory experiments under constant-amplitude cyclic loading, the CAFL is established for each categoryin the AASHTO design specifications as presented in Table 1: nofatigue cracks appear if the applied stress cycles have the con-stant-amplitude smaller than the corresponding CAFL. For the fieldmonitored variable-amplitude stress cycles, the upper limitthreshold may be typically as high as a quarter of the CAFL [8].When the number of cycles corresponding to lower stress-rangesis considered, it has been demonstrated that the cumulative fatiguedamage by the calculated effective stress-range becomes asymp-totic to the applicable S–N curve [17]. Therefore, stress-rangecut-off levels can be assigned a lower limit. Thus, upper and lowerlimits of thresholds can be defined. Sensitivity studies on the pre-defined thresholds will be conducted in this study by using stress-range cut-off levels for various percentages of the CAFL.

For effectively assessing fatigue performance of steel bridges, itis necessary to integrate a long-term SHM program [20] into the fa-tigue reliability evaluation. SHM can easily provide the histogramsof the collected daily number of stress cycles and correspondingdaily number of passages of the heavy vehicle traffic during themonitoring period Tshm; Rshm can be used to represent the ratio ofthese two daily numbers. The PDF gN (n) that is used to fit the his-togram of the collected daily number of stress cycles representsgN (n,t) within Tshm. The time adjustment factor n(t) taking intoaccount Tshm, Rshm, and the annual average daily traffic (AADT)during the period T, is considered in gN (n,t). Thus, Eq. (4) can bere-written as

Nt ¼Z TS

0

Z 1

0ðn � gNðnÞÞdn

� �� nðtÞdtþ

Z T

TS

Z 1

0ðn � gNðnÞÞdn

� �� nðtÞdt

¼ �Nshm �Z TS

0nðtÞdtþ

Z T

TnðtÞdt

� �ð7Þ

S

where �Nshm = mean value of the collected daily number of stress cy-cles within Tshm; n(t) quantifies the variability of �Nshm during the en-tire period T, and �Nshm must be obtained from the field monitoreddata. However, this is impossible for almost all existing bridgessince there was no daily number of stress cycles collected beforeSHM was used. On the other hand, AADT is well documented formany existing bridges so that it is logical to develop n(t) with AADT,although the estimation errors may be introduced because the fati-gue stress cycles may be caused by the events other than the pas-sages of heavy vehicle traffic. For example, wind loads mayinitiate free vibrations of a bridge that contribute to the numberof the fatigue stress cycles �Nshm.

It should be also pointed out that the cumulative fatigue dam-ages are caused by the heavy vehicles in weight greater than89 kN (20 kips) only [13,4]. Thus, AADT must to be converted tothe annual average daily truck traffic (AADTT) as

AADTTðiÞ ¼ pðiÞ � AADTðiÞ ð8Þ

where p(i) = percentage of the heavy vehicles with weight greaterthan 89 kN (20 kips) in AADT in the ith year, which is a time-variantparameter obtained from traffic data.

As a result, the limit state equation for the fatigue reliabilityassessment with the field monitored data can be derived fromEq. (5) as

A � De � SB

reff

� �Nshm �XT

i¼1

ðpðiÞ � AADTðiÞÞ" #

� Rshm ¼ 0 ð9Þ

where B = 3, Rshm = ratio of collected daily number of stress cycles todaily heavy truck traffic during Tshm, and e = measurement error,which may be considered as a lognormal distributed random vari-able with 1.0 as its mean value and 4% as its COV, a typical measure-ment error in SHM [18]. Table 2 summarizes the probabilitydistributions of the random variables and the deterministic param-eters in Eq. (9). In case of absence of p(i) and AADT(i), Bayesianupdating may be applied to obtain Nt directly from the collecteddaily number of stress cycles within Tshm, where the duration ofthe monitoring period Tshm is critical to ensure the validation ofthe fatigue reliability assessment from Eq. (9). If the monitored dataare considered to be reliable enough, as is the case herein [9], thePDFs derived from the data can be applied without consideringBayesian updating.

In general, Tshm is also very important in the estimations of Sreff

and �Nshm from the field monitored data. The longer the monitoringperiod Tshm is, the more reliable the estimated Sreff and �Nshm are.Although the computed Sreff and �Nshm from the field monitoreddata, which are collected during about two to four weeks of contin-uous monitoring, may converge or stabilize [8], it should beemphasized herein that the achieved stabilities of the estimatedSreff and �Nshm within Tshm are primarily due to the improved capac-ity of a continuous SHM to capture the actual loading conditionsonly. In other words, the actual variability of the bridge loadingconditions may be almost completely observed within a continu-ous period of about two to four weeks. On the other hand, muchslower processes of increasing Sreff due to corrosion deteriorationsare typically undetectable within such a relative short monitoringperiod. Consequently, it is essential to perform the proposed fati-gue reliability assessments with the field monitored data severaltimes during the lifetime of a steel bridge, where Tshm may be usedto determine the maximum allowable time intervals. The proposedapproach should be repeated within the maximum allowable timeintervals or occurrences of any physical damage and/or significantchange of the loading conditions. Ultimately, further studies ontime-variant Sreff and �Nshm with the availability of a large numberof long-term field monitored data may greatly improve the appli-cability of the proposed approach.

Table 2Parameters for fatigue reliability assessment with field monitored data.

Parameter Notation Distribution Source

Fatigue–strength coefficient A Lognormal (see Table 1) Zhao et al. [28]Damage-accumulation index D Lognormal, LN (1.0, 0.3) Wirsching [26]Measurement error e Lognormal, LN (1.0, 0.04) Frangopol et al. [18]Effective stress-range Sreff Random Based on field monitored dataMean of collected daily number of stress cycles during Tshm Nshm Deterministic Based on field monitored dataBridge life in years T Deterministic Bridge dataPercentage of heavy truck traffic p(i) Deterministic Traffic dataAnnual average daily traffic AADT(i) Deterministic Traffic dataRatio of collected daily number of stress cycles to daily heavy truck traffic during Tshm Rshm Random Based on field monitored dataSpatial adjustment factor SAF Random Based on FEM


4. Application

The proposed approach is illustrated by using a bridge moni-tored from October to December 2003, by the ATLSS Center. Thisbridge was built in 1976, and the fatigue cracks had been foundin nearly all of the transverse floor-beams at the connections to

STRINGER

WELD

FLOOR BEAM

STIFFENER

CA

TYPICAL CINITIATED W

(a)

(CH-2)

(CH-7

STRINGER

WELD

FLOOR BEAM

STIFFENER

(b)

Note: Sensors CH-2, CH-7 & CH-11 located a

Fig. 1. Connection detail between a floor-beam and a tie g

the tie girders several years before 2003 [9]. The softening connec-tion retrofit method was applied in 2003, and SHM was installedimmediately after retrofitting. Connor et al. [9] described the mon-itoring process as follows. The monitoring was performed to eval-uate fatigue performance of the retrofitted connection details. Ininstrumentation plan, a total of 32 weldable type strain gages were

ONNECTIONNGLE

HIGH STRENGTHBOLT

TIE GIRDER

RACKITHIN WEB GAP

CL-I

)(CH-11)

CONNECTIONANGLE

CL-II CL-III

HIGH STRENGTHBOLT

TIE GIRDER

t 6.4 mm (0.25 in) away from edges of cut

irder: (a) before retrofitting, and (b) after retrofitting.


installed and used to measure stress that produce cracking. In or-der to collect reliable and noise-free data, analog and digital filter-ing was employed in recording. The field monitored data werecontinuously collected for a total of almost 40 days during themonitoring period. The collected monitoring data (i.e. stress vs.time in seconds) by using the rain-flow cycle counting method[12] were converted to establish the stress-range histogram (i.e.stress-range vs. number of cycles) that is commonly used to calcu-late effective stress-range and to find maximum stress-range infatigue performance assessment. Figs. 1a and b present the connec-tion details before and after retrofitting, respectively. The observedfatigue cracks before retrofitting and SHM sensor locations afterretrofitting are also displayed in Figs. 1a and b, respectively [9].The actual cut-off size was 0.52 m (20.5 in.) in length and 0.30 m(11.75 in.) in depth, and the floor-beams have a depth of 2.84 m(112 in.) as shown in Figs. 2 and 3. The detailed information onthe bridge, such as material properties, bridge geometries and con-nection details between the floor-beams and the tie girders, can befound in Connor et al. [9].

As shown in Fig. 1b, the potential fatigue cracking re-initiationafter retrofitting is identified at the three critical locations (i.e. CL-I,CL-II, and CL-III). Sensor CH-2 collected the fatigue stress-rangedata at the intersection of the top flange and web of the floor-beam(i.e. CL-I). According to the AASHTO standard fatigue specifications,CL-I can be classified as category C. Similarly, sensors CH-7 and CH-11 collected the fatigue stress-range data near the bottom of thecut-off (i.e. CL-II) and at the connection angles (i.e. CL-III), respec-tively. CL-II and CL-III can be classified as categories A and B,

0.711m1.676 m

1.676 m1.676 m

1.676 m1.62

0.019 m (0.063 ft)

16.066 m (52.71 ft)

2.33 ft5.50 ft

5.50 ft5.50 ft

5.50 ft

Fig. 2. Finite element mode

respectively. It should be noted that all sensor locations are about6.4 mm (0.25 in.) away from the edges of the cut-off. Therefore, theoriginal readings from these sensors are modified in this study byusing SAF from the validated FEM in order to truly represent thefatigue stress-ranges at critical locations CL-I, CL-II, and CL-III.

4.1. Development and validation of FEM

Linear-elastic 3-D FEM was developed for the connection detailswith the software ABAQUS, version 6.7-1 [3], and the relevantinformation including all dimensions and material properties areobtained from either ‘‘as-built” blue prints or actual field measure-ments. Fig. 2 presents the FEM before retrofitting along with all ofthe modeling dimensions, while Fig. 3 presents the FEM after ret-rofitting and all loading conditions (i.e. LC-I, LC-II and LC-III) con-sidered in this study. The solid elements are used to model allthe components such as the flanges and web of the floor-beam,the connection angles and the stiffeners at the stringer locations.In regions around the cut-off location, 20-node reduced-integratedsolid elements are used, while the rest of the elements are 8-nodereduced-integrated solid elements. Various mesh sizes are alsoadopted for efficient computations. For the same reason (i.e. effi-cient computations), only half of the floor-beam is included inthe developed FEM with the assumption that the geometry, mate-rial properties and loading conditions are symmetrical. There are atotal of over 34,000 elements and 120,000 nodes in the developedFEM. The loading conditions considered in this study include bothout-of-plane relative displacement (i.e. LC-I) and in-plane traffic

6 m1.626 m

1.626 m1.626 m

1.626 m 0.521 m

5.33 ft5.33 ft

5.33 ft5.33 ft

5.33 ft (1.71 ft)

CL

CL

711.2 mm (28 in)

STIFFENER: 203.2 x 12.7 mm

(8 x 0.5 in)

1422

.4 m

m (5

6.0

in)

THICKNESS OF WEB:

12.7 mm (0.5 in)

ling before retrofitting.

CL-I : FIELD = 44.8 MPa (6.5 ksi)FEM1 = 41.2 MPa (6.0 ksi) FEM2 = 35.8 MPa (5.2 ksi)

CL-II : FIELD = 37.2 MPa (5.4 ksi)FEM1 = 36.8 MPa (5.3 ksi) FEM2 = 35.8 MPa (5.2 ksi)

CL-III : FIELD = 48.3 MPa (7.0 ksi)FEM1 = 57.9 MPa (8.4 ksi) FEM2 = 49.3 MPa (7.1 ksi)

CL-I

0.5CL-II

CL-III

Note: FEM2 results are reported in Connor et al. (2004)

Fig. 4. Validation of FEM after retrofitting at critical locations CL-I, CL-II and CL-III.


loading, where the vertical traffic loading may be applied at eitheroutside (i.e. LC-II) or inside (i.e. LC-III) traffic lane as shown inFig. 3. The out-of-plane relative displacement (i.e. LC-I) is longitu-dinally applied at each of the stringer locations as shown in Fig. 3as well. The magnitudes of both in-plane and out-of-plane loadingsare based on the field monitored data and/or previous experiences.In addition, although the connection details are designed as thetrue ‘‘pin” connections, the actual behaviors of the connection de-tails based on the field monitored data reveal that the connectionsprovide some restraints to the rotations of the web of the floor-beam [9]. Therefore, two FEMs are created in this study, one forthe fixed end condition (Model I), the other for the true ‘‘pin”end condition that allows the free in-plane rotations of the webof the floor-beam (Model II). It is believed that the actual fatiguestress-ranges generated at the connection details must be thosebetween these two extreme conditions. Nevertheless, the effectsof the boundary conditions on the collected fatigue stress-rangedata will be discussed further in this paper for both in-plane trafficloadings and out-of-plane relative displacements at each of theidentified critical locations. Furthermore, it is recognized that thestringers sitting at the top of the floor-beam may somewhat pre-vent the free movements of the floor-beam at the stringer bearingareas. However, the comparisons of the analytical results from theFEM with and without the considerations of the stringer’s re-straints indicate that the FEM results without the restraints alwaysprovide the larger fatigue stress concentrations around the cut-off

Note:

= 2.54 mm (0.1 in)LC – I: HORIZONTAL DISPLACEMENT,

LC – II: VERTICAL LOAD, P = 66.7 kN (15 kips)

LC – III: VERTICAL LOAD, Q = 66.7 kN (15 kips)

P P

10.687 m

2.407 m 1.676 m( 7.90 ft) (5.50 ft)

(35.06 ft)

LC - II

LC - I

Fig. 3. Finite element modeling after retrofittin

areas, regardless of the cut-off sizes and the loading conditions.This is because the restraints from the stringer bearings tend todistribute portions of the entire applied loads to the stringersand other bridge components nearby, resulting in the smaller

520.7 mm (20.50 in)

298.

45 m

m

(11.

75 in

)

CUT-OUT AREA

Q Q

1.626 m(5.33 ft)

–LC – III

g, and load conditions LC-I, LC-II and LC-III.

Table 3Computed ryy and rzz from FEM models before fatigue cracking initiation.

Model Stress Loading condition

LC-IMPa (ksi)

LC-IIMPa (ksi)

LC-IIIMPa (ksi)

I ryy 108.7(15.8)

97.0(14.1)

250.0(36.3)

rzz 77.7(11.3)

152.0(22.0)

379.0(55.0)

II ryy 108.7(15.8)

4.1(0.60)

6.0(0.87)

rzz 77.7(11.3)

2.2(0.32)

4.5(0.65)

CL-I

CL-IICL-III

CONNECTIONANGLE

max AROUND CL-I: 46.2 MPa (6.7 ksi)

Note: The stresses are computed by FEM under LC-I

max AROUND CL-III: 61.9 MPa (9.0 ksi)

0.5 AROUND CL-II: 45.6 MPa (6.6 ksi)max

AT CL-I: 41.2 MPa (6.0 ksi)

0.5 AT CL-II: 36.8 MPa (5.3 ksi)

AT CL-III: 57.9 MPa (8.4 ksi)

Fig. 6. Computed stresses at critical locations CL-I, CL-II, and CL-III under LC-I.


fatigue stresses around the cut-off areas. For conservative consid-erations, the analytical results presented in this paper are associ-ated with the FEM without the considerations of the stringer’srestraints.

The two developed FEMs in this study (i.e. Models I and II) havebeen validated by comparing the analytical results from the FEMwith the field measurements from the monitoring as well withthe analytical results from the previous FEM computations [9].The largest peak longitudinal relative displacement recorded inthe field monitored data was about 2.54 mm (0.1 in.), and thecorresponding stress-ranges were 44.8 MPa (6.5 ksi), 37.2 MPa(5.4 ksi), and 48.3 MPa (7.0 ksi) at critical locations CL-I, CL-II,and CL-III, respectively. These results were used to validate theprevious FEM created by the ATLSS Center and the Models I andII. It should be noted that Models I and II produce the same stressesunder load condition LC-I. Therefore, the fixed and pinned end con-ditions have no effects on the resulting fatigue stresses under theout-of-plane relative displacement loading condition. This is be-cause the out-of-plane relative displacements do not induce anyin-plane rotations of the web of the floor-beam. Fig. 4 shows thatreasonable agreements are obtained at critical locations CL-I, CL-II, and CL-III from the newly developed FEM and Connor et al.[9]. This indicates that Models I and II can be used as a basis for fa-tigue reliability assessment at the identified critical locations. Figs.5a and b present the FEM-based stress diagrams distributed alongthe small web gap before retrofitting and along the edges of thecut-off after retrofitting, respectively. Fig. 5b indicates that the bot-tom of the cut-off region provides weak restraints to the out-of-plane movements of the web of the floor-beam after retrofitting.As a matter of fact, the structural behavior of the web of thefloor-beam under load condition LC-I can be represented by thevirtual beams with the fixed–fixed ends (before retrofitting) andthe fixed-pinned ends (after retrofitting), respectively. Before ret-rofitting, the small web gap between the ends of the connectionangles and the flanges of the floor-beam is strongly restrained bythe flange at the top and by the connection angles at the bottom,

FLANGE

Δ

N.A.

STRESS DIAGRAM

A

B

A

B

CONNECTION ANGLEWEB

yyWEB

(a)

AB

CONNECTION ANGLE

FLANGE

A

ΔSTRESS DIAGRAM

B

CCONNECTION ANGLE

A

B C

yy

zz

WEB

PIN PIN

WEB

A

BC

WEB

STIFFENERFLANGE

CONNECTION ANGLE

(b)FLANGE

Fig. 5. Modeling of structural behavior of the web of the floor-beam: (a) before retrofitting, and (b) after retrofitting.


resulting in the double curvature shown in Fig. 5a. After retrofit-ting, as shown in Fig. 5b, the vertical edge of the cut-off is re-strained by the flange at the top and by portion of the web at thebottom, while the horizontal edge of the cut-off is restrained bythe connection angles at one end and by portion of the web atthe other end. Since the restraints provided by the portion of theweb are relatively weak, the pinned end may be assigned atthe bottom of the cut-off to reproduce the computed stressdistributions.

4.2. Primary causes of fatigue cracking at connection details

The primary cause of fatigue cracking at the connection detailscan be easily investigated and identified by using Models I and II.Table 3 summarizes the computed stresses at the small web gapbetween the ends of the connection angles and the top flange ofthe floor-beam by using Models I and II before fatigue cracking.The out-of-plane relative displacement associated with load condi-tion LC-I is assigned as 0.1 mm (0.004 in.), which is a typical valuefor the floor-beam to girder connection details before fatiguecracking [16]. Each of the two in-plane point loads that space about1.68 m (5.5 ft) apart in load conditions LC-II and LC-III is imposedas 66.7 kN (15 kips) to approximate a wheel load. It is recognizedthat the actual loading conditions may be different from the typical

(a)

CL-I

CL-IICL-III

CONNECTIONANGLE

AROUND CL-I: 1.2 MPa (0.17 ksi)

AROUND CL-III: 27.3 MPa (3.96 ksi)

+ )0.5 AROUND CL-II: 43.0 MPa (6.24 ksi)max


0.5 AT CL-II: 19.2 MPa (2.78 ksi)


MODEL - I

Note: The stresses are computed by FEM under LC-II

(b)

CL-I

CL-IICL-III

CONNECTIONANGLE





0.5 AT CL-II: 15.0 MPa (2.18 ksi)


MODEL - II

+ )

+ )

+ )

Fig. 7. Computed stresses at critical locations CL-I, CL-II, and CL-III under LC-II: (a)Model I, and (b) Model II.

ones above, but the relative importance of the out-of-plane and in-plane loading conditions in initiating the fatigue cracks at the con-nection details can be identified by using these typical values. Asshown in Table 3, the out-of-plane relative displacement of0.1 mm (0.004 in.) induces the vertical tensile stress (ryy) of about108.7 MPa (15.8 ksi), and the horizontal tensile stress (rzz) of about77.7 MPa (11.3 ksi) at the small web gap, respectively. As men-tioned previously, Models I and II yield the exactly same ryy andrzz under load condition LC-I. On the other hand, the in-plane traf-fic loads induce significant ryy and rzz in Model I, but small valuesof ryy and rzz in Model II. Moreover, the transverse positions of thein-plane traffic loads as shown in load conditions LC-II and LC-IIIgreatly affect the resulting stresses ryy and rzz from Model I. There-fore, the in-plane traffic loads, depending on their transverse posi-tions, may theoretically make the same level of contributions asthe out-of-plane relative displacements to the fatigue cracking ini-tiation at the fixed connection details. However, the field moni-tored data has suggested that the end fixity of the connectiondetails seems low, if not truly ‘‘pinned” [9]. Consequently, it canbe concluded that the out-of-plane relative displacement inducedby different movements between the tie girder and top flange ofthe floor-beam, even on the order of only 0.1 mm (0.004 in.), is pri-marily responsible for the fatigue cracking initiation at the connec-tion details. Nevertheless, it is important to realize that the effectsof the in-plane traffic loads on the fatigue cracking initiation can

(a)

CL-I

CL-IICL-III

CONNECTIONANGLE

( )max AROUND CL-I: 5.9 MPa (0.86 ksi)


0.5 AROUND CL-II: 152 MPa (22.06 ksi)max


0.5 AT CL-II: 68.8 MPa (9.98 ksi)


MODEL - I

CL-I

CL-IICL-III

CONNECTIONANGLE





0.5 AT CL-II: 16.2 MPa (2.35 ksi)


MODEL - II

(b)

Note: The stresses are computed by FEM under LC-III

( )

Fig. 8. Computed stresses at critical locations CL-I, CL-II, and CL-III under LC-III: (a)Model I, and (b) Model II.

Table 4Computed spatial adjustment factor (SAF) at critical locations CL-I, CL-II, and CL-III.

Model Criticallocation

Loading condition

LC-I LC-II LC-III

SAF(ryy) SAF(rzz) SAF(ryy) SAF(rzz) SAF(ryy) SAF(rzz)

I CL-I 1.12 CS* CS*

CL-II 1.29 1.22 2.53 2.19 2.32 2.18CL-III 1.07 1.24 1.19

II CL-I 1.12 CS* CS*

CL-II 1.29 1.22 CS* CS* CS* CS*

CL-III 1.07 1.59 1.64

* Compression stress: no SAF is calculated when only compression stress isobtained from FEM.

STRESS RANGE, S (MPa)

NU

MB

ER

OF

ST

RE

SS

CY

CLE

S, N

0

1500

3000

4500

6000

7500

9000

STRESS RANGE, S (ksi)

CH - 2: WITHOUT SAF

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.01.5 2.5 3.5 4.5 5.5 6.5 7.5

0.0 13.8 20.7 27.6 34.5 41.4 48.3 55.210.3 17.2 24.1 31.0 37.9 44.8 51.7

0.0 0.5

3.46.9

TOTAL NUMBER OF OBSERVATIONS = 12265

TOTAL CUT-OFF NUMBER OF CYCLES = 46501

INTERVAL = 3.45 MPa (0.50 ksi)

7735

2609

1550

245 83 35 5 1 2

Sreff = 11.51 MPa (1.67 ksi)

Nshm = 307 CYCLES per DAY

Sreff = 11.51 MPa (1.67 ksi)


(a)

CL - I: WITH SAF

TOTAL NUMBER OF OBSERVATIONS = 8420

TOTAL CUT-OFF NUMBER OF CYCLES = 50346

INTERVAL = 3.45 MPa (0.50 ksi)

4738

1977

1102

402123 48 18 9 1 1 1

Sreff = 15.70 MPa (2.28 ksi)


Sreff = 15.70 MPa (2.28 ksi)


(b)

NU

MB

ER

OF

ST

RE

SS

CY

CLE

S, N

0

1500

3000

4500

6000

7500

9000

STRESS RANGE, S (MPa)

0.0 13.8 20.7 27.6 34.5 41.4 48.3 55.210.3 17.2 24.1 31.0 37.9 44.8 51.73.4

6.9

STRESS RANGE, S (ksi)1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.01.5 2.5 3.5 4.5 5.5 6.5 7.50.0 0.5

Fig. 9. Stress-range histograms at critical location CL-I: (a) based on sensor datafrom CH-2, and (b) based on simulated data with SAF.


not be ignored, if the strong end fixity exists at the connectiondetails.

The FEM analyses are also performed for the connection detailsafter retrofitting. The computed stresses ryy and rzz under loadcondition LC-I at both sensor and maximum stress locationsaround critical locations CL-I, CL-II, and CL-III are presented inFig. 6. Based on the potential fatigue cracking re-initiation at theseidentified critical locations, only ryy at CL-I and rzz at CL-III are pre-sented in Fig. 6, respectively, while both ryy and rzz at CL-II are in-cluded in Fig. 6. All computed stress-ranges are found to be lowerthan the corresponding CAFL at both sensor and maximum stresslocations around critical locations CL-I, CL-II, and CL-III. Theout-of-plane relative displacement after retrofitting under loadcondition LC-I was measured up to 2.54 mm (0.1 in.) as reportedin Connor et al. [9]. Similarly, Figs. 7 and 8 present the computedstresses ryy and rzz under load conditions LC-II and LC-III, respec-tively. The in-plane point loads under load conditions LC-II and LC-III keep the same 66.7 kN (15 kips) as before retrofitting. It isclearly demonstrated that the potential fatigue cracking re-initia-tion at critical location CL-I is completely caused by the out-of-plane relative displacements (i.e. load condition LC-I), since onlysmall compressive stresses are obtained at critical location CL-I un-der load conditions LC-II and LC-III. Moreover, load condition LC-IIin Model II yields the same compressive ryy and rzz at critical loca-tion CL-II and small tensile rzz at critical location CL-III under loadcondition LC-III (see Fig. 8b). In other words, the transverse posi-tions of the in-plane traffic loads in Model II do not affect theresulting stresses ryy and rzz at the identified critical locationsCL-II and CL-III. This is because the truly ‘‘pinned” connection inModel II results in no reaction moment under both load conditionsLC-II and LC-III at the ‘‘pinned” end, where critical locations CL-I,CL-II, and CL-III are located nearby. On the other hand, the com-puted stresses ryy and rzz at critical locations CL-II and CL-III fromModel I have been confirmed that the in-plane traffic loads,depending on their transverse positions, may make the same levelof contributions as the out-of-plane relative displacements (i.e.load condition LC-I) to the fatigue cracking re-initiation at criticallocations CL-II and CL-III after retrofitting, if the strong end fixityexists at the modified connection details.

4.3. Fatigue reliability assessment at critical locations

Based on the computed stresses from the validated FEM and thefield monitored data from sensors CH-2, CH-7, and CH-11, the fati-gue reliabilities at critical locations CL-I, CL-II, and CL-III can beestimated. As mentioned previously, the original field monitoreddata must be modified by the spatial adjustment factor (SAF) be-cause the maximum stress locations are different from the sensorlocations. Since SAF is defined as the ratio of the computed stressesat the maximum stress locations around critical locations CL-I, CL-II, and CL-III to those at the corresponding sensor locations, thecomputed stresses ryy and rzz in Figs. 6–8 can be used to calculateSAF. The calculated SAF is summarized in Table 4. Underconsideration of boundary conditions (i.e. Models I and II) and loadconditions (i.e. LC-I, LC-II, and LC-III), the calculated SAF can be ex-pressed as a set at each critical location: SAF at CL-I = {1.0, 1.12,1.5}; SAF at CL-II = {1.22, 1.29, 2.18, 2.19, 2.32, 2.53}; and SAF atCL-III = {1.07, 1.19, 1.24, 1.59, 1.64}. It is noted that the minimumSAF and maximum SAF at CL-I are assumed based on the only valueof 1.12 obtained from FEM. As a result, the SAF in this study is trea-ted as a random variable that has a triangular PDF with differentminimum, mode, and maximum values for critical locations CL-I,CL-II, and CL-III.

Because the original monitored data only contain the stress-range histograms at sensor locations directly obtained by therain-flow cycle counting method, but the SAF needs to be applied

to individual stress-ranges, the random number generator has tobe adopted to reproduce the individual stress-ranges in the modi-fication procedures. Since the typical stress-range bin in the rain-flow cycle counting method is rather narrow, for example,3.45 MPa (0.5 ksi) in this study, the uniform distribution of theindividual stress-ranges in the corresponding stress-range bincan be assumed. Moreover, only stress-ranges greater than3.45 MPa (0.5 ksi) in the original monitored data are used in themodification procedures for computation efficiency. Fig. 9 com-pares the stress-range histogram from CH-2 sensor data (see

1.0

11.5

100.0

1000.0

1 10 102 103 104 105 106 107

NUMBER OF CYCLES, N t

1.672.28

145.0

10.014.5

10 % CAFL STRESS CUT-OUT

108

69.0

15.7

CAFL

AASHTO S-N CURVE CATEGORY C


CL-I WITH SAFCH-2 WITHOUT SAFCL-I WITH SAFCH-2 WITHOUT SAF

(a)

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

MPa

)

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

ksi)

1000.0

1.0

10.0

100.0

1.45

3.73

145.0

24.014.5


165.5

25.7

CAFL23 % CAFL STRESS CUT-OUT

54.9 7.97

AASHTO S-N CURVE CATEGORY ACL-II WITH SAF

CH-7 WITHOUT SAFCL-II WITH SAF CH-7 WITHOUT SAF

(b)

1 10 102 103 104 105 106 107


108

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

MPa

)

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

ksi)

1.452.20

145.0

16.0


CAFL10 % CAFL STRESS CUT-OUT

3.08

1.0

10.0

110.3

1000.0

15.121.3

AASHTO S-N CURVE CATEGORY B

CL-III WITH SAFCH-11 WITHOUT SAFCL-III WITH SAFCH-11 WITHOUT SAF

(c)

1 10 102 103 104 105 106 107

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

MPa

)

EFFE

CTI

VE S

TRES

S R

ANG

E,S r

eff(

ksi)


108

Fig. 10. Effects of the predefined threshold on Sreff and Nt: (a) at CL-I, (b) at CL-II, and (c) CL-III.

Table 5Probabilistic characteristics of SAF, Sreff and �Nshm , and estimated remaining fatigue life.

Critical location CL-I(CH-2)

CL-II(CH-7)

CL-III(CH-11)

Fatigue category C A BCAFL, MPa (ksi) 69.0 (10.0) 165.0 (24.0) 110.0 (16.0)SAF (*triangular PDF) {1.00, 1.12, 1.50} {1.22, 2.18, 2.53} {1.07, 1.24, 1.64}With SAF Sreff (lognormal PDF) Mean

MPa (ksi)15.7 (2.28) 54.9 (7.97) 21.3 (3.08)

COV 0.272 0.264 0.235�Nshm 210 143 1190

Without SAF Sreff (lognormal PDF) MeanMPa (ksi)

11.5 (1.67) 25.7 (3.73) 15.1 (2.20)

COV 0.278 0.259 0.228�Nshm 307 178 1403

**Remaining fatigue life (in years) >50 50 >50

* The PDF values (a, b, c) represent (minimum, mode and maximum), respectively.** Based on the targeted minimum reliability index b = 3.72, and annual increase of number of stress cycles of 5%.


Fig. 9a) with that from the simulated data by using SAF (see Fig. 9b)at critical location CL-I. By using Eq (6), individual effective stress-

ranges are calculated based on the truncated stress-range histo-grams according to cut-off thresholds, and the mean value and


COV of the Sreff are computed. Sreff is assumed to follow a lognormaldistribution. Fig. 10 presents the computed Sreff with different pre-defined thresholds, based on both original field monitored data andsimulated data with SAF. It has been evidenced that the higher pre-defined thresholds yield the higher Sreff but much lower Nshm.Moreover, it has been demonstrated that 10% of the CAFL as thepredefined threshold is appropriate for the original field monitoreddata. This is because the curves representing the relationships be-tween the computed Sreff and Nshm in Fig. 10 become asymptotic tothe applicable S–N curves after the predefined threshold is set to belower than 10% of the CAFL for the original field monitored data.On the other hand, the predefined threshold can be increased upto 15% and 23% of the corresponding CAFL at critical locationsCL-I and CL-II, respectively, when the simulated data with SAFare used. Table 5 summarizes the mean value and COV used in thisstudy for the lognormal distributed Sreff as well as the deterministicNshm. Fig. 11 presents the predicted number of stress cycles at crit-ical locations CL-I, CL-II, and CL-III in the next fifty years, based onthe estimated Nshm with and without SAF as well as the assumedannual increase rate of 2% and 5%. The number of stress-range cy-

0

5

10

15

20

25

30

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054

NU

MB

ER

OF

CY

CLE

S,

N t

(MIL

LIO

N)

TIME (YEAR)

WITHOUT SAF

WITH SAF

α = 0 %

α = 2 %

α = 5 %

CL - Iα = INCREASE OF NUMBER OF CYCLES per YEAR

(a)

0

4

8

12

16

20

WITHOUT SAF

WITH SAF

α = 0 %

α = 2 %

α = 5 %

(b)

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054

NU

MB

ER

OF

CY

CLE

S,

N t

(MIL

LIO

N)

TIME (YEAR)

CL - IIα = INCREASE OF NUMBER OF CYCLES per YEAR

0

20

40

60

80

100

120

WITHOUT SAF

WITH SAF

α = 0 %

α = 2 %

α = 5 %

(c)

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054

NU

MB

ER

OF

CY

CLE

S,

N t

(MIL

LIO

N)

TIME (YEAR)

CL - IIIα = INCREASE OF NUMBER OF CYCLES per YEAR

Fig. 11. Comparisons of the predicted total number of stress cycles with andwithout SAF: (a) at CL-I, (b) at CL-II, and (c) CL-III.

cles per year with SAF tends to decrease, when comparing withthat estimated by the original monitored data. It is noted that Eq.(7) instead of AADT is used herein to estimate Nt, where n(t) canbe expressed as

nðtÞ ¼ 365 � ð1þ aÞt ð10Þ

where a = annual increase rate in Nshm; and t = time in years. There-fore, Eq. (9) can be re-written as

A � De � S3

reff

� 365 � �Nshm �Z TS

0ð1þ aÞtdt þ

Z T

TS

ð1þ aÞtdt� �

¼ 0 ð11Þ

As described previously, the time adjustment factor n(t) (see Eq.(10)) considering Rshm and AADTT during the period T is used. In thisstudy, all fatigue reliability assessments are performed based on Eq.(11). It is recognized that Sreff may be also time-variant due to pos-sible increase of the weights of future heavy vehicles as well asdeterioration of bridge components with time. However, Sreff is di-rectly related to the distributions of the axle loads instead of theweights of heavy vehicles. The increase of the axle loads in future

0

2

4

6

8

10

12

14

FA

TIG

UE

RE

LIA

BIL

ITY

IND

EX

,

WITHOUT SAF

WITH SAF

α = 0 %α = 2 %α = 5 %

α = 0 %α = 2 %α = 5 %

target

CL - I(a)

α = INCREASE OF NUMBER OF CYCLES per YEAR

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054TIME (YEAR)

CAFL= 7.03

WITHOUT SAF

WITH SAF

α = 0 %α = 2 %α = 5 %

α = 0 %α = 2 %α = 5 %

(b)

0

2

4

6

8

10

12

14

target

CL - II

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054TIME (YEAR)


FA

TIG

UE

RE

LIA

BIL

ITY

IND

EX

,

WITHOUT SAF

WITH SAF

α = 5 %α = 2 %α = 0 %

α = 0 %α = 2 %α = 5 %

(c)

0

2

4

6

8

10

12

14

target

CL - III

2004 2009 2014 2019 2024 2029 2034 2039 2044 2049 2054TIME (YEAR)


FA

TIG

UE

RE

LIA

BIL

ITY

IND

EX

,

Fig. 12. Comparisons of the time-variant reliability index b with and without SAF:(a) at CL-I, (b) at CL-II, and (c) CL-III.


heavy vehicle designs is expected to be limited, while the distribu-tions of the axle loads may change with time, depending on trafficpatterns on the bridge. The crawl deterioration of bridge compo-nents is a very complex process and is dependent on bridge sites,and the locations of the maximum stress-ranges that cause the po-tential fatigue cracking in the future. Therefore, the importance offurther studies on time-variant Sreff and Nshm has been demon-strated herein once again.

Fig. 12 compares the time-variant reliability index b directly ob-tained from sensor data with those obtained from the simulateddata with SAF at critical locations CL-I, CL-II, and CL-III, wherethe importance of modifying the original field monitored data totruly represent the fatigue stress-ranges at the identified criticallocations has been observed. The computations of b are based onEq. (11) with the computer program CalREL [21]. In this paper,the accumulated number of stress cycles, Nt, is considered as de-mand for fatigue reliability assessment. Therefore, the reliabilityindex corresponds to the accumulated probability of failure. It isnoted that the fatigue damages at critical locations CL-I, CL-II,and CL-III started in 2003 after retrofitting since the stress levelsbefore retrofitting are very low at these identified critical locationsafter retrofitting, which can be confirmed by the validated FEM.The solid and dash lines in Fig. 12 represent the computed b withand without SAF, respectively. In particular, bCAFL corresponding toNS is presented in Fig. 12a since the maximum stress-range at CL-Idoes not exceed the CAFL, indicating a theoretical infinite fatiguelife at CL-I. The remaining fatigue life of the connection details afterretrofitting is defined as the period from immediately after retrofit-ting to the time when b in Fig. 12 reaches the targeted minimumbtarget = 3.72 [5]. In other words, the predefined target reliability in-dex is used to predict the remaining fatigue lifetime of the criticallocations. As shown in Table 5, critical location CL-II has the short-est remaining fatigue life among the three identified critical loca-tions, which indicates that the potential fatigue cracking re-initiation after retrofitting will most likely occur at this criticallocation firstly. This implies that the softening connection retrofit-ting method may shift the most vulnerable locations to fatiguedamages under load condition LC-I from the intersections of thetop flange and web of the floor-beam to the web of the floor-beamalone after retrofitting. Thus, it may be concluded that SAF derivedfrom the validated FEM offers the possibility to more conserva-tively assessing the fatigue performance of steel bridges throughcreating new stress-range histograms associated with potentialcritical locations other than sensor locations.

5. Conclusions

This paper presented an efficient approach to evaluating theeffectiveness of retrofitting distortion-induced fatigue cracking insteel bridges. Both analytical results from 3-D finite element mod-els (FEM) and the field monitored data from structural health mon-itoring (SHM) were used to estimate the fatigue reliability of theconnection details after retrofitting, based on the AASHTO fatigueevaluation methodology. The new concept of the spatial adjust-ment factor (SAF) was introduced and quantified with the devel-oped FEM in order to truly represent the fatigue stress-ranges atthe identified critical locations after retrofitting. The field moni-tored stress-range data from an existing bridge, which was moni-tored in 2003 by the ATLSS Center, were analyzed by using theproposed approach. The following conclusions can be drawn fromthis study.

(1) The developed limit state equation for fatigue reliabilityassessment can integrate the field monitored data fromSHM into the evaluation of the softening connection retrofit-

ting method. The field monitored data herein cover bothstress-range data to estimate effective stress-ranges andtraffic data such as AADT and AADTT to predict the numberof stress-range cycles. The importance of modifying the ori-ginal field monitored data to truly represent the fatiguestress-ranges at the identified critical locations has beenobserved from the example presented.

(2) The proposed SAF that can be quantified through the vali-dated FEM results in a reasonable estimation of the fatiguestress-ranges at the identified critical locations after retrofit-ting. However, further studies on time-variant Sreff and Nshm

based on a large number of long-term field monitored dataare extremely important in validating and improving theapplicability of the proposed approach. Also, future researchis needed on the integration of monitored data in fatiguereliability assessment of retrofitted steel bridges usingBayesian statistical techniques.

(3) The primary cause of the observed fatigue cracks before ret-rofitting and potential fatigue cracking re-initiation after ret-rofitting is the out-of-plane relative displacement inducedby different movements between the tie girder and floor-beam. It is important to realize that the effects of the in-plane traffic loads on the fatigue cracking initiation beforeretrofitting and re-initiation after retrofitting can not beignored, if the strong end fixity exists at the connectiondetails. This implies the importance in specifying the appro-priate ratios of the design stiffness between the floor-beamand tie girders in order to prevent the end fixity occurrences.

Acknowledgements

The support by grants from the Commonwealth of Pennsylva-nia, Department of Community and Economic Development,through the Pennsylvania Infrastructure Technology Alliance(PITA) is gratefully acknowledged. The support of the US NationalScience Foundation through grant CMS-0639428 is also gratefullyacknowledged. Finally, the writers gratefully acknowledge the sup-port from the Federal Highway Administration under cooperativeagreement DTFH61-07-H-0040, and the support from the Officeof Naval Research under award N-00014-08-0188. The opinionsand conclusions presented in this paper are those of the writersand do not necessarily reflect the views of the sponsoringorganizations.

References

[1] AASHTO. Guide specifications for fatigue evaluation of existing steelbridges. Washington, DC: American Association of State Highway andTransportation Officials; 1990.

[2] AASHTO. Standard specifications for highway bridges. Washington,DC: American Association of State Highway and Transportation Officials; 2002.

[3] ABAQUS. ABAQUS version 6.7-1: users manual. Pawtucket, RI: Hibbitt,Karlsson and Sorensen, Inc.; 2007.

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Fatigue reliability assessment of retrofitted steel bridges integrating monitored data

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