SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL HIGHWAY BRIDGES

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1 CIVIL AND ENVIRONMENTAL ENGINEERING REPORTS No SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL HIGHWAY BRIDGES Adam WYSOKOWSKI Univesity of Zielona Góa, of. Z. Szafana St. 2, Zielona Góa, Poland In this ae, the detemination in a theoetical way by the simulation analysis - of the stess secta and stength, as well the sevice life aametes fo the stuctual elements of oad bidges is esented. The analytically obtained esults have been veified by the esults fom field tests of selected elements of steel oad bidges unde the sevice load. Fo the simulation calculations, a linea load model of oad bidges (esented in the ae) has been alied, deived fom autho s measuements on stuctues and concening taffic on the Polish uban and county oads and bidges. The esented studies and theoetical analyses have oved that the fatigue hazad exists also in oad bidges and they have indicated what tyes of bidges, which bidge elements in what static schemes, which sans and unde what sevice loads, ae endangeed by the fatigue hazad. Keywods: highway bidges, steel stuctues, fatigue, simulation analysis 1. INTRODUCTION Because steel highway bidges wok in vey advese weathe conditions and ae subjected to changing loads of diffeent magnitude and duation, a elatively lage numbe of them needs enovation and ebuilding and many of them ae nea the end of thei life. Cuently in Poland about 35% oad bidges, of the total 35 thousand, ugently need to be elaced. In these conditions any stuctual failue oses a hazad to the bidge if no emedial measues ae taken. The oblem becomes inceasingly essing as each yea the numbe of old steel highway bidges inceases. Theefoe decisions should be made whethe these bidges can emain Univesity of Zielona Góa Pess, Zielona Góa 2005 ISBN

2 274 Adam WYSOKOWSKI in sevice afte thei design life is ove o, deending on thei imotance, they should be einfoced o elaced. All those factos and the aid develoment of the oad tansot in Cental Euoe ceate an ugent need to detemine, comehensively, the effect of moving loads on the fatigue stength of the stuctual elements of highway bidges. Since the field investigations of bidge stuctues ae time-consuming and laboious, it would be extemely difficult to cay out a lage numbe of them to obtain the esults eesentative fo vaious kinds of bidge elements, having diffeent static schemes and coss-sections, located on the outes of oads having diffeent volumes and stuctue of taffic. Futhemoe, the esults following fom field investigations ae influenced by a numbe of othe factos. Theefoe, it seems sensible to aly simulation methods in conjunction with some exeience gained fom the investigations of actual bidge stuctues fo a wide analysis of the highway bidge fatigue oblem. 2. SIMULATION METHOD OF DETERMINING THE STRESS SPECTRA Simulation is usually defined as the eoduction of the oeties of an actual object, henomenon o ocess by means of its model. With advances in comute technology the use of simulation methods has become moe univesal. The investigation of a model instead of the actual ocess is simle, less laboious and cheae and it can be conducted eeatedly in a shot time. In the autho s oinion, comute simulation in which a deteministic sequence of foces eesents the axle loads of vehicles seems to be the most sensible solution. The vehicles move indeendently of one anothe within a given taffic lane, and they ae saced aat by at least the length of the influence-line banch fo the consideed element (fo one taffic lane). A simila aoach to this oblem was adoted in (Baus and Buls 1981). If the oblem of the sevice duability oblem is to be handled comehensively, all ossible aametes should be taken into account, aticulaly in the case of highway bidges with a elatively high taffic volume o a lage numbe of taffic lanes. Then the likelihood of the simultaneous assage of seveal vehicles ove the bidge should be taken into consideation. Thus it seems logical to teat the movement of vehicles within a bidge as a andom ocess. A geneal flow chat illustating the simulation method of detemining the stess o the eaction ange secta fo bidge stuctual elements is shown in fig. 1.

3 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 275 Fig. 1. Pocedue fo detemining the stess anges o eactions by comute simulation The values of bending moments (eactions) geneated by the sequences of foces moving ove an influence line can be calculated using the following geneal fomula ( R) = q x) 1 M ( y( x) dx (2.1) k o whee the notations ae: q k tansvese load-distibution coefficient fo lane k, (x) foce values along the length of an influence line, y(x) odinates of the influence line of a static quantity (moment o a eaction) fo a consideed cosssection.

4 276 Adam WYSOKOWSKI The aticula values in the static magnitude oscillogams ae calculated fo a case when any of the foces, moving ove the influence line, is situated ove its chaacteistic oint, i.e. a zeo, maximum o minimum value, as well as ove the inflexion oint. An exession fo the next value σ i in the stess oscillogam can be witten as follows σ = a ϕ, (2.2) i M i actual with the notations: a static stength constant, ϕ actual dynamic actual-load stess-ange coefficient. Similaly, the stess ange geneated by standad load σ n is σ = a ϕ (2.3) n M n s tan d. whee ϕ stand. is the standad-load stess ange dynamic coefficient. Substituting the exessions (2.2) and (2.3), the elative stess value will be σ M ϕ i i actual α i = =, (2.4) σ n M n ϕ s tan d Fo the analysis of a single bidge stuctue one can use the exession (2.4), substituting the actual test o exeimental values of the dynamic coefficient ϕ actual, and the aoiately assumed (deending on a static scheme) values of the standad-load stess ange dynamic coefficient ϕ stand. It would be extemely difficult to adot this aoach in the case of comlex analysis of diffeent tyes of bidge stuctues done on EMC. The eason is that the actual dynamic coefficient ϕ actual assumes wide-anging values deending on the kind of bidge stuctue, the kind and condition of its suface, the way in which vehicles ass ove the bidge and the tye of the vehicles. Most often ϕ actual ϕ stand. Theefoe it is advisable to assume fo futhe consideations that ϕ actual = ϕ stand which will standadize the calculations and make ossible a final comaative analysis of the sevice stength and the duability fo diffeent tyes of bidge stuctues. Thus, the calculations done unde this assumtion ae on the safe side. Taking this into account, the elative stess-ange values can be descibed by the following elation M i α i =, (2.5) M n

5 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 277 Anothe imotant asect of the assessment of the sevice stength of steel bidges, in cases of comlex no stationay stess secta, is the counting of stess-cycle anges. This oblem aeas when the obtained stess oscillogams ae conveted into the stess-ange histogams (fig. 1). Thee ae many methods of counting the stess cycles, e.g. Level Cossing Count, Positive Peak o Range Pais, which ae descibed in detail in (Baus and Buls 1981). Neithe of these methods takes into account the ode in which the cycles occu in a sectum. The most univesal method of cycle counting is the Rain Flow method descibed in (Hit 1977) and (Stie et al. 1982). The advantage which it has ove the othe methods is that it can be alied diectly, i.e. it is not necessay fist to eject the shot stess cycles. The method is ecommended by many intenational oganizations such as ISO (Intenational Oganization fo Standadization) (Hit 1983), IIW (Intenational Institute of Welding) (IIW 1981), Euocode (Euoean Code fo Steel Constuction) and ECCS (Euoean Convention fo Constuctional Steelwok) (Caena 1982). A modification of the Rain Flow method is the Resevoi method ecommended fo the calculation of oad and ailway bidges by the Bitish standad (B.S. 1980). Nevetheless, the Rain Flow method has significant limitations when alied to wide analyses, since it enables only the counting of stess cycles fo secta beaing the same sign, i.e. basically it can be alied only to simlesuoted elements (Hit 1983). In this ae the autho has solved this oblem by intoducing a useful innovation enabling the counting of cycles fo elements with continuous static schemes (changing stess signs), which consists in the io tansfomation of the stess sectum. The autho used this modified Rain Flow method to detemine the sevice stength by simulation. The ocedue in the case of vaiable stess signs is shown in fig. 1. Fist, the stess-oscillogam values σ min ae sought. Then all the stess values to the left of the minimum value should be moved to the end of the stess oscillogam. Next, the absolute value of the minimum stess σ min should be added to all the stess values. The futhe stes ae simila as in the conventional Rain Flow method, beaing in mind that a full stess-ange cycle is made u of two single half-cycles. The diection in which cycles ae counted is indicated by an aow in fig. 1. Then a histogam of the stess anges is comiled fom the obtained individual stess-ange values. In this way, the stess-ange sectum fo a single tye of vehicle is obtained. The same ocedue is alied to othe vehicle tyes. The final esult is a comehensive sevice-stess-ange sectum fo a consideed coss-section of a stuctual element, assembled fom the individual-vehicle secta, which takes into account tansvese-load-distibution coefficients fo the aticula taffic lanes, numbe-of-vehicles-e-taffic-lane distibution func-

6 278 Adam WYSOKOWSKI tions and the andom model of vehicle taffic on the bidge descibed in chate 3. The model takes account of, among othe things, the effect of the simultaneous assage of vehicles ove the bidge on the obtained static magnitudes and the numbe of cycles. The sevice stength and duability aametes wee calculated using the fomulas and elations comiled below. Fo the simulation analysis of the stess secta and the assessment of sevice stength and duability of stuctual elements, the autho has develoed a softwae ogam named TE-MD (Sevice Duability of Road Bidges in Polish). Because of the comlexity of the oblem, thee ogams in all have been develoed. They fom a unifom system having the following functions: 1. PROGRAM LW fo enteing influence lines, 2. PROGRAM MOMN fo enteing standad moments, 3. PROGRAM TE-MD detemination of sets of stess oscillogams, counting of stess-ange cycles, detemination of stess-ange histogams and calculation of sevice stength and duability aametes, intout of esults. If the stess sectum is nonstationay (fig. 2) and all its anges σ i cause atial damage, it can be elaced (on the basis of, among othes, Mine s hyothesis) by a stationay sectum with the equivalent stess ange σ e and the actual numbe of cycles N = n i i= 1, (2.6) which occued in the sectum [5], and the equivalent stess ange σ e 1 m m ni ( σ i ) i= 1 σ e =, (2.7) i= 1 n i o it can be elaced by a stationay sectum with on equivalent numbe of cycles N n and standad-load stess ange σ n N σ i n = n i i= 1 σ n m, (2.8) whee is the numbe of stess-ange levels in the consideed no stationay sectum.

7 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 279 Fig. 2. The elationshis between stess secta aametes In the simulation stess analysis, it is moe advantageous to use the elative stess-ange level α i efeed to the standad-load stess ange σ n (2.4) and the elative fequency f i of the numbe of cycles n i with level α i, efeed to the total numbe of cycles in the sectum f i = n i= 1 i n i, (2.9) If the elations (2.4) and (2.9) ae used, the fomulas (2.7) and (2.8) become the following foms: and 1 m m σ e = σ n f iα i (2.10) i= 1 N n = n i i= 1 ( ) m α. (2.11) Using the above elations, the following aametes, descibing the sevice-stess sectum fo a stuctual element, ae obtained: Maximum-stess-ange utilization coefficient fo the whole sectum i

8 280 Adam WYSOKOWSKI σ α max =, (2.12) max σ n whee σ max is the maximum stess ange in the sectum. Equivalent-substitute-stess-ange coefficient α σ e e =, (2.13) σ n which exesses the sevice utilization of the stess anges fo the whole sectum with the actual numbe of cycles N. By using the elation (2.10) fo σ e we get the following fomula fo α e 1 m m α e = f i α i. (2.14) i= 1 Single-vehicle-stess-ange-utilization coection coefficient (measue of the sevice single-vehicle hamfulness) 1 m σ e N α, (2.15) = σ n N whee N is the numbe of vehicles geneating the N stess cycles. This coefficient was intoduced into the Swiss egulations SIA 161 (SIA 1979) and into the Euoean code UIC (CODE UIC 1981). It is a measue of the fatigue hamfulness of a vehicle deending on its ofile (numbe of axles, thei configuation and loads), the oad taffic aametes, and the kind of bidge element (static scheme, san, numbe of taffic lanes). If the elation (2.10) is used, this coefficient takes the following fom 1 1 m m m N α = f iα i, (2.16) i 1 N = o if the elation (2.13) is substituted fo the fist tem, one gets 1 m N α = α e. (2.17) N

9 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 281 Equivalent substitute numbe of cycles fo the sectum with the standadload stess ange σ n N n = N i= 1 f i ( ) obtained by eaanging the exessions (2.11) and (2.9). Actual single-vehicle numbe of cycles N i m α, (2.18) N =. (2.19) N 3. HIGHWAY BRIDGE LOAD MODEL To cay out the simulation analysis of the sevice stength of highway bidges one must know the taffic stuctue and its volume on national highways and bidges. To the autho s knowledge, no study of the sevice loads acting on the highway bidges in Poland has been made. Besides, that the esults of studies and exeiments caied out in othe counties ae not diectly alicable to Polish bidges because of the much diffeent load conditions (tyes of vehicles, taffic density, egulations concening vehicle technical secifications). Fo this eason the autho decided to efom own measuements of the oad taffic stuctue and loads, in ode to develo a model of the actual sevice loading of the oad bidges in Poland. Fist, it was necessay to acquie and classify the technical data of vehicles that occu in the actual taffic steam. A classification of all the basic (distinguishable) tyes of tucks is esented in table 1. Fig. 3 shows the taffic stuctue (including the loading of the aticula tyes of vehicles) on county bidges, uban bidges, and bidges in geneal. Fig. 4. shows the histogams of total weights Q c of tucks fo county bidges, uban bidges and oad bidges in geneal, lotted on the basis of the efomed measuements. It also shows the vehicle weight Q c distibution obability density function deived by the autho fo oad bidges in Poland.

10 282 Adam WYSOKOWSKI Table 1. Tyes of vehicles Taking into account the geomety and the constuction of tucks, the autho divided all the consideed tyes of vehicles into thee main gous (tab. 1): two-axle single vehicles 2, two-axle vehicles with a two-axle taile 2P2, two-axle vehicles with a single-axle 2N1, fo which the aoiate substitute vehicles, constituting the basis fo the taffic model and thus eesenting the whole actual oad taffic, wee develoed.

11 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 283 Fig. 3. Vehicula taffic stuctue on highway bidges Fig. 4. The distibution of total weights of vehicles on highway bidges

12 284 Adam WYSOKOWSKI The chaacteistics of the thee tyes of substitute vehicles ae given in table 2. The vehicles eesent the cuent national oad taffic and so they can be taken as a model of the sevice loads acting on bidge stuctues located on the outes of the main national oads in Poland. Othe factos which should be taken into account when descibing the vehicula taffic on oad bidges (on the county to ailway bidges) ae: the aths of movement of vehicles within the lane, the tansvese distibution of taffic volume and its stuctue ove the taffic lanes (fo the detemined 24h taffic volumes), the seed of the vehicles and thei sacing and the fact that vehicles ass each othe on the bidge. Because of the natue of the oad taffic, the above aametes ae difficult to define ecisely even on the basis of a lage numbe of suveys. Theefoe, the autho had to make cetain assumtions based on his own eseaches, descibed in a moe detailed manne in (Wysokowski 1990). 4. DESCRIPTION OF ANALYZED BRIDGE ELEMENTS Using the TE-MD softwae, one can detemine the stess secta and the sevice duability aametes fo actically any stuctual element of any bidge with any static schemes, sans, coss-sections, numbe of taffic lanes, numbe of vehicle tyes, taffic volumes and vehicle seeds. Because of the esent ae s limited size, the autho cannot analyze all the above cases. Theefoe the calculations and the analyses of the stess secta and the sevice duability ae limited to classical cases, egading the bidge constuction (static schemes, the numbe of taffic lanes, the numbe of beams), the tyes of vehicles, thei 24h flows and seeds of tavel ove the bidge. The aticula cases wee so selected

13 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 285 as to make ossible the assessment of the effect of diffeent aametes on the sevice duability and to oint at the stuctual elements which ae most sensitive to fatigue. Fig.5 shows the bidge-element coss-sections and the static schemes assumed fo the calculations. Fig. 5. Stuctual bidge elements coss-sections and static schemes assumed fo calculations

14 286 Adam WYSOKOWSKI As examles, a two-beam bidge, with both two and fou taffic lanes, whose coss-section is shown in fig. 5, was analyzed. The two-beam cosssection eesents the most disadvantageous configuation of the main beams, since each vehicle assing ove the bidge geneates at least one load cycle in thei coss-section, egadless of the taffic lane on which it moves. The calculations wee done fo the following stuctual elements: main gides, coss-beams, longitudinal ibs, deck lates. Fig. 5 shows the coss-beams, static scheme assumed fo the calculations. This is the most disadvantageous scheme fo coss-bas since each vehicle assing ove the bidge geneates at least one stess cycle in them. Values R, in both two-lane and fou-lane bidges, eesent the eaction fom the aticula taffic lanes, tansfeed onto the ibs, including the deck s flexibility. The coss-ba s cental coss-section was adoted fo the calculation. The stess anges ae calculated taking into account vehicle assing at diffeent taffic volumes. 5. RESULTS OF SIMULATION ANALYSIS 5.1. Stess secta Since a vey lage numbe of calculation esults wee obtained, the autho had to select only the most significant ones and to esent them in the most concise fom to fit them into the esent ae. Figs 6a,b,c and 7 show the samle comutational stess secta and thei distibution functions (10, 50, 90 and 100%) fo the selected main-gide systems of the two-lane bidge. Fo comaison, fig. 8 shows the samle stessange secta fo the simle-suoted main gides of the fou-lane bidge. Comutational stess-ange secta fo the deck elements, at diffeent flexibility coefficients γ and diffeent coss-beam sacing t, ae esented in fig. 9 (fo longitudinal ibs) and fig. 10 (fo coss-beams) Calculated sevice stength aametes Besides the stess secta, the comute simulation analysis yielded the sevice stength aametes fo the consecutive numbes of main-gide and deckelement systems. The aametes fo the aticula systems deend on the san length, the 24h taffic volume and the diving seed of vehicles.

15 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 287 Fig. 6. Comutational stess-ange secta fo main gides of two-lane bidge

16 288 Adam WYSOKOWSKI Fig. 7. Comutational stess-ange secta fo main gides of two-lane bidge Fig. 8. Comutational stess-ange secta fo main gides of fou-lane bidge

17 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 289 Fig. 9. Comutational stess-ange secta fo longitudinal ibs fo diffeent flexibility coefficients γ Fig. 10. Comutational stess-ange secta fo coss-beams at diffeent flexibility coefficients γ Figs 11a and b, 12a and b and 13a and b show gahs of the equivalent substitute stess-ange coefficient α e and the maximum stess-ange utilization coefficient α max, fo the aticula main-gide systems of the two-lane bidge, deending on san length l at 24h taffic volume seed v = 13.9 m/s. N = 1 and 7000 vehicles and d

18 290 Adam WYSOKOWSKI Fig. 11. Gahs of equivalent substitute stess-ange coefficient α e and maximum stessange utilization coefficient α max fo main gides of two-lane bidge Fig. 12. Gahs of equivalent substitute stess-ange coefficient α e and maximum stessange utilization coefficient α max fo main gides of two-lane bidge

19 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 291 Fig. 13. Gahs of equivalent substitute stess-ange coefficient α e and maximum stess-ange utilization coefficient α max fo main gides of two-lane bidge Fo comaison, fig. 14 shows the gahs of coefficients α e and α max fo the simly suoted main gides of the fou-lane bidge at 24h taffic volume N = 1 and vehicles and seed v = 13.9 m/s. d Fig. 14. Gahs of equivalent substitute stess-ange coefficient α e and maximum stessange utilization coefficient α max fo main gides of fou -lane bidge

20 292 Adam WYSOKOWSKI Gahs of coefficients α e and α max fo the deck elements ae shown in figs 15a and b (fo longitudinal ibs and deck lates) and fig. 16 (fo coss-beams). Fig. 15. Gahs of equivalent substitute stess-ange coefficient α e and maximum stessange utilization coefficient α max fo longitudinal ibs (a, b) and deck lates (b) Fig. 16. Gahs of equivalent substitute stess-ange coefficient α e and maximum stessange utilization coefficient α max fo coss-beams Figs 17a, b, c, d and 18a, b show taces of the single-vehicle-stess-angeutilization coection coefficient α (a measue of single-vehicle sevice hamfulness) fo the aticula numbes of the main-beam systems of the two-lane bidge vesus san length l fo 24h taffic volume and seed v = 13.9 m/s. N = 1 and 7000 vehicles d

21 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 293 Fig. 17. Gahs of single-vehicle stess-ange utilization coection coefficient α fo main gides of two-lane bidge

22 294 Adam WYSOKOWSKI Fig. 18. Gahs of single-vehicle stess-ange utilization coection coefficient α fo main gides of two-lane bidge Fig. 19 shows the tace of the coefficient α fo the simle-suoted main gides of the fou-lane bidge at 24h taffic volume N = 1 and vehicles and seed v = 13.9 m/s. d Fig. 19. Gahs of single-vehicle stess-ange utilization coection coefficient α fo main gides of fou-lane bidge

23 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 295 Gahs of the coection coefficient α fo a deck element ae shown in fig. 20a,b (fo longitudinal ibs and deck lates) and fig. 21 (fo coss-beams). Fig. 20. Gahs of single-vehicle stess-ange utilization coection coefficient α fo longitudinal ibs (a, b) and deck lates (b) at diffeent flexibility coefficients γ Fig. 21. Gahs of single-vehicle stess-ange utilization coection coefficient α fo coss-beams at diffeent flexibility coefficients γ Figs 22a,b, 23a,b and 24a,b show gahs of actual numbes of singlevehicle cycles N fo the aticula main-beam systems of the two-lane bidge vesus san length l fo 24h taffic volume v = 13.9 m/s. Fig. 25 shows gahs of values of the fou-lane bidge at 24h taffic volume seed v = 13.9 m/s. d N = 1 and 7000 vehicles and seed N fo the simly suoted main gides d N = 1 and vehicles and

24 296 Adam WYSOKOWSKI Fig. 22. Gahs of actual numbes of cycles geneated by one vehicle N fo main gides of two-lane bidge

25 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 297 Fig. 23. Gahs of actual numbes of cycles geneated by one vehicle fo main gides of two-lane bidge N

26 298 Adam WYSOKOWSKI Fig. 24. Gahs of actual numbes of cycles geneated by one vehicle N fo main gides of two-lane bidge Gahs of the aamete N fo a deck element deending on longitudinal sacing of coss-beams (t) at diffeent flexibility coefficients γ, ae shown in fig. 26a,b (fo longitudinal ibs and deck lates) and fig. 27 (fo coss-beams).

27 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 299 Fig. 25. Gahs of actual numbes of cycles geneated by one vehicle fo main gides of fou-lane bidge N Fig. 26. Gahs of actual numbes of cycles geneated by one vehicle fo longitudinal ibs (a, b) and deck lates (b) N Fig. 27. Gahs of actual numbes

28 300 Adam WYSOKOWSKI 6. ANALYSIS OF SERVICE STRENGTH RESULTS Examining the collected stess-ange secta fo the simly suoted main gides of the two-lane bidge (fig. 6), one can notice that thei eak anges α i occu at shot san lengths (1 = 2-6 m) and then at m. Thei lowest values and the highest fequency of occuence of naow-stess-ange cycles ae obseved at san lengths of 6-15 m. In addition, the tace of the 50% distibution function indicates that ove 50% of the sevice cycles have stess anges naowe than 10% of the standad-load stess ange σ n. A comaison of the stess-ange secta fo the mid-san coss-section with those fo the suot coss-section of the continuous two-san gide of the two-lane bidge (fig. 6) shows that fo the san lengths shote than 4 m and san lengths longe than 15 m the stess-ange values fo the mid-san cosssection ae highe than fo the suot coss-section wheeas fo san lengths of 4-15 m the oosite is tue. A simila comaative analysis fo the continuous thee-san gide suot coss-section fo diffeent lengths of the cental san (1.01 and 1.31) (fig. 6) shows that highe stess-ange values wee obtained fo the gide having all sans of equal length. Fo the two thee-san schemes ove 50% of the total numbe of sevice cycles has a stess ange naowe than 5% of the stess ange σ n (geneated by standad load). The highest stess-ange values in a sectum wee obtained fo the san coss-section of a gide with an infinite numbe of sans (fig.7) on igid suots, with the san length of 2.0 m (α =75%). Fo a beam with an infinite numbe of sans on flexible suots the obtained stess-ange values ae on the whole lowe than fo the beam on igid suots. A comaison of the stess-ange sectum fo the single-san simly suoted main gide of the fou-lane bidge (fig. 8) with the coesonding secta fo the two-lane bidge (fig. 6) shows that the stess-ange values in the fome case ae lowe than in the latte case. In addition, the fequency of the occuence of lowe stess-ange values, in the case of the main gide of the fou-lane bidge, is highe than the coesonding fequency fo the main gide of the two-lane bidge, as indicated by, among othe things, the taces of elevant distibution functions (10% and 50%). The highest stess-ange-sectum values ae aoximately constant, iesective of the longitudinal sacing t and the flexibility γ of the coss-bas, fo both the longitudinal ib suot coss-section (fig. 9) and the deck coss-beams (fig. 10). Only fo the longitudinal ib suot coss-section at the flexibility γ = 0.05 (most common in the case of othotoic bidge lates) fo the naow

29 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 301 sacing of coss-beams: t = m, the stess-ange values ae highe (40-50%). The analysis of gahs of the equivalent substitute stess ange coefficient α e and maximum stess ange utilization coefficient α max fo the aticula bidge elements, esented in figs 11-16, shows that: The values of coefficients α max and α e deend on san length l fo the main gide o sacing of the coss-beams t fo the deck elements. The α max gahs un oughly aallel to the α e gahs in the case of both the main beams and the deck elements. The highest value of the sevice stess ange utilization coefficient α e fo the main beams was ecoded fo the mid-san section of the thee-san gide with equal san lengths (system no. 5): α e = at l = 2.0 m. Sevice stess ange utilization α e fo the coss-bas is on the whole indeendent of thei flexibility and thei sacing in the deck t. The highest sevice stess ange utilization α e occus in the case of the deck lates and the longitudinal ib mid-san coss-section when the coss-bas ae igid (system no. 9) and saced longitudinally at t = 2.0 m α e = Fo the above scheme and aametes, the highest maximum stess-ange value α max = was ecoded fo the deck. The gahs of the single-vehicle-stess-ange-utilization coection coefficient α, shown in figs fo the aticula oad bidge elements, have a close similaity to the esective gahs of the equivalent substitute stess ange α e. The similaity follows fom, among othe things, the fom of fomula (2.17). In the case of simly suoted gide of shote sans (aticulaly with continuous static schemes) and deck elements, the actual numbe of cycles is, as a ule, much lage than the numbe of vehicles that ass ove the bidge. Thus, in such cases, the values α ae aoiately highe than the values α e. The analysis of the gahs of the actual single-vehicle numbe of cycles N esented in figues fo the aticula stuctual elements shows that: Numbe of cycles N deends on the coss-section and the static scheme, as well as on the san length l - in case of the main gides, and on the longitudinal sacing of coss-beams t - fo the deck elements. In case of the main gides, the gahs of the actual numbe of cycles e vehicle N goes down to N = 1.0; 2.0 o 3.0, as the san length inceases to 20.0 m (fo mid-san coss-sections) and 30.0 m (fo suot cosssections), deending on the system. As the san length l inceases futhe, the gahs couse uns hoizontally (constant numbe of cycles N ). This shae of the gahs and thei local inflexions ae detemined by the axle base

30 302 Adam WYSOKOWSKI and the system elative to the length of the influence-line banches deending on a static scheme and the san length. Geneally, the aveage numbe of stess-ange cycles e vehicle is lage fo suot coss-sections than fo the mid-san coss-sections. Fo the thee-san continuous gide with sans of equal length (fo both the suot coss-section and the mid-san coss-section), the actual singlevehicle numbe of stess-ange cycles N is slightly lage than fo the scheme in which the cental san is longe than the side ones. A noticeable effect of vehicles assing each othe on the bidge on the numbe of stess-ange cycles e vehicle N is ecoded in the case of the main gides fo san lengths ove m, and as the san length and 24h taffic volume N incease, the actual numbe of cycles e vehicle d N deceases (due to the simultaneous geneation of stess-ange cycles by the vehicles assing each othe). Taking into account the most common main gide san lengths and the tyical longitudinal sacing of coss-bas in the othotoic late tyical fo oad bidge stuctues, it can be stated that lage numbes of single-vehicle stessange cycles N occu fo the deck elements than fo the main gides. 7. COMPARATIVE ANALYSIS OF SIMULATION CALCULATION RESULTS AND IN SITU TEST RESULTS To comae the field test esults with the simulation esults, the autho calculated, using the fomulas given in chate 2, the sevice stength and duability aametes fo the elements of tested bidge stuctues. The calculations wee based on the stess-ange histogams obtained fom the tests caied out on actual bidge stuctues. The field test esults fo the elements of the steel highway bidges wee lotted on the gahs of the coesonding aametes, obtained fom comute simulations. The field-test values of the equivalent substitute stess-ange coefficient α e and the maximum stess-ange utilization coefficient α max fo the elements wee lotted in figs 11-13, 15 and 16. The field-test values of the single-vehicle stess-ange utilization coection coefficient α fo the elements wee lotted on the gahs shown in figs 17, 18, 20 and 21. The field-test values of the actual single-vehicle numbe of cycles N fo the elements wee lotted on the gahs in figs 22-24, 26 and 27.

31 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL 303 To sum u, the field-test values of the sevice stength fo the consideed elements have oved the assumtions made fo the theoetical calculations and the obtained ageement between the test esults and the calculation esults confims the comute simulation analyses of the sevice duability. 8. CONCLUSIONS On the basis of the analysis made in this ae the following geneal conclusions can be dawn: 1. The simulation comute softwae develoed by the autho makes it ossible to ente the data necessay fo calculations and to obtain esults in a concise fom, on the basis of which a quick assessment of the sevice stength and duability can be made fo any tye of bidges with diffeent static schemes, san lengths, coss-sections, numbe of taffic lanes, fo any numbe of tyes of vehicles and thei chaacteistics fo diffeent taffic volumes and vehicle tavelling seeds. 2. The substitute-vehicle stess-ange values yielded by the comute simulation, similaly as those obtained fom field tests on actual bidges, ae much lowe (by 15-75% deending on the kind of element and its san) than the calculated standad-moving-load stess-ange values. 3. The field-test values of sevice stength obtained fo the consideed elements have oved the assumtions made fo the theoetical calculations and thus they have validated sevice duability analyses made by comute simulation. 4. All the stuctual elements of the steel sans of highway bidges ae sensitive to fatigue and this hazad deends on such factos as: the kind of element, the static scheme, the flexibility of the suots, the length of the sans, the 24h taffic volume, the tavelling seed of vehicles, the kind of steel and the tye of notch. 5. Geneally, elements with continuous static schemes ae moe sensitive to fatigue than the simly suoted single-san elements. 6. In the case of the main gides it has been found that: The fatigue hazad is diectly linked to the length of the sans and it is the geatest fo san lengths below 8.0 m and then fo san lengths in the ange of m. Fo san lengths m and ove 40.0 m the fatigue hazad is elatively small. In case of continuous thee-san gides, as the length of the cental san inceases elative to the side sans, the fatigue hazad to of gides eases.

32 304 Adam WYSOKOWSKI The main gides of bidges with two taffic lanes ae moe exosed to fatigue hazad than the ones in bidges with a lage numbe of taffic lanes. The fatigue hazad fo the mid-san coss-sections is geate than to suot section, egadless of the numbe of sans. As the flexibility of the main gides inceases, thei fatigue hazad deceases. 7. Geneally, the deck elements ae moe sensitive to fatigue than the main gides. Futhemoe, it has been established that: As the longitudinal sacing of the coss-bas in the gate deceases, so does the fatigue hazad fo the longitudinal ibs. As the flexibility of the coss-beams inceases, the fatigue hazad fo the deck elements (deck lates, longitudinal ibs and coss-bas) deceases. The most seious fatigue hazad occus in the longitudinal ibs. Unde the cuent maximum 24h taffic volume N = 4000, egadless of the kind of steel and the tye of notch, one should be concened about the fatigue in the longitudinal ibs fo any sacing of coss-bas in the deck and any flexibilities of the coss-bas. The simulation EMC softwae develoed by the autho and the esented aoach to the assessment of the sevice stength and duability of highway bidges can be alied, without any limitations, to simila analyses of the san elements of one-tack and two-tack ailway bidges with any static schemes. d REFERENCES 1. Baus R., Buls A. (1981). Etude du comotement des onts en acie sous l action du tafic outie. Détemination des actions ou le calcul statique et le calcul à la fatigue. Cente de echeches scientifiques et techniques de l industie des fabications métalliques, CRIF, Buxelles. 2. B. S. (1980). 5400: Steel, Concete and Comosite Bidges, Pat 10: Code of Pactice fo Fatigue. Bitish Standads Institution, London. 3. Caena A. (1982). Fatigue Design Concet of the ECCS. IABSE Colloquium Fatigue of Steel and Concete Stuctues, Laussanne, Switzeland, IABSE Poceedings, vol. 37, Code UIC 778 1R. (1981). Recommandations elatives aux facteus de fatigue a considee los du dimensionnement des onts métalliques de chemin de fe. UIC, Pais. 5. Czudek H., Pietaszek T. (1980). Twałość stalowych konstukcji mostowych zy obciążeniach zmiennych. WKiŁ, Waszawa.

33 SIMULATION ANALYSIS OF FATIGUE STRENGTH IN STEEL Hit M. A. (1977). Neue Ekenntnisse auf dem Gebiet de Emüdung und deen Beücksichtigung bei de Bemessung von Eisenbahnbücken. Bauingenieu J. 52, H. 7, Hit M. A. (1983). Réseve de duée de vie des onts. Constuction Métallique, no.1, IIW (1981). Intenational Institute of Welding, Design Recommendations fo Cyclic Loaded Welded Steel Stuctues. Document JWG XII XV-53-81, Pais, Fance. 9. SIA 161 (1979). Nome, Constuctions Métalliques. Societe Suisse des Ingénieus et des achitectes /SIA/, Zuich, Switzeland. 10. Stie W., Steinhadt O., Valtinat G., Kosteas D. (1982). Residual Fatigue Life of Railway Bidges. IABSE Colloquium Fatigue of Steel and Concete Stuctues Lausanne, Switzeland, IABSE Poceedings, vol. 37, Wysokowski A. (1990). Wływ losowości uchu dogowego na wytzymałość eksloatacyjną mostów. Pace Instytutu Badawczego Dóg i Mostów, no 2, Wysokowski A. (1989). Duability Asects in the Design of Steel Highway Bidges. IABSE Symosium "Duability of Stuctues" Lisbon, Potugal. IABSE Poceedings, vol. 57/2, Wysokowski A. (1985). Oeational Stength of Steel Sans of Road Bidges,(in Polish), Doctoal Thesis,, Raoty Instytutu Inżynieii Lądowej Politechniki Wocławskiej, PRE n Wysokowski A., Łęgosz A. (1995). Monitoing of Bidge Conditions fo Duability Evaluation in Poland. IABSE Symosium Extending the Lifesan of Stuctues San Fancisco Wysokowski A. (1985). Estimation of the Oeational Duability of Steel Highway Bidges by the Simulation Method. Poceedings of ICOSSAR 85, the 4 th Intenational Confeence on Stuctual Society and Reliability. The Intenational Confeence Cente Kobe, Jaan May Wysokowski A. (1990). Oveview of Some Reseach Pojects on Duability of Bidges. Poceedings Bidge Engineeing Reseach in Pogess. National Science Foundation and Civil Engineeing Deatment Univesity of Nevada, Reno.

34 306 Adam WYSOKOWSKI NOTATION f i elative fequency of numbe of cycles n i numbe of stess-ange levels in the consideed no stationay sectum (x) foce values along the length of an influence line, q k tansvese load-distibution coefficient fo lane k, t diffeent coss-bas sacing y(x) odinates of the influence line of a static quantity (moment o a eaction) fo a consideed coss-section N N n N Q c actual numbe of cycles equivalent numbe of cycles numbe of vehicles geneating N stess cycles total weights of tucks fo county bidges a static stength constant, α e equivalent substitute stess-ange coefficient α max maximum stess-ange utilization coefficient α coection coefficient fo a deck element γ diffeent flexibility coefficients σ e equivalent stess ange σ i exession fo the next value of stess σ n standad-load stess ange σ min absolute value of minimum stess σ max maximum stess ange in the sectum ϕ actual dynamic actual-load stess-ange coefficient ϕ stand. standad-load stess ange dynamic coefficient