Innovative method for deriving residual stress distribution in hot-rolled steel beams An inverse analysis based on finite element modelling and strain data measured with optical fibres Master’s thesis in Structural Engineering Filip Karlström Borik Ronnby DEPARTMENT OF ARCHITECTURE AND CIVIL ENGINEERING CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2024 www.chalmers.se www.chalmers.se Master’s thesis 2024 Innovative method for deriving residual stress distribution in hot-rolled steel beams An inverse analysis based on finite element modelling and strain data measured with optical fibres Filip Karlström Borik Ronnby Department of Architecture and Civil Engineering Division of Structural Engineering Chalmers University of Technology Gothenburg, Sweden 2024 Innovative method for deriving residual stress distribution in hot-rolled steel beams An inverse analysis based on finite element modelling and strain data measured with optical fibres Filip Karlström Borik Ronnby © Filip Karlström 2024. © Borik Ronnby 2024. Examiner: Mohammad al-Ermrani, Division of Structural Engineering Supervisors: Farshid Zamiri, AFRY Infastructure Carlos Gil Berrocal, Division of Structural Engineering Master’s Thesis 2024 Department of Architecture and Civil Engineering Division of Structual Enginering Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX, template by Magnus Gustaver Gothenburg, Sweden 2024 iv Innovative method for deriving residual stress distribution in hot-rolled steel beams Filip Karlström & Borik Ronnby Department of Architecture and Civil Engineering Chalmers University of Technology Abstract In this masters project, an innovative method of determining residual stresses in steel beams has been developed. The method is based on an inverse analysis method which utilises finite element modelling and strain data collected with optical fibres to estimate the variables describing the residual stress distribution. The problem solved is of a non-linear type since the material behaviour during loading is non- linear, resulting in that a direct-search method was used in the inverse analysis. The variables describing the residual stress distribution are therefore derived using a Nelder-Mead optimisation algorithm where the difference between the finite ele- ment model’s strains and strains obtained from testing is minimised. The method was tested by performing four point bending tests on six hot-rolled profiles split between three HEA200 and three HEB200 beams. It can be concluded that the method is highly dependent on the quality and amount of test data that can be used. Especially the ability to capture data after the point of yielding is crucial in order to accurately mirror the non-linear behaviour of the material. However, if the data is good enough the method works as expected and will find a suitable residual stress distribution of the specimen’s tested. The accuracy of the method in its current state is however questionable as simplifi- cations made in the FE-model might have significantly affected the result. Several improvements of the method is thereby proposed to make it more accurate and reliable. Keywords: Residual stress distribution, Inverse analysis, Optical fibres, Hot-rolled steel beam, Nelder-Mead v Innovativ metod för att bestämma egenspänningsfördelningen i varmvalsade balkar Filip Karlström & Borik Ronnby Institutionen för Arkitektur och Samhällsbyggnadsteknik Chalmers Tekniska Högskola Sammanfattning I detta mastersprojekt har en innovativ metod för att bestämma egenspänningar i stålbalkar framtagits. Metoden är baserad på en inversanalys som använder sig av finit elementmodellering och töjningsdata från optiska fibrer för att estimera variablerna som beskriver egenspänningarna. Problemet som löses är icke-linjärt eftersom materialets beteende under lastning är icke-linjärt, vilket resulterade i att en direkt sökmetod användes i inversanalysen. Variablerna för egenspänningsfördel- ningen är framtagna genom en Nelder-Mead optimeringsmetod där skillnaden mellan töjningsdatan från den finita element modellen och de fysiska testerna minimeras. Metoden var testad genom att utföra fyrpunktsböjningstester på sex varmvalsade profiler, varav tre var av typ HEA200 och tre var av typ HEB200. Det kunde kon- stateras att metoden är ytterst beroende av kvalitén och mängden testdata som kan användas. Framför allt möjligheten att mäta data efter att stålet flyter är viktigt för att noggrant kunna modellera det icke-linjära beteendet hos materialet. Om datan är bra nog så fungerar dock metoden som väntat och den finner en passande egenspänningsfördelning för den testade provkroppen. Träffsäkerheten hos metoden i sitt nuvarande skede är dock ifrågasättbar eftersom förenklingar av balkmodellen kan ha avsevärt påverkat resultatet. Flertalet för- bättringar av metoden är därmed föreslagna för att göra den mer träffsäker och trovärdig. Nyckelord: Egenspänningsfördelning, Inversanalys, Optiska fibrer, Varmvalsad stål- balk, Nelder-Mead vi Preface This Master’s thesis was a collaboration between Chalmers University of Technology, Department of Architecture and Civil Engineering and AFRY AB’s, department of Inspection and investigation. The work on this thesis was carried out at AFRY AB and at Chalmers University of Technology’s Structural lab from January 2024 to June 2024. It is part of a pilot study examining the ability to use optical fibres to measure strain in steel members. The authors want to express their deepest gratitude to the projects supervisors, Ph.D Farshid Zamiri at AFRY AB and Senior Lecturer Carlos Gil Berrocal at the Department of Architecture and Civil engineering, as well as to examiner Prof. Mohammad al-Emrani at the Department of Architecture and Civil Engineering. Thanks for your constant support and guidance throughout the project. Farshid, for your constant help and deep understanding of the topic at hand. Thanks, it has been a pleasure. Carlos, for selflessly spending your time to advance this project. Thanks for you time and expertise using optical fibres. Mohammad, for your constant enthusiasm and dedication to the project as well as our development and understanding. Thanks, it has been a privilege to have you as our examiner. A special thanks to Sebastian Almfeldt and Anders Karlsson at Chalmers Struc- tural lab, without your engagement and expertise this project would not have been possible. Filip Karlström & Borik Ronnby, Gothenburg, 2024/06/27 viii x Table of Contents List of Figures xiii List of Tables xix Nomenclature xxi 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Method overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.5 Societal, ethical, and ecological aspects . . . . . . . . . . . . . . . . . 2 2 Literature review 3 2.1 Residual stress in hot-rolled profiles . . . . . . . . . . . . . . . . . . . 3 2.2 Methods of measuring residual stresses . . . . . . . . . . . . . . . . . 4 2.3 Models of residual stress distribution . . . . . . . . . . . . . . . . . . 5 2.4 Optimisation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Strain measurement using optical fibres . . . . . . . . . . . . . . . . . 12 3 Methodology 13 3.1 Laboratory tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Tensile strength test of the material . . . . . . . . . . . . . . . . . . . 19 3.3 Python script for Abaqus-model and post processing . . . . . . . . . 20 3.4 Optimising routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Validation of method . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Results 31 4.1 Beam dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Test data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Evaluation of coefficients through time . . . . . . . . . . . . . . . . . 36 4.5 Optimisations results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.6 Comparison between finite element and real strain . . . . . . . . . . . 40 5 Discussion 47 5.1 Test preparations and execution . . . . . . . . . . . . . . . . . . . . . 47 xi Table of Contents 5.2 Determination of material properties from tensile tests . . . . . . . . 49 5.3 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 Post processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.5 Optimisation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Conclusion 59 6.1 Thoughts and reflections . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 References 61 A Matlab script for extracting and formatting data from physical tests 65 B Original and cut strain curves 71 C Tensile test calculations script 91 D Script for running abaqus and calculating error 95 E Steel specification from supplier 121 F Optimisation script 125 G Beam dimensions 127 H Comparison of final strain curves 133 I Full run strain comparison of HEB1 161 xii List of Figures 2.1 American model of residual stress distribution. . . . . . . . . . . . . . 6 2.2 Young’s model of residual stress distribution. . . . . . . . . . . . . . . 7 2.3 ECCS’s model of residual stress distribution. . . . . . . . . . . . . . . 8 2.4 Skiadopoulos model of residual stress distribution . . . . . . . . . . . 9 2.5 Illustration of simplex in two-dimensions. . . . . . . . . . . . . . . . . 10 2.6 Illustration of the reflection step in the Nelder-Mead procedure for a two-dimensional space. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Illustration of the expansion step in the Nelder-Mead procedure for a two-dimensional space. . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8 Illustration of the contraction step in the Nelder-Mead procedure for a two-dimensional space. . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.9 Illustration of the shrinkage step in the Nelder-Mead procedure for a two-dimensional space. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1 Sketch of test setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Picture of the distribution beam. . . . . . . . . . . . . . . . . . . . . 14 3.3 Beams in different stages of surface corrosion. Top two beams are of type HEB200 and bottom two are of type HEA200. . . . . . . . . . . 14 3.4 Points for measurements on flanges. . . . . . . . . . . . . . . . . . . . 15 3.5 Fibre pattern taped to beam to guide fibre placement. . . . . . . . . . 16 3.6 Fibre patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.7 Lateral support configuration . . . . . . . . . . . . . . . . . . . . . . 18 3.8 Example of cut data from HEA1 bottom flange. . . . . . . . . . . . . 19 3.9 Simplified flow chart of Abaqus and error calculation script . . . . . . 21 3.10 Shell and solid assembly with visible supports and symmetry conditions. 22 3.11 Example view of an element set for residual stress application. . . . . 23 3.12 Residual stress distribution. . . . . . . . . . . . . . . . . . . . . . . . 24 3.13 Location of paths from where longitudinal strain is extracted in Abaqus marked in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.14 Example of data reshaping and error calculation for one location. The dimensioning coordinates are circled in blue and dimensioning time frames are circles in red. After reshaping the data sets are of the same dimensions. The strains used for calculating the error are underlined in green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.15 Flow chart of the inverse analysis . . . . . . . . . . . . . . . . . . . . 27 xiii List of Figures 3.16 Coverage of starting simplexes with residual coefficient c on the x-axis and residual coefficient d on the y-axis. . . . . . . . . . . . . . . . . . 27 3.17 Validation of model simulations, coefficients through time . . . . . . . 28 4.1 HEA web strain-stress curve . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 HEA flange strain-stress curve . . . . . . . . . . . . . . . . . . . . . . 33 4.3 HEB web strain-stress curve . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 HEB flange strain-stress curve . . . . . . . . . . . . . . . . . . . . . . 34 4.5 Force-deflection curves for all six beams. Deflection is the calculated deflection according to equation 3.1. . . . . . . . . . . . . . . . . . . . 35 4.6 Example of cut data from HEA1 top flange. . . . . . . . . . . . . . . 35 4.7 Simulation of HEA1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.8 Simulation of HEA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.9 Simulation of HEA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.10 Simulation of HEB1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.11 Simulation of HEB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.12 Simulation of HEB3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.13 Residual stress distribution of HEA based on mean coefficients in Table 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.14 Residual stress distribution of HEB based on mean coefficients in Table 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.15 HEB1 deflection over time for Abaqus and physical test. . . . . . . . 41 4.16 Final strain comparison for HEB1 top flange . . . . . . . . . . . . . . 41 4.17 Final strain comparison for HEB1 left inner top flange . . . . . . . . 42 4.18 Final strain comparison for HEB1 right inner top flange . . . . . . . . 42 4.19 Final strain comparison for HEB1 left web . . . . . . . . . . . . . . . 43 4.20 Final strain comparison for HEB1 right web . . . . . . . . . . . . . . 43 4.21 Final strain comparison for HEB1 left inner bot flange . . . . . . . . 44 4.22 Final strain comparison for HEB1 right inner bot flange . . . . . . . . 44 4.23 Final strain comparison for HEB1 bot flange . . . . . . . . . . . . . . 45 5.1 Pit corrosion located on the flanges of one of the HEB beams. . . . . 48 5.2 Principal sketch of error in measuring the deflection at the roller support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Original model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Mesh generation HEB with edge seed 100mm, 9817 shell elements . . 52 5.5 Mesh generation HEB with edge seed 50mm, 7057 shell elements . . . 52 5.6 Resulting model for the HEB compared to existing models. . . . . . . 56 5.7 HEB1 left web for full test duration with Abaqus data based on the final residual stress coefficients. Abaqus was run up until the point where the physical test was unloaded. . . . . . . . . . . . . . . . . . . 57 5.8 HEB1 strains over the height for certain time frames. . . . . . . . . . 58 B.1 HEA1 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 HEA1 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.3 HEA1 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.4 HEA1 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 72 xiv List of Figures B.5 HEA1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 73 B.6 HEA1 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B.7 HEA1 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 73 B.8 HEA1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 74 B.9 HEA2 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 B.10 HEA2 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.11 HEA2 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 B.12 HEA2 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 75 B.13 HEA2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 76 B.14 HEA2 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 B.15 HEA2 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 76 B.16 HEA2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 77 B.17 HEA3 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 B.18 HEA3 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 B.19 HEA3 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 B.20 HEA3 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 78 B.21 HEA3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 79 B.22 HEA3 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.23 HEA3 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 79 B.24 HEA3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 80 B.25 HEB1 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 B.26 HEB1 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.27 HEB1 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.28 HEB1 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 81 B.29 HEB1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 82 B.30 HEB1 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 B.31 HEB1 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 82 B.32 HEB1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 83 B.33 HEB2 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 B.34 HEB2 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.35 HEB2 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 B.36 HEB2 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 84 B.37 HEB2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 85 B.38 HEB2 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 B.39 HEB2 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 85 B.40 HEB2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 86 B.41 HEB3 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 B.42 HEB3 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.43 HEB3 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.44 HEB3 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 87 B.45 HEB3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 88 B.46 HEB3 bottom flange . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 B.47 HEB3 left inner bottom flange . . . . . . . . . . . . . . . . . . . . . . 88 B.48 HEB3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 89 H.1 HEA1 deflection in abaqus and physical test . . . . . . . . . . . . . . 133 xv List of Figures H.2 HEA1 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 H.3 HEA1 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 134 H.4 HEA1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 135 H.5 HEA1 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 H.6 HEA1 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 H.7 HEA1 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 136 H.8 HEA1 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 137 H.9 HEA1 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 H.10 HEA2 deflection in abaqus and physical test . . . . . . . . . . . . . . 138 H.11 HEA2 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 H.12 HEA2 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 139 H.13 HEA2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 139 H.14 HEA2 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 H.15 HEA2 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 H.16 HEA2 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 141 H.17 HEA2 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 141 H.18 HEA2 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 H.19 HEA3 deflection in abaqus and physical test . . . . . . . . . . . . . . 142 H.20 HEA3 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 H.21 HEA3 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 143 H.22 HEA3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 144 H.23 HEA3 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 H.24 HEA3 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 H.25 HEA3 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 145 H.26 HEA3 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 146 H.27 HEA3 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 H.28 HEB1 deflection in abaqus and physical test . . . . . . . . . . . . . . 147 H.29 HEB1 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 H.30 HEB1 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 148 H.31 HEB1 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 148 H.32 HEB1 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 H.33 HEB1 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 H.34 HEB1 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 150 H.35 HEB1 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 150 H.36 HEB1 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 H.37 HEB2 deflection in abaqus and physical test . . . . . . . . . . . . . . 151 H.38 HEB2 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 H.39 HEB2 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 152 H.40 HEB2 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 153 H.41 HEB2 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 H.42 HEB2 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 H.43 HEB2 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 154 H.44 HEB2 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 155 H.45 HEB2 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 H.46 HEB3 deflection in abaqus and physical test . . . . . . . . . . . . . . 156 H.47 HEB3 top flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 xvi List of Figures H.48 HEB3 left inner top flange . . . . . . . . . . . . . . . . . . . . . . . . 157 H.49 HEB3 right inner top flange . . . . . . . . . . . . . . . . . . . . . . . 157 H.50 HEB3 left web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 H.51 HEB3 right web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 H.52 HEB3 left inner bot flange . . . . . . . . . . . . . . . . . . . . . . . . 159 H.53 HEB3 right inner bot flange . . . . . . . . . . . . . . . . . . . . . . . 159 H.54 HEB3 bot flange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 I.1 HEB1 Extended run top flange . . . . . . . . . . . . . . . . . . . . . 161 I.2 HEB1 Extended run left inner top flange . . . . . . . . . . . . . . . . 162 I.3 HEB1 Extended run right inner top flange . . . . . . . . . . . . . . . 162 I.4 HEB1 Extended run left web . . . . . . . . . . . . . . . . . . . . . . . 163 I.5 HEB1 Extended run right web . . . . . . . . . . . . . . . . . . . . . . 163 I.6 HEB1 Extended run left inner bot flange . . . . . . . . . . . . . . . . 164 I.7 HEB1 Extended run right inner bot flange . . . . . . . . . . . . . . . 164 I.8 HEB1 Extended run bot flange . . . . . . . . . . . . . . . . . . . . . 165 xvii List of Figures xviii List of Tables 3.1 Plastic behaviour used when creating the simulated data. . . . . . . . 28 3.2 Summary of the validation runs with the three simplexes . . . . . . . 29 4.1 Average thickness of the flanges and web for all tested beams. . . . . 31 4.2 Measurements of tensile test specimens. A denotes a HEA-profile, B a HEB-profile and F or W represent flange or web respectively. . . . . 32 4.3 Young’s modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 Time of data cut for the different beams and the deflection at the cut time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Results of simulation for HEA-profiles . . . . . . . . . . . . . . . . . . 38 4.6 Results of simulation for HEB-profiles . . . . . . . . . . . . . . . . . . 39 xix List of Tables xx Nomenclature Greek σc Compressive stress σf Flange stress σt Tensile stress σw Web stress Latin B/b Width of cross section bf Flange width d Depth dAbaqus Deflection used in Abaqus dP ointLoad Deflection at point load dSupport Deflection at support fy Material yield limit h Height of cross section tf Flange thickness tw Web thickness xxi Nomenclature xxii 1 Introduction This section describes the background of the project and what its aim and limitations are. 1.1 Background Residual stresses have a large influence on the fatigue life of manufactured steel products (James et al., 2007). Since the mean stress is inversely related to the fatigue capacity, an increase in the mean stress in the form of residual stress will lower the fatigue life of the product. In addition, residual stresses affect when parts of the cross-section yield, which in compression make the products more sensitive to buckling (Young, 1975). Because of these effects, it is of interest to be able to accurately determine the residual stresses of steel members. There are several methods to determine the residual stresses, and they can be cate- gorised into three types: destructive, semi-destructive, and non-destructive (Rossini et al., 2012). Destructive and semi-destructive methods are usually easy to perform and can be used for any type of material, but they leave the specimen damaged after the procedure. In addition, errors are introduced in the process of drilling or cutting the specimen for the measurements, and the data from the tests could be difficult to interpret (Rossini et al., 2012). Methods that are non-destructive, on the other hand, leave the specimen unchanged but usually require lab conditions and tweaking for the specific specimen to give proper results (Rossini et al., 2012). In search of an alternative way of measuring residual stresses, a method based on inverse analysis of strains obtained using optical fibre sensors has been proposed. The fibres allow for high amounts of strain measurements along the length of the cable compared to traditional strain gauges (Sabri et al., 2013), which could improve the accuracy of the residual stress profile. In this thesis, the method described above was tested and then evaluated. 1.2 Method overview The method compares strain data from physical testing with that of a numerical model. By assuming a residual stress distribution and evaluating each assumption the method derives the initial residual stress distribution of the beam. This is a type of inverse analysis were the result is known, in this case the strain data, the 1 1. Introduction initial state of the problem is then to be solved, in this case the residual stress distribution. To solve the problem, an optimisation algorithm is used which makes new assumptions based on the result of the previous assumption, until it reaches a set convergence criteria. Measurements are made using optical fibres glued to a specimen subjected to loading, with at least one section going beyond yielding. The fibres allow for high amounts of strain measurements along the length of the cable compared to traditional strain gauges (Sabri et al., 2013), which could improve the accuracy of the residual stress profile. 1.3 Aim The aim of this master project was to develop and evaluate a method of deriving the residual stress distribution of steel beams based on strain data measured with optical fibres. 1.4 Limitations For the physical tests conducted, no cyclic loading was performed. This decreased the number of variables that could affect the result, leading to a reduced work load in calibrating the model based on the test results. Only hot-rolled H-profiles were tested, and their sizes were limited by the fact that the maximum available force in four-point bending was 200 kN and that plasticity had to be reached in the cross-section. Because of the limited available force, the steel strength was also limited, not including high strength steels. Only a steel grade of S355J2+M was studied. The residual stress distribution that is analysed in this thesis is only the residual stress distribution of the macroscopic scale in the axial direction. The microstructure stress distribution and the transverse stress are disregarded. 1.5 Societal, ethical, and ecological aspects This thesis does not include discussions regarding societal or ethical aspects as it focuses on the behaviour of steel, not its implementation in the real world regarding sourcing, production, etc. There is a connection to the ecological aspect in the sense that it might be possible to use the results of the investigation to optimise cross- sections of beams, which would result in less material being used. This would in turn reduce carbon emissions from steel constructions built using this optimisation. 2 2 Literature review There are three types of residual stresses that are categorised by either the scale on which they are in self-equilibrium or by the scale on which they are measured (Withers & Bhadeshia, 2001b). The three types are: macrostresses (type I), which are on the scale of structures, intergranual stresses (type II), which are on the scale of a small number of grains, and atomic scale stresses (type III), which are on the scale smaller than a singular grain (Withers & Bhadeshia, 2001a). For this thesis, only axial residual stress of type I is of interest since the stress distri- bution of interest is the one over the whole cross-section. In addition, only hot-rolled profiles are investigated in this thesis. Therefore, the following text mainly focuses on information relevant to hot-rolled profiles and type I axial residual stresses. 2.1 Residual stress in hot-rolled profiles The residual stresses in hot-rolled beams are introduced during cooling of the beam after the rolling process. The stresses appear due to different points of the cross- section cooling at different rates, the parts that cool first will end up in compression, and the ones that cool later will end up in tension (Young, 1975). The parts that cool first depend on the geometry of the section and the cooling conditions, such as how close together the beams lay on the cooling bed and in which orientation the individual beams are placed (Alpsten, 1968). If the beam is allowed to radiate heat freely, the orientation of the beam has far less impact, and the tips of the flanges as well as the web will end up in compression while the area around the web-flange junctions will end up in tension (Alpsten, 1968). However, if the beams on the cooling bed are placed close together, the orientation of the beam matters greatly as it determines which surfaces face each other and their ability to radiate heat (Alpsten, 1968). According to Alpsten’s study (1968), the whole web can end up being in tension if the beams are placed close enough with the webs in the vertical direction. If the webs are instead placed horizontally, the study shows that the stress distribution differs much less from the case with free heat transfer compared to the case with vertical webs (Alpsten, 1968). The stress distribution for the case with horizontal webs seems to follow the same pattern as the case with free heat radiation, but with an increased magnitude of compression and tension in the web and flange respec- 3 2. Literature review tively. After the cooling process, the beams usually have to be straightened since they curve when cooling (Alpsten, 1975). Straightening is done either in a rotorizing machine or in a gag press. The process of straightening the beam causes plastic deformations which alter the residual stress pattern in the beam to a sort of asymmetric zig-zag pattern (Alpsten, 1975). Furthermore, in Alpstens study of rotorized beams, the average residual stress in the flanges of all the rotorized beams was significantly reduced while the residual stresses in the web were just slightly changed. Thereby, straightening could be seen as a sort of stress relief (Young, 1975) However, there is no guarantee that the beams are straightened. Based on measure- ments, many delivered products have a simple residual stress pattern from cooling, also called thermal residual stress pattern. (Alpsten, 1968). For this reason, assum- ing that the beam’s residual stress pattern is the thermal residual stress pattern in the design results in a design that is on the safe side (Young, 1975) (Alpsten, 1975). The resulting residual stress, from cooling and possible straightening of the beam, will negatively affect its fatigue life, susceptibility to stress corrosion, and buckling strength (Young, 1975). Areas subjected to residual tension will have a higher mean stress during cyclic loading, which results in a lower fatigue life for the whole beam (James et al., 2007), while areas subjected to residual compression will yield earlier when loaded in compression, increasing the risk of in-elastic buckling (Young, 1975). In addition, the areas in residual tension are at risk of stress corrosion cracking even without any external load (Ghosh et al., 2011). 2.2 Methods of measuring residual stresses As mentioned in Chapter 1, there are several methods used for measuring residual stresses ranging from destructive to non-destructive methods, with some in between (Rossini et al., 2012). In general, destructive and semi-destructive methods are eas- ier to apply and are valid for a wider range of materials. The drawback is that the method will permanently damage the object studied as well as that the data gathered are usually hard to interpret as errors can easily occur during the processes (Rossini et al., 2012). Non-destructive methods revolve around measuring physical properties. For example, electromagnetism or the spacing between the crystalline structure of the steel, usually referred to as diffraction methods (Rossini et al., 2012). Some of the most common destructive and non-destructive methods are further de- scribed below. Hole-drilling is one of the most used methods today and is classified as a semi- destructive method (Withers & Bhadeshia, 2001b). Drilling a hole in a region with residual stresses allows the material to release or relax its in-plane stresses in the area around the insertion. During the relaxation of the stresses, the material around the hole will deform, allowing the strain to be measured (Withers & Bhadeshia, 2001b). Important to note is the reduction of accuracy as the depth of the hole approaches 4 2. Literature review and later surpasses its diameter. Also, high residual stresses in an area may cause local yielding, disrupting the relaxation and resulting in incorrect measurements. The effects of this will begin to appear as the residual stresses exceed half of the yield stress (Withers & Bhadeshia, 2001b). Sectioning is another method that has frequently been used for measuring residual stresses in metallic materials (Rossini et al., 2012). The method involves cutting the specimen into strips while measuring the strain changes, from which the residual stresses along the cutting lines can be calculated (Rossini et al., 2012). It is impor- tant that no plasticity or heat is introduced during cutting to achieve accurate results (Rossini et al., 2012). Overall, the method is easy to perform, economical, and gives accurate results (Tebedge et al., 1972), but it is destructive and cannot be performed on the site of the structure or object. Two diffraction methods in regular use are X-ray and Neutron diffraction. X-ray diffraction has a small penetration depth in steel, allowing for the assumption of plane-strain in the studied area (Withers & Bhadeshia, 2001b). The geometries of elements are an important factor in the availability of x-ray diffraction, as they need to allow the rays to hit the desired measurement area and then diffract directly onto the detector (Rossini et al., 2012). Neutron diffraction is, at large, very similar to X-ray diffraction, but it has some distinctions. The main advantage is the higher penetration depth that allows for measurements up to 60 mm (Fitzpatrick & Lo- dini, 2003). Drawbacks today are largely the cost and lack of portability (Liu et al., 2023), making it unsuitable for any type of field work. The ultrasonic method is also a non-destructive technique that has gained traction due to its ease of use, cheap procurement of equipment, and lightweight, making it suitable for on-site inspection (Withers & Bhadeshia, 2001b). Compared to x-ray diffraction, the penetration depth of ultrasonic waves is higher but cannot give as high a resolution of the data collected. Another limitation of the technique is mainly that, for thicker plates only the average stress can be measured (Rossini et al., 2012), requiring supplementary methods to get a better picture of the residual stress dis- tribution. 2.3 Models of residual stress distribution Several different models of residual stress distribution have been proposed through- out the 20th century. One of the earliest models widely applied was developed by Galaambos and Ketter in the late 1950s (Galambos & Ketter, 1959). Based on results from tests conducted at Fritz Engineering Laboratory, Leigh University in Bethlehem, Pennsylvania (Huber & Beedle, 1953), was applicable only to hot-rolled beams. The model developed based on the measurements assumes a bi-linear distribution of the residual stresses in the flanges, with the outer-flanges in compression and the 5 2. Literature review Figure 2.1: American model of residual stress distribution. middle part in tension (Abambres & Quach, 2016). In the web, a uniform distri- bution is assumed with tension matching the magnitude of the middle part of the flange, the model is illustrated in Figure 2.1. Equations 2.1 are the equations used to describe the stress distribution according to the model (Abambres & Quach, 2016). σc = 0.3fy (2.1a) σt = σc · bf tf bf tf + tw(h − 2tf ) (2.1b) In the mid-1970s, B. W. Young at the University of Sussex performed his own tests and examined experimental data done in previous decades regarding distribution of residual stresses (Young, 1975). Young found that in most cases, the distribution of residual stresses could be described with a parabolic curve for both the web and flanges, where the peak stresses are functions of only the geometry (Abambres & Quach, 2016), see Figure 2.2 and Equations 2.2a-c. σc1 = 165 · ( 1 − h · tw 2.4 · Btf ) (2.2a) σc2 = 100 · ( 1.5 + h · tw 2.4 · Btf ) (2.2b) σt = 100 · ( 0.7 + h · tw 2 · Btf ) (2.2c) 6 2. Literature review Figure 2.2: Young’s model of residual stress distribution. Certain column sections tested in the USA showed a lean towards a bi-linear distri- bution over the flanges, a distribution that could be recreated, although the distri- bution in the web differed. For the specimen tested in the USA, a uniform tensile stress was observed in the web, a phenomenon that was not observed in other studies or tests recreated in the UK (Young, 1975). These tendencies to a more bi-linear distribution are the results used in the model described in Figure 2.1 . Young rea- sons against some of those assumptions in that, due to the process that generates the residual stress, a bi-linear distribution does not seem reasonable as it introduces a discontinuity in the residual stress (Young, 1975). Introducing a rounded middle point or a heavily weighted parabolic curve would instead be more fitting and follow the results from other experiments according to Young. Another proposed model is the one from the European Convention of Constructional Steelwork (ECCS), which suggests that the distribution is bi-linear in both web and flanges (Abambres & Quach, 2016). ECCS’s model of the residual stress distribution depends on the yield stress of the material and the ratio between the height and the width of the cross section. The distribution is described by Equation 2.3 and Figure 2.3 (Abambres & Quach, 2016).  h b ≤ 1.2 ⇒ α = 0.5 h b > 1.2 ⇒ α = 0.3 (2.3) More recently, a model with quadratic distributions was proposed by Skiadopoulos et al. (2023). The model was developed by applying a constrained least square method to measured residual stresses, with an assumption of a quadratic residual 7 2. Literature review Figure 2.3: ECCS’s model of residual stress distribution. stress distribution according to Equations 2.4a and 2.4b (Skiadopoulos et al., 2023) and Figure 2.4. The constraints in the least square problem were the criteria of force equilibrium in the cross section and stress continuity in the web-flange junction according to Equations 2.4c and 2.4d. An important thing to note is that the fillet radii is not included in the equilibrium condition, which simplifies the equation. σ0,f (x) = a + b ( x − bf 2 )2 (2.4a) σ0,w(y) = c + d ( y − h 2 )2 (2.4b) (2tfbf )a + ( 2tfb3 f 12 ) b + [tw(h − 2tf )]c + [ tw(h − 2tf )3 12 ] d = 0 (2.4c) a = c + d (h − tf )2 4 (2.4d) The proposed values for coefficients a and c were 37 MPa and 81 MPa, respectively. With the values of a and c, coefficients b and d are determinable by Equations 2.4c and 2.4d (Skiadopoulos et al., 2023). 2.4 Optimisation procedure To solve the inverse analysis problem, the use of an optimisation algorithm is needed. Several types of algorithms exist to solve this: direct search, gradient, or higher-order derivative methods. For this thesis, a direct search method is to be implemented, a type of heuristic method. Contrasted with derivative methods that uses gradients and derivatives of a function to find a solution, direct search use only the function 8 2. Literature review Figure 2.4: Skiadopoulos model of residual stress distribution value itself to minimise the function. The particular variation to be implemented is called the Nelder-Mead method (Nelder & Mead, 1965) and it is usually used in non-linear problem, such as the one presented in this thesis. First published in the mid-1960s, Nelder-Mead, as well as a lot of other direct search algorithms, use a simplex to derive an approximate solution (Kolda et al., 2003). A simplex can be described as a geometric object with n+1 vertices in an n-dimensional space and is used in the Nelder-Mead procedure as a tool that is manipulated to find a solution. The initial simplex, either assigned to or created by the algorithm, is of great importance since if the initial simplex is small, the algorithm cannot easily find its way out of a potential local minima. It is therefore recommended to pick a large initial simplex that almost covers the whole area where the solution is expected to be found (Wessing, 2019). It is also recommended to normalise the search space (Wessing, 2019). To describe the algorithm in an intuitive way, the use of a visual example is com- monly used, as in Lagarias et al. (1998). Considering a two-variable problem, the simplex then becomes a triangle. Below, the algorithm is described in six steps, with corresponding figures explaining the process. The algorithm procedure consist of six steps: • Order Sort the vertices so that the value of the function follows f(x1) < f(x2) < ... < f(xn+1) 9 2. Literature review Figure 2.5: Illustration of simplex in two-dimensions. • Reflect Calculate the reflection point xr and evaluate f(xr). If f(x1) < f(xr) < f(xn) accept xr as the new point and terminate the iteration. Figure 2.6: Illustration of the reflection step in the Nelder-Mead procedure for a two-dimensional space. • Expand If f(xr) < f(x1) a guess expanding along the same line as xr is made and a expansion point is calculated xe. If f(xe) < f(xr) chose xe and end the iteration else choose xr. 10 2. Literature review Figure 2.7: Illustration of the expansion step in the Nelder-Mead procedure for a two-dimensional space. • Contract If f(xr) ≥ f(xn) calculate the contraction points xc1 or xc2 located on the line formed by xn and xr. Figure 2.8: Illustration of the contraction step in the Nelder-Mead procedure for a two-dimensional space. • Shrink If it follows that neither point xc1 or xc2 generates a value lower than f(xn) the simplex is shrunk towards the vertices that has the lowest value. 11 2. Literature review Figure 2.9: Illustration of the shrinkage step in the Nelder-Mead procedure for a two-dimensional space. • Convergence The iteration continues until a set convergence criteria is met. 2.5 Strain measurement using optical fibres Optical fibres can be used to measure several physical parameters, where the most common are strain or temperature changes. The scattering of electromagnetic waves is measured inside the fibres which is divided into three parts where each part contains a different type of spectral data, Rayleigh, Raman and Brillouin scatter- ing.(Jansson, 2024). In essence, the majority of methods compare the current state of the light in the fibre with a base state and the shift in wavelength/frequency is then multiplied by an empirically determined factor to obtain the strain (Kreger et al., 2016). Compared to traditional strain gauges, distributed optical fibre sensors have a higher sensitivity (Sabri et al., 2013) and they are able to measure at several densely spaced points along their length, allowing more measurements to be done more easily (Kreger et al., 2016) compared to traditional gauges. These traits of the optical fibres result in that the strain field can be measured with high detail and accuracy. 12 3 Methodology The first part of the thesis work was to establish a good understanding of the con- cepts regarding residual stress: how it is created, measured, taken into consideration and modelled. The result of this literature study is presented in Chapter 2. The work after the literature study was composed of physical testing of beams to extract strain data, the development of a numerical model of the test, and the implemen- tation of a optimisation algorithm to find suitable variables describing the residual stress pattern. 3.1 Laboratory tests Based on the literature findings presented in Chapter 2 and limitations of the avail- able lab equipment, it was concluded that performing a four-point bending test was the most suitable alternative. For a steel grade of S355J2+M and a lever arm of 1.5 m, the largest viable beam profile was calculated to be a HEB220. To ensure a margin of error regarding equipment capacities and other uncertainties, the profiles HEA200 and HEB200 were chosen to be used in the tests. Two beams with a length of 12.1 m were cut into three pieces of roughly 4 m in length by the supplier. Based on this length, the span length was chosen to be 3.6 m to make sure the beams would not roll off the supports during the tests. The distance between the loading points was 600 mm, out of which the middle 200 mm were chosen for strain measurements. An illustration of the setup can be seen in Figure 3.1. For load application, a hydraulic jack was used. To distribute the load from the hydraulic jack to the two load points, a strengthened rectangular hollow steel beam was used with two rollers attached to the bottom. The rectangular section was 200 mm high and 120 mm wide with a steel thickness of 10 mm. At the bottom of it, a 140 mm wide steel plate with a thickness of 10 mm was spot welded to the rectangular hollow section, reinforcing the beam as well as providing an area for the rollers to be attached to, see Figure 3.2 for a picture of the distribution beam. 13 3. Methodology Figure 3.1: Sketch of test setup. Figure 3.2: Picture of the distribution beam. Figure 3.3: Beams in different stages of surface corrosion. Top two beams are of type HEB200 and bottom two are of type HEA200. 14 3. Methodology Figure 3.4: Points for measurements on flanges. Testing and the necessary preparations for it were performed at Chalmers Struc- tural Lab. When delivered to the lab, the six beams were labelled according to their cross-section type, followed by a number between 1 and 3, e.g., HEA3. The beams were delivered in different stages of surface corrosion, as seen in Figure 3.3, which resulted in different amounts of preparation needed for each beam. Overall, each beam was prepared according to the following procedure: Using a wire brush the centre of the beam was cleaned of any loose debris and then vacuumed. The area where the fibres were to be attached was marked out, and a margin of 50 mm in each direction along the beam was added. With a combina- tion of manual labour, a drill with an attachment for sandpaper, as well as an angle grinder with the same configuration as the drill, the marked area was cleaned from both rust and scaling. The thickness of the flange in the cleaned region was measured using a thickness gauge. It was measured from left to right along six points of the cross-section over three section, 100 mm before the middle section, middle section and then 100 mm after. For a reference see Figure 3.4. Inside the area of raw steel a lot of dust, debris from the sanding and other oily residue that were not previously removed were wiped clean using paper towels and acetone. In cleaning the area with acetone, each swipe of the towel was done in only one direction before a new and clean part of the towel was used. This was done to ensure that the grime was removed from the area and not wiped onto the sanded surface again. Templates showing the pattern of the fibres were cut out and taped on the beam as 15 3. Methodology seen in Figure 3.5. This made it easy to tape down the fibres accurately along the cross-section with ten strands along the outside of the flanges, four on each part of the inside of the flanges and seven every side of the web. See Figure 3.6 for a more detailed visual of the fibre patterns. Figure 3.5: Fibre pattern taped to beam to guide fibre placement. To be able to extract information from the fibres, a connection and an end terminal were needed to be welded on to the fibres. Beginning by preparing the connection end, the protective layer was scaled off and then the inner coating was removed as well by using a wire stripper. The remaining glass fibre were then cleaned with an anti-static wipe and isopropanol. Placed inside a holding block, it was cut to length and then placed into the welding machine on its corresponding side. For the other fibre, a melting sleeve was placed over the fibre before being pre- pared. Its only coating, corresponding to the same one as their inner one on the connection, was scrapped off, cleaned, cut to length and placed in the welder as the connection. With both fibres placed in the welder, they were joined together under an arc created by two electrodes. Tested to see if the joining was successful before it was removed from the welding area, the sleeve was aligned over the joint and then heated in a small oven to encapsulate the weld and keep it from breaking during future handling. Preparing the end terminal, in this case a solid glass fibre with similar coating as the main fibre is done in the same way as the other and then welded onto the other end of the fibres to create a closed loop. Before gluing the fibres to the beam, they were connected to a sensor interrogator of type Luna Odisi 6100 to be configured and checked. In the software they were labelled and assigned the right refraction index of 1.4610. By configuring each loop of fibre or channel before they were glued down, any damaged or disturbed fibre could be repaired beforehand. Right before permanently attaching the fibres to the beam using 3M DP-190, a two- part thermoplastic glue, the surface was wiped clean a final time using anti-static wipes and a mixture of isopropanol and acetone. The glue was applied as evenly as 16 3. Methodology (a) Fibre pattern for webs. (b) Fibre pattern for outer flanges where the measurement specifies each fibre’s distance from the left flange edge. (c) Fibre pattern for inner flanges. Figure 3.6: Fibre patterns possible using a manual glue-gun along each strand, stopping before reaching the tape holding down the fibres in place. Finally a cotton swab was used to push down the glue and fibres to the surface of the steel to ensure a good bond. The glued up beam was then left to cure for a minimum of four days before it was tested. After the glue had dried, the ends of the sections of interest were marked using a touch-to-locate method in which the program can locate and mark gauges along the length of the fibres. It works by applying a local pressure with to the end of the area that is to be measured on a unloaded fibre which causes a local spike in the strain data that the program will notice and mark as an reference point. In cases where the software had issues with marking the location due to disturbances in the fibre, the gauge was selected manually through visual inspection of the strain graph. In order to prevent lateral-torsional buckling, a set of brackets were manufactured and placed centred over the loading points. A chain was attached from each side of the bracket onto the testing rig itself and adjusted in tension to restrain lateral movement, see Figure 3.7. 17 3. Methodology Figure 3.7: Lateral support configuration Placed upon two cylindrical rollers, the beam was measured and adjusted to follow the specifications of the four-point bending test. Deflections were measured using linear variable differential transformers (LVDT) that were placed on the inside of the bottom flange at one side of the supports and at the outside of the bottom flange centred under the loading points. Once the beam was placed and adjusted in the rig, the fibres were routed away from the beam and connected to the sensor interrogator. In total, four channels were connected, each one checked to be in good condition before being tared to remove any strains generated from the handling of the beam as well as hardening of the glue. When everything was set up and working correctly, the recording of the strains was started and the test began. The strain data was measured with a frequency of 12.5 Hz and a gauge length of 5.2mm and after each test the data was exported as text files for further post-processing. The data retrieved was loaded into Matlab and reduced using a Matlab script pro- vided by the institution which averaged every 12 time step together, resulting in roughly 1 value per second for every gauge. From the reduced data, the strain data between the previously marked gauges of each strand were extracted using a Matlab script shown in Appendix A. For each strand, the median strain for each time step was calculated from the extracted data and saved as a row vector with the first ele- ment in the vector being the strands x- or y-coordinate, depending on if the surface it was on was horizontal or vertical. The row vectors for each strand were then saved in arrays with strands of similar location on the cross section, e.g., left side of the web or topside of the top flange. Furthermore, the end of the data in time was cut off to exclude noisy data as well as the strain measurements from unloading, after which the time was normalised again. The cut-off-time was the same for every location on the same beam. See Figure 3.8 for an example and Appendix B for all original and cut versions of the strain data. 18 3. Methodology The deflection at the cut-time step to be used in Abaqus was calculated and noted by taking the average deflection below the two point loads and subtracting the average deflection of the vertical supports from it according to Equation 3.1. (a) Full data (b) Cut data Figure 3.8: Example of cut data from HEA1 bottom flange. dAbaqus = (dPointLoad1 + dPointLoad2) 2 − (dSupport1 + dSupport2) 2 (3.1) 3.2 Tensile strength test of the material In order to implement the strain data obtained from testing, the corresponding ma- terial properties for each beam were needed. As each set of profiles, HEA or HEB, all were from the same original beam only one of each beam was used to gather the testing stock. Sections of the flange and web located inside the supports were cut out using a plasma cutter. From the testing stock each test specimen or dog-bone was cut out using a water jet cutter. During the cutting process the water expands out of the nozzle causing a wider cut as it cuts through the material, resulting in a rectangular cross-section that will turn into a parallelogram. After the specimens were cut out, they were measured using digital calipers to calculate the true cross- sectional area, as well as to see if the other dimensions were according to standard. The tensile test was performed according to SS-EN ISO 6892-1:2019 with an exten- someter with a gauge of 50 mm. HEA flange specimen 1 was tested first and it was determined that the rust on the grip areas of the specimen had to be removed before testing, to ensure a proper grip in the machine. Therefore, the results from flange specimen 1 were ignored. For the other tests the strain and force data were collected with strain given in mil- limetres. To get the strain, the displacement data was divided by the extensometer’s gauge length, and to get the stress, the force was divided by the measured cross- sectional area. The mean of the strain-stress relation for the specimens of the same 19 3. Methodology type were then calculated and based on the mean relation, Young’s modulus and the plastic behaviour were determined. See Appendix C for the calculation script. 3.3 Python script for Abaqus-model and post pro- cessing The creation of the Abaqus model and the post processing was done through a Python script, which can be seen in Appendix D. This was done so that later changes of the parameters of the beam, more specifically the parameters describing the resid- ual stress field could be changed without recreating the whole model manually. In addition, using a script allowed Abaqus to run without a graphical user interface (GUI), which reduced the time of the analysis. The structure of the script can be divided in to two sections: creation and execution of functions for analysis, and comparison of data. The former section follows the overall structure of an analysis in Abaqus, beginning by creating the parts, mate- rial and instances, moving on to assembly and later loads. Finally the instances in the assembly were seeded for the mesh and element type was decided before being meshed and analysed. In the latter section of the script the data from the analysis is compared to the data from the physical tests and an "error" is calculated as the sum of the absolute value of the difference in strain for the measured points. In Figure 3.9 a simplified flowchart of the script(s) are visualised . For the digital model some simplifications were made. Half of the beam was mod- elled and a symmetric boundary condition was applied to what would normally be the mid-section of the beam. All displacements and rotations out of the symmetry plane were constrained and the rest were set free. The parts of the beam outside the vertical supports were not modelled and the loads were applied over a line rather than an area as in the physical tests. Additionally, the vertical supports were only modelled as a vertical constraint rather than modelling the rollers with contact points. The lateral support were also simplified by modelling it as a fixed horizontal constraint in a line at the ends of the top flange at the location of the point loads, when in reality the beam were allowed to move slightly and the support had a rect- angular contact area with the beam. Furthermore, the cross section was modelled as an ideal beam with dimensions based on the average measured dimensions where both flanges were modelled with the same thickness. See Figure 3.10 for a visual representation of the model. The beam was modelled with a combination of shell elements and solid elements. The parts where plastic behaviour was expected to occur were modelled with solid elements, while the parts outside the point loads and towards the supports were modelled with shell elements. This was done to increase the level of detail through out the thickness of model in the area of interest while having lower resolution in other areas by using shell elements. 20 3. Methodology Figure 3.9: Simplified flow chart of Abaqus and error calculation script 21 3. Methodology Figure 3.10: Shell and solid assembly with visible supports and symmetry condi- tions. Elements used in the solid section were hexahedral with 8 nodes (C3D8) with a seed size of 5 mm whereas the shell elements were triangular (S3) with a varying seed size. At the edge between the solid and the shell sections the seed size was 5 mm while at the edge by the support the seed size was set to 50 mm. This resulted in a gradually increasing element size from the solid edge to the support. No rigorous sensitivity analysis were performed of the model, instead simulations for different seed sizes in the beginning of the development together with the need for a non- linear relationship between strain at the outside and inside of the flanges quickly guided the model to the seed sizes used. The residual stress was applied through predefined stress fields and only to the solid elements. Instead of going through each element and applying a value, sets of elements containing the same x- and y-coordinates were created to reduce the computational load during data entry. To determine the residual stress for each set, modified versions of Skiadopoulos equations, mentioned in Section 2.3 were used without the set values for residual stress coefficients a and c. The equations were modified such that the coordinates origin was in the middle of the cross section which resulted in Equations (3.2) where a, b, c and d are constants. Instead of the values for a and c given by Skiadopoulos, residual stress coefficients c and d were assumed to be known, wherefrom a and b could be determined using Equations (3.2c) and (3.2d). σ0,f (x) = a + bx2 (3.2a) 22 3. Methodology σ0,w(y) = c + dy2 (3.2b) (2tfbf )a + ( 2tfb3 f 12 ) b + [tw(h − 2tf )]c + [ tw(h − 2tf )3 12 ] d = 0 (3.2c) a = c + d (h − tf )2 4 (3.2d) With all constants being known, each set of elements were given their residual stress based on their coordinates and Equations (3.2a) and (3.2b). In Figure 3.11 and 3.12 the principal idea for the sets for residual stress are shown as well as the residual stress state of the beam right after application. Figure 3.11: Example view of an element set for residual stress application. 23 3. Methodology Figure 3.12: Residual stress distribution. The analysis was split into three steps. In the first step the applied residual stress were allowed to reach self equilibrium before any loading was applied. In step two the gravitational load was applied using the gravity type load which is based on the density assigned to the material. The densities of the beams were calculated to 7.845 · 10−9 tonnes mm3 and 7.850 · 10−9 tonnes mm3 for the HEA and HEB beams respectively, based on the specifications given by the supplier, showwn in Appendix E. The den- sity was calculated to tonnes mm3 as to match with the units used in Abaqus. During the last step the two line loads were applied using deformation control. The dis- placements applied to the loading lines were applied linearly until the deflection at the lines were equal to the deflection from the physical test specified by Equation 3.1. After each simulation, the strain data of interest had to be retrieved. The data was retrieved using 8 paths along the web and flanges at the symmetry edge, i.e. in the middle of the full beam, see Figure 3.13. From the paths the longitudinal strains (E33) were extracted for each time frame in the loading step and the strain from the steps before were then subtracted from the extracted strains to simulate the tare made in the physical tests. The strain data was then compiled in arrays, one for each path. The arrays’ first columns held the X- or Y-coordinates along each path and the following columns held the respective strain for each time frame. Whether it held X- or Y-coordinates were dependant on if the surface was horizontal of vertical. To determine how well the modelled beam with its assumed residual stress distri- bution depicted the real behaviour, the strain data was compared to the physical test data. To compare the two data sets, the data first had to be formatted such that the two matrices in each location had the same number of coordinates and time frames. The set with the highest resolution was reduced to match the one with lower resolution by firstly checking for direct matches of coordinates/time frames and otherwise interpolating the data between two nearby values to match the coor- 24 3. Methodology Figure 3.13: Location of paths from where longitudinal strain is extracted in Abaqus marked in red. dinate/time frame. This resulted in that the final dimensions were based on Abaqus time frames and the physical tests coordinates. When the data sets had the same dimensions, an error was calculated for every location by first subtracting the two matrices for the location and taking the ab- solute value of each element. The sum of all the elements in the resulting matrix was then calculated and the resulting number was the error of the location. The eight locations’ error were summed up and saved in a text file. The error calculation could be simply described by Equation 3.3. For an example of the reshaping and error calculation process see Figure 3.14, where dimensioning coordinates are circled in blue, dimensioning time frames are circled in red, and strain data used for error calculation is underlined in green. Error(c,d) = ∑ i ∥εmeasured.i − εF EM.i(c,d)∥ (3.3) 3.4 Optimising routine In order to obtain the residual stress distribution for the beams that were tested, an optimisation routine was implemented using a Nelder-Mead algorithm. This routine was created in a separate script from the one implementing and running the Abaqus model as well as post processing the results. The script is shown in Appendix F. Using SciPy Library’s minimise function, the script extracts the error from com- paring FE strain data and real strain data from the Abaqus script as well as the 25 3. Methodology Figure 3.14: Example of data reshaping and error calculation for one location. The dimensioning coordinates are circled in blue and dimensioning time frames are circles in red. After reshaping the data sets are of the same dimensions. The strains used for calculating the error are underlined in green. residual stress coefficients used for each simulation. The algorithm tries to minimise the error by changing the residual stress coefficients as described in Section 2.4. The tolerances for the Nelder-Mead algorithm were set to 0.01 for input values and 0.0001 for the error which meant that the change in input and error from the pre- vious assumption had to be lower than its respective tolerance to consider that the optimisation algorithm had converged. When both criteria were met, the final resid- ual stress coefficients were assumed to describe the true residual stress distribution. A flowchart of the procedure is shown in Figure 3.15. 26 3. Methodology Figure 3.15: Flow chart of the inverse analysis To reduce the possibility that the algorithm closes in on a local minima or not ex- panding its search from an initial guess of the residual stress coefficients, a staring simplex was implemented in the script. By forcing the initial search area to cover a larger area, it will force the algorithm to change the coefficients in a more volatile fashion for each simplex node, compared to a single initial guess. In Figure 3.16 a visualisation of the coverage obtained from using starting simplexes is illustrated. In total, three starting simplexes were implemented for each beam to further reduce the potential of the script finding local minima. Figure 3.16: Coverage of starting simplexes with residual coefficient c on the x- axis and residual coefficient d on the y-axis. 27 3. Methodology 3.5 Validation of method To validate the developed method of estimating the residual stress coefficients, sim- ulated test data were generated for an ideal beam and then the routine was run once for each simplex mentioned in Section 3.4. If the routine would be able to reach the residual stress parameters used for creating the simulated test data multiple simplex, the routine could be considered valid. The simulated test data was created with the measurements of an ideal HEB200 beam loaded until 30 mm deflection at the point loads and the residual stress parameters c and d were set to -88.75 and 0.02 respectively. For web and flanges Young’s modulus was set to 200 GPa and the plastic behaviour was described by Table 3.1. Stress [Mpa] 355 420 570 610 640 675 Plastic strain [-] 0 0.2 0.4 0.6 0.8 1 Table 3.1: Plastic behaviour used when creating the simulated data. For the first simplex the routine successfully found the original values ending the optimisation at c = −88.7538 and d = 0.020001. Using the second simplex the origi- nal values were also found, ending at c = −88.7469 and d = 0.019999. The run with the third simplex was also successful and the optimisation ended at c = −88.7542 and d = 0.020001. In Figure 3.17 the changes of the residual stress coefficients and error throughout the iterations for the three simplexes are shown and in Table 3.2 a summary of the runs are shown. Figure 3.17: Validation of model simulations, coefficients through time According to the validation attempts, the method works. It finds the original values of c and d even with starting simplexes that does not initially contain the optimal value. 28 3. Methodology Beam Simplex Number of iterations c d Total error Validation run 1 69 -88.7538 0.020001 0.001148 2 67 -88.7469 0.019999 0.001147 3 67 -88.7542 0.020001 0.00115 Mean 67.667 -88.7516 0.02 0.001148 Table 3.2: Summary of the validation runs with the three simplexes 29 3. Methodology 30 4 Results In this chapter the results from all parts of the project will be presented. 4.1 Beam dimensions In Table 4.1 the average thicknesses of the beams are shown. For all measurements see Appendix G. Beam Average flange thickness [mm] Average web thickness [mm] HEA1 9.522 6.775 HEA2 9.529 6.708 HEA3 9.500 6.750 HEB1 14.943 9.025 HEB2 14.953 14.958 HEB3 14.965 9.050 Table 4.1: Average thickness of the flanges and web for all tested beams. 4.2 Material model Based on the data obtained from the tensile tests described in Section 3.2, four stress-strain curves were obtained that are used to describe the material properties of the test specimen. The specimens cross-sectional dimensions are shown in Table 4.2 and the stress-strain curves are shown in Figures 4.1-4.4, where the specimens’ curves are shown as dashed lines and their mean value, which is used for analysis in Abaqus, as a black line. For the HEB flange, the second specimen was removed from the mean calculation since it differed greatly from the other two specimen due to issues during the testing procedure. The determined Young’s modulus for the four mean curves are shown in Table 4.3. 31 4. Results Tensile test specimen Cross-sectional area, units in milimeters [mm] b1 b2 h1 h2 b_mean h_mean Area [mm2] AF1 17.94 18.08 9.89 10.06 18.01 9.975 179.650 AF2 18.16 18.07 9.86 9.93 18.115 9.895 179.248 AF3 17.66 17.87 9.56 9.36 17.765 9.46 168.057 AF4 17.77 17.93 9.35 9.24 17.85 9.295 165.916 AW1 20.41 20.43 6.78 6.82 20.42 6.8 138.856 AW2 20.39 20.47 6.83 6.68 20.43 6.755 138.005 AW3 20.37 20.47 6.89 6.73 20.42 6.81 139.060 BF1 11.78 11.9 15.21 15.2 11.84 15.205 180.027 BF2 11.79 11.86 15.22 15.21 11.825 15.215 179.917 BF3 11.7 11.95 15.2 15.23 11.825 15.215 179.917 BW1 18.87 18.91 9.21 9.22 18.89 9.215 174.071 BW2 18.9 19.03 9.14 9.14 18.965 9.14 173.340 BW3 18.85 19.03 9.08 9.11 18.94 9.095 172.259 Table 4.2: Measurements of tensile test specimens. A denotes a HEA-profile, B a HEB-profile and F or W represent flange or web respectively. Cross-section part Young’s modulus, E [GPa] HEA web 201.132 HEA flange 203.580 HEB web 184.166 HEB flange 220.328 Table 4.3: Young’s modulus Figure 4.1: HEA web strain-stress curve 32 4. Results Figure 4.2: HEA flange strain-stress curve Figure 4.3: HEB web strain-stress curve 33 4. Results Figure 4.4: HEB flange strain-stress curve 4.3 Test data The force-displacement curves from the four point bending tests of the six beams are shown in Figure 4.5. 34 4. Results Figure 4.5: Force-deflection curves for all six beams. Deflection is the calculated deflection according to equation 3.1. As earlier mentioned, the results from the physical tests had to be cut to remove unreasonable data. An example of full and cut strain data from the tests is shown in Figure 4.6, the rest of the strain curves are show in Appendix B. The time of the cut and what the calculated deflection at the cut-time was, for each beam, is shown in Table 4.4. This deflection is the the deflection assigned as a displacment controlled load to the Abaqus model. (a) Full data (b) Cut data Figure 4.6: Example of cut data from HEA1 top flange. 35 4. Results Beam name Time of data cut [s] Deflection at time of cut [mm] HEA1 500 33.907 HEA2 440 29.360 HEA3 460 30.287 HEB1 480 29.193 HEB2 520 32.454 HEB3 500 28.530 Table 4.4: Time of data cut for the different beams and the deflection at the cut time. 4.4 Evaluation of coefficients through time Results from the optimisation through time is shown in Figures 4.7-4.12, where the x-axis is the evaluation number of the Nelder-Mead algorithm. This is used in evaluating the effectiveness of the algorithm and the impact of using different starting simplexes. Figure 4.7: Simulation of HEA1 Figure 4.8: Simulation of HEA2 36 4. Results Figure 4.9: Simulation of HEA3 Figure 4.10: Simulation of HEB1 Figure 4.11: Simulation of HEB2 37 4. Results Figure 4.12: Simulation of HEB3 4.5 Optimisations results In the Tables 4.5 and 4.6 the results from the simulations of the final routine is shown. Several attempts for each beam were done using different initial assumptions as shown in Figure 3.16 in a effort to avoid a local minimum. Standard deviation is given as a value and not a percentage of deviation. Error is given as the sum of all errors in time and space. Beam Simplex a b c d Total error HEA1 1 5.393 0.01197 -247.241 0.02952 0.08301 2 139.564 -0.02622 -337.165 0.05570 0.08001 3 139.503 -0.02620 -337.150 0.05687 0.08002 Mean 94.820 -0.01348 -307.185 0.04736 0.08101 HEA2 1 -0.998 0.02990 -467.428 0.05728 0.04073 2 -0.842 0.02982 -467.042 0.05726 0.04073 3 -0.840 0.02982 -467.074 0.05726 0.04073 Mean -0.893 0.02985 -467.181 0.05727 0.04073 HEA3 1 -43.643 0.03094 -303.734 0.03036 0.06652 2 -44.553 0.03116 -303.713 0.03028 0.06652 3 -43.988 0.03098 -303.731 0.03035 0.06652 Mean -44.061 0.03103 -303.724 0.03032 0.06652 HEA total mean 16.622 0.01580 -359.363 0.04498 0.06275 HEA standard deviation 71.078 0.02536 93.389 0.01363 0.02040 Table 4.5: Results of simulation for HEA-profiles 38 4. Results Beam Simplex a b c d Total error HEB1 1 118.806 -0.03105 -129.496 0.02901 0.02749 2 118.818 -0.03105 -129.427 0.02900 0.02749 3 118.771 -0.03105 -129.312 0.02898 0.02749 Mean 118.798 -0.03105 -129.412 0.02900 0.02749 HEB2 1 61.691 -0.01601 -67.899 0.01514 0.03996 2 61.660 -0.01607 -67.918 0.01514 0.03996 3 61.663 -0.01608 -67.890 0.01514 0.03996 Mean 61.671 -0.01605 -67.902 0.01514 0.03996 HEB3 1 151.933 -0.04120 -138.555 0.03394 0.03830 2 152.137 -0.04123 -139.159 0.03403 0.03830 3 151.847 -0.04118 -138.379 0.03391 0.03830 Mean 151.972 -0.04121 -138.697 0.03396 0.03830 HEB total mean 110.814 -0.02944 -112.004 0.02603 0.03525 HEB standard deviation 45.677 0.01265 38.474 0.00975 0.00677 Table 4.6: Results of simulation for HEB-profiles Figure 4.13: Residual stress distribution of HEA based on mean coefficients in Table 4.5 . 39 4. Results Figure 4.14: Residual stress distribution of HEB based on mean coefficients in Table 4.6 . 4.6 Comparison between finite element and real strain Strain data obtained from Abaqus simulation using the final residual stress coeffi- cients presented in Table 4.5 and 4.6 is compared with its corresponding data from testing in Figures 4.16-4.23. The curves in these figures were the ones used for the final error calculation of HEB1. In Appendix H, the curves for all the beams can be seen. As seen in the figures, the x-axis is the normalised time. To see how the deflection over time was for the physical test compared to Abaqus, see Figure 4.15. All data acquired from testing starts at a strain of zero, as fibres are tared before testing is started, see Chapter 3 for further explanation. As time moves forward, it can be observed that the curves increasing in strain will at a certain point decrease or stay at the current strain level for a while until the strain rate will increase once again. This happens as the beam goes from elastic to plastic behaviour and stress redistribution occurs throughout the cross-section. 40 4. Results Figure 4.15: HEB1 deflection over time for Abaqus and physical test. Figure 4.16: Final strain comparison for HEB1 top flange 41 4. Results Figure 4.17: Final strain comparison for HEB1 left inner top flange Figure 4.18: Final strain comparison for HEB1 right inner top flange 42 4. Results Figure 4.19: Final strain comparison for HEB1 left web Figure 4.20: Final strain comparison for HEB1 right web 43 4. Results Figure 4.21: Final strain comparison for HEB1 left inner bot flange Figure 4.22: Final strain comparison for HEB1 right inner bot flange 44 4. Results Figure 4.23: Final strain comparison for HEB1 bot flange 45 4. Results 46 5 Discussion In this chapter the choices and observations made in preparing, testing and modelling the beam are discussed as well as how they might have affected the results. In addition, the resulting residual stress distributions are discussed and analysed. 5.1 Test preparations and execution In preparing the beams for testing, several actions and decisions may have impacted the final results obtained during testing. From the level of rust removal, thickness variations, to the cleanliness of the fibres and the tools used. One of the main concerns during preparations was how to not impact the level and distribution of residual stresses in the beams before the tests were performed. Therefore, some criteria were set up beforehand: the use of angle grinders with a grinding wheel was not to be used and final sanding was to be done by hand. In the end, time constraints forced the abandonment of those criteria in order to move on to fibre preparations due to the long hardening times of the two-part glue used to attach the fibres to the beams. However based on the strain data obtained from the testing it is hard to say if the rushed cleaning process on four of the beams caused in any way a shift in results compared to the two first beams that were cleaned (HEA3 and HEB3). A far greater impact could be argued was the over- all dimensional differences in the beams regarding different flange thicknesses and flange shape of each beam. The beams delivered to Chalmers structural lab were all delivered at various stages of corrosion. The main impact was that the cleaning of the HEA beams that were delivered took far longer than anticipated as well as the extra dust generated during the process contaminated a large area of the lab. Furthermore, local occurrences of pit corrosion on the outside of the flanges forced the need to remove more material than needed to create a flat and clean surface to attach the optical fibres. In Fig- ure 5.1 pit corrosion on both the flanges as well as some areas of the web that are impacted are shown. Due to the complexity of modelling the effects from the preparations of the beams, the potential loss or redistribution of residual stresses caused by it has not been measured. During rust removal the use of power tools was needed to complete the process in time to perform the test. This caused in certain situations heating of local 47 5. Discussion Figure 5.1: Pit corrosion located on the flanges of one of the HEB beams. areas as the machines gripped the surfaces, dug down and in other ways generated heat that could have altered the residual stress. Another problem created by the way the beams were prepared was the changing thickness of the flanges and in smaller part the thickness of the web in the centre of the beam due to material removal. Described further in the next section discussing the creation of the model, the mate- rial removal caused the need to measure the new dimensions to be used in the model. As previously mentioned, three of the beams tested (HEA2, HEA3, and HEB3) did buckle to some extent during testing, forcing the tests to be terminated early due to risk of damage to equipment or personnel at or near the test rig. This inherent insta- bility was known beforehand but estimated through calculations and FE-simulations to occur later than they appeared during the tests. It would be easy to assume that if the chains that held the bracket for lateral stability were not tensioned as they were after the first test, all subsequently beams would have followed the same tra- jectory. For the second test the beam did not buckle, it was in fact during the third test and fourth test that the beam started to show signs of lateral-torsional-buckling again and the tests were forced to terminate early. Several reasons can be given as to why buckling may have happened. Lower accu- racy during the installation of the first tested beams in the test rig is one and the asymmetry of the beams is another potential reason. For all the tests where the beam did not buckle it was observed that the bracket were taking up some amount of lateral force as it was a notable tension in the chains for some of the beams. One exception was the last HEB tested (HEB1) that did not seem to move or affect the tension of the chains connected to the bracket at all, instead it deformed straight 48 5. Discussion along its vertical axis until the maximum deflection was reached. It is interesting to note that during all the other tests, regardless if the beams buckled or the forces were taken up by the bracket, all beams tended to move or buckle in the same direction of its axis. It is speculated that this happened because of the impact of one or several of the following factors: the small tapper of the HEA beams on the flanges being different in size between each side, the angle of attack during the cleaning process, difference in weight of the chains connected the bracket, small imperfections of the load cell forcing the beams to the right. However the most plausible explanations is just that all the beams were slightly off the centre during the tests. To remove the issue of LT-buckling more robust lateral supports could have been made which could restrain the bottom flange as well. During the tests it was noticed that after an area had yielded, the strain data in that area were not behaving according to expectations for steel during loading. This could potentially have some thing to do with the glue used, that it lost adhesion to the beam when the steel plastically deformed or that it was too stiff, resulting in damaged fibres. There was however no visual sign of the glue losing its adhesion to the beam during or after the tests. Regardless, the result of it was that the full data could not be used and instead had to be cut. Further experiments regarding different glues or other methods of attaching the fibres would have been necessary in order to evaluate if the attachment method, described in Section 3.1, was adequate for the task. The deflections at the supports measured during the experiment might not be too accurate since the LVDT sensors were placed at the rollers. For the support which had the fastened steel cylinder the accuracy of the deflection was only dependant on how centred the placement of the sensor was, but the other support’s steel cylinder were allowed to move which introduced another error source. The movement of the cylinder would increase the deflection measured since the point of the LVDT sensor would slide down the side of the cylinder, giving inaccurate results. An illustration of the issue is shown in Figure 5.2. However, the error does not have a large impact on the final result as the deflection at the supports were 60-100 times smaller than the deflection at the load positions. In addition, the error’s influence is reduced further by the fact that only the average support deflection is used for calculating the deflection used in Abaqus. 5.2 Determination of material properties from ten- sile tests Uncertainty in determining the cross sectional area of the dog bones may have af- fected the calculated stress and in turn the Young’s modulus. The measurements were made with digital calipers which had a precision of ±0.01 mm but due to the uneven surfaces of the cross-section the measurements had greater error margins. To improve the accuracy, more measurements of the specimens could have been taken or another more precise way of measuring them, like 3D-scanning, could have been 49 5. Discussion Figure 5.2: Principal sketch of error in measuring the deflection at the roller support. carried out. When determining the Young’s modulus the result also depended greatly on what two points were chosen as reference points for the calculations. The Young’s mod- ulus for two different choices could differ by as much as tens of gigapascal. For the HEB profile the difference between the flange and webs elastic moduli were mea- sured to be 40 GPa, a difference that seems too large to be attributed to variability of the material. The cause of this could be that the specimens were not properly gripped by the tensile test machine, as the flange parts of the HEB profile were so thick that they had to be machined down around half a millimetre to fit in the wise. Other reasons could be down to the accuracy of the cross-sectional measurements or attachment of the exstensometer. 5.3 Numerical model In this section the process from the first model to the one that was used in the final calculations and the choices that were done along the way are discussed and examined. The first iteration of the model consisted of one solid part, a single sketch of the cross-section extruded to cover the length between the supports then assigned a simple material model, see Figure 5.3. Partitioned to apply a displace- ment controlled boundary condition, and a single time step was created. Coding wise the model was simple, only consisting of a few lines of code allowing it to be easily changed and understood. However it had its drawbacks, the model was far too slow to be practical, created out of only solid elements with a seed size of 5mm it consisted of 196000 elements. 50 5. Discussion Figure 5.3: Original model To alleviate the load of the calculation created by the massive amount of elements in the first model some changes were needed. The first step was to make use of symmetry and halving the beam in two. From there it was decided to take the element reduction even further by implementing the use of shell elements. Applying residual stress were done per element and to reduce computing times the use of sets was possible to apply residual stress to several elements at the same time. As the stress for a certain position in the cross-section will be the same throughout the length of the beam, using solid hexahedral mesh elements allows the script that is based upon element numbering to select the elements with the same cross-section coordinates in one go and then assign a pre-calculated residual stress value to that set. If the solid section had been made with a tetrahedral mesh the mesh generation changes to a a more unstructured pattern and the script would break. Regarding the mesh size for the central section, i.e the solid meshed part, the amount of residual stress information that can be applied and extracted from the model is based on the number of elements. The choice was made so that at least two elements were needed through the thickness of the web as well as two elements throughout the flanges. In doing this it was possible the extract and compare data from the model with the data collected from the physical tests without having a linear relationship of the strain and stress between the two sides of the flange and web respectively. For the shell part of the beam, focus was on matching the seed size to that of the solid and the other end was then manipulated to generate the least amount of ele- ments. In this case it would be easy to say that the size of the first element should have a size of half the flange width. However, this generates more elements com- pared to the starting seed with a quarter of the flange width. For the HEB model a seed size of 100mm generated 9817 elements for the shell part whereas using a seed size of 50mm generated 7057 shell elements, in Figure 5.4 and 5.5 the pattern of mesh generation for both scenarios is visualised. 51 5. Discussion Figure 5.4: Mesh generation HEB with edge seed 100mm, 9817 shell elements Figure 5.5: Mesh generation HEB with edge seed 50mm, 7057 shell elements This happens as the edge with the larger starting seed creates such a high constraint for large elements, combining that with rules for the angles between each side of the element results in that too large elements is forced to be created and leaving a lot of empty space in between. Without breaking the rules of the angles the mesh will simply not create one or two elements to fill the void, instead it will slowly approach the edge of the part with smaller and smaller elements. The seed size was optimised for the HEB model and was not changed for the HEA model. This resulted in that the HEA model had less elements than the HEB model which could have impacted the HEA results negatively. However, it still has two elements through the thickness of the web and flanges which should still yield good results. Running the established model is a balancing act between the size of the mesh ele- ments and the maximum size of each time step during simulation. If the time-steps 52 5. Discussion are too large, in the best case data will be lost or buckling will appear. In the worst case the simulation will terminate prematurely due to not finding convergence in the iterations. Using a seed of 5 millimetre for the solid section, the maximum time- step that was able to run without crashing was a step of 0.05 seconds. However, a lot of individual steps were needed to be recalculated resulting in the overall time of the run increasing. A simplified explanation of this process is as follows: if the simulation could not calculate the step it would first try to recalculate it for a set number of times to try and find convergence before in would decrease the size of the time step and repeat the previously step. This process continues until the minimum step size is reached, where it either finds a solution or terminates the simulation. In the end it was decided to run the optimisation routine with the step size of 0.05 to have a shorter run time for each simulation. If one would want a more accurate solution the step size could be decreased, especially in the expected plastic region. The same could be said for the mesh size, that for a more accurate result the mesh size could be reduced at cost of the run time. As for the dimensions of the beam in the model, the average measured thickness of the web and the combined average of the flanges was used to model the respective parts. The observed change of thickness, specifically for the flanges, was ignored in the model as the equations used to describe the residual stress distribution only are valid for double symmetric cross sections with uniform thickness. In assuming an ideal cross-section the accuracy of the results will differ from a beam with true dimensions where both deformation for the same load as well as residual stress dis- tribution are impacted. Better results could have been produced if a more complex model of the residual stress, that would be able to describe the residual stress distribution in non-ideal beams, was used. In addition, even for an ideal beam the assumption of a parabolic relation might not be correct. As mentioned in Section 2.3 there are several models for the residual stress. In the Abaqus model the deflection applied at the supports was set to be linear up to the maximum deflection calculated. In reality the hydraulic jack had a linear deflection while the load positions actual deflection were dependant on the supports deflections as well. This results in a close to linear but nonetheless non-linear de- flection over time, shown in the Figure 4.15. For a better result the time-deflection measured during the physical test could have been used in Abaqus, to ensure that the deflection in the model changes in the same way as in the tests. 5.4 Post processing In processing the data from the simulations run in Abaqus and the data collected during testing as presented in Chapter 3 certain simplification or truncation of the data collected were necessary in order to easily be able to review and work with it. Looking at a single strand, for every 5.2 mm along its length a strain measurement 53 5. Discussion was collected at a sampling rate of 12.5 Hz, which for a 10 minute test means that a single 200 mm long strand will collect over 288 thousand data points. Extending this calculation to all the strands for all the beams and the amount of data to be handled would be far too much. The first step in reducing the amount of data was done by selecting the largest gauge length available in the Odisi software of 5.2 mm. After that the data was reduced in time by taking the average of every 12th time frame. The choice of taking every 12th frame was made since the sample 12.5 Hz and by averaging every 12th frame it would the result in roughly one time step per second. Due to the uncertainty if data could be collected for every point along the strand, be it caused from bad connection to the surface during gluing, release during yielding or in any other way stopped capturing strain, it was decided to take the median strain value instead of the mean strain value from each strand. Discrepancies in the data had therefor less impact on the final results since the mean value is more sensitive to outliers a than the median value. As clearly visualised in Figure 3.14 there was a need to match the strain data ob- tained from testing with the one extracted from Abaqus. Each data set consists of two matrices, were every row corresponds to a spatial position on along the cross- section and every column corresponds to time-step. The time here is normalised for each beam so the first and final column in both the data from Abaqus and the tests are the same, the only difference is the amount of steps from start to end. In the script the data is processed using interpolation to create two matrices that are of the same size. With interpolation it is assumed that between two existing positions there is a linear relationship, increasing time steps would then lead to that the strain behaves in a linear fashion which it does as long as it is in the elastic zone. However, when strain moves into the plastic zone and the material starts to yield, the relationship changes which results in that the interpolated data will create an approximation error between the real measured points. In the same way interpolating spatial positions over the cross-section will assume a behaviour for sections of the beam that does not mirror reality. The plastic redis- tribution that occurs during deformation of a member is not strictly linear through the cross-section. Taking this into consideration, it was of interest to interpolate as short distances as possible in both time and space when modifying the size of the data. Hence, the dimensions were modified to match the spatial and time data to the matrix size with the least amount of data for each category. This resulted in, as previously mentioned, that the spatial data collected from Abaqus is reduced into points matching the position of the strands on the beams and the data through time collected from the fibres is reduced to match the time for each step that the Abaqus analysis created. 54 5. Discussion 5.5 Optimisation procedure There is no guarantee that the global minimum has been found since the Nelder- Mead method finds local minima as well. To be more confident in the results, more initial simplex would have to be run through the routine. A major drawback of the chosen optimisation algorithm for this thesis is, as dis- cussed earlier in this chapter, the fact that the basic Nedler-Mead simplex lacks a ’kickback’ feature. The ’kickback’ is a part of an algorithm where it once in a while it takes a step back and makes a wild guess outside of the now established simplex to see if it has worked itself into a local minima or in any other way forced itself into a locked position. The process of the kick is not more different than the reflect step discussed in Chapter 2 but the change in values or put in other words the distance it travels, is far greater than that original step. Comparing all of the data points in time and space to determine how well the esti- mation describes reality is somewhat redundant since any estimation of the residual stress, results in the same behaviour in the elastic zone. The method should work equally good by only checking the strains for one time frame in the plastic region. If the last time frame would be compared, there would be no need for temporal interpolation which would slightly speed up the error calculation. No boundaries for the residual stress coefficients were given to the Nelder-Mead algorithm which resulted in that the algorithm could produce coefficients