Advanced methodologies for the fatigue analysis of representative details of metallic bridges
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ADVANCED METHODOLOGIES FOR THE FATIGUE ANALYSIS OF REPRESENTATIVE DETAILS OF METALLIC BRIDGES António Luís Lima da Silva 2015 Thesis presented to the Faculty of Engineering of the University of Porto for the Doctor Degree in Mechanical Engineering Supervisors António Augsto Fernandes Full Professor Abílio Manuel Pinho de Jesus Assistant Professor
To my wife Andreia and my daughter Áurea.
V ACKNOWLEDGMENTS This thesis was possible due to the assistance received from many individuals and organizations. Acknowledgments have to be addressed to a large number of persons and institutions. The words and feelings expressed in these acknowledgments express my genuine gratitude to those who have provided important assistance and support during my studies. The order of appearance of individuals and organizations does not reflect in any way the level of importance of their contribution to the success of the research presented in this thesis. • To my supervisors, Professor António Augusto Fernandes and Professor Abílio Manuel Pinho de Jesus, who gave me far more help and assistance than that I could expect. Their guidance and discussion about the structure of the thesis and several theoretical and methodological aspects of the research is recognized; their assistance on pointing out me the right direction regarding the experimental and numerical analysis were also enthusiastically appreciated; • To the Portuguese Foundation for Science and Technology (FCT) and the European Commission, respectively in the context of the doctoral scholarship SFRH/BD/72434/2010 and FADLESS project (RFSR-CT-2009-00027), that provided the necessary funds to carry out the work; • To the University of Trás-os-Montes e Alto Douro for providing the access to their laboratories; • To the INSA-Lyon and particularly to Prof. Maigre and Prof. Gravouil for their support and help, during a 3 months internship, that allowed me to understand the XFEM methodology;
VI • To Dr. José Xavier for his unconditional support and help in the development of experimental procedures and also several algorithms used in this thesis, in particular what concerns the use of DIC; • To my collegues from the Civil Engineering Department of the Faculty of Engineering of the University of Porto, Carlos Albuquerque and Fernando Marques, for their sympathy, partnership and collaboration; • To Professors Rui Calçada and Álvaro Cunha for their leadership in the FADLESS project context and for their key contributions for the presented research; • To Engº Miguel Figueiredo from the Mechanical Department of the Faculty of Engineering of the University of Porto and responsible for LET, Dr. Cristóvão Santos, Prof. Paula Braga both from the University de Trás-os-Montes and Alto Douro for their support in their respective work domain, particularity concerning the experimental activities; • My fraternal thanks to João Pereira and Fábio Pereira; without their friendship and help, this PhD would be more burdensome; • Lastly, but certainly not least at all, I am very grateful to my wife Andreia for her love and continuous understanding and support.
VII Abstract Advanced methodologies for the fatigue analysis of metallic bridges are investigated in this thesis. Local approaches to fatigue are proposed for complex bridge details leading to multi-scale problems requiring efficient tools to allow feasible solutions for real bridge components under actual traffic conditions. The current doctoral thesis was prepared within the framework of the FADLESS European project (Fatigue Damage Control and Assessment for Railways Bridges) where two bridge case studies were investigated, and selected representative details supported the analyses presented in this thesis. Trezói and Alcácer do Sal Railway Bridges, respectively riveted and welded bridges, motivated a significant experimental study aiming the characterization of the fatigue behaviour of riveted and welded joints, as well as the fatigue characterisation of respective materials (S235 and S355 structural steels). The generated experimental data was important to provide basic material data for models identification as well as for the models validation. The down-scale fatigue testing of riveted and welded components and the mixed-mode fatigue crack propagation tests assisted by Digital Image Correlation could be emphasized as noteworthy contributions from the experimental program. This doctoral research also included significant numerical fatigue investigation. Both local approaches to fatigue and Fracture Mechanics were applied to simulate the fatigue crack initiation and propagation of tested details. Several finite element models of tested riveted and welded connections were developed in order to validate fatigue life prediction models. Distinct finite element approaches were used to simulate the fatigue crack propagation. This included the finite element method and the extended finite method, the latter being attractive since it does not require remeshing operations.
VIII In general, it was observed a good compromise between simulations and experimental results. Finally, numerical fatigue studies were developed for selected structural details of the Trezói and Alcácer do Sal railway bridges. With respect to the first case study, it was recognized the necessity to develop multiaxial fatigue analyses. Also, nonproportionality of stress states and their random time history nature were verified, demanding specialized damage models. For riveted joints, the simulation of rivets using contact elements is crucial to obtain reliable critical locations. An innovative approach for the fatigue damage assessment based on modal stress intensity factors superposition was also addressed in this doctoral thesis. The key objective of this methodology aims an efficient fatigue damage analysis considering the concept of modal stress intensity factors. It was found that the approach was able to simulate the residual fatigue life of a welded detail of the Alcácer do Sal Railway Bridge, for a significant traffic sample from the monitoring system.
IX RESUMO Metodologias avançadas para análise à fadiga de pontes metálicas são objeto de estudo nesta dissertação. As abordagens locais de fadiga propostas para detalhes complexos de pontes conduzem a problemas multiescala que requerem ferramentas eficientes de modo a permitir soluções confiáveis para detalhes de pontes reais sob a ação de condições de tráfego representativas. A presente tese foi realizada no contexto do projeto Europeu FADLESS onde dois casos de estudo foram investigados, tendo-se selecionado dois detalhes estruturais representativos para suportar as análises realizadas nesta dissertação. As pontes ferroviárias de Trezói e Alcácer do Sal, respetivamente rebitadas e soldadas, motivaram um programa experimental visando a caracterização do comportamento à fadiga de ligações rebitadas e soldadas, assim como a caracterização à fadiga dos respetivos materiais (aços estruturais S235 e S355). Os resultados experimentais providenciaram informação essencial relativa aos materiais, necessária à identificação dos modelos, assim como a informação para a validação dos mesmos modelos. Merecem destaque os ensais de fadiga em ligações rebitadas e soldadas, construídas numa escala reduzida dos detalhes reais das pontes, assim como os ensaios de propagação de fendas em modo misto, assistidos por técnicas de correlação digital de imagem. A presente tese de doutoramento também contempla uma componente numérica importante que visa a simulação do comportamento à fadiga de detalhes estruturais recorrendo a modelos de iniciação e de propagação de fendas, baseados em abordagens locais da fadiga e Mecânica da Fratura. Foram propostos vários modelos de elementos finitos dos componentes rebitados e soldados testados, no sentido de validar modelos de previsão de vida à fadiga. Neste domínio, a presente tese de doutoramento abrange diversas metodologias numéricas para a modelação de propagação de fendas de fadiga. A metodologia XFEM foi considerada para modelar a propagação de fendas de fadiga em ligações, sem haver a necessidade de refazer a
XVI Chapter VII - Fatigue modelling of a relevant detail from the Trezói bridge 7.1. Introduction 7.2 7.2. Overview of the proposed approach 7.3 7.3. Finite element model of a structural detail from Trezói bridge 7.8 7.4. Results and discussion 7.15 7.4.1. Continuous finite element model of the structural detail of the Trezói bridge: results analysis 7.18 7.4.2. Riveted finite element model of a structural detail of the Trezói bridge: results analysis 7.30 7.5. Concluding remarks 7.38 7.6. References 7.40 Chapter VIII - Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.1. Introduction 8.2 8.2. Theoretical background 8.3 8.2.1. Dynamic analysis using modal superposition 8.3 8.2.2. Fatigue model 8.5 8.2.3. Modal superposition of stress intensity factors 8.7 8.3. Proposed workflow for residual fatigue life assessment of bridge details 8.9 8.4. Application of the proposed methodology to a case study 8.13 8.4.1. Monitoring system 8.13 8.4.2. The global numerical model of the bridge 8.15 8.4.3. The local numerical model of the selected detail and shell-tosolid sub-modelling technique 8.16
XVII 8.4.4. Fatigue model assumptions 8.18 8.5. Results 8.19 8.5.1. Experimental Validation 8.19 8.5.2. Comparison of stress intensity factors computation techniques 8.22 8.5.3. Residual fatigue life computation 8.22 8.6. Concluding remarks 8.26 8.7. References 8.27 Chapter IX - Final conclusion and future works 9.1. Overview of main results 9.2 9.2. Future works 9.8 9.3. References 9.9
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XIX LIST OF FIGURES 1.1 Location of the Trezói and Alcácer do Sal Railway Bridges. 1.4 1.2 General view of the Trezói Bridge [1]. 1.6 1.3 Technical representation of the Trezói Bridge: elevation and plan view [1]. 1.6 1.4 Partial view of the upper structure. 1.7 1.5 Selected structural detail. Connection between the cross girder and the main chord [1]. 1.7 1.6 Overview of the bridge of the new railway crossing of the river Sado (adapted from [2]). 1.8 1.7 Diaphragm 54 of the Alcácer do Sal Bridge: global overview of the diaphragm; b) detail of the welded joint. 1.9 2.1 Schematic S-N curve. 2.6 2.2 Schematic da/dN versus K curve. 2.8 2.3 Collected fatigue damage cases listed according to the type of detail in which they were met [37]. 2.13 2.4 Stringer-to-cross-girder assembly tested by Abouelmaaty et al. [44]. 2.16 2.5 Stringer-to-cross-girder assembly tested by Al-Emrani [45] (dimensions in mm). 2.16 2.6 Finite element model of a stringer-to-cross-girder assembly developed by Al-Emrani and Kliger [54]. 2.19 2.7 3-D finite element model developed by Al-Emrani et al. [55]. 2.19 2.8 Global-local FE models of a riveted bridge module proposed by Imam [58]. 2.21
XX 2.9 FE model of the substructure elements from the local analysis proposed by Pantoli et al. [60]. 2.22 2.10 FE model of the substructure elements according Ghafoori et al. [62]. 2.22 2.11 FE model of a single rivet connection from the Trezói railway bridge proposed by Correia et al. [65]. 2.23 2.12 Finite element models of a single riveted connection: a) Solid finite element models; b) Shell finite element model (Correia et al. [65]). 2.24 2.13 Finite element models of a multiple rivets connections: a) solid finite element models; b) shell finite element models (Rodrigues et al. [66]). 2.24 2.14 Small ICOM specimens tested by Frýba [68]: a) specimen with circular cut-outs; b) specimens with cut-outs with apple form. 2.25 2.15 Truss girder tested by Schumacher, general geometric configuration and dimensions [70]. 2.27 2.16 Transverse stiffener specimen geometries tested by Ghahremani et al.: a) straight geometry; b) variable width geometry [72]. 2.27 2.17 Finite element model of a welded joint investigated by Xiao et al [73]. 2.28 2.18 Finite element models of an investigated welded joint by Heshmati [74]. 2.30 2.19 Welded detail investigated by Aygül [75]: a) Fatigue test specimen loading and investigated hot spot points; b) solid element model with an element size of 4 mm. 2.30 2.20 Finite element models of a welded connection investigated in [78]. 2.31 3.1 Typical microstructures of the S355 and S235 structural steels: a) S355, 200x magnification; b) S235, 200x magnification; c) S355, 500x magnification; d) S235, 500x magnification; e) S355, 1000x magnification; f) S235, 1000x magnification. 3.4 3.2 Experimental setup: a) INSTRON 8801 servo-hydraulic machine; b) INSTRON clip gauge, model 2670-602. 3.5 3.3 Smooth plane fatigue specimen (S355 steel): a) dimension in mm; b) photo of a smooth plane specimen. 3.7 3.4 Stabilized stress–strain hysteresis loops (S355 steel). 3.7
XXI 3.5 Stress range evolution with the number of cycles (S355 steel grade). 3.8 3.6 Cyclic curve of the S355 steel grade. 3.9 3.7 Smooth plane fatigue specimen (S235 steel) (dimension in mm). 3.10 3.8 Fatigue hysteresis loop of the S235 steel. 3.11 3.9 Stress range evolution with the number of cycles (S235 steel). 3.11 3.10 Cyclic curve of the S235 steel grade. 3.12 3.11 Strain-life data obtained for the S355 steel grade: a) elastic strain– life data; b) plastic strain–life data; c) total strain–life data. 3.13 3.12 Strain-life data for the S235 steel grade: a) elastic strain–life data; b) plastic strain–life data; c) total strain–life data. 3.15 3.13 Comparison of the cyclic curves obtained for the S235 and S355 steel grades. 3.16 3.14 Comparison of strain-life data between S235, S355 and S690 steel grades: a) elastic strain–life data; b) plastic strain–life data; c) total strain–life data. 3.17 3.15 Experimental setup used for fatigue crack growth testing: a) crack tip measurement using optical microscopes inspecting both surfaces of the CT specimen; b) crack tip measurement using both optical microscope and DIC inspecting opposite surfaces of the CT specimen; c) setup overview for mixed mode fatigue crack growth testing. 3.21 3.16 CT specimens made of S355 structural steel: a) thickness of 4mm; a) thickness of 8mm. 3.21 3.17 Experimental fatigue crack propagation rates obtained for the S355 steel grade, with a thickness of 4 mm and correlated using the Paris law: a) R =0.01; b) R =0.5; c) stress ratio effects; d) global correlation with the Paris relation. 3.23 3.18 Experimental fatigue crack propagation rates obtained for the S355 steel grade, with a thickness of 8mm and correlated using the Paris law: a) R =0.01; b) R =0.5; c) stress ratio effects; d) global correlation with Paris relation. 3.25 3.19 Experimental fatigue crack propagation rates for the S355 steel grade: a) thickness effects; b) global Paris correlation; c) global Walker correlation. 3.27
XXII 3.20 Compact tension specimens made of S235 steel: a) geometry of CT specimens; b) geometry of modified CT specimens; c) photo of CT specimens; d) photo of modified CT specimens. 3.29 3.21 Pure mode I fatigue crack propagation rates obtained for the S235 steel and correlations with Paris relation: a) R =0.01; b) R =0.5; c) stress ratio effects; d) global correlation with the Paris relation. 3.30 3.22 Correlation of pure mode I fatigue crack propagation rate using the Walker relation. 3.31 3.23 Schematic representation of the variation of relative position between adjacent subsets: a) subsets grid; b) adjacent subset displacements. 3.34 3.24 Speckle pattern applied to one face of the modified CT specimen. 3.35 3.25 A(x,y) map for the cracked S235_I+II_01_01 specimen at distinct stages of the crack propagation: a) for measurement stage 20; b) for measurement stage 40. 3.36 3.26 A(x,y) map for the cracked S235_I+II_01_01 specimen with the application of the threshold: a) for measurement stage 20; b) for measurement stage 40. 3.36 3.27 Comparison of assessed crack paths for S235_I+II_01_01 specimen: a) evolution of the X coordinate of the crack tip; b) evolution of the Y coordinate of the crack tip; c) crack path. 3.37 3.28 Comparison of assessed crack paths for S235_I+II_01_02 specimen: a) evolution of the X coordinate of the crack tip b) evolution of the Y coordinate of the crack tip; c) crack path. 3.38 3.29 Comparison of assessed crack paths for S235_I+II_01_03 specimen: a) evolution of the X coordinate of the crack tip b) evolution of the Y coordinate of the crack tip; c) crack path. 3.40 3.30 Comparison of assessed crack paths for S235_I+II_01_04 specimen: a) evolution of the X coordinate of the crack tip b) evolution of the Y coordinate of the crack tip; c) crack path. 3.41 3.31 Comparison of crack paths evaluated from direct optical measurements, DIC computation and from polynomial and exponential fittings of DIC results: a) S235_I+II_01_01; b) S235_I+II_01_02; c) S235_I+II_01_03 and d) S235_I+II_01_04. 3.42 3.32 Finite element mesh of a cracked modified CT specimen. 3.45
XXIII 3.33 Comparison of fatigue crack path predictions and experimental based crack paths: a) S235_I+II_01_01 and S235_I+II_01_02 specimens; b) S235_I+II_01_03 specimen; c) S235_I+II_01_04 and S235_I+II_01_05 specimens. 3.46 3.34 Nomenclature for Virtual Crack Closure Technique applied to 8noded plane finite elements. 3.51 3.35 Comparison of equivalent stress intensity factor range evolutions with the fatigue crack size: a) S235_I+II_01_01; b) S235_I+II_01_02; c) S235_I+II_01_03; d) S235_I+II_01_04. 3.54 3.36 Mixed mode fatigue crack propagation rates measured for the S235 steel grade: combine pure mode I (R =0) and Mixed mode fatigue crack growth rate data. 3.56 3.37 Comparison of pure mode I fatigue crack propagation rates for several structural steels. 3.57 3.38 Fracture surfaces of CT specimens tested under pure mode I fatigue loading, obtained from SEM: a) S355, 5000x magnification; b) S235, 5000x magnification; c) S355, 1000x magnification; d) S235, 1000x magnification. 3.58 4.1 Location of the potential fatigue critical node in the riveted Trezói railway bridge. 4.4 4.2 Actual photo of the potential fatigue critical structural node of the Trezói railway bridge (node 6). 4.4 4.3 Down-scale riveted specimen, R1 series. 4.5 4.4 Down-scale riveted specimen geometry (R1 series). 4.6 4.5 Riveted down-scale specimen mounted in a rigid frame and 10 tons servo-hydraulic actuator. 4.8 4.6 Instrumentation of the riveted beams with strain gauges placed along the beam longitudinal direction, at remote position (flange of the beam) (CEA-06-250UW-350) and at local position in between the two top most rivets (CEA-06-062UW-350). 4.9 4.7 Single line double shear riveted splices (R2 series), instrumented with strain gauges: 9 specimens with remote strain gauges (CEA-06250UW-350) and 4 specimens with strain gauges placed in between first and second rivet (CEA-06-062UW-350). 4.9
XXIV 4.8 Double line double shear riveted splices, R3 series: a) photo of the complete series; b) geometry of the specimens (dimensions in mm); c) instrumentation of the specimen with strain gauges (3 specimens instrumented; remote strain gauge (1) (CEA-06-250UW-350) and local strain gauges (2), (3) and (4) (CEA-06-062UW-350). 4.10 4.9 Servo-hydraulic machines used in the fatigue tests of the R2 and R3 riveted series: a) 10 tons Instron testing machine used in the R2 specimens; b) 25 tons MTS machine used in the R3 specimens. 4.10 4.10 Hardness measurement locations assessed on a cross section of a R2 riveted specimen. 4.13 4.11 Hardness values on a cross section of a R2 riveted specimen. 4.14 4.12 Micrograph of the rivet/plate interface: rivet and plate (S235) microstructures. 4.14 4.13 S-N curves of riveted specimens: R1 series. 4.18 4.14 S-N curves of riveted specimens: R2 series. 4.18 4.15 S-N curves of riveted specimens: R3 series. 4.18 4.16 Typical failure modes of riveted specimens of R1 series: a) fatigue cracks propagated from riveted holes; b) fatigue crack propagated from corner of angles. 4.19 4.17 Typical failure surfaces of riveted tested series: a) R2 series; b) R3 series. 4.20 4.18 Fatigue striations representative of the fracture surfaces of R2 riveted specimens observed with SEM (magnification 5000 x). 4.21 4.19 Fatigue striations representative of the fracture surfaces of R2 riveted specimens observed with SEM (magnification 10000 x). 4.21 4.20 Fatigue striations representative of the fracture surfaces of R2 riveted specimens observed with SEM (magnification 30000 x). 4.22 4.21 Comparison of experimental S-N data of all tested riveted series. 4.24 4.22 S-N curves for R1 and R3 series jointed tested data. 4.25 4.23 S-N curves of jointed riveted test data. 4.25 4.24 Typical strain measurements for riveted specimens: a) R1 series; b) R2 series; c) R3 series. 4.27
XXV 4.25 Experimental S-N data for riveted joints from several sources. 4.30 4.26 Experimental S-N data for riveted joints from several sources. 4.31 4.27 Alcácer do Sal bridge box girder: a) inside view with reinforcing diaphragms; b) local view of the connection between the diagonal and the hanger gusset; c) welded connection investigated between the gusset the diagonal 4.33 4.28 Welded specimens designed according actual geometry used in the Alcácer do Sal bridge. 4.34 4.29 Welded specimens designed according to EC3 [1]. 4.34 4.30 Welded specimens geometries for laboratory fatigue tests: a) same weld configuration of the bridge detail and a 5 mm thickness (W1 series); b) weld configuration according to EC3 and a 5 mm thickness (W2 series); c) same weld configuration of the bridge detail and a 12 mm thickness (W3 series); d) weld configuration according to EC3 and a 12 mm thickness (W4 series). 4.38 4.31 Hardness measurements performed in a W3 welded specimens. 4.40 4.32 Hardness values in the W3 welded specimens. 4.41 4.33 Hardness measurements performed in a W1 welded specimen. 4.41 4.34 Hardness values in the W1 welded specimens. 4.42 4.35 Microstructures of the steel: a) microstructure of the BM zone; b) microstructure of the weld toe (10x); c) microstructure of the weld toe (20x);d) microstructure of the weld toe (50x). 4.43 4.36 Experimental S-N curves of W1 welded series, using remote stress definition. 4.44 4.37 Experimental S-N curves of W2 welded series, using remote stress definition. 4.45 4.38 Experimental S-N curves of W3 welded series, using remote stress definition. 4.45 4.39 Experimental S-N curves of W4 welded series, using remote stress definition. 4.46 4.40 Experimental S-N curves of W1 welded series, using net stress definition. 4.47
XXXII 6.33 R1 riveted specimen modelled with XFEM and crack emanating from angle corner: a) global finite element mesh; b) local finite element mesh and XFEM crack location; c) z stress field 6.50 6.34 Finite element model of the R2 riveted specimen: a) global finite element mesh; b) local finite element mesh; c) y stress field for a particular fatigue crack. 6.51 6.35 Finite element model for the R3 riveted specimen. 6.52 6.36 Comparison between R1 riveted specimen experimental S-N curve with numerical results of the model with a crack propagating at the web, from a rivet hole. 6.54 6.37 Comparison between R1 riveted specimen experimental S-N curve and the numerical results obtained with the FEM model and XFEM with a crack at angle corner. 6.55 6.38 Normal stress range evolution at the net section of the angle: a) applied load of 18kN; b) applied load of 25kN. 6.55 6.39 Effective stress intensity factor evolution computed using FEM and VCCT, and XFEM and contour integral method of a propagating crack at the angle corner ( P=18 kN). 6.56 6.40 Comparison between R2 riveted specimen experimental curve with the R2 specimen numerical model: a) Clamping stress of 30MPa; b) Clamping stress of 80MPa. 6.57 6.41 Comparison between R3 riveted specimen experimental curve with the R3 specimen numerical model: a) Clamping stress of 30MPa; b) Clamping stress of 80MPa. 6.58 7.1 Bannantine and Socie methodology [5]. 7.7 7.2 Location of the potential critical node of the Trezói bridge to be fatigue assessed using a detailed local model. 7.9 7.3 Potential critical node (node 6): actual photograph. 7.9 7.4 Geometry and dimensions of node 6 of the Trezói bridge. 7.10 7.5 Local finite element models: continuous model. 7.11 7.6 Local finite element model: combined riveted and continuous model. 7.12 7.7 Cross sections of the local models where boundary conditions were applied. 7.13
XXXIII 7.8 Locomotive-hauled passenger train. 7.14 7.9 Displacements histories at sections 1 to 9 of node 6 from Trezói bridge. 7.14 7.10 Rotations histories at sections 1 to 9 of node 6 from Trezói bridge 7.15 7.11 Verification of the multiaxiality of the stress-time history: a) principal stresses evolution; b) direction cosines of the first principal stress direction. 7.16 7.12 Selected node for the verification of the multiaxiality of the stresstime history (full continuous model). 7.17 7.13 Location of strain gauges used to monitor the strains at the node 6 of the bridge model. 7.19 7.14 Experimental vs numerical stress measurements at locations illustrated in Figure 7.13. 7.20 7.15 Damage computed at each surface FE node of the local continuous model, for each passenger train crossing, using the Von Mises equivalent stress criterion. 7.20 7.16 Von Mises stress distribution (continuous model) (step 10). 7.21 7.17 Angular coordinates defining an arbitrary plane thru its normal. 7.22 7.18 Critical plane assessment for the node located at location 1 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.24 7.19 Critical plane assessment for the node located at location 2 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.26 7.20 Critical plane assessment for the node located at location 3 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.28 7.21 Fatigue damage computed at each surface FE node of the local combined continuous/riveted model, for each passenger train crossing. 7.31 7.22 Von Mises stress distribution (combined continuous/riveted model) (step 10): a) local model overview; b) detail of riveted cross girder with some highlighted hot spot locations at rivet holes. 7.31
XXXIV 7.23 Critical plane fatigue damage assessment for the location 1 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.32 7.24 Critical plane fatigue damage assessment for the location 2 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.34 7.25 Critical plane fatigue damage assessment for the location 3 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. 7.36 8.1 Workflow of the proposed analysis: pre-processing of the input data (1st step). 8.10 8.2 Workflow of the proposed analysis: crack propagation simulation (2nd step). 8.11 8.3 New Alcácer do Sal railway bridge. 8.13 8.4 Location of diaphragms 51 and 54 on bridge. 8.13 8.5 Location of strain gauges at diaphragm 51: a) global and local strain gauges locations; b) detail of local strain locations. 8.14 8.6 Some details of the global numerical modal of the bridge: a) deck, hangers and arch; b) diaphragm 51 and corresponding diagonals 8.16 8.7 Local finite element model of the welded joint of the Alcácer do Sal bridge 8.17 8.8 Shell-to-solid sub-modelling, location of the local FE model with respect the shell model: a) front view; b) side view; c) top view. 8.17 8.9 Experimental vs. numerical strain measurements: a) strain gauge 1; b) strain gauge 2; c) strain gauge 3; d) strain gauge 4; e) strain gauge 5. 8.20 8.10 Fatigue crack initiation spot: a) numerical simulation of real detail; b) experimental evidence from fatigue tests of small-scale fatigue tests 8.21 8.11 Figure 8.11 – VCCT vs. DE: a) KI; b) KII. 8.22 8.12 Fatigue crack propagation path simulated for the welded detail of the Alcácer do Sal bridge. 8.23
XXXV 8.13 Types of trains crossing the bridge: a) Alfa Pendular passengers train; b) intercity passengers train; c) freight train 8.23 8.14 Characteristics of simulated trains: a) Trains loads per unit length: b) Trains speeds; c) Trains lengths. 8.24 8.15 Crack propagation length as a function of cumulative traffic for current traffic volumes. 8.25 8.16 Comparison of crack propagation for distinct traffic scenarios. 8.26
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XXXVII LIST OF TABLES 3.1 Fatigue experimental program of smooth plane specimens made of S355 steel. 3.6 3.2 Fatigue experimental program of smooth plane specimens made of S235 steel. 3.10 3.3 Summary of cyclic elastoplastic and fatigue properties for structural steels. 3.18 3.4 Overview of fatigue crack growth testing adopted for the S355 steel. 3.22 3.5 Overview of fatigue crack growth testing adopted for the S235 steel. 3.28 4.1 Comparison between the structural node of the Trezói Bridge and the suggested riveted down-scale simplified specimens. 4.8 4.2 Chemical composition of the S235JR steel and rivet material used in the fabrication of the riveted specimens (% weight). 4.11 4.3 Tensile properties of the S235JR steel and rivet material used in the fabrication of the riveted specimens. 4.11 4.4 Experimental fatigue test data of R1 riveted specimens. 4.11 4.5 Experimental fatigue test data of R2 riveted specimens. 4.12 4.6 Experimental fatigue test data of R3 riveted specimens. 4.12 4.7 Summary of linear regression data for each tested riveted series. 4.22 4.8 Confidence intervals for regression parameters A and B given in Table 4.7. 4.23 4.9 Summary of linear regression data for joint data from R1 and R3 series. 4.26 4.10 Confidence intervals for S-N curve parameters resulting from R1 and R3 series joint test data. 4.26
XXXVIII 4.11 Summary of linear regression data for joint data from all test series. 4.26 4.12 Confidence intervals for S-N curve parameters resulting from all joint test data. 4.26 4.13 Riveted connection categorization proposed by Taras & Greiner [4]. 4.30 4.14 S-N curves constants, referring to experimental data presented in Figure 4.25. 4.31 4.15 Summary of linear regression parameters for S-N data of riveted joints from several sources illustrated in Figures 4.25 and 4.26. 4.32 4.16 Confidence intervals for S-N curve parameters resulting from all joint test data shown in Figures 4.25 and 4.26. 4.32 4.17 Fillet weld lap joint configuration and respective strength according to the EC3. 4.33 4.18 Welding procedure specifications. 4.35 4.19 Experimental data of the fatigue tests of W1 welded specimens. 4.36 4.20 Experimental data of the fatigue tests of W2 welded specimens. 4.36 4.21 Experimental data of the fatigue tests of W3 welded specimens. 4.37 4.22 Experimental data of the fatigue tests of W4 welded specimens. 4.37 4.23 Chemical composition of S355J2+N steel, S355NL1 and welded material (% weight). 4.39 4.24 Tensile strength properties of the S355 and welded materials. 4.39 4.25 Summary of linear regression parameters for all welded series (using of remote stresses). 4.46 4.26 Confidence intervals for regression parameters A and B given in Table 4.25. 4.47 4.27 Summary of linear regression parameters for all welded series (using of net stresses). 4.49 4.28 Confidence intervals for regression parameters A and B given in Table 4.27. 4.50 4.29 Summary of linear regression parameters resulting from W1 and W3 welded data analyzed together. 4.52
XXXIX 4.30 Confidence intervals for regression parameters A and B given in Table 4.29. 4.53 4.31 Summary of linear regression parameters resulting from W2 and W4 welded data analyzed together. 4.54 4.32 Confidence intervals for regression parameters A and B given in Table 4.31. 4.54 4.33 Summary of linear regression parameters resulting from W1 and W2 welded data analyzed together (5 mm thick plates). 4.55 4.34 Confidence intervals for regression parameters A and B given in Table 4.33. 4.55 4.35 Summary of linear regression parameters resulting from W3 and W4 welded data analyzed together (12 mm thick plates). 4.56 4.36 Confidence intervals for regression parameters A and B given in Table 4.35. 4.57 5.1 Finite element details for the verification example 1. 5.13 5.2 Stress range values at 2E6 cycles. 5.32 6.1 Finite element refinement used for the verification example 1 (FEM approach). 6.24 6.2 Finite element refinement used for the verification example 1 (FEM approach). 6.25 6.3 Finite element size used for the verification example 2 using XFEM approach. 6.32 6.4 Paris law material constant for the S355 steel considering thicknesses of 4mm and 8mm (R=0.01). 6.39 6.5 Constants of the strain-life Morrow relation for S235 steel. 6.44 6.6 Constants of the Paris equation for S235 steel. 6.44 7.1 Fatemi and Socie model constants [8]. 7.22 7.2 Findley model constants [8]. 7.22 7.3 Summary of the number of passenger train crossings until failure, computed using several fatigue assessment models, based on a continuous finite element model. 7.29
XL 7.4 Summary of the number of passenger train crossings to failure computed using several fatigue assessment models, based on a combined continuous/riveted finite element model. 7.38 8.1 Adopted fracture mechanics and fatigue parameters for the fatigue analysis performed using the welded model of the Alcácer do Sal bridge. 8.19
XLI NOMENCLATURE Latin a Fatigue crack size B Pre-logarithmic energy factor b Fatigue strength exponent b γ torsional cyclic fatigue strength exponent C Paris law’s coeficient c Fatigue ductility exponent D Fatigue damage D Damping matrix da dN Fatigue crack growth rate dA Infinitesimal area segment E Young modolus Fmax Maximum load in a load range Fmin Minimum load in a load range f Body force per volume unit j f Modal forces, for the jth mode of vibration G Elastic shear modulus J Energy release rate calculated with respect to a finite segment Jaux J-integral related to the auxiliary field
Introduction 1.2 1.1. INTRODUCTION Bridges are very complex structures that are built to support dynamic loads resulting from traffic actions (e.g. cars, trucks, trains, etc). The action of the dynamic loads over these structures is responsible for their progressive damaging due to fatigue. This is a major concern of designers and owners of the structures since they need to guarantee significant operational safety levels in order to avoid catastrophic collapses with strong social and economic impact. The awareness of fatigue behaviour of bridges did not appear with the construction of the first metallic riveted bridges at the end of XIX century. Only some decades ago, the fatigue design became a concern of engineers. This happened with the development of the first design rules that included a fatigue design section. Nowadays there is a full awareness for the need of a fatigue design of metallic bridges (e.g. bolted, welded, composite bridges) and design codes are strictly observed. The traffic intensity (e.g. axle loads, speed and frequency) on modern bridges is very high and with an increasing trend which requires constant attention in order to avoid fatigue cracks and in case they cannot be avoided, maintenance plans supported on residual life calculations need to be implemented. Although fatigue modelling has suffered significant advances for mechanical engineering applications, the fatigue methodologies preconized in design codes for bridges have a very simple form, despite the enormous complexity of the generality of those structures. The question that may be advanced is why the state-of-the art fatigue models are not being explored for bridge fatigue assessment? One answer that could be formulated is the fact that the most advanced fatigue models are local models that recognize the local nature of the fatigue damage. The application of those local models to bridges is cumbersome since a multi-scale problem needs to be faced. A global model of the structure is required to account conveniently for the traffic loads, but local information (e.g. strains, stresses, stress intensity factors) at critical hot spots is also
Chapter I 1.3 required to provide the required input information to the fatigue models. The conciliation of these two scales of analyses is currently an open problem that requires significant contributions in other to allow the progress of current design code approaches which are based on global S-N approaches. 1.2. AIM OF THE THESIS The main goal of the present dissertation was to develop advanced methodologies to be applied in the fatigue assessment of structural details from metallic bridges, including welded and riveted joints. Local models for the fatigue analysis were preferably followed in the proposed research, in alternative to traditional global S-N approaches, and strategies proposed to overcome the difficulties introduced by the multi-scale problem that this kind of fatigue models introduces. The present research was performed within the framework of the FADLESS European project, entitled Fatigue Damage Control and Assessment for Railways Bridges [1]. This European project aimed at the development of methodologies for the assessment of the fatigue behaviour of critical details, to be demonstrated for several European railway bridges, selected as case studies. Regarding the Portuguese case studies, two railway bridges were selected, namely the Trezói Railway Bridge and the new Alcácer do Sal Railway Bridge (see Figure 1.1).
Introduction 1.4 Figure 1.1 - Location of the Trezói and Alcácer do Sal Railway Bridges. The research proposed in this PhD work was focused on the local fatigue assessment methodologies for bridge details, the global structural dynamic behaviour of the bridges, another important counterpart of a fatigue assessment procedure, was out of scope of this PhD work. However, information from global dynamic analysis, such as displacement/member forces histories required for the local fatigue approaches, were made available from the other partners involved in the FADLESS project. The research proposed in this PhD work also benefitted from field monitoring data that was gathered during the FADLESS project, in particular for the two selected Portuguese case studies, which allowed the validation of the proposed models in terms of reproducibility of the local stresses and strains. The Portuguese cases studies selected in the FADLESS project involved two types of bridge construction, namely riveted and welded construction. Therefore the proposed research focused on both types of details. In addition to the fatigue simulation activities, this PhD research also aimed at the development of a significant experimental program focused on base materials and riveted/welded joints, representative of the two Portuguese case studies investigated within the FADLESS project. The generated fatigue data was used as an input for the numerical models. Also, it was used to allow the application of assessment procedures under controlled
Chapter I 1.5 laboratory conditions, where fatigue damage was possible to be generated and controlled. This is an important issue of the research, since the selected bridge case studies did not exhibit any visible fatigue damage, disabling the possibility of testing the models reliability for fatigue simulation in the case studies. However, the case studies are still very important since they represent the complexity of real bridge details, requiring several strategies for the mitigation of the important computational costs this kind of problems poses to the structural analysts. Also, stress fields in real bridge details are not fully represented using laboratory specimens. Concerning the fatigue simulation strategies, both crack initiation and crack propagation fatigue damages should be addressed. While fatigue crack propagation has been understood as the governing damage process for welded joints, fatigue crack initiation may not be neglected for riveted joints (unwelded material/details). Recognising the central role of the FADLESS case studies in this research, the following subsections give an overview of those case studies and the respective selected details that deserved major attention in this PhD dissertation. 1.2.1. Trezói Railway Bridge case study The Trezói Bridge (Figure 1.2) is located in the international Beira Alta railway line that links Portugal to Spain. This bridge is located at the km 62, north of Mortágua in the village of Trezói. The bridge was constructed as part of a project aiming the replacement of existing bridges in the railway line, carried out during the decade of 50 of last century, and was opened to traffic in August 1956. The project was funded by the Marshall Plan and the conception, manufacturing and erection, together with 6 other bridges of larger span on the same line, were of the responsibility of the German House Fried Krupp. The metallic riveted bridge is formed by three spans of 39 m, 48 m and 39 m, representing a total length of 126 m (see Figure 1.3). The two inverted Warren truss girders that compose the metallic deck of the bridge are 5.68 m height. The girder panels are 6.50 m wide in the central span and 6.00 m in the end spans. Two trapezoidal shape trusses acting as columns and two granite masonry abutments transmit the loads
Introduction 1.6 carried out by the structure to the foundation. The bridge has a constant width of 4.40m throughout its length. Figure 1.2 - General view of the Trezói Bridge [1]. Figure 1.3 - Technical representation of the Trezói Bridge: elevation and plan view [1].
Chapter I 1.7 The cross girders, as well as the stringers resting on them, were built using “I-shaped” sections, as shown in Figure 1.4. The cross girders are 71 cm height and are connected to the lateral vertical elements with riveted plates. The chords and diagonals of the truss girders are formed by double “U-shape” sections. The bearing supports of the superstructure are metallic and allow free rotations in the structure plane. At the east support, the longitudinal displacements are constrained, while at the west support displacements caused by longitudinal horizontal forces (thermal actions, braking, etc.) are allowed. Figure 1.5 illustrates the selected node of the bridge that was investigated in this research. This is a representative node of the bridge that was demonstrated in the FADLESS project as the one most fatigue sensitive node [1]. The cross girder receives directly the traffic load action and transfers it to the two Warren truss girders. Figure 1.4 - Partial view of the upper structure Figure 1.5 – Selected structural detail. Connection between the cross girder and the main chord [1].
Introduction 1.8 1.2.2. Alcácer do Sal Railway Bridge case study The new Alcácer do Sal railway bridge is located in the Lisbon – Algarve railway line, in south of Portugal (Figure 1.6). It was designed for passenger trains (as the Alfa Pendular tilting train and the Intercity train) running at speeds up to 220 km/h. Since it is also part of the main railway connection to the sea port of Sines, it is also designed for freight trains with up to 25 tons per axle. The bridge has a continuous composite deck that covers three spans (160 m per span). The deck is suspended from 3 arches. Each arch holds one span of the deck, with 18 vertical hangers, 8 m apart from each other. The structure is preceded by the north and south access viaducts (with 1115 m and 1140 m, respectively). Both viaducts and bridge were designed to hold two ballasted tracks even if only the upstream track is on service, at the current time. A reinforced concrete slab, above a U-shaped steel box, forms the deck. The 15.85 m wide concrete slab has a maximum thickness of 0.43 m. The steel box, with sloped webs, was built up by welding steel plates. Maximum plate thickness is 120 mm. A diaphragm reinforces the deck at each hanger connection. Two diagonals per hanger distribute the suspension force into the deck (see Figure 1.7). The arches present a hexagonal hollow section with variable height and width. Figure 1.6 - Overview of the bridge of the new railway crossing of the river Sado (adapted from [2]).
Chapter I 1.9 a) b) Figure 1.7 - Diaphragm 54 of the Alcácer do Sal Bridge: global overview of the diaphragm; b) detail of the welded joint at the diaphragm to central gusset plate. The welded connection between the diaphragm diagonals and the vertical gusset (Figures 1.7a) and b)) were selected for the fatigue assessment. Stress analysis revealed relatively small stress levels in the steel structure which is an indication of the absence of fatigue damage concerning this structure. Nevertheless, the referred welded joint exhibited a design deviation with respect to existing recommendations such the Eurocode 3 fatigue classes. Therefore, the proposed research concerning the Alcácer do Sal Bridge is twofold: i) to assess the influence of the design variation of the detail on fatigue behaviour; ii) to propose an efficient methodology to assess such a detail in the context of a very complex structure. 1.3. OUTLINE OF THE THESIS A brief outline of the thesis is given in this section. This dissertation is organized into nine chapters, including the present one. Chapter II presents a review about fatigue assessment methodologies for bridge details. A brief introduction to the fatigue of steel structures is also presented in the first part of chapter. Then, a review of experimental and numerical research concerning riveted and welded joints is performed. Chapter III presents the results of an experimental work that was performed within this PhD work, in order to characterize the fatigue behaviours of the structural steels S235 and S355. Elastoplastic cyclic behaviours, fatigue behaviours based on smooth
Introduction 1.10 specimens and fatigue crack propagation rates were assessed for both materials. Concerning the fatigue crack propagation, special mixed-mode fatigue crack propagation tests were performed and digital image correlation (DIC) used in data reduction. The investigated steels were used to produce riveted and welded connections that were also fatigue tested and results presented in Chapter IV. Chapter IV deals with an experimental campaign of riveted and welded joints. Regarding riveted specimens, 1 down-scale specimen and 2 small-scale specimens were considered. Code-based S-N curves were validated. About the welded specimens, 4 small-scale specimens were tested, aiming to represent distinct weld/geometry variations. While the riveted specimens were motivated by the Trezói bridge case study, the welded specimens were motivated by the Alcácer do Sal bridge. In Chapter V the master S-N curve concept presented by Dong [3] is applied to the investigated welded joints. The equivalent structural stress concept is computed for each investigated welded joint and respective experimental results where then compared with the master S-N curve, based on existing literature. Chapter VI presents a numerical analysis of the specimens presented in Chapter IV. Finite elements models of riveted and welded were described. The fatigue lives are assessed using local strain and Fracture Mechanics approaches. The fatigue crack propagation is simulated using both standard finite element method and extended finite elements method. Chapter VII presents a numerical analysis of the selected riveted detail from the Trezói bridge. Both continuous and riveted models were considered in the analysis. Submodels of the node were proposed and validated using monitoring fatigue data. Multiaxial local fatigue criteria were applied in the analysis of the detail aiming the fatigue crack initiation simulation. Chapter VIII was dedicated to the analysis of a selected welded detail of the Alcácer do Sal bridge. A new and efficient computational method to assess stress intensity factors is presented, based on modal superposition analysis. The proposed approach is demonstrated for a residual fatigue life assessment. Finally, Chapter VIII summaries the main conclusions of each chapter and also suggests some future works.
Chapter I 1.11 1.4. REFERENCES [1] Lippi, F., Salvatore, W., Braconi, A., Finetto, M., Wenzel, H., De Roeck, G., Peeters, B., Könke C., Zabel, V., Cunha, A.,“Fatigue damage control and assessment for railway bridge”, Research Fund for Coal and Steel, Directorate-General for Research and Innovation, 2014. [2] Rede Ferroviária Nacional, R.E., Variante de Alcácer, Lisboa: REFER, 187 p., 2010. [3] Dong, P., Hong, J. K., Cao, Z., “Structural stress based master SN curve for welded joints”, IIW Doc. XIII -1930-02/XV-1119-02, pp. 24, 2002.
A review of fatigue assessment methodologies for bridge details 2.6 Figure 2. 1 – Schematic S-N curve. The use of smooth specimens based S-N curves in conjunction with appropriate kt to predict the fatigue life of notched details may be implemented. However fatigue lives of notched specimens generally yield longer fatigue lives than the ones predicted using the kt-corrected stresses. In fact, the actual fatigue strength reduction, which can be expressed by the fatigue notch factor kf, is usually less than the reduction obtained by using kt. The notch sensitivity of a material can be defined by 1 1 f t k qk (2.3) In Equation (2.3), q = 0 indicates that the notch does not impact the fatigue life, whereas q = 1 suggests that the notch cause the maximum possible effect ( t k = f k ). The statistical size effect is another well-known influent phenomenon on fatigue behaviour. The reason for the existence of such effect is generally justified by the fact that larger volumes of stressed material contains a higher number of defects which results in a larger probability of finding crack initiation defects. Log or /2 Log N Log 0 m 1
Chapter II 2.7 2.2.2. Fracture mechanics based fatigue approach The fatigue life prediction for components with incipient cracks can be evaluated using a Fracture Mechanics approach. Generally, the Fracture Mechanics approach is performed as a complement to the stress-based approach, the latter being adopted to model the fatigue crack initiation. The Fracture Mechanics is usually divided into two main groups, namely the Linear Elastic Fracture Mechanic (LEFM) and Elastoplastic Fracture Mechanic (EPFM). Irwin [11] proposed the stress intensity factor to describe the severity of the stresses around a crack tip. The stress intensity factor can be defined as: K Y a (2.4) where is the remotely applied stress, a is the crack length and Y is the stress intensity magnification factor which depends on the geometry and loading conditions. The stress intensity factor solutions are available in the literature for a wide range of cracked geometries [11-12]. An effort has been made in order to relate the fatigue crack growth with the stress intensity factor range, K, at the crack tip. Many relations between the fatigue crack growth rates and the stress intensity factor range have been proposed in the literature, such as the well-known Paris law [14]: m da CK dN (2.5) where m and C are experimentally measured material constants. On a log-log plot, Equation (2.5) yields to a linear relation, known as Region II of the typical fatigue crack propagation curve, shown in Figure 2.2. Two more distinct regions can be point out in the Figure 2.2, namely the asymptotic Region I, which is associated to a fatigue threshold below which fatigue cracks are non-propagating and the Region III, where fatigue cracks accelerates toward an unstable propagation.
A review of fatigue assessment methodologies for bridge details 2.8 Figure 2.2 – Schematic da/dN versus K curve. 2.2.3. Variable amplitude stress conditions Large structures such as metallic bridges are usually subjected to variable amplitude loading histories. In order to perform fatigue assessments on such structures, several models have been proposed to compute cumulative damage under variable loading stress ranges [15]. The most common method is based on the linear damage accumulation assumption. This assumption does not consider the loading sequence effects, assuming that the fatigue crack growth during a given cycle is not affected by prior loading history. However, Skorupa [16] refers some sequence load effects such as overloads or underloads which can retard or accelerate fatigue crack growth. However, in the case of very irregular random stress histories, the retardation and acceleration effects may cancel out each other effects. Therefore, linear damage accumulation may be a considered a proper choice. The linear Palmgren-Miner’s rule is the most widely used model for calculating fatigue damage due to variable amplitude loading [17]. The Miner sum is expressed as: i fi n DN (2.6) where D is the total damage, i n is the applied number of cycles at a specific stress amplitude or stress range, and fi N is the fatigue life corresponding to the same stress
Chapter II 2.9 amplitude or stress range. The Palmgren-Miner’s rule postulates that failure occurs when D =1. 2.2.4. Multiaxial fatigue The majority of the studies in fatigue have been performed under uniaxial loading conditions, and this is the basis of the main approaches being proposed in the literature, including the design codes for bridge structures. However, in some cases, the local stresses developed at critical details of metallic bridges are multiaxial. Besides the stress multiaxiality, the principal stress directions may change over time, defining nonproportional stress histories. In addition to the non-proportional multiaxial stress histories, variable amplitude stress histories are frequent [18]. Although over the years, different methods have been proposed for fatigue life prediction under multiaxial stress states, currently, there is no universally accepted approach to the problem [19]. Due to its simplicity, one of the most common approaches is the transformation of the multiaxial stress state into an equivalent uniaxial stress state with the same fatigue life characteristics [18, 20-22]. The transformation is usually performed using the well-known Von Mises or Tresca equivalent stress criteria and replacing the principal stresses with their ranges. However, this approach has been found to give non-conservative results for the cases of non-proportional loading [20, 22-23]. Another type of approaches that appears to be more successful in multiaxial fatigue data description [19] is the critical plane based approaches, on which fatigue damage is assumed to take place. Combinations of shear and normal stresses on the critical plane are assumed to govern the fatigue behaviour [21-22]. For proportional and constant amplitude loading, the critical planes are usually identified as being the ones experiencing the maximum normal or shear stress. On the other hand, for multiaxial variable amplitude loading, the critical plane is identified, having investigated different plane orientations, as the one experiencing the highest value of different shear and normal stress combinations [19, 24]. Further, the complex multiaxial loading histories can be resolved into individual cycles by using modified forms of the rainflow counting method and then applying Miner’s rule [18, 25-28].
A review of fatigue assessment methodologies for bridge details 2.10 2.2.5. Code-based S-N fatigue curves for structural details Methods for fatigue analysis available in structural design codes of practice are based on S-N fatigue curves which are applicable to distinct details such as, for example, riveted, bolted and welded joints. A family of parallel S-N curves is proposed, with the same slope (m), each one representing the fatigue resistance of a group of details, often called fatigue class [1]. For constant amplitude loading, a fatigue limit 0 is specified for a given number of cycles (e.g. 7 10N [7]). Below this fatigue limit no damage occurs, if constant amplitude loading is applied. For variable amplitude loading conditions, a second S-N segment with a slope m+2 is proposed, for stress ranges below constant amplitude fatigue limit. The fatigue curves are assumed independent of the stress ratio, R [31]. The detrimental effects of positive mean stresses are included in the S-N curves. Usually, the S-N curves slope is equal to 3, this value being based on Fracture Mechanics principles [10]. The BS5400 standard [29] presents 9 fatigue classes, designated as B, C, D, E, F, F2, G, W and S. This standard allows different target failure probabilities by parallel shifting of the S-N curves. EC3 [1] considers 14 fatigue strength curves, characterizing 14 fatigue classes. The fatigue class designation results from the fatigue strength at 6 2 10N cycles. For variable amplitude loading, and for all fatigue classes, a S-N curve slope of m=5 is assumed beyond 6 5 10N . The American Association of State Highway Traffic Officials (AASHTO) code [30] proposes the use of 7 detail categories (A, B, B0, C, D, E, and E) with various modifications. Variable amplitude loading is assessed by lowering the cutoff constant amplitude fatigue limit for each fatigue category, calculating an effective constant amplitude stress range and comparing it with the modified fatigue limit. The American Railway Engineering Association (AREA) standard [31] is very similar to the AASHTO code, the only differences being the S-N curves suggested for riveted details.
Chapter II 2.11 The UK railway assessment code [32] adopts the BS5400 fatigue classes but it provides two additional fatigue classes for plain and riveted wrought-iron details (class WI-rivet). Furthermore, cut-off limits at 8 10N cycles are suggested for all detail classes. Riveted details are considered in the BS5400, AASHTO, AREA and UK railway assessment codes but are not explicitly considered in Eurocode 3. BS5400 suggests the use of its Class D for lapped or spliced riveted connections. In AASHTO, category D is suggested for riveted connections. On the other hand, the AREA code suggests the use of its category D with a distinction being made between riveted connections with high or low clamping force and with drilled or punched holes. However riveted joints are not explicitly referred in the EC3. Meanwhile some recommendations [33] suggest the use of the Class 71 S-N curve from EC3 for any type of riveted joint. Recent studies conducted by Taras and Greiner [34] suggested an alternative approach to the universal slopes proposed by existing codes, and several fatigue classes of riveted joints. The assessment of riveted joints using code-based S-N curves should be based on net stresses which are nominal stresses computed at cross sections containing the riveted holes. Concerning the welded joints, several types of stress categories may be used in conjunction with the proposed code-based S-N curves. The nominal stress approach is the basic stress form proposed to be used with some types of bridge details in many codes, including the EC3 and the International Institute of Welding (IIW) recommendations [35]. Structural or geometric stresses are also proposed for use with the with the S-N curves of some details. In many cases structural details are assessed on the basis of the maximum principal stress range in the section where potential fatigue cracking is expected. However, maximum shear stress range is required for shear loaded details. Distinct S-N curves are provided for normal or shear loaded welds.
A review of fatigue assessment methodologies for bridge details 2.12 2.3. OVERVIEW OF EXISTING RESEARCH ON RIVETED MEMBERS AND CONNECTIONS A significant number of fatigue assessment studies for riveted railway bridges have been proposed in the past. An investigation about the fatigue performance of existing steel and composite bridges collected fatigue damage cases for various bridge types and details [36]. Reza Haghani [37] reported more than 100 damage cases which were studied and categorized according to the type of detail and/or the mechanism behind the observed fatigue cracking. The results of this study showed that more than 90% of all reported cases are caused by secondary effects, so-called deformation-induced cracking. This type of fatigue damage is often the result of secondary restraining forces generated by overlooked interaction between different members in the bridge. Poor detailing, along with unstiffened gaps and abrupt changes in stiffness at the connections between different members, also contributes to fatigue cracking in most details. Design codes and evaluation methods generally provide very little guidance on how fatigue damage should be accounted for or prevented. It is the responsibility of the bridge designer to ensure—through good detailing—that these secondary effects and the kind of fatigue damage associated with them are avoided. Figure 2.3 shows collected damage cases categorized according to detail type. The most common types of deformation-induced fatigue damage can be found in the connections between stringers and floor beams, between the latter and the main load-carrying elements in the bridge and at the connections of diaphragms and cross-bracings.
Chapter II 2.13 Figure 2.3 – Collected fatigue damage cases listed according to the type of detail in which they were met [37]. Brühwiler et al. [38] presented a fatigue research comprising full scale tests on three different girder types, four rolled girders with an extra cover plate riveted to the lower flange and six built up girders, and finally three lattice girders made of wrought iron. The author concluded that the corrosion of riveted girders did not provide lower fatigue life than the one non-corroded. A corrosion loss of ~10 % of the cross section did not give a combined effect worse than the conditions of rivet holes. Wrought iron elements showed fatigue strength similar to steel. The failures in the lattice girders were always in the rivets due to shear stresses component. EC3 fatigue class 71 provided a reasonable estimation of the fatigue life. The constant amplitude fatigue limit of riveted wrought iron girders was estimated to be 70 MPa. For mild steel as well as for girders with punched holes the level was believed to be lower. The shear resistance of rivets may be the governing failure mode for connections. Adamson et al. [39] investigated the fatigue behaviour of stringers retrieved from a bridge built in 1911. From the load history and strain measurements, it was concluded that the accumulated fatigue damage was negligible. Presence of corrosion on the
A review of fatigue assessment methodologies for bridge details 2.14 stringers was also believed to have a negligible effect on the fatigue performance. The investigation included five full scale tests on stringers. Non-bearing riveted details showed a tendency of having fatigue resistance higher than bearing details. The results of the fatigue endurance of the stringers were covered by the AASHTO fatigue class category. Helmerich et al. [40] investigated the fatigue life of girders from three bridges. The results from the tests were used to develop a non-destructive inspection technique, for identification of cracks in bridges. Nine full scale tests were performed, and the results indicated that the EC3 fatigue class 71 could be used to assess the fatigue life of riveted bridges. The influence of corroded material impact damages and structural defects were covered by the fatigue class. The fatigue endurance of wrought iron was not worse than that of mild steel. Considering appropriate values of Young’s modulus and the yield strength, wrought iron bridges can be assessed as steel bridges accordingly to the author. DiBattista et al. [41] made fatigue tests on seven full-scale tension members. The tension members were retrieved from the same bridge as the tests of Adamson et al. [39]. A uniform corrosion was observed on all tension members. The tests showed that the fatigue resistance of the diagonals and their connections to the bottom chord could be evaluated by AREA detail fatigue class D, depending on definition of net section area. Non-bearing riveted details showed a tendency of having fatigue resistance higher than bearing details. Repairing cracked connections between tension members and the gusset plate, with preloaded bolts, extended the life of the connections significantly. Out et al. [42] investigated the fatigue resistance of four riveted stringers. The tests focused on corroded girders. Measurements conducted on the girders while still in service showed that 1 % of the stress cycles exceeded 48 MPa, thus the cumulative fatigue damage from service was believed to be negligible. The resistances of the corroded sections were between AASHTO fatigue class E and C (EC3 fatigue class 56 and 80) depending on the loss of cross section. The riveted beams showed redundancy, the stresses being redistributed to nearby parts when cracks were formed. Tests performed at reduced temperatures did not result in unstable crack growth. Zhou et al. [43] investigated the effect of hole preparation methods on the fatigue life of riveted structures. Investigations concerning the fatigue limit were also performed.
Chapter II 2.15 Fatigue tests were carried out at stress ranges between 44 MPa to 54 MPa. A total of 20 tests were performed, 12 at constant amplitude and 8 with a variable load spectra. The result showed that rivet holes were the most frequent origin for crack initiation and was believed to be an effect of the surface condition of the holes. Girders with punched holes provided lower fatigue endurance than drilled or sub punched and reamed. Five tests reached 1x108 cycles, after which the tests were terminated and examination of the girders showed that no fatigue cracking had occurred. The fatigue limit was determined to be 41 MPa. The AREA fatigue class D1 was believed to provide a lower bound for riveted girders in general. The investigations also showed that wrought iron girders exhibited lower fatigue endurance than steel. Abouelmaaty et al. [44] carried out constant amplitude fatigue tests on two deliberately constructed full-scale models of a typical stringer-to-cross-girder steel connection (Figure 2.4). Due to stringer rotation, both test specimens failed at the cross-girder web with fatigue cracks initiating at the locations where the upper and lower flanges of the stringer came into contact with the cross-girder web. More cracks were also detected in the cross-girder web around the double-angle connection and in the stringer web. A considerable amount of bending moment (approximately 8.5% of the stringer fixed-end moment) was found to be carried by the double-angle connection. Al-Emrani [45] fatigue tested three full-scale bridge parts, which were extracted from an old mild steel railway bridge. Each part consisted of four riveted built-up stringers connected through double angles to three riveted built-up cross-girders (Figure 2.5). Some connection angles were found to contain fatigue cracks prior to testing. The specimens failed either by fatigue cracking in the connection angle, due to out-of-plane distortion of the outstanding legs, or fatigue cracking of the rivets connecting the outstanding legs of the angles to the web of the cross-girder. Fatigue cracks were always found to initiate near the angle fillet at the upper row and after slowly vertical propagation, they were self-arrested due to the gradual reduction in the rotational stiffness of the connections. Fatigue failures of rivets were attributed to combined bending and tensile stresses being present in the rivets, the flexure of the outstanding legs and the stress concentration between the shank and the head. It was found that the double-angle connections were capable of developing up to 67% of the corresponding moment of a fully continuous beam.
A review of fatigue assessment methodologies for bridge details 2.22 a) b) Figure 2. 9 – FE model of the substructure elements from the local analysis proposed by Pantoli et al. [60]. The effect of un-bonded carbon fibre reinforced polymer post-tensioning retrofit system on the fatigue vulnerability of an existing bridge in Münchenstein, Switzerland, is explored by E. Ghafoori et al. [62]. Finite elements models were considered (Figure 2.10) in order to determine critical fatigue locations and to investigate the relative effects of the post-tensioning retrofit on these locations. Fatigue analyses with and without the retrofit are considered. Submodeling was considered in order to assess this multi-scale problem. Figure 2.10 – FE model of the substructure elements according Ghafoori et al. [62].
Chapter II 2.23 Regarding Portuguese research about the fatigue resistance assessment of bridge details, several works were performed in order to model S-N curves of riveted connections. Correia et al. [63, 65] presented a numerical and experimental procedure aiming the simulation and validation of S-N curves for an original simple riveted joint from a Portuguese railway bridge (see Figure 2.11). Crack initiation and crack propagation phases were considered. Several parameters such as the evolution of the stress concentration factors with the clamping stress were investigated. Concerning the crack propagation phase, through thickness fatigue cracks were simulated emanating from rivet holes. Correia et al. [65] has also investigated the single riveted joint presented in [63], using both solid and shell finite elements (Figure 2.12). The second modelling approach may be convenient if a reduction in computation effort is required. Several numerical parameters were investigated such the friction coefficient, the clamping stress, the stress concentration factor, the normal penalty stiffness factor (FKN) and the penetration tolerance (FTOLN). Jesus et al. [64] presented the results of an experimental program aiming the evaluation of the fatigue behaviour of two types of joints (single and double shear) with preloaded resin-injected bolts. Results are compared with test data obtained with standard preloaded bolts, revealing a systematic fatigue strength reduction when preloaded resin-injected bolts were used. Figure 2.11 – FE model of a single rivet connection from the Trezói railway bridge proposed by Correia et al. [65].
A review of fatigue assessment methodologies for bridge details 2.24 a) b) Figure 2.12 – Finite element models of a single riveted connection: a) Solid finite element models; b) Shell finite element model (Correia et al. [65]). a) b) Figure 2.13 – Finite element models of a multiple rivets connections: a) solid finite element models; b) shell finite element models (Rodrigues et al. [66]). Connections with multiple rivets were investigated by Rodrigues et al. [66] (Figure 2.13). The author also proposes a comparison of two alternative finite element modelling strategies for riveted connections, which may be used for fatigue assessments. In the first approach, the plates of the connection are modelled using finite solid elements; the second approach uses finite shell elements. Based on proposed finite element models, some riveted connections are analysed to assess the local stresses, namely the elastic stress concentration factors for uncracked geometries and the stress intensity factors,
Chapter II 2.25 for cracked geometries. The stress intensity factors are evaluated using the virtual crack closure technique (VCCT technique) (Krueger [67]). The effects of friction and clamping stresses on rivets are accounted in the models. 2.4. OVERVIEW OF EXISTING RESEARCH ON WELDED MEMBERS AND CONNECTIONS The research on fatigue behaviour of welded details for generic applications is too wide to be covered in this section. Therefore this section presents a selection of research works performed on welded members and connections for applications in bridge structures. Frýba [68] investigated the fatigue properties of orthotropic decks of steel railway bridges with open ribs. Being fully welded, the orthotropic decks are sensitive to fatigue. Fatigue tests were carried out on several series of specimens (see Figure 2.14). The shape of cut-outs in the web of the cross-girder, the effect of shear forces and the weld penetration appeared to be very important factors influencing the fatigue behaviour. Figure 2.14 – Small ICOM specimens tested by Frýba [68]: a) specimen with circular cut-outs; b) specimens with cut-outs with apple form.
A review of fatigue assessment methodologies for bridge details 2.26 Clubley et al. [69] presented the consequences of a fatigue analysis on the structural integrity of the River Mardle Viaduct. The River Mardle Viaduct is a curved, twin box girder bridge, spanning a total length of 177 m. Four continuous spans carry the A38 trunk road over the River Mardle and Old Totnes Road. The bridge forms an important part of the heavy loaded network in and out of the tourist areas of Devon and Cornwall. In September 2001, an initial inspection of the site splice welds joining the box girders detected a series of large imperfections. These weld defects were confirmed by a further independent inspection 6 months later and the results subsequently analysed using detailed fracture mechanics. Schumacher [70] presented fatigue tests carried out on welded circular hollow section K-joints typical on bridges. The tests specimens were large-scale (approximately 9 m long and 2 m high) trusses loaded in the plane of the truss (see Figure 2.15). Measured member stresses showed that a significant proportion of the load in a truss member may be due to bending, underlining the importance of considering correctly this load case in the design of these structures. Measured hot-spot stresses in the joints were compared with hot-spot stresses calculated using the current design guidelines. The authors found that the measured values are considerably lower than the calculated values, calling into question the applicability of the design guidelines to these types of (bridge) structures. The S-N fatigue results from the current study, on the other hand, showed that the fatigue resistance of the joints that were tested is lower than the corresponding S-N design curves. This means that when the considerably higher calculated hot-spot stress range is applied to the corresponding design curve, the predicted resistance is similar to the resistance predicted using the lower measured hotspot stresses in combination with the lower measured S-N curve too. This has highlighted the importance of relating hot-spot stresses to the appropriate, corresponding S-N curves. Evidence from the fatigue tests has clearly demonstrated the effect of size on the fatigue strength of welded tubular joints. A comparison of fatigue S–N results from smaller and larger welded circular hollow section joints has shown the same trend indicated in design specifications: a thicker failed member results in lower fatigue strength. The size correction factor integrated into the S-N design curves of the specifications, however, does not seem to represent this significant effect in a fairly manner.
Chapter II 2.27 Righiniotis et al. [71] considered the application of a probabilistic fracture mechanics approach to predict the fatigue life of welded steel details in the presence of cracks under bridge spectrum loading, based on a recently proposed bi-linear relationship to model fatigue crack growth. The fatigue model incorporates a failure criterion to describe the interaction between fracture and plastic collapse. Ghahremani et al. [72] presented fatigue tests results of needle peened structural steel weld specimens loaded under simulated in-service loading histories typical of highway bridges (see Figure 2.16). A strain-based fracture mechanics model was then validated by comparison with the test results and used to perform additional studies. Similar welds were analysed under loading histories encompassing a wider range of influence lines and bridge spans. Ghahremani et al. concluded that the applied model is well suited for studying the effects of peening on the fatigue performance of highway bridge welds under in-service loading conditions. The consideration of tensile dead load stresses and periodic overload trucks is seen to decrease the predicted benefit of peening. This benefit can still be substantial, however, for a wide range of loading conditions likely to occur in highway bridges. Figure 2.15 –Truss girder tested by Schumacher, general geometric configuration and dimensions [70]. Figure 2.16 – Transverse stiffener specimen geometries tested by Ghahremani et al.: a) straight geometry; b) variable width geometry [72].
A review of fatigue assessment methodologies for bridge details 2.28 The transverse distribution of wheel loads in orthotropic decks generates significant out-of-plane bending moments in the deck plate and rib wall at the rib-to-deck joint. Due to the relatively small thickness of both the deck plate and rib wall, the out-ofplane bending moments result in high local flexural stresses causing fatigue cracks to develop at the joints. Xiao et al. [73] investigated the transverse stresses within the joint region that arise under the action of wheel loads are investigated using finite element analyses (see Figure 2.17). Based on the stress results and the basic theories of linear elastic fracture mechanics, the design fatigue strength was determined for the investigated joint. Factors affecting the stress range were also studied. The authors reveal that the surface stresses in the deck plate are much larger than those in the rib wall, in the case of 75% weld penetration into the rib wall, indicating that the fatigue strength of the joint is governed by the fatigue cracks propagating into the thickness of deck plate. Figure 2.17 – Finite element model of a welded joint investigated by Xiao et al [73].
Chapter II 2.29 The EC3 is the governing design code for steel structures in Europe. In this design code, the nominal stress method is the predominant approach for fatigue design. However, the limitations of this method along with new advanced computational possibilities, have allowed the way for more accurate stress based fatigue design approaches. The structural hot spot stress (SHSS) and the effective notch stress (ENS) methods are among those which have widely drawn engineers’ attention since their advent. These approaches take advantage of new computational modelling possibilities and designate the basic stress by taking into account the geometrical variations of the detail at the expected fatigue crack initiation location (hot spot). As a result, a higher variation of constructional details can be assessed more accurately by these methods. Although the SHSS and ENS approaches have been extensively exploited in other industries, both methods are considered inexperienced in the field of bridge engineering. Heshmati [74] assessed the fatigue strengths of the welded details using the structural hot spot stress and the effective notch stress methods. Finite elements models were carried out using different modelling techniques mainly based on the well-known IIW modelling instructions (see Figure 2.18). Furthermore, the thickness effects, weld shape and overall member geometry on the computed stress concentration factor were studied and discussed by the author. A large database including available fatigue test was built up and used to produce S-N curves using structural hot spot stress and effective notch stress definitions. The effects of altering the geometric shape of the cope-holes were also investigated using finite element analysis. The fatigue strength enhancement of cope-hole details by means of post-weld treatment is experimentally evaluated by conducting constant amplitude fatigue tests. Based on the test results, a new fatigue strength category for improved cope-hole details is proposed by Heshmati.
A review of fatigue assessment methodologies for bridge details 2.30 Figure 2.18 – Finite element models of an investigated welded joint by Heshmati [74]. Aygül [75] studied the fatigue life estimation of orthotropic steel bridge decks using the finite element method (see Figure 2.19). The application of the structural hot spot stress approach or the effective notch stress approach to a welded joint with cut-out holes in orthotropic bridge decks was made. Aygül compared the results of the finite element calculations with the results of the fatigue tests which were carried out on full-scale specimens. The results of the finite element analyses revealed that the structural hot spot stresses obtained from the shell element models were unrealistically high when the welds were omitted. The author also underlined that the way in which the welds were represented had a substantial influence on the magnitude of the hot spot stress. The results of the analysis when using the effective notch stress approach showed that the agreement between the estimated fatigue life using this approach and the fatigue life obtained from the fatigue tests was good. a) b) Figure 2.19 – Welded detail investigated by Aygül [75]: a) Fatigue test specimen loading and investigated hot spot points; b) solid element model with an element size of 4 mm.
Chapter II 2.31 Taras et al. [76] presented an investigation regarding the effects of thickness steps or transitions of weld joints, which are often used in flanges of bridge girders in order to adapt the bending resistance of the cross-sections to variable bending moment distributions. Taras et al. [76] focused his work primarily on the stress raising effects for the longitudinal stresses near the butt weld, at the thickness transition. Klinger et al. [77] investigated fatigue cracks on specific steel components and joints of a railway bridge over the Elbe River at Lutherstadt Wittenberg, Germany, due to wind induced vibrations. The authors adopted Fracture Mechanic calculations to assess the remaining service life of the welded joints. Binhua Wang [78] presents a comparison study of specimens with two types of cut-out holes, namely the circular arc transition configuration and the vertical transition configuration. The fatigue life estimation of specimens was investigated with the application of the structural hot spot stress approach by using finite element analyses (see Figure 2.20). Structural stresses were computed using the Dong method [79]. The author observed that compared with the measured results, the solid element models, both using contact elements and non-contact elements, presented a good estimation of the hot spot stresses. Aiming the simplification, the solid element model with noncontact elements is recommended. The results that were obtained through the shell increased thickness model were not in good agreement with the test result. The Dong’s method yielded good results for the investigated points because the calculated stress was obtained from stress distributions obtained using the solid FE analysis. Figure 2.20 – Finite element models of a welded connection investigated in [78].
A review of fatigue assessment methodologies for bridge details 2.38 Safety, reliability and risk of structures, infrastructures and engineering systems; ICOSSAR2009;p. 545, 2010. [59] Taylor, D., Wang, G,. “The validation of some methods of notch fatigue analysis”, Fatigue and Fracture of Engineering Materials and Structures 23(5), pp. 387-394, 2000. [60] Pantoli, E., Vincenzi, L., Savoia, M., Testa, R., “The effect of local vibrations on fatigue in old steel riveted bridges. A case study: the Manhattan Bridge”, Proceedings of the 8th International Conference on Structural Dynamics, EURODYN, Leuven, Belgium, 4-6 July 2011 [61] Mayer, L., Yanev, B., Olson, L.D., Smyth, A., “Monitoring of the Manhattan Bridge for Vertical and Torsional Performance with GPS and Interferometric Radar Systems”, Transportation Research Board 89th Annual Meeting, 2010. [62] Ghafoori, E., Prinz, G.S., Mayor, E., Nussbaumer, A., Herwig, M.M.A., Fontana, M., “Finite Element Analysis for Fatigue Damage Reduction in Metallic Riveted Bridges Using Pre-Stressed CFRP Plates”, doi:10.3390/polym6041096, Polymers 6, pp.10961118, 2014. [63] José, A.F.O.C., Jesus A.M.P., Figueiredo, M.A.V., Ribeiro, A.S., Fernandes, A.A., “Fatigue Assessment of Riveted Railway Bridge Connections. Part I: Experimental Investigations”, 7th International Conference on Steel Bridges, Guimarães, Portugal, 4-6 June 2008. [64] Jesus, A.M.P., Silva, J.F.N., Figueiredo, M.A.V., Ribeiro, A.S., Fernandes, A.A., José, A.F.O.C., Silva, A.L.L., Maeiro, J.M.C., “Fatigue Behaviour of Resin-Injected Bolts: An Experimental Approach”, berian Conference on Fracture and Structural Integrity 2010 - CIFIE'2010, FEUP, Porto, Portugal, 2010. [65] Correia, J.A.F.O., Jesus, A.M.P. Silva, A.L.L., “Simulação por Elementos Finitos de Curvas S-N de Ligações Rebitadas”, 8º Congresso Nacional de Mecânica Experimental – APAET 2010, Univ. Minho, Guimarães, 21-23 Abril, 2010 [66] Rodrigues, M.P.G., Jesus, A.M.P., Silva, A.L.L., “Comparison Between Alternative FE Modelling Strategies for Riveted Connections Concerning Fatigue Assessments”, 8º Congresso Nacional de Mecânica Experimental Guimarães, 21-23 de Abril, 2010. [67] Krueger, R., “Virtual crack closure technique: History, approach, and applications”, Applied Mechanics Reviews, p. 109-143, 2004.
Chapter II 2.39 [68] Frýba, L., Gajdoš, L.,“Fatigue properties of orthotropic decks on railway bridges”, Engineering Structures 21, pp. 639–652, 1999. [69] Simon, K., Clubley, Stephen N., “On the fatigue and fracture of site splice welds at the River Mardle Viaduct”, Engineering Failure Analysis 10, pp. 593–604, 2003 [70] Schumacher, A., Nussbaumer, A., “Experimental study on the fatigue behaviour of welded tubular K-joints for bridges”, Engineering Structures 28, pp. 745–755, 2006. [71] Righiniotis, T. D., Chryssanthopoulos, M.K., “Fatigue and fracture simulation of welded bridge details through a bi-linear crack growth law”, Structural Safety 26, pp. 141–158, 2004. [72] Ghahremani, K., Walbridge, S., “Fatigue testing and analysis of peened highway bridge welds under in-service variable amplitude loading conditions”, International Journal of Fatigue 33, pp. 300–312, 2011. [73] Xiao, Z.G., Yamada, K., Samol, Y., Zhao, X.L., “Stress analyses and fatigue evaluation of rib-to-deck joints in steel orthotropic decks”, International Journal of Fatigue 30, pp. 1387–1397, 2008. [74] Heshmati, M., “Fatigue Life Assessment of Bridge Details Using Finite Element Method”, Master’s thesis in the Master’s Programme Structural Engineering and Building Performance Design, Department of Civil and Environmental Engineering Division of Structural Engineering Steel and Timber Structures CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2012 [75] Aygül, M., Al-Emrani, M., Urushadze, S., “Modelling and fatigue life assessment of orthotropic bridge deck details using FEM”, International Journal of Fatigue 40, pp. 129–142, 2010 [76] Taras, A., Unterweger, H., “Proposal for a stress modification factor for the fatigue design of flange thickness transitions in welded girders”, Engineering Structures 56, pp. 1758–1774, 2013. [77] Klinger, C., Michael, T., Bettge,D., “Fatigue cracks in railway bridge hangers due to wind induced vibrations – Failure analysis, measures and remaining service life estimation, Engineering Failure Analysis, 2014. [78] Wang, B., Lu, P., Shao, Y., “Fatigue Test and Simulation Research of Rib-toDiaphragm Welded Connection”, Journal of Convergence Information Technology (JCIT) Volume8, Number9, May 2013
A review of fatigue assessment methodologies for bridge details 2.40 [79] Dong, P., “A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints”, International Journal of Fatigue 23, pp. 865-876, 2001
Chapter III Fatigue characterization of structural steels
Fatigue characterization of structural steels 3.2 3.1. INTRODUCTION This chapter presents an experimental campaign that was carried out in order to characterize the fatigue behaviour of two current structural steels. The investigated materials were used in the production of welded and riveted joints fatigue tested within this research, the respective results being presented in Chapter IV. The materials under consideration are the S355 and S235 structural steels, which were used respectively in the manufacturing of welded and riveted joints. Concerning the riveted joints, the number of cycles required to initiate a crack may be significant, when compared with the total fatigue life of the structural component. In order to be able to appropriately characterize the crack initiation behaviour, strain-controlled fatigue tests were performed on smooth specimens according to the ASTM E606 [1] standard. These tests were performed for both structural steels. Classically, fatigue behaviour of welded joints is understood as a process of fatigue crack propagation. It is assumed that welded joints show intrinsic defects whose cyclic growth controls the fatigue life of the welded component. Fracture Mechanics is commonly used to model the fatigue crack propagation, allowing the assessment of the failure life of the component. In order to facilitate the application of the Fracture Mechanics to the assessment of the crack propagation fatigue live of structural components, fatigue crack propagation tests were carried out according to the ASTM E647 standard [2], on the referred structural streels, in order to allow the derivation of the fatigue crack growth rates. Besides the pure mode I fatigue crack propagation tests, mixed mode (I+II) fatigue crack propagation tests were also performed for the S235 material. This kind of fatigue data is very important, since for real details the mode mixity is very common. The results analysis for the mixed mode fatigue crack propagation tests was performed using Digital Image Correlation (DIC). DIC was used for assessing both the crack path as well as the stress intensity factors.
Chapter III 3.3 3.2. MICROSTRUCTURES OF THE S355 AND S235 STRUCTURAL STEELS The Figure 3.1 presents the microstructures of the S355 and S235 structural steels. The observations were performed using an optical microscope on previously polished and etched surfaces. Several magnifications were considered: 200x (Figure 3.1a) and b)), 500x (Figure 3.1c) and d)) and 100x (Figure 3.1e) and f)) for both S355 and S235 steels. A global analysis of Figure 3.1 allows the following considerations: i) Both materials show an aligned grain microstructure which results from the rolling process; ii) Both materials show a ferrite/perlite microstructure with is typical of carbon steels; iii) The S355 steel shows higher perlite zones which results from higher carbon content; iv) The S355 steel exhibits smaller grain sizes than the S235 steel. 3.3. ASSESSMENT OF CYCLIC ELASTOPLASTIC AND FATIGUE BEHAVIOURS OF THE S355 AND S235 STRUCTURAL STEELS This section presents a cyclic elastoplastic characterization of the S355 and S235 structural steels. Strain-controlled cyclic tests were performed for both materials using smooth specimens, following the recommendations of the ASTM E606 standard [1]. Besides the fatigue behaviour, these tests allowed the assessment of the cyclic elastoplatic behaviours of the structural steels under consideration. All specimens were tested under strain control, with null strain R-ratio (R=0.0), on a servo-hydraulic machine rated to 100kN (see Figure 3.2a)). Test frequencies ranged between 0.4 and 1.6Hz, depending on the strain ranges applied to the specimens. An average strain rate of 0.8 %/s was kept constant in the test programme. The strain control was guaranteed by means of a clip gauge with a maximum displacement of ±2.5mm, as shown in Figure 3.2b).
Fatigue characterization of structural steels 3.4 a) b) c) d) e) f) Figure 3.1 – Typical microstructures of the S355 and S235 structural steels: a) S355, 200x magnification; b) S235, 200x magnification; c) S355, 500x magnification; d) S235, 500x magnification; e) S355, 1000x magnification; f) S235, 1000x magnification.
Chapter III 3.5 a) b) Figure 3.2 - Experimental setup: a) INSTRON 8801 servo-hydraulic machine; b) INSTRON clip gauge, model 2670-602. 3.3.1. Cyclic elastoplastic behaviour of the S355 steel Concerning the S355 structural steel, a total of 13 smooth plane specimens were tested under strain-controlled conditions, according to the programme depicted in Table 3.1. This table includes the central section (gauge) dimensions, for each specimen. Figure 3.3a) shows the nominal dimensions of the flat dog-bone specimens and Figure 3.3b) exhibits a photo of such specimens. All specimens were manufactured from undamaged base material removed from a remote part of the welded specimens that will be presented in the next chapter. Figure 3.4 presents the stabilized cyclic stress–strain hysteresis loops obtained for the S355 steel. Some relative scatter is observed in the Figure 3.4, mainly when the amount of cyclic plasticity decreases. These hysteresis loops allow the assessment of the Masing behaviour of the material [3]. The Masing behaviour is observed when the upper branches of the hysteresis loops are coincident. For a material obeying the Masing behaviour, the cyclic curve of the material may be used to describe the shape of the hysteresis loops.
Fatigue characterization of structural steels 3.6 Concerning the S355 steel, some degree of deviation from the Masing behaviour is observed. The material may be then characterized as a non-Masing material. The hysteresis loops presented in Figure 3.4 were determined using a half-life criterion. This criterion may coincide with a cyclic stabilized behaviour criterion, for those tests that showed stabilization. However, for this material some tests did not show cyclic stabilization and, therefore, this criterion represented a pseudo-stabilized behaviour. The evolution of the stress amplitude with the number of cycles is plotted in Figure 3.5, for the S355 steel. No clear stabilization of the cyclic behaviour is observed for the S355 steel grade. For the highest tested strain ranges, the material suffers some hardening after initial softening. In this case, the global response may be considered coarsely stable. For lower strain ranges, the material tends to soften. Some specimens experienced accentuated cyclic softening, showing an uncharacteristic cyclic behaviour. Table 3.1 - Fatigue experimental program of smooth plane specimens made of S355 steel. Specimens L1 L2 St R d /dt f [mm] [mm] [mm] [-] [%] [%/s] [Hz] S355-200-01 6.19 5.03 31.14 0.01 1.00 0.80 0.40 S355-100-01 6.18 5.04 31.15 0.01 1.00 0.80 0.40 S355-200-02 6.13 5.03 30.83 0.01 2.00 0.80 0.20 S355-100-02 6.12 5.04 30.84 0.01 2.00 0.80 0.20 S355-0.50-01 6.15 5.04 31.00 0.01 0.50 0.80 0.80 S355-0.50-02 6.12 5.04 30.84 -0.01 0.50 0.80 0.80 S355-0.40-01 6.08 5.04 30.64 0.01 0.40 0.80 1.00 S355-0.40-02 6.07 5.04 30.59 0.01 0.40 0.80 1.00 S355-0.30-01 6.07 5.10 30.96 0.01 0.30 0.80 1.33 S355-0.30-02 6.21 5.04 31.30 0.01 0.40 0.80 1.00 S355-0.25-01 6.18 5.04 31.15 0.01 0.30 0.80 1.33 S355-0.75-01 6.10 5.04 30.74 0.01 0.25 0.80 1.60 S355-0.75-02 6.19 5.04 31.20 0.01 0.25 0.80 1.60
Chapter III 3.7 a) b) Figure 3.3 - Smooth plane fatigue specimen (S355 steel): a) dimension in mm; b) photo of a smooth plane specimen. Figure 3.4 - Stabilized stress–strain hysteresis loops (S355 steel). 15 4xR12 100 6 5 20 20 0 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 S355_200_01 S355_200_02 S355_100_01 S355_100_02 S355_05_01 S355_05_02 S355_04_01 S355_04_02 S355_03_01 S355_075_01 S355_075_02 S355_025_01 Strain, [%] Stress, s[MPa]
Fatigue characterization of structural steels 3.14 b) c) Figure 3.11 - Strain-life data obtained for the S355 steel grade: a) elastic strain–life data; b) plastic strain– life data; c) total strain–life data (2/2). 3.3.4. Strain-life behaviour of the S235 steel Regarding to the strain-life analysis of the S235 steel, Figures 3.12a) and 3.12b) present, respectively, the evolution of the elastic and plastic strain amplitudes with the number of reversals to failure. Figure 3.12c) represents the Morrow strain-life relation, which results from the superposition of the results from the elastic and plastic strain-life relations. The strain-life fatigue data presented in this section resulted from the cyclic tests performed on the smooth plane dog-bone specimens, as described in Section 3.3.2. 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Plastic strain amplitude, P/2 [-] Number of reversals to failure, 2Nf P/2 = 0.608x(2Nf) -0.616 R2=0.9195 5.E-04 5.E-03 5.E-02 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 strain amplitude, /2[-] Number of reversals to failure, 2Nf /2 =0.608x(2N f ) -0.616 +0.0048x(2N f ) -0.090
Chapter III 3.15 a) b) c) Figure 3.12 - Strain-life data for the S235 steel grade: a) elastic strain–life data; b) plastic strain–life data; c) total strain–life data. 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Elastic strain amplitude, E/2 [-] Number of reversals to failure, 2Nf E /2 = 0.00347x(2Nf) -0.081 R 2 =0.8340 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Plastic strain amplitude, P/2 [-] Number of reversals to failure, 2Nf P/2 = 1.348x(2Nf) -0.740 R2=0.9336 5.E-04 5.E-03 5.E-02 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Total strain amplitude, /2[-] Number of reversals to failure, 2Nf /2 =1.348x(2N f ) -0.740 +0.0035x(2N f ) -0.081
Fatigue characterization of structural steels 3.16 3.3.5. Comparison of the cyclic elastoplastic and fatigue behaviours of the tested structural steels This section compares the cyclic curves and the strain-life behaviours of the S355 and S235 steel grades. Figure 3.13 compares the cyclic curves obtained for the S355 and S235 materials. The comparison of the two cyclic curves shows that materials exhibit very similar cyclic strain hardening behaviours (cyclic strain hardening coefficient). Concerning the slopes of the cyclic curves, they are essentially parallel, which means very similar slopes (cyclic strain hardening exponent). Concerning the comparison of the strain-life relations, three distinct steel grades were considered: the S235 and S355 investigated in this work, and the S355 and S690 whose results were made available in reference [9]. Figure 3.14 compares the elastic, plastic and total strain amplitude versus reversals to failure, for all considered materials. Regarding to the elastic strain-life relations, the S690 steel grade shows significantly higher fatigue resistance, when compared with the other lower strength steel grades. Figure 3.13 - Comparison of the cyclic curves obtained for the S235 and S355 steel grades. 1.E+02 1.E+03 1.E-05 1.E-04 1.E-03 1.E-02 Stress amplitude, s/2 [MPa] Plastic strain amplitude, P/2 [-] S355 S235 s/2=590.881( p /2) 0.089 R2=0.6493 s/2=555.520( p /2) 0.077 R2=0.7771
Chapter III 3.17 a) b) c) Figure 3.14 - Comparison of strain-life data between S235, S355 and S690 steel grades: a) elastic strain– life data; b) plastic strain–life data; c) total strain–life data. 5.E-04 5.E-03 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Elastic strain amplitude, E/2 [-] Number of riversals to failures, 2Nf S235 S355 S355 [9] S690 [9] E/2=0.0041 x (2Nf)-0.089 E/2=0.0067 x (2Nf)-0.087 E/2=0.0035 x (2Nf)-0.081 E/2=0.0048 x (2Nf)-0.090 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Plastic strain amplitude, p/2 [-] Number of riversals to failures, 2Nf S235 S355 S355 [9] S690 [9] P/2=0.737x(2Nf)-0.664 P/2=0.740x(2Nf)-0.809 P/2=1.348x(2Nf)-0.740 P/2=0.608x(2Nf)-0.616 5.E-04 5.E-03 5.E-02 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Total strain amplitude, /2 [-] Number of riversals to failures, 2Nf S235 S355 S355 [9] S690 [9] /2=0.0041 x (2Nf)-0.089+0.737x(2Nf)-0.664 /2=0.0067 x (2Nf)-0.087+0.740x(2Nf)-0.809 /2=0.0035 x (2Nf)-0.081+1.348x(2Nf)-0.740 /2=0.0048 x (2Nf)-0.090+0.608x(2Nf)-0.616
Fatigue characterization of structural steels 3.18 The fatigue resistance increases with the increasing static strength of the steel. Nevertheless, the slopes of the elastic strain-life relations remain similar for all steel grades. Considering the plastic strain-life relations, the S690 steel shows the lower fatigue ductility. An analysis of the total strain-life relations show that the S690 steel show better fatigue performance for high-cycle fatigue; however, for low-cycle fatigue the better fatigue performance is attributed to the lower strength steels. The fatigue curves rotate counter-clockwise around approximately 104 reversals, with the increase of the fatigue strength of the structural steels. Table 3.3 summarizes the cyclic elastoplastic constants as well as the strain-life parameters for all investigated structural steels. Table 3.3 also presents the constants proposed for the S355 and S690 steel grades given in references [9] and [10], as well as for structural steels specified in ASTM standards [11]: a high performance steel (HPS 485W) and one carbon structural steel (A7). Table 3.3 – Summary of cyclic elastoplastic and fatigue properties for structural steels. Steel grade K´ n´ ´ f s b ´ f c 2NT [MPa] [-] [MPa] [-] [-] [-] [reversals] S235 555.52 0.079 720 -0.076 1.348 -0.740 6822 S355 590.88 0.089 857 -0.090 0.608 -0.616 4054 S355 [9] 595.85 0.076 952 -0.089 0.737 -0.664 7095 S690 [9] 1282.65 0.092 1403 -0.087 0.740 -0.809 675 S690 [10] - - 1191 -0.090 0.911 -0.674 5809 HPS 485W [11] 956 0.113 851 -0.069 0.775 -0.701 3686 A7 [11] 1139 0.248 760 -0.121 0.196 -0.486 50119 3.4. ASSESSMENT OF FATIGUE CRACK GROWTH RATES OF THE S355 AND S235 STRUCTURAL STEELS This section presents an investigation on fatigue crack growth behaviour of the S355 and S235 structural steels. Concerning the S355 steel, pure mode I fatigue crack growth tests
Chapter III 3.19 were performed according to the ASTM E647 [2] standard. Compact Tension (CT) specimen geometries were adopted and the effect of the thickness in the fatigue crack growth rates was investigated. For this purpose, two distinct thicknesses were considered: 4mm and 8mm. Regarding the S235 steel, pure mode I and mixed mode (mode I and mode II) fatigue crack growth behaviours were studied. Pure mode I fatigue tests were conducted according to the ASTM E647 [2] standard, using the CT geometry. Mixed mode crack propagation tests were performed using modified CT geometries. The experimental data assessment for the mixed mode tests was performed using Digital Image Correlation (DIC). DIC was used with two purposes: i) crack path evaluation and ii) stress intensity factors computation. A sequential two-steps approach was followed. The crack path is first assessed and then the stress intensity factors are computed. The experimental crack propagation data was correlated using the power relation between the fatigue crack propagation rates and the stress intensity factor ranges, as proposed Paris and Erdogan [12]: m da CK dN (3.5) Equation (3.5) does not account for the stress ratio effects on fatigue crack growth rates. In order to allow stress ratio effects to be accounted conveniently on fatigue crack growths rates, the Walker model [13] was considered: s 1 1 m m da K C K C dN R (3.6) where C and m are the same constants as presented in the Paris law and is an additional material constant. Concerning the pure mode I fatigue crack propagation tests, the fatigue crack growth rates were computed using the seven point incremental polynomial technique, as proposed in the ASTM E647 standard [2]. The stress intensity factor ranges were computed using the formulation also proposed in the ASTM E647 standard for CT specimens:
Fatigue characterization of structural steels 3.20 2 3 4 32 20.886 4.64 13.32 14.72 5.6 1 F KtW (3.7) where: aW , a is the fatigue crack size, t is the thickness of the CT specimen (t=4mm or t=8mm), W is the width of the specimen (W=40 mm for t=4mm; W=50 for t=8mm) and ΔF is the applied load range. For pure mode I fatigue crack propagation, the crack tip position was evaluated as the average value of two surface measurements performed on both faces of the CT specimens. These measurements were performed using two travelling microscopes with an accuracy of 0.001mm (Figure 3.15a)). The crack tip measurement for modified CT specimens tested under mixed mode conditions was accomplished using two alternative systems: the previous referred optical microscope was used to inspect one surface of the CT specimens; and DIC method was considered to observe the other surface of the CT specimen (Figure 3.15b). Figure 3.15c) shows the experimental setup used for the mixed mode fatigue crack propagation tests. The direct optical observations performed on one face of the modified CT specimens were used for the calibration of the DIC technique which was applied on the other face of the modified CT specimen. All fatigue crack propagation tests were carried out in load control, with constant amplitude loading and assuming a load ratio of 0.01. All tests were also performed in air condition, at room temperature, under sinusoidal waveform with a maximum frequency of 20 Hz. 3.4.1. Fatigue crack propagation rates of the S355 steel The thickness effect on pure mode I fatigue crack propagation rates was investigated for the S355 steel grade. A total of 8 CT specimens were manufactured, from welded joints presented in the Chapter IV, according to the ASTM E647 standard [2] (see Figure 3.15). Table 3.4 presents the details of the fatigue crack growth testing program for the S355 steel, which includes the nomenclature adopted to identify each CT specimen, the main dimensions and the tested stress ratios, R s . Thicknesses of 4 mm and 8 mm were taken
Chapter III 3.21 into account. An initial stress intensity factor of 475.34 Nmm1.5 was applied to all tested CT specimens. Since the tests were performed under constant load amplitude, stress intensity factors increased during the crack growth. a) b) c) Figure 3.15 - Experimental setup used for fatigue crack growth testing: a) crack tip measurement using optical microscopes inspecting both surfaces of the CT specimen; b) crack tip measurement using both optical microscope and DIC inspecting opposite surfaces of the CT specimen; c) setup overview for mixed mode fatigue crack growth testing. a) b) Figure 3.16 – CT specimens made of S355 structural steel: a) thickness of 4mm; a) thickness of 8mm.
Fatigue characterization of structural steels 3.22 Table 3.4 – Overview of fatigue crack growth testing adopted for the S355 steel. Specimens W Thickness R s = s min/ s max [mm] [mm] [-] S355_T4_01_01 40 4 0.01 S355_T4_01_02 40 4 0.01 S355_T4_05_01 40 4 0.50 S355_T4_05_02 40 4 0.50 S355_T8_01_01 50 8 0.01 S355_T8_01_02 50 8 0.01 S355_T8_05_01 50 8 0.50 S355_T8_05_02 50 8 0.50 Figure 3.17 plots the experimental fatigue crack propagation rates measured for the S355 steel grade with a thickness of 4 mm tested for a stress ratio, R s =0.01 (see Figure 3.17a) and R s =0.5 (see Figure 3.17b). The fatigue crack growth rates, da/dN, are plotted as a function of the applied stress intensity factor ranges, ΔK. For each stress ratio, the Paris relation is fitted to the experimental data, a very high determination coefficient being found. Figure 3.17c) compares the Paris relation lines for each stress ratio and one may conclude that the material only exhibits a minor dependency on stress ratio. A slight increase of the crack propagation rate is observed with the increase in the stress ratio, which is a well-known behaviour in fatigue. Figure 3.17d) plots the experimental data for the two tested stress ratios and a global correlation is proposed using the Paris relation. A very high determination coefficient still is observed in this case. Similarly, Figure 3.18 plots the experimental fatigue crack growth rates obtained for the S355 steel grade with a thickness of 8 mm, which were tested for a stress ratio, R s =0.01 (see Figure 3.18a) and R s =0.50 (see Figure 3.18b). Figure 3.18c) shows that for the thickness of 8 mm there is a slight increase of the stress ratio effects on the fatigue crack growth rates. Consequently, the overall correlation of the data using the Paris relation resulted in a reduction of the determination coefficient, when compared with the lower thicknesses.
Chapter III 3.23 a) b) c) Figure 3.17 - Experimental fatigue crack propagation rates obtained for the S355 steel grade, with a thickness of 4 mm and correlated using the Paris law: a) Rs=0.01; b) Rs=0.5; c) stress ratio effects; d) global correlation with the Paris relation. (1/2) 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S355_T4_01_01 S355_T4_01_02 da/dN=1.01E-14xK 3.3926 R2=0.996 R s =0.01 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S355_T4_05_01 S355_T4_05_02 da/dN=7.13E-14xK 3.1435 R2=0.991 R s =0.5 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] Series4 Series3 R s =0.01 R s =0.5 da/dN=7.13E-14xK3.1435 R2=0.991 da/dN=1.01E-14xK3.3926 R2=0.996
Fatigue characterization of structural steels 3.30 a) b) c) Figure 3.21 - Pure mode I fatigue crack propagation rates obtained for the S235 steel and correlations with Paris relation: a) R s =0.01; b) R s =0.5; c) stress ratio effects; d) global correlation with the Paris relation. (1/2) 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S235_01_01 S235_01_02 da/dN2,21E-16xK 4.0035 R 2 =0.988 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S235_05_01 S235_05_02 da/dN=2,12E-15xK 3.71 R 2 =0.991 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] Series4 Series3 da/dN2,21E-16xK 4.0035 R2=0.988 da/dN=2,12E-15xK 3.71 R2=0.991 R s =0.01 R s =0.5
Chapter III 3.31 d) Figure 3.21 - Pure mode I fatigue crack propagation rates obtained for the S235 steel and correlations with Paris relation: a) R s =0.01; b) R s =0.5; c) stress ratio effects; d) global correlation with the Paris relation. (2/2) Figure 3.22 – Correlation of pure mode I fatigue crack propagation rate using the Walker relation. The mode I and mode II stress intensity factors computation from experimental displacement fields may be a challenging task, since experimental displacement fields near the crack tip may contain measurement errors due to significant amount of plastic deformation, out-of-plane displacement and high displacement gradients [15]. However, works have been done in order to directly determine stress intensity factors from experimental displacement fields [16–18], using an extrapolation method. This 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S235_01_01 S235_01_02 S235_05_01 S235_05_02 da/dN=7.48E-16xK 3.84 R 2 =0.974 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 100 1000 Crack growth rate, da/dN [mm/cycle] Stress intensity factor range, K [N.mm-1.5] S235_01_01 S235_01_02 S235_05_01 S235_05_02 da/dN=3.47E-16x[(1-R)0.896K]3.936 R2=0.987
Fatigue characterization of structural steels 3.32 displacement extrapolation method is also frequently used with finite element analyses [19-20]. The least-squares regression method has been widely used to extract stress intensity factors using experimental full-field displacement data [21-26]. With this technique, analytical solutions for the displacement fields nearby the crack tip will be fitted to the experimental full-field displacement data. The set of parameters of the analytical solution to be identified by the least-squares technique will include the mode I and mode II stress intensity factors. Yoneyama [15] proposed a one-step method to assess both stress intensity factors and crack tip location. H0wever, this methodology can be time consuming for a large displacement fields. In this chapter, a two-step approach is followed, as follows: firstly the crack tip is located from the displacement field resulting from Digital Image Correlation (DIC). Once located the crack tip, the stress intensity factors will be estimated using linear least squares method to match the analytical solution for the displacement field and the experimental full-field DIC data. 3.4.2.2.1. Digital image correlation The DIC technique was mainly developed by Sutton et al. [27-28]. The method allows the assessment of the full-field displacements for an object surface. It is assumed that the spatial distribution of pixel grey levels within each subset gives a distinctive fingerprint of the surface with suitable contrast and isotropy [29]. These image patterns, before and after deformation, are then digitized and stored in a computer. The digitized images are then compared and subsets are matched between one image and the other. The initial image representing the body before motion is a discrete function f(x,y) that is transformed into another discrete function f*(x*,y*) after motion (deformation/distortion or displacement). The theoretical relation between the two discrete functions can be written as:
Chapter III 3.33 * * * ( , ) ( ( , ),y ( , )) 0 xy f x y f x u x y u x y (3.8) where ux(x,y) and uy(x,y) represent the displacement field for a pattern. Since an independent value of displacement is measured per subset, the size of the subsets must be carefully defined and compromised between correlation and interpolation errors. A sub-pixel correlation algorithm calculates the position (centre) of each feature on the deformed configuration, therefore it is determined the displacement field across the region of interest. In this study, the ARAMIS® 2D DIC system by GOM was used to store periodically a sequence of two images at maximum and minimum cycle loads (see setup represented in Figure 15). One face of each modified CT specimen was provided with a speckledpattern, as illustrated in Figure 3.24. The size of the correlation windows used in the proposed method was 19x19 pixels, whilst the subset on the Aramis was defined as 15x15 pixels (with a step size of 15x15pixels). A telecentric lens TC 23 36 from Opto-Engineering, with a field of view of 26.3x22.1 mm2 was used. The proposed algorithm for the crack path assessment and stress intensity factors assessment was programmed in Matlab code. However, the GOM Aramis 2D DIC commercial software [30], v6.0.2, was also used to compute displacement fields for comparison purposes with the ones from Matlab code. 3.4.2.2.2. Crack tip location assessment The algorithm used to identify the crack tip location in the CT specimens was proposed by Xavier et al. [31]. The method is based on the variation of the relative position among adjacent subsets [32-34], as illustrated in the Figure 3.23. A mapping scalar function A(x,y) is introduced as the maximum norm of the relative position vectors: ( , ) max( u u ; u u ) i k j l A x y (3.9)
Fatigue characterization of structural steels 3.34 where u represents the displacement vector of data point (subsets) (with =i, j, k and l). A mapping mask is then defined assuming threshold segmentation according to the following inequalities: ( , ) 1 if ( , ) ( , ) 0 if ( , ) ( , ) 1 if ( , ) nodata M x y A x y A M x y A x y A M x y A x y (3.10) where A is the average of ( , )A x y and is a scalar representing a threshold, as used for instance in image segmentation. This mask function M(x,y) (Equation 3.10) allows the classification of the measuring region according to the following description: • M(x,y) = 1 represents the damaged region (assumed here as the region around the crack tip); • M(x,y) = 0 corresponds to the region where the material is undamaged; • M(x,y) = -1 represents the region where the material is completely damaged and no information is available using digital image correlation (boundary of discontinuities). The crack tip is then estimated from the mask function for locations with M(x,y)=1. A key issue of this algorithm, however, is the calibration of the threshold (Equation 3.10). In this work, the threshold parameter was calibrated crossing direct xy optical observations on one face of the specimens and the DIC results from the opposite face of the specimen. This calibration process is not the most accurate one is the crack deviates across the thickness. a) b) Figure 3.23 - Schematic representation of the variation of relative position between adjacent subsets: a) subsets grid; b) adjacent subset displacements.
Chapter III 3.35 Figure 3.24 - Speckle pattern applied to one face of the modified CT specimen. In order to illustrate the process for crack tip identification, Figure3.25 plots the scalar function A(x,y) for the S235_I+II_01_01 specimen, for two distinct stages of the crack propagation process (stage 20 and 40). This scalar function may be understood as a damage map. Once assessed the A(x,y) map, the crack geometry are then defined using a cut plane located at A . Figure 3.26 plots the transformed map after the application of the cutting threshold, again for the S235_I+II_01_01 specimen, at stages 20 and 40. The crack paths assessed for S235_I+II_01_01, S235_I+II_01_02, S235_I+II_01_03 and S235_I+II_01_04 specimens are plotted in Figures 3.27, 3.28, 3.29 and 3.30, respectively. The Figures 3.27a) and b) plot the comparison between the X coordinate (horizontal direction) (Figure 3.27a) and Y coordinate (vertical direction) (Figure 3.27b), both assessed along the number of cycles using both optical microscope observations and DIC approach, for the S235_I+II_01_01 modified CT specimen. Figure 3.27 shows that both approaches yield similar crack paths, mainly regarding the X coordinate of the crack tip evolution. The deviation between approaches was higher when the comparison was based on the Y coordinate (see Figure 3.27b)). Figure 3.27c) points out a maximum difference between the trajectories of the crack tip of about 0.5 mm for the S235_I+II_01_01 specimen. It is important to stress that direct optical observations and DIC approach were applied to opposite specimen faces which may result in distinct observations due to cracks non-perpendicular to the side faces.
Fatigue characterization of structural steels 3.36 a) b) Figure 3.25 - A(x,y) map for the cracked S235_I+II_01_01 specimen at distinct stages of the crack propagation: a) for measurement stage 20; b) for measurement stage 40. a) b) Figure 3.26 - A(x,y) map for the cracked S235_I+II_01_01 specimen with the application of the threshold: a) for measurement stage 20; b) for measurement stage 40. Regarding the analysis of the Figure 3.28, it is observed that the X coordinate measured between both approaches presents a very good match (Figure 3.28a). However, the comparison between the Y coordinates (Figure 3.28b) exhibits a maximum difference of about 1 mm between both approaches. The deviation between approaches tends to reduce when the Y coordinate is higher than 1 mm. Figure 3.28c) shows the global crack path of the fatigue crack observed for S235_I+II_01_02, resulting from both experimental measurements.
Chapter III 3.37 a) b) c) Figure 3.27 – Comparison of assessed crack paths for S235_I+II_01_01 specimen: a) evolution of the X coordinate of the crack tip; b) evolution of the Y coordinate of the crack tip; c) crack path. 0 2 4 6 8 10 12 0 50 100 150 200 250 300 X coordinate [mm] Number of cycles XX coordinate assessed using DIC XX coordinate assessed using OM -0.5 0 0.5 1 1.5 2 2.5 3 3.5 050 100 150 200 250 300 Y coordinate [mm] Number of cycles YY coordinate assessed using DIC YY coordinate assessed using OM -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 Y coordinate [mm] X coordinate [mm] a_DIC (S235_I+II_01_01)
Fatigue characterization of structural steels 3.38 a) b) c) Figure 3.28 – Comparison of assessed crack paths for S235_I+II_01_02 specimen: a) evolution of the X coordinate of the crack tip b) evolution of the Y coordinate of the crack tip; c) crack path. 0 2 4 6 8 10 12 0 50 100 150 200 250 300 350 X coordinate [mm] Number of cycles XX coordinate assessed using DIC XX coordinate assessed using OM -0.5 0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 250 300 350 Y coordinate [mm] Number of cycles YY coordinate assessed using DIC YY coordinate assessed using OM -0.5 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 Y coordinate [mm] X coordinate [mm] a_DIC (S235_I+II_01_02) a_REF (S235_I+II_01_02)
Chapter III 3.39 The crack path assessments regarding the S235_I+II_o1_03 specimen, using both applied methods, are plotted in Figure 3.29. Figure 3.29a) compares the evolution of the X coordinate evaluated using the proposed two approaches, a very good agreement between approaches being observed. However, the comparison of the evolution of the Y coordinate (Figure 3.29b) exhibits a maximum deviation of about 1mm between both methodologies. However this deviation tends to reduce as the Y coordinate increases to a value higher than 1mm. The fatigue crack path of the modified CT specimen (S235_I+II_01_03) is presented in Figure 3.29c). The Figure 3.30 plots the measured crack paths for the S235_I+II_01_04 specimen. Again the absolute deviations were registered for the Y coordinate computation. For this specimen the crack propagated horizontally for a few millimetres and only deviated latter when the measured fatigue crack length increased above 4mm (Figure 3.30b). The measured fatigue crack path of the S235_I+II_01_04 specimen is presented in Figure 3.30c).
Fatigue characterization of structural steels 3.46 a) b) c) Figure 3.33 - Comparison of fatigue crack path predictions and experimental based crack paths: a) S235_I+II_01_01 and S235_I+II_01_02 specimens; b) S235_I+II_01_03 specimen; c) S235_I+II_01_04 and S235_I+II_01_05 specimens. -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 Y coordinate [mm] X coordinate [mm] a_DIC (S235_I+II_01_01) a_REF (S235_I+II_01_01) a_DIC (S235_I+II_01_02) a_REF (S235_I+II_01_02) MTS MCS MERR -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 Y coordinate [mm] X coordinate [mm] a_DIC (S235_I+II_01_03) a_REF (S235_I+II_01_03) MTS MCS MERR -0.5 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 Y coordinate [mm] X coordinate [mm] a_DIC (S235_I+II_01_04) a_REF (S235_I+II_01_04) a_REF (S235_I+II_01_05) MTS MCS MERR
Chapter III 3.47 The discrepancies in the experimentally based crack paths verified in some test repetitions may be attributed to geometric variations within general manufacturing tolerances. However, these specimens showed great sensitivity to the side hole position, therefore the tolerances respecting the side hole position and size should be reduced. 3.4.2.2.3. Stress intensity factors assessment for experimentally based crack paths In this section, the mode I and mode II stress intensity factors were evaluated for the tested modified CT specimens, along the experimentally based crack paths. This task was performed using the displacement fields from the DIC (displacement based technique). In addition, a numerical approach was followed based on finite element analysis. In this second approach, the experimentally based crack paths (from DIC analysis) were introduced in the finite element model and the respective stress intensity factors were computed using the virtual crack closure technique and the displacement extrapolation techniques. 3.4.2.2.3.1. SIFs computation using full-field DIC displacements DIC was applied previously to compute crack tip location. Once determined the crack tip location, the corresponding stress intensity factors are computed using a displacement based approach. For mixed-mode (mode I and mode II) loaded cracks, the analytical displacement fields around the crack tip are expressed using the Williams series, as follows [38]: 2 1 2 1 cos cos 2 1 cos 2 2 2 2 2 2 sin sin 2 1 sin 2 2 2 2 2 2 nn In x n nn IIn n An n n n n ur G An n n n n r G (3.11) 2 1 2 1 sin sin 2 1 sin 2 2 2 2 2 2 cos cos 2 1 cos 2 2 2 2 2 2 nn In y n nn IIn n An n n n n ur G An n n n n r G (3.12)
Fatigue characterization of structural steels 3.48 where ux and uy are the displacement components, G represents the elastic shear modulus, is equal to 3l for plane stress conditions and is equal to 34 for plane strain conditions, is the Poisson ratio and r and are respectively the polar coordinates cantered at crack tip. In the previous series the first terms AI1 and AII1 are relate to the mode I and mode II stress intensity factors, KI and KII according to the following relations: 12 I I K A (3.13) 12 II II K A (3.14) The dominant displacement component, concerning mixed-mode crack problems is unknown. Luo and Huang [25] and Yoneyama et al. [39] used the radial or circumferential displacement component on the polar coordinate system transformed from displacements on Cartesian coordinates. In the proposed method, both displacement components on the Cartesian coordinates are used simultaneously for the determination of the mixed-mode stress intensity factors. The displacement fields in Equation (3.11) and Equation (3.12) can be rewritten in the following compact form: 11 ,, x In In IIn IIn nn u A f r A f r (3.15) 11 ,, y In In IIn IIn nn u A g r A g r (3.16) where fI, fII, gI and gII are known functions of the coordinates r and and characteristic properties of the material, as shown in Equations (3.11) and (3.12). Considering the possibility of rigid body displacements, the displacement field data obtained by experimental or numerical methods can be expressed as:
Chapter III 3.49 11 ,, NN xk In In k k IIn IIn k k x k nn u A f r A f r T Ry (3.17) 11 ,, NN yk In In k k IIn IIn k k y k nn u A g r A g r T Rx (3.18) where Tx and Ty express the rigid body translation along the x and y directions, R is the rigid body rotation. In previous equations the series were truncated to a finite number of N terms. The subscript k (k =1,2,..., M, M: total number of displacement data points) represents a point with polar coordinates , kk r at which the displacement values uxk and uyk are expressed using the truncated series. The polar coordinates of a given point at crack tip vicinity can be computed using the following relations: 22 00k k k r x x y y (3.19) 1 tan ko k ko yy xx (3.20) where x0 and y0 are the location of a crack tip relative to an arbitrary cartesian coordinate system and xk and yk are the Cartesian coordinates of the k point under consideration. As the crack tip coordinate is already known. The unknown coefficients AIn, AIIn, Tx, Ty and R may be computed using a system of linear equations that results from the application of the least-squares technique to fit the analytical relations (3.17) and (3.18) to the experimental DIC displacement fields. All experimental data points were considered leading to an over-determined set of simultaneous equations. In this work the Williams series were truncated to 10 terms. In this case, not only the coefficients of the Williams series were computed, including the stress intensity factors, but also the rigid body translations and rotation were estimated.
Fatigue characterization of structural steels 3.50 3.4.2.2.3.2. Stress intensity computation using FEM Besides the simulation of the fatigue crack paths, finite element modelling was also used to compute the history of stress intensity factors for the experimentally-based crack paths. The same finite element models of the modified CT specimens described previously were also used to this purpose. The experimental fatigue crack paths assessed using the DIC approach were introduced into the finite element models of the modified CT specimens and the stress intensity factors afterwards computed using numerical approaches. The computed numerical stress intensity factors were then compared with the stress intensity factors computed from the full DIC approach. A typical finite element mesh of a modified CT specimen was already presented in Figure 3.32. The mesh presented in this figure corresponds to the mesh proposed for the S235_I+II_01_01 specimen and the presented crack corresponds already to an imposed crack path from DIC evaluation. It is interesting to refer that regular quadratic quadrilateral finite elements were considered around the crack tip, in order to facilitate the stress intensity factors computation using the Virtual Crack Closure Technique, as proposed by Krueger [36]. A coordinate system was adopted at the crack tip, oriented along the crack tip faces direction, in order to allow the appropriate computation of the nodal displacements and forces. The displacement extrapolation procedure was also used to compute the stress intensity factors, as an alternative approach. The stress intensity factors computation may be achieved using several alternative techniques, as the referred Virtual Crack Closure Technique (VCCT) [36], the Displacement Extrapolation method (DE), the contour integral method [40], among others. However, it was decided to limit the study presented in this section to the referred two techniques. The Virtual Crack Closure Technique The virtual crack closure technique is based on the assumption that the energy E released when the crack is extended by a from a+ a (node i) to a+2 a (node k) is identical to the energy required to close the crack. A modified version of the virtual crack closure technique [36] considers both nodal forces and displacement for the same step, to obtain the energy release rate. On effect, it is assumed that a crack extension from a+ a to a+2 a does not significantly modify the state at the crack tip. Therefore the
Chapter III 3.51 displacements behind the crack tip are approximately equal to the displacements behind the original crack tip [36]. For 8-noded 2D elements (see Figure 3.34), the equations to calculate the mode I and mode II strain energy release rates are: ** 1 2 I i l j m lm G Z w w Z w w a (3.21) ** 1 2 II i l j m lm G X u u X u u a (3.22) where: a is the length of the elements at the crack front; X and Z are nodal forces; u and w are nodal displacements. Previous relations were proposed for a unit thick plate. For a plate with thickness t, the relations must be divided by this parameter. The strain energy release rates can be converted into stress intensity factors using well-known fracture mechanics relations. Figure 3.34 – Nomenclature for Virtual Crack Closure Technique applied to 8-noded plane finite elements. Displacement extrapolation method The displacement extrapolation method uses a fit of the nodal displacements in the vicinity of the crack tip. The actual displacements near the crack tip, for linear elastic materials, may be computed using the following relations, which correspond to the truncated Williams series [41]: Ul a a Wl Um Wm Ul* Wl* Um* Wm* Xj ZiZj a crack closed x,u,X z,w,Z
Fatigue characterization of structural steels 3.52 33 2 1 cos cos 2 3 sin sin 4 2 2 2 4 2 2 2 I II x KK rr uGG (3.23) 33 2 1 sin sin 2 3 cos cos 4 2 2 2 4 2 2 2 I II y KK rr uGG (3.24) where ux and uy are the nodal displacements, assessed in a local Cartesian coordinate system located at the crack tip, r and are polar coordinates defined with respect to a origin at the crack tip, G is the shear modulus and is defined according to the following relation: 34 3 1 if planestrain if planestress (3.25) where is the Poisson’s ratio. Considering 180 , Equations (3.23) and (3.24) result into the following expressions: 1 22 II x Kr uG (3.26) 1 22 I y Kr uG (3.27) Equations (3.26) and (3.27) can be applied for some points at crack faces, located at distinct distances, r, from the crack tip. The stress intensity factors can then be computed by extrapolation the K(r) results for r=0. By this process, the numerical and truncation errors of the proposed procedure are minimized. Stress intensity factors values Figure 3.35 plots the equivalent stress intensity factors range evolution with the fatigue crack size. The equivalent stress intensity factor was computed using the Tanaka relation
Chapter III 3.53 42 Four series were presented in the graphs of Figure 3.35. In one series, the stress intensity factors were computed using a full two-steps approach. Firstly the crack path was assessed using DIC and then the stress intensity factors were computed using fullfield displacements from DIC. This original approach led to significant scatter in the data. In order to reduce this scatter, the crack path was then fitted using a third degree polynomial function and the stress intensity factors were afterwards computed using this smoothed crack path, which resulted in significantly less scatter. Stress intensity factors computed with the numerical model, for the imposed experimental crack path, were also plotted in Figure 3.35. The smoothed crack path from DIC analysis was used as input in the numerical model for the stress intensity factors computation. Figure 3.35a) presents the experimental and numerical stress intensity factor results for the S235_I+II_01_01 modified CT specimen. Figure 3.35a) allows the observation of a good match between the experimental and numerical equivalent stress intensity factors, at least until a crack of 7 mm. The full DIC results, without crack path filtering, shows very oscillating responses. The numerical does not exhibits this kind of scatter because a smoothed crack path was used in the simulations. Both numerical techniques explored for stress intensity factors computation produced the same results. The same analysis was made for the S235_I+II_01_02 specimen (Figure 3.35b)) and a reasonable agreement between both numerical and experimental stress intensity evolution curves was also observed. The Figure 3.35c) presents the results for the S235_I+II_01_03 specimens. A very good agreement may be observed for crack sizes until 8 mm. Above this value, some deviations are found between the numerical approaches and the experimental approaches. Figure 3.35d) exhibits the comparisons of the numerical and experimental equivalent stress intensity factor evolutions for the S235_I+II_01_04 specimen. In this case, only above crack sizes of 8 mm, the numerical/experimental approaches deviate significantly.
Fatigue characterization of structural steels 3.54 a) b) c) Figure 3.35 - Comparison of equivalent stress intensity factor range evolutions with the fatigue crack size: a) S235_I+II_01_01; b) S235_I+II_01_02; c) S235_I+II_01_03; d) S235_I+II_01_04. (1/2) 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 12 Equivalent stress intensity factor range, Keq [Nmm-1.5] Crack size [mm] Experimental SIF (DIC crack path) Experimental SIF (fitted crack path) FEM + DE FEM + VCCT 0 500 1000 1500 2000 2500 0 2 4 6 8 10 12 Equivalent stress intensity factor range, Keq [Nmm-1.5] Crack size [mm] Experimental SIF (DIC crack path) Experimental SIF (fiited DIC crack path) FEM + DE FEM + VCCT 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 2 4 6 8 10 12 Equivalent stress intensity factor range, Keq [Nmm-1.5] Crack size [mm] Experimental SIF (DIC crack path) Experimental SIF (fitted DIC crack path) FEM + DE FEM + VCCT
Chapter III 3.55 d) Figure 3.35 - Comparison of equivalent stress intensity factor range evolutions with the fatigue crack size: a) S235_I+II_01_01; b) S235_I+II_01_02; c) S235_I+II_01_03; d) S235_I+II_01_04. (2/2) 3.4.2.2.4. Correlation of pure mode I and mixed mode crack propagation data In this section, the pure mode I and mixed-mode fatigue crack propagation rates are correlated for the S235 steel. The mixed-mode fatigue crack propagation data used in this section came from DIC data reduction analysis applied to mixed-mode fatigue crack propagation tests, with the crack path fitted with a 3rd degree polynomial. Figure 3.36a) presents the fatigue crack growth rates obtained for the S235 steel grade, combining pure mode I fatigue crack growth data for a stress ratio R s =0.01, and mixed-mode (I+II) fatigue crack propagation data. The data is correlated using the Paris relation, a determination coefficient of 0.916 being observed, representing a satisfactory value, taking into account the distinct data sources. The equivalent stress intensity factor, as proposed by Tanaka [42], is demonstrated to be adequate to correlate mixed-mode fatigue crack propagation data. 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 12 Equivalent stress intensity factor range, Keq [Nmm-1.5] Crack size [mm] Experimental SIF (DIC crack path) Experimental SIF (fitted crack path) FEM + DE FEM + VCCT
Welded joint fatigue behaviour assessment using mesh insensitive procedures and S-N master curve concepts 5.24 d) Figure 5.18 - Membrane stress assessment using nodal forces approach and stress integration procedure for: a) W1 welded specimen; b) W2 welded specimen; c) W3 welded specimen and d) W4 welded specimen (1MPa applied remotely in the thicker plate). (2/2) Membrane and bending stresses were computed using both stress integration and nodal forces approaches, considering an applied uniform remote stress of 1 MPa in the thicker plate of the specimens. The stress integration was performed along the crack path direction at plate mid-thickness line. Figure 5.18 plots the membrane stress using the nodal forces approach and stress integration procedure for W1 welded specimen (Figure 5.18a), for W2 welded specimen (Figure 5.1 b), for W3 welded specimen (Figure 5.18c) and for W4 welded specimen (Figure 5.18d). A global analysis of the Figure 5.18 allows concluding that both procedures to compute the membrane stress components yield similar results, with the stress integration approach resulting in slightly higher structural stress values. However the deviations between approaches used to compute the membrane stresses increase slightly for the thicker specimens (W3 and W4). 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 mesh 1 mesh 2 mesh 3 W4 Membrane stress, sm[MPa] Nodal forces approach Stress integration approach
Chapter V 5.25 a) b) c) Figure 5.19 - Bending stress assessment using nodal forces approach and through thickness stress integration approach procedure for: a) W1 welded specimen; b) W2 welded specimen; c) W3 welded specimen and d) W4 welded specimen (1MPa applied remotely in the thicker plate). (1/2) 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 mesh 1 mesh 2 mesh 3 W1 Bending stress, sb[MPa] Nodal forces approach Stress integration approach 0.0 2.0 4.0 6.0 8.0 10.0 mesh 1 mesh 2 mesh 3 W2 Bending stress, sb[MPa] Nodal forces approach Stress integration approach 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 mesh 1 mesh 2 mesh 3 W3 Bending stress, sb[MPa] Nodal forces approach Stress integration approach
Welded joint fatigue behaviour assessment using mesh insensitive procedures and S-N master curve concepts 5.26 d) Figure 5.19 - Bending stress assessment using nodal forces approach and through thickness stress integration approach procedure for: a) W1 welded specimen; b) W2 welded specimen; c) W3 welded specimen and d) W4 welded specimen (1MPa applied remotely in the thicker plate). (2/2) Figure 5.19 plots the bending stress results for W1 welded specimens (Figure 5.19a), for W2 welded specimens (Figure 5.19b), for W3 welded specimens (Figure 5.19c) and for W4 welded specimens (Figure 5.19d). Both investigated procedures yield similar results for the thinner specimens. However for the thicker specimens both approaches yield distinct results, the stress integration approach yielding the higher values. This observation may be explained by the fact that the stress integration was performed along the mid-thickness plate line. With this process it is assumed that the stress distribution is uniform across the plate thickness which could not correspond to the reality. Structural stresses from the nodal forces approach showed greater mesh size insensitivity. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 mesh 1 mesh 2 mesh 3 W4 Bending stress, sb[MPa] Nodal forces approach Stress integration approach
Chapter V 5.27 a) b) c) Figure 5.20 - Comparison between structural stresses computed using through thickness stress integration and nodal forces approaches and notch stress (1MPa applied remotely in the thicker plate). (1/2) 0 10 20 30 40 50 60 70 80 mesh 1 mesh 2 mesh 3 W1 Stress, s[MPa] Equivalent structural stress (through thickness stress approach) Equivalent structural stress (Nodal Forces) Local stress 0 2 4 6 8 10 12 14 16 mesh 1 mesh 2 mesh 3 W2 Stress, s[MPa] Equivalent structural stress (through thickness stress approach) Equivalent structural stress (Nodal Forces) Local stress 0 5 10 15 20 25 30 35 40 mesh 1 mesh 2 mesh 3 W3 Stress, s[MPa] Equivalent structural stress (through thickness stress approach) Equivalent structural stress (Nodal Forces) Local stress
Welded joint fatigue behaviour assessment using mesh insensitive procedures and S-N master curve concepts 5.28 d) Figure 5.20 - Comparison between structural stresses computed using through thickness stress integration and nodal forces approaches and notch stress (1MPa applied remotely in the thicker plate). (2/2) Figure 5.21 - Stress concentration factors resulting from the ratio between the equivalent structural stress computations, based on Equation (5.18), and the remote uniform applied stress. Figure 5.20 compares the structural stress assessed using both aforementioned approaches and the local notch stresses. Figure 5.20 illustrates that the structural stress assessment is insensitive to the mesh size change, unlike the local notch stresses. Both structural stress approaches provide similar results, except for the W4 welded specimen, mainly due to differences in the bending stress components. Figure 5.21 presents the stress concentration factor resulting from the ratio between the equivalent structural stress computations, based on Equation (5.18), and the remote applied stress. This information is important since it allows the computation of the equivalent structural stress for the welded specimens taking into account the distinct 0 5 10 15 20 25 30 35 40 mesh 1 mesh 2 mesh 3 W4 Stress, s[MPa] Equivalent structural stress (through thickness stress approach) Equivalent structural stress (Nodal Forces) Local stress 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 W1 W2 W3 W4 Stress concentration factor Nodal forces approach through thickness structural stress
Chapter V 5.29 tested remote stresses. The structural stresses were based on the mean results from all considered FEM meshes. Figure 5.22 shows the individual experimental S-N curves, assessed for all welded specimens using the equivalent structural stresses based on through thickness stress integration approach. Figure 5.23 gives the S-N curves based on structural stresses computed using the nodal forces approach. Comparing the S-N curves resulting from both approaches, the main difference lies on C parameter, which means a translation of the S-N curves. The slope of the S-N curves remains the same for both approaches. a) b) Figure 5.22 - Experimental S-N curves assessed using through thickness structural stress procedure for welded specimens: a) W1; b) W2; c) W3 and d) W4. (1/2) 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W1 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 63.97 MPa @ 2.0E+6 s=2.30E+12Nf3.29 R2=0.9824 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W2 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 71.56 MPa @ 2.0E+6 s=1.64E+12Nf3.73 R2=0.9656
Welded joint fatigue behaviour assessment using mesh insensitive procedures and S-N master curve concepts 5.30 c) d) Figure 5.22 - Experimental S-N curves assessed using through thickness structural stress procedure for welded specimens: a) W1; b) W2; c) W3 and d) W4. (2/2) a) Figure 5.23 - Experimental S-N curve assessed using nodal forces assessment for welded specimens: a) W1; b) W2; c) W3 and d) W4. (1/2) 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W3 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 72.53 MPa @ 2.0E+6 s=2.22E+12Nf3.25 R2=0.9936 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W4 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 76.59 MPa @ 2.0E+6 s=4.26E+12Nf3.36 R2=0.9926 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W1 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 70.20 MPa @ 2.0E+6 s=2.30E+12Nf3.29 R2=0.9824
Chapter V 5.31 b) c) d) Figure 5.23 - Experimental S-N curve assessed using nodal forces assessment for welded specimens: a) W1; b) W2; c) W3 and d) W4. (2/2) 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W2 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 70.57 MPa @ 2.0E+6 s=1.56E+13Nf3.73 R2=0.9656 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W3 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 68.87 MPa @ 2.0E+6 s=1.87E+12Nf3.25 R2=0.9936 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W4 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 68.07 MPa @ 2.0E+6 s=2.86E+12Nf3.36 R2=0.9926
Welded joint fatigue behaviour assessment using mesh insensitive procedures and S-N master curve concepts 5.32 Figure 5.24 shows the master S-N curve that results from the combination of all welded specimens fatigue tested in this work. The master S-N curve was assessed using the equivalent structural stress parameters, computed using the through thickness stress integration approach. Figure 5.24 plots the master S-N curve for all welded specimens, using the equivalent structural stress based on nodal force based approach. Correlations of Figures 5.24 and 5.25 show a higher determination coefficient, than observed in Figure 4.50 for the S-N curve based on nominal stress definition. However, the S-N curve assessed using the nominal stress definition yield a higher stress range value for a life of 2E6 cycles as summarized in Table 5.2. Table 5.2 - Stress range values at 2E6 cycles. Stress definition s [MPa] Nominal stress 76.50 Equivalent structural stress (through thickness stress integration approach) 71.09 Equivalent structural stress (nodal forces approach) 69.37 Figure 5.24 - S-N master curve assessed using through thickness structural stress procedure of welded specimens. 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W1 Welded specimen - W2 Welded specimen - W3 Welded specimen - W4 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 71.09 MPa @ 2.0E+6 s=4.44E+12Nf3.43 R2=0.96
Chapter V 5.33 Figure 5.25 - S-N master curve assessed using nodal forces data, for all welded specimens fatigue tested. Figures 5.26 and 5.27 compares the S-N fatigue data from this research with the fatigue data from several literature sources. The objective is to validate the S-N master curve concept for a wider range of fatigue data. Figure 5.25 includes the fatigue data from this research assessed using the through thickness stress integration approach. Figure 5.26 includes the fatigue data from this research assessed using the nodal forces approach. In both cases a very significant determination coefficient was observed. However, this determination coefficient was marginally higher in Figure 5.25 that resulted from the through thickness integration approach. These graphs illustrate clearly the validity of the S-N master curve concept, given the diversity of data sources and the high determination coefficients observed in the correlations. 10 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 Equivalent structural stress range, ΔSs[MPa] Number of cycles to failure, Nf Welded specimen - W1 Welded specimen - W2 Welded specimen - W3 Welded specimen - W4 Mean S-N curve Mean S-N curve ± 2 standard deviation Mean S-N curve ± 3 standard deviation EC3-9 - Detail 45 69.37 MPa @ 2.0E+6 s=4.44E+12Nf3.43 R2=0.98
Finite element modelling of experimental details 6.2 6.1. INTRODUCTION This chapter presents numerical models for the connection details presented in the Chapter IV. Several numerical approaches were considered in order to perform fatigue life predictions. Two main finite element tools were used: the standard Finite Element Method (FEM) and the eXtended Finite Element Method (XFEM). However, the XFEM approach was mainly limited to the welded joints simulation since the XFEM available in the ABAQUS® code [1] used for this purpose, did not allow the use of contact elements simultaneously with XFEM. Thus, the riveted specimens were essentially modelled using the FEM and the welded specimens were modelled using both FEM and XFEM approaches. The aim of the simulations was to derive numerical S-N curves for the joints. Fatigue life predictions were performed taking into account both fatigue crack initiation and fatigue crack propagation phases. The number of cycles to initiate a fatigue crack was computed using local notch strain approach and the number of cycles to propagate the fatigue crack was computed using a Fracture Mechanics based approach. To assist the Fracture Mechanics approach, both ANSYS® and ABAQUS® commercial codes were used to build numerical models in order to assess the stress intensity factor range. Contour integral method using a 3D XFEM (implemented in ABAQUS®), as well as the virtual crack closure technique (VCCT), as described by Krueger [2], with standard FEM (implemented in ANSYS®) were considered. The fatigue crack growth was modelled using Linear Elastic Fracture Mechanics (LEFM) together with the Paris Law [3]. Numerical S-N curves were simulated including both fatigue crack initiation and fatigue crack propagation phases, for riveted joints, and considering the crack propagation phase, for the welded joints.
Chapter VI 6.3 6.2. CRACK INITIATION ASSESSMENT The number of cycles to initiate a fatigue crack was computed using a local notch strain approach. The required cyclic elastoplastic strains were computed using the Neuber [4] and Ramberg-Osgood [5] relations, which may be combined in the following system of simultaneous equations: 1 22' 1' 22' 22' n t nom loc loc loc n loc loc loc k E E K EK (6.1) where kt is the elastic stress concentration factor, K´ and n´ are, respectively, the cyclic strain hardening coefficient and exponent; nom is the nominal stress range computed at the net area near the hot spot; loc and loc are respectively the local strain and stress ranges. The stress concentration factor was computed using the finite element model of the riveted joint. The number of cycles for crack initiation was computed thru the Morrow relation [6]: c if b i f loc NN E22 2 , , (6.2) where loc is the local elastoplastic strain range; , f and b are, respectively, the cyclic fatigue strength coefficient and exponent; , f and c are, respectively, the fatigue ductility coefficient and exponent; E is the Young modulus.
Finite element modelling of experimental details 6.4 6.3. FATIGUE CRACK PROPAGATION ASSESSMENT Once assessed the number of cycles to start a fatigue crack, the number of cycles to propagate the respective fatigue crack is performed using a Linear Elastic Fracture Mechanics (LEFM) based approach. The stress intensity factor (K) is a key parameter to compute the number of cycles to propagate a fatigue crack. In this section, two techniques available in the literature for the computation of the stress intensity factors, namely the contour integral method and the Virtual Crack Closure Technique will be described. 6.3.1. Fatigue crack growth simulation A well-established approach for fatigue crack propagation simulation consists of relating the fatigue crack growth with stress intensity factor range K at the crack tip. One can find many approaches in the literature for the relationship between the crack growth rate and the stress intensity factor range [7]. In this work, the well-known Paris law will be used due to the availability of experimental data [3]: m da CK dN (6.3) where C and m are material constants. Despite mode I fatigue crack propagation is dominant for almost all modelled specimens, for some models a mixed mode fatigue crack growth criterion was considered, therefore both KI and KII stress intensity factors were computed. A crack branching procedure was then considered [8], with the new crack increment angle computed using the maximum hoop stress criterion: 22 2242 1 9 83 cos III IIIIII KK KKKK (6.4)
Chapter VI 6.5 In order to allow the application of the Paris relation for mixed mode conditions, an equivalent stress intensity factor was computed using the relation proposed by Tanaka [9]: 44 48 eq I II K K K (6.5) The number of cycles spent during the fatigue crack propagation may be computed thru the integration of the Paris crack propagation relation [3]. 6.3.2. Stress intensity factor computation Irwin [10-12] developed a method for the evaluation of the amount of energy available for the crack propagation. It is worthy to introduce the three basic crack deformation/loading modes, which are the mode I, mode II and mode III, as illustrated in Figure 6.1. The mode I fatigue crack propagation is the opening mode, which corresponds to the opening of the crack faces under a tensile load. Mode II is the in-plane shear/sliding loading mode. The applied shear stresses act parallel to the plane of the crack and perpendicular to the crack front. Mode III is classified as out of plane tearing mode. Loads are applied parallel to both the plane of the crack and the crack front. The method evaluates the energy release rate based on the loading and geometry conditions. The stress field for a linear elastic solid in the neighbourhood of crack tip, in its generic form, is given as: ( , ) ( , ) ( , ) I II III K r K r K r I II III f f f (6.6) where KI, KII and KIII are the stress intensity factors for the respective modes I, II and III, 0 is a finite stress, r is the distance from the crack tip and is the angle from the crack tip and fI, fII, and fIII are functions of and r, being proportional to 1 √𝑟 (see Equations (6.7) and (6.8)).
Finite element modelling of experimental details 6.6 Figure 6.1 - Three pure crack loading modes: a) mode I; b) mode II and c) mode III. Figure 6.2 - A three-dimensional coordinate system describing the stresses near the crack front. 13 ( ) cos 1 sin sin 2 2 2 2 xx a a a r I f (6.7) 13 ( ) cos 1 sin sin 2 2 2 2 yy a a a r I f (6.8) The terms represented in Equations (6.7) and (6.8) are singular when r tends to zero. Other terms may be included in the series which can be found in reference [13]. A schematic definition of the stress field near the crack tip can be observed in Figure 6.2. The stress intensity factors, for the three loading modes, are defined as: 0,2 lim 0 rrK yy r I (6.9) X Y Z X Y ZX Y Z
Chapter VI 6.7 0,2 lim 0 rrK yx r II (6.10) 0,2 lim 0 rrK yz r III (6.11) where ij is a particular stress component. 6.3.2.1. Contour integral evaluation A way used in this work to compute the stress intensity factors considers the J-integral together with the so-called interaction integral method [14]. The J-integral is a contour integral that allows the assessment of the strain energy release rate during a fracture process [15]. This parameter is important in Fracture Mechanics since the energy can be related to the crack growth. The interaction integral method is an extension of the Jintegral that allows the computation of the energy release rates for modes I and II, where the J-integral only allows pure modes. 6.3.2.2. J-integral The J-integral is originally defined for a two dimensional contour integral, as shown in Figure 6.3a), but it can also be used, for computational purposes, considering a closed two dimensional contour integral, as exhibited in Figure 6.3b). This method can be extended to a 3D problem, which is currently linked to the interaction integral method to extract the stress intensity factors [14]. For a two dimensional quasi-static analysis, the J-integral may be defined as [14]: 0 lim Jdn H q (6.12)
Finite element modelling of experimental details 6.8 a) b) Figure 6.3 - Contour Integral definitions: a) 2D contour integral (open contour); b) 2D closed contour integral. where is the contour around the crack tip, d is the arc increment on , n is the outwards normal of the contour, q is the unit vector in the virtual crack extension direction. Adopting the contour integral illustrated in Figure 6.3a), it is assumed that the contour is connected to the two crack faces sealing the crack tip. The contour is reduced so that it only includes the crack tip ( in Equation (6.12)). The outwards pointing normal is located along the whole contour and the unit vector in the virtual crack extension direction q is located at the crack tip. The contour does not have to shrink until the crack tip, but can be specified anywhere enclosing the crack tip, since the J-integral is path-independent for elastic material in the absence of body forces and tractions on the crack surfaces [14]. A regular 2D contour integral can be rewritten as a 2D closed contour integral [14]: C C C Jdm H q (6.13) where the integral segments are defined as a closed contour that is extended from , (see Figure 6.3b), C and C are contours along the crack faces respectively and encloses the contour from and over the crack tip, m is a unit outwards-pointing normal defined as m=- n. The weighting function q has been introduced as the unit vector in the virtual crack extension direction, qq on and vanishes on C. The J-integral can now be transformed into a domain integral with the divergence theorem [14]: X Y X Y q n crack crack n m m AC C C + - X Y X Y q n crack crack n m m AC C C + -
Chapter VI 6.9 A J dA xHq (6.14) where A is the area domain enclosed by the closed contour, and dA the infinitesimal area segment. Considering the equilibrium equation, the gradient of the strain energy for a homogenous material with constant material parameters, the 2D J-integral can be rewritten in its final form [14]: : A J dA xx qu H f q (6.15) where f is the body force per volume unit. Thermal influence is neglected here. The two dimensional J-integral procedure (Equation (6.15)) can be extended to a three dimensional crack front where the J-integral is a function of the variable s along the crack front, J(s), as illustrated in Figure 6.4a) [14]. The three dimensional calculations are performed in a similar manner as the two dimensional case, but the energy release rate is calculated with respect to a finite segment of the crack front, denoted as J . This is then used to obtain energy release rate J(s) for each node located along the crack tip. This procedure is done by defining a parametric variable s along the crack front with a local coordinate system, as represented in Figure 6.4a). The axis, z, runs tangentially to the crack, y is defined perpendicular to the crack plane, and x normal to the crack front. a) b) Figure 6.4 - A three-dimensional coordinate system (a) used to define the 3D domain integral at the crack front (b). x Aends Aends At A0 V Crack Acrack y z s m z x Aends Aends At A0 V Crack Acrack y z s m z
Finite element modelling of experimental details 6.10 In three dimension analysis, the energy release for a unit crack advance over a finite segment of the crack front, J , is defined as [14]: : V J dV xx qu H f q (6.16) The three dimensional case is a volume integral for the domain V shown in Figure 6.4b). This is a tubular domain for a closed contour along a finite segment of the crack front. The three dimensional integral presents the inner tube surface, At, the outer tube surface, A0, the two surfaces along the crack face, Acrack and lastly the two surfaces at the ends, Aends, in accordance with the closed contour domain. Noting that still 0 which means that 0 t A . The two dimensional area domain along the crack front in the x yplane is called contour domain in this document. 6.3.2.3. Stress intensity factor computation For linear elastic materials, the J-integral is related to the stress intensity factors by the following relationship [16]: 1 1 8 T JK B K (6.17) where T I II III K K KK and B is the pre-logarithmic energy factor. For a homogeneous and isotropic material, the equation can be rewritten as: 222 2 11 IIIIIIK G KK E J (6.18) where EE for plane stress and 2 1 E E for plane strain, axisymmetric and three dimensional problems. Furthermore, under pure mode I loading, the relation between the J-integral and KI for three dimensions is:
Chapter VI 6.11 E KJ I 2 2 1 1 (6.19) To evaluate mixed-mode stress intensity factors, the interaction integral method can be used. The interaction integral method uses auxiliary fields superimposed to the actual fields. The auxiliary field consists of stresses or strains fields applied around the crack tip. The J-integral of the actual field is denoted as J, the J-integral related to the auxiliary field as Jaux and the J-integral from the interaction integral as Jint. These three terms joined together define the total J-integral, tot J , that is: int JJJJ auxtot (6.19) By choosing the auxiliary fields wisely, the interaction integral for mode ( =I, II and III) can be expressed as int tot aux J J J J , which is used to extract the individual stress intensity factors. The stress intensity factor extraction for mode I is obtained expanding equation (6.17). The relation between the J-integral and the stress intensity factors is given as: 1 1 1 11 12 13 I 12 2 terms without K 8I I I II I III J K B K K B K K B K (6.20) The J-integral for an auxiliary field, mode I crack tip field with 1 k as stress intensity factor, is chosen as: 1 11 1 8 I aux I I J k B k (6.21) Superposition of the auxiliary field and the real field gives: 1 1 1 11 12 13 II 122 8 terms without K andk I tot I I I I I I II I I III J K k B K k K k B K K k B K (6.22)
Finite element modelling of experimental details 6.18 Asymptotic near crack tip functions When the finite element is not completely cut by the crack, the Heaviside function cannot be used to approximate the displacement field over the entire element domain, since the element contains the crack tip. Fleming et al. [22] proved that the displacement field from LEFM theory is included within the span of the following four functions, expressed in terms of the local crack tip coordinate system (r, ): sin 2 cossin 2 sin 2 sin 2 cos),( 4 1rrrrrF i i (6.30) These enrichment functions are added together with four degrees of freedom in each direction, for each node, in addition to those related with the standard finite element discretization. Based on the two types of enrichment functions presented above, the following expression for the XFEM displacements approximation can be formulated: 12 12 44 12 11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h FEM ENRICHED ll i i j j k k l k k l i I j J k K l k K l u x u x u x N x u N H x a N x b F x N x b F x (6.31) where, J indicates the set of nodes belonging to the domain which is completely cut by the crack and enriched with the Heaviside function H(x), K1 and K2 are the sets of nodes associated with the crack tips 1 and 2, and respectively enriched with the )( 1xFl and )( 2xFl functions. Moreover, ui are the standard degrees of freedom, while aj, 1 l k b and 2 l k b represent the additional nodal degrees of freedom introduced for modelling crack faces and two crack tips, respectively. 6.5.2. Level set method for modelling discontinuities The modelling and tracking of time-varying objects, such as a fatigue propagating crack, is particularly cumbersome. The Level Set Method [23] is a numerical technique that
Chapter VI 6.19 allows overcoming these difficulties. This method represents the crack as a zero level set function and to fully characterize a crack, two different level set functions are required: A normal level set function )(x ; A tangential level set function )(x . Figure 6.9 - Level set functions used for crack definition (crack path and crack tip). In order to evaluated the ()x function, the signed distance functions that define the location of any point x to the crack, has to be assess. Consider the crack surface, c , of the Figure 6.9. In the same manner of what was achieved in the previous subsection, the normal level set function might be defined as: ()x x x n (6.32) The tangential level set function )(x is computed by finding the minimum signed distance to the normal at the crack tip; in case of an interior crack, two different functions can be formulated, but a unique tangential level set function can be defined as: )()(max)( 21 xxx (6.33)
Finite element modelling of experimental details 6.20 Figure 6.10 - Standard, enriched and blending elements in a XFEM domain discretization. 6.5.3. Blending elements The finite elements having all nodes enriched may be named as reproducing elements since they reproduce the enrichment functions exactly. The blending elements allow to blend the enriched sub-domain with the rest of the domain, where standard finite elements are employed. Only some of the nodes in blending elements are enriched. Enriched finite elements, blending elements and standard finite elements defines the whole domain in three different parts, namely an enriched domain, a blending domain and finally a standard domain, as illustrated in Figure 6.10. It is worthwhile to refer that two important drawbacks affect blending elements [24]: Enrichment functions cannot be reproduced exactly in blending elements, since the partition of unity property is not satisfied within them. These elements produce unwanted terms in the approximation, which cannot be compensated by the finite element part. For instance, if the enrichment introduces non-linear terms, a linear function can no longer be approximated within blending elements. A significant reduction of the convergence rate for general enrichment functions [25] is observed. Thus, suboptimal rate of convergence in XFEM may be caused by problems in blending elements [26]. Crack Enriched nodes Blending elements
Chapter VI 6.21 Figure 6.11 – Summary of the proposed simulations for each verification example, for stress intensity factors computation. 6.6. VERIFICATION EXAMPLES OF STRESS INTENSITY FACTORS COMPUTATION This section presents some verification examples concerning the computation of the stress intensity factors using the XFEM procedure, combined with the contour integral method. The resulting stress intensity factors are compared with stress intensity factors computed using the VCCT approach, as well as using available analytical solutions (benchmark solutions). Several finite element mesh refinements are considered in order to observe the mesh sensitivity of the adopted approaches. Figure 6.11 presents an overview of the performed simulations for each verification example. Only thru thickness (constant depth) cracks were considered. Concerning the standard finite element modelling (FEM), the ANSYS® program [27] was used to model 3D geometries using parametric models. Linear and quadratic finite elements were considered. A special care was dedicated to the model construction in order to control the mesh size at the crack tip. A refined mesh was built at the crack tip. The sensitivity of the stress intensity factors to the mesh size was evaluated, using four mesh refinements (mesh 1 to 4). SIF FEM XFEM VCCT Contours integral mesh1 mesh2 mesh3 mesh4 mesh1 mesh2 mesh3 mesh4 Linear FE Second order FE Linear FE All contour (1 to 5) Contour 2 to 5 Contour 3 to 5 Contour 4 and 5
Finite element modelling of experimental details 6.22 Figure 6.12 - Contours used in the computation of the stress intensity factors with the XFEM approach. The XFEM is a methodology incorporated in the ABAQUS® software [1]. Stress intensity factors were computed using the contour integral method available in ABAQUS®. In order to evaluate accurately stress intensity factors, several contours were used (see Figure 6.12). As a first approach, five contours around the crack tip element were used to compute the stress intensity factors. The second approach to assess the stress intensity factors did not consider contour 1. The third approach only used contours 3, 4 and 5 to compute stress intensity factors and the last approach only considered contours 4 and 5. In order to allow more accurate computations of the stress intensity factors, the initial crack was considered long enough to assure the contours to be moved away from the plate edge. In order to investigate the mesh size effect on the numerical results, several finite element meshes were considered, referred in Figure 6.11 as mesh 1, 2, 3, and 4. Only linear finite element was considered in the analyses, since second order finite elements are not allowed to be used with the XFEM methodology in ABAQUS® [1]. 6.6.1. Verification example 1 The first verification example consists of a central through-thickness crack located in a finite rectangular plate, as illustrated in Figure 6.12. As referred above, stress intensity factors were computed according to procedures depicted in Figure 6.11. The numerical results were then compared with analytical solutions available in BS 7910 standard [28]: Contour 1 Contour 2 Contour 3 Contour 4 Contour 5 Crack tip FE XFEM crack
Chapter VI 6.23 2 1 42 sec96.001.01 W a W a W a aPKI (6.34) where P is the applied load (equivalent to a uniformly distributed stress), 2a is the crack length and W a geometrical parameter (refer to Figure 6.13). Only ¼ of the geometry was modelled. Figure 6.14 illustrates the FEM model and plots three distinct mesh refinements modelled around the crack tip. For a fixed volume around the crack tip (box), the number of finite elements is progressively increased. Also, the element size through the thickness direction was also progressively increased. The information regarding the mesh refinement is summarized in Table 6.1, for the FEM model. Figure 6.13 - Central through-thickness crack located in a finite plate. a) b) c) Figure 6.14 - Finite element meshes for the verification example 1: a) coarse FE mesh; b) intermediate FE mesh; c) fine FE mesh. 2a B W P P ne2 ne2 ne1 ne2 ne2 ne1 ne2 ne2 ne1
Finite element modelling of experimental details 6.24 Table 6.1 - Finite element refinement used for the verification example 1 (FEM approach). ne1 – Number of FE ne2 – Number of FE mesh1 3 2 4 8 mesh2 3 3 4 8 mesh3 3 4 4 8 mesh4 3 8 4 8 Figure 6.15 illustrates the enriched finite elements zone used in the XFEM approach with two distinct mesh refinements. Table 6.2 summarizes the number of elements used in four distinct mesh refinements. Figure 6.16 to 6.19 plot the ratio between the numerical stress intensity factors and the analytical stress intensity factors for distinct crack lengths. In particular, a) figures present the results from finite element computation and VCCT approach; b) figures show the results from the XFEM approach with the contour integral method. a) b) Figure 6.15 – Examples of enriched finite elements area used in the computation of the stress intensity factors with the XFEM approach, for verification example 1: a) coarse FE mesh; b) fine FE mesh. ne3 ne1 ne2 ne3 ne1 ne2 ne3 ne1 ne2 ne3 ne1 ne2
Chapter VI 6.25 Table 6.2 - Finite element refinement used for the verification example 1 (FEM approach). ne1 – Number of FE ne2 – Number of FE ne3 – Size of FE [mm] mesh1 3 5 1 4 8 mesh2 3 5 0.2 4 8 mesh3 3 50 1 4 8 mesh4 3 50 0.2 4 8 Analyzing the stress intensity values computed using XFEM (Figure 6.16 b), Figure 6.17 b), Figure 6.18 b) and Figure 6.19 b)), the numerical/analytical ratio was always greater than unity, which means that the computed numerical stress intensity factors are higher than the analytical stress intensity factors. The VCCT plus FEM results (Figure 6.16 a), Figure 6.17 a), Figure 6.18 a) and Figure 6.19 a)) are, in general, more close to the analytical results than XFEM approach and the accuracy of the first method increases with the increase of the crack length. For higher crack sizes, the VCCT method produces very accurate results. Concerning the VCCT approach, the higher refined mesh combined with quadratic elements produced less accurate results. For this approach, the use of quadratic elements did not result in more accurate predictions. The XFEM with the contour integral approach gives better results if contour 1 is excluded from the analysis. We observe that the mesh refinement in the plate plane did not impact significantly the convergence of the numerical solution to the analytical solution. However the mesh refinement in the thickness direction showed important effect on stress intensity factors, mainly for smaller crack sizes.
Finite element modelling of experimental details 6.26 a) b) Figure 6.16 - Ratio between the numerical stress intensity factor and analytical stress intensity factor versus the crack length computed for the mesh 1 (verification example 1): a) FEM analysis; b) XFEM analysis. 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 2 4 6 8 10 SIFnum/SIFanalytical Crack length (mm) FEM and VCCT (2 order FE) - ne1=3 FEM and VCCT (2 order FE) - ne1=4 FEM and VCCT (2 order FE) - ne1=8 FEM and VCCT (linear FE) - ne1=3 FEM and VCCT (linear FE) - ne1=4 FEM and VCCT (linear FE) - ne1=8 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 2 4 6 8 10 SIFnum/SIFanalytical Crack length (mm) XFEM and Contour Integral Method (all contour) - ne1=3 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=3 XFEM and Contour Integral Method (all contour) - ne1=4 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=4 XFEM and Contour Integral Method (all contour) - ne1=8 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=8
Chapter VI 6.27 a) b) Figure 6.17 - Ratio between the numerical stress intensity factor and analytical stress intensity factor versus the crack length computed for the mesh 2 (verification example 1): a) FEM analysis; b) XFEM analysis. 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 2 4 6 8 10 SIFnum/SIFanalytical Crack length (mm) FEM and VCCT (2 order FE) - ne1=3 FEM and VCCT (2 order FE) - ne1=4 FEM and VCCT (2 order FE) - ne1=8 FEM and VCCT (linear FE) - ne1=3 FEM and VCCT (linear FE) - ne1=4 FEM and VCCT (linear FE) - ne1=8 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 2 4 6 8 10 SIFnum/SIFanalytical Crack length (mm) XFEM and Contour Integral Method (all contour) - ne1=3 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=3 XFEM and Contour Integral Method (all contour) - ne1=4 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=4 XFEM and Contour Integral Method (all contour) - ne1=8 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=8
Finite element modelling of experimental details 6.34 a) b) Figure 6.25 - Ratio between the numerical stress intensity factor and analytical stress intensity factor versus the crack length computed for the mesh 3 (verification example 2): a) FEM analysis; b) XFEM analysis. 0.8 0.85 0.9 0.95 1 1.05 1.1 0.5 0.7 0.9 1.1 1.3 SIFnum/SIFanalytical Título FEM and VCCT (2 order FE) - ne1=3 FEM and VCCT (2 order FE) - ne1=4 FEM and VCCT (2 order FE) - ne1=8 FEM and VCCT (linear FE) - ne1=3 FEM and VCCT (linear FE) - ne1=4 FEM and VCCT (linear FE) - ne1=8 0.8 0.9 1 1.1 0.5 0.7 0.9 1.1 1.3 SIFnum/SIFanalytical Crack length (mm) XFEM and Contour Integral Method (all contour) - ne1=3 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=3 XFEM and Contour Integral Method (all contour) - ne1=4 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=4 XFEM and Contour Integral Method (all contour) - ne1=8 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=8
Chapter VI 6.35 a) b) Figure 6.26 - Ratio between the numerical stress intensity factor and analytical stress intensity factor versus the crack length computed for the mesh 4 (verification example 2): a) FEM analysis; b) XFEM analysis. 0.8 0.85 0.9 0.95 1 1.05 1.1 0.5 0.7 0.9 1.1 1.3 SIFnum/SIFanalytical Título FEM and VCCT (2 order FE) - ne1=3 FEM and VCCT (2 order FE) - ne1=4 FEM and VCCT (2 order FE) - ne1=8 FEM and VCCT (linear FE) - ne1=3 FEM and VCCT (linear FE) - ne1=4 FEM and VCCT (linear FE) - ne1=8 0.8 0.9 1 1.1 0.5 0.7 0.9 1.1 1.3 SIFnum/SIFanalytical Crack length (mm) XFEM and Contour Integral Method (all contour) - ne1=3 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=3 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=3 XFEM and Contour Integral Method (all contour) - ne1=4 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=4 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=4 XFEM and Contour Integral Method (all contour) - ne1=8 XFEM and Contour Integral Method (Contour 2,3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 3, 4 and 5) - ne1=8 XFEM and Contour Integral Method (Contour 4 and 5)) - ne1=8
Finite element modelling of experimental details 6.36 6.7. NUMERICAL SIMULATION OF WELDED SPECIMENS This section presents a numerical investigation aiming the assessment of S-N curves for the welded specimens presented in the Chapter IV. The proposed assessment was performed considering both standard finite element method and extended finite element method. Numerical S-N curves were computed considering the contribution of fatigue crack growth (LEFM based approach). There is a generally accepted understanding that fatigue crack propagation is the major damage mechanism in welded joints. Existing code based design curves are supported by this assumption, their slopes being established from the fatigue crack propagation data. Fatigue crack initiation may be surpassed by existing defects in the welds. Concerning the standard FE models, performed in ANSYS®, only ¼ of specimen geometries were modelled, taking into account existing symmetries (see Figure 6.27). Materials were considered linear elastic and isotropic (E=210 GPa; =0.27). Hexahedral 20-noded finite elements were used. The crack path was defined during the finite element mesh generation phase. Both the initial crack length and the crack increments were defined with 0.5mm. Stress intensity factor ranges were computed using the Virtual Crack Closure Technique, as proposed by Krueger [1]. The applied loading conditions result in dominant pure mode I, which means that only mode I stress intensity factors were computed at the crack tip. Figure 6.27 shows the finite element meshes of W1 welded specimens (Figure 6.27a), W2 welded specimens (Figure 6.27b), W3 welded specimens (Figure 6.27c) and W4 welded specimens (Figure 6.27d). Figure 6.27 also exhibits y (y=loading direction) stress field for an arbitrary crack.
Chapter VI 6.37 a) b) c) d) Figure 6.27 - FE models of welded specimens obtained using ANSYS® and y stress field illustration for an arbitrary crack: a) W1 series; b) W2 series; c) W3 series; d) W4 series. Finite element models were also built using ABAQUS® code aiming the application of the XFEM approach, implemented in ABAQUS®. As referred for standard FEM analysis, only ¼ of specimen geometries were modelled. Linear hexahedral 8-noded finite elements were used. Finite element formulation with reduced integration method was considered. Figure 6.28 illustrates the finite element model of the W1 welded specimen. The finite
Finite element modelling of experimental details 6.38 element mesh used may be observed in Figures 6.28a) and b). Figure 6.28 b) illustrates the mesh region where finite elements were enriched with additional degrees of freedom. Figure 6.28 c) plots the y stress field at the fatigue crack domain, for an arbitrary applied load. It is clear the asymptotic stress field introduced by the crack. Figures 6.28 d) and f) exhibit the and level set functions given by ABAQUS® for a given generic crack. Stress intensity factors were computed using the contour integral method available in ABAQUS®, several contours being considered. Stress intensity factors were computed considering 3 contours, namely contours 3, 4 and 5. Contours 1 and 2 were excluded since they led to instabilities and deviation from the expected theoretical solution as illustrated in the verification examples presented before. a) b) c) d) e) Figure 6.28 - W1 welded specimen modelled using 3D XFEM: a) overview of the welded specimen finite element mesh; b) detail of the enriched finite elements; c) illustration of yy stress field for an arbitrary crack; d) level set function; e) level set function.
Chapter VI 6.39 Table 6.4 – Paris law material constant for the S355 steel considering thicknesses of 4mm and 8mm (R=0.01). Welded specimens C* m W1 (t=4mm) 1.01E-14 3.39 W2 (t=4mm) W3 (t=8mm) 2.55E-14 3.33 W4 (t=8mm) *da/dN in mm/cycle and K in N.mm-3.5 Stress intensity factors range computed using FEM plus VCCT and X-FEM plus Contour Integral, for W1, W2, W3 and W4 welded specimens, are plotted, respectively, in Figures 6.29a), b), c) and d). Figure 6.29a) presents the stress intensity factors computed for W1 welded specimens loaded to 50kN (R=0.01). Figure 6.29 b) exhibits the stress intensity factors computed for a crack propagating in W2 welded specimens, loaded at 25kN (R=0.01). Figure 6.29 c) shows the stress intensity factors resulting from a crack propagating at the W3 welded specimens, loaded by 50kN load (R=0.1), and the Figure 6.29 d) illustrates the stress intensity factor history computed for a crack growing in the W4 welded specimens, loaded by a 75kN load (R=0.1). The grey curve corresponds to the numerical stress intensity factors results from ANSYS® plus VCCT computations (pure mode I). The other curve (black) was derived by means of the XFEM method together with the contour integral method, computed using several contours. The stress intensity factor computed using the XFEM method was elaborated considering both KI and KII stress intensity factors. An equivalent stress intensity factor was thus computed in order to compare both numerical approaches. The maximum crack size was defined by the maximum stress intensity factor registered in the small-scale fatigue crack propagation tests, using compact tension specimens, tested for the same stress ratio (see Chapter III). The analysis of Figure 6.29 reveals that the FEM procedure gives always higher stress intensity factors, than the X-FEM procedure. It is important to refer that the finite element mesh used in FEM analysis was coarser than in X-FEM models, which may justify some discrepancies in the stress intensity values. The deviation between the two approaches is small for specimens W1 and W4. A higher deviation between FEM and XFEM results was observed for the W2 and W4 specimens that are characterized by a significant discontinuity due to the fillet weld end. The XFEM plus contour integral
Finite element modelling of experimental details 6.40 approach has more difficulties to simulate the stress intensity factors for small cracks since the proximity of the contours around the crack tip to the boundaries of the plate or geometric discontinuities may lead to spurious values. This was particularly the case of the W3 and W4 series, where some instability was observed as well as some divergent results. a) . b) Figure 6.29 - Stress intensity factors range computed using FEM/VCCT and XFEM approaches: a) W1 welded specimens loaded at 50kN; b) W2 welded specimens loaded at 25kN; c) W3 welded specimens loaded at 50kN; d) W4 welded specimens loaded at 75kN. (1/2) 0 500 1000 1500 0 5 10 15 20 25 30 Equivalente stress intensity factor range Keq [Nmm 3/2] Crack length, a [mm] FEM and VCCT X-FEM and Contour Integral Method (contour 3, 4 and 5) Kmax 0 500 1000 1500 0 5 10 15 20 25 30 Equivalente stress intensity factor range Keq [Nmm 3/2] Crack length, a [mm] FEM and VCCT X-FEM and Contour Integral Method (Contour 3, 4 and 5) Kmax
Chapter VI 6.41 c) d) Figure 6.29 - Stress intensity factors range computed using FEM/VCCT and XFEM approaches: a) W1 welded specimens loaded at 50kN; b) W2 welded specimens loaded at 25kN; c) W3 welded specimens loaded at 50kN; d) W4 welded specimens loaded at 75kN. (2/2) 0 500 1000 1500 0 5 10 15 20 25 30 Equivalente stress intensity factor range Keq [Nmm 3/2] Crack length, a [mm] FEM and VCCT X-FEM and Contour Integral Method (Contour 3, 4 and 5) Kmax 0 500 1000 1500 0 5 10 15 20 25 Equivalente stress intensity factor range Keq [Nmm 3/2] Crack length, a [mm] FEM and VCCT X-FEM and Contour Integral Method (Contour 3, 4 and 5) Kmax
Finite element modelling of experimental details 6.42 a) b) c) Figure 6.30 - Comparison between experimental S-N data and numerical S-N curves computed using FEM/VCCT and XFEM analysis: a) W1 series; b) W2 series; c) W3 series; d) W4 series. (1/2) 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Remote stress range, Δσrem [MPa] Number of cycles to failure, Nf[cycles] Welded specimens, W1 Mean S-N curve FEM and VCCT XFEM and Contour Integral Method (R2=0.9834) 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Remote stress range, Δσrem [MPa] Number of cycles to failure, Nf[cycles] Welded specimens, W2 Mean S-N curve FEM and VCCT XFEM and Contour Integral Method (R2=0.9659) 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Remote stress range, Δσrem [MPa] Number of cycles to failure, Nf[cycles] Welded specimens, W3 Mean S-N curve FEM and VCCT XFEM and Contour Integral Method (R2=0.9930)
Chapter VI 6.43 d) Figure 6.30 - Comparison between experimental S-N data and numerical S-N curves computed using FEM/VCCT and XFEM analysis: a) W1 series; b) W2 series; c) W3 series; d) W4 series. (2/2) Figure 6.30 compares the experimental S-N data with numerical S-N curves resulting from crack initiation and propagation modelling, supported by FEM/VCCT and XFEM analysis. The analysis of the Figure 6.30 reveals a very good agreement between the numerical S-N curves and the experimental fatigue data for all welded specimens. Also, it is visible a very good agreement between the two proposed numerical S-N curves. The discrepancy in the observed stress intensity factors did not result in significant discrepancies in the predicted S-N curves. The slopes of the simulated S-N curves were very similar to the slope of the mean experimental S-N curve, which is justified by the fact that the crack propagation curves were correlated with a Paris relation with an exponent that is in very close agreement with the slope of the experimental S-N curve. The crack was propagated according constant increments of 0.5 mm for both FEM/VCCT and XFEM/J Integral approaches. For the first case, an initial crack of 0.5 mm was simulated. For the second case, if a similar initial crack size was simulated and taking into account the higher stress intensity factors that were simulated, lower fatigue lives would result from the XFEM approach. The similar values presented in the Figure 6.30 where obtained with an initial crack size of 0.8 mm for the XFEM simulation. Both values of possible initial crack sizes of 0.5 and 0.8 mm are within the usual range adopted in welded joints simulation (0.1-1.0mm). 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Remote stress range, Δσrem [MPa] Number of cycles to failure, Nf[cycles] Welded specimens, W4 Mean S-N curve FEM and VCCT XFEM and Contour Integral Method (R2=0.9929)
Finite element modelling of experimental details 6.50 a) b) c) Figure 6.33 - R1 riveted specimen modelled with XFEM and crack emanating from angle corner: a) global finite element mesh; b) local finite element mesh and XFEM crack location; c) z stress field XFEM Crack Crack tip Enriched finite element zone
Chapter VI 6.51 a) b) c) Figure 6.34 – Finite element model of the R2 riveted specimen: a) global finite element mesh; b) local finite element mesh; c) y stress field for a particular fatigue crack.
Finite element modelling of experimental details 6.52 Figure 6.35 - Finite element model for the R3 riveted specimen. The R3 riveted specimens were modelled considering two cracks (a1 and a2) emanating from the first rivet hole, as observed in the experimental tests (Figure 6.35). The crack a1 propagated towards the half width of the plate. The crack a2 also started at the rivet hole however it propagated towards the plate edge of the riveted specimen model. Mixedmode fatigue crack propagation conditions were assumed. A fatigue crack branching criterion was selected (maximum tangential stress criteria) and the kink angle assessed from computed KI and KII, using the Tanaka [9] crack branching criterion. The crack increment was imposed to the crack a1 being equal to 0.5mm. Given this crack increment, the corresponding number of cycles was assessed using an effective stress intensity factor and the Paris’s law [2]. With these cycles, the crack increment for the crack a2 was computed taking into account the effective stress intensity factor computed with KI and KII assessed at the a2 crack tip. The fatigue life predictions were based on a procedure that accounts separately the crack initiation and crack propagation phases. The number of cycles to failure was, therefore, computed as the summation of the number of cycles required to initiate a crack and the number of cycles to propagate the crack until the failure of the component. The criterion for the crack initiation was the development of a macroscopic crack of 0.5 mm depth. The Crack a1 Crack a2 Crack a1 Crack a2
Chapter VI 6.53 crack initiation was modelled using the Morrow’s relation [5] which requires the local elastoplastic strains at the crack initiation site, which were computed using the Neuber’s approach. Since the stress concentration at the two crack initiation spots considered in this example were very close to each other, both initial cracks were postulated at the beginning of the crack simulation process. Figures 6.36 compares the experimental mean S-N curves with numerical fatigue life predictions obtained for the R1 riveted specimens using the finite element model simulating a fatigue crack emanating from the rivet hole. The number of cycles required to initiate a fatigue crack (black line), the number of cycles required for the fatigue crack propagation (dashed black line) and the total number of cycles to failure (grey continuous line) are plotted in the figure. The experimental mean S-N curve is also presented as a dotted line. The simulations confirmed a dominant fatigue crack propagation phase. The simulations overestimated the total fatigue lives and the mean experimental S-N curve for medium to high cycle fatigue. The numerical S-N curve slope, m, is higher than the experimental S-N curve slope. However, the experimental slope is not characteristic of the failure mode characterised by the crack propagation from the rivet hole. If the two experimental points tested at the lowest stress ranges, corresponding to another failure mode (failure at the angle corner), are removed from the analysis, the resulting experimental mean S-N curve will show a slope very similar to the simulated S-N curve. The overestimation of the simulation S-N curve will be lower if the consistent failure mode is considered. It is very likely that the crack initiation simulation was overestimated in this process. In fact, the surface quality of the rivet holes was not controlled during the manufacturing process; a poor quality is expectable which may justify lower crack initiation lives. It must be emphasized that crack initiation was simulated using smooth/polished specimens fatigue data. Figures 6.37 compares the experimental mean S-N curve with numerical S-N curves computed for R1 riveted specimens, assuming a fatigue crack propagating at the angle corner. Numerical results from standard finite element method and extended finite element models are presented. It can be observed that the total predicted fatigue life computed using the XEFM model provides a numerical S-N curve (grey continuous line) similar to the experimental S-N curve and only slightly above of the two experimental data points corresponding to the angle cracking failure mode. Figure 6.37 also illustrates
Finite element modelling of experimental details 6.54 that the two numerical approaches resulted in a similar fatigue crack propagation curves. However the number of cycles computed using both numerical approaches to predict the fatigue crack initiation was very distinct. This is only justified by distinct stress estimates at the uncracked angle corner. The assumptions adopted for the finite element models to be used with XFEM included continuity between the rivets shoulders and the hole surfaces, at the angle leg and at the beam web. This condition led to higher stresses at uncracked angle corners, resulting significantly lower fatigue lives, but significantly more consistent with the experimental data. This result emphasizes an overestimation of the load transfer by friction (preload/friction coefficient not fully tuned) in the multi-body approach using contact finite elements (FEM approach). As observed in Figure 6.36, the propagating phase is not dominant as observed before for the failure mode consisting of a fatigue crack emanating from the rivet hole (see Figure 6.35). Figure 6.36 - Comparison between R1 riveted specimen experimental S-N curve with numerical results of the model with a crack propagating at the web, from a rivet hole. 30 300 3000 2.E+04 2.E+05 2.E+06 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles] Experimental data - Riveted specimens R1 Experimental mean S-N curve Number of cycles to failure Number of cycles of crack propagation Number of cycles of crack inicitation Fatigue crack was propagate in the angle
Chapter VI 6.55 Figure 6.37 - Comparison between R1 riveted specimen experimental S-N curve and the numerical results obtained with the FEM model and XFEM with a crack propagating at angle corner. a) b) Figure 6.38 - Normal stress range evolution at the net section of the angle: a) applied load of 18kN; b) applied load of 25kN. In order to get a clear indication of the effects of the two modelling assumptions on stress fields, Figure 6.38 plots the normal to the crack surface stress evolution for both numerical models used to simulate the crack propagation at the angle corner. It is shown that the numerical model built using the XFEM methodology exhibits higher stress values at the crack initiation spot, which is consistent with the above-mentioned crack initiation predictions. The model used to implement the XFEM approach gives a stress distribution that is compatible with a higher moment transferred thru the angles. Figure 6.39 compares the effective stress intensity range evolution, computed using contour integral method (XFEM) and VCCT (FEM) for an applied load range of 18 kN and load ratio of 0.1. Figure 6.39 exhibits that both simulation approaches yield similar stress intensity factors, with the XFEM slightly 30 300 3000 2.E+04 2.E+05 2.E+06 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles] Experimental data - Riveted specimens R1 Experimental mean S-N curve Number of cycles to failure (FEM) Number of cycles of crack propagation (FEM) Number of cycles of crack inicitation (FEM) Number of cycles to failure (XFEM) Number of cycles of crack propagation (XFEM) Number of cycles of crack inicitation (XFEM) Fatigue crack was propagate in the angle -300 -200 -100 0 100 200 300 025 50 75 100 125 150 Normal stress range [MPa] Distance from crack initiation spot [mm] FEM+VCCT XFEM+J -400 -300 -200 -100 0 100 200 300 400 025 50 75 100 125 150 Normal stress range [MPa] Distance from crack initiation spot [mm] FEM+VCCT XFEM+J
Finite element modelling of experimental details 6.56 Figure 6.39 - Effective stress intensity factor range evolution computed using FEM plus VCCT and XFEM plus contour integral method, for a propagating crack at the angle corner (P=18 kN). Figure 6.40 shows numerical S-N results for the R2 riveted specimens which were simulated using two distinct clamping stresses, namely 30MPa and 80MPa. In both cases the predictions were conservative, i.e., fatigue lives were underestimated for the complete range of tested stresses. The simulated crack initiation and propagation phases demonstrated to be more balanced, without a significant domination of one phenomenon over the other. The clamping stresses have a significant effect on predictions. The consideration of a higher clamping stress in the rivets enhanced the fatigue predictions, including the slope of the resulting numerical S-N curve. The actual values on riveted specimens are not known, the simulations suggesting clamping stresses even higher than 80MPa. The numerical model of the R2 specimens was based on the ideal geometry of the joint, which leads consequently to a symmetrical crack propagation, which was not fully observed in the tests due to small misalignments. A two cracks (multi-site damage) model would be preferable, but it would lead to similar results if the ideal specimen geometry still is considered. A way to introduce random asymmetries in the specimens would be required. 0 500 1000 1500 0.0 5.0 10.0 15.0 Equivalent stress intensity factor range Keq [Nmm 3/2] Crack length, a [mm] FEM and VCCT XFEM and Contour Integral Method (contour 3, 4 and 5) Kmax
Chapter VI 6.57 a) b) Figure 6.40 - Comparison between R2 riveted specimen experimental S-N data and simulation results: a) clamping stress of 30MPa; b) clamping stress of 80MPa. 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles Experimental data - Riveted specimens - R2 Experimental mean S-N curve Number of cycles to failure Number of cycles of crack propagation Number of cycles of crack initiation 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles] Experimental data - Riveted specimens R2 Experimental mean S-N curve Number of cycles to failure Number of cycles of crack propagation Number of cycles of crack initiation
Finite element modelling of experimental details 6.58 a) b) Figure 6.41 - Comparison between R3 riveted specimen experimental S-N data and simulation results: a) clamping stress of 30MPa; b) clamping stress of 80MPa. 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles] Experimental data - Riveted specimens R3 Experimental mean S-N curve Number of cycles to failure Number of cycles of crack propagation Number of cycles to crack initiation 100 1000 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 Net stress range, net [MPa] Number of cycles to failure, Nf [cycles] Experimental data - Riveted specimens R3 Experimental mean S-N curve Number of cycles to failure Number of cycles of crack propagation Number of cycles to crack initiation
Chapter VI 6.59 Figures 6.41a) and b) plot the numerical S-N curves vs. experimental S-N data for the R3 riveted specimens. The same two clamping stresses used for the R2 specimens, were also adopted for this series. The computed number of cycles corresponding to the fatigue crack propagation is significantly higher than the number of cycles necessary to initiate a fatigue crack. The analysis of the global predictions illustrated in the Figure 6.41 shows very accurate predictions for the R3 riveted joints, including a fair description of the S-N slopes. In is interesting to note that the clamping stress variation did not resulted in significant variation in the predicted S-N curves as observed for the R2 riveted series. 6.9. CONCLUDING REMARKS This chapter presented the fatigue simulation results for the welded and riveted specimens tested in the present research. For the riveted specimens, a fatigue model based exclusively on the fatigue crack propagation phase produced very consistent results. In the simulation of the riveted specimens, both crack initiation and crack propagation phases were considered since for non-welded material fatigue crack initiation could represent an important contribution to the total fatigue life. Well established approaches were used for both fatigue crack initiation and fatigue crack propagation simulations, respectively the local strain approach and the fracture mechanics-based Paris propagation law. Concerning the welded specimens, the required stress intensity factors for fatigue crack propagation simulation were computed based on two alternative techniques. A standard finite element method was considered with the VCCT technique. However, this could represent a cumbersome approach to apply in 3D problems with complex geometries, once the crack path must be compatible with the finite element mesh. An alternative method to compute the stress intensity factors, using the extended finite element method, was adopted. This method allows the fatigue crack modelling, independently of the finite element mesh. Both methods were applied to compute the stress intensity evolution, and slight differences were verified in the stress intensity values but these
Fatigue modelling of a relevant detail from the Trezói bridge 7.2 7.1. INTRODUCTION This chapter presents a fatigue analysis of a representative detail from the Trezói riveted railway bridge. This bridge has been selected as a case study for some research projects in the last decade [1-2]. Also, several academic studies about the fatigue behaviour of the bridge has been published [3-4].Therefore, there is available significant information about the bridge, particularly about the mechanical properties of their materials and also about the typical load spectra the bridge has been subjected. In addition, global finite element models of the bridge were developed aiming dynamic simulation of the bridge [2]. Due to the available information about the bridge and also due to the collaboration of the author on the FADLESS European project, it was decided to develop a detailed local numerical model of a selected node of the bridge that was assumed as a potential critical node of the bridge [2].This local model was subjected to typical boundary condition histories obtained from global numerical models of the bridge. Also actual experimental strain data obtained from the bridge monitoring was used to validate both the local model and the boundary conditions. Finally, the fatigue damage assessment of the node was performed testing distinct levels of approaches. Since the bridge is a riveted bridge (no welds used), and no visible damage (cracks) on the bridge were observed during inspections performed to the bridge [2], it was decided to apply a local notch approach for fatigue crack initiation simulation. It is assumed that the crack initiation phase is a significant fatigue damage mechanism for riveted construction in opposition to the classical assumption for welded construction, where crack initiation may be neglected with respect to the fatigue crack propagation.
Chapter VII 7.3 7.2. OVERVIEW OF THE PROPOSED APPROACH A numerical finite element approach for the stress analysis of a representative detail (node) of the Trezói bridge is proposed in this chapter. Several levels of approximation will be simulated for the node, all involving 3D solid modelling. The node experiences a complex loading history, which results in complex multiaxial stress/strain histories. In particular non-proportional time-history stresses are expected. The principal stress directions rotate with time and there is not a clearly defined cycle (random stress spectra). The topic of non-proportional fatigue has deserved a lot of attention by scientists concerning the study of mechanical components [5]. However, the use of non-proportional multiaxial fatigue damage theories for the analysis of bridge details may be considered an innovative aspect of this work. The topic of non-proportional random fatigue is not completely solved, despite some available attempts, some of them being followed in this chapter. The main difficulty is concerned with the cycle counting (cycle definition) under non-proportional loading conditions, i.e., for situations that both the principal stress directions and their intensities vary with the time. For mechanical components these conditions have been investigated assuming that the loading history is formed by the repetition of small variable amplitude blocks. For these small blocks one equivalent cycle is extracted. For bridges, the block corresponds to a train crossing that may be repeated along the bridge life. The application of standard approaches for non-proportional fatigue would lead to one stress cycle for each train crossing which could be a non-conservative approach since other damaging cycles are most likely being disregarded. Typically non-proportional multiaxial fatigue damage approaches try to identify the critical planes were fatigue cracking (damage) is most likely to occur. This is the base of the critical plane models, one of the important approaches in multiaxial fatigue, aside with the energy based approaches. For arbitrary multiaxial stress fields and complex details, the critical plane models application would require the scanning over an
Fatigue modelling of a relevant detail from the Trezói bridge 7.4 important number of planes, at every point of the structure. Also the stress analysis has to be performed for each load step considered in the simulation of the train crossing. If several distinct train crossings are to be considered in the fatigue analysis, this task concerning a fatigue assessment of a real bridge detail could reach quickly a practical impossibility. Finally an important aspect of riveted joints simulation is the interaction of rivets with the plates, which must account for clamping effects and friction between elements. These interaction effects are responsible for non-linear finite element analyses. For the aforementioned reasons, the fatigue damage assessment of real riveted bridge details is very time consuming, requiring a strategy for mitigating the computational costs. Therefore, a preliminary fatigue damage assessment of the structural node based on Von Mises equivalent stress is proposed to identify the most fatigue critical locations that will be subjected to a subsequent more refined non-proportional multiaxial fatigue damage assessment. In this simplified approach, the principal stress histories are derived at each node. The cycle counting is performed using the first principal stress, to establish the stress cycles ( 1,i, ni). The Rainflow cycle counting was considered in this investigation [6]. Assuming an approximation of proportional loading, the other principal stress ranges ( 2,i, 3,i) are also defined by means of the same times (load steps) used to extract the first principal stress ranges. Then, equivalent stress ranges are computed using the Von Mises criterion as given by the following equation: 2 2 2 , 1, 2, 1, 3, 2, 3, 1 2 eq i i i i i i i (7.1) Once the bridge detail is expected to show an elastic behaviour, one may apply the Basquin relation to derive the number of cycles to failure, Ni, for each equivalent Von Mises stress range, eq i, : ,(2 ) 2 eq i b fi N (7.2)
Chapter VII 7.5 A linear fatigue damage accumulation model, known as Miner’s relation [7] may be considered to account for the different stress cycles: i i n DN (7.3) After the preliminary fatigue assessment is performed and most critical locations identified, a more refined non-proportional multiaxial fatigue assessment approach is proposed [5]. In the current work five distinct approaches are considered. A crucial problem regarding non-proportional multiaxial fatigue analysis is the cycle counting procedure definition. It is crucial to define the cycle counting strategy to be used for each approach. The first three approaches are strain-based models, namely the SmithWatson-Topper (SWT) model, the Fatemi and Socie model and the Bannantine and Socie models. The Findley model, which corresponds to a stress-based approach [8], was also used following two formulations, particularly the original Findley and a modified Findley approach. Smith et al. [9] proposed a fatigue damage parameter that relates the cyclic strain amplitude with the maximum stress. This parameter, commonly referred to as SWT parameter, was originally developed for uniaxial loading conditions to account for mean stress effects [8]. The SWT parameter has also being used in the analysis of both proportionally and non-proportionally loaded components constructed from materials that fail primarily due to mode I, tensile cracking [9]. The damage relation assumes the following form: 2 2 1 ,max '(2 ) ' ' (2 ) 2 fb b c n f f f f NN E (7.4) where 1 is the maximum principal strain range and ,maxn is the maximum direct stress acting on the first principal strain range plane assessed during principal strain cycle counting procedure; 'f and b are respectively the tensile cyclic fatigue strength
Fatigue modelling of a relevant detail from the Trezói bridge 7.6 coefficient and exponent; 'f and c are respectively the tensile cyclic fatigue ductility coefficient and exponent; E is the Young modulus and 2Nf is number of cycles to failure. Fatemi and Socie (FS) [10] suggested a damage parameter that results from the combination of the shear strain range and the normal stress: ,max ' 1 (2 ) ' (2 ) 2 bc f n f f f y k N N G (7.5) where, for a given plane, is the shear strain range, ,maxn is the maximum normal stress value assessed during the cycle counting procedure applied to the shear strain; 'f and b are respectively the torsional cyclic fatigue strength coefficient and exponent; 'f and c are respectively the torsional cyclic fatigue ductility coefficient and exponent; G is the shear modulus. This model assumes that mode II shear cracking is the dominant fatigue damage process. During shear loading, the irregularly shaped crack surface results in friction forces that will reduce crack tip stresses, thus hindering crack growth and increasing the fatigue life [8]. Tensile stresses and strains will separate the crack surfaces reducing the frictional forces. Bannantine and Socie (BS) [8] proposed a multiaxial cumulative damage criterion based on the combination of critical plane damage parameters, rainflow cycle counting method and Miner’s rule. The main concept behind BS criterion is to assess both the FS and SWT parameters for each plane. For this purpose the rainflow cycle counting technique is applied separately to the axial (normal) and shear strains. Then the FS and SWT damage parameters are computed independently and the damage associated to a specific plane is established as the greatest value given by the two alternative criteria (see Figure 7.1).
Chapter VII 7.7 Figure 7.1 - Bannantine and Socie methodology [5]. Findley [11-14] criterion was also used in this investigation. This criteria was one of the first critical plane criteria being proposed. It considers that the shear stress amplitude 2 is one of the most damaging parameters acting on a certain plane. The normal stress to that plain, ,maxn , has a secondary but not negligible effect on the fatigue damage. The Findley relation may be represented by the following equation [8]: * ,max () 2 b n f f kN (7.3) where k is a constant computed as 2 1 1.04k , * f is computed from the torsional fatigue strength coefficient *2 1 ff k . The maximum normal stress value, ,maxn is computed from the maximum of two load points (load steps) that were used to define the shear stress range, when the Rainflow procedure is applied. Another version of the Findley approach also considered in this work consisted in the definition of an equivalent shear stress parameter, ,max 2 eq n k , that is computed for a specific plane along the time history. Then the cycle counting is applied directly to this equivalent shear stress parameter. This latter procedure is name here as modified Findley model.
Fatigue modelling of a relevant detail from the Trezói bridge 7.8 7.3. FINITE ELEMENT MODEL OF A STRUCTURAL DETAIL FROM TREZÓI BRIDGE Preliminary fatigue studies performed on Trezói bridge [2-3] pointed out the node 6 (see Figures 7.2 and 7.3) as one of the most fatigue critical detail of the bridge. In the previous fatigue studies, the S-N approach combined with 3D beam models of the bridge have been used. These global approaches are useful to identify in an efficient way the potential critical details of the bridge. However, they do not provide detailed information about the damage mechanisms. Also, the application of the global S-N approach to complex details such as the one selected in this work is not a straightforward task since the required nominal stresses may be not easily defined. Existing code based S-N curves for riveted details are mainly based on tensile stresses on members. Therefore, secondary bending dynamic effects that are frequent in bridges are not properly accounted using such a global S-N approach. A local detailed finite element model of the structural node 6 was proposed using 3D solid quadratic finite elements. Materials were assumed linear elastic and isotropic (E=210 GPa; =0.27). The members connected in the node were modelled with 900 mm in length (see Figure 7.4). Two versions of the local model were constructed. The first model was built assuming full continuity between the components forming the node (Figure 7.5). The second finite element model was built assuming contact between the cross-girder and the remaining of the structural (Figure 7.6). Rivets were modelled between the cross-girder flange and the horizontal gusset (12 rivets) and between the cross-girder web and the angles (7 rivets). For the remaining components, full continuity is assumed. The model represented in Figure 7.5 has the same geometry of the model of Figure 7.6, including some rivets. However, in the model of Figure 7.5 the volumes were merged before meshing; when performing the mesh, a fully continuous model is obtained. The simulation of all rivets actually used in the joint it is not possible since the model becomes too large for the simulation of a complete train crossing. Therefore, the
Chapter VII 7.9 following strategy may be used to mitigate the high computational cost associated to a model with all rivets: several models can be developed, each one to assess a specific riveted part of a complex riveted detail, assuming continuity elsewhere. It is assumed that the behaviour of a specific riveted location is not influenced by the type of modelling approach used for other adjacent parts (riveting or continuity). Figure 7.2 - Location of the potential critical node of the Trezói bridge to be fatigue assessed using a detailed local model (node 6). Figure 7.3 - Potential critical node (node 6): actual photograph.
Fatigue modelling of a relevant detail from the Trezói bridge 7.10 Figure 7.4 – Geometry and dimensions of node 6 of the Trezói bridge. Local models of node 6 were constructed using the ANSYS® code. The contact between plates and rivets was modelled using contact finite elements available in ANSYS®, using a surface-to-surface option. In particular, the CONTA174 and TARGE170 elements were used to model, respectively, the contact and target surfaces, forming the so-called contact pairs. Both surfaces in contact were assumed flexible. The contact simulation was carried out using the augmented Lagrange algorithm available in the ANSYS®, together with the Coulomb friction model. A similar approach that was applied for the riveted specimens simulated in Chapter VI was also followed for this local node. One key aspect of the numerical simulation using local models is the accurate definition of the boundary conditions to apply at the ends of the members. Multi-point constraints (MPC’s) were used to impose the boundary conditions. With this technique, pilot nodes (reference points) were placed at the end of each member, at the centroid position of the cross section of the members. 100 3000 400 1700 1500 328 900 14 14 12 900 360 360 568
Chapter VII 7.11 Figure 7.5 - Local finite element models: continuous model.
Fatigue modelling of a relevant detail from the Trezói bridge 7.18 7.4.1. Continuous finite element model of the structural detail of the Trezói bridge: results analysis In this section, fatigue assessment results are presented using the nodal stress histories from the continuous finite element model of the structural node of the bridge. Only surface nodes of the structural joint were analysed since fatigue damage is commonly associated to surface crack initiation. In order to select a limited set of nodes most susceptive to fatigue damage, Equations (7.1) to (7.3) were applied to all surface nodes. The required material constants were obtained from Table 6.5 for the S235 steel grade. Figure 7.15 presents the fatigue damage calculated for each surface node, resulting from the above referred passenger train (refer to Figure 7.8) crossing the bridge. Figure 7.15 also presents a cut-off line used to select the set of 10 most damaged nodes. Those 10 nodes belong to three distinct locations (see Figure 7.16), that will be further investigated using multiaxial fatigue damage theories presented in Section 7.2. The number of train crossings until failure (D=1) using the Von Mises criterion are also presented in Figure 7.16. The Von Mises criterion suggested that the most critical locations are between the cross-girder and node connection parts. Damage levels for the simulated train crossing are very small which is consistent with the type of traffic considered (type of train/train speed).
Chapter VII 7.19 Figure 7.13 – Location of strain gauges used to monitor the strains at the node 6 of the bridge model. Strain gauge direction Strain gauges SG 2 Strain gauges Strain gauge direction SG 1 SG 3
Fatigue modelling of a relevant detail from the Trezói bridge 7.20 Figure 7.14 - Experimental vs numerical stress measurements at locations illustrated in Figure 7.13. Figure 7.15 – Damage computed at each surface FE node of the local continuous model, for each passenger train crossing, using the Von Mises equivalent stress criterion. -30 -20 -10 0 10 20 30 40 50 60 0 5 10 15 Stress, [MPa] times [s] Continuous finite element model - SG1 Riveted finite element model - SG1 SG1 -50 -40 -30 -20 -10 0 10 20 30 0 5 10 15 Stress, [MPa] times [s] Continuous finite element model - SG2 Riveted finite element model - SG2 SG2 -30 -20 -10 0 10 20 30 40 50 0 5 10 15 Stress, [MPa] times [s] Continuous finite element model - SG3 Riveted finite element model - SG3 SG3 1.E-40 1.E-34 1.E-28 1.E-22 1.E-16 1.E-10 1.E-04 0 5000 10000 15000 20000 25000 Damage Node Selected node set where multiaxial fatigue assessment is applied
Chapter VII 7.21 Figure 7.16 – Von Mises stress distribution (continuous model) (step 10). The multiaxial fatigue criteria presented in Section 7.2, particularly the Bannantine and Socie, Fatemi and Socie, Smith-Watson-Topper, Findley and the modified Findley proposals were applied to the three most stressed locations identified previously (see locations 1 to 3 in Figure 7.16). For each location, the most stressed node was selected. Since the referred multiaxial damage criteria are critical plane based approaches, the critical plane was searched thru the variation of two angular coordinates (, ) as illustrated in Figure 7.17. Tables 6.5, 7.1 and 7.2 summarize the constants required for the multiaxial damage models. Constants of Tables 7.1 and 7.2 resulted from empirical approximations instead of being computed using actual multiaxial fatigue data from materials testing. Multiaxial fatigue testing was considered out of scope of this research, but in future it should be considered for bridge materials. train crossings= 2.3627 x1008 train crossings= 2.788 x1014 train crossings= 3.2699 x 1014 Location 2 Location 3 Location 1
Fatigue modelling of a relevant detail from the Trezói bridge 7.22 Figure 7.17 – Angular coordinates defining an arbitrary plane thru its normal. Table 7.1 – Fatemi and Socie model constants [8]. 3 f f [MPa] bb 3 ff cc G [GPa] k 415.9 -0.081 2.335 -0.74 80 1 Table 7.2 – Findley model constants [8]. k ( 2 1 1.04k ) *2 1' ff k [MPa] 0.286 432.5 Figures 7.18 to 7.20 present for the three locations identified in the previous Von Mises fatigue analysis, a map of the damage as a function of the plane orientation. The damage map was firstly computed for one single passenger train crossing. Then, the damage was linearly accumulated using the same train crossing until damage reaches the unity in a specific plane (critical plane). When this condition is verified, the corresponding number of train crossings required to failure is obtained and the respective value is plotted over the damage maps of the Figures 7.18 to 7.20. In x z y t
Chapter VII 7.23 particular, Figure 7.18 presents the damage map computed for the planes at the location between the horizontal gusset and the vertical gusset (location 1). Figure 7.18a) presents the fatigue damage map assessed using the Bannantine and Socie relation; the Fatemi and Socie relation produced the damage map of Figure 7.18b); the SmithWatson-Topper relation is responsible for the damage map of Figure 7.18c); the Findley and modified Findley relations resulted in the damage maps of Figures 7.18d) and e), respectively. The analysis of the Figure 7.18 reveals that the strain-based approaches result in more conservative results than the stress-based methods. In particular, the BS approach leads to the lower fatigue live values. Figure 7.19 presents the results corresponding to the node located between the column and the lower support of cross girder (location 2), using the same approaches that were considered for the previous location. As observed in Figure 7.18, the strain-based approach results into the more conservative fatigue lives. Finally, Figure 7.20 presents the accumulated fatigue damage for the node located at the cross-girder/ horizontal gusset junction (location 3). The strain-based approach also produced more conservative results. Table 7.3 summarizes the fatigue life computed using the continuous finite element model. It can be observed that the stress-based approaches such as Findley and modified Findley approaches result in the less conservative solutions. It is interesting to note that all critical plane approaches led to less conservative predictions than resulted with the application of the Von Mises criterion, which makes the last criterion for this kind of structures which are not fatigue critical. In all the cases the number of train crossings required to start a fatigue crack is very high as a result of a very low damage induced by a single passengers train crossing. The actual structure does not exhibit any signal of fatigue damage. Therefore, the results shown are consistent with the observations performed over the bridge. The proposed methodology can be also applied to more damaging traffic as the one from freight trains crossing the bridge at higher velocities, as soon as the respective global analysis of the bridge is performed to derive the local sections displacements histories. The selection of a passengers train was motivated by the available experimental strain measurements at the node of the bridge.
Fatigue modelling of a relevant detail from the Trezói bridge 7.24 a) b) c) Figure 7.18 – Critical plane assessment for the node located at location 1 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (1/2) 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 Damage Train crossings=1.4190x1025 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 Damage Train crossings=4.8190x1028 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 Damage Train crossings=1.4190x1025
Chapter VII 7.25 d) e) Figure 7.18 – Critical plane assessment for the node located at location 1 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (2/2) 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 1 0 0.5 Train crossings=3.5634x1025 Damage 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 1 0 0.5 Train crossings=3.5634x1025 Damage
Fatigue modelling of a relevant detail from the Trezói bridge 7.26 a) b) c) Figure 7.19 – Critical plane assessment for the node located at location 2 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (1/2) 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 Damage 1 0 0.5 Train crossings=4.5540x1022 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 1 0 0.5 Train crossings=2.4286x1025 Damage 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 Damage Train crossings=4.5540x1022
Chapter VII 7.27 d) e) Figure 7.19 – Critical plane assessment for the node located at location 2 (continuous FEM model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (2/2) 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 Train crossings=3.5782x1023 Damage 0 360 360 [rad] [rad] 180 27090 90 180 270 1 0 0.5 1 0 0.5 Train crossings=3.5782x1023 Damage
Fatigue modelling of a relevant detail from the Trezói bridge 7.34 a) b) c) Figure 7.24 – Critical plane fatigue damage assessment for the location 2 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (1/2) 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=6.18561032 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=4.6286x1038 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=6.18561032
Chapter VII 7.35 d) e) Figure 7.24 – Critical plane fatigue damage assessment for the location 2 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (2/2) 0 360 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=7.5421x1034 0 360 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=7.5421x1034
Fatigue modelling of a relevant detail from the Trezói bridge 7.36 a) b) c) Figure 7.25 – Critical plane fatigue damage assessment for the location 3 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (1/2) 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=6.3700x1027 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 270 Train crossings=4.3225x1031 0 360 [rad] [rad] 180 27090 90 180 270 360 Damage 1 0 0.5 Train crossings=6.3700x1027
Chapter VII 7.37 e) f) Figure 7.25 – Critical plane fatigue damage assessment for the location 3 (combined continuous/riveted model): a) BS approach; b) FS approach; c) SWT approach; d) Findley approach and e) modified Findley approach. (2/2) Table 7.4 summarises the fatigue life results computed for the selected three locations identified in the riveted cross-girder. It can be observed that Von Mises approach was the most conservative one. Considering the critical plane approaches, significantly higher fatigue lives were obtained. The SWT approach was consistently the less conservative approach among the critical plane models, followed by Findley approaches and finally the FS approach, which was the less conservative one. The comparison of Tables 7.3 and 7.4 shows that the fully continuous model produces lower fatigue lives (more conservative estimations of 2.3x1008 vs. 3.5x1017 train 0 360 [rad] 18090 90 180 270 360 Damage 1 0 0.5 Train crossings=1.6938x1029 270 [rad] 0 360 [rad] 18090 90 180 270 360 Damage 1 0 0.5 Train crossings=1.6938x1029 270 [rad]
Fatigue modelling of a relevant detail from the Trezói bridge 7.38 crossings), which is an indication of the higher stress estimations using the continuous model. In fact, the continuous model introduces more severe geometric discontinuities than the riveted model. Table 7.4 – Summary of the number of passenger train crossings to failure computed using several fatigue assessment models, based on a combined continuous/riveted finite element model. Node location Von Mises BS approach FS approach SWT approach Findley Modified Findley Location 1 6.5x1017 6.3x1025 8.3 x1029 6.3x1025 8.1x1027 8.1x1027 Location 2 1.0x1018 6.2x1032 4.6x1038 6.2x1032 7.5x1034 7.5x1034 Location 3 1.39x1018 6.4x1027 4.3x1031 6.4x1027 1.7x1029 1.7x1029 7.5. CONCLUDING REMARKS In this chapter a numerical strategy for the fatigue analysis of riveted details from bridges was proposed and illustrated for a node of the Trezói Bridge. The proposed strategy mitigates the very high computational costs that a local fatigue analysis of real complex details may demand. The application of local approaches to fatigue in real bridge details leads to a multi-scale problem that, taking into account the state-of-the art computational resources, cannot be solved with a unique model that encompasses both scales. Therefore, the proposed methodology suggested a multi model approach to overcome the multi-scale problem, according the following sequence: - Creation of global models of the bridge using beam elements where dynamic effects of traffic should be conveniently accounted. With this model a prior selection of potential critical details is performed using an S-N approach. - Development of local models of the selected potential critical location, using solid elements. This model will be simulated using a quasi-static analysis but with displacement histories from the global dynamic analysis of the bridge that will be applied with sub-modelling techniques (e.g. multi-point constraints).
Chapter VII 7.39 - If the details are riveted ones, it would be impossible to simulate all the rivets that for real bridge nodes could reach the value of hundreds. Therefore, various local models can be proposed to simulate specific riveted parts interaction from the whole detail. This was the case of the problem presented in this chapter to investigate the fatigue behaviour of the riveted connection between the crossgirder and the node. The study presented in this chapter demonstrated that the use of full continuous models to simulate the fatigue behaviour of riveted details leads to distinct potential failure locations of the ones obtained using riveted models, with contact finite elements. In the latter case, the rivet holes revealed as the most critical locations. The use of rivets in the numerical modelling also allowed a better correlation of the available experimental strain-life data, which is a clear indication of the advantage of the proposed local model. The analysis of the stress histories for a real traffic stress history revealed that the stress states are multiaxial and the stress histories are non-proportional at the nodes of the local model. Therefore, it was recommended the use of non-proportional multiaxial fatigue approaches for the model. This is actually an open problem for variable amplitude (random) load spectra, but some existing literature approaches were applied in the fatigue analysis of the node. The critical planes approaches are very popular approaches in multiaxial fatigue but they are very time consuming since they require the scanning of the multiplicity of damage planes. This represents a difficulty for 3D problems, which is the case of the complex bridge details, since the number of planes to be search is very high. In order to mitigate the increased computational cost resulting from the critical plane approaches, a preliminary assessment based on Von Mises stresses was proposed to identify the most critical locations at the local model for further analysis using the more accurate damage models. Despite the multiaxial fatigue models requires constants to be identified with specific multiaxial material fatigue tests, in this chapter empirical approximations were adopted. The critical plane damage models were less conservative than the Von Mises approach, resulting in higher number of train crossings. Therefore the Von Mises approach could be a valuable approach for the prior identification of critical locations.
Fatigue modelling of a relevant detail from the Trezói bridge 7.40 The different critical plane damage models resulted in some cases in very distinct life results. This is justified by the fact that only empirical estimations of their parameters were used and the model reliability depends greatly on these parameters. In addition, since the damage levels of one train crossing is very low, the models are working in the fatigue endurance limit domain. Therefore, small perturbations in the computed fatigue damage parameters or on model constants may lead to significant distinct results. The SWT damage approach was the one leading to less conservative approaches within the critical planes approaches. It is worth mentioning that the data required for this model was assessed using fatigue tests performed in this work with tension/compression smooth specimens. 7.6. REFERENCES [1] Cunha, A., “Avaliação da Integridade Estrutural de Pontes Metálicas Ferroviárias/ Assessment of the Structural Integrity of Metalic Railway Bridges”, POCTI/ECM/57286/2004, FEUP, 2005-2008. [2] Lippi, F., Salvatore, W., Braconi, A., Finetto, M., Wenzel, H., De Roeck, G., Peeters, B., Könke C., Zabel, V., Cunha, A.,“Fatigue damage control and assessment for railway bridge”, Research Fund for Coal and Steel, Directorate-General for Research and Innovation, 2009-2012. [3] Marques, F.M., “Avaliação do comportamento estrutural e análise de fadiga em pontes metálicas ferroviárias”, Dissertação de mestrado, FEUP, 2006. [4] Correia, J.A.F.O., “Desenvolvimento de Modelos de Previsão da Vida à Fadiga de Ligações Rebitadas”, Dissertação de mestrado, UTAD, 2008. [5] Anes, V., Reis, L., Li, B., Fonte, M., de Freitas, M., “New Approach for Analysis of Complex Multiaxial Loading Paths”, International Journal of Fatigue 62, pp. 21–33, 2014. [6] ASTM - American Society for Testing and Materials, “E1049-85: Standard Practices for Cycle Counting in Fatigue Analysis”, Vol. 03.01, West Conshohocken, PA, pp. 5-8, 1999. [7] Miner, M.A., ”Cumulative Damage in Fatigue”, Journal of Applied Mechanics 67, pp. A159-A169, 1945. [8] Socie, D.F., Marquis, G.B., “Multiaxial Fatigue”, Society of Automotive Engineers, Inc. Warrendale, Pa., ISBN 0-7680-0453-5, 2000.
Chapter VII 7.41 [9] Smith, R.N., Watson, P., Topper, T.H., “A Stress-Strain Parameter for the Fatigue of Metals”, Journal of Materials 5 (4), pp. 767-778, 1970. [10] Socie, D.F., Shield, T.W., “Mean Stress Effects in Biaxial Fatigue of Inconel 718”, ASME Journal of Engineering Materials and Technology 106, pp. 227-232, 1983. [11] Findley, W.N., “Combined Fatigue Strength of 76S-T61 Aluminum Alloy with Superimposed Mean Stresses and Correction for Yielding”, NACATN 3934, National Advisory Committee for Aeronautics, Washington, DC, p. 90, 1953. [12] Findley, W.N., “A Theory for the Effect of Mean Stress on Fatigue Metals Under Combined Torsion and Axial Load or Bending”, Journal of Engineering Industry, pp. 301306, 1959. [13] Findley, W.N., “Modified Theories of Fatigue Failure Under Combined Stress”, Proceedings of the Society of Experimental Stress Analysis 14(1), pp. 35-46, 1956. [14] Findley, W.N., Coleman, J.J., Hanley, B.C., “Theory for Combined Bending and Torsion Fatigue with Data for SAE 4340 Steel”, Proceedings of the International Conference on Fatigue of Metals, Institute of Mechanical Engineers, London, pp. 150156, 1956. [15] Marques, F., Cunha, A., Fernandes, A.A., Caetano, E., Magalhães, F., “Evaluation of dynamic effects and fatigue assessment of a metallic railway bridge”, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance 6(5), pp. 635–646, 2009.
Fatigue modelling of a relevant detail from the Trezói bridge 7.42
Chapter VIII Finite element modelling of a relevant detail of the Alcácer do Sal Bridge
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.8 Nevertheless, K*total is not yet the final total stress intensity factor, as it requires a “signal” correction to be performed. This correction is detailed in later paragraphs. By applying the modal superposition method, σdyn(t) can be expressed as: dyn j j j t Y t (8.12) In Equation (8.12), the j subscript refers to the number of each mode of vibration, σj is the nominal stress in the jth mode shape and Yj(t) is the modal coordinate of the jth mode of vibration [21]. Replacing Equation (8.12), in Equation (8.10), the SIF due to the dynamic loading becomes: dyn n n j j n n j j j j j j j K t C a Y t C a Y t K Y t (8.13) where Kj is the modal stress intensity factor, or in other words, the stress intensity factor obtained with the configuration of the jth mode shape. Finally, the total stress intensity factor can be expressed as: * total sta j j j K t K K Y t (8.14) As detailed in [19], the computed * total Kt can have a negative sign during some periods of time. In the opening mode that situation would correspond to crack closure. From a practical perspective, when the computed * total Kt assumes negative values it will be converted to a null value to represent a closed crack. Thus, the final expression for mode I stress intensity factor computation becomes: , , , , ,, 0 00 I sta I j j I sta I j j jj I I sta I j j j K K Y t K K Y t Kt K K Y t (8.15)
Chapter VIII 8.9 If mixed mode crack propagation is assumed, besides I K also II K needs to be computed. That is done, in a similar way, using the following equation: ,,II II sta II j j j K t K K Y t (8.16) In the case of mode II, negative values of stress intensity factor are allowed. The change of signal of KII corresponds to the reversal of sliding movement between the faces of the crack with consequence change in direction propagation. As defined by Equations (8.14) to (8.16), the mode I and mode II stress intensity factors time histories may be obtained if the modal stress intensity factors for an adequate number of vibration modes, j K , plus the stress intensity factor regarding to the static load (e.g. self-weight), sta K are known in advance. Also the modal coordinates j Yt associated to each mode of vibration are also required. 8.3. Proposed workflow for residual fatigue life assessment of bridge details Based on the methodology and fatigue model described previously, a workflow was developed aiming the simulation of the fatigue crack propagation in critical details of bridges, subjected to a diversity of traffic. The workflow is divided in two main steps. The first step (Figure 8.1) consists of pre-processing activities involving the following inputs: global numerical model of the bridge; local numerical model of the detail under analyses; traffic information, such as axle loads, axles spacing and trains speed. The traffic scenario may come from monitoring systems or from other sources such as design codes.
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.10 The modal analysis performed on the global numerical model of the structure allows the computation of the modal displacement fields, j , and dynamic properties, j and j m . The required modal damping ratios, j , can be estimated based on design codes or field measurements [22]. On the other hand, the displacement field originated by the static loading, stat , is obtained after a static analysis using the same model. The above mentioned displacement fields are then extrapolated to the boundary nodes of the local numerical model of the detail, through a submodelling process. The boundary nodal displacements associated to the static loading are stored in the file BDCOsta. Additionally, the displacements associated to each jth mode of vibration are stored in the file BDCOj. Finally, the traffic information available in the monitoring system database (trains speeds, axle loads and axles spacing) is combined with the dynamic properties ( j , j , j and j m ) of the global model in order to obtain the modal coordinates time histories, j Yt . That is achieved by solving Equation (8.2) for each traffic event and for each mode of vibration [22]. Figure 8.1 – Workflow of the proposed analysis: pre-processing of the input data (1st step).
Chapter VIII 8.11 Therefore, the outputs of the first step of the workflow (pre-processing of the input data) will be: • the time history of the modal coordinates, for each mode of vibration of the structure and for each traffic event considered; • the nodal displacements to be applied to the boundaries of the local model of the detail in order to replicate the static and modal displacement fields. One key advantage of the proposed method is that the referred outputs only need to be computed once. They then become the only inputs needed for the second step of the workflow (Figure 8.2), which consists on the crack propagation simulation. Figure 8.2 – Workflow of the proposed analysis: crack propagation simulation (2nd step).
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.12 The simulation starts with an initial crack in the local model of the detail. Applying the previously stored boundary conditions (BDCOsta and BDCOj), the static and modal stress intensity factors, Ksta and Kj are computed, using an appropriate numerical technique, such as the virtual crack closure technique (VCCT) [30], the J-integral [31] or the displacements extrapolation (DE) methods [32]. For each train stored in the monitoring system database, the corresponding modal coordinates are loaded allowing to compute KI(t) and KII(t), by application of Equation (8.15) and (8.16), respectively. Equivalent stress intensity factors, Keq(t) (Equation (8.4)), and the kink angle ()t (Equation (8.6)) are computed next. At this point, if the maximum observed value of Keq(t) exceeds the material toughness, KIc, that means the crack would propagate in an instable way, the simulation stops and current crack length is considered the final crack length before the failure of the detail. Otherwise, if the maximum observed value of Keq(t) does not surpass the adopted material toughness, the crack propagation associated to the train passage is computed. The rainflow method is applied to the Keq(t) time history to obtain the histogram of stress intensity factor ranges. Then, Equations (8.5) and (8.7) are applied to compute the fatigue crack increment , Δa, and corresponding direction for that train passage. The contribution of all trains of an existing database is summed up to obtain a final crack increment, Δan. For computational efficiency reasons, it is useful to adopt constant crack length increments, Δainc. In this case, the crack increment corresponding to a loading block needs to be scaled by Bn=Δainc/Δan. Bn represents the number of loading blocks, i.e. the number of the trains database need to be applied in order to achieve the Δainc increment. After computing the crack increment for all the trains of the monitoring system database, the total crack length and direction is updated in the local model and the process is repeated. In the next section, the proposed approach is demonstrated for a welded detail from the Alcácer do Sal railway bridge.
Chapter VIII 8.13 Figure 8.3 – New Alcácer do Sal railway bridge. Figure 8.4 – Location of diaphragms 51 and 54 on bridge. 8.4. APPLICATION OF THE PROPOSED METHODOLOGY TO A CASE STUDY The case study selected to demonstrate and validate the proposed methodology was a welded detail of the composite bridge of the new railway crossing of river Sado, which is located in the Lisboa-Algarve connection, in Portugal (Figure 8.3). 8.4.1. Monitoring system Due to the paramount relevance of this structure, a monitoring system was installed to allow the real time traffic characterization and structural behaviour evaluation [20]. The structural behaviour was assessed by means of strain gauges installed at the diagonals of diaphragm 51 and 54 (Figure 8.4). In addition, diaphragm 51 was monitored
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.14 with 5 additional strain gauges that were installed at the top of the diagonal in order to better assess the stress field near the detail (local strain gauges) (Figure 8.5). Figure 8.5 – Location of strain gauges at diaphragm 51: a) global and local strain gauges locations; b) detail of local strain locations. The traffic was characterized using shear strain gauges welded to the rails and instrumented rail pad sensors. Each train crossing the bridge is totally characterized (train speed and direction, axle loads and axles spacing), through routines developed for that purpose, in MATLAB®, and the corresponding information stored in a database also in MATLAB® format. The information has been sent, through a 3G connection, to a server at the Faculty of Engineering of University of Porto. Therefore, an extensive database, including real strain measurements and trains characteristics is available and was used in the present work. The monitoring system development was out of scope of the current dissertation, therefore the respective details can be found elsewhere [33, 34]. Only the local strain gauges instrumentation was performed with the works leading to this document. 40 80 120 50 50 Diagonal 1 2 3 4 5
Chapter VIII 8.15 8.4.2. The global numerical model of the bridge The global numerical model of the bridge was developed using the ANSYS® code [35] and its ANSYS® Parametric Design Language (APDL). This was performed within the context of FADLESS project and only general information will be given in this chapter for the sake of completeness [22-34]. The concrete slab and the steel box girder were modelled using linear 4-noded shell elements (SHELL63 of ANSYS® library) while most of the diaphragms and diagonals, the arches and the hangers were modelled with linear beam elements (BEAM44 of ANSYS® library [32]). The connection between the concrete slab and the upper flanges of the steel box was achieved by means of multipoint constraint procedures, assuming rigid connections (MPC184 of ANSYS® library). The diaphragms 51 and 54 and corresponding diagonals were modelled with a fine mesh of shell elements, with maximum dimension of 0.1m, in order to facilitate the comparison of the numerical results with the strain gauge measurements captured with the permanent monitoring system [33]). The rest of the deck used a mesh size of 2m, as per the results of a sensitivity analysis. The numerical model was calibrated, based on the results of an Ambient Vibration Test and a load test performed in the bridge [36]. After calibration of the global numerical model, a modal analysis was performed and the corresponding modal properties ( j , j , j and j m ) were computed and exported to MATLAB®. In MATLAB®, those modal properties are combined with the traffic information already stored in the database, ()Ft , and the dynamic response of the structure is simulated, using the modal superposition technique (Equation 8.2) and adopting a total of 1500 modes of vibration. At this stage, the modal coordinates, () j Yt , are computed and stored.
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.16 a) b) Figure 8.6 – Some details of the global numerical modal of the bridge: a) deck, hangers and arch; b) diaphragm 51 and corresponding diagonals. 8.4.3. The local numerical model of the selected detail and shell-to-solid submodelling technique A local finite element model of a selected welded detail was built. Its geometry was defined using the commercial software SOLIDWORKS® and then exported to the finite element software ANSYS®. All the properties of the finite element model were defined in a parametric format, using ANSYS® APDL language. The parametric code was prepared for the simulation of a fatigue crack with curved generic crack path or to simulate an uncracked detail. An uncracked geometry was considered first, aiming the validation of the submodelling process and the submodel itself, by comparing numerical results with the experimental data from the local strain gauges placed near the welded joint (Figure 8.5). Furthermore, the uncracked finite element model allowed detecting potential fatigue cracking hot spots. It must be emphasized that the bridge does not show any visible signal of fatigue damage, in particular fatigue cracks. Therefore, the monitoring system gives information regarding the uncracked bridge. Also, the bridge was inaugurated very recently which means that it could be considered in virgin state. Once the potential cracking location was defined, the APDL routine developed allowed the simulation of an initial postulated crack explicitly in the finite element model of the welded detail. Moreover, the APDL routine was also used to update the fatigue crack path after each running of the workflow. 1 File: Diafragma54_v3 JAN 11 2014 13:10:59 ELEMENTS REAL NUM
Chapter VIII 8.17 Figure 8.7 – Local finite element model of the welded joint of the Alcácer do Sal bridge. a) b) c) Figure 8.8 – Shell-to-solid sub-modelling, location of the local FE model with respect the shell model: a) front view; b) side view; c) top view. Figure 8.7 shows the local finite element model with an initial crack. The bulk of the welded detail was modelled using tetrahedral quadratic finite elements. However, at the crack tip, a refined region was modelled with hexahedral quadratic finite elements, in order to accurately assess the stress intensity factors. The transition between the refined and coarse regions is achieved by means of pyramidal finite elements. In order to compute the boundary conditions to be applied to the local finite element model of the detail, displacements resulting from the global finite element model of the bridge were interpolated for the whole set of boundary nodes of the local finite element model. This was accomplished using the shell-to-solid submodeling procedure which was an automatic procedure available in ANSYS® [32]. In order this procedure work properly, the shell planes of the global model should fit the mid thickness of the plates of the local model. That was achieved in this case, as illustrated in Figure 8.8.
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.24 a) b) c) Figure 8.14 – Characteristics of simulated trains: a) Trains loads per unit length: b) Trains speeds; c) Trains lengths. Freq. Load of the train per unit length [kN/m] Cum. Rel. Freq. Freq. Speed of the train [km/h] Cum. Rel. Freq. Freq. Cum. Rel. Freq. Length of the train [m]
Chapter VIII 8.25 The evolution of the total crack length as a function of the cumulative traffic volume is shown in Figure 8.15. It should be noted that the changes in crack propagation rates and in the crack path observed over time are affected by the length of the finite crack increments adopted for the crack simulation (Δainc=5mm). Smaller crack increments would provide more accurate results, but with associated costs in terms of computational time. If the current traffic volumes on the bridge were kept stable in the remaining life of the bridge, the hypothetical crack would propagate during approximately 95 years (see Figure 8.16). Nevertheless, it should be noted that the fatigue traffic mixes present in the standards usually assume higher traffic volumes. In the Eurocodes, for instance, an annual traffic volume of 25 MM t is considered as reference. Therefore, a scenario of 25 MM t per year was also included in the analysis. In that case, the simulated crack would take approximately 12 years to propagate (Figure 8.16). Figure 8.15 – Crack propagation length as a function of cumulative traffic for current traffic volumes. 0 15 30 45 60 75 90 105 120 135 150 165 180 050 100 150 200 250 300 Total crack length [mm] Cummulative traffic volume [MM t]
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.26 Figure 8.16 – Comparison of crack propagation for distinct traffic scenarios. It should also be noted that these results were achieved with a highly conservative assumption on the initial crack dimension (15mm). The hypothetical crack initiation period and/or propagation period up to that 15mm crack dimension have been therefore disregarded. The employment of the Paris Law is also considered a conservative assumption for the range of stress intensity factors observed in this analysis, since almost no crack propagation threshold Kth was adopted. 8.6. CONCLUDING REMARKS This chapter presented a new efficient method for the fatigue assessment of railway bridges. This method was validated by application to a case study, the bridge of the new railway crossing of river Sado. The main conclusions withdrawn from the current work are as follow: - The simulation of fatigue crack propagation in critical details can be achieved using minimal computational resources; 12 95 0 10 20 30 40 50 60 70 80 90 100 15 35 55 75 95 115 135 155 175 Time [Years] Total crack length [mm] 25 MM t/year traffic scenario Current traffic
Chapter VIII 8.27 - Shell-to-solid submodelling proves to be an effective way to address multiplescale structural problems, such as localized fatigue crack propagation in large structures; - Modal superposition of stress intensity factors confirms to be an adequate and efficient method when multiple and complex load histories are considered, such as multiple traffic events on bridges; - For the current case study and selected detail, the assumption of mixed mode fatigue crack propagation has impact on crack propagation path; - The proposed methodology allows the quick assessment of different traffic scenarios, e.g, the consideration of an increase of annual traffic volume over time. Finally, it should be underlined that the simulation can be optimized, by the implementation of parallel computing, further increasing the computational efficiency. In that case it will be possible to incorporate more complex fatigue crack propagation models, for example to foreseen complex geometries for the fatigue cracks. 8.7. REFERENCES [1] Kühn, B., Lukić, M., Nussbaumer, A., Günther, H.P., Helmerich, R., Herion, S., Kolstein, M.H., Walbridge, S., Androic, B., Dijkstra, O., Bucak, Ö., “Assessment of Existing Steel Structures: Recommendations for Estimation of Remaining Fatigue Life”, in: G. Sedlacek, F. Bijlaard, M. Géradin, A. Pinto, S. Dimova (Eds.) JRC Scientific and Technical Reports, European Commission, Joint Research Centre, 2008. [2] Haghani, R., Al-Emrani, M., Heshmati, M., “Fatigue-Prone Details in Steel Bridges”, Buildings 2, pp.456-476, 2012. [3] Miki, C., “Retrofitting Engineering for Fatigue Damaged Steel Structures”, International Institute of Welding, pp. 70, 2010. [4] Fisher, W., Roy, S., “Fatigue of steel bridge infrastructure”, Structure and Infrastructure Engineering 7, pp.457-475, 2010.
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.28 [5] Zhou, H., Liu, K., Shi, G., Wang, Y.Q., Shi, Y.J., De Roeck, G., “Fatigue assessment of a composite railway bridge for high speed trains. Part I: Modeling and fatigue critical details”, Journal of Constructional Steel Research 82, pp.234-245, 2013. [6] Guo, T., Chen, Y.W., “Fatigue reliability analysis of steel bridge details based on field-monitored data and linear elastic fracture mechanics”, Structure and Infrastructure Engineering 9, pp.496-505, 2013. [7] Cremona, C., Eichler, B., Johansson, B., Larsson, T., “Improved Assessment Methods for Static and Fatigue Resistance of Old Metallic Railway Bridges”, Journal of Bridge Engineering 18, pp.1164-1173, 2013. [8] Vincenzi, L., Savoia, M., Salvatore, W., “Experimental modal analysis and fatigue assessment on the Lagoscuro viaduct”, Crc Press-Taylor & Francis Group, Boca Raton, 2012. [9] Lippi, F.V., Orlando, M., Salvatore, W., “Assessment of the dynamic and fatigue behaviour of the Panaro railway steel bridge”, Structure and Infrastructure Engineering 9, pp.834-848, 2011. [10] Caglayan, B., O., Ozakgul, K., Tezer, O., “Fatigue life evaluation of a through-girder steel railway bridge, Engineering Failure Analysis”, 16, pp. 765-774, 2009. [11] Zhou, Y., “Assessment of Bridge Remaining Fatigue Life through Field Strain Measurement”, Journal of Bridge Engineering 11, pp. 737-744, 2006. [12] Marques, F., Cunha, Á., Fernandes, A.A., Caetano, E., Magalhães, F., “Evaluation of dynamic effects and fatigue assessment of a metallic railway bridge”, Structure and Infrastructure Engineering 6, pp.635-646, 2009. [13] Righiniotis, T.D., Chryssanthopoulos, M.K., “Fatigue and fracture simulation of welded bridge details through a bi-linear crack growth law”, Structural Safety 26, pp. 141-158, 2004. [14] Imam, B.M., Righiniotis, T.D., Chryssanthopoulos, M.K.,” Numerical modelling of riveted railway bridge connections for fatigue evaluation”, Engineering Structures 29, pp. 3071-3081, 2007. [15] Righiniotis, T.D., Imam, B.M., Chryssanthopoulos, M.K., “Fatigue analysis of riveted railway bridge connections using the theory of critical distances”, Engineering Structures 30, pp.2707-2715, 2008.
Chapter VIII 8.29 [16] Radaj, D., Sonsino, C.M., Fricke, W., “Recent developments in local concepts of fatigue assessment of welded joints”, International Journal of Fatigue 31, pp.2-11, 2009. [17] Chan, T.H.T., Guo, L., Li, Z.X., “Finite element modelling for fatigue stress analysis of large suspension bridges, Journal of Sound and Vibration”, 261, pp.443-464, 2003. [18] Li, Z.X., Zhou, T.Q., Chan, T.H.T., Yu, Y., “Multi-scale numerical analysis on dynamic response and local damage in long-span bridges”, Engineering Structures 29, pp. 1507-1524, 2007. [19] Albuquerque, C., de Castro, P.M.S.T., Calçada, R., “Efficient crack analysis of dynamically loaded structures using a modal superposition of stress intensity factors”, Engineering Fracture Mechanics 93, pp.75-91, 2012. [20] Albuquerque, C., Calçada, R., de Castro, P.M.S.T., “Fatigue assessment of a bowstring railway bridge”, P.B.H.V. Topping (Ed.) CST 2012, Civil-Comp Press, Dubrovnik, Croatia, 2012. [21] Clough, R., W., Penzien, J., “Dynamics of structures”, McGraw-Hill, 1975. [22] Albuquerque, C., da Silva, A.L.L., de Jesus, A.M.P., Calçada, R., “An Efficient Methodology for Fatigue Damage Assessment of Bridge Details Using Modal Superposition of Stress Intensity Factors”, International Journal of Fatigue, 2014. [23] Broek, D., “Elementary engineering fracture mechanics”, 4th rev. ed., Martinus Nijhoff Publishers, Dordrecht, 1987. [24] Tada, H., Paris, P.C., Irwin, G.R,. “The stress analysis of cracks handbook”, 3rd ed ed., ASME Press Professional Engineering Publishing, New York, 2000. [25] Paris, P.C., Gomez, M.P., Anderson, W.E., “A Rational Theory of Fatigue”, The Trend in Engineering 13, pp.9-14, 1961. [26] Tanaka, K., “Fatigue crack propagation from a crack inclined to the cyclic tensile axis”, Engineering Fracture Mechanics 6, pp.493-507, 1974. [27] Matsuishi, M., Endo, T., “Fatigue of metals subjected to varying stress”, Japan Society of Mechanical Engineers, 1968. [28] Erdogan, F., Sih, G.C., “On the Crack Extension in Plates Under Plane Loading and Transverse Shear”, Journal of Fluids Engineering 85, pp.519-525, 1963. [29] I. Fracture Analysis Consultants, Franc3D Reference Manual, in, New York, 2011. [30] Krueger, R., “The Virtual Crack Closure Technique: History, Approach and Applications”, NASA, Hampton, Virginia, USA, 2002.
Finite element modelling of a relevant detail of the Alcácer do Sal Bridge 8.30 [31] Rice, J.R., “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks”, Journal of Applied Mechanics 35, pp.379-386, 1968. [32] I. Advanced Analysis Techniques Guide, in: Release 12.0 Documentation for ANSYS, ANSYS, Inc., 2009. [33] Albuquerque, C., “Advanced methodologies for the assessment of the fatigue behaviour of railway bridges", ongoing. [34] Lippi, F., Salvatore, W., Braconi, A., Finetto, M., Wenzel, H., De Roeck, G., Peeters, B., Könke C., Zabel, V., Cunha, A.,“Fatigue damage control and assessment for railway bridge”, Research Fund for Coal and Steel, Directorate-General for Research and Innovation, 2009-2012. [35] SAS, ANSYS, Version 12.0, Swanson Analysis Systems Inc., Houston, 2010. [36] Albuquerque, C.M.C., Pinto, N.M.P., Calçada, R.A.B., Gabriel, J., “Experimental characterization of the dynamic behaviour of the new railway bridge over the river Sado”, U.o. L'Aquila, P.d. Milano (Eds.) EVACES 2011, University of L'Aquila, Politecnico di Milano, Varenna, Italy, 2011. [37] Albuquerque, C.M.C., Miranda, R.M.C., Richter-Trummer, V., Figueiredo, M.A.V.d, Calçada, R., “Fatigue crack propagation behaviour in thick steel weldments”, International Journal of Structural Integrity, 2012. [38] Carvalho, D., da Silva, A.L.L., Jesus, A.M.P., Fernandes, A.A., “Fatigue Behaviour of Structural Steels. Comparison of Strain-Life And Fatigue Crack Propagation Data”, in: 9º Congresso Nacional de Mecânica Experimental, Aveiro, Portugal, 2014. [39] da Silva, A.L.L., Jesus, A.M.P., Fernandes, A.A., Figueiredo, M., Calçada, R., “Análise do comportamento à fadiga de ligações soldadas com base no conceito das tensões estruturais”, 3º Congresso Nacional sobre Segurança e Conservação de Pontes - ASCP'2013, Porto, 2013.
Chapter IX Final conclusions and future works
Final conclusions and future works 9.2 9.1. OVERVIEW OF MAIN CONCLUSIONS Fatigue is a cause of progressive damage of bridges, particularly of steel and composite steel/concrete railway bridges [1-4]. The increasing number of numerical and experimental fatigue damage assessments being performed on railway bridges [5-12] is a clear indication of the relevance of this topic in structural engineering. A number of projects covering this topic have been performed in the last years. This was the case of the FADLESS European project, entitled Fatigue Damage Control and Assessment for Railways Bridges [13], which was the background for the research performed within this PhD work. The referred European project aimed the development of methodologies for the fatigue assessment of critical details, to be demonstrated for several European railway bridges, selected as case studies. Regarding the Portuguese case studies, two railway bridges were selected, namely the Trezói Railway Bridge and the new Alcácer do Sal Railway Bridge. The presented dissertation presented contributions towards the development of advanced methodologies to be applied in the fatigue assessment of structural details from metallic bridges, including welded and riveted joints. Local models for the fatigue analysis were preferably followed in the proposed research, in alternative to traditional global S-N approaches. Strategies were proposed to overcome the difficulties introduced by the multi-scale problem that this kind of local fatigue models introduces. In the following paragraphs, the main conclusions of the performed work are presented, following the chapters structure of the thesis. Chapter II presented an overview of important works in the field of experimental fatigue testing and numerical simulation of bridge details. Both fields need further developments, but the field of numerical simulation of bridge details (riveted and welded) seems to be the one that needs substantial work in order to provide reliable alternative tools to the conventional blind and over conservative S-N approaches.
Chapter IX 9.3 A fatigue characterization of two structural mild steels, the S235 and S355 steel grades, was presented in Chapter III. These materials are common options for bridge construction. The investigated materials were extracted from welded and riveted joints that were fatigue tested in the Chapter IV. Fatigue tests using smooth specimens were performed on these steels since this data is important to assess fatigue crack initiation on structural components. Additionally, fatigue crack propagation tests were performed in order to measure fatigue crack propagation rates that were used to model the fatigue crack propagation in structural components. Besides the pure mode I fatigue crack propagation tests, mixed mode fatigue crack propagation tests were also considered for the S235 steel. Digital Image Correlation technique was used to measure the fatigue crack path and to compute stress intensity factors directly from field information. The following set of conclusions can be formulated: i) The two investigated steels show similar cyclic elastoplastic and fatigue properties. ii) Both investigated materials were compared to the S690 high strength structural steel. Clear distinct behaviours between both steel categories were observed. iii) Mild steels showed lower fatigue crack propagation rates than the high strength steel and the stress ratio effects were more relevant for the high strength steel than for the mild steels. iv) The thickness effect on fatigue crack propagation was investigated for the S355 steel grade. Similar fatigue crack growth rates were observed, when tested thicknesses range from 4 to 30 mm. v) Pure mode I and mixed mode (mode I + mode II) fatigue tests were performed for the S235 steel. A two-step methodology to compute stress intensity factors using the displacements fields from DIC data was proposed. The approach allowed the assessment of the fatigue crack paths and the stress intensity factors. Further algorithm developments may be required to reduce scatter in the experimental data. vi) Mixed-mode fatigue crack propagation data was satisfactorily correlated with pure mode I fatigue crack propagation data using the effective stress intensity factor range as proposed by Tanaka for the S235 steel.