Fatigue Reliability Analysis of Multi-loading Suspension Bridges Considering Nonlinear Accumulative Damage

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1 JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY Vol 49 No Apr DOI /j issn U448 7TU311 3 A Fatigue Reliability Analysis of Multi-loading Suspension Bridges Considering Nonlinear Accumulative Damage CHEN Zhiwei 1 XU Youlin 3 1 Department of Civil EngineeringXiamen UniversityXiamen China Key Laboratory of Engineering Mechanics of Jiangsu ProvinceNanjing 10096China3 Department of Civil and Structural EngineeringThe Hong Kong Polytechnic UniversityHong KongChina AbstractTaking into account uncertainties in the nonlinear process of fatigue damage accumulation for fatigue reliability analysis of multi-loading long-span bridgesa fatigue reliability assessment method for a long suspension bridge under combined highway railwayand wind loadings was proposed using a continuum damage model CDM Firstthe CDM based on continuum damage mechanics was briefly introducedand main model parameters were analyzed Thena simplified CDM was proposed for further application to bridge structures A limit state function for fatigue reliability analysis based on CDM was defined by introducing proper random variables into the CDM The Monte Carlo simulation MCSwas adopted to generate the random variables and to calculate the failure probability Finally the Tsing Ma Bridge in Hong Kong was taken as a case studyand the failure probabilities of the bridge at the end of 10 years were estimated for different loading scenarios The results indicate that the health condition of the bridge in fatigue is satisfactory under the current traffic conditionsbut attentions should be paid to future traffic growth because it will largely accelerate the damage growth NSFC cezhiwei@ xmu edu cn J

2 14 49 Key wordsfatiguesuspension bridgeswindrailwayhighwaynonlinear damagereliability D = S - S~ 1 S S ~ S D = σ σ - σ 珚 β' B'1 - D? σ? α Miner σ 3-4 Miner σ珚 σ 珚 = σ m α β'b' 5? McCauley xx > 0 6-7?x? = { 0x < σ 珚 = D α δd δn = σ β' + 3 a B'β D α 3 σ a β 3 10 δd δn = σ β' + 3/ r + σ m σ r 4 Bβ' D α B = β' + B' m 4σ m = 0 σ r α D = 1 - [ 1 - σβ' r N Bβ' + 3 ] 1 /α S-N - B β'α S-N S-N σ r N f = K σ - m r 6 5

3 15 N f = D f α + 1 Bβ' + 3 σ α β' + 3 r 7 D1 = 1 - D f ( 1 - σm 1 /α r 1 K ) 1 1 D = 1 - [ 1 - D1 α σm r α K ] D f α β' + 3 = m } Dk = 1 - Bβ' [ 1 - Dk - 1 α k σm r k K ] = K α D = 1 - [ 1-1 σ m r N K ] 1 /α α 11 α σ r = 80 MPa 13 = m = 3 K 9 α α = Dkk D0= 0 1 /α /α k α 1 1 α α = 0 Miner α 11 S-N σ re α k N 1 N σ r i 1 α Fig 1 Fatigue damage accumulation for different α N r k k 9 m S-N m = 3 0 Dk= 1 - α e [ 1 - Dk - 1α k σm 1 /α k + 1 r k K ] 10 α α 10 σ r k α k k k α α e k σ re k α e k σ re k = ( N 1 σ r 0 i = 1 1 N r ( N 1 n i σ m r i + i = 1 k r) i ) n i σ m+ 1 /m 1 n i σ r i k

16 49 10 k Dk= 1 - [ 1 - Dk - 1 α e k + 1 - N r kσ m re k K ] 1 /α e k + 1 13 Dk Dk - 1 7 333 13 1 σ re = 6 0 MPa 69 1 MPa α e = 9 1 1113 3 m S mr k k 1 N b N b Δ K Fig 3 Fig A sample stress time

4 k Dk= 1 - [ 1 - Dk - 1 α e k N r kσ m re k K ] 1 /α e k Dk Dk σ re = 6 0 MPa 69 1 MPa α e = m S mr k k 1 N b N b Δ K Fig 3 Fig A sample stress time history under multiple types of loading α e k gx= Dk- Dk - 1 Dk= 1 - [ 1 - Dk - 1 α e k S mr k K Δ ] S mr k N 1 = 14 1 /α e k P f Monte Carlo Fatigue damage accumulation in a daily block i = 1 n i σ m r i + 1 N σ r 0 i = 1 n i σ m+ r i = N r k σ m re k 16 S mr k m S mr k = N r k σ m re k Δ μ Δ = 1 0 σ Δ = K 11 A

17 0 N Z 3 P f 1 I gx N N i < 0 i = 1 I gx i < 0= { 0 1gX i< 0 17 gx i i X i n 1 4 X i = X i 1 X i X i n X i 1 X i Δ K 3 3 m X i 3 ~ X i n S mr k 3 m 3 15 1 4 13 3 3 4

5 17 0 N Z 3 P f 1 I gx N N i < 0 i = 1 I gx i < 0= { 0 1gX i< 0 17 gx i i X i n 1 4 X i = X i 1 X i X i n X i 1 X i Δ K 3 3 m X i 3 ~ X i n S mr k 3 m Fig 4 Z 4 Z 5 T Z S mr S mr 1 30% Ⅰ 10% 100% Ⅱ Tsing Ma Bridge under railwayhighway Ⅰ and wind loading Ⅱ S mr 5 Z 1 S mr Z 1 S mr Z Ⅰ Ⅱ 3

6 18 49 T r 15 T r T t α g 3 4 X = X 0t T r { X α g t t > T r 18 K Δ S mr k 5 m 3 Fig 5 Probability density function of 14 gx the daily sum of m-power stress ranges i gx i < 0 N f 1 N N f /N P f Ⅰ Ⅰ Ⅱ X = X 0t T r 1 19 { X nα g t /nt r t > T r 3% 11 Ⅱ X 0 t T r X = X 0 { nα g [ 1 - ( 1 + nα ) g t > T r t /nt r ] } 6 1 K μ K = σ K = Δ μ Δ = 1 0 σ Δ = 0 3 S mr 10 S mr k 15 α e α e Z m Tab Fatigue failure probabilities at the end of 10 years at the fatigue critical locations Z 1 Z Z 3 Z 4 Z 5 Z 6 C C F 1 Ⅰ Ⅱ C F Ⅰ Ⅱ C F 1 ⅠF Ⅱ

7 19 4 for bridge under traffic loadingapplication made to Tsing Ma Bridge J Theoretical and Applied Fracture Mechanics XU Y LLIU T TZHANG W Set al Buffetinginduced fatigue damage assessment of a long suspension bridge J International Journal of Fatigue LI Z XCHAN T H TKO J M Fatigue analysis and life prediction of bridges with structural health monitoring datapart Imethodology and strategy J International Journal of Fatigue British Standard Institute BSI BS5400part 10 steelconcrete and composite bridgescode of practice 1American Society of Civil Engineering ASCE Committee on fatigue and fracture reliabilitystructural safety and reliability of the structural divisionfatigue reliability J Journal of Structural Engineering-ASCE WIRSCHING P H Probabilistic structural mechanics handbookprobabilistic fatigue analysism New YorkChapman and Hall CHUNG H C Fatigue reliability and optimal inspection strategies for steel bridgesd AustinUniversity of Texas at Austin004 4ZHOU Y Assessment of bridge remaining fatigue life through field strain measurement J Journal of Bridge Engineering-ASCE LEMAITRE J A course of damage mechanicsm BerlinSpringer-Verlag KACHANOV L M Introduction to continuum damage mechanicsm Dordrecht Martinus Nijhoff KRAJCINOVIC DLEMAITRE J Continuum damage mechanics theory and applicationsm Vienna Springer LI Z XCHAN T H TKO J M Fatigue damage model for fatigues LondonBSI1980 1BHATTACHARYA BELLINGWOOD B Continuum damage mechanics analysis of fatigue crack initiation J International Journal of Fatigue CHEN Z WXU Y LXIA Yet al Fatigue analysis of long-span suspension bridges under multiple loading case study J Engineering Structures WIRSCHING P H Fatigue reliability for offshore structures J Journal of Structural Engineering- ASCE CHEN Z W Fatigue and reliability analyses of multiload suspension bridges with WASHMSD Hong KongThe Hong Kong Polytechnic University010 16RIGHINIOTIS T D Effects of increasing traffic loads on the fatigue reliability of a typical welded bridge detail J International Journal of Fatigue

Copyright. Hsin-Yang Chung

Copyright. Hsin-Yang Chung Copyright by Hsin-Yang Chung 2004 The Dissertation Committee for Hsin-Yang Chung Certifies that this is the approved version of the following dissertation: FATIGUE RELIABILITY AND OPTIMAL INSPECTION STRATEGIES