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Lateral-Torsional Buckling of Steel Beams with Open Cross Section Lateral-Torsional Buckling of Steel Beams with Open Cross Section Elastic Critical Moment Study and Software Application Master of Science Thesis in the Master’s Programme Structural Engineering and Building Technology HERMANN ÞÓR HAUKSSON JÓN BJÖRN VILHJÁLMSSON Department of Civil and Environmental Engineering Division of Structural Engineering Steel and Timber Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2014 Master’s Thesis 2014:28 MASTER’S THESIS 2014:28 Lateral-Torsional Buckling of Steel Beams with Open Cross Section Elastic Critical Moment Study and Software Application Master of Science Thesis in the Master’s Programme Structural Engineering and Building Technology HERMANN ÞÓR HAUKSSON JÓN BJÖRN VILHJÁLMSSON Department of Civil and Environmental Engineering Division of Structural Engineering Steel and Timber Structures CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2014 Lateral-Torsional Buckling of Steel Beams with Open Cross Section Elastic Critical Moment Study and Software Application Master of Science Thesis in the Master’s Programme Structural Engineering and Building Technology HERMANN ÞÓR HAUKSSON JÓN BJÖRN VILHJÁLMSSON © HERMANN ÞÓR HAUKSSON & JÓN BJÖRN VILHJÁLMSSON, 2014 Examensarbete / Institutionen för bygg- och miljöteknik, Chalmers tekniska högskola 2014:28 Department of Civil and Environmental Engineering Division of Structural Engineering Steel and Timber Structures Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: + 46 (0)31-772 1000 Cover: Lateral-torsional buckling of a monosymmetric I-beam, as illustrated by the commercial structural engineering software SAP2000 Chalmers Reproservice Göteborg, Sweden 2014 Lateral-Torsional Buckling of Steel Beams with Open Cross Section Elastic Critical Moment Study and Software Application Master of Science Thesis in the Master’s Programme Structural Engineering and Building Technology HERMANN ÞÓR HAUKSSON JÓN BJÖRN VILHJÁLMSSON Department of Civil and Environmental Engineering Division of Structural Engineering Steel and Timber Structures Chalmers University of Technology ABSTRACT The elastic critical moment is an important design parameter when it comes to lateral-torsional buckling of steel beams. Lateral-torsional buckling is influenced by number of factors and thus structural engineering software often produce different results, based on different assumptions. A monosymmetric I-beam was studied considering different load heights and different degree of end-restraint for rotation about the minor axis and warping. Beams with a channel cross section were studied considering different beam lengths, different types of load and different eccentricities of the loads, including centric loading. was calculated using various software and compared to values assessed by an analytical expression called the 3-factor formula. To obtain the elastic critical moment for eccentrically loaded channel beams, a number of methods were used. The difference between the software tools studied was based on different methods and assumptions they use to calculate . For example not all of them take the monosymmetry of the cross section into account while others only count on the beam bending moment diagram. Some use the finite element method while others use analytical expressions. The methods implemented in the current thesis were not successful in obtaining the elastic critical moment for eccentrically loaded channel beams. No clear instability behaviour was detected unless the beams were loaded in the vertical plane through its shear centre. The software tools used in this thesis are ADINA, Colbeam, LTBeam, SAP2000 and STAAD.Pro. Key words: Lateral-torsional buckling, elastic critical moment, 3-factor formula, warping, torsion, fork supports, monosymmetric I-section, channel section, ADINA, Colbeam, LTBeam, SAP2000, STAAD.Pro I II Contents ABSTRACT I CONTENTS III PREFACE VII NOTATIONS VIII 1 INTRODUCTION 1 1.1 Aim of the thesis 2 1.2 Objectives 2 1.3 Method 2 1.4 Limitations 3 2 THEORY 4 2.1 Beam modelling properties 4 2.1.1 Coordinate system 4 2.1.2 Degrees of freedom 4 2.1.3 Centre of gravity 5 2.1.4 Centroid 5 2.1.5 Shear centre 6 2.1.6 Torsion centre 8 2.1.7 Point of load application 8 2.1.8 Support conditions 9 2.1.9 Torsion 11 2.1.10 Warping 15 2.2 Buckling 19 2.2.1 Stability 19 2.2.2 Behaviour of ideal beams and real beams in bending 21 2.2.3 Local buckling 22 2.2.4 Distortional buckling 23 2.2.5 Global buckling 23 2.3 Energy methods 26 2.4 Elastic critical moment, Mcr 27 2.4.1 I-beams with doubly symmetric cross section 29 2.4.2 I-beams with monosymmetric cross section 29 2.4.3 Beams with channel cross section 32 2.5 Design of beams in bending for LT-buckling 33 2.5.1 Design methods in Eurocode 3 33 2.5.2 Beams with channel sections 40 3 SOFTWARE INTRODUCTION 42 3.1 ADINA 42 3.1.1 Elastic critical moment 43 3.1.2 Calculation of sectional parameters 46 CHALMERS Civil and Environmental Engineering, Master’s Thesis 2014:28 III 3.2 Colbeam 48 3.2.1 Elastic critical moment 49 3.3 LTBeam 52 3.3.1 Simple input mode 53 3.3.2 File input mode 54 3.3.3 Elastic critical moment 55 3.4 SAP2000 56 3.4.1 Elastic critical moment 56 3.5 STAAD.Pro 57 3.5.1 Elastic critical moment 59 4 SOFTWARE APPLICATION STUDY 63 4.1 Cases studied 63 4.1.1 Monosymmetric I-beam 63 4.1.2 Channel beams 66 4.2 Modelling methods 70 4.2.1 Modelling in Colbeam 70 4.2.2 Modelling in LTBeam 72 4.2.3 Modelling in SAP2000 75 4.2.4 Modelling in STAAD.Pro 78 4.2.5 Modelling with beam elements in ADINA 82 4.2.6 Modelling with shell elements in ADINA 91 4.2.7 Comparison of sectional parameters in different software 99 5 PRESENTATION AND ANALYSIS OF RESULTS 100 5.1 Centrically loaded beams 100 5.1.1 Monosymmetric I-beams 100 5.1.2 Channel beams 103 5.2 Eccentrically loaded channel beams 106 5.2.1 Expression of results 106 5.2.2 Relation between bending moment and different kind of displacements 107 5.2.3 Influence of the beam length 109 5.2.4 Influence of different load eccentricities 110 5.2.5 Influence of different types of load 111 6 FINAL REMARKS 113 6.1 Conclusions 113 6.2 Discussion 114 6.3 Further studies 116 7 REFERENCES 117 APPENDIX A NUMERICAL RESULTS 120 CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:28 IV A.1 Monosymmetric I-beam 120 A.2 Centrically loaded channel beams 121 A.3 Eccentrically loaded channel beams 122 A.3.1 Influence of different load eccentricities 123 A.3.2 Influence of different types of load 127 A.3.3 Influence of the beam length 129 APPENDIX B ADINA-IN COMMAND FILES 131 B.1 Monosymmetric I-beam 131 B.2 Centrically loaded channel beams 134 B.2.1 Concentrated load 134 B.2.2 Uniformly distributed load 138 B.3 Eccentrically loaded channel beams 142 B.3.1 Concentrated load with an eccentricity of yg = 5 mm 142 B.3.2 Uniformly distributed load with an eccentricity of yg = 5 mm 151 CHALMERS Civil and Environmental Engineering, Master’s Thesis 2014:28 V CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:28 VI Preface This master’s thesis project was carried out in 2014 between January and June at Reinertsen Sweden AB in cooperation with the Department of Civil and Environmental Engineering, Division of Structural Engineering, Steel and Timber Structures at Chalmers University of Technology in Göteborg, Sweden. Professor Robert Kliger at the Division of Structural Engineering was the examiner of the thesis. Supervisors of the thesis were Martin Gustafsson, Structural Engineer at Reinertsen Sweden AB and Professor emeritus Bo Edlund at the Division of Structural Engineering. To them we would like to give thanks for their continuous support and guidance during the thesis project. We also thank our opponents Bengt Lundquist and Guðni Ellert Edvardsson for the discussions they had with us about the subject and providing thoughtful comments, as well as proofreading the thesis. At last we thank the staff at Reinertsen Sweden AB for providing us a comfortable working environment and support, especially concerning the various commercial software tools employed in this thesis. Göteborg June 2014 Hermann Þór Hauksson & Jón Björn Vilhjálmsson CHALMERS Civil and Environmental Engineering, Master’s Thesis 2014:28 VII Notations Abbreviations CISC Canadian Institute of Steel Construction CTICM Centre Technique Industriel de la Construction Métallique DOFs Degrees Of Freedom ECCS European Convention for Constructional Steelwork FEM Finite Element Method GC Centre of Gravity LDC Load-Displacement-Control LT-buckling Lateral-torsional buckling NCCI Non Contradictory Complementary Information OP Origin of coordinate system PLA Point of Load Application SC Shear Centre TC Torsion Centre Roman upper case letters Area of the cross section Value to determine the factor using the closed form expression to calculate Wagner effect constant Value to determine the factor using the closed form expression to calculate Wagner effect constant Wagner effect constant Wagner effect constant Wagner effect constant Area of the flange in compression Area of the part of the web which is in compression Bi-moment Factor depending on the moment diagram and the end restraints Factor depending on the moment diagram and the end restraints, related to the vertical position of loading Factor depending on the moment diagram and the end restraints, related to the monosymmetry of the beam CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2014:28 VIII Value to determine the Wagner’s coefficient Value to determine the Wagner’s coefficient Young’s modulus of
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