Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3247by individual loading are generated based on the databases. 9. Compute the cumulative fatigue damage Dk = Dk−1 + DkThe corresponding stress time histories due to the combined and the cumulative service time tk = tk−1 + t in the kth timeaction of multiple loading are compiled. Fatigue analysis is then step; andperformed to compute the cumulative fatigue damage over the 10. Move to the next time step and go from step (7) to the end ofdesign life of 120 years. The cumulative fatigue damage induced by the design life.individual loading and the damage magnification due to multiple Fatigue damage accumulated in the time step can be calculatedloading are finally investigated. using the Palmgren–Miner’s rule based on the two-slope S–N curves in 2. Establishment of framework D= DH + DL (1) To establish a framework for fatigue analysis of long-span wheresuspension bridges under combined action of railway, highway,and wind loading, some key issues need to be considered. First, − ni σrmi N1 ,given that a great number of stress time histories caused by DH = if σr ,i ≥ σr ,0 and (2) K2multiple loading are required for a complete fatigue assessment i=1of a long-span suspension bridge, it is desirable to develop a − ni σrmi+2 N2 ,computationally efficient engineering approach for dynamic stress DL = if σr ,i < σr ,0 (3)analysis. Second, as a long-span suspension bridge consists of a i=1 K2 σr2 0 ,large number of bridge components, it is not only impossible butalso unnecessary to carry out fatigue analysis for all the structural where K2 and m are constants relevant to the fatigue detail; K2 iscomponents. The fatigue-critical locations should be properly determined from constant amplitude experiments correspondingdetermined for fatigue analysis. Third, the design life of the bridge to a probability of failure of 2.3%; ni is the applied number of stressconcerned should be designated before the calculation of wind- cycles at the stress range level σr ,i ; N1 and N2 are the number ofinduced stress responses for fatigue analysis, because the wind stress range levels σr ,i in the stress time histories above and belowintensity taken into consideration in the fatigue analysis is related σr ,0 , respectively, and σr ,0 is the constant amplitude fatigue limit,to the design life. Fourth, databases should be established in order which is defined as N = 107 .to generate the stress response time histories of the bridge over itsdesign life. Databases of railway, highway, and wind loading shall 3. An engineering approach for dynamic stress analysisbe built in different ways because of different properties of loadingtype. Wind-induced stress responses are computed in one hour 3.1. Simplifications used in the engineering approachto build a database for fatigue analysis. As urban passenger trainsoften follow a regular timetable that is similar on different days, In the previous work, the authors proposed a coupled dynamicrailway-induced stress time histories are computed in one day, and approach for dynamic stress analysis of long-span suspensiondaily time histories are used to compose the database. The database bridges under combined railway, highway, and wind loading .of highway stress time histories is also composed of daily time Though the coupled dynamic approach provides an accuratehistories, as highway traffic conditions among different days are estimation of bridge dynamic stresses, the complexity of thefound to be similar. Fifth, multiple loading-induced fatigue damage framework makes computation very time consuming. It isshould be calculated based on the stress responses induced by impractical to apply the coupled dynamic approach to fatiguemultiple types of loading rather than the summation of damage analysis of a long-span suspension bridge. In this regard, twoinduced by individual loading. Fatigue analysis shall therefore major simplifications are adopted here to simplify the coupledbe applied directly to the multiple loading-induced stress time dynamic approach and lead to the engineering approach basedhistories, which is the superposition of stress responses induced by on the features and properties of long-span suspension bridgesthree individual loadings. Finally, it is recommended that the data under normal operation condition with a trade-off betweenmeasured be adopted in the computation of fatigue damage as far computational accuracy and efficiency.as possible to represent better the real conditions of the bridge. The first major simplification is to neglect coupled effects ofTaking the above issues into consideration, a framework for the multiple load-induced dynamic stresses. This is because windfatigue analysis of a long-span suspension bridge under multiple speed is normally not too high when vehicles are running on thetypes of loading within the design life is proposed and summarized bridge. Under extreme wind conditions, such as when a strongas follows: typhoon is blowing, bridge traffic management systems shall come 1. Develop a computationally efficient engineering approach for into effect and the bridge will be closed to traffic. Therefore, it is dynamic stress analysis; reasonable to assume that the coupled effects of dynamic stress 2. Designate the design life of the suspension bridge concerned; responses of the bridge induced by railway, highway, and wind 3. Determine the fatigue-critical locations of key structural loading are insignificant under normal operation condition and components of the bridge; that the bridge motions induced by railway, highway, and wind 4. Establish databases of the dynamic stress responses at the loading are considered to be independent of each other. As a fatigue-critical locations induced by railway, highway, and result, the bridge stress response at a given point induced by the wind loading, respectively, using an engineering approach; combined effects of three types of loading can be approximately 5. Generate the multiple loading-induced dynamic stress time obtained by the synchronous superposition of stress responses histories at the fatigue-critical locations within the design life induced by individual loadings. based on the databases established in step (4); σb = σrb + σhb + σwb (4) 6. Set the initial damage D0 = 0 and time step t; 7. Count the number of stress cycles at different stress range where σrb , σhb , and σwb are the bridge stress responses induced by levels from the multiple load-induced stress time history in the railway, highway, and wind loading, respectively. kth time step using the rainflow counting method ; Another major simplification is to neglect the dynamic 8. Compute the increase in the level of fatigue damage Dk in the magnification related to vehicle dynamics. This is because the self- kth time step for a given fatigue-critical location; weight of a long-span suspension bridge carrying both trains and
3248 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256road vehicles is much larger than the weight of a train and/ora series of road vehicles. Furthermore, this study concerns thedynamic stress response of a long-span suspension bridge ratherthan the safety of vehicles. As a result, trains and road vehiclescan be simplified as a series of moving forces on the bridge deck.Moreover, through the analysis of three resonance conditions, itis found that the impact factor of a long-span cable-supportedbridge under a series of moving forces is often small . Basedon the above reasons, the bridge stress responses induced bytrains and road vehicles can be calculated based on a series of Fig. 1. Distribution of dynamic strain gauges and anemometers in Tsing Ma Bridge (unit: m).static forces and stress influence lines. Therefore, bridge stressresponses induced by railway and highway vehicles are calculatedusing the stress influence lines without considering dynamic 5. Calculate the railway load-induced stress σrb by the triplemagnification in this study, but wind-induced stress responses of summation of the product of the stress influence coefficienta long-span suspension bridge are, however, computed based on Φk,ij and axle load fk,ij ; andthe aerodynamic analysis. 6. Move to the next time instant and go from step (2) to the end of the given duration of stress responses.3.2. Dynamic stress analysis using the engineering approach To consider the dynamic stresses induced by road vehicles running along the bridge on different traffic lanes, highway loading Based on the two major simplifications proposed in the stress influence line for each traffic lane should be established, andpreceding section, the engineering approach for dynamic stress highway loading should be determined based on the measuredanalysis of long-span suspension bridges under multiple loading road vehicle data. For instance, the highway loading of a typicalcan be implemented by the following four steps: (1) analysis of four-axle road vehicle can be represented by four vertical forcesrailway-induced bridge dynamic stress based on stress influence with each load coming from one axle. To determine not onlylines; (2) analysis of highway-induced bridge dynamic stress the highway loading due to a given road vehicle but also thebased on stress influence lines, (3) analysis of wind-induced road vehicle flow running along the bridge, the highway loadingdynamic stress using buffeting theory; and (4) combination of information should include at least the number and types of roadthe stress responses induced by multiple types of loading by the vehicles, traffic lane in use, arrival time, heading direction, runningsuperposition method in the time domain. The procedures for the speed, axle number, axle weight, and axle spacing. Underlyingfirst three steps are presented as follows. assumptions include a constant speed and no switching of the To determine railway-induced dynamic stress responses of a traffic lane for a given road vehicle running along the bridge. Thelong-span suspension bridge, the stress influence lines should be computational procedure of the stress time history under highwayestablished. To derive the stress influence lines for a given fatigue- loading can be derived in a similar way to that under railwaycritical location, the stress response at the designated location due loading.to a unit vertical force moving along the railway tracks from one Long-span suspension bridges that are built in wind-proneend of the bridge to the other end is computed. The abscissa of regions suffer considerable buffeting-induced vibration. Therefore,the stress influence line denotes the position of the unit vertical wind-induced dynamic stress responses should also be considered.force in the longitudinal direction of the bridge, and the ordinate of Wind-induced dynamic stress response time histories can bethe stress influence line, the so-called stress influence coefficient, computed using a step-by-step procedure. In the first step, windΦ , is the stress response induced by the unit vertical force at characteristics in a given time period, such as one hour or tenthe corresponding position. Railway loading is then determined minutes, are identified from wind data collected by anemometersin terms of a series of moving vertical forces. For example, the installed at the bridge site. In the second step, the stochasticrailway loading for an eight-car train with 32 wheel-sets can be wind velocity at the simulation points along the bridge deck andrepresented by 32 vertical forces, with each force coming from one the normal mean wind speed in the time period of concern arewheel-set. The railway loading information is used to determine generated based on the wind characteristics acquired from thethe railway loading for a given train and to simulate the railway measured wind data. The buffeting and self-exciting forces on thevehicle flow running along the bridge. Such information can be surface of the bridge deck are then computed . In the third step,obtained from the train data recorded at the bridge site. The the wind-induced stress responses in the time period of concerninformation includes at least the number of trains, the number are computed at the given stress output points using an integrationand types of railway vehicles in a train, arrival instant, running method. The procedure then moves to the next time period andspeed, heading direction, railway track in use, number of bogies, goes from the first step to the end of the given duration of stressbogie weight, and bogie spacing. Underlying assumptions include responses.a constant speed of a typical train running across the bridge ona given railway track. The computational procedure of the stress 3.3. Verification of the engineering approachtime history under railway loading is summarized as follows.1. Establish the database of railway loading stress influence lines The Tsing Ma Bridge in Hong Kong is a suspension bridge with for a given stress output point; an overall length of 2132 m (see Fig. 1). The bridge deck is a2. Update the train information at the instant t, which includes the hybrid steel structure and carries a dual three-lane highway on number of railway vehicles of a train and wheel-set locations of the top deck and two railway tracks and two carriageways on a the railway vehicle; lower level within the bridge deck. The dynamic stress responses3. Determine the vertical loading fk,ij due to the ith wheel-set in of the Tsing Ma Bridge are mainly induced by railway, highway, and the jth railway vehicle on the kth railway track using the train wind loading. Both loading conditions and bridge responses are information obtained; monitored by the WASHMS installed on the bridge. Therefore, this4. Determine the stress influence coefficient Φk,ij due to the ith bridge is taken as an example to verify the computational accuracy wheel-set in the jth railway vehicle on the kth railway track and efficiency of the engineering approach. The information on the using the stress influence line database; train was converted from the strain response time history recorded
Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3249 The normal mean wind speed is 11.91 m/s. The standard deviation of turbulent wind is 1.310 m/s in the horizontal direction and 0.679 m/s in the vertical direction. The integral length scales are 256.7 m in the horizontal direction and 40.8 m in the vertical direction. To use the engineering approach to compute 140-s dynamic stress time history, the railway and highway loading stress influence lines of the Tsing Ma Bridge should be established. For each of the two strain gauges specified, two stress influence lines corresponding to the two railway tracks are established. Six Fig. 2. Two strain gauges installed at Section L and used in case study. stress influence lines corresponding to the six highway traffic lanes are also established. All stress influence lines are generated by structural analyses based on the finite element model of the bridge. The details on how to generate these influence lines are not given because of the limitation of paper length. In the computation of stress response, the train and road vehicle information is updated at each time step, and the length of the time step t is 0.02 s. The acquired wind characteristics are adopted to generate the stochastic wind velocity field of the entire bridge deck, and Fig. 3. 3-D Finite element model of Tsing Ma Bridge . then the buffeting and self-excited forces on the bridge deck are estimated. The stress time histories under wind loading at theby a special set of strain gauges arranged under the railway beams. locations concerned are computed. Based on the stress responsesThe information on heavy road vehicles was recorded by dynamic induced by railway, highway, and wind loading, respectively, theweigh-in-motion (WIM) stations. Wind data were collected by the multiple load-induced stress responses are computed using theanemometers installed at both bridge deck and towers. There are a superposition method. The result computed by the engineeringtotal of 110 dynamic strain gauges installed at three deck sections method is also plotted in Fig. 4 for comparison. The figure shows(see Fig. 1). Two of them installed at Section L are selected in this that the stress time histories computed using the engineeringstudy (see Fig. 2). approach match well with those from the coupled dynamic Considering the requirement of stress analysis of local bridge approach. The relative differences in the peak-to-peak stresscomponents, a complex structural health monitoring oriented responses (the response obtained by the coupled dynamic methodfinite element model (FEM) of the Tsing Ma Bridge was established minus that by the engineering approach, divided by one predictedand shown in Fig. 3 . The bridge is modelled using a series by the coupled dynamic method) at the location of strain gaugesof beam elements, plate elements, shell elements, and others. The SP-TLS-02 and SS-TLS-12 are 16.1% and 5.4%, respectively. Thefinite element model contains 12,898 nodes, 21,946 elements and 16.1% error is the worst case and this error will not exaggerate4788 Multi-Point Connections (MPC). The finite element model the final fatigue damage because fatigue damage depends onwas also updated using the first 18 measured natural frequencies a large number of stress ranges rather than peak stresses. Theand mode shapes of the bridge from the WASHMS. It turned out results demonstrate that the level of computational accuracy of thethat the updated complex finite element model could provide engineering approach is acceptable.comparable and credible structural dynamic modal characteristics. In addition to the computational accuracy, the computational To validate the computational accuracy of the engineering efficiency is also an important factor for the engineering approach.approach, the stress responses induced by multiple types of Most of the trains running across the bridge follow a timetable on aloading computed using the engineering approach are compared daily basis; thus the cycle of railway loading is close to one day. Aswith those calculated using the coupled dynamic approach. A 140-s hundreds of trains and thousands of road vehicles run across thedynamic stress time history was computed by the coupled dynamic bridge every day, the computational efficiency of the engineeringmethod , as shown in Fig. 4. It is used as a reference for approach is tested for one day only. The measured train, roadcomparison. During this period, there was one train running on vehicle, wind, and strain data collected on 19 November 2005 arethe north track heading towards the Hong Kong Island and 29 used for dynamic stress computation and comparison. This day wasroad vehicles weighing over four tons running along the bridge. chosen as the wind was relatively strong and the traffic was heavy. a 8 b 10 Coupled dynamic method Coupled dynamic method 6 Engineering method Engineering method 5 4 2 0 Stress (MPa) Stress (MPa) 0 –5 –2 –10 –4 –6 –15 –8 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Time(s) Time (s) Fig. 4. Stress time histories under railway, highway, and wind loading: (a) SP-TLS-02; (b) SS-TLS-12.
3250 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 a 20 a 10 10 0 Stress (MPa) Stress (MPa) 0 –10 –10 –20 –20 –30 0 5 10 15 20 0 5 10 15 20 b 20 b 10 10 Stress (MPa) 0 Stress (MPa) 0 –10 –10 –20 –20 0 5 10 15 20 –30 0 5 10 15 20 Time (hour) Time (hour)Fig. 5. Daily stress time histories under multiple types of loading at SP-TLS-02: Fig. 6. Daily stress time histories under multiple types of loading at SS-TLS-12:(a) Calculated; (b) measured. (a) Calculated; (b) measured.The gross train weight (GTW) ranges from 282.7 to 402.2 tons. locations shall be determined for the key structural componentsTheir running speed ranges from 25 to 38 m/s. On the same day, of a long-span suspension bridge. Given that the main structural8225 and 8623 heavy road vehicles weighing over 30 kN ran across components of and loadings on one long-span suspension bridgethe bridge using the north and south three-lane carriageway, can be quite different from another, the determination of fatigue-respectively. The gross vehicle weight (GVW) ranges from 4 to 54 critical locations is case-dependent. The Tsing Ma suspensiontons. The mean wind speed and direction are obtained from wind bridge is taken as an example for illustration. The key structuraldata recorded by the anemometers installed at the middle of the components of the Tsing Ma Bridge can be classified into 55bridge deck. The hourly mean wind speed perpendicular to the components in 15 groups. The details of the classification of thebridge axis ranges from 2 to 13 m/s on that day. The turbulence components in each group are given in Table 1 . The fatigue-intensities are taken as 24% and 17% in the horizontal and verticaldirections by considering the most turbulent cases in the field. The critical locations are determined through the stress analysis of eachintegral length scales are taken as 251 and 56 m in the horizontal component. To make sure that the size of the FEM is not too largeand vertical directions, respectively. As wind-induced dynamic to be used for dynamic analysis, some types of bridge componentsstress responses are dominated by vibration modes of a relatively cannot be modelled exactly. For instance, if the orthotropic decklow frequency, only the first 153 modes of vibration up to 2 Hz of the bridge were modelled using shell elements, the size of theare included in this case for the stress response computation. FEM would be too large to be used. Therefore, the orthotropic deck24-h time periods of the railway-, highway-, and wind-induced between the two adjacent cross frames at an interval of 4.5 m wasstress responses are calculated using the engineering approach. simply modelled by a plate element that was fixed to the two crossBased on the daily stress responses induced by the three individual frames at its two ends in the longitudinal direction and free on theloadings, the multiple load-induced stress response is obtained other two sides in the lateral direction. Such a modelling makesby superposition. The daily multiple load-induced stress time it impossible to obtain actual stresses of the orthotropic deck.histories computed using the engineering approach at the location Apart from these components, some other types are neglectedof strain gauges SS-TLS-12 and SP-TLS-02 are shown in Figs. 5(a) because they are not critical to fatigue in practice. The bridgeand 6(a), respectively. The measured ones are shown in Figs. 5(b) components taken into consideration for fatigue analysis in thisand 6(b) for comparison. It can be seen that the computed stress study are highlighted in grey in Table 1.time histories agree well with the measured ones. The relative As the fatigue damage of the Tsing Ma Bridge is induced bydifferences in the root mean square (RMS) of the stress responses the combined effect of railway, highway, and wind loading, theare calculated to determine the relative differences (the measured fatigue-critical locations should be determined based on the mul-value minus the computed one, divided by the measured one) at tiple types of loading. However, it is very difficult because so manythe two typical locations. The relative differences in the RMSs of stress analyses are required for a great number of structural com-the stress responses are 12.9% and 8.4% for strain gauges SS-TLS- ponents under a large number of loading combinations in which12 and SP-TLS-02, respectively. The coupled dynamic approach is different intensities of the three loadings shall be considered. Someactually not applicable for the computation of the daily dynamic simplifications are therefore necessary. The fatigue damage in-stress responses as it takes an intolerably long time, whereas only duced by railway and highway loading was separately investi-several minutes are required for the engineering approach if thestress influence lines are available. The small relative differences gated, and it was found that for most bridge components exceptbetween the computed and measured time histories and a short for the upper deck, the fatigue damage is mainly caused by mov-computation time for the engineering approach demonstrate the ing trains, and that the contribution of moving road vehicles ishigh level of computational efficiency and acceptable level of small. In addition, wind-induced fatigue damage to the bridge iscomputational accuracy. not significant . Therefore, railway loading is a dominant fac- tor for the fatigue damage of the bridge. Given that almost all of the4. Determination of fatigue-critical locations trains running across the Tsing Ma Bridge since November 2005 are 16-bogie trains, a standard train is defined to represent all 16- The above section proposes an engineering approach for bogie trains by taking the weight of each bogie as the mean weightdynamic stress analysis. In the next step, the fatigue-critical of the relevant bogies of all 16-bogie trains in November 2005.
Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3251Table 1Classification of the structural components of Tsing Ma Bridge . Name of group Name of component Group no. Component no. Serial no. Main cables (a) 1 Strand shoes (b) 2 Suspension cables Shoe anchor rods 1 (c) 3 Anchor bolts (d) 4 Cable clamps & bands (e) 5 Hangers (a) 6 Suspenders Hanger connections: Stiffeners 2 (b) 7 Hanger connections: Bearing plates (c) 8 Legs (a) 9 Towers Portals 3 (b) 10 Saddles (c) 11 Chambers (a) 12 Anchorages Prestressing anchors 4 (b) 13 Saddles (c) 14 Legs (a) 15 Piers: M1, M2, T 1, T 2, T 3 5 Cross beams (b) 16 Top chord (a) 17 Outer-longitudinal trusses Diagonal 6 (b) 18 Vertical post (c) 19 Bottom chord (d) 20 Top chord (a) 21 Inner-longitudinal trusses Diagonal 7 (b) 22 Vertical post (c) 23 Bottom chord (d) 24 Top web (a) 25 Main cross frames Sloping web 8 (b) 26 Bottom web (c) 27 Bottom chord (d) 28 Top web (a) 29 Intermediate cross frames Sloping web 9 (b) 30 Bottom web (c) 31 Bottom chord (d) 32 Upper deck (a) 33 Plan bracings 10 Lower deck (b) 34 Troughs (a) 35 Deck 11 Plates (b) 36 T -sections (a) 37 Railway beams Top flanges 12 (b) 38 Connections (c) 39 Rocker bearings at Ma Wan tower (a) 40 PTFE bearings at Tsing Yi tower (b) 41 PTFE bearings at Pier T 1 (c) 42 PTFE bearings at Pier T 2 (d) 43 Bearings PTFE bearings at Pier T 3 13 (e) 44 PTFE bearings at Tsing Yi Anchorage (f) 45 Rocker bearings at M2 (g) 46 PTFE bearings at M1 (h) 47 Hinge bearing at Lantau Anchorage (i) 48 Highway movement joint (a) 49 Movement joints 14 Railway movement joint (b) 50 Top chord (a) 51 Diagonal (b) 52 Tsing Yi approach deck 15 Vertical post (c) 53 Bottom chord (d) 54 Diagonals (K -bracings) (e) 55The standard train has a fixed configuration, and the railway load- the criteria for determining fatigue-critical location, an assumptioning of the train is represented by 32 vertical forces. The standard is adopted that the number of stress cycles induced by a standardtrain is then adopted to compute the railway-induced dynamic train is almost the same for all components of the same type, andstress responses of members in a given type of bridge component, difference only exists in the stress range level. This assumption isand then the responses are compared to each other to determine acceptable because the stress fluctuations at all components of thethe fatigue-critical members and locations. same type induced by the standard train running across are found Eqs. (2) and (3) indicate that fatigue damage is the function of to have a similar pattern. In addition, the equations demonstratethe stress range level σr and number of stress cycles n. To simplify that the damage is most sensitive to the maximum stress range
3252 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 Fig. 7. Major structural components of bridge deck. of the member at the two ends. Similar procedures are applied to 50 the other bridge components to determine the potential fatigue- critical locations. Furthermore, the net stresses at the potential 40 fatigue-critical locations are also checked for determining the final fatigue-critical locations. The results demonstrate that the fatigue- critical sections of the bridge deck are around the bridge towers. 30 Within the fatigue-critical sections, six of the strain points are chosen for fatigue analysis, that is, the elements E32123 (T ) at the top flange of the outer-longitudinal diagonal member close to 20 the Ma Wan Tower, E34415 (B) at the bottom flange of the outer- longitudinal bottom chord of the Tsing Yi Tower, E40056 (T ) at the 10 top flange of the inner-longitudinal top chord of the Tsing Yi Tower, E40906 (B) at the bottom flange of the inner-longitudinal bottom chord of the Tsing Yi Tower, E39417 (B) at the bottom flange of the 0 T -section of the railway beam of the Tsing Yi Tower, and E55406 0 500 1000 1500 2000 (T ) at the top flange of the bottom web of the cross frame close to the Tsing Yi Tower.Fig. 8. Maximum stress ranges of diagonal members in north outer-longitudinaltrusses. 5. Databases of dynamic stress responses to different loadings σmax because fatigue damage is a function of σrm or σrm+2 . σmax In this section, the databases of dynamic stress responsesis therefore selected as the index for determining the fatigue- induced by railway, highway, and wind loading at the criticalcritical locations of bridge components of the same type. To make locations of the Tsing Ma Bridge are established based on thethe problem manageable, σmax is approximately decided by using loading information recorded by the WASHMS.the difference of the maximum and minimum stress in the stresstime history based on the principle of the level crossing method. As 5.1. Database of wind-induced dynamic stress responsefatigue is critical to the tension and reversal members, additionalstructural analysis should be performed to check the net stress in Long-span suspension bridges built in wind-prone regionsthe member due to the dead and super-imposed dead loads plus suffer from considerable wind-induced vibration, which appearsan extreme live load. If it is positive, then the member is finally within a wide range of wind speeds and lasts for almost the wholedefined as a fatigue-sensitive member. design life of the bridge. A joint probability distribution function Let us take the diagonal members of outer-longitudinal trusses of the mean wind speed and direction is utilized to describe windas an example to illustrate the determination of fatigue-critical intensity at the bridge site . The distribution of wind speedlocations (see Fig. 7). Given the symmetry of the cross sections for any given wind direction is assumed to follow the Weibullof the bridge, the standard train is supposed to run on the north distribution. The parameters in the distribution are determinedrailway track and accordingly only the outer-longitudinal truss from monsoon wind records of hourly mean wind speed andon the north needs to be considered. The stress time histories at direction during the period from 1 January 2000 to 31 Decemberthe stress output points of all of the diagonal members of the 2005, which were collected by an anemometer installed on thenorth outer-longitudinal truss are computed based on the standard top of the Ma Wan tower. Given that the measured typhoon windtrain running across the bridge on the north railway track. The records are not enough to establish a reliable joint probabilitymaximum stress ranges are subsequently estimated. Fig. 8 shows distribution, only monsoon wind is of concern in this study. Thethe maximum stress ranges of the diagonal members of the north maximum wind speed at the top of the tower in each windouter-longitudinal truss due to the standard train running on direction within the 120-year design life is then obtained from thethe north side of the bridge deck. The potential fatigue-critical joint probability distribution. The maximum wind speed obtainedlocations in the diagonal members of the north outer-longitudinal at the top of the tower is converted to the average deck level of thetruss can be determined from the figure: the diagonal member bridge. The maximum hourly mean wind speed at the deck levelE32123 (T ) close to the Ma Wan tower in the main span, and the is 25.89 m/s in the north direction for winds over the over-landdiagonal member E32403 (T ) close to the Tsing Yi tower in the fetch, and 15.47 m/s in the south direction for winds over the open-main span. ‘‘T ’’ or ‘‘B’’ in brackets denotes that the potential fatigue- sea fetch . Finally, a database of hourly wind-induced dynamiccritical location is at the top or bottom flange of the cross section stress responses at the fatigue-critical locations is established:
Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3253from 5 to 26 m/s at an interval of 1 m/s for winds over the over- 60 Stress time history (MPa)land fetch, and from 5 to 16 m/s at an interval of 1 m/s for winds 40over the open-sea fetch. The stress fluctuations induced by windof a normal hourly mean wind speed less than 5 m/s are neglected 20as they contribute little to fatigue. The database includes a totalof 34 one-hour time histories for each fatigue-critical location. 0The nominal stress of each fatigue-critical member is computed –20based on the stresses at five points of the two ends of the member. 0 5 10 15 20The hot-spot stresses, which reflect the stress concentration at Time (hour)welded joints, should be considered in fatigue analysis . Thehot-spot stresses at the fatigue-critical locations are determined Fig. 9. A sample stress time history due to multiple types of loading.by multiplying the nominal stresses by the stress concentrationfactor (SCF). It is noted that the value of SCF depends largely on railway, highway, and wind loading need to be generated forthe local geometry of the connection details. Nevertheless, there fatigue analysis. To generate them, the hourly mean wind speedsare a number of fatigue-critical locations in this bridge, and the and directions for 120 years should be first obtained. A two-steplocal geometry of the connection details at these locations is quite Monte Carlo simulation (MCS) method is adopted to draw outdifferent. Considering that the SCF of 1.4 was used in the design of 1051,200 (120 × 365 × 24) pairs of hourly mean wind speed andthe bridge concerned for almost all connections, this number is also direction for 120 years. In the first step, the mean wind directionused for the six identified fatigue-critical locations in this study. is extracted through MCS according to the relative frequency ofThe fatigue damage at fatigue-critical locations refers hereafter to the mean wind direction without considering wind speed. In thethe fatigue damage at these hot spots. second step, the mean wind speed at the top of the tower is drawn out through MCS according to the probability distribution5.2. Database of railway-induced dynamic stress response of the mean wind speed at the given mean wind direction under the condition that it is not larger than the maximum wind speed Since it is almost impossible to predict railway traffic volume in this direction. Finally, the mean wind speed and direction arein the distant future for the Tsing Ma Bridge, one month of paired after two steps of MCS, and then converted into the hourlyrailway traffic that is close to the current traffic conditions is normal mean wind speed to generate a sequence of 120 years.adopted here to establish the database of railway-induced stress As the monsoon wind in Hong Kong normally is southerly (from 90° to 270°) in summer and northerly (from 270° to 90°) inresponses at the fatigue-critical locations for fatigue analysis. winter, the sequence of hourly normal mean wind speeds in eachMonthly railway traffic in November 2005 is selected to establish year is adjusted to consider this. For each hourly normal meanthe database, and more than 90% of trains are of the 16-bogie wind speed in the sequence, the corresponding wind-inducedtype. In addition, the daily average number of trains in this month dynamic stress response can be found in the database establishedreaches the maximum in the time period of concern. The daily in the previous section. As wind blowing in two directions is oftime histories of railway-induced stress responses at the fatigue- concern, the mean wind direction in each hour of the sequencecritical locations are computed using the stress influence lines for is adopted to determine whether the wind is blowing in therailway loading and the railway loading information measured in direction of the over-land fetch or open-sea fetch. As the databaseeach day of November 2005. No large difference can be found in is established for different levels of mean wind speed at an intervalthe stress time histories among these days. The railway loading of 1 m/s, rounding towards infinity is adopted to handle the hourlyinformation in each day of November 2005 is adopted to compute normal mean wind speeds in the sequence. Finally, 1,051,200 h of30 daily railway-induced stress time histories at the fatigue-critical wind-induced dynamic stress time histories are generated at eachlocations, to compile the database of railway-induced dynamic fatigue-critical location to compose a time history of 120 years.stress responses. In addition, 120 years of railway-induced stress time histories are generated at the fatigue-critical locations based on the5.3. Database of highway-induced dynamic stress response database of 30 daily time histories. An integer between one and thirty is randomly drawn out for each day to generate a random The highway traffic on the Tsing Ma Bridge has been monitored number sequence of 120 years. For each one in the sequence,through seven dynamic weigh-in-motion (WIM) stations installed the corresponding daily railway-induced dynamic stress responsesnear the Lantau Administration Building since August 1998. The can be found in the database established in the aforementionedroad vehicle data in November 2005 are adopted to build a section. Finally, 43,800 daily time histories at each fatigue-criticaldatabase of highway-induced stress responses because this month location are used to compose 120 years of railway-inducedreached a maximum number of monthly vehicles and other dynamic stresses. As the database of highway-induced stressmonths had slightly less vehicle numbers. Highway-induced stress responses is also based on 30 daily time histories, a similartime histories are also computed in one-day units. The daily processing method is applied to obtain 120 years of highway-time histories of highway-induced stress responses at the fatigue- induced dynamic stresses at the fatigue-critical locations.critical locations are computed using the stress influence lines for Based on the engineering approach proposed in the previoushighway loading and the highway loading information measured section, the stress responses at the critical locations induced byin each day of November 2005. No large differences can be found the combined effects of railway, highway, and wind loading canin the stress time histories among these days. Finally, 30 daily be approximately obtained from the three responses induced bystress response time histories at the fatigue-critical locations are individual loadings by superposition. Therefore, a 120-year timecomputed to create the database of highway-induced dynamic history of the stress induced by multiple types of loading isstress response. determined from those induced by railway, highway, and wind loading individually. It should be noted that the bridge is closed to6. Multiple load-induced dynamic stress time histories in traffic when the mean wind speed recorded on site is very high;design life therefore, the bridge stress responses under this condition are induced by wind loading only. Fig. 9 shows a sample daily multiple The design life of the Tsing Ma Bridge is 120 years; therefore, load-induced hot-spot stress time history at the critical location120 years of time histories of the dynamic stresses induced by E32123.
3254 Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 Fig. 10. Multiple load-related spectra: (a) Stress range spectrum; (b) fatigue damage spectrum.7. Fatigue analysis at fatigue-critical locations The rainflow counting method is applied to the 120-yearmultiple load-induced stress time history, and the number of stresscycles in different stress range levels can be obtained. A stressrange spectrum is defined as the percentage of the number ofstress ranges in each stress range set to the total number in allsets. Fig. 10(a) displays the stress range spectrum at E32123. Itdemonstrates that most of the stress ranges are at the low levels,as 92.0% are less than 8 MPa. Because fatigue damage is much moresensitive to high level stress range rather than low level one, astress range of large amplitude may make a great contribution tofatigue damage although it occurs less frequently. Fatigue damagein each stress range bin is computed using Eq. (2) or Eq. (3). Thetype of welded connection at the six fatigue-critical locations inthis study is classified as F according to British Standard  withσr ,0 = 40 MPa, K2 = 6.3 × 1011 , and m = 3. A fatigue damage Fig. 11. Cumulative fatigue damage curves at fatigue-critical locations.spectrum is defined as the percentage of fatigue damage in eachstress range set to the total damage in all sets. Fig. 10(b) displays loading is greater than that due to wind loading at some locationsthe fatigue damage spectrum at E32123. The figure shows that the whereas other locations are in reverse. It is also found that fatiguecontribution of stress ranges in the low levels (less than 8 MPa) to damage due to combined effects of railway, highway, and windfatigue damage is small, and that the greatest fatigue damage is in loading is larger than the sum of fatigue damage due to each ofthe stress range of 36–44 MPa. individual loadings, for fatigue damage is the function of m-power Based on the multiple load-induced stress time histories over stress range (nonlinear relationship), and stress ranges induced bythe period of 120 years and the time step t = 1/365 year, multiple loading are larger than those caused by individual loading.the curves of cumulative fatigue damage within 120 years at the In addition, the fatigue damage spectra of railway, highway, andfatigue-critical locations can be computed. The cumulative fatigue wind loading are investigated based on the 120-year time histories,damage Dk in the kth day is calculated based on the daily stress and the results are shown in Fig. 12(a–c). The figure shows thattime history using Eqs. (1)–(3), and the cumulative fatigue damage the spectra are quite different. For example, the greatest fatigueDk is updated by adding the new damage on this day. Fig. 11 damage induced by railway loading is in the stress range ofshows the cumulative fatigue damage curves at the fatigue-critical 32–40 MPa, that induced by highway loading is in the range oflocations within a design life of 120 years. It is noted that the 0–4 and 8–24 MPa, and that induced by wind loading in thestructure is in danger when the cumulative fatigue damage is range of 0–12 MPa. To study the combined effect of multiple types of loading on fatigue damage, a multiple load magnificationgreater than one. The maximum of the 120 years of cumulative factor is defined as the ratio of the fatigue damage due to thefatigue damage at the fatigue-critical locations of the Tsing Ma combined effect of the three loadings to the sum of the damageBridge is very close to one, which implies that the health condition due to each individual loading. The factors at the six fatigue-criticalof the bridge is satisfactory. In addition, the cumulative damage locations are computed and range from 1.06 to 1.35. The maximumcurves seem to be very linear. That is because Miner’s model is factor is at critical locations E32123 and E34415, at which thea linear damage model, and traffic loading is assumed to remain fatigue damage induced by highway and wind loading is muchstable over the design life closer to that induced by railway loading than at the other critical In addition to the fatigue damage induced by multiple types of locations. The results indicate that the combined effect of multipleloading, the fatigue damage induced by each individual loading loads must be considered in a bridge subject to multiple typestype is also investigated. The 120 years of cumulative fatigue of loading, especially in the case in which the contributions ofdamage induced by railway, highway, and wind loading are different loadings to fatigue damage are close.respectively computed based on the three stress responses underthe different loadings. The results of the damage at different 8. Conclusionsfatigue-critical locations are listed in Table 2. It is found thatrailway loading plays a dominant role in the fatigue damage of A general framework has been proposed for fatigue analysisthe Tsing Ma Bridge, and that the damage induced by highway of a long-span suspension bridge under multiple loading over its
Z.W. Chen et al. / Engineering Structures 33 (2011) 3246–3256 3255Table 2120 years of cumulative fatigue damage under different loading types. Fatigue-critical locations Loading types Railway (R) Highway (H) Wind (W ) R+H +W E32123 (T ) 0.70 0.048 0.011 1.02 E34415 (B) 0.66 0.044 0.0092 0.96 E40056 (T ) 0.52 0.0022 0.0057 0.68 E40906 (B) 0.42 0.0025 0.0052 0.54 E55406 (T ) 0.34 0.0037 0.0016 0.41 E39417 (B) 0.48 0.0020 0.0074 0.52 Fig. 12. Fatigue damage spectra: (a) Railway; (b) highway; (c) wind.design life in this paper. The framework was applied to the Tsing in other locations. Furthermore, it is necessary to consider theMa suspension bridge in Hong Kong. An engineering approach combined effect of multiple types of loading in the fatigue analysisfor dynamic stress analysis of a long-span suspension bridge of long-span suspension bridges. In reality, uncertainties exist inunder multiple types of loading was first proposed. The Tsing external loadings, structural modelling, and structural parameterMa Bridge and the measurement data recorded by the WASHMS in fatigue assessment. The fatigue reliability analysis of multi-were employed to verify the feasibility of the proposed approach. loading bridges deserves further study.The fatigue-critical locations of the bridge were determined forthe key structural components. Based on the measurement data Acknowledgementsrecorded by the WASHMS installed on the bridge, the databasesof wind, railway, and highway loadings as well as stress responses The authors wish to acknowledge the financial support from theat the fatigue-critical locations were established to generate 120- Research Grants Council of Hong Kong (PolyU 5327/08E), the Hongyear time histories of multiple loading-induced stress responses. Kong Polytechnic University (PolyU-1-BB68), and the NationalFinally, the fatigue analysis based on the 120-year stress time Natural Science Foundation of China (NSFC-50830203 and NSFC-histories was performed to compute the cumulative fatigue 51108395). Sincere thanks go to the Highways Department of Hongdamage over the bridge’s design life. The results indicate that Kong for providing the authors with the field measurement data.the health condition of the bridge is satisfactory. The cumulative Any opinions and concluding remarks presented in this paper arefatigue damage induced by individual loading and the damage entirely those of the authors.magnification due to the combined action of three types of loadingswere also investigated. The results show that railway loading Referencesplays the dominant role in the fatigue damage of the bridge. The  ASCE. Committee on fatigue and fracture reliability of the committee ondamage induced by highway loading is greater than that due to structural safety and reliability of the structural division, fatigue reliability:wind loading at some locations, whereas the reverse is the case 1–4. J Struct Eng ASCE, 1982; 108: p. 3–88.
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