Fatigue assessment of multi-loading suspension bridges using continuum damage model

International Journal of Fatigue 40 (2012) 27–35

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Fatigue assessment of multi-loading suspension bridges using continuum damage model You-Lin Xu a,⇑, Zhi-Wei Chen a,b, Yong Xia a a b

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Department of Civil Engineering, Xiamen University, Xiamen, China

a r t i c l e

i n f o

Article history: Received 21 August 2011 Received in revised form 26 December 2011 Accepted 4 January 2012 Available online 24 January 2012 Keywords: Fatigue assessment Suspension bridges Wind loading Railway loading Highway loading

a b s t r a c t Long-span steel suspension bridges carrying both highway and railway have been built in wind-prone regions. The fatigue assessment of such bridges under the combined action of railway, highway, and wind loading represents a challenging task in consideration of uncertainties in both fatigue loading and fatigue resistance. This paper presents a framework for fatigue assessment of a long-span suspension bridge under combined highway, railway, and wind loadings using a continuum damage model. The continuum damage model (CDM) is first established based on continuum damage mechanics with an effective stress range and an effective nonlinear accumulative parameter to represent all of the stress ranges within a daily block of stress time history of the bridge. The CDM is then applied to estimate damage accumulation of the Tsing Ma suspension bridge at fatigue-critical locations, and the results are compared with those estimated by the linear Miner’s model. A limit state function for fatigue reliability analysis based on CDM is also defined by introducing proper random variables into CDM. The Monte Carlo simulation (MCS) is then adopted to generate the random variables and to calculate failure probability. Finally, the failure probabilities of the Tsing Ma Bridge at the end of 120 years are estimated for different loading scenarios. The results demonstrate that the fatigue condition of the Tsing Ma Bridge at the end of its design life depends on loading scenarios. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Long-span steel suspension bridges carrying both highway and railway have been built in wind-prone regions. The fatigue assessment of such bridges under the combined action of railway, highway, and wind loading represents a necessary but challenging task. The fatigue assessment of a multi-loading suspension bridge shall be based on multiple loading-induced stress response time histories rather than the simple summation of fatigue damage induced by individual loading. As a result, databases of railway, highway, and wind loading shall be built and the corresponding stress time histories shall be generated in different ways because of different properties of loading type. Also, given that a great number of multiple loading-induced stress response time histories are required for a complete fatigue assessment, a computationally efficient approach for dynamic stress analysis and an engineering procedure for determining fatigue-critical locations shall be developed. Furthermore, fatigue damage accumulation involving fatigue crack initiation and growth is actually a nonlinear process during the service life of a bridge [1,2]. The fatigue damage accumulation also ⇑ Corresponding author. Tel.: +852 2766 6050; fax: +852 2365 9291. E-mail addresses: [email protected] (Y.-L. Xu), [email protected] (Z.-W. Chen), [email protected] (Y. Xia). 0142-1123/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2012.01.015

involves many uncertainties in both fatigue loading and fatigue resistance [3,4]. For a bridge subjected to multiple types of loading, uncertainties become more complicated, making the fatigue assessment of such a bridge difficult to be performed. The Miner’s rule is widely used in civil engineering for fatigue damage and reliability analysis of steel structures [4,5]. However, the Miner’s rule is a linear damage model and does not address actual physical mechanism of fatigue crack initiation and growth. The Miner’s rule also does not consider the fatigue loading sequence effect, leading to either overly optimistic or pessimistic results [6]. Fatigue damage models based on continuum damage mechanics have been recently proposed in the field of engineering mechanics to deal with the mechanical behavior of a deteriorating medium at the continuum scale [7–9]. These fatigue damage models are highly nonlinear in terms of damage evolution. Some of these models were also tried by some researchers to estimate fatigue damage of long-span suspension bridges under single type of loading [10,11], but none of these investigations refer to fatigue damage assessment of suspension bridges under multiple types of loading and fatigue reliability analysis with uncertainties. In this connection, this paper presents a framework for fatigue assessment of a long-span suspension bridge under combined highway, railway, and wind loadings using a continuum damage model. The continuum damage model (CDM) is first established

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based on continuum damage mechanics with an effective stress range and an effective nonlinear accumulative parameter to represent all of the stress ranges within a daily block of stress time history of the bridge. The CDM is then applied to estimate damage accumulation of the Tsing Ma suspension bridge at fatigue-critical locations, and the results are compared with those estimated by the linear Miner’s model. A limit state function for fatigue reliability analysis based on CDM is also defined by introducing proper random variables into CDM. One of the random variables is the daily sum of m-power stress ranges, and its probability distribution is determined based on the measurement data recorded by the structural health monitoring system installed in the bridge. The Monte Carlo simulation (MCS) is then adopted to generate the random variables and to calculate failure probability. Finally, the failure probabilities of the Tsing Ma Bridge at the end of 120 years are estimated for different loading scenarios. 2. Continuum damage model

2.1. Basic theory of continuum damage mechanics The damage growth of a material is considered to consist of a progressive internal deterioration that causes some loss in the effective cross-section area that carries loads. In the continuum damage mechanics, the damage index D for isotropic damage is often defined as

Se S S

ð1Þ

where S is the nominal cross-section area and e S is the effective cross-section area when loss due to damage is taken into account. Based on thermodynamics and dissipation potential, the rate of damage for high-cycle fatigue can be expressed as a function of the accumulated micro-plastic strain, the strain energy density release rate, and the current state of damage [8]. The micro-plastic strain, which is often neglected in a low-cycle fatigue problems, and its accumulation must be considered for high-cycle fatigue damage, even if macro-plastic strain is not present [12]. In a onedimensional situation, the equation for the rate of fatigue damage D_ can be given as b0

r jr  r j _ hri D_ ¼ 0 B ð1  DÞa 2

ð2Þ

 ¼ rm is the mean stress over the stress cycle; the symbol hi where r denotes the McCauley brackets, where hxi = x for x > 0 and hxi = 0 for x < 0; and a, b0 , and B0 are the material parameters. Eq. (2) is a general constitutive model for high-cycle fatigue and can be integrated over time for cycles with different mean stresses and stress ranges.  ¼ 0 and the variation of (1  D)a in a single stress For instance, if r cycle is neglected, then integrating Eq. (2) over the cycle yields 0

dD 2rab þ3 ¼ 0 dN B ðb þ 3Þð1  DÞa

ð4Þ

0

where B ¼ 2b þ2 B0 . Integrating this equation over N stress cycles, in which rm = 0 and rr = constant, yields the damage accumulation. 1  ðaþ1Þ 0 ða þ 1Þ D¼1 1 ðrr Þðb þ3Þ N 0 Bðb þ 3Þ

ð5Þ

Many experiments have been conducted on the fatigue failure of structural details under different constant stress ranges, and S–N curves have been established based on these experimental results [14]. Given that far few experiments have been conducted to determine the parameters B, b0 , and a in Eq. (5), they can be expressed by using parameters of the S–N curves [10]. The number of stress cycles to failure Nf under the stress range rr (rr P rr ; 0;rr,0 is the fatigue limit) can be determined based on the British S–N curves [14].

Nf ¼ K 2 ðrr Þm

The continuum damage model (CDM) proposed in this study for fatigue assessment of long-span suspension bridges under multiple loadings is based on continuum damage mechanics with an effective stress range and an effective nonlinear accumulative parameter to represent all of the stress ranges within a daily block of stress time history of the bridge. Thus, the basic theory of the continuum damage mechanics is briefly introduced first for the sake of completion.

D¼

ðb0 þ3Þ

dD ½ðrr þ 2rm Þrr  2 ¼ dN Bðb0 þ 3Þð1  DÞa

ð3Þ

where ra is the amplitude of the stress cycle. By considering the mean stress effect [13], Eq. (3) can be rewritten as [12]:

ð6Þ

Eq. (5) can then be written as

h i Bðb0 þ 3Þ 0 Nf ¼ 1  ð1  Df Þðaþ1Þ rðb þ3Þ ða þ 1Þ r

ð7Þ

where the damage at failure, Df, is an intrinsic material property that is dependent on the durability of the material [1,15]. As the values of a adopted for different amplitudes of stress ranges occurring on a bridge are deemed to be sufficiently large to make ½1  ð1  Df Þðaþ1Þ  very close to one, the parameters B, b0 , and a can be expressed by m and K2 by comparing Eqs. (6) and (7).

(

b0 þ 3 ¼ m Bðb0 þ3Þ aþ1

ð8Þ

¼ K2

Substituting this equation into Eq. (5) gives 1  aþ1 1 D ¼ 1  1  ðrr Þm N K2

ð9Þ

To extend the approach to fatigue analysis under variableamplitude loading, Eq. (9) is expressed as

h ia 1þ1 DðkÞ ¼ 1  ð1  Dðk  1ÞÞðak þ1Þ  ðrr;k Þm =K 2 k

ð10Þ

where rr,k and ak are the stress range and nonlinear accumulative parameter for the kth stress cycle, respectively, and D(k) is the cumulative fatigue damage after the kth stress cycle. The equation can also be derived using stepwise iteration from the initial damage D(0) = 0.

8 > > > > > > < > > > > > > :

  1 Dð1Þ ¼ 1  1  ðrr ; 1Þm =K 2 a1 þ1 h ia 1þ1 Dð2Þ ¼ 1  ð1  Dð1ÞÞða2 þ1Þ  ðrr ; 2Þm =K 2 2 h



DðkÞ ¼ 1  ð1  Dðk  1ÞÞðak þ1Þ  ðrr;k Þm =K 2

ð11Þ

ia 1þ1 k

2.2. Nonlinear properties of fatigue damage accumulation Fatigue damage accumulation is nonlinear because it is induced by fatigue crack initiation at a very slow pace, then fatigue crack growth at a relatively fast pace, and finally fatigue failure suddenly. In the continuum damage mechanics-based fatigue damage model, the nonlinear accumulative parameter a is the parameter that controls the nonlinearity. Numerical simulations are carried out at this point to study the sensitivity of a to fatigue damage accumulation. In the simulations, the constant stress range rr = 80 MPa is used,

Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

K2 = 6.3  1011 and m = 3 are adopted for detail class F in British Standard [14], and different a (a = 0, 20, 60, and 100) are tried to compute the fatigue damage accumulation over a certain number of cycles until failure by using Eq. (9). Fig. 1 shows the fatigue damage accumulation curves for different a. It shows that the cumulative damage increases with the number of stress cycles applied. The cumulative rate of damage is small at the initial stage, but increases at late stages, showing a strong nonlinear trend. The nonlinearity of the cumulative damage curve increases with the growth of a. It is noted that when a = 0 is adopted, the nonlinear damage model simply becomes the linear Miner’s model. Therefore, a low a represents a conservative estimate of damage accumulation. Fig. 1 also shows that the number of cycles to failure is the same for different values of a. This is because the parameters in Eq. (9) are determined by the S–N curves.

Long-span suspension bridges carrying railway and highway in wind prone regions undergo variable-amplitude loading induced by the combined effect of multiple loads. Hence, cycle-by-cycle computation may be required to estimate the damage accumulation by using Eq. (11). However, given that the design life of a long-span suspension bridge is normally a 100 years or more, it is difficult to compute so many stress cycles occurring over such a long period using the cycle-by-cycle approach. To simplify the computation and make it feasible for practical application, the effective parameters shall be derived for a given block of stress time history, and the cumulative fatigue damage in the block can then be computed using these effective parameters rather than the complete set of parameters for all of the stress cycles in the block. For example, urban passenger trains often follow a regular timetable that is similar on different days, and highway traffic conditions are also similar across days. Thus, railway and highway traffic operates almost on a 1-day cycle, and the daily stress time history can thus be considered as one block. The effective stress range rre can be defined to represent all of the stress ranges in the daily block according to the two-slope S–N curves defined in British Standard [14], in which if the stress range level rr,i is less than the fatigue limit rr,0, it will diminish proportionally. The effective stress range in the kth block can thus be expressed as

rre;k ¼

(" N1 X

1

N2 X

ðrr ; 0Þ2

i¼1

m

ni ðrr;i Þ þ

i¼1

#

)m1

m

ni ðrr;i Þ þ 2 =Nr;k

ð12Þ

Cumulative fatigue damage

1.0 α = 20 α = 60 α = 100 α =0

0.8

where ni is the number of stress cycles applied at the stress range level rr,i, which is determined from the stress response time history in the kth block using the rainflow counting method. N1 and N2 are the number of stress range levels above rr,i and below rr,0, respectively. Nr,k is the total number of cycles in the kth block. m is a parameter in the S–N curve and it is taken as 3.0. The nonlinear accumulative parameter ae is another effective parameter that is also determined from the stress cycles in the daily block. As the daily block contains a large number of cycles in different stress range amplitudes, the value of a varies for different cycles. Given that the damage estimation would tend to conservative side if a low parameter a is adopted, the effective accumulative parameter ae,k is thus adopted as the minimum a in the kth block for simplicity, and it can be determined by the maximum stress range in the block according to the function given below [12]:

a ¼ ka rr þ a0

2.3. Continuum damage model used in this study

0.6

0.4

0.2

29

ð13Þ

where ka and a0 are the material parameters determined using the Woehler curves, which were obtained through uniaxial periodic fatigue tests under strain-controlled conditions and direct measurements of the material damage. ka and a0 are taken as 0.135 and 101.4 MPa in [12]. By introducing the effective parameters rre,k and ae,k into Eq. (10), the damage accumulation in the kth block can be estimated by the following continuum damage model.

h ia 1þ1 DðkÞ ¼ 1  ð1  Dðk  1ÞÞðae;k þ1Þ  Nr;k ðrre;k Þm =K 2 e;k

ð14Þ

2.4. Tsing Ma Bridge and FEM model The Tsing Ma Bridge in Hong Kong is selected for the case study. It is a suspension bridge with a main span of 1377 m, carrying a dual three-lane highway on the upper level of the bridge deck and two railway tracks on the lower level within the bridge deck (see Fig. 2). A Wind and Structure Health Monitoring System (WASHMS) has been installed and operated by Highways Department (HyD) of the Government of Hong Kong Special Administrative Region since 1997. A huge amount of data has been collected by the WASHMS. Considering the requirement of stress analysis of local bridge components, a complex structural health monitoring oriented finite element model (FEM) of the Tsing Ma Bridge was established and shown in Fig. 3 [16]. The bridge was modeled using a series of beam elements, plate elements, shell elements, and others. The FEM contains 12,898 nodes, 21,946 elements (2906 plate elements and 19,040 beam elements) and 4788 Multi-Point Connections. The FEM was also updated using the first 18 measured natural frequencies and mode shapes of the bridge from the WASHMS. It turned out that the updated complex FEM could provide comparable and credible structural dynamic modal characteristics. Based on the FEM, a computationally efficient engineering approach was also proposed for dynamic stress analysis of the bridge under railway, highway and wind loading. A computer program was written to implement the approach. The approach and the computer program were then validated through the comparison with typical daily strain responses measured at the critical locations by the WASHMS. Details can be found in the Ref. [19]. 2.5. Verification of continuum damage model

0.0 0

200

400

600

Stress cycle

800

1000

1200 3 (× 10 )

Fig. 1. Fatigue damage accumulation for different values of a .

Before the continuum damage model (CDM) expressed by Eq. (14) can be applied for fatigue assessment of long-span suspension bridge under multiple loadings, some verification is necessary. Since a Wind and Structural Health Monitoring System (WASHMS) has been installed in the Tsing Ma Bridge since 1997, there are

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Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

(a) Tsing Ma Bridge under railway, highway, and wind loading 455.0

300.0

1377.0 Ane Anemometer (6) Str -L Linear Strain Gauge (106) Str -R Rosette Strain Gauge (4)

Section B Ane (2) Ane (1)

Section P Ane (1)

Tsing Yi

Ma Wan Section D Section E: Str -L(29) Str -L(30) Str-R (2)

Section J: Ane (2)

Section L: Str -L(47) Str -R(2)

(b) Locations of strain gauges and anemometers Fig. 2. Tsing Ma Bridge and WASHMS.

Fig. 3. 3-D finite element model of Tsing Ma Bridge.

many stress time histories recorded by the WASHMS. As done by other researchers [17,18], this study also used the measurement data recorded by the WASHMS. Fig. 4 shows a typical daily multiple load-induced hot-spot stress time history recorded at one of the fatigue-critical locations of the bridge. The fatigue damage accumulation within this daily block is then computed using the cycle-by-cycle method and the effective parameter method, respectively, based on Eq. (11). The obtained results are compared with each other to see if the effective parameter method can yield reasonable damage estimation. Fig. 5 shows the stress range sequence determined from the daily stress time history (see Fig. 4) using the rainflow counting method. There are a total of 72,333 stress cycles in the sequence in this time history. The effective

stress range rre = 6.0 MPa is calculated from these cycles using Eq. (12). The maximum stress range in the block is 69.1 MPa, and the corresponding effective nonlinear accumulative parameter ae = 92.1. The fatigue damage accumulation within the block is calculated by using Eq. (11) with the step-by-step method and the effective parameter method. The results are shown in Fig. 6 and demonstrate that the final cumulative damage values calculated with the two methods are very close, although the result from the effective parameter method is slightly more conservative than that from the step-by-step method. This indicates that the effective parameter method, neglecting the load sequence effect on damage accumulation within a daily block, is reasonable and effective for estimating damage accumulation for a daily stress cycle interval.

Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

31

Fig. 4. A typical daily multiple load-induced stress time history.

70

Stress range (MPa)

60 50 40 30 20 10 0 0

10

20

30

40

50

60

70 (×10 ) 3

Stress cycle Fig. 5. Typical daily stress range sequence.

−6

(×10 )

Cumulative fatigue damage

0.30 Cycle-by-cycle method

0.25

Effective parameter method

0.20 0.15 0.10 0.05 0.00 0

10

20

30

40

50

60

3 70 (×10 )

Stress cycle Fig. 6. Fatigue damage accumulation within a daily block.

The effective parameter method or the continuum damage model (CDM) makes it possible to estimate the fatigue damage accumulation of a long-span suspension bridge under multiple loading.

3. Fatigue damage accumulation of Tsing Ma Bridge The fatigue damage accumulation over the design life of the Tsing Ma Bridge is an important determinant of inspection and maintenance regimes. For the Tsing Ma Bridge installed with the WASHMS (see Fig. 2), the loading conditions can be determined from measurement data over a long period. This provides an opportunity to identify the different loading conditions over the

service history of the bridge. The loading sequence across the entire design life of the bridge can thus be generated based on the actual measurement data for past loading and the assumptions of future loading. The aforementioned nonlinear damage model can then be used to estimate the fatigue damage accumulation of the Tsing Ma Bridge. As the design life of the Tsing Ma Bridge is 120 years, 120 years of dynamic stress time histories induced by railway, highway, and wind loadings were generated at six fatigue-critical locations based on the databases of railway, highway, and wind loadings reported by Chen et al. [19]. Fig. 7a defines the main components of the bridge deck while Fig. 7b shows one of the critical sections of the bridge deck where the critical Element E32123 is located. The six locations are Element E32123 at the top flange of the outer-longitudinal diagonal member close to the Ma Wan Tower, Element E34415 at the bottom flange of the outer-longitudinal bottom chord of the Tsing Yi Tower, Element E40056 at the top flange of the inner-longitudinal top chord of the Tsing Yi Tower, Element E40906 at the bottom flange of the inner-longitudinal bottom chord of the Tsing Yi Tower, Element E39417 at the bottom flange of the T-section of the railway beam of the Tsing Yi Tower, and Element E55406 at the top flange of the bottom web of the cross frame close to the Tsing Yi Tower. More detailed information can be found in [19]. A stable traffic loading was first assumed in this section, with no increases or decreases over the 120 years, and the linear and exponent traffic growth cases will be discussed in Section 4. Based on these time histories, the Miner’s model and the CDM model are applied to compute the fatigue damage accumulations over the design life of the bridge at daily intervals, and the results are compared with each other. The cumulative damage is updated after each daily block and computed using Eq. (14). The parameters of Nr, rre, and ae for each daily block are determined from the corresponding daily stress time history. The other parameters are taken as K2 = 6.3  1011 and m = 3. The fatigue damage begins with D(0) = 0. Fig. 8 shows the fatigue damage accumulation curves computed at one of the fatigue-critical locations by using the Miner’s model and the CDM model. The main difference between the two curves is that the cumulative damage curve calculated by the CDM model is nonlinear, with a low cumulative rate initially but a very high cumulative rate at the end of the design life, whereas that from the Miner’s model is linear. Given that structural deterioration leads to a loss of effective cross-section area and accelerates damage accumulation, the nonlinear cumulative damage curve is more reasonable than the linear curve. The fatigue life (cycles to failure) estimated by the two models is, however, almost the same. This is because the same S–N curves are introduced into the two models, which makes the numbers of cycles to failure (or fatigue lives) under a constant stress range are same, as shown in Fig. 1. Furthermore, under the condition of no traffic growth, the effective stress ranges from daily stress

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Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

Innerlongitudinal top chord

Top web cross frame

Diagonal truss Vertical post Top chord

Cross bracings (bottom outer)

Outerlongitudinal vertical post

Railway beam Bottom web cross frame Cross bracings (bottom center)

Outerlongitudinal bottom chord

Innerlongitudinal bottom chord

Outerlongitudinal Diagonal truss

(a) Bridge deck components

455.0

50

300.0

1377.0

E32403(T)

E32123(T)

Maximum stress range (MPa)

40

Tsing Yi

Ma Wan 30

Diagonal member of outer-longitudinal trusses

20

10

0 0

500

1000

1500

2000

Location (m)

(b) Maximum stress ranges of diagonal members in north outer-longitudinal trusses Fig. 7. Bridge deck components and critical sections.

responses are also close in different days. The fatigue life at the fatigue-critical location is very close to the bridge’s design life. Similar conclusions can be made from the cumulative fatigue damage curves for the other fatigue-critical locations. Table 1 lists the 120-year cumulative fatigue damage estimated at the six fatigue-critical locations using the CDM model and the Miner’s model, and shows that there is a significant difference between the results estimated by two models except for location E32123, at which the condition of fatigue failure has been reached at the end of 120 years. At the other fatigue-critical locations, the damage estimated by the CDM model is much smaller than that estimated by the Miner’s model. This is because the damage D is a ratio of the service life to the final fatigue life in the Miner’s model, but is the ratio of the lost area due to damage to the nominal cross-section area in

the CDM model. This means that the damage calculated by the CDM model is closer to the actual fatigue state.

4. Fatigue reliability analysis based on CDM 4.1. Framework To consider the uncertainties exist in the nonlinear process of damage accumulation, this section proposes a framework for fatigue reliability analysis based on the CDM model. The first step of fatigue reliability analysis is to define a limit state function that adequately describes the relationship between fatigue resistance and fatigue loading for a fatigue-sensitive structural member. As

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Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

Smr;k ¼

1.0

Cumulative fatigue damage

Miner's model

0.6

0.4

0.2

0.0 0

20

ni ðrr ; iÞm þ

i¼1

CDM model

0.8

N1 X

40

60

80

100

120

Time (year) Fig. 8. Fatigue damage accumulation curves for location E32123.

Table 1 120-year cumulative damage at the fatigue-critical locations. Fatigue damage model

Fatigue-critical location E32123

E34415

E40056

E40906

E55406

E39417

CDM Miner’s

1.00 1.02

0.27 0.96

0.10 0.68

0.06 0.54

0.05 0.41

0.06 0.58

gðXÞ ¼ DðkÞ  Dðk  1Þ

ð15Þ

h ia 1þ1 DðkÞ ¼ 1  ð1  Dðk  1ÞÞae;k þ1  Smr;k =ðK DÞ e;k

ð16Þ

N2 X

ðrr ; 0Þ2

i¼1

ni ðrr ; iÞmþ2 ¼ Nr;k ðrre;k Þm

ð17Þ

where Smr;k ¼ Nr ; kðrre ; kÞm is the daily sum of m-power stress ranges. The fatigue damage accumulation index, D, is regarded as a random variable modeled by a lognormal distribution with a mean value lD of 1.0 and a standard deviation rD of 0.3 [20]. This random variable actually accounts for the uncertainty associated with the damage to failure when fatigue model is applied to deal with problems involving variable-amplitude stress ranges. The fatigue detail coefficient K is assumed to be a lognormal distribution, and its mean value and standard deviation for different fatigue detail classes are obtained from the relevant information provided in Appendix A of British Standard [14]. It shall be noted that although the British Standards BS5400 and BS76058 are outdated, they are still referred by this study because they were used in original design of the Tsing Ma Bridge and some information could be accessed by the authors. Based on the defined limit state function, the failure probability Pf of a structural member or system can be estimated. In this connection, the Monte Carlo simulation (MCS) is adopted, which involves the repeated drawings of random samples for all of the random variables involved in the limit state function and simply checks whether a new ‘‘failure’’ or ‘‘non-failure’’ has resulted (or, equivalently, whether the limit state function has a value of less than zero). Base on N simulations, Pf can be estimated using the following equation:

Pf  indicated by Eq. (14), the fatigue limit state can be defined as the point at which the cumulative fatigue damage stops increasing when more loads are applied, such as D(k) 6 D(k  1). In other words, it is the point at which the structure cannot suffer any further damage. For computational simplicity, the continuous process of damage accumulation is reduced to a discrete process and updated at daily intervals, as expressed in Eq. (14). Distinct from deterministic fatigue analysis, the randomness in both the fatigue loading and fatigue resistance shall be considered in fatigue reliability analysis. From fatigue loading aspect, given that the multiple loading-induced stress time history is a stochastic process, the number of stress cycles at a given stress range level is random. Furthermore, the operation of urban passenger trains often follows a daily timetable, the cycle of railway and highway traffic is close to 1 day. As a result, the daily sum of m-power stress ranges Smr,k (k = 1, . . ., Nb, where Nb is the total number of days in the time period concerned) is treated as a random variable in terms of fatigue loading. From fatigue resistance aspect, the fatigue damage accumulation index D and the fatigue detail coefficient K are regarded as random variables to replace the parameter K2 = KD. According to the findings of the previous numerical simulations, different ae should be used to consider the nonlinear damage accumulation. The loading condition is assumed to remain constant over a given period, and daily stochastic stress time histories from this period are generated based on a constant probabilistic loading model. Thus, ae,k is assumed to remain constant during the same period but to change across different periods. Taking these random variables and constants into account, the limit state function for fatigue reliability analysis based on the CDM model is defined as

1

N 1 X I½gðXi Þ < 0 N i¼1

ð18Þ

where g(Xi) is the value of the limit state function in the ith simulation. I[g(Xi) < 0] = 1 if g(Xi) < 0, and I[g(Xi) < 0] = 0 otherwise; Xi = {Xi,1, Xi,2, . . ., Xi,n} is a vector of n random variables; Xi,1 and X i,2 are the random variables D and K; and Xi,3  Xi,n are the random variables Smr,k (k = 1, . . ., Nb, where Nb is the number of daily blocks). 4.2. Probability distribution of the daily sum of m-power stress ranges The probability distribution of the daily sum of m-power stress ranges can be estimated for a long-span suspension bridge installed with the WASHMS and under multiple loading through the following three steps: 1. Establish probabilistic models of railway, highway, and wind loads based on the measured load data. 2. Generate multiple loading-induced daily stochastic stress responses at fatigue-critical locations by using the finite element method and the MCS together with the loading probabilistic models established. 3. Estimate the probability distribution of Smr from the samples, which are computed based on the generated daily stochastic stress responses, and then estimate its distribution in the period concerned based on the assumed future loadings and traffic growth patterns. Following the above three steps, Chen et al. [21] estimate the probability distributions of Smr at all the fatigue-critical locations of the Tsing Ma Bridge under the combined action of railway, highway, and wind loadings. Fig. 9 shows the histogram and theoretical distributions of Smr at the fatigue-critical location E32123. The histogram was estimated based on the 200 generated samples, and the theoretical density function was modeled as a normal distribution. The theoretical density function matches the histogram quite well. The mean value and standard deviation of the fitted normal distribution are 1.15  1012 and 3.22  108. No traffic growth and

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Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

Another two growth patterns are also assumed, both of which are exponential types [22]. Simple algebraic considerations lead to the following expressions.

value and STD of D are lD = 1.0 and rD = 0.3. The probability distributions of Smr remained constant within a given month, but changed across different months for the three traffic growth patterns. The distribution parameters of Smr,k (k = 1, . . ., 120  365) in different months across 120 years at the various fatigue-critical locations under different traffic growth patterns were determined by Chen et al. [21]. Correspondingly, a constant ae is adopted here for the no traffic growth pattern, whereas ae changes across months for the three growth patterns. Secondly, the random variables of K, D, and Smr,k are randomly sampled from the corresponding distributions to yield a random variable vector. Thirdly, the vector is substituted into the limit state function to calculate the value of g(Xi) in Eq. (15). Fourthly, Nf is increased by one if g(Xi) < 0. Finally, the steps are repeated N times, and the probability of failure Pf is computed by Nf/N. Note that N should be sufficiently large for a good estimate of Pf. Nf P 10 is adopted for this estimation. Using this procedure, the fatigue failure probabilities at the end of 120 years are estimated at the six fatigue-critical locations of the Tsing Ma Bridge, and the results are listed in Table 2. The highest fatigue failure probability is at E32123 and the lowest is at E55406. A failure probability of 2.3% is recommended in British Standard [14], above which the concerned structural components are regarded as in danger. The highest failure probabilities at the end of 120 years at the six fatigue-critical locations under the current traffic conditions without growth are close to the reference (design) failure probability, which implies the health condition of the bridge to be satisfactory in terms of fatigue. Special attention should be paid to future traffic growth, because it may lead to a failure probability at the end of 120 years that is much greater than the reference level.

X ¼ X 0 t 6 T r ; X ¼ X 0 ð1 þ nag Þt=ðnT r Þ t > T r

5. Concluding remarks

-7

x 10

Computed data

5

Probability density

Theoretical probability density

4

3

2

1

0

1.4

1.5

1.6

1.7

1.8

Daily sum of m-power stress ranges

1.9 7

x 10

Fig. 9. Probability density function of the daily sum of m-power stress ranges.

three traffic growth patterns (Linear, Exp-1, and Exp-2) were investigated. The first growth pattern is assumed to take a linear pattern in which no growth takes place in the first Tr years, but does take place from Tr to Tt in a linear fashion at a constant growth rate ag. The growth function is given as

X ¼ X 0 t 6 T r ; X ¼ X 0 ð1 þ ag tÞ t > T r

(

ðLinearÞ

"

X ¼ X 0 t 6 T r ; X ¼ X 0 1 þ ð1 þ nag Þ 1  t > Tr

ð19Þ

ðExp  1Þ



1 1 þ nag

ð20Þ

t=ðnT r Þ #)

ðExp  2Þ

ð21Þ

Based on the assumed future loadings and traffic growth patterns, the probability distribution of the sum of m-power stress ranges within the 120 year deign life of the Tsing Ma Bridge was estimated. The further details can be found in Chen et al. [21]. 4.3. Reliability analysis results In this section, the fatigue reliability (or failure probability) at the six fatigue-critical locations of the Tsing Ma Bridge are solved. Firstly, the probability distributions of the random variables are determined for the reliability analysis. The mean value and STD of K for detail F are lK = 1.73  1012 and rK = 0.52  1012. The mean

The framework for fatigue damage accumulation and reliability analysis of a long-span suspension bridge under combined highway, railway, and wind loadings by using a continuum damage model has been established and applied to the Tsing Ma Bridge in Hong Kong. Major conclusions drawn from this investigation can be summarized as follows: (1) The continuum damage model (CDM) proposed in this study based on continuum damage mechanics with an effective stress range and an effective nonlinear accumulative parameter to represent all of the stress ranges within a daily block of stress time history of the bridge was verified through the comparison with field measurement data. (2) The application of the CDM to estimate the fatigue damage accumulation of the Tsing Ma Bridge at its fatigue critical locations shows that the main difference between the CDM

Table 2 Fatigue failure probabilities at the end of 120 years at the fatigue-critical locations. Load case

Growth pattern

Fatigue-critical location E32123

E34415

E40056

E40906

E55406

E39417

CL

Constant

0.024

0.017

6  103

5  103

8  106

2  105

CL ? FL1

Linear Exp-1 Exp-2

0.21 0.17 0.24

0.19 0.15 0.23

0.03 0.02 0.05

0.03 0.02 0.04

4  103 2  103 8  103

6  103 3  103 0.01

CL ? FL2

Linear Exp-1 Exp-2

0.21 0.17 0.24

0.12 0.10 0.15

5  103 5  103 6  103

6  103 5  103 8  103

3  104 3  104 5  104

3  104 3  104 4  104

Note: CL – current loading, FL1 – future loading 1, FL2 – future loading 2.

Y.-L. Xu et al. / International Journal of Fatigue 40 (2012) 27–35

and the Miner’s model is that the cumulative damage curve calculated by the CDM model is nonlinear, with a low cumulative rate initially but a very high cumulative rate at the end of the design life, whereas that from the Miner’s model is linear. The fatigue life (cycles to failure) estimated by the two models is, however, almost the same. This is because the same S–N curves are introduced into the two models, which makes the numbers of cycles to failure (or fatigue lives) under a constant stress range are same. (3) A framework for fatigue reliability analysis based on the CDM model has been established for long-span suspension bridges under multiple loadings. The results show that the highest failure probabilities at the end of 120 years at the six fatigue-critical locations under the current traffic conditions without growth are close to the reference (design) failure probability. Special attention should be paid to future traffic growth, because it may lead to a failure probability at the end of 120 years that is greater than the reference level.

Acknowledgements The authors wish to acknowledge the financial supports from the Research Grants Council of the Hong Kong (PolyU 5327/08E), The Hong Kong Polytechnic University (PolyU-1-BB68), and the National Natural Science Foundation of China (NSFC-50830203 and NSFC-51108395). Sincere thanks should go to the Highways Department of Hong Kong for providing the authors with the field measurement data. Any opinions and concluding remarks presented in this paper are entirely those of the authors. References [1] ASCE. Committee on fatigue and fracture reliability of the committee on structural safety and reliability of the structural division, fatigue reliability: 1– 4. J Struct Eng (ASCE) 1982; 108: 3–88.

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