Journal of Constructional Steel Research 53 (2000) 283–305 www.elsevier.com/locate/jcsr Fatigue in steel structures under random loading Henning Agerskov * Department of Structural Engineering and Materials, Technical University of Denmark, Lyngby, Denmark Received 16 July 1998; received in revised form July 1999; accepted July 1999 Abstract Fatigue damage accumulation in steel structures under random loading is studied The fatigue life of welded joints has been determined both experimentally and from a fracture mechanics analysis In the experimental part of the investigation, fatigue test series have been carried through on various types of welded plate test specimens and full-scale offshore tubular joints The materials that have been used are either conventional structural steel with a yield stress of fyෂ360–410 MPa or high-strength steel with a yield stress of fyෂ810–1010 MPa The fatigue tests and the fracture mechanics analyses have been carried out using load histories, which are realistic in relation to the types of structures studied, i.e primarily bridges, offshore structures and chimneys In general, the test series carried through show a significant difference between constant amplitude and variable amplitude fatigue test results Both the fracture mechanics analysis and the fatigue test results indicate that Miner’s rule, which is normally used in the design against fatigue in steel structures, may give results, which are unconservative, and that the validity of the results obtained from Miner’s rule will depend on the distribution of the load history in tension and compression  2000 Elsevier Science Ltd All rights reserved Keywords: Steel structures; Fatigue; Random loading; Variable amplitude fatigue Introduction In the design of steel structures against fatigue, one of the problems that has attracted increasing attention in recent years is the problem of fatigue damage * Tel.: +45 45251706; fax: +45 45883282 E-mail address: ha@bkm.dtu.dk (H Agerskov) 0143-974X/00/$ - see front matter  2000 Elsevier Science Ltd All rights reserved PII: S - X ( 9 ) 0 - 284 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Nomenclature A fu fy I M m N n R ⌬s ⌬se r constant ultimate tensile strength yield stress irregularity factor Miner sum slope of S–N line total number of cycles number of cycles stress ratio stress range equivalent constant amplitude stress range correlation coefficient accumulation Codes and specifications normally give simple rules, using a Miner summation and based on the results of constant amplitude fatigue tests Over the years, fatigue test series have been carried through using different types of block loadings, and for these types of loading, Miner’s rule has in many cases been found to give reasonable results, see e.g [1–5] However, in a real steel structure the loading normally does not consist of loading blocks, but the structure is subjected to a stochastic loading, due to traffic, wind, waves, etc Thus, the need for a better understanding of the fatigue behaviour under more realistic fatigue loading conditions is obvious The question of the validity of Miner’s rule is the background for a series of research projects on fatigue in steel structures, carried out at the Department of Structural Engineering and Materials of the Technical University of Denmark over the last eight years The main purpose of these projects is to study the fatigue life of steel structures, primarily bridges, offshore structures and chimneys, under various types of stochastic loading that are realistic in relation to these types of structures The fatigue tests in these investigations have been carried out on various types of welded plate test specimens and full-scale offshore tubular joints The test specimens have been fabricated either in conventional structural steel with a yield stress of fyෂ360–410 MPa or in high-strength steel with a yield stress of fyෂ810–1010 MPa Besides the fatigue tests, these projects also include analytical determination of the fatigue life under the actual types of random loading by use of fracture mechanics, to be able to compare experimentally and theoretically determined fatigue lives The present paper gives an overview of the experimental and analytical investigations carried out, the types of loading used in the fatigue tests, and in the fracture mechanics analysis, and the main results obtained in the fatigue tests and in the analytical investigations H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 285 Experimental investigations 2.1 Welded plate test specimens The majority of the fatigue test series have been carried through using welded plate test specimens with transverse attachments These test specimens consist of a 40 or 90 mm wide main plate with two transverse secondary plates welded to the main plate by means of full penetration butt welds Two different dimensions have been used to study size effects The applied loading is a central normal force in the main plate The test specimens are shown in Fig For the plate test specimens with transverse attachments, both conventional structural steel with a yield stress of fy=400–409 MPa and an ultimate tensile strength, Fig Welded plate test specimens with transverse attachments 286 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 fu=537–575 MPa, and high-strength steel with fy=810–840 MPa and fu=845–875 MPa have been used [6,7] Four test series have been carried through on welded plate test specimens with longitudinal attachments For these test specimens, an 80 mm wide main plate is used, and two longitudinal attachments are welded to the main plate by means of fillet welds Also in this case, two different dimensions are used to study size effects [7] The material used for the plate test specimens with longitudinal attachments is high-strength steel with a yield stress, fy=965–1010 MPa and an ultimate tensile strength, fu=995–1045 MPa These test specimens are shown in Fig A third type of plate test specimen has been used in two test series This test specimen is a 40×8 mm plate with a transverse partial penetration butt weld A penetration of about 2/3 of the plate thickness has been used The material has a yield stress of fy=405 MPa and an ultimate tensile strength, fu=566 MPa [8] 2.2 Tubular joint test specimens The dimensions that have been chosen for the tubular joints, see Fig 3, correspond to a large number of the joints in the platforms of the Tyra Field in the North Sea Compared with the largest joints in these platforms, the actual test joints are approximately half size The test specimens are carried out as double T–joints The test joints are loaded in in-plane bending Each test series comprises three test specimens with two tubular joints in each of them In the investigation on joints in conventional offshore structural steel, the material used has a yield stress of fy=363–381 MPa and an ultimate tensile strength of fu=506– 548 MPa [6] In the test series on joints in high-strength steel, the material used in the fabrication of the test specimens is a quenched and tempered high-strength steel The yield stress of the material used is fy=823–830 MPa, and the ultimate tensile strength is fu=853– 863 MPa [9] After the first series of fatigue tests on the tubular joints in conventional offshore structural steel, the fatigue cracks in the joints were repair-welded, and the fatigue tests were repeated In the repair-welding, it has been emphasised that the welding procedures correspond as precisely as possible to procedures used presently in the North Sea in repair-welding of fatigue cracks in tubular structures After both series of tests had been carried through, the fatigue life of the repair-welded tubular joints could be compared with the fatigue life of the original joints [10] 2.3 Test equipment and test procedure 2.3.1 Plate test specimens The tests on the plate specimens have been carried out in two fixed test frames, one with a capacity of ±100 kN and the other with a capacity of ±500 kN The equipment — actuators, computers, valves, etc — have been chosen so that a high frequency is possible in the tests of these series H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Fig Welded plate test specimens with longitudinal attachments 287 288 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Fig Tubular joint test specimen Small eccentricities due to the welding of the test specimens are inevitable in these test series This results in additional secondary bending stresses at the joint Strain gages are used on all test specimens in these series to determine the resulting stresses from normal force and eccentricity moment 2.3.2 Tubular joint test specimens In the test equipment used in the investigation on the tubular joints, the test specimen has a fixed support in the central plane, whereas the rest of the test specimen is free to move The test joint is loaded in in-plane bending, using a 125 kN servocontrolled hydraulic actuator between the two secondary tubes Strain gages are used to determine the stresses in the test specimens Furthermore, the stresses in the most critical areas with respect to fatigue have been determined from finite element analysis and by use of the thermoelastic technique (SPATE) This is an experimental stress analysis technique based on the measurement of infrared radiant flux emitting from the surface of a body under cyclic stress [11–13] Fatigue crack propagation during the test is determined by use of the AC-potential drop technique A computer is used to store sequences of recorded maxima and minima of the load history H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 289 Variable amplitude loading Various types of random loading have been used in the fatigue tests and in the fracture mechanics determination of the fatigue life The load histories applied correspond to offshore structures, highway bridges and chimneys 3.1 Offshore structures In the investigations on offshore structures, five different types of load histories have been used These load histories are generated by a computer program, developed at the Department of Structural Engineering and Materials of the Technical University of Denmark [14] The program simulates a stationary Gaussian stochastic process in real time Only the extremes of the process are needed, since the load course between consecutive extremes is considered unimportant In the load simulation, a one-step Markov model is used In this load model, the next extreme to be generated will depend only on the present extreme, and not on the preceding load history, i.e it has a one-step memory Each column and each row in the Markov matrix contains cumulated transition probabilities The matrix element to be chosen, given a certain load level, is determined by use of a random number generator The elements in the matrix have been determined numerically from the spectral density function of the wave elevation spectrum [15,16] The load histories used in the investigations on offshore structures are equally in tension and compression and with irregularity factors, I, varying from 0.745 to 0.987 The irregularity factor is defined as the number of positive-going mean-value crossings divided by the number of maxima of the load history For narrow band loading, the irregularity factor will be close to unity Typical load histories for fixed offshore structures will be more broad banded, with irregularity factors in the range from ෂ0.6 to 0.8 Details of the load simulation procedures and the main characteristics of the various load histories used may be found in [6,17] Fig shows examples of typical load histories, generated by use of the matrices BROAD64 and PMMOD64 BROAD64 was evaluated from a truncated white noise spectral density function, and PMMOD64 from a modified Pierson–Moscowitz wave elevation spectrum 3.2 Highway bridges The variable amplitude loading that was used in the investigation on highway bridges has been determined from strain gage measurements on the orthotropic steel deck structure of the Farø Bridges in Denmark The measurements were carried out during the months of May and November [18,19] The load histories correspond to one week’s traffic loading Strain gage measurements were taken at 10 different locations in the orthotropic deck The load histories that have been used in the present investigation were measured by two strain gages, both placed on the bottom of one of the trapezoidal longitudinal stiffeners of the deck plate The stiffener chosen is located under the most heavily loaded lane of the motorway Strain gage No is 290 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Fig Examples of load histories from investigations on offshore structures 150 extremes generated by use of the matrices BROAD64 and PMMOD64, respectively placed in the middle of the longitudinal stiffener span, which has a length of m Strain gage No is placed at a distance of 0.5 m from one of the transverse diaphragms This means that the stresses measured by strain gage No are primarily tensile stresses, whereas the stresses registered by strain gage No are almost equal in tension and compression For strain gage No 1, the stress measurements were taken during the month of May, whereas for strain gage No 5, load histories were measured in both May and November Figs and show examples of typical load histories based on the measurements from strain gages No and 5, respectively In both cases, 200 extremes are included in the load history shown The load history based on strain gage No has an irregularity factor, I=0.617, while the load histories from strain gage No have I=0.793– 0.834 Further details of these load histories may be found in [19] Furthermore, one fatigue test series has been carried through, in which a cantilever bridge girder of a cable-stayed bridge during construction is studied The bridge girder is subjected to vertical oscillations due to transverse wind The stress history has been generated from the bridge girder response, simulating a 50-year storm with H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Fig Fig 291 Example of load history 200 extremes based on the measurements from strain gage No Example of load history 200 extremes based on the measurements from strain gage No 5, May five days duration This stress history is equally in tension and compression, and has an irregularity factor of I=0.888 [8] 3.3 Chimneys The variable amplitude loading that was used in the investigation on chimneys has been determined from wind tunnel tests An undamped chimney model, subject to transverse oscillations due to vortex shedding was studied Two load histories have been used, both determined from strain gage measurements on the model These load histories are equal in tension and compression and very narrow-banded, with an irregularity factor of I=0.998 [20] 292 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Fatigue test results In the following is given an overview of the main results that have been obtained in the various fatigue test series on the plate test specimens and the tubular joints A total of 520 fatigue tests on welded plate specimens have been carried out in these investigations, and 18 full-scale tubular joints have been tested In all investigations, initial test series with constant amplitude loading were carried out as a reference, and also — for the plate test specimens — to obtain the actual value of the exponent m for calculation of the equivalent stress ranges of the tests with variable amplitude loading, cf Eq (1) In the results from the variable amplitude tests, the stress parameter used is the equivalent constant amplitude stress range, ⌬se, defined as: ⌬seϭ ΄ ͸ i N m ΅ (ni·⌬smi ) (1) in which ni=number of cycles of stress range ⌬si; ⌬si=variable amplitude stress range; N=total number of cycles (=⌺ini); and m=slope of corresponding constant amplitude S–N line The cycle counting method that has been chosen for the analysis of the stress history is in all investigations “rainflow counting” This method is usually recommended for the analysis of random loading histories in steel structures [21] Examples of the results that have been obtained in the various test series are shown in the S–N diagrams in Figs 7–9 Both the linear regression S–N line from the Fig Results obtained from variable amplitude tests with BROAD64 spectrum Small plate specimens with transverse attachments Conventional structural steel H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 293 Fig Results obtained from variable amplitude tests with BROAD64 spectrum and constant amplitude tests Small plate specimens with transverse attachments High-strength steel Fig Results obtained from variable amplitude tests with load history from strain gage No 5, May Small plate specimens with transverse attachments Conventional structural steel 294 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 variable amplitude test series and the S–N line obtained in the corresponding constant amplitude test series are shown in Figs 7–9 Figs and show the results of the test series on the small plate specimens with transverse attachments and the offshore broad-band spectrum, BROAD64, for conventional structural steel and high-strength steel, respectively In the S–N diagrams, points marked with a star have not been included in the regression analysis These points correspond to a number of cycles to failure of 107 or more, and they are thus expected to be situated on the transition curve from the linear relationship up to about 107 cycles to a possible horizontal line at a high number of cycles Points marked with an arrow correspond to a test with a non-broken test specimen For the test results obtained in each test series on the plate specimens, the data are fitted to an expression: log Nϭ log AϪm· log ⌬s (2) by the method of least squares In Eq (2), m and A are constants, N is the number of cycles to failure, and ⌬s is the stress range Fig shows the results obtained in the test series on the small plate specimens with transverse attachments in conventional structural steel, and with highway bridge loading, strain gage No 5, May Details of the results obtained in the individual test series of the various investigations may be found in [7,17,19] When comparing the results obtained in the various test series, it appears that there is in general a significant difference between constant amplitude and variable amplitude fatigue test results This was specially pronounced in the investigations with stochastic loading corresponding to offshore structures, but the same tendency was observed in the other investigations The difference in fatigue life between constant amplitude and variable amplitude test results has in these investigations been quantified by the Miner sum, M, determined as the number of cycles to failure at variable amplitude loading, Nva, divided by the number of cycles to failure at constant amplitude loading, Nca, at the same equivalent stress range level When the slope of the linear regression S–N lines from variable amplitude and constant amplitude tests are not identical, M will be a function of the stress range level Miner’s rule, which is the cumulative damage rule generally used today in the design of steel structures against fatigue, assumes that fracture occurs for a Miner sum of M equal to The main observations with respect to the values of the Miner sum, M, obtained in the fatigue test series of the various investigations are given in the following 4.1 Offshore structures Conventional structural steel For the investigations on variable amplitude fatigue in offshore structures in conventional structural steel, the values of the Miner sum obtained are given in [22,23], for the test series with the various load histories investigated These values are based on approximately 160 fatigue tests H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 295 In Table 1, values of the Miner sum calculated at different stress range levels from the regression S–N lines that have been obtained, are given for the test series on the plate specimens with transverse attachments Four different offshore load histories, PM32, BROAD64, PMMOD64, and NARROW64 were investigated As may be seen from Table 1, the Miner sum corresponding to failure in the variable amplitude test series, varies in the range ෂ0.40–0.85, in all but one case The value for the small plate specimens and the load history PMMOD64 at ⌬se=100 MPa is given in brackets and should be taken with some precaution A value of M=1.35 is obtained, when the linear regression S–N lines are used directly However, the test results indicate that the transition curve for the constant amplitude test series starts already at 5–6×106 cycles and continues down to a fatigue limit of approximately 100 MPa at a high number of cycles Taking this into consideration, it seems appropriate to estimate the value of M to ෂ1.0 at ⌬se=100 MPa for this test series This is in good agreement with the value obtained in the fracture mechanics analysis For the tubular joints, the number of test results in the present investigation is too limited to make possible any significant statistical analysis If best fit S–N lines are determined, assuming a slope of m=3 for both constant amplitude and variable amplitude tests, a Miner sum of Mෂ0.6–0.8 is found for the variable amplitude tests on the tubular joints in conventional offshore structural steel With an irregularity factor, I=0.82–0.84 for the load histories used in the variable amplitude tests on the tubular joints, these results are in good agreement with the observations from the test series on the welded plate test specimens The fatigue cracks that developed during this first series of tests on the tubular joints were repair-welded according to specifications used presently in the North Sea in repair-welding of fatigue cracks in tubular structures After this, the fatigue tests were repeated It was expected beforehand that the repair-welded joints would have fatigue lives that were shorter than those of the original joints However, a comparison of the results obtained showed that the repair-welded joints had fatigue lives of 1.9–5.0 times the life of the original joints Furthermore, the fatigue crack initiation Table Values of Miner sum, M, at different equivalent stress range levels, ⌬se Test series on plate specimens with transverse attachments Conventional structural steel Load history PM 32 BROAD64 PMMOD64 NARROW64 Plate test specimen M Small Large Small Large Small Large Small Large ⌬se=100 MPa ⌬se=200 MPa ⌬se=300 MPa 0.46 0.62 0.51 0.38 (1.35) 0.38 0.84 0.53 0.49 0.66 0.44 0.41 0.67 0.56 0.82 0.62 0.51 0.69 0.41 0.43 0.45 0.71 0.80 0.68 296 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 in general moved from the chord wall in the first series of tests to the branch wall in the test series on the repair-welded joints The main reason for these observations is assumed to be the differences in weld shape and thus also in stress concentrations in the two cases Details of the results obtained in the investigation on repair-welded tubular joints may be found in [10] 4.2 Offshore structures High-strength steel In the investigations on offshore structures in high-strength steel, the values of the Miner sum obtained may be found in [7,24], for the various test series investigated A total of 170 fatigue tests were carried out in these test series Table gives values of the Miner sum calculated at different stress range levels from the regression S–N lines obtained, for the test series on the plate specimens with longitudinal and transverse attachments In these investigations, two different offshore load histories, BROAD64 and PMMOD64 were applied For the test series on both small and large plate specimens with longitudinal attachments, little difference in fatigue life was found between the series with constant amplitude loading and with the broad-band offshore load history, BROAD64, at an equivalent stress range of about 160–180 MPa At higher stress range levels, longer fatigue lives were obtained for the stochastic loading, while the constant amplitude loading at lower stress ranges resulted in longer fatigue lives, for both small and large test specimens In Table 2, one of the values of the Miner sum — at ⌬se=300 MPa — is given in brackets The reason for this is that the regression S–N line for the corresponding test series (BROAD64, small plate specimens) is based on results obtained in the stress range interval ⌬se=100–200 MPa For the test series on plate specimens with transverse attachments, there is in all series a significant difference between constant amplitude and variable amplitude fatigue test results For the test series on both small and large plate specimens, and Table Values of Miner sum, M, at different equivalent stress range levels, ⌬se Test series on plate specimens High-strength steel Type of test specimens Longitudinal attachments Transverse attachments Load history Size of test specimens M ⌬se=100 MPa ⌬se=200 MPa ⌬se=300 MPa Small 0.49 1.24 (2.13) Large 0.48 1.22 2.12 BROAD64 Small 0.50 0.52 0.54 PMMOD64 Large Small Large 0.63 0.55 0.56 0.66 0.61 0.81 0.68 0.65 1.01 BROAD64 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 297 with the two different load histories investigated, there is a clear indication, that the fatigue life is shorter with variable amplitude loading than with constant amplitude loading for the same stress range level It appears from Table that the Miner sum corresponding to failure in the variable amplitude test series, varies in the range ෂ0.50–0.80, in all but one case With respect to the tubular joints, the number of tests carried out is not sufficient to make possible any significant statistical analysis However, if best fit S–N lines are determined, assuming a slope of m=3 for both constant and variable amplitude tests, a Miner sum of Mෂ0.75 is obtained for the variable amplitude tests on the tubular joints in high-strength steel The irregularity factor is I=0.82 for the load history used in the variable amplitude tests on the tubular joints, and thus this result agrees well with the observations from the test series on the plate specimens 4.3 Highway bridges For the investigations with bridge traffic loading, determined from strain gage measurements on the Farø bridges, the main results obtained may be found in [19,25] The test series on the small and the large plate specimens gave Miner sums of 0.5– 1.0, corresponding to failure in the variable amplitude tests with the load history based on strain gage No These values of M correspond to stress range levels of approximately 90–200 MPa For the test series on the small plate specimens with the load history based on strain gage No 1, the interval of the equivalent constant amplitude stress range covered by the fatigue tests carried out is approximately 80–150 MPa In the corresponding constant amplitude test series with stress ratio, R=smin/smax=Ϫ1/5, fatigue failure occurred for stress ranges in the interval 100–275 MPa For the stress range area covered by both the test series with constant amplitude loading, R=Ϫ1/5, and the series with the load history based on strain gage No 1, values of the Miner sum of Mෂ1.2–1.8 were obtained Thus, in this case Miner’s rule apparently was found to give conservative predictions of the fatigue life However, the values of the Miner sum, MϾ1, obtained in the test series with the load history from strain gage No should be taken with some precaution due to the following reasons: The correlation coefficient for the linear regression in this test series is r=Ϫ0.69, which is a quite bad value, compared to the values of r obtained in the other test series (Ϫ0.82 to Ϫ0.98) Furthermore, the stress range interval covered by both test series (constant amplitude loading, R=Ϫ1/5, and load history from strain gage No 1) is small, from 100 to 150 MPa Finally, the test results indicate that the transition curve for the constant amplitude test series starts already at 2–3×106 cycles and continues down to a fatigue limit of approximately 105 MPa at a high number of cycles Taking the above into consideration, it seems appropriate to estimate the values of the Miner sum for the test series with the load history from strain gage No to ෂ1.0 This is in good agreement with the corresponding values of Mෂ0.8–0.9, obtained in the fracture mechanics analysis In the investigation, in which a cantilever bridge girder during construction is studied, and where the load history is a simulated 50-year storm of five days duration, 298 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 values of the Miner sum of Mෂ0.32–0.51 have been obtained for a stress range interval of ⌬se=100–400 MPa [8] 4.4 Chimneys In the investigation on chimneys, two test series were carried through with very narrow-banded loading In these test series, values of the Miner sum of Mෂ0.85– 1.16 were obtained The stress range interval covered was ⌬seෂ130–270 MPa Further details of these test series and the results obtained may be found in [20] Fracture mechanics prediction of fatigue life Besides the fatigue tests, the present investigations also include analytical determination of the fatigue life under the actual types of random loading by use of fracture mechanics, to be able to compare experimentally and theoretically determined fatigue lives Of special importance for the validity of the results that are obtained from the fracture mechanics analysis is the consideration of crack closure in the analytical model The crack growth analysis model used in the present investigations is based on the Dugdale–Barenblatt strip yielding assumption, with modifications to allow plastically deformed material to be left along the crack surfaces as the crack grows The crack closure model accounts for load interaction effects, such as retardation and acceleration, under variable amplitude loading The model may be used to simulate fatigue crack growth under both constant amplitude and variable amplitude loading, taking into account the influence of crack closure upon fatigue crack growth Furthermore, in the determination of the crack growth life the effects of stress concentrations and welding residual stresses are included More details of the crack growth model used may be found in [26–28] The fatigue lives have been calculated by use of fracture mechanics in two investigations: Offshore structures in conventional structural steel, and Highway bridges with load histories from Farø bridges S–N curves have been determined for both constant amplitude loading and variable amplitude loading In the investigation on offshore structures, a fracture mechanics determination of the fatigue life was carried out for both the small and the large plate test specimens with transverse attachments and with the three different load histories, BROAD64, PMMOD64, and NARROW64 Fig 10 shows the analytical results obtained for the small plate test specimens with these three load histories, together with the analytical results for constant amplitude loading [26] In the investigation on highway bridges, the fatigue lives have been calculated for the small plate test specimens with transverse attachments in conventional structural steel for constant amplitude loading and for variable amplitude loading using the load histories measured by strain gages No and No 5, May Fig 11 shows a comparison between the results obtained at constant amplitude loading and with the load history from strain gage No 5, May, for the small plate test specimens [28] H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 299 Fig 10 Comparison of analytical results for small plate test specimens with transverse attachments Conventional structural steel Offshore load histories and constant amplitude loading [26] Fig 11 Comparison of analytical results for small plate test specimens with transverse attachments Conventional structural steel Highway bridge load history and constant amplitude loading [28] 300 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 Table Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics analysis Small plate test specimens with transverse attachments Conventional structural steel [27] Load history BROAD64 PMMOD64 NARROW64 M ⌬se=120 MPa ⌬se=200 MPa ⌬se=250 MPa 0.75 0.88 0.95 0.38 0.47 0.76 0.34 0.41 0.65 Also in the analytical investigations, the values of the Miner sum, M, corresponding to failure in the variable amplitude series, have been determined The main observations with respect to the values of the Miner sum obtained in these investigations are given in the following 5.1 Offshore structures Conventional structural steel In the investigation on fracture mechanics determination of the fatigue life of offshore structures in conventional structural steel, the values of the Miner sum that have been obtained may be found in [26,27], for the various load histories investigated In Tables and 4, values of the Miner sum calculated at different stress range levels are given for three of the offshore load histories used in the test series on the small and the large plate specimens, respectively As may be seen from Tables and 4, the Miner sums corresponding to failure for the various offshore load histories studied varies in the range ෂ0.35–0.95 5.2 Highway bridges Load histories from Farø bridges The values of the Miner sum that were obtained in the investigation on fracture mechanics determination of the fatigue life of highway bridges, may be found in [19,28], for the load histories investigated Table Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics analysis Large plate test specimens with transverse attachments Conventional structural steel [27] Load history BROAD64 PMMOD64 NARROW64 M ⌬se=150 MPa ⌬se=200 MPa ⌬se=250 MPa 0.74 0.83 0.93 0.55 0.63 0.81 0.42 0.49 0.69 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 301 Table gives the values of the Miner sum that have been calculated at different stress range levels for two of the load histories, determined from strain gage measurements on the Farø bridges It appears from Table that the Miner sum corresponding to failure for the highway bridge load histories investigated varies in the range ෂ0.30–0.95 Observations When comparing the results obtained in the experimental investigations, it appears that there is in general a significant difference between constant amplitude and variable amplitude fatigue test results This observation was confirmed by the results obtained in the fracture mechanics analyses The main reason for these differences in fatigue behaviour is crack growth acceleration and/or retardation due to the high tensile and compressive loads of the variable amplitude load histories Important factors in this connection are crack closure mechanisms, the size of the welding residual stresses, the size of the stress concentrations, and the yield stress of the material Both the experimental and analytical investigations carried through show that acceleration effects dominate the fatigue crack growth for the load histories studied For the test series on both plate specimens and tubular joints, and with the different load histories investigated, it was found that the fatigue life in general is shorter with variable amplitude loading than with constant amplitude loading at the same equivalent stress range level, for load histories which are by and large equally in tension and compression This was observed both in the experimental investigations and in the fracture mechanics determination of the fatigue life In all the fracture mechanics analyses carried out, higher values of the Miner sum, M, were obtained at the lower equivalent stress range levels The reason for this is that the crack growth acceleration effects at the variable amplitude loading in general are less important at the lower stress levels This observation was not similarly clear for the test results In about 1/3 of the test series higher values of M were obtained at lower stress range levels, ⌬se; in about 1/3 of the test series, generally the same values of M were obtained for all values of ⌬se considered; and in about 1/3 of the test series, higher values of M were obtained at higher stress range levels Table Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics analysis Small plate test specimens with transverse attachments Conventional structural steel [19] Load history M Strain gage No ⌬se=100 MPa 0.94 ⌬se=150 MPa 0.81 ⌬se=200 MPa 0.68 ⌬se=250 MPa 0.61 Strain gage No 5, ⌬se=100 MPa May 0.77 ⌬se=200 MPa ⌬se=300 MPa ⌬se=400 MPa 0.38 0.30 0.33 302 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 In two cases, the test series with the bridge load history based on strain gage No and with the stresses primarily in tension, and the test series on plate specimens with longitudinal attachments in high-strength steel, the results showed — generally — the same fatigue life at variable and constant amplitude loading For the plate specimens with longitudinal attachments, values of MϽ1 were obtained at lower stress range levels, while at higher stress range levels, MϾ1 was found Thus, in this case the combination of a rather irregular load history (BROAD64), a high stress level, high stress concentrations, and a high-strength material resulted in the crack growth retardation effects being dominating, compared with the corresponding constant amplitude tests The results obtained in the various investigations carried through show that there is a clear tendency that the value of the Miner sum, M, corresponding to failure, decreases with the irregularity factor of the load history For the offshore load histories studied, which are equally in tension and compression and with irregularity factors ranging from Iෂ0.7 to 1.0, the results indicate that, taking the uncertainties into consideration, the use of a value of M=2•IϪ1, corresponding to failure, might be appropriate [9] The results obtained — both experimental and theoretical — show that the distribution of the load history in tension and compression has a significant influence on the validity of the results, which are obtained by use of Miner’s rule Conclusions A series of research projects on fatigue in steel structures, primarily bridges, offshore structures and chimneys, has been carried out at the Department of Structural Engineering and Materials of the Technical University of Denmark The main purpose of these projects has been to study the fatigue life under various types of random loading, which are realistic in relation to the types of steel structures investigated Comparisons between experimental results, results of fracture mechanics analysis, and results obtained using current codes and specifications, i.e Miner’s rule, are given In the experimental investigations, test series with a total of approximately 540 fatigue tests on welded plate specimens and full-scale tubular joints have been carried through The materials used are either conventional structural steel with a yield stress of fyෂ360–410 MPa or high-strength steel with a yield stress, fyෂ810–1010 MPa The experimental investigations in general show a significant difference between constant amplitude and variable amplitude fatigue test results For the variable amplitude tests, the stress parameter used is the equivalent constant amplitude stress range, ⌬se, and the difference in fatigue life between constant amplitude and variable amplitude test results is quantified by the Miner sum, M, determined as the number of cycles to failure at variable amplitude loading divided by the number of cycles to failure at constant amplitude loading, at the same equivalent stress range level The values of the Miner sum that were obtained in the variable amplitude test series, generally vary in the range ෂ0.40–0.85 for the test series with the offshore H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 303 load histories In the investigations on highway bridges and chimneys, values of the Miner sum of 0.50–ෂ1.0 were obtained The fatigue lives of the welded joints under both constant amplitude and variable amplitude loading have been determined theoretically by use of fracture mechanics These investigations have shown that the crack closure mechanisms are of paramount importance for the physical understanding and explanation of the fatigue crack growth under variable amplitude loading The main reason for the difference between constant amplitude and variable amplitude fatigue behaviour is crack growth acceleration and/or retardation due to the high tensile and compressive loads of the variable amplitude load histories The most important factors in this connection are the crack closure mechanisms, the size of the welding residual stresses, the size of the stress concentrations, and the yield stress of the material Both the experimental and analytical investigations carried through show that acceleration effects dominate the fatigue crack growth for the load histories studied A comparison of the experimental results and the results of the fracture mechanics calculations in general shows good agreement, when the calculations are based on the estimated values of the actual welding residual stresses and crack closure is included The results of the fracture mechanics calculations also show that Miner’s rule may give unconservative predictions of the fatigue life, since the Miner sums, which were obtained from the calculations in the present investigations, all are less than In most of the investigations carried through, e.g in all the fracture mechanics analyses, greater values of the Miner sum, M, were obtained at the lower equivalent stress range levels The reason for this is believed to be that the crack growth acceleration effects at the variable amplitude loading in general are less important at the lower stress levels The results obtained in the various investigations show that there is a clear tendency that the value of the Miner sum, M, corresponding to failure, decreases with the irregularity factor of the load history For the test series on both plate specimens and tubular joints, and with the different load histories investigated, it was found that the fatigue life in general is shorter with variable amplitude loading than with constant amplitude loading at the same equivalent stress range level, for load histories which are by and large equally in tension and compression This was observed both in the experimental investigations and in the fracture mechanics determination of the fatigue life The results obtained demonstrate that Miner’s rule, which is normally used in the design against fatigue in steel structures, may give quite unconservative predictions of the fatigue life, and that the distribution of the load history in tension and compression has a significant influence on the validity of the results, which are obtained by use of Miner’s rule On the basis of the results obtained in the present investigations, the following general recommendations can be given: For rather broad-banded types of random loading, which are by and large equal in tension and compression, a value of the Miner sum, corresponding to failure, of Mෂ1/3–1/2 should be used For rather nar- 304 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 row-banded types of random loading, which are primarily in tension, a value of M of ෂ0.8–1.0 seems appropriate However, it should be emphasized that in special cases both higher and lower values of M may be obtained Acknowledgements The funding for the various investigations carried out has been provided by the Danish Technical Research Council, the Nordic Fund for Technology and Industrial Development, the Technical University of Denmark, and SSAB Oxelo¨sund AB, Sweden, who are gratefully acknowledged The permission from the Road Directorate, Danish Ministry of Transport to carry out the strain gage measurements on the Farø Bridges is greatly appreciated References [1] Gurney TR Fatigue of welded structures 2nd ed Cambridge: University Press, 1979 [2] Maddox SJ Fatigue strength of welded structures 2nd ed Cambridge: Abington Publishing, 1991 [3] Blom AF editor Fatigue under spectrum loading and in corrosive environments Proc of the Nordic Conference on Fatigue West Midlands, UK: EMAS Publishers, 1993 [4] International Institute of Welding Fatigue Design Recommendations IIW Doc XIII-1539-94 Cambridge: Abington Publishing, 1994 [5] Maddox SJ, Prager M editors Performance of dynamically loaded welded structures Proc of the IIW 50th Annual Assembly Conference New York: Welding Research Council, 1997 [6] Ibsø JB, Agerskov H Fatigue Life of Offshore Steel Structures under Stochastic Loading Report No R 299 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1992 [7] Petersen RI, Agerskov H, Lopez Martinez L, Askegaard V Fatigue Life of High-Strength Steel Plate Elements under Stochastic Loading Report No R 320 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1995 [8] Carlsen R Udmattelse i Sta˚lbrodæk Udsat for Stokastisk Last, (Fatigue in Steel Bridge Deck Subjected to Stochastic Loading), in Danish, M.Sc thesis Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1994 [9] Petersen RI, Agerskov H, Lopez Martinez L Fatigue Life of High-Strength Steel Offshore Tubular Joints Report No R Dept of Struct Engrg and Materials, Techn Univ of Denmark, Lyngby, Denmark, 1996 [10] Agerskov H, Ibsø JB An Investigation on Fatigue in Repair-Welded Tubular Joints in Offshore Structures IIW Doc XIII-1525-94 International Institute of Welding, Annual Assembly, Beijing, China, 1994 [11] Askegaard V Prediction of Initial Crack Location in Welded Fatigue Test Specimens by the Thermoelastic Stress Analysis Technique Report No R 276 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1991 [12] Stanley P, Chan WK Quantitative stress analysis by means of the thermoelastic effect J of Strain Analysis 1985;20(3):129–37 [13] Stanley P, Chan WK A new experimental stress analysis technique of wide application Proc of the VIII’th Int Conf on Exptl Stress Analysis, Amsterdam, The Netherlands, 1986 [14] Aarkrog P A Computer Program for Servo Controlled Fatigue Testing Documentation and User Guide Report No R 253 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1990 [15] Gluver H One Step Markov Model for Extremes of Gaussian Processes Report No R 261 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1990 H Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305 305 [16] Krenk S, Gluver H A Markov Matrix for Fatigue Load Simulation and Rainflow Range Evaluation Proc of the Symposium on Stochastic Structural Dynamics Urbana, Illinois, USA, 1988 [17] Agerskov H, Ibsø JB Fatigue Life of Plate Elements with Welded Transverse Attachments Subjected to Stochastic Loading Blom AF, editor Proc of the Nordic Conference on Fatigue West Midlands, UK: EMAS Publishers, 1993 [18] Vejrum T, Nielsen JA Udmattelse i Sta˚lkonstruktioner Udsat for Stokastisk Last Brolast, (Fatigue in Steel Structures Subjected to Stochastic Loading Bridge Loading), in Danish, M.Sc thesis Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1993 [19] Nielsen JA, Agerskov H, Vejrum T Fatigue in Steel Highway Bridges under Random Loading Report No R 15 Dept of Struct Engrg and Materials, Techn Univ of Denmark, Lyngby, Denmark, 1997 [20] Esdahl S, Jacobsen KL Dynamik og Udmattelse i Sta˚lskorstene, (Dynamics and Fatigue of Steel Chimneys), in Danish, M.Sc thesis Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1995 [21] Almar-Næss A, editor Fatigue Handbook Offshore Steel Structures Trondheim, Norway: Tapir Publishers, 1985 [22] Agerskov H, Pedersen NT Fatigue life of offshore steel structures under stochastic loading J Struct Engng, ASCE 1992;118(8):2101–17 [23] Ibsø JB, Agerskov H Fatigue Life Prediction of Offshore Tubular Structures under Stochastic Loading Blom AF editor Proc of the Nordic Conference on Fatigue West Midlands, UK: EMAS Publishers, 1993 [24] Agerskov H, Petersen RI, Lopez Martinez L An Investigation on Fatigue in High-Strength Steel Offshore Structures IIW Doc XIII-1670-97 International Institute of Welding, Annual Assembly, San Francisco, USA, 1997 [25] Ha˚kansson S, Hansen PS Spændingsbestemmelse og Udmattelse i Sta˚lbrodæk, (Stress Analysis and Fatigue in Steel Bridge Decks), in Danish, M.Sc thesis Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1995 [26] Ibsø JB Fatigue Life Prediction of Welded Joints Based on Fracture Mechanics and Crack Closure Ph.D thesis Report No R 322 Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1995 [27] Ibsø JB, Agerskov H An analytical model for fatigue life prediction based on fracture mechanics and crack closure J Construct Steel Res 1996;37(3):229–61 [28] Agerskov H, Nielsen JA Fatigue in steel highway bridges under random loading J Struct Engng, ASCE 1999;125(2):152–62 [...]... equal to 1 The main observations with respect to the values of the Miner sum, M, obtained in the fatigue test series of the various investigations are given in the following 4.1 Offshore structures Conventional structural steel For the investigations on variable amplitude fatigue in offshore structures in conventional structural steel, the values of the Miner sum obtained are given in [22,23], for the... Also in the analytical investigations, the values of the Miner sum, M, corresponding to failure in the variable amplitude series, have been determined The main observations with respect to the values of the Miner sum obtained in these investigations are given in the following 5.1 Offshore structures Conventional structural steel In the investigation on fracture mechanics determination of the fatigue. .. branch wall in the test series on the repair-welded joints The main reason for these observations is assumed to be the differences in weld shape and thus also in stress concentrations in the two cases Details of the results obtained in the investigation on repair-welded tubular joints may be found in [10] 4.2 Offshore structures High-strength steel In the investigations on offshore structures in high-strength... research projects on fatigue in steel structures, primarily bridges, offshore structures and chimneys, has been carried out at the Department of Structural Engineering and Materials of the Technical University of Denmark The main purpose of these projects has been to study the fatigue life under various types of random loading, which are realistic in relation to the types of steel structures investigated Comparisons... found in [26–28] The fatigue lives have been calculated by use of fracture mechanics in two investigations: 1 Offshore structures in conventional structural steel, and 2 Highway bridges with load histories from Farø bridges S–N curves have been determined for both constant amplitude loading and variable amplitude loading In the investigation on offshore structures, a fracture mechanics determination... References [1] Gurney TR Fatigue of welded structures 2nd ed Cambridge: University Press, 1979 [2] Maddox SJ Fatigue strength of welded structures 2nd ed Cambridge: Abington Publishing, 1991 [3] Blom AF editor Fatigue under spectrum loading and in corrosive environments Proc of the Nordic Conference on Fatigue West Midlands, UK: EMAS Publishers, 1993 [4] International Institute of Welding Fatigue Design Recommendations... Martinez L Fatigue Life of High-Strength Steel Offshore Tubular Joints Report No R 1 Dept of Struct Engrg and Materials, Techn Univ of Denmark, Lyngby, Denmark, 1996 [10] Agerskov H, Ibsø JB An Investigation on Fatigue in Repair-Welded Tubular Joints in Offshore Structures IIW Doc XIII-1525-94 International Institute of Welding, Annual Assembly, Beijing, China, 1994 [11] Askegaard V Prediction of Initial... JA Udmattelse i Sta˚lkonstruktioner Udsat for Stokastisk Last Brolast, (Fatigue in Steel Structures Subjected to Stochastic Loading Bridge Loading) , in Danish, M.Sc thesis Dept of Struct Engrg., Techn Univ of Denmark, Lyngby, Denmark, 1993 [19] Nielsen JA, Agerskov H, Vejrum T Fatigue in Steel Highway Bridges under Random Loading Report No R 15 Dept of Struct Engrg and Materials, Techn Univ of Denmark,... is normally used in the design against fatigue in steel structures, may give quite unconservative predictions of the fatigue life, and that the distribution of the load history in tension and compression has a significant influence on the validity of the results, which are obtained by use of Miner’s rule On the basis of the results obtained in the present investigations, the following general recommendations... retardation and acceleration, under variable amplitude loading The model may be used to simulate fatigue crack growth under both constant amplitude and variable amplitude loading, taking into account the influence of crack closure upon fatigue crack growth Furthermore, in the determination of the crack growth life the effects of stress concentrations and welding residual stresses are included More details