Microsoft Word - numero_42_art_15.docx Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 136 Focused on Mechanical Fatigue of Metals Statistical analysis of fatigue crack propagation data of materials from ancient portuguese metallic bridges J.A.F.O. Correia, A.M.P. De Jesus, R. Calçada Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal. jacorreia@inegi.up.pt, http://orcid.org/0000-0002-4148-9426 ajesus@fe.up.pt, http://orcid.org/0000-0002-1059-715X ruiabc@fe.up.pt, http://orcid.org/0000-0002-2375-7685 B. Pedrosa, C. Rebelo, Luis Simões da Silva ISISE, Department of Civil Engineering, University of Coimbra, Rua Luís Reis Santos, Pólo II, 3030-788 Coimbra, Portugal. brunoalex_pedrosa@hotmail.com, crebelo@dec.uc.pt G. Lesiuk Faculty of Mechanical Engineering, Department of Mechanics, Material Science and Engineering, Wrocław University of Science and Technology, Smoluchowskiego 25, 50-370 Wrocław, Poland Grzegorz.Lesiuk@pwr.edu.pl, https://orcid.org/0000-0003-3553-6107 ABSTRACT. In Portugal there is a number of old metallic riveted railway and highway bridges that were erected by the end of the 19th century and beginning of the 20th century, and are still in operation, requiring inspections and remediation measures to overcome fatigue damage. Residual fatigue life predictions should be based on actual fatigue data from bridge materials which is scarce due to the material specificities. Fatigue crack propagation data of materials from representative Portuguese riveted bridges, namely the Pinhão and Luiz I road bridges, the Viana road/railway bridge, the Fão road bridge and the Trezói railway bridge were considered in this study. The fatigue crack growth rates were correlated using the Paris’s law. Also, a statistical analysis of the pure mode I fatigue crack growth (FCG) data available for the materials from the ancient riveted metallic bridges is presented. Based on this analysis, design FCG curves are proposed and compared with BS7910 standard proposal, for the Paris region, which is one important fatigue regime concerning the application of the Fracture Mechanics approaches, to predict the remnant fatigue life of structural details. KEYWORDS. Fracture Mechanics; Fatigue Crack Growth; Statistical Analysis; Old Steels; Ancient Bridges; BS7910. Citation: Correia, J.A.F.O., De Jesus, A.M.P., Calçada, R., Pedrosa, B., Rebelo, C., Simões da Silva, L., Lesiuk, G., Statistical Analysis of Fatigue Crack Propagation Data of Materials from Ancient Portuguese Metallic Bridges, Frattura ed Integrità Strutturale, 42 (2017) 136-146. Received: 08.06.2017 Accepted: 17.06.2017 Published: 01.10.2017 Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 137 INTRODUCTION n Portugal, the major concern of governmental agencies is related with the maintenance and safety of centenaries riveted steel bridges. These old riveted road and railway bridges were fabricated and placed into service during the 19th century and beginning of 20th century. The traffic conditions both in terms of vehicle gross weight and frequency, are today completely different from those considered in the design phase. Additionally, the original design procedures of these bridges did not account for fatigue, since fatigue understanding only achieved the maturity after the design of those structures. In the 19th century the designer engineers were not aware of some important damage phenomena including fatigue. Fatigue was only intensively studied in the 20th century. In order to maintain the high safety levels of old riveted steel bridges, road and railway authorities have to invest heavily in inspection, maintenance and retrofitting, with those activities supported by fatigue assessment studies, including remnant life calculations [1]. The crack propagation data is essential to perform fatigue life predictions according to the Linear Elastic Fracture Mechanics (LEFM), which is an important alternative to the usual code-based S-N curve procedures, mainly in what concerns residual life estimations. In this perspective, the knowledge about design fatigue crack growth curves for materials from ancient Portuguese steel bridges is extremely appropriate [1]. The present paper reports research work carried out to determine the design FCG curves of materials from ancient Portuguese riveted bridges, namely the Pinhão, Fão and Luiz I road bridges, built in 1906, 1891 and 1886, respectively, the Eiffel road/railway bridge (inaugurated in 1878) and the Trezói railway bridge (inaugurated in 1956). The design FCG curves were obtained using the procedure proposed by Gallegos Mayorga et al. [2] and a comparison with design crack propagation curves proposed by BS7910 standard [3] is made. The authors have investigated the mechanical behaviour of those bridge materials, such as, metallographic, chemical composition, monotonic and fatigue behaviours of the materials of the old riveted steel bridges have been characterized [4-7]. Others authors have performed similar work for other centenary bridges [8,9]. Correia et al. [6,10,11] and Bogdanov et al. [12] have proposed probabilistic curves for the fatigue crack growth propagation curves for several materials. Correia et al. [6,10,11] proposed the probabilistic fields for fatigue crack growth using the probabilistic fatigue local approaches using old materials from riveted steel bridges and current steels. Meanwhile, Bogdanov et al. [12] proposed a probabilistic analysis of the fatigue crack growth rates based on the Monte-Carlo method applied to the Unigrow model. FATIGUE LIFE EVALUATION BASED ON FRACTURE MECHANICS ypically, the fatigue life predictions of structural details based on Fracture Mechanics are used for residual fatigue life assessments containing initially known defects acting as cracks and are typically used as evaluation criteria for planning in-service inspections [13]. De Jesus et al. [13] evaluated the residual fatigue life of an ancient riveted steel road bridge using Fracture Mechanics approach based on experimental fatigue crack propagation data obtained for the old material from the Pinhão bridge. Thus, fatigue crack propagation data are fundamental to be used in fatigue life prediction approaches using Fracture Mechanics. In this sense, design fatigue crack growth curves are extremely important to establish and maintain the safety levels of the structural details and consequently of the structures. In Fig. 1, a schematic bi-logarithm representation of the three fatigue crack growth regimes, between /da dN (crack growth rate) and K (stress intensity factor range), is showed. This behaviour shows two vertical asymptotes: one on the left, at thK K   , indicates that K values below this threshold level are too low to cause macro crack growth; and the right asymptote occurs for a ∆K cycle with max cK K , which leads to complete failure of the specimen. These three regimes can be denominated as: I – the threshold region; II – the Paris region; III - the unstable tearing crack growth region [14]. Paris and Erdogan [15] established a power function (Eq. (1)) to describe the relation between /da dN and K :   mda C K dN    (1) where C and m are material parameters. This law is used to describe the so-called Paris region and the experimental data follows a linear relation when using bi-logarithmic scales are used, as shown in Fig. 1. I T Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 138 Figure 1: Three crack growth regimes for ( /da dN ) versus K [11]. The integration of the fatigue crack propagation laws should be made between the initial crack size, ia , and the final crack size, fa , as indicated by Eq. (2): 1 f i a f m a N da C K   (2) The final crack size, , is established by unstable crack propagation, dictated by material toughness, or plastic failure at the net section. The initial crack size, , is assumed around 0.25 to 1 mm for metals underestimating fatigue life of the component [16-18]. Furthermore, a crack depth of 0.5 mm can be assumed in Fracture Mechanics analysis if it is not indicated by the available standards [3]. The stress intensity factor, , can be evaluated using the weight functions [19] and finite element analysis or using finite element analysis [20] to calculate the stresses and displacements on the crack front followed by the implementation of the virtual crack closure technique (VCCT) [21]. Alternatively, this parameter can be obtained by analytical relations that were established by several authors depending of the geometry. The definition of a design fatigue crack propagation curve for current steels and old materials is of great importance for obtaining safe residual fatigue lives of structural details from old metallic bridges. STATISTICAL EVALUATION OF EXPERIMENTAL FATIGUE CRACK PROPAGATION DATA Statistical procedure n this paper, the statistical procedure to determine the design curves for the experimental fatigue crack propagation data used was proposed by Gallegos Mayorga et al. [2]. This procedure follows the same recommendations that were proposed by ASTM E739-91 standard [22]. This statistical procedure is based on the linear Paris law that is described by Eq. (3), where da dN is the fatigue crack growth (FCG) rates, K is the applied stress intensity factor range, C and m are material constants, * log da da dN dN            ,  * logC C , *m m and  * logK K   .   * ** *da C m K dN          (3) I Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 139 This procedure considers that the fatigue crack growth (FCG) data pertain to a random sample where all * i da dN       are independent and there are no run-outs or suspended tests for the entire range of  *K . Furthermore, the linear model for the Paris relation can be rewritten as Eq. (4) shows, where * i da dN       is the estimative (estimator) for the values of FCG rates.    * ** * i i da C m K dN          (4) The characterization parameters of the statistical analysis, such as the variance and the standard deviation must be defined. Regarding the variance of the log-normal distribution, it is constant and maximum likelihood estimators of *C and *m are, respectively, defined by Eqs. (5) and (6).     * * * ** * * 1 1 k k i i i i da K dadN C m m K k k dN                     (5)           * * * * 1 * 2 * * 1 k ii i k ii da da K K dN dN m K K                         (6) where * da dN       is the average values of * i da dN       ,  *K is the average value of  * i K and k is the total number of readings during the test by specimen. The average values * da dN       and  *K are determined as shown in Eq. (7).     * * * * ;   k k i i i i da K da dN K k dN k                (7) where  * i K represents the computed during the test of the stress intensity factor ranges and * i da dN       represents the readings during the test of the FCG rates. Concerning the standard deviation of the normal distribution for  log K , it is defined through the Eq. (8). * * 1 2 k i i i da da dN dN S k                  (8) Aiming the definition of a design FCG curve, rectilinear confident bands were defined as Eq. (9) shows, where  is an integer. Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 140       * * ** * * *da C m K S C S m K dN                   (9) In this analysis, it is assumed that = 2 which means that the confident band will cover approximately 95%. Design FCG curve is defined through the upper boundary of this confident band. The material parameters and are defined by Eqs. (10) and (11).  * 2 10 C S C   (10) *m m (11) This statistical procedure can be completed for several slopes in the FCG law using the notes proposed by Bogdanov et al. [12]. In these notes, the FCG law may have three or four slopes identified [12,23,24], hence three or four pairs of  * *,i iC m coefficients that are needed to fit the experimental fatigue crack propagation data. Each pair { * *,i iC m } corresponds to a segment with linear behaviour between  log /da dN and  log K values. Slopes  *i im m are obtained using Eq. (6). The materials constants of fatigue crack growth { * *,i iC m } and standard deviations can be obtained using Eqs. (5), (6) and (8). Several authors have discussed the evaluation of the fatigue crack propagation rates using statistical assumptions [6,7,10, 12,23,24], demonstrating the importance that the subject raises in the scientific community and engineers. Experimental data The experimental fatigue crack growth data from the old riveted metallic bridges are collected for the statistical analysis proposed by Gallegos Mayorca et al. [2] aiming at obtaining the design curves for these materials. The experimental fatigue crack propagation data was derived accordingly ASTM E647 standard procedures [25]. This standard establishes the geometry of Compact Tension specimens – CT specimens and Middle Tension specimens – MT specimens. CT specimens were used for materials from Eiffel ( W = 40 mm; B = 4.35 mm), Fão ( W = 50 mm; B = 8 mm), Pinhão ( W = 40 mm; B = 4.35 mm) and Trezói ( W = 50 mm; B = 8 mm) bridges. Specimens from Luiz I bridge were manufactured as MT specimens ( W = 40 mm; B = 10 mm). All tests were carried out under a sinusoidal waveform with a frequency of 20 Hz except for Luiz I bridge specimens that were tested at a frequency of 10 Hz. Two travelling microscopes with accuracy of 0.001 mm were used to measure the crack growth on both faces of the specimens by direct visual inspection. Regarding the number of tested specimens, five were manufactured from the Eiffel bridge (four according to the transverse direction and one according to the longitudinal direction), twelve from the Fão bridge, thirteen from the Pinhão bridge (six from a diagonal and seven from a bracing), eight from the Trezói bridge and four specimens from the Luiz I bridge [1]. The following stress ratios were investigated for each material: ‐ Eiffel bridge: R = 0.1 and R = 0.5; ‐ Luiz I bridge: R = 0.1; ‐ Fão bridge: R = 0.1; ‐ Pinhão bridge: R = 0.0, R = 0.1 and R = 0.5; ‐ Trezói bridge: R = 0.0, R = 0.25 and R = 0.5. Experimental results from all tested specimens are presented in Fig. 2. In each case, fatigue crack growth data is correlated using the previously referred power law developed by Paris and Erdogan [15] (see Eq. (1)). Application and discussion The statistical analysis described in the research work proposed by Gallegos Mayorca et al. [2] was used to estimate the probabilistic field of the FCG data for all old materials from the ancient riveted metallic bridges. The C and m parameters for the materials from the Eiffel and Fão bridges were estimated by Gallegos Mayorca et al. [2]. The FCG constants of the material from Eiffel bridge using the statistical procedure are the following: * 17.614C   Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 141 ( 171.199 10C   ), *m =4.69 (m=4.69) and 0.3463S  . For the material from the Fão bridge, the FCG constants are: * 15.126C   ( 151.237 10C   ), *m =4.03 ( 4.03m  ) and 0.2378S  . All FCG constants were obtained with /da dN in mm/cycle and K in N.mm-1.5. a) b) c) d) e) Figure 2. Crack propagation data correlated with the Paris law: (a) Eiffel; (b) Luiz I; (c) Fão; (d) Pinhão; (e) Trezói. Figs. 3 to 8 show the experimental FCG data, mean curve and the mean curve 2S (5% and 95% of probability of failure). All experimental results and probabilistic fields for the fatigue crack propagation data were compared with the design FCG curve of the Stage B (mean curve 2S , with 2.88m  and 136.77 10C   where /da dN is in /mm cycle and K in 1.5.N mm  ) proposed by the BS7910 standard [3]. It should be noted that the design FCG curve proposed in Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 142 the BS7910 standard [3] is indicated for current structural steels. This standard use the mean curve 2S to describe the design curve for the fatigue crack growth rates. In Tab. 1 the C and m parameters are presented for all materials from Portuguese metallic bridges and also the standard deviation, S , which is used to define the mean FCG curve 2S . This design curve is important to be used in rehabilitation studies of historical bridges (to analyse the fatigue residual life of structural details). The design FCG curve proposed by BS7910 standard when compared with each material from the Eiffel, Luiz I and Fão bridges revealed not be representative of these materials. However, when compared to the experimental FCG data of the Pinhão and Trezói bridges, the design FCG curve proposed by BS7910 standard, seems to be more representative of these old materials. It should be noted that the materials of the Pinhão and Trezói bridges are more recent when compared to the other materials. In the Fig. 8, the statistical analysis using the mean FCG curve 2S applied to the experimental data for all materials from ancient riveted bridges proved to be reasonably satisfactory, however the use of the mean FCG curve 3S to cover all experimental data is suggested. The slopes of the FCG curves of the materials from the Eiffel, Luiz I and Fão bridges revealed to be similar, indicating the tendency exhibited for old materials. However, the slopes of the FCG curves for the materials from Pinhão and Trezói bridges exhibited a consistent behaviour with the slope of the design FCG curve proposed by BS 7910 standard [3]. The statistical analysis applied to FCG data proved to be efficient. Material *C S C *m m Eiffel 17.614 0.3463 171.199 10 4.69 Luiz I 19.543 0.2548 209.243 10 5.50 Fão 15.126 0.2378 152.237 10 4.03 Pinhão 14.539 0.1082 154.757 10 3.62 Trezói 14.278 0.1317 159.660 10 3.54 All materials 13.998 0.2967 143.940 10 3.47 Table 1: Constants of the mean FCG curve for all materials from the Portuguese old metallic bridges, with /da dN in /mm cycle and K in 1.5.N mm  . Figure 3: Statistical analysis of the FCG data for the material from the Eiffel bridge. 1.0E‐8 1.0E‐7 100 500 1000 1500 1.0E‐6 1.0E‐5 1.0E‐4 1.0E‐3 1.0E‐2 1.0E‐1 [N.mm‐1.5] [m m /c y cl e ] Mean curve ‐ Exp. Data mean FCG curve + 2S mean FCG curve ‐ 2S Mean curve + 2S: Stage B; BS7910 Mean curve: Stage B; BS7910 2T1 (R=0.1) 2T2 (R=0.1) 2T3 (R=0.5) 2T4 (R=0.5) 2L1 (R=0.1) Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 143 Figure 4: Statistical analysis of the FCG data for the material from the Luiz I Bridge. Figure 5: Statistical analysis of the FCG data for the material from the Fão Bridge. Figure 6: Statistical analysis of the FCG data for the material from the Pinhão Bridge. S1 (R=0.0) S2 (R=0.0) S3 (R=0.0) S4 (R=0.0) Mean curve +2S: Satge B; BS7910 Mean curve: Stage B; BS 7910 Mean curve ‐ Exp. Data mean FCG curve + 2S mean FCG curve ‐ 2S 100 500 1000 1.0E‐6 1.0E‐5 1.0E‐4 1.0E‐3 1.0E‐2 [N.mm‐1.5] [m m /c y cl e ] 1.0E‐8 1.0E‐7 100 500 1000 1500 1.0E‐6 1.0E‐5 1.0E‐4 1.0E‐3 1.0E‐2 1.0E‐1 [N.mm‐1.5] [m m /c yc le ] P‐00 (R=0.0) P‐25 (R=0.25) P‐05 (R=0.5) P‐75 R=0.75) Mean curve ‐ Exp. Data mean FCG curve + 2S mean FCG curve ‐ 2S Mean curve + 2S: Stage B; BS7910 Mean curve: Stage B; BS7910 B1 (R=0.0) B2 (R=0.0) B3 (R=0.0) B4 (R=0.0) D1 (R=0.0) D2 (R=0.0) B1 (R=0.1) D1 (R=0.1) D2 (R=0.1) B1 (R=0.5) B2 (R=0.5) D1 (R=0.5) D2 (R=0.5) Mean curve ‐ Exp. Data mean FCG curve ‐ 2S mean FCG curve + 2S Mean curve +2S: Stage B; BS7910 Mean curve: Stage B; BS7910 300 500 1000 1500 1.0E‐6 1.0E‐5 1.0E‐4 1.0E‐3 1.0E‐2 [N.mm‐1.5] [m m /c y cl e ] Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 144 Figure 7: Statistical analysis of the FCG data for the material from the Trezói Bridge. Figure 8: Statistical analysis of the FCG data for all materials from ancient riveted bridges. CONCLUSIONS he statistical procedure used to analyse the experimental FCG data of the materials from the Portuguese ancient metallic bridges proved to be efficient. The design FCG curves proposed by BS7910 standard are not representative of all materials from old metallic bridges. The slopes of the design curves for the fatigue crack growth data of the materials from Eiffel, Luiz I and Fão bridges revealed to be similar. However, the slopes of the design curves for the FCG data of the materials from Pinhão and Trezói bridges exhibited a consistent behaviour and similar when compared with the slope of the design FCG curve proposed by BS 7910 standard. It should be noted that the latter materials are more recent and have mechanical properties similar to current constructional steels. Further statistical analysis of the experimental FCG data from old materials is therefore necessary to better represent their behaviour. A comparison between this statistical analysis for FCG curve using several pairs { *iC , * im } and the probabilistic approaches for evaluating the FCG rates using local approaches should also be performed. ACKNOWLEDGEMENTS he authors of this paper thank the SciTech-Science and Technology for Competitive and Sustainable Industries, R&D project NORTE-01-0145-FEDER-000022 co-financed by Programme Operational Regional do Norte ("NORTE2020") through Fundo Europe de Desenvolvimento Regional (FEDER) and the Portuguese Science R=0.0 R=0.25 R=0.5 Mean curve ‐ Exp. Data mean FCG curve + 2S mean FCG curve ‐ 2S Mean curve +2S: Stage B; BS7910 Mean curve: Stage B; BS7910 100 500 1000 1500 1.0E‐6 1.0E‐5 1.0E‐4 1.0E‐3 1.0E‐2 [N.mm‐1.5] [m m /c y cl e ] 2000 Luiz I Pinhão Eiffel Trezói Fão Mean curve ‐ Exp. Data mean FCG curve + 2S mean FCG curve ‐ 2S Mean curve +2S: Satge B; BS7910 Mean curve: Satge B; BS790 100 2000 1.0E‐7 1.0E‐6 1.0E‐5 1.0E‐3 1.0E‐2 [N.mm‐1.5] [m m /c y cl e ] 1.0E‐4 1000500 1500 T T Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 145 Foundation (FCT) through the post-doctoral grant SFRH/BPD/107825/2015 the for their collaboration, financial and technical support during these research works. The authors gratefully also acknowledge to the Institute for Sustainability and Innovation in Structural Engineering (ISISE) by financial support through of the European project which is named of ProLife - Prolonging Life Time of Old Steel and Steel-Concrete Bridges (RFSR-CT-2015-00025) by Research Fund for Coal and Steel (RFCS). REFERENCES [1] Correia, J.A.F.O., Jesus, A.M.P., Figueiredo, M.A.V., Ribeiro, A.S., Fernandes, A.A., Variability analysis of fatigue crack growth rates of materials from ancient Portuguese steel bridges. Proceedings of the 4th International Conference on Bridge Maintenance, Safety and Management, (2008) 290-291. [2] Gallegos Mayorga, L., Sire, S., Correia, J.A.F.O., De Jesus, A.M.P., Rebelo, C., Fernández-Canteli, A., Ragueneau, M., Plu, B., Statistical evaluation of fatigue strength of double shear riveted connections and crack growth rates of materials from old bridges, Engineering Fracture Mechanics, (in press). [3] BS7910:2005, Guidance on Methods for Assessing the Acceptability of Flaws in Metallic Structures, BSI, (2005). [4] De Jesus, A.M.P., Da Silva, A.L.L., Correia, J.A.F.O., Fatigue of riveted and bolted joints made of puddle iron— A numerical approach, Journal of Constructional Steel Research, 102 (2014) 164–177. [5] De Jesus, A.M.P., Silva, A.L.L., Figueiredo, M.V., Correia, J.A.F.O., Ribeiro, A.S., Fernandes, A.A., Strain-life and crack propagation fatigue data from several Portuguese old metallic riveted bridges, Engineering Failure Analysis, 17 (2010) 1495–1499. [6] Correia, J.A.F.O., De Jesus, A.M.P., Fernández-Canteli, A., A procedure to derive probabilistic fatigue crack propagation data, International Journal of Structural Integrity, 3(2) (2012) 158-183. [7] Sampayo, L.M.C.M.V., Monteiro, P.M.F., Correia, J.A.F.O., Xavier, J.M.C., De Jesus, A.M.P., Fernandez-Canteli, A., Calçada, R.A.B., Probabilistic S-N Field Assessment for a Notched Plate Made of Puddle Iron from the Eiffel Bridge with an Elliptical Hole, Procedia Engineering, 114 (2015) 691-698. [8] Lesiuk, G., Szata, M., Bocian, M., The mechanical properties and the microstructural degradation effect in an old low carbon steels after 100-years operating time, Archives of Civil and Mechanical Engineering, 15 (4) (2015) 786-797. [9] Kucharski, P., Lesiuk, G., Szata, M., Skibicki, D., Description of fatigue crack growth in steel structural components using energy approach-Influence of the microstructure on the FCGR, AIP Conference Proceedings, 1780 (1) (2016) 050003. [10] Correia, J.A.F.O., De Jesus, A.M.P., Fernández-Canteli, A., Local unified probabilistic model for fatigue crack initiation and propagation: Application to a notched geometry. Engineering Structures, 52 (2013) 394-407. [11] Correia, J.A.F.O., De Jesus, A.M.P., Fernández-Canteli, A., Calçada, R.A.B., Modelling probabilistic fatigue crack propagation rates for a mild structural steel, Frattura ed Integrita Strutturale, 31 (2015) 80-96. [12] Bogdanov, S., Mikheevskiy, S., Glinka, G., Probabilistic Analysis of the Fatigue Crack Growth Based on the Application of the Monte-Carlo Method to Unigrow Model, Materials Performance and Characterization, 3(3) (2014) 214–231. [13] De Jesus, A.M.P., Figueiredo, M.A.V., Ribeiro, A.S., de Castro, P.M.S.T., Fernandes, A.A., Residual Lifetime Assessment of an Ancient Riveted Steel Road Bridge, Strain: An International Journal for Experimental Mechanics, 47(1) (2011) 402–415. [14] Schijve, J., Fatigue of Structures and Materials, Kluwer Academic Publishers, New York, (2004). [15] Paris, P., Erdogan, F., A critical analysis of crack propagation laws, Trans. ASME, Series D, 85 (1963) 523-535. [16] Joint service specification guide aircraft structures, JSSG-2006. United States of America: Department of Defense; (1998). [17] Gallagher, J.P., Berens, A.P., Engle Jr, R.M., USAF damage tolerant design handbook: guidelines for the analysis and design of damage tolerant aircraft structures, Final report. (1984). [18] Merati, A., Eastaugh, G., Determination of fatigue related discontinuity state of 7000 series of aerospace aluminum alloys. Eng Failure Anal, 14(4) (2007) 673–685. [19] Hafezi, M.H., Abdullah, N.N., Correia, J.A.F.O., De Jesus, A.M.P., An assessment of a strain-life approach for fatigue crack growth, Int. J. Struct. Integrity, 3(4) (2012) 344–376. [20] Correia, J.A.F.O., Blasón, S., De Jesus, A.M.P., Canteli, A.F., Moreira, P.M.G.P., Tavares, P.J., Fatigue life prediction based on an equivalent initial flaw size approach and a new normalized fatigue crack growth model, Engineering Failure Analysis, 69 (2016) 15-28. Correia et alii, Frattura ed Integrità Strutturale, 42 (2017) 136-146; DOI: 10.3221/IGF-ESIS.42.15 146 [21] Krueger, R., Virtual crack closure technique: history, approach, and applications, Appl Mech Rev, 57(2) (2004) 109– 43. [22] ASTM E739-91: Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain Life (ε- N) Fatigue Data. Annual Book of ASTM Standards, American Society for Testing and Materials, (1991) 597-603. [23] Correia, J.A.F.O., Blasón, S., Arcari, A., Calvente, M., Apetre, N., Moreira, P.M.G.P., De Jesus, A.M.P., Canteli, A.F., Modified CCS fatigue crack growth model for the AA2019-T851 based on plasticity-induced crack-closure. Theoretical and Applied Fracture Mechanics, 85(1) (2016) 26-36. [24] Castillo, E., Fernández-Canteli, A., Siegele, D., Obtaining S-N curves from crack growth curves: An alternative to self-similarity. International Journal of Fracture, 187 (1) (2014) 159-172. [25] ASTM – American Society for Testing and Materials, ASTM E647: standard test method for measurement of fatigue crack growth rates, in: Annual book of ASTM standards, ASTM – American Society for Testing and Materials, West Conshohocken, PA, 03.01 (1999) 591-630. << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.) /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.) /NOR /POL /PTB /RUM /RUS /SKY /SLV /SUO /SVE /TUR /UKR /ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.) >> /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ << /AsReaderSpreads false /CropImagesToFrames true /ErrorControl /WarnAndContinue /FlattenerIgnoreSpreadOverrides false /IncludeGuidesGrids false /IncludeNonPrinting false /IncludeSlug false /Namespace [ (Adobe) (InDesign) (4.0) ] /OmitPlacedBitmaps false /OmitPlacedEPS false /OmitPlacedPDF false /SimulateOverprint /Legacy >> << /AddBleedMarks false /AddColorBars false /AddCropMarks false /AddPageInfo false /AddRegMarks false /ConvertColors /ConvertToCMYK /DestinationProfileName () /DestinationProfileSelector /DocumentCMYK /Downsample16BitImages true /FlattenerPreset << /PresetSelector /MediumResolution >> /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ] >> setdistillerparams << /HWResolution [2400 2400] /PageSize [612.000 792.000] >> setpagedevice