Engineering Structures 32 (2010) 3467–3477

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Optimization of retrofitting distortion-induced fatigue cracking of steel bridgesusing monitored data under uncertainty

Ming Liu 1, Dan M. Frangopol ∗,2, Kihyon Kwon 3

Department of Civil & Environmental Engineering, Advanced Technology for Large Structural Systems (ATLSS) Engineering Research Center, Lehigh University, Bethlehem,PA 18015-4729, USA

a r t i c l e i n f o

Article history:Received 22 May 2009Received in revised form8 July 2010Accepted 9 July 2010Available online 9 August 2010

Keywords:Fatigue reliabilityOptimizationDistortion-induced fatigue crackingCut-off size adjustment factorField monitored dataSteel bridges

a b s t r a c t

This paper focuses on optimization of retrofitting distortion-induced fatigue cracking of steel bridgesusing monitored data under uncertainty. The optimization problem has two competing objectives: (i)maximization of the fatigue reliability of the connection details after retrofitting and (ii) minimizationof the cut-off area. The geometrical restrictions, predefined maximum tensile stresses, and minimumremaining fatigue life of the connection details after retrofitting are all taken into account as constraints.The fatigue reliability assessment with monitored data is based on the formulation used in the AASHTOspecifications. The original monitored data may be modified by using a proposed cut-off size adjustmentfactor (SAF) to represent the fatigue stress ranges at the identified critical locations after retrofitting.The nonlinear relationships between the cut-off size and SAF are established. The proposed approachis illustrated by using an existing steel tied-arch bridge monitored by the Advanced Technology for LargeStructural Systems (ATLSS) Engineering Research Center at Lehigh University.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In the United States, many existing steel highway bridges werebuilt from the late 1950s through the 1970s. Due to the lack of in-depth research on the fatigue performance of connection details[1], a considerable number of these bridges have developed fatiguecracks caused by out-of-plane distortion. Thus, the connectiondetails of steel bridges subjected to out-of-plane distortions arerecognized as the largest category of fatigue cracking nationwide[2–4]. Even if the magnitude of out-of-plane distortions is only0.5 mm (0.02 in.), it may induce high cyclic stress ranges up to 276MPa (40 ksi) in small welded web gaps [4]. Due to tensile stressconcentrations, the fatigue cracking initiated in the small web gapspropagates parallel to the flange along the flange-web connectionof the floor beam [5].

The typical retrofit methods include (a) drilling a crack arresthole at the crack tip to stop the crack propagation, and (b)softening the connection by cutting off portions of its upper end

∗ Corresponding author. Tel.: +1 610 758 6103; fax: +1 610 758 4115.E-mail address: dan.frangopol@lehigh.edu (D.M. Frangopol).

1 Research Associate.2 Professor and the Fazlur R. Khan Endowed Chair of Structural Engineering and

Architecture.3 Graduate Research Assistant.

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.07.016

to allow the relative movements to effectively occur over a longerlength of the web without the restraint of the top flange andthe connection plate or angles (see Fig. 1, [6]). While drilling acrack arrest hole only provides a temporary solution because re-initiation of the distortion-induced fatigue cracks often occursaround the drilled hole, the softening connection retrofittingmethod is a cost-efficient and technical effective option. Thissoftening connection retrofitting has been used in existing steelbridges such as the Des Moines Bridge [2], the Midland CountyBridge [7], and the Birmingham Bridge [8]. It is the shape and sizeof the cut-off portion that hold the key for a successful retrofittingoperation, under consideration of anticipated fatigue life. For thisreason, optimization problems regarding shape and/or size canbe formulated to provide optimal solutions associated with thesoftening retrofit strategies.

Shape optimization may be used to find the optimal shape ofthe cut-off in terms of the required stress field after retrofitting,while cut-off size (area) optimization may be used to find theoptimal size of the retrofitting considering remaining service life.In this study, the cut-off size optimization for a rectangular shapeused in an existing bridge, the Birmingham Bridge (see Fig. 2), isconsidered in order to (a) determine optimal sizes according toanticipated service life of the bridge after retrofitting, (b) use themonitoring data collected from the rectangular cut-off retrofittingfor fatigue reliability evaluation, and (c) compare the optimizedareas with the actual cut-off area. The rectangular shape, obtained

3468 M. Liu et al. / Engineering Structures 32 (2010) 3467–3477

Fig. 1. Schematic distortion of a floor beam small welded web gap.Source: Adapted from [6].

by using a plasma or saw for cutting, is recommended as themost common and economic for the efficient dispersion of stresseslocally concentrated in potential critical locations of the specifiedretrofitting detail [8]. The rectangular cut-off region is usuallysmoothed in the corners to increase fatigue strength by providinga smooth transition with grinding hole edges. The transitionradius associated with fatigue details can be determined basedon the American Association of State Highway and TransportationOfficials (AASHTO) specifications [27].

This paper focuses on the size optimization of the cut-off area ofthe softening connection retrofitting, considering (a) two compet-ing objectives including the maximization of the fatigue reliabilityof the connection details after retrofitting and the minimization ofthe cut-off area, and (b)multiple constraints including geometricalrestrictions, predefined maximum tensile stresses, and minimumremaining fatigue life of the connection details after retrofitting.According to [6], the retrofitting cut-off size has to be determinedbased on finite element (FE) analysis and the AASHTO constant am-plitude fatigue limit (CAFL). To preserve bridge performance, thefatigue limit criterion (i.e., maximum stresses developed at poten-tial critical locations after retrofitting should not exceed the CAFL)has to be satisfied. However, under uncertainties, bridge remaininglifetime after retrofitting can be overestimated or underestimated.The proposed optimization approach taking into consideration un-certainties and the constraint related to the remaining fatigue liferestriction provides a more realistic and cost-effective methodfor determining the retrofitting cut-off size of steel bridges underdistortion-induced cracking.

In this study, the fatigue reliability of the connection detail isevaluated, based on the field monitored data and the approachused in the AASHTO Guide Specifications for Fatigue Evaluation ofExisting Steel Bridges [9] and Standard Specifications for HighwayBridges [10]. The originalmonitored datamay bemodified by usinga cut-off size adjustment factor (SAF) to represent the fatiguestress ranges at the identified critical locations after retrofitting[14]. This is similar to the method of applying a scale factor tothe stress ranges in order to produce a new stress-range binhistogram for finite fatigue life of a detail [11]. The nonlinearrelationships between the cut-off size and its corresponding SAFare established. The optimal solutions are obtained by using thedesign optimization software [12]. The proposed approach isillustrated on an existing steel tied-arch bridge monitored in 2003by the Advanced Technology for Large Structural Systems (ATLSS)Engineering Research Center at Lehigh University.

Table 1Random variables for fatigue reliability analysis.

Parameter Notation Distribution Reference

Miner’s critical damageaccumulation index

∆ Lognormal(1.0, 0.3)

[16]

Fatigue detail coefficient A LognormalCOV(A) = 0.45(see Table 2)

[17]

Measurement error factor e Lognormal(1.0, 0.04)

[18]

Product of N(t) and S3reff Ns(t) LognormalCOV(Ns(t)) = 0.30

[14]

2. Fatigue reliability assessment with field monitored data

The AASHTO approach to fatigue reliability assessment is basedon the S–N curves in the AASHTO Specifications [10] and theMiner’s rule [13]. When integrated with the field monitored data,the limit-state equation, g(X) = 0, where X is a vector of randomvariables, can be expressed for fatigue reliability analysis of theconnection details as [14,15]

g(X) = ∆ − e × D = 0 (1a)

where

D = [N(t) × (Sreff )m]/A = Ns(t)/A. (1b)

In Eq. (1a), ∆ = Miner’s critical damage accumulation indexwhich is assumed as a lognormal distributed random variable withparameters λ∆ = ln∆ and ζ∆ = COV(ln∆) × ln∆ representingthe mean value and standard deviation of ln ∆, respectively(see Table 1, [16]), e = measurement error factor in structuralhealth monitoring (SHM) which may be considered as lognormal[18], and D = Miner’s damage accumulation index in terms ofloading. In Eq. (1b), A = fatigue–strength coefficient, a lognormaldistributed random variable with the mean value, A, that is basedon the category of the connection details under consideration,and the standard deviation of ln A, σ(ln A) = 0.429 [17], andNS(t) = product of N(t) and Smreff where N(t) is the total number ofstress cycles within a period of time T under consideration, Sreff isthe equivalent constant-amplitude stress range during T which canbe estimated from field monitored data, andm = 3.0 is a materialconstant representing the slope of the S–N curve [10]. Thus, NS(t)is a stochastic process. According to previous studies based on fieldmonitored data [14], NS(t) can be treated, in a simplified way, asa lognormal random variable with a coefficient of variation (COV)usually less than 0.30. The random variables for fatigue reliabilityanalysis are presented in Table 1.

The time-dependent reliability index β(t) associated withEq. (1a) is used to estimate fatigue life. The random variables,∆, A, e, and Ns(t), are assumed to be statistically independent.Correlations can, of course, be taken into account if data areavailable. Unfortunately, this is not the case. Reliability of astructural component or system is defined as the probability of safeperformance, P(g(X) > 0). Based on the performance function,g(X) = R − S, including resistance, R, and load effect, S, the limit-state formulation (see Eqs. (1a) and (1b)), will have R = ∆ × Aand S = e × Ns(t). Since ∆, A, e, and Ns(t) are considered to belognormal random variables, the equivalent performance functionadopted is expressed as

g(X) = ln(R/S) = ln R − ln S = ln(∆ × A) − ln[e × Ns(t)]

= ln∆ + ln A − ln e − lnNS(t). (2)

Therefore, the time-dependent reliability index β(t), defined asthe mean value of g(X) divided by the standard deviation of

M. Liu et al. / Engineering Structures 32 (2010) 3467–3477 3469

b

a

Fig. 2. Floor beam connection details after retrofitting (a) FE modeling and critical locations, and (b) sensor locations.

g(X), is

β(t) =(λ∆ + λA) − (λe + λNs(t))

ς2∆ + ς2

A + ς2e + ς2

Ns(t)

= 1.67 ln[

A1.10Ns(t)

], (3)

where λ and ζ denote the mean value and standard deviationof each random variable, and NS(t) denotes the mean value of

NS(t). This value may be obtained from the original monitoredsensor data which may be modified by using a SAF. This factorrepresents the fatigue stress ranges and corresponding number ofstress cycles at the identified critical locations after retrofitting.To identify critical locations in the retrofitting detail (i.e., a cut-offregion of rectangular shape), the AASHTO category correspondingto each location along the edges is first defined and classified. Then,

3470 M. Liu et al. / Engineering Structures 32 (2010) 3467–3477

Table 2Basic information at critical locations CL-I, CL-II, and CL-III.

Critical location CL-I (CH-2) CL-II (CH-7) CL-III (CH-11)Fatigue category C A B

Mean value of coefficient Aa MPa3 14.4 × 1012 82.0 × 1012 39.3 × 1012

(ksi3) (44.0 × 109) (250.0 × 109) (120.0×109)

CAFL MPa (ksi) 69.0 (10.0) 165.0 (24.0) 110.0 (16.0)

Predefined threshold MPa (ksi) 6.9 (1.0) 16.5 (2.4) 11.0 (1.6)

C1 in Eq. (14a)b MPa m2 10.89 (σ2 in MPa) (σ3 in MPa)(ksi in2) (2.45 × 103) (σ2 in ksi) (σ3 in ksi)

C21 in Eq. (14b) h in m (in.) (σ1 in MPa) 2.59(0.066) × h (σ3 in MPa)

C23 in Eq. (14b) h in m (in.) (σ1 in MPa) 1.69(0.043) × h (σ3 in MPa)

C3 in Eq. (14c)b MPa m−1 (σ1 in MPa) (σ2 in MPa) 0.52(ksi in−1) (σ1 in ksi) (σ2 in ksi) (0.002)

a See [17] for computation procedures.b Values are based on the out-of-plane displacement of 2.54 mm (0.1 in.) applied to the top of the floor beam flange only.

the most critical location within the same category is identified byusing FE modeling.

3. Optimization

As mentioned previously, the softening connection detail iscost-efficient and technical effective for retrofitting distortion-induced fatigue cracks in steel bridges. A successful retrofittingmay depend on the size of the cut-off, when a rectangular shapeis adopted (see Fig. 2). Too small cut-off size may result in re-initiation of the fatigue cracks soon after retrofitting, as evidencedon the Poplar Street Bridge in East St. Louis [6]. Conversely, toolarge cut-off size may significantly reduce the shear capacity of theconnection details. This reduction could potentially increase themagnitude of out-of-plane distortions and corresponding tensilestresses after retrofitting, eventually resulting in re-initiation ofthe fatigue cracks. Consequently, an optimization problem withthe design variables of the cut-off height (h) and length (l)is necessary to be formulated, where the objective functionsare the maximization of the computed fatigue reliability of theconnection details after retrofitting and the minimization of thecut-off area. In many practical optimization applications, two ormore objective functions can be optimized at the same time.These are referred to, respectively, as biobjective or multiobjectiveoptimization problems [19]. Accordingly, the proposed approachis a biobjective optimization since both objectives have to beachieved simultaneously under predefined constraints. This isdifferent from the classical optimization under uncertainty in thata single objective, usually the expected total cost, is consideredand the decision maker has a single choice (to implement theoptimum solution). If the cost associated with the optimumsolution is not affordable, the decisionmaker has to choose anothernon-optimal solution. Alternatively, if a biobjective optimizationapproach is used, it will offer multiple optimal solutions fordecision makers. Therefore, two objective functions are hereinconsidered to providemultiple optimal cut-off sizes for retrofittingof the bridge connection details while satisfying all pre-imposedconstraints.

The responses at the softening connection details are fullydependent when a relative horizontal displacement is appliedto the floor beam. Therefore, it is assumed that the failuremodes at the critical locations are perfectly correlated. Using thisassumption, the computed fatigue reliability of the connectiondetails after retrofitting is defined as the minimum of the fatiguereliabilities of the identified critical locations. The constraintsassociated with the optimization problem include the geometricalrestraints, predefined maximum tensile stresses at each of theidentified critical locations, andminimum remaining fatigue life of

the connection details after retrofitting. As a result, the biobjectiveoptimization problem can be formulated as follows:Find the design variables: h and l

Objective functions: (i) maximize {minimum (β1, β2, . . . , βp)} (4)(ii) minimize h × l (5)

Subjected to hmin ≤ h ≤ hmax (6a)lmin ≤ l ≤ lmax (6b)

σi ≤ σmax,i (i = 1, 2, . . . , p) (7)

Ti ≥ Tmin (i = 1, 2, . . . , p), (8)

where βi = fatigue reliability index at the ith identified criticallocation (i = 1, 2, . . . , p); p = number of the identified criticallocations after retrofitting; hmin and hmax = minimum andmaximum cut-off height due to the geometrical restrictions,respectively; lmin and lmax = minimum and maximum cut-offlength associated with the geometrical restrictions, respectively;σi and Ti = tensile stress and remaining fatigue life at theith identified critical location, respectively; σmax,i = predefinedmaximum tensile stress at the ith identified critical location,and Tmin = predefined minimum remaining fatigue life of theconnection details after retrofitting. It is noted that σmax,i shouldbe related to the fatigue category classified by the AASHTOspecifications. Thus, σmax,i may vary at different critical locations.

The procedure for solving the optimization problem formulatedby Eqs. (4)–(8) includes the following steps.Step 1. Identifying the critical locations after retrofitting

The critical locations for potential re-initiation of fatiguecracking after retrofitting may be identified by developingvalidated FE modeling and/or from field monitored data.Step 2. Collecting the stress range data at the identified criticallocations

Since the optimal cut-off size is not available at this step,the sensors for collecting fatigue stress ranges and correspondingnumber of stress cyclesmay be installed for a trial cut-off size only.However, the collected stress range data at these sensor locationsmay bemodified by using the proposed SAF to estimate the fatiguestress range data at the identified critical locations after optimalretrofitting.Step 3. Establishing the relationship between SAF and NS(t) at theidentified critical locations

Theproposed SAFi at the ith identified critical location is definedas the ratio of σi to the stress ranges collected at the correspondingsensor location. Themean values ofNS(t) associated with different

M. Liu et al. / Engineering Structures 32 (2010) 3467–3477 3471

a

c

b

Fig. 3. Stress-range bin histograms at critical locations CL-I, CL-II, and CL-III (a) at CL-I (CH-2), (b) at CL-II (CH-7), and (c) at CL-III (CH-11).

Table 3Validation of Eq. (14) by FE modelinga .

Cut-off size l (m) × h (m) Stress by Eq. (14) (MPa) Stress by FE modeling (MPa) RatioCL-I CL-II CL-III CL-I CL-II CL-III CL-I CL-II CL-III

0.61 × 0.43 7.886 32.372 42.598 10.247 36.365 51.541 0.770 0.890 0.8260.61 × 0.41 8.495 31.962 44.589 8.670 37.260 51.700 0.980 0.858 0.8620.61 × 0.30 12.984 33.050 62.304 13.000 44.341 53.000 0.999 0.745 1.1760.52 × 0.41 23.943 43.922 52.202 23.858 37.207 56.999 1.004 1.180 0.9160.52 × 0.30 42.376 49.995 74.571 45.000 45.320 62.100 0.942 1.103 1.2010.41 × 0.30 73.373 73.941 93.456 78.013 63.949 94.094 0.941 1.156 0.9930.41 × 0.25 100.735 81.779 110.742 95.905 73.348 104.543 1.050 1.115 1.0590.30 × 0.30 94.509 97.542 127.391 111.783 86.841 160.895 0.845 1.123 0.7920.30 × 0.25 130.163 107.673 150.954 144.936 103.632 186.284 0.898 1.039 0.8100.30 × 0.20 202.902 125.535 190.220 201.257 136.075 222.599 1.008 0.923 0.8550.30 × 0.15 360.039 157.049 255.242 318.288 170.979 301.395 1.131 0.919 0.847a Values are based on the out-of-plane displacement of 2.54 mm (0.1 in.) applied to the top of the floor beam flange only.

values of SAFi may be obtained from the collected stress rangedata [14].

Step 4. Developing the formulation to calculate σi based on designvariables h and l

The formulation to calculate σi based on design variables hand l is developed at each of the identified critical locations (seeAppendix). These developed formulations are validated by com-paring the computed results with those from the corresponding FEmodeling (see Table 3).

Step 5. Re-formulating the developed optimization problem inEqs. (4)–(8)

Eqs. (4)–(8) can be re-formulated by using the results fromSteps 1–4, where only design variables h and l are explicitlyincluded in both objective functions and constraints.Step 6. Solving the optimization problem

The design optimization software, such as [12], may be used tosolve the re-formulated optimization problem.

4. Application example

The proposed approach is illustrated by using an existing steeltied-arch bridge monitored by the ATLSS Engineering ResearchCenter [6]. The trial (actual) cut-off sizewas h0 = 0.30m (11.75 in.)in height and l0 = 0.52m (20.5 in.) in length, while the floor beam

3472 M. Liu et al. / Engineering Structures 32 (2010) 3467–3477

is 2.85 m (112.0 in.) in height and 32.13 m (1265.0 in.) in length.The field monitored data were continuously collected for a total of39.95 days immediately after retrofitting [6]. The rain-flow cyclecounting method [20] was adopted to obtain the fatigue stress-range bin histograms at the sensor locations that are displayed inFig. 2(b). The detailed information on the bridge, includingmaterialproperties, bridge geometries, and connection details between thefloor beams and the tie girders, can be found in [6].Steps 1 and 2. As shown in Fig. 2(a), the potential fatigue crackingre-initiation after retrofitting is identified at the three criticallocations (i.e., CL-I, CL-II, and CL-III) based on the FEmodeling stressresults and the defined AASHTO category at each location. Threesensors (CH-2, CH-7, and CH-11) are identified in Fig. 2(b) [6].Sensor CH-2 collected the fatigue stress range data (i.e., σ1 = σyy,1)at the intersection of the top flange and web of the floor beam (i.e.,CL-I). According to the AASHTO specifications, CL-I can be classifiedas S–N category C. Similarly, sensors CH-7 and CH-11 collectedthe fatigue stress range data σ2 =

σ 2yy,2 + σ 2

zz,2 and σ3 = σzz,3,respectively, near the bottom of the cut-off (i.e., CL-II) and at webnear the connection angles (i.e., CL-III). CL-II and CL-III can beclassified as S–N categories A and B, respectively. Fig. 3(a)–(c)present the original monitored stress-range bin histograms atcritical locations CL-I, CL-II, and CL-III, respectively, where 12,265stress range cycles greater than 6.89 MPa (1.0 ksi) and 23,587stress range cycles greater than 10.34 MPa (1.5 ksi) were collectedat critical locations CL-I (CH-2) and CL-III (CH-11), respectively,while 7120 stress range cycles greater than 17.24MPa (2.5 ksi) wasrecorded at critical location CL-II (CH-7) [6].Step 3. Fig. 4 presents the mean values of NS(t) associated withdifferent values of SAF at critical location CL-I. Because the originalmonitored data only contain the stress-range bin histograms atsensor locations directly obtained by the rain-flow cycle countingmethod, but the SAF needs to be applied to individual stress ranges,the random number generator has to be adopted to reproducethe individual stress ranges in the modification procedures [14].Since the typical stress-range bin in the rain-flow cycle countingmethod is rather narrow, for example, 3.45 MPa (0.5 ksi) in thisstudy, the uniform distribution of the individual stress ranges inthe corresponding stress-range bin can be assumed. Only stressranges greater than 3.45 MPa (0.5 ksi) in the original monitoreddata are used because those less than 3.45 MPa (0.5 ksi) makeno contributions to accumulated fatigue damages [21]. The effectof the annual increase rate of the number of stress cycles, α, onthe mean values of NS(t) is also indicated in Fig. 4. The newlygenerated stress-range bin histograms based on the randomlysimulated individual stress ranges and SAF are used to calculatethe corresponding Sreff . This is equal to the cubic root of the meancube (rmc) of all stress ranges, Srj [13,9]:

Sreff =

[− nj

NtotalS3rj

]1/3

, (9)

where nj = number of stress range cycles in the predefined stress-range bin Srj andNtotal = total number of stress range cycles greaterthan the predefined stress range threshold. Therefore, the meanvalue of NS(t) with SAF during the monitoring period, Nshm, can beexpressed as

Nshm = Ntotal × S3reff =

−(nj × S3rj). (10)

It is emphasized that the predefined stress range threshold mustbe established in the computation of Nshm by using Eq. (10).This is because the low magnitude stress cycles make nocontributions to accumulated fatigue damages, but, when includedin Eq. (10), yield larger values of Nshm which results in unnecessaryconservative estimations of β(t) by using Eq. (3). This is in

Fig. 4. Relationship between NS(t) and cut-off size adjustment factor (SAF).

contrast to the estimations of the fatigue resistance capacitiesof the connection details with field monitored data, where thehigher predefined stress range thresholds result in higher Sreff ,and lower (conservative) fatigue resistance capacities from thecorresponding S–N curves in the AASHTO specifications [22,23].From a large number of laboratory experiments under constantamplitude cyclic loading, the constant amplitude fatigue limit(CAFL) is established for each category as presented in Table 2,indicating that no fatigue cracks appear if the applied stresscycles have the constant amplitude smaller than the correspondingCAFL. For the field monitored variable-amplitude stress cycles, thepredefined stress range thresholds may be lowered to a quarterof the CAFL [24]. In this study, 10% of the corresponding CAFL isused as the predefined stress range threshold (see Table 2). Thisis because the curves representing the relationships between thecomputed Sreff and corresponding Ntotal become asymptotic to theapplicable S–N curves after the predefined threshold is set to belower than 10% of the CAFL [6,14]. As a result, the relationshipsbetween SAF (i.e., S1, S2, . . ., and Sp) and Nshm can be established byusing the regressionmodels of the q-order polynomial functions as

Nshm,i =

q−j=0

aij × S ji (i = 1, 2, . . . , p) (11)

where aij = coefficients that can be obtained from the fieldmonitored data. The quadratic polynomial functions (i.e., q = 2)are adopted in this study, where the regression models of Nshm inMPa can be described as

Nshm,1 = (1.03S21 − 1.21S1 + 0.38) × 108 (12a)

Nshm,2 = (7.87S22 − 10.50S2 + 3.66) × 108 (12b)

Nshm,3 = (9.69S23 − 12.80S3 + 4.56) × 108. (12c)

Furthermore, the regressionmodelswith the quadratic polynomialfunctions for any targeted time period Tg in years, that is,Nsi(Tg) (i = 1, 2, 3), can be expressed as

NSi(Tg) =365Tshm

× Nshm,i ×

∫ Tg

0(1 + α)tdt, (13)

where Tshm =monitoring period in days (i.e., Tshm = 40),N shm,i = asshown in Eq. (11) or (12), and α = annual increase rate of thenumber of the fatigue stress cycles.Step 4. As shown in Fig. 5, the structural behavior of the web ofthe floor beam after retrofitting, under out-of-plane displacementloading condition, can be represented by the virtual beams wherethe vertical edge of the cut-off is restrained by the flange at the topand by portion of the web at the bottom, while the horizontal edge

M. Liu et al. / Engineering Structures 32 (2010) 3467–3477 3473

Fig. 5. Modeling the structural behavior of the web of the floor beam after retrofitting.

of the cut-off is restrained by the connection angles at one end andbyportion of theweb at the other end. Since the restraints providedby the portion of the web are relatively weak, the pinned end maybe assigned at the bottom of the cut-off as shown in Fig. 5 [6,14].Consequently, the analytical formulations to calculate σ1, σ2 andσ3 can be expressed as

σ1 = C1H3

× L2c − (H + 2h) × (H − h)2 × l2

H3 × L2c × h2(14a)

σ2 =

(C21 · σ1)2 + (C23 · σ3)2 (14b)

σ3 = C3(H + 2h) × (H − h)2

l × h, (14c)

where C1, C21, C23 and C3 = constants that are independent on hand l as listed in Table 2, H = height of the floor beam, that isH = 2.85 m (112.0 in.), and Lc = length of the floor beam affectedby the end constraints under out-of-plane loading conditionwhichmay be obtained from the FE modeling as Lc = 0.64 m (25.0 in.).

Table 3 compares the computed stresses σ1, σ2, and σ3 fromEq. (14) with those from FE modeling [14], where reasonableagreements can be observed for validating Eq. (14). Therefore, thenonlinear relationships between SAF (i.e., S1, S2, and S3) and thecut-off size h and l can be established as

S1 =H3

× L2c − (H + 2h) × (H − h)2 × l2

H3 × L2c − (H + 2h0) × (H − h0)2 × l20×

h20

h2(15a)

S2 =h

50.0×

6.71σ 2

1 + 2.86σ 23 (15b)

S3 =(H + 2h) × (H − h)2

(H + 2h0) × (H − h0)2×

l0 × h0

l × h. (15c)

Step 5. Based on the analytical results from Steps 1–4 and Table 2,the reliability indicesβ1(t),β2(t), andβ3(t) at critical locations CL-I (Category C), CL-II (Category A), and CL-III (Category B), for anytargeted time period Tg in years, can be expressed as

β1(t) = 1.67

× ln

1.44 × 104

1.03S21 − 1.21S1 + 0.38·

∫ Tg

0(1 + α)tdt

−1

(16a)

β2(t) = 1.67

× ln

8.20 × 104

7.87S22 − 10.50S2 + 3.66·

∫ Tg

0(1 + α)tdt

−1

(16b)

β3(t) = 1.67

× ln

3.93 × 104

9.69S23 − 12.80S3 + 4.56·

∫ Tg

0(1 + α)tdt

−1

(16c)

where

S1 =9.29 − (2.85 + 2h) × (2.85 − h)2 × l2

36.3h2(17a)

S2 = h ×

4.82S21 + 6.36S23 (17b)

S3 =(2.85 + 2h) × (2.85 − h)2

143.6(h × l). (17c)

The geometrical constraints in this application example are hmin =

0.10 m (4.0 in.), hmax = 0.43 m (17.0 in.), lmin = 0.31 m (12 in.),and lmax = 0.61 (24 in.). The maximum tensile stresses afterretrofitting are predefined as the corresponding CAFL at each ofthe identified critical locations, that is, σmax,1 = 69 MPa (10 ksi),σmax,2 = 165 MPa (24 ksi), and σmax,3 = 110 MPa (16 ksi).Moreover, the remaining fatigue life of the connection details afterretrofitting is defined as the period from the start of the fatiguedamage to the time when the reliability index β(t) in Eq. (16)reaches the targeted minimum βtarget = 3.72 [25]. Based onEqs. (3) and (11) or (12), Eq. (8) can be expressed for i = 1, 2, and3 at critical locations CL-I, CL-II, and CL-III, respectively:

β(Tmin) = 1.67 ln[

A1.10NS(Tmin)

]≥ 3.72 (18)

where

NS(Tmin) =365Tshm

× Nshm,i ×

∫ Tmin

0(1 + α)tdt. (19)

Therefore∫ Tmin

0(1 + α)tdt × Nshm,i ≤ 0.0107A. (20)

3474 M. Liu et al. / Engineering Structures 32 (2010) 3467–3477

Fig. 6. Flowchart for solving the proposed optimization problem.

Moreover, Eq. (7) can be re-formulated considering Eq. (14) as

σ1 = C1 ×9.29 − (2.85 + 2h) × (2.85 − h)2 × l2

9.29h2≤ 69.0 (21a)

σ2 = h ×

6.71σ 2

1 + 2.86σ 23 ≤ 165.0 (21b)

σ3 = C3 ×(2.85 + 2h) × (2.85 − h)2

l × h≤ 110.0. (21c)

Consequently, the optimization problem in Eqs. (4)–(8) can be re-formulated for any targeted time period Tg in years as follows:Find the design variables: h and lObjective functions:(i) maximize {β(h, l)} (22)

(ii) minimize h × l (23)Subjected to(i) geometrical constraints0.102 ≤ h ≤ 0.432 (24a)0.305 ≤ l ≤ 0.610 (24b)(ii) stress constraints

for the critical location CL-I,

9.29C1 − 641.0h2

C1 × (2h3 − 8.54h2 + 23.0)≤ l2 (25a)

for the critical location CL-II,C1 ×

9.29 − (2.85 + 2h) × (2.85 − h)2 × l2

66.05h

+

C3 ×

(2.85 + 2h) · (2.85 − h)2

10.89l

2

≤ 80.272 (25b)

for the critical location CL-III,

C3 ×2h3

− 8.54h2+ 23.0

110.0h≤ l (25c)

(iii) fatigue reliability constraintsfor the critical location CL-I,

1.03S21 − 1.21S1 + 0.38 ≤ 1.54 × 103

×

∫ Tmin

0(1 + α)tdt

−1

(26a)

for the critical location CL-II,

7.87S21 − 10.50S1 + 3.66 ≤ 8.77 × 103

×

∫ Tmin

0(1 + α)tdt

−1

(26b)

for the critical location CL-III,

9.69S21 − 12.80S1 + 4.56 ≤ 4.21 × 103

×

∫ Tmin

0(1 + α)tdt

−1

(26c)

where

β(h, l) = min

β1(h, l)β2(h, l)β3(h, l)

. (27)

Step 6. The biobjective optimization problem in Eqs. (22)–(27) issolved by using the design optimization software [12], accordingto the flowchart presented in Fig. 6.

Fig. 7(a)–(d) present the feasible regions of the cut-off sizes fordifferent values of the out-of-plane displacement, ∆h, the annual

M. Liu et al. / Engineering Structures 32 (2010) 3467–3477 3475

a b

c d

Fig. 7. Feasible region and optimal cut-off sizes for different values: (a) ∆h = 2.54 mm (0.10 in.) and α = 2% for a target life of 50 years, (b) ∆h = 3.81 mm (0.15 in.) andα = 2% for a target life of 50 years, (c) ∆h = 2.54 mm (0.10 in.) and α = 5% for a target life of 50 years, and (d) ∆h = 2.54 mm (0.10 in.) and α = 5% for a target life of 100years.

a b

Fig. 8. Pareto optimal solutions with ∆h = 2.54 mm (0.1 in.) and α = 5%, (a) objective space and (b) design space.

increase rate of the stress cycles, α, and the minimum requiredfatigue life (i.e., 50 and 100 years). The trial and optimal cut-offsizes are also indicated in Fig. 7, where the three optimal cut-offsizes (i.e., optimal points 1, 2, and 3) are based on (i) maximizationof the computed fatigue reliability of the connection details afterretrofitting (see optimal point 1), (ii) minimization of the cut-off

area (see optimal point 2), and (iii) the combined objective functionwith equal weights on (i) and (ii) (see optimal point 3),respectively.

For a given target life of 50 years, Fig. 7(a)–(c) compare theeffects of ∆h and α on the feasible regions and optimal solutionsof the cut-off sizes. It can be concluded that the feasible regions

3476 M. Liu et al. / Engineering Structures 32 (2010) 3467–3477

Fig. 9. Time-dependent maximum reliability indices for all Pareto optimalsolutions indicated in Fig. 8.

and optimal solutions of the cut-off sizes change significantlywith different out-of-plane displacements (see Fig. 7(a) and (b)),whereas they are not sensitive to the annual increase rate of thestress cycles up to α = 5% (see Fig. 7(a) and (c)). It is noted thatthe trial cut-off size is in the infeasible region due to the constraintσ3 > σmax,3, when the out-of-plane displacement is 3.81 mm(0.15 in.) as shown in Fig. 7(b). Therefore, it is critical to verify theactual out-of-plane displacements before retrofitting connectiondetails. In addition, the effects of theminimum required fatigue lifeon the feasible regions and optimal solutions of the cut-off sizes areinvestigated. As shown in Fig. 7(c) and (d), the active lower boundconstraints of the feasible regions are σ1 < σmax,1 and σ3 < σmax,3for theminimum required fatigue life of 50 years, while T2 > Tmin,2and T3 > Tmin,3 become the active lower bound constraints for theminimum required fatigue life of 100 years. It is interesting to notethat the trial (actual) cut-off size is always in the feasible region.

Fig. 8(a) and (b) present the Pareto optimal solutions in bothobjective and design variable spaces, respectively, consideringthe out-of-plane displacement of 2.54 mm (0.1 in.). Accordingto [26], the dominant solution concept in defining solutions forbiobjective or multiobjective optimization problems is that ofPareto optimality. A point x∗ in the feasible design space (FDS)is called Pareto optimal if there is no other point x in FDS, thatreduces at least one objective function without increasing anotherone. As shown in Fig. 8(a), the Pareto frontiers are clearly shownin the objective space, where the targeted levels of the objectivefunctions can be determined according to the decision maker’spreferences. The optimal cut-off areas can be easily found in thecorresponding design variable space. It is very interesting to notethat the trial cut-off area is always on the Pareto frontiers fordifferent minimum required fatigue lives up to 100 years, whichimplies that no future re-retrofitting is necessary. However, itshould be emphasized herein that this conclusion is based onthe out-of-plane displacement of 2.54 mm (0.1 in.). Finally, Fig. 9presents the time-dependent maximum fatigue reliability indicesassociated with the Pareto optimal solutions indicated in Fig. 8.

5. Conclusions

This paper presented a novel approach to finding the optimalcut-off size of the connection details for retrofitting distortion-induced cracking in steel bridges using monitored data underuncertainty. Two competing objectives indicating minimizationof the cut-off area and maximization of the fatigue reliability ofthe connection details were used. The concept of the cut-off SAFwas introduced. This factor was used to develop the nonlinear

relationship with respect to the cut-off size. The optimal cut-off size was found by using the stress range data of an existingbridge monitored by the ATLSS Engineering Research Center. Thefollowing conclusions can be drawn from this paper.

1. The cut-off size of the connection details can be found from theproposed optimization formulation using the field monitoreddata.

2. The monitored data can be used (a) to represent the fatiguestress ranges at the identified critical locations after retrofittingbased on the proposed SAF and (b) to find the mean values ofNS(t).

3. The developed stress formulations can be validated by FEmodeling.

4. The geometrical constraints on connection details, stress con-straints associated with AASHTO CAFL, and fatigue reliabilityconstraints defining structural service life after retrofitting haveto be used in order to provide practical solutions.

5. The optimization approach proposed herein can be generallyapplied for finding the optimal cut-off size of connection detailsunder uncertainty.

6. Further research is needed to develop the proposed ap-proach for cost-oriented reliability-based shape optimization ofretrofitting distortion-induced fatigue cracking. Also, for imple-mentation purposes simplified procedures are needed.

Acknowledgements

The support by grants from the Commonwealth of Pennsyl-vania, Department of Community and Economic Development,through the Pennsylvania Infrastructure Technology Alliance(PITA) is gratefully acknowledges. The support of the US NationalScience Foundation through grant CMS-0639428 is also grate-fully acknowledged. Finally, the writers gratefully acknowledgethe support from the Federal Highway Administration under co-operative agreement DTFH61-07-H-0040, and the support fromthe Office of Naval Research under award N-00014-08-0188. Theopinions and conclusions presented in this paper are those of thewriters and do not necessarily reflect the views of the sponsoringorganizations.

Appendix. Derivations of σ1, σ2, and σ3

According to Fig. 5, the structural behavior of the web ofthe floor beam after retrofitting, under constant out-of-planedisplacement, ∆top, applied at the top of the web only, thecomputed stress, σ1 at critical location CL-I can be expressed as

σ1 = σyy =Myy

Syy=

3(E × Iyy)Syy × h2

× (∆top − ∆h,1), (A.1)

where E, Iyy, and Syy = constants related to the material and crosssection properties of the web of the floor beam after retrofitting;∆h,1 = out-of-plane displacement at height h and length l, for σ1(see Fig. A.1), that is

∆h,1 = ∆h ×

lLc

2

= ∆top ×

3

H − hH

2

− 2H − hH

3

×

lLc

2

(A.2)

where Lc = length of the floor beam affected by the end constraintsunder ∆top, which may be obtained from the FE modeling as Lc =

0.635 m (25.0 in.) in this study.

M. Liu et al. / Engineering Structures 32 (2010) 3467–3477 3477

Fig. A.1. Derivations of σ1, σ2 , and σ3 .

Let C1 =3(E×Iyy×∆top)

Syy, then

σ1 =C1

h2×

1 −

3

H − hH

2

− 2H − hH

3

×

lLc

2

= C1H3

× L2c − (H + 2h) × (H − h)2 × l2

H3 × L2c × h2. (A.3)

Similarly, the computed stress, σ3 at critical location CL-III, can beexpressed as

σ3 = σzz =Mzz

Szz=

3(E × Izz)Szz × l2

× ∆h,3, (A.4)

where E, Izz and Szz = constants related to the material and crosssection properties of the web of the floor beam after retrofitting;∆h,3 = out-of-plane displacement at height h and length l, for σ3(see Fig. A.1), that is

∆h,3 = ∆h ×

lh

= ∆top ×

3

H − hH

2

− 2H − hH

3

×

lh

. (A.5)

Let C3 =3(E×Izz×∆top)

Szz×H3 , then

σ3 =C3 × H3

l2×

3

H − hH

2

− 2H − hH

3

×

lh

= C3(H + 2h) × (H − h)2

l × h. (A.6)

It should be noted that the effects of the cut-off length l on ∆h,1and ∆h,3 are different, that is, (l/Lc)2 for ∆h,1 and (l/h) for ∆h,3. Inaddition, σ2 can be derived by using the regression model that isrelated to σ1 and σ3 (see Eq. (12)), which have been validated bythe FE modeling as presented in Table 3.

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