BEHAVIOUR OF STAINLESS STEEL AND COLD FORMED C SECTIONS WITH LARGE WEB OPENINGS IN BENDING AND SHEAR. Antoine Basta

Transcription

1 BEHAVIOUR OF STAINLESS STEEL AND COLD FORMED C SECTIONS WITH LARGE WEB OPENINGS IN BENDING AND SHEAR Antoine Basta Submission in partial fulfillment of te requirement for te Doctor of Pilosopy degree in Structural Engineering Department of Civil and Environment Engineering University of Surrey Febrauary 017 Antoine Basta (601514) Supervised by: R M Lawson Co-Supervisor: GAR Park

2 ABSTRACT Tis researc addresses te use of tin steel C sections wit large circular, exagonal and diamond web openings in applications were service integration in beams is required or were termal bridging of members crossing te building envelope as to be minimised. Te effect of tese openings in terms of combined bending and sear effects on stainless steel and tin steel C sections is evaluated. Te beaviour of tin steel C sections wit large web openings in sear is a new subject and te knowledge gap is mainly concerned wit te local buckling around and between openings wic affect te ability of te perforated section to resist sear. Simply supported beams were considered in tis researc. Te main part of te researc was divided into various parts as follows: Simple teories were developed for te design of tin steel wit circular web openings (Tangential Stress Metod) and wit exagonal web openings. Te Tangential Stress Metod is a metod in wic te tangential stresses around te web openings are determined as a function of te applied sear force. A total of 16 tests on pairs of C sections wit web openings was carried out. Tree groups of beams were tested as follows: 1- Stainless steel C sections of 10 mm dept and 70 mm widt wit 150 mm diameter openings at 50, 100 and 50 mm edge distances were tested. Two groups of stainless steel were tested; Austenitic and Lean Duplex (LDX) grades of and 3 mm tickness. For beams wit isolated and widely spaced web openings, Vierendeel bending associated wit local buckling was te mode of failure. Beams wit closely spaced web openings failed by web-post buckling. - Galvanized steel sections of 50 mm dept and 63mm flange widt wit 150 and 180 mm diameter web openings at 60 and 90 mm edge distances were tested in 1.5 and 1.8 mm steel ticknesses. Te failure of te C sections wit isolated web openings was due to Vierendeel bending associated wit local buckling. For closely spaced web openings, te failure was due to web-post buckling and twisting of te top flange. 3- Galvanized C sections wit diamond and exagonal web openings were tested to investigate te sape of te web openings effect on te beaviour of te beams subject to sear. Te diamond-saped openings were 180 mm deep and exagonal openings were 167 mm deep. Te failure of beams wit isolated diamond-saped web openings was due to buckling of te un-supported web next to te openings. For te C sections wit pairs of openings, it was due to web-post buckling and twisting of te top flange. Te bending resistances of te two C sections were ten predicted from te parametric study and were compared wit te design resistance calculated using section properties to BS EN and BS EN Te tangential stresses using te metod presented in Capter 4 were calculated I

3 at te failure loads in all tests for beams wit circular web openings and compared to te measured steel strengts f y at 0.% strain. Te ratio of te direct tangential stress to steel proof strengt σ/f y varied between 0.70 and 1.1 for te stainless steel beams and between 0.5 and 0.8 for te galvanized steel beams. Tis sows tat te Tangential Stress Metod is reasonably accurate. Linear and non-linear finite element (FE) models were defined to investigate te beaviour of te tin C sections. ABAQUS software was used for te finite element analysis. An extensive parametric study was conducted to study te effects of opening diameter, opening spacing, and span to dept ratio of te beams. Te failure load for eac beam was determined using te Riks Analysis Metod (explained in Capter 8) ignoring te effect of web imperfection wic was found to ave a little effect of on te failure load. Te comparison between buckling analysis wit different imperfection values and te Riks analysis for te two beams sections is presented in Capters 10 and 11 for te various openings configurations. Te results from te FEA were in good agreement wit te test results and sowed te effect of te opening dept to te beam dept ratio ( 0 /), opening spacing (s o) and te tickness of te web (t w) on te section resistance. Te section resistance obtained from te finite element analysis for all models was in good agreement wit te test results and te proposed teory. In te final part of tis researc, te additional deflection due to te loss of te sear and bending stiffness at te position of web openings was investigated and simple formulas were developed in Capter 6 to predict te additional deflection of perforated beams. Linear finite element analysis was considered for comparison and te results were in good agreement wit te proposed teory. II

4 ACKNOWLEDGEMENTS I am deeply indebted to my supervisor Professor R Mark Lawson wo introduced me to te subject of tis tesis and allowed me to be part of tis project. I also tank im for is elpful stimulating suggestions and encouragement during tis project and te writing of tis tesis. Also, I would like to express my gratitude to Dr Stefan Szyniszewski and Dr Boulent Imam for teir valuable comment on my confirmation report and teir guidance and recommendations on te project. My eartfelt appreciation also goes to all tose wo gave me te possibility to complete tis tesis especially for my colleagues for teir elp, support, interest and valuable ints and also I would like to express my tanks to Outokumpu Stainless Steel Researc Foundation for supporting tis researc and for providing te test specimens. Special tanks to my director at work, Alan Pemberton for is appreciation to my researc and for giving me te time needed off work for my testing and meetings wit my supervisor. Last, but not least, I would like to express my eartfelt tanks to my beloved fater, Mr George Basta, my dear broter, Micel Basta and wo ever directly or indirectly involved in te successful completion of tis project. Antoine Basta Postgraduate Student University of Surrey Guildford GU 7XH. III

5 TABLE OF CONTENT CHAPTER 1 INTRODUCTION TO THIN WALLED SECTIONS WITH WEB OPENINGS 1.1. Introduction Design of beams wit large web openings Tests on galvanised cold formed C sections wit web openings Tests on stainless steel in structural applications Beaviour of beams wit web openings in sear and bending Design metods for cold-formed steel members Effective widt metod AISI Direct Strengt Metod (DSM) Significance, Novelty and Knowledge gap Significance Novelty Knowledge gap Material caracteristics Stainless steel sections Manufacture of cold formed steel sections Difference between cold formed and stainless steel Opening configurations Common sape of openings Design cases Scope of work Outline of te tesis CHAPTER LITERATURE REVIEW ON COLD FORMED AS A STRUCTURAL MATERIAL.1. Cold Formed Steel Structural Applications Introduction Type of sections Design of cold formed steel sections to BS and BS EN Design of tin walled sections according to BS EN1993-1: (Eurocode 3) Elastic buckling Post-critical beaviour Effective widt metod in BS EN IV

6 ..5. Influence of stiffeners Effective of edge stiffeners in BS EN Beaviour of webs in sear and bending Beaviour of Members in Bending Previous researc on Cold formed steel beams wit web openings Tests of tin-webbed beams wit unreinforced oles by Redwood, Baranda, and Daly (1978) and Buckling of webs wit openings, Redwood and Uenoya Investigation on te beaviour of web elements wit openings subjected to bending, sear and te combination of bending and sear by Yang, La Boube and Yu Bending and torsion of cold-formed cannel beams by Put, Pi, and Traair, Distortional Buckling Tests on Cold-Formed Steel Beams by Ceng Yu and Scafer Local Buckling Tests on Cold-Formed Steel Beams by Ceng Yu and Scafer Direct Strengt Design of Cold-Formed Steel Members wit Perforations by Moen, Numerical Study of Cold-Formed Steel Beams Subject to Lateral-Torsional Buckling by Kankanamge, Nirosa, Maendran and Maen Experiments on Cold-Formed Steel C-Section Joists wit Unstiffened Web Openings by Moen, Scudlic; and Heyden Experiments on Cold-Formed Steel Beams wit Holes by Soroori Rad, D.Moen, Wollmann and E. Cousins Experimental studies of te sear beaviour and strengt of lipped cannel beams wit web openings by Keertan, Maendran Summary of te general review on previous papers and researc CHAPTER 3 STAINLESS STEEL AS A STRUCTURAL MATERIAL 3.1. Stainless Steel Structural Applications Introduction Applications of stainless steel Basic stress-strain beaviour Design of stainless steel beams to BS EN Classification of cross-sections Effective widts of elements in class 4 cross-section Flange curling Resistance of cross section subject to sear Stainless Steel Design Manual s recommendation for te design of stainless steel beams (SCI Publication P91) Lateral-torsional buckling Sear resistance V

7 3.3.3 Web crusing Determination of deflection of stainless steel beams Previous researc on stainless steel beams wit large web openings Te Lateral Torsional Buckling Strengt of Cold-Formed Stainless Steel Beams by Breden & Den Berg Louis Distortional Buckling of Cold-Formed Stainless Steel Sections: Experimental Investigation by Lecce1 and Rasmussen Distortional Buckling of Cold-Formed Stainless Steel Sections: Finite-Element Modelling and Design by Lecce1 and Rasmussen Experimental and Numerical Studies of Lean Duplex Stainless Steel Beams by Teofanous and Gardner Sear Design Recommendations for Stainless Steel Plate Girders by Saliba, Real and Gardner Web Crippling Design of Cold-Formed Duplex Stainless Steel Lipped Cannel Sections wit Web Openings under End One Flange Loading Condition By Yousefi, Lim, Lianc Uzzamanb, Cliftona and Young Summary of te general review on previous papers and researc. 48 CHAPTER 4 DESIGN OF BEAMS WITH CIRCULAR WEB OPENINGS 4.1. Introduction Literature Review of te design of beams wit web openings to BS EN Pure bending of beams wit web openings Vierendeel bending of beams wit web openings Effective lengt of web openings Sear beaviour of beams wit web openings Horizontal sear force in beams wit web openings Deflection of beams wit web openings Proposed Tangential Stress Merod Equilibrium of forces around closely and widely spaced circular openings Comparison wit Vierendeel bending for equivalent rectangular opening wit te Tangential Stress Metod Treatment of web-post buckling in Class 3 or 4 Sections Conclusion VI

8 CHAPTER 5 LITERATURE REVIEW AND DESIGN OF CASTELLATED BEAMS 5.1. Introduction Castellated beams profile Applications of castellated beams Manufacture of castellated beams Failure modes of castellated beams Literature Review Experimental Investigation of Open Web Beams by Toprac and Cooke Plastic Beaviour of Castellated Beams by Serbourne Limit Analysis of Castellated Steel Beams by Halleux Tests on Castellated Beams by Bazile and Texier Failure of Castellated Beams due to Rupture of Welded Joints by Husain and Speirs Experiments on Castellated Beams by Husain and Speirs Optimum Expansion ratio of Castellated Steel Beams, Galambos, Husain and Speirs Design of Castellated Beams by Knowles Web Buckling in Tin Webbed Castellated Beams by Zaarour Castellated Beam Web Buckling in Sear by Redwood and Demirdjian Stability of Castellated beam Webs by Sevak Demirdjian Design of non-composite beams wit large openings Sear resistance of perforated steel section Vierendeel bending resistances Plastic bending resistance of Tees Elastic bending resistance in te absence of axial force Web-posts between openings Design of exagonal web openings Sear resistance Bending resistance of castellated beam Web-post buckling Deflection of perforated beams Design of beams wit diamond saped web openings Web-post buckling of closely spaced openings Web-post buckling of widely spaced openings Comparison between te section resistances obtained from te proposed teory and te failure loads for four castellated beams tested by Ricard Redwood and Demirdjian Conclusion VII

9 CHAPTER 6 ADDITIONAL DEFLECTION OF BEAMS DUE TO RECTANGULAR AND CIRCULAR WEB OPENINGS 6.1. Introduction Additional deflection of C section beams due to rectangular web openings Additional deflection due to bending curvature at an opening Additional sear deflection Combined deflections for a C section wit rectangular openings subject to uniform loading Combined deflections for a C section beam wit a single rectangular opening subject to uniform loading Proposed simplified formula for te additional deflection of a C section beam wit rectangular openings and subject to uniform loading Elastic finite element analysis on beams wit rectangular web openings subject to uniform loadings Additional deflection of C section beams due to circular web openings Combined deflections for a C section beam wit a single circular opening subject to uniform loading Combined deflections for a C section beam wit circular openings subject to point loading Comparison wit additional deflection for circular openings from FEA CHAPTER 7 TESTS ON COLD FORMED C SECTION BEAMS WITH DIFFERENT SHAPES OF WEB OPENINGS 7.1. Introduction Scedule of Tests Testing arrangement Testing procedure Loading sequence Local deformation of flanges Tests and results on cold formed C sections wit circular web openings Test 1: Isolated 150 mm diameter web openings Test : Isolated 180 mm diameter web openings Test 3: Pairs of 180 mm diameter web openings at 90 mm edge distance Test 4: Pairs of 180 mm diameter web openings at 60 mm edge distance VIII

10 Test 5: Pairs of 180 mm diameter web openings at 90mm edge distance Test 6: Isolated 180 mm diameter web openings Test 7: Isolated stiffened elongated openings Test 8: Pairs of stiffened elongated openings Tests and results on cold formed C sections wit diamond and exagonal web openings Test 1: Isolated diamond saped opening Test : Pairs of diamond saped openings Test 3: Pairs of exagonal web openings (1.53 mm tick) Test 4: Pairs of exagonal web openings (1.93mm tick) Tensile test results for te steel used in te beam tests Tests and results on stainless steel C sections wit circular web openings Tests 1&9: mm Austenitic steel and isolated web openings Test : 3 mm Austenitic steel wit 50mm edge spacing Test 3: mm Austenitic steel wit 100mm edge spacing Test 4: mm Lean Duplex steel wit 100mm edge spacing Test 5: 3 mm Lean Duplex steel wit 100 mm edge spacing Test 6&8: mm Lean Duplex steel wit 50 mm edge spacing Test 7: mm Austenitic steel wit 50 mm edge spacing Discussion of results of tests on cold formed beams Test on beams wit circular web openings Tests on beams wiit elongated stiffened web openings Tests on beams wit diamond and exagonal web openings Discussion of te results of tests on stainless steel beams CHAPTER 8 FINITE ELEMENT ANALYSIS ON THIN WALLED STEEL BEAMS WITH WEB OPENINGS 8.1. Introduction to Finite Element Analysis Linear and nonlinear analysis Geometric Nonlinearity Boundary Nonlinearity Material Nonlinearity Buckling and post buckling Analysis Linear buckling analysis Non-linear buckling analysis Eigenvalue and Riks Metods IX

11 Eigenvalue buckling Te Riks metod Effect of imperfections and effect of plastification of deformable elements Modelling stainless steel beams wit circular web openings Using ABAQUS Mes generation Material caracteristics Boundary conditions Loading Finite Element Analysis results CHAPTER 9 FINITE ELEMENT RESULTS FOR COLD FORMED BEAMS WITH DIFFERENT WEB OPENINGS 9.1. Introduction Mes Sensitivity Study on Cold Formed C Sections Linear Analysis of cold formed C sections wit Circular web openings Model 1: 1.8 mm tick beam wit 150 mm diameter isolated web opening Model : 1.80 mm tick beams wit 180 mm diameter web openings Model 3: 1.5 mm tick beams wit 180 mm web openings at 90 mm edge distance Model 4: 1.5 mm tick beams wit 180 mm web openings at 60 mm edge distance Model 5: 1.8 mm tick beam wit 180 mm web openings at 90 mm edge distance Model 6&7: 1.8 mm tick beams wit 180 mm isolated elongated stiffened web opening and web openings at 80 mm edge distance Stresses from FEA models compared wit tangential stress metod for cold formed steel beams wit web openings at different edge distances Beams wit isolated web openings (Model 1 and ) Beams wit closely spaced web openings (Models 3, 4 and 5) Non-Linear Analysis of Cold Formed C Sections Imperfection Sensitivity Study Riks Analysis on cold formed C sections wit web openings Finite Element results of cold formed sections using te Riks metod Model 1: 1.8 mm tick beams wit isolated 150 mm diameter web openings Model : 1.8 mm tick beams wit isolated 180 mm diameter web openings Model 3:1.5 mm tick beams wit 180 mm diameter web openings at 60 mm edge distance X

12 Model 4:1.5 mm tick beams wit 180mm diameter web openings at 90 mm edge distance Model 5:1.8 mm tick beams wit 180 mm web openings at 90 mm edge distance Analysis of results and comparison between te tests and Finite Element Analysis Comparison of te proposed teory on additional deflection due to web openings as presented in Capter 6 wit tests on sort span cold formed C sections Linear analysis of C sections wit diamond and exagonal web openings Model 1: 50 mm deep x 1. mm tick C section wit isolated diamond saped web opening Model : 50 mm deep x 1. mm tick C section wit double diamond web openings at 9 mm edge distance Model 3: 50 mm deep x 1.53 mm tick C section wit 167 mm deep exagonal web opening at 45 mm edge distance Model 4: 50 mm deep x 1.93 mm tick C section wit 167 mm deep exagonal web openings at 45 mm edge spacing Imperfection sensitivity study on C sections wit diamond web openings mm deep x 1. mm tick C section wit diamond web openings at 9 mm edge distance mm deep x 1.53 mm tick C section wit exagonal web openings at 45 mm edge distance Non-linear FE results of C sections wit diamond and exagonal web openings Model 1: 50 mm deep x 63 mm wide x 1. mm tick C section wit diamond saped web openings Model : 50 mm deep x 1. mm tick C section wit diamond saped web openings at 9mm edge distance Model 3: 50 mm deep x 1.53 mm tick beam wit exagonal web openings at 45 mm edge Model 4: 50 mm deep x1.93 mm tick beam wit exagonal web openings at 45 mm edge distance Discussion of results of FE analysis CHAPTER 10 FINITE ELEMENT ANALYSIS RESULTS FOR STAINLESS STEEL BEAMS WITH CIRCULAR WEB OPENINGS Introduction XI

13 10.. Mes sensitivity study on stainless steel C sections Linear finite element analysis Principal stresses Linear FEA results of stainless steel C sections Model 1: mm tick Austenitic Stainless steel C section wit web openings at 50 mm edge distance Model : 3 mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Model 3: mm tick Austenitic stainless steel C section wit web openings at 100mm edge distance Model 4: mm tick Lean Duplex stainless steel C section wit web opening at 100 mm edge distance Model 5: 3 mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Model 6: mm tick Lean Duplex Stainless steel C section wit web openings at 50 mm edge distance Model 7: mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Stresses from FEA models compared wit direct stress metod for stainless steel beams Beams wit isolated web openings Beams wit closely spaced web openings Non-Linear finite element analysis on stainless steel beams wit circular web openings Imperfection sensitivity study FEA results of stainless steel C sections using te Riks metod Model 1: mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Model : 3 mm tick Austenitic stainless steel C section wit150 mm web openings at 50 mm edge distance : Model 3: mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Model 4: mm tick Lean Duplex stainless steel C section wit 150 mm web opening at 100 mm edge distance Model 5: 3mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Model 6: mm tick Lean Duplex stainless steel C section wit 150 mm web opening at 50 mm edge distance Model 7: mm tick Austenitic stainless steel C section wit 150 mm web opening at 50mm edge distance Analysis of te results and comparison between te test and Finite Element Analysis...19 XII

14 Comparison of te proposed teory on additional deflection due to web openings as presented in Capter 6 wit tests on stainless steel C sections CHAPTER 11 CONCLUSIONS OF THE RESEARCH AND PROPOSALS FOR FUTURE WORK Proposed Tangential Stress Metod Tests on cold formed steel beams wit circular and elongated stiffened web openings Tests on stainless steel beams wit circular web openings Tests on cold formed beams wit diamond and castellated web openings Design recommendations for C sections wit castellated and diamond-saped openings Additional deflections due to web openings Future Work REFERENCES APPENDIX A A.1. Effective section properties of cold formed C-Sections at ULS A.1.1. Effective section properties of a 50 x 63 x 1. cold formed steel C section in bending.39 A.1.. Summary of effective section properties of C-sections in cold formed steel sections in bending A.. Section Properties of stainless steel C-sections at ultimate State A..1. Effective section properties of a 10 x 70 x mm stainless steel C-section in bending...46 A.3.Effective Section Properties of C-Sections at SLS for Deflection Calculations...5 A.3.1: Effective section properties of a 50 x 63 x 1. C-section in cold formed steel section in bending at SLS A.3.: Effective section properties of c-section in cold formed and stainless steel in bending at SLS APPENDIX B B.1.Vierendeel bending resistance based on gross section properties.. 58 B mm tick C-section wit isolated 150 mm web opening B mm tick C-section wit isolated 180 mm web opening B mm tick C-section wit 180mm web opening at 60 mm edge distance B mm tick C-section wit 180mm web opening at 90 mm edge distance B mm tick C-section wit 180mm web opening at 90 mm edge distance B.. Section resistance of cold formed c-section wit circular web opening based on te proposed Tangential Stress Metod XIII

15 B mm deep x 1.5 mm tick C section wit 180mm web openings at 90 mm edge distance B mm deep x 1.5 mm tick C section wit 180mm web openings at 60 mm edge distance B mm deep x 1.8mm tick C section wit 180mm web openings at 90mm edge distance B mm deep x 1.8 mm tick C section wit 180 mm isolated openings...74 B mm deep x 1.8 mm tick C section wit 150 mm isolated opening...74 B mm deep x 1.8 mm tick C section wit 180 mm Isolated web opening...75 B.3. Section resistance of beams wit elongated stiffened web openings c-sections based on tangential stress metod B mm x 1.5 mm tick C section wit 160 mm x 40 mm isolated elongated stiffened opening...77 B mm x 1.5 mm tick C section wit 160 mm x 40 mm elongated stiffened opening at 80 mm edge distance B.4. Section Resistance of cold formed c-sections wit exagonal web openings based on Tangential Stress Metod B mm deep x 1.49 mm tick C-section wit 167mm deep exagonal openings at 45 mm edge distance B mm deep x 1.49 mm tick C-section wit 167 mm deep exagonal openings at 45 mm edge distance APPENDIX C C.1. Sear and Bending Resistance Based on Gross Section Properties.. 81 C.1.1. Sear and bending resistance of Austenitic stainless steel C-section C.1.. Sear and bending resistance of Steel x 3mm C-section C.1.3. Sear and bending resistance of Lean Duplex stainless steel x mm c-section C.. Sear and bending resistance of stainless steel C-section wit circular web openings based on te proposed tangential stress metod C..1: Test 1: mm tick Austenitic C section wit 150 mm web openings at 50 mm edge distance C..: Test : 3 mm tick Austenitic C section wit 150 mm diameter web openings at 50 mm edge distance C..3: Test 3: mm Austenitic C section wit150 mm diameter web openings at 100 mm edge distance C..4: Test 4: mm tick Lean duplex C section wit 150 mm web openings at 100 mm edge distance XIV

16 C..5: Test 5: 3 mm tick Austenitic C section wit 150 mm web openings at 100 mm edge distance.. 95 C..6: Test 6&8: mm Lean Duplex C section wit 150 mm web openings at 50 mm edge distance C..7: Test 7: mm tick Austenitic C section wit 150 mm web openings at 50 mm edge distance APPENDIX D D.1. Sear and bending resistance of one beam tested by Redwood...98 D.. Section Resistance of C-Sections wit exagonal web openings D mm tick C-section wit exagonal web openings at 45mm edge distance D mm tick C-section wit exagonal web openings at 45mm edge distance D.3. Section resistance of C-Sections wit diamond-saped web openings D mm Tick C-Section wit isolated diamond web openings D mm tick C-section wit diamond-saped web openings at 45 mm edge distance..311 Appendix E E.1. Relevant part of te tecnical paper on te additional deflection of c-section beams due to circular web openings E.1.1. Additional deflection due to bending curvature at an opening E.1.. Additional sear deflection E Pure sear deflection due to circular openings E.1... Sear deflection due to Vierendeel bending E Web-post sear deflection E Web-post bending deflection E.1.3. Combined deflections for a C-Section beam wit a single circular opening subject to uniform loading E.1.4. Combined deflections for a C-Section beam wit circular openings subject to point loading XV

17 LIST OF FIGURES CHAPTER 1 Figure 1.1: Failure modes by distortional buckling for one of te Specimens tested by Moen...03 Figure 1.: Failure modes of stub columns tested by Teofanous and Gardner in Figure 1.3: Failure modes of lipped cannel beams tested by Keertan and Maendran Figure 1.4: Illustration of transverse restraints in te buckling of te stiffened elements...06 Figure 1.5: Effective widt concept for cold-formed steel beams Figure 1.6: Carbon steel and stainless steel stress-strain beaviour...10 Figure 1.7: Various forms of web openings in cold formed sections Figure 1.8: Tests on an equivalent beam to simulate te beaviour of cantilevers subject to a point load by applying te load at mid-span at mid-span CHAPTER Figure.1: Typical ligt gauge steel frame...15 Figure.: Infill wall in structural steel frame...16 Figure.3: Worksop Building, Mongolia was built using cold formed steel sections as teir main structural components Figure.4: Different sapes of C and Z sections Figure.5: Local buckling of plates wit different boundary conditions Figure.6: Illustration of effective widt of compression plate Figure.7: Types of element and stiffeners...1 Figure.8: Spring stiffness of edge stiffener..... Figure.9: Effective dept of webs in bending to BS EN Figure.10: Lateral-torsional buckling of a simply supported beam CHAPTER 3 Figure 3.1: Classification of stainless steel according to nickel and cromium content...35 Figure 3.: Stainless steel beams and columns Figure 3.3: Crysler Building in New York and Petronas Twin Towers sows stainless steel cladding Figure 3.4: Bp pedestrian bridge in Cicago, USA...38 Figure 3.5: Double Helix Bridge was built using stainless steel tubes Figure 3.6: Stress strain beaviour of stainless steel and carbon steel XVI

18 CHAPTER 4 Figure 4.1: Yielding due to ig bending of te Tee sections above and below openings...50 Figure 4.: Generation of plastic inges around openings due to Vierendeel bending under sear force Figure 4.3: Equivalent rectangular opening for te circular openings...,..51 Figure 4.4: Horizontal sear at te mid web-post between two openings...5 Figure 4.5: Equilibrium of forces acting on plane wen tan -1 (s/)...55 Figure 4.6: Equilibrium of forces acting on plane wen > tan -1 (s/)...57 Figure 4.7: Variation of tangential stress around te openings for widely and closely spaced openings for various opening sizes and spacings CHAPTER 5 Figure 5.1: Typical narrow and wide profiled castellated beam section Figure 5.: Use of castellated beams in enclosures and car parks...66 Figure 5.3: Use of castellated beams as a igway bridge Figure 5.4: Cutting of castellated beams Figure 5.5: Laterally braced flexural failure of castellated beams Figure 5.6: Lateral-torsional buckling of castellated beams Figure 5.7: Vierendeel mecanism caused by sear transfer troug perforated web zone..68 Figure 5.8: Rupture of a welded joint in orizontal sear...69 Figure 5.9: Web-post buckling below a concentrated load Figure 5.10: Definition of te sear area of rolled and welded sections to EN Figure 5.11: Compression and tension action in web-post between exagonal openings...81 Figure 5.1: Buckled waves and boundary conditions next to exagonal opening...83 Figure 5.13: Compression and tension action in web-post between diamond openings. 85 Figure 5.14: Configuration of test beams tested by Redwood and Demirdjian CHAPTER 6 Figure 6.1: Additional deflection due to bending at a large web opening Figure 6.: Distribution of bending stresses around a rectangular web opening. 91 Figure 6.3: Sear deflection due to a web opening Figure 6.4: Layout of beam 1 wit rectangular web openings...96 Figure 6.5: Deflection of alf a beam wit 8 no. rectangular web openings subject to a unifrom load of 6.6 kn/m as obtained from te elastic FE analysis on ABAQUS Figure 6.6: Layout of beam wit recatngular web openings...97 XVII

19 Figure 6.7: Deflection of alf a beam wit 8 no. rectangular web opening subject to a uniform load of 4 kn/m as obtained from te elastic FE analysis on ABAQUS...97 Figure 6.8: Layout of beam 1 wit circula web openings Figure 6.9: Deflection of alf a beam wit 8 circular web opening subject to a uniform load of 6.6kN/m as obtained from te elastic FEA Figure 6.10: Layout of beam wit circular web openings.. 10 Figure 6.11: Deflection of alf a beam wit 9 circular web openings subject to a uniform load of 5kN/m as obtained from te elastic FEA CHAPTER 7 Figure 7.1: Details of circular opening positions in te test cold formed steel C-section Figure 7.: Details of diamond saped and exagonal opening positions in te test cold formed steel C-sections Figure 7.3: Details of circular opening positions in te tests on stainless steel C section Figure 7.4: Details of test arrangement on pairs of C sections Figure 7.5: Loading arrangement via a central jack and Steel block Figure 7.6: Test arrangement sowing pair of beams connected using steel blocks Figure 7.7: Deformation of te stiffened flanges in bending due to te deep stiffeners and flexible flanges Figure 7.8: Test arrangement for Test 1 wit isolated 150 mm web openings Figure 7.9: Failure mode of Test 1 by bending of top flange and local failure at connection between beams and steel blocks Figure 7.10: load-deflection curve for Test 1 wit isolated 150 mm web openings Figure 7.11: Arrangement for Test wit isolated 180 mm we opening Figure 7.1: Failure mode of Test by local buckling due to te Vierendeel Bending Figure 7.13: load-deflection curve for Test wit isolated 180 mm web opening Figure 7.14: Test arrangement for Test 3 wit 180 mm web openings at 90 mm spacing Figure 7.15: Failure mode of test 3 by web-post buckling between web openings Figure 7.16: Load-deflection curve for Test 3 wit 180 mm web opening at 90 mm spacing Figure 7.17: Test arrangement for Test 4 wit 180 mm web openings at 60 mm spacing Figure 7.18: Failure mode of Test 4 by web-post buckling between web openings Figure 7.19: Load-deflection curve for Test 4 wit 180 mm web opening at 60 mm spacing Figure 7.0: Test arrangement for Test 5 wit 180 mm web openings at 60 mm edge distance 117 Figure 7.1: Failure mode of Test 5 by web-post buckling between web openings Figure 7.: Load-deflection curve for Test 5 wit 180 mm web opening at 90 mm spacing Figure 7.3: Test arrangement for Test 6 wit isolated 180 mm web opening Figure 7.4: Failure mode of Test 6 by local buckling and Vierendeel bending Figure 7.5: Load-deflection curve for Test 6 wit isolated 180 mm web opening XVIII

20 Figure 7.6: Arrangement for Test 7 wit single elongated stiffened web opening Figure 7.7: Failure mode of Test 7 by local bending of te top flange and web at connection point Figure 7.8: Load-deflection curve for Test 7 wit single elongated stiffened elongated stiffened web opening Figure 7.9: Arrangement for Test 8 wit elongated stiffened web opening at 80 mm edge distance Figure 7.30: Failure mode of te beam in Test 8 by bending of te top flanges at te connection point Figure 7.31: load-deflection curve for Test 8 wit elongated stiffened web opening at 80 mm edge distance Figure 7.3: Test arrangement for Test 9 wit isolated diamond web opening Figure 7.33: Failure mode of Test 9 by web buckling troug te diamond opening Figure 7.34: Load-deflection curve for Test 9 wit isolated diamond web opening...15 Figure 7.35: Arrangement for Test 10 wit two diamond openings at 9 mm edge distanc...15 Figure 7.36: Failure mode of Test 10 by web-post buckling Figure 7.37: Load-deflection curve for Test 10 wit diamond opening at 9mm edge distance...16 Figure 7.38: Test arrangement for Test 11 wit pairs of exagonal opening at 45 mm edge distance Figure 7.39: Failure mode of Test 11 by web-post buckling Figure 7.40: Load-deflection curve for Test 11 wit 167 mm deep exagonal web openings at 45 mm edge distance Figure 7.41: Test arrangement for Test 1 wit exagonal web openings at 45 mm edge distance Figure 7.4: Failure mode for Test 1 by web-post buckling Figure 7.43: Load-deflection curve for Test 1 wit 167 mm deep exagonal web opening at 45mm edge distance Figure 7.44: Tensile test arrangement and failure of specimen in te tensile test Figure 7.45: Test arrangement for Tests 1and 9 wit openings at 50 mm edge distance.131 Figure 7.46: Failure mode of te mm tick Austenitic stainless steel C section wit circular web opening at 50 mm edge distance by Vierendeel bending Figure 7.47: Variation of load wit displacement for Test 1 wit isolated circular web opening...13 Figure 7.48: Test arrangements of te beam for Test wit circular web openings at 50mm edge distance Figure 7.49: Failure mode of te 3mm tick Austenitic stainless steel C section beam wit circular web opening at 50 mm edge distance Figure 7.50: Variation of load wit displacement for Test wit circular web openings at 50 mm edge distance by orizontal sear and buckling of te web-post XIX

21 Figure 7.51: Stresses around te openings in Test Figure 7.5: Test arrangements of Test 3 wit circular openings at 100 mm edge distance.135 Figure 7.53: Variation of load wit displacement for Test 3 wit circular web openings at 100 mm edge spacing Figure 7.54: Failure mode of te mm tick Austenitic stainless C section wit circular opening at 100 mm edge distance due to Vierendeel bending and buckling of te top flange Figure 7.55: Test arrangements of te beam for Test 4 wit circular web openings at 100 mm edge distance 136 Figure 7.56: Variation of load wit displacement for Test 4 wit circular web openings at 100 mm edge distance Figure 7.57: Failure mode of te mm tick Lean Duplex stainless steel C section wit circular web openings at 100 mm edge distance by buckling at load point Figure 7.58: Test arrangements of Test 5 wit circular web openings at 100 mm edge distance.138 Figure 7.59: Variation of load wit displacement for Test 5 wit circular web openings at 100 mm edge distance Figure 7.60: Failure mode of te 3 mm tick Austenitic stainless steel wit circular web opening at 100 mm edge distance Figure 7.61: Test arrangement of Test 6 and 8 wit circular web openings at 50mm edge distance.139 Figure 7.6: Variation of load wit displacement for Test 6&8 wit circular web openings at 50 mm edge distance Figure 7.63: Failure mode of te mm tick Lean Duplex steel C section wit web openings at 50 mm edge distance by web-post buckling Figure 7.64: Test arrangement of Test 7 wit circular web openings at 50mm edge distance Figure 7.65: Variation of load wit displacement for Test 7 wit circular web openings at 50 mm edge distance. 141 Figure 7.66: Failure mode of te mm Austenitic stainless steel wit circular web openings at 50 mm edge distances by web-post buckling CHAPTER 8 Figure 8.1: A cantilever beam taken as an example to sow te boundary nonlinearity Figure 8.: Curve sowing te linear and nonlinear stress/strain relationsip Figure 8.3: Proportional loading wit unstable response Figure 8.4: Finite element model of alf beam as modelled using ABAQUS Figure 8.5: Typical finite element model sowing its mes in ABAQUS Figure 8.6: Stress-strain curve for mm tick Austenitic stainless steel.156 Figure 8.7: Stress-strain curve for mm tick Lean Duplex stainless steel Figure 8.8: Stress-strain curve for 3 mm tick Austenitic stainless steel Figure 8.9: Model wit support arrangement in ABAQUS XX

22 CHAPTER 9 Figure 9.1: Failure load for different mes size comparing to te test failure load for 1.5mm tick C section wit 180 mm web opening at 60mm edge distance Figure 9.: Sear force applied to te web of te beam as sell load in ABAQUS Figure 9.3: Maximum principal stresses of 430N/mm around web opening for Model 1 subject to a sear force of 0.5 kn Figure 9.4: Principal stresses around circular openings for Model 1 wit single 180mm web opening 163 Figure 9.5: Maximum principal stresses of 507 N/mm around te web opening for Model subject to a sear force of kn Figure 9.6: Principal stresses around circular openings for Model wit isolated 180mm diameter web openings..164 Figure 9.7: Maximum principal stresses of 340N/mm around te opening for Model 3 subject to a sear force of 7. kn Figure 9.8: Principal stresses around openings for Model Figure 9.9: Maximum principal stresses of 310 N/mm around te opening for Model 4 subject to a sear force of 7.kN Figure 9.10: Principal stresses around te web openings for Model Figure 9.11: Maximum principal stress of 450N/mm around te web opening for Model 5 subject to a sear force of kn Figure 9.1: Principal stresses around circular openings for Model Figure 9.13: Maximum principal stresses of 367 N/mm around web opening for Model 6 subject to a sear force of kn Figure 9.14: Maximum principal and bending stresses of 376N/mm around te openings for Model 7 subject to a sear force of kn Figure 9.15: Comparison between te principal stresses around te openings from te FEA at te Test failure load and from te design metod for Model Figure 9.16: Comparison between te principal stresses around te openings from te FEA at te Test failure load and from te design metod for Model Figure 9.17: Comparison between te principal stresses around te openings from te FEA at te Test failure load and from te design metod for Model Figure 9.18: Comparison between te principal stresses around te openings from te FEA at te Test failure load and from te design metod for Model Figure 9.19: Comparison between te principal stresses around te openings from te FEA at te Test failure load and from te design metod for Model Figure 9.0: Comparison between te test failure load and te failure load for different imperfection values for 50 mm x1.5 mm tick beam wit 180mm web openings at 90 mm edge distance XXI

23 Figure 9.1: Deformation and maximum principal stresses based on te non-linear FE analysis of Model Figure 9.: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x1.8 mm tick C sections wit 150 mm diameter web opening Figure 9.3: Deformation and maximum principal stresses based on te non-linear FE analysis of Model..174 Figure 9.4: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x1.8 mm tick C sections wit isolated 180 mm diameter web opening Figure 9.5: Deformation and maximum principal stresses based on te non-linear FE analysis of Model Figure 9.6: Failure load for two beams based on te non-linear FE analysis compared to te test failure load for two 50 mm deep x 1.5 mm tick C section wit 180 mm web openings at 60 mm edge distances Figure 9.7: Deformation and maximum principal stresses based on te non-linear FE analysis of Model Figure 9.8: Failure load for two beams based on te non-linear FE analysis compared to te test failure load for 50mm deep x1.5mm tick C section wit 180 mm web openings at 90mm and 60mm edge distances Figure 9.9: Deformation and maximum principal stresses based on te non-linear FE analysis of Model Figure 9.30: Failure load for two beams obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x 1.8 mm tick C sections wit 180 mm isolated web openings and web openings at 90mm edge distance Figure 9.31: Deflection of 1.8 mm tick C section wit isolated 150 mm web opening subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.3: Deflection of 1.8 mm tick C section wit isolated 180 mm web opening subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.33: Deflection of 1.8 mm tick C section wit 180 mm web openings at 90 mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.34: Deflection of 1.5 mm tick C section wit 180 mm web openings at 60 mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.35: Deflection of 1.5 mm tick C section wit 180 mm web opening at 90 mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.36: Maximum principal stresses of 87N/mm around te opening based on te linear FE analysis of Model 1 subject to a sear force of 8.1 kn Figure 9.37: Maximum principal stress of 77 N/mm around te opening near te load point based on te linear FE analysis of Model subject to a sear force of 7. kn 184 XXII

24 Figure 9.38: Maximum principal stress of 400 N/mm around te opening for linear analysis of Model 3 subject to sear force of 10.5 kn 184 Figure 9.39: Maximum principal stress of 485 N/mm around te web opening based on te linear analysis of model 4 subject to a sear force of 16.5 kn Figure 9.40: Comparison between test failure load and failure loads obtained from te FE analysis for different imperfection values for beam wit diamond web openings at 9mm edge distance Figure 9.41: Comparison between test failure load and failure loads obtained from te FE analysis for different imperfection values for 1.53mm tick beam wit exagonal web openings at 45mm edge distance Figure 9.4: Comparison between te test mode of failure and two different mode of failure based on te FE analysis for a 50mm deep x 1.53mm tick beam wit exagonal web openings at 45mm edge distance Figure 9.43: Deformation and maximum principal stresses based on te FE analysis of Model Figure 9.44: Failure load from FE analysis for two beams compared to te test failure load of two 50 mm deep C sections wit isolated diamond saped web opening..189 Figure 9.45: Deformation and maximum principal stresses based on te FE analysis of Model..190 Figure 9.46: Failure load from te FE analysis for beams compared to te test failure load of two 50 mm deep x 63mm wide C sections wit a diamond saped web openings at 9mm edge distance Figure 9.47: Deformation and maximum principal stresses based on te FE analysis of Model Figure 9.48: Failure load based on te FE analysis compared to te test failure load of two 50 mm deep x 1.53 mm tick C section wit exagonal openings at 45mm edge distance.19 Figure 9.49: Deformation and maximum principal stresses based on te FE analysis of Model Figure 9.50: Failure load based on te FE analysis compared to te test failure load of two 50 mm deep x1.9 mm tick beam wit castellated web openings at 45 mm edge distance.193 CHAPTER 10 Figure 10.1: Comparison between te test failure load and te failure loads for different mes Sizes Figure 10.: Mes comparison of 10mm x 70mm x mm tick Lean Duplex stainless steel C Section wit 150mm web openings at 50 mm edge distance Figure 10.3: Maximum principal stresses of 35 N/mm based on te linear FE analysis of Model 1 subject to sear force of 15.4 kn Figure 10.4: Principal stresses around te web openings of Model 1 subject to a sear force of 15.4kN. 198 Figure 10.5: Maximum principal stress of 359 N/mm around web openings based on te linear FE analysis of Model subjected to a sear force of 1kN.199 XXIII

25 Figure 10.6: Principal stresses around te web openings of Model subject to a sear force of 1 kn Figure 10.7: Maximum principal stresses of 35 N/mm around web openings based on te linear FE analysis of Model 3 subject to a sear force of 11.1kN..00 Figure 10.8: Principal stresses around te web openings of Model 3 subject to a sear force of 11.1 kn Figure 10.9: Maximum principal stress of 580 N/mm around te web opening based on te linear FE analysis of Model 4 subject to a sear force of 17.8 kn Figure 10.10: Principal stresses around te web openings of Model 4 subject to a sear force of 17.8 kn Figure 10.11: Maximum principal stresses of 349 N/mm based on te linear Fe analysis of Model 5 subject to a sear force of 1.5kN Figure 10.1: Principal stresses around te web openings of Model 5 subjected to a sear force of 1.5kN Figure 10.13: Maximum principal stresses of 500N/mm around te opening based on te linear analysis of model 6 subject to a sear force of 13.7 kn Figure 10.14: Principal stresses around web openings of Model 6 subject to a sear force of 13.7 kn Figure 10.15: Maximum principal stresses of 35 N/mm around te web openings based on te linear analysis of Model 7 subject to a sear force of 9.8 kn. 04 Figure 10.16: Tangential stresses around te web openings of Model 7 subject to a sear force of 9.8 kn...04 Figure 10.17: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model Figure 10.18: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model...06 Figure 10.19: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model Figure 10.0: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model Figure 10.1: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model Figure 10.: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model 6 08 Figure 10.3: Comparison between teprincipal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model Figure 10.4: Comparison between test failure load and failure loads obtained from te FE analysis for different imperfection values XXIV

26 Figure 10.5: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.6: Failure load for two beams as obtained from te FE non-linear analysis compared to te test failure load for two 10 mm x mm tick Austenitic stainless steel C sections wit 150 mm web openings at 50 mm edge distance... 1 Figure 10.7: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode...1 Figure 10.8: Failure load as obtained from te non-linear FE analysis for two 10 mm deep x 3 mm tick Austenitic stainless steel C sections wit web openings at 50 mm edge distance.13 Figure 10.9: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.30: Failure load for beams as obtained from te FE non-linear analysis compared to te test failure load for two 10 mm deep x mm tick Austenitic stainless steel C sections wit 150 mm web openings at 100 mm edge distance Figure 10.31: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.3: Failure load from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick Lean Duplex stainless steel C section wit web openings at100 mm edge distance Figure 10.33: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.34: Failure load from te non-linear FE analysis compared to te test failure load for two 10 mm deep x 3 mm tick, Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Figure 10.35: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.36: Failure load from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick, Lean Duplex stainless steel C sections wit 150 mm web openings at 50 mm edge distance Figure 10.37: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Mode Figure 10.38: Failure load from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick, Austenitic stainless steel C section wit 150 mm web openings at 50 mm edge distance Figure 10.39:Deflection of mm tick Austenitic stainless steel beam wit web openings at 50 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA XXV

27 Figure 10.40: Deflection of mm tick Austenitic stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA...0 Figure 10.41: Deflection of mm tick Austenitic stainless steel beams wit web openings at 50 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA...0 Figure 10.4: Deflection of 3 mm tick Austenitic stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA...1 Figure 10.43: Deflection of mm tick Lean Duplex stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10kN as obtained from te elastic FEA.1 Figure 10.44: Deflection of mm Tick Lean Duplex stainless steel beams wit web openings at 50 mm edge distance subject to a sear force of 10kN as obtained from te elastic FEA...1 Appendix E Figure E1: Additional deflection due to bending at a large web opening Figure E: Variation of bending stresses in te web-post between circular openings Figure E3: Sear deflection due to a web opening Figure E4: Opening dept y as a function of for pure sear deflection calculation XXVI

28 LIST OF TABLES CHAPTER Table.1: Effective widt of webs to BS EN using S80 steel CHAPTER 3 Table 3.1: Grades of stainless steel included in BS EN Table 3.: Representative values of stress-strain caracteristics for materials in te annealed condition Table 3.3: Internal compression elements for uniform and non-uniform stress....4 CHAPTER 4 Table 4.1: Maximum compression stress for ɵ > 45 o as influenced by te spacing of te opening cases controlled by orizontal sear in te web-post are saded...59 Table 4.: Variation of tangential stresses around te opening as ratio of te sear stress V or H 61 Table 4.3: Comparison of Vierendeel bending stresses and tangential stresses for an equivalent rectangular opening Table 4.4: Maximum tangential stress around circular openings to prevent local buckling as a function of o/t w and steel grade CHAPTER 5 Table 5.1: Failure loads for te four beams tested by Redwood and Demirdjian Table 5.: Sections sear capacity based on section properties and proposed teory Table 5.3: Relationsip between te failure loads and te section resistance for te tested beams...88 CHAPTER 6 Table 6.1: Additional deflection of C sections due to rectangular openings based on proposed metod Table 6.: Comparison wit additional deflection for rectangular openings as obtained from te FEA and as calculated from te proposed teory Table 6.3: Comparison of additional deflection obtained from te FEA and te teory for a beam wit span: dept ratio of Table 6.4: Comparison of additional deflection obtained from te FEA and te teory for a beam wit span: dept ratio of XXVII

29 CHAPTER 7 Table 7.1: Scedule of cold formed steel C sections wit circular web openings (50 mm deep x 63 mm wide) Table 7.: Scedule of cold formed C sections wit diamond and exagonal web openings (50 mm deep x 63 mm wide) Table 7.3: Scedule of specimens for stainless steel beams tested as pairs of C sections..106 Table 7.4: Stress-strain curves for te steel used in beam tests Table 7.5: Summary of test results for cold formed C sections wit circular web openings 143 Table 7.6: Tangential stress σ at edge of openings calculated for test failure loads Table 7.7: Summary of te test results compared to sections capacity obtained from te gross section properties Table 7.8: Summary of test results for cold formed C sections wit diamond and exagonal web openings Table 7.9: Summary of te test results compared to sections resistance obtained from te teory Table 7.10: Comparison between applied stresses and compressive strengt of te tested beams 146 Table 7.11: Test series, failure loads and mode of failure of stainless steel C sections wit 150 mm diameter web openings Table7.1: Failure loads compared to bending and sear resistances calculated using gross properties of te section CHAPTER 8 Table 8.1: Tensile test results for stainless steel C sections Table 8.: Tensile test results for galvanized steel C sections Table 8.3: Boundary conditions in FE Model CHAPTER 9 Table 9.1: Cases analysed in finite element models for cold formed steel C sections wit circular web openings Table 9.: Cases of finite element models of cold formed steel C sections wit diamond and exagonal web openings Table 9.3: Maximum principal and mid-span deflection for eac model obtained from te linear elastic finite element analysis Table 9.4: Failure load for different imperfection values compared to te test failure load Table 9.5: Comparison between te test failure loads and te failure loads obtained from te Riks analysis Table 9.6: Comparison between te test failure loads and te failure loads obtained from te FE analysis for two beams Table 9.7: Comparison of test deflections for cold formed steel beams XXVIII

30 Table 9.8: Deflections of 50 mm deep cold formed steel sections according to te proposed teory in Capter Table 9.9: Comparison of test deflections, FEA and teory for 50 mm deep cold formed C section...18 Table 9.10: Summary of te linear analysis of cold formed C sections wit diamond and exagonal web openings.183 Table 9.11: Comparison between te test failure load and te failure loads obtained from te finite element Riks and buckling analysis wit different imperfection values for a 50 mm x 1. mm tick beam wit diamond web openings at 9 mm edge distance Table 9.1: Comparison between te test failure load and te failure loads obtained from te finite element Riks and buckling analysis wit different imperfection values for 50 mm deep x 1.53 mm tick beam wit exagonal web openings at 45 mm edge distance Table 9.13: Comparison between testing sear forces at failure and te sear resistance for eac C section as obtained from te FE analysis and te corresponding deflection Table 9.14: Comparison between test failure loads and FE failure loads CHAPTER 10 Table 10.1: Cases analysed in Finite element models of stainless steel beams Table 10.: Applied sear force and te corresponding maximum principal stress of te linear finite element analysis of te stainless steel models Table 10.3: Comparison between te test failure sear forces and te sear force obtained from te Riks analysis and te corresponding deflection Table 10.4: Comparison between te test failure load and te failure loads obtained from te finite element Riks and buckling analysis wit different imperfection values.. 10 Table 10.5: Comparison between te test failure loads and te failure loads obtained from te FE analysis for te stainless steel tests Table 10.6: Deflections of 10 mm deep stainless steel sections according to te proposed teory Table 10.7: Comparison of test deflections for stainless steel beams... Table 10.8: Comparison of test deflections, FEA and teory for stainless steel beam 3 CHAPTER 11 Table 11.1: Comparison of failure loads and failure loads obtained from te tangential stress metod and te FEA for te cold formed C section Table 11.: Comparison between test failure loads and failure load obtained from te FEA and te proposed teory for eac stainless steel C section XXIX

31 Table 11.3: Comparison between te test failure loads for eac C section wit diamond and Castellate web openings as te failure loads obtained from te finite element analysis and te developed teory. 9 APPENDIX A Table A.1: Effective section properties of cold formed C-sections at ULS Table A.: Effective section properties of cold formed C-sections at SLS APPENDIX E Table E1: Comparison of te algoritm for additional deflection calculations and a simplified formula XXX

32 A A 0 cross-sectional area original cross-section area List of Symbols A eff A eff,f A eff,w,c A f A f,net A net A v A V A w b c d E e N effective area of a cross section effective area of compression flange effective area of compressed part of web area of te tension flange net area of te tension flange net area of a cross section sear area area of one post (or transverse element) of a built-up column area of a web widt of a cross section widt or dept of a part of a cross section dept of straigt portion of a web modulus of elasticity sift of te centroid of te effective area A eff relative to te centre of gravity of te gross cross-section e N,y sift of te centroid of te effective area A eff relative to te centre of gravity of te gross cross section (y-y axis) e N,z sift of te centroid of te effective area A eff relative to te centre of gravity of te gross cross section (z-z axis) f f y G o eff I modification factor for χ LT yield strengt sear modulus dept of a cross section dept of web opening effective dept of cross section second moment of area XXXI

33 i I eff,f I s k l o l eff L L c M b,rd M c,rd M cre M Ed M el M N,,Rd radius of gyration about te relevant axis effective second moment of area of compression flange about te minor axis effective second moment of area of stiffener buckling factor lengt of web opening equivalent lengt of web opening member lengt lengt between lateral restraints design buckling resistance moment design resistance for bending about one principal axis of a cross-section critical elastic lateral-torsional buckling load design bending moment elastic bending moment resistance of web-post reduced design values of te resistance to bending moments making allowance for Presence of normal force M Rk M V,,Rd caracteristic moment resistance of te critical cross-section reduced design values of te resistance to bending moments making allowance for te presence of sear forces M y,ed M y,rd M y,rk M z,ed M z,rd M z,rk M Ed design bending moment, y-y axis design values of te resistance to bending moments, y-y axis caracteristic value of resistance to bending moments about y-y axis design bending moment, z-z axis design values of te resistance to bending moments, z-z axis caracteristic value of resistance to bending moments about z-z axis additional moment from sift of te centroid of te effective area A eff relative to te centre of gravity of te gross cross section M y,ed M z,ed moments due to te sift of te centroidal y-y axis moments due to te sift of te centroidal z-z axis XXXII

34 n N b,rd N c,rd N cr number of openings in a member or part of te member design buckling resistance of a compression member design resistance to normal forces of te cross-section for uniform compression elastic critical force for te relevant buckling mode based on te gross cross sectional properties N Ed N net,rd N pl,rd N Rd N Rk N Rk N w N f r,r 1 R d s s o t t f t w T w, Ed V V c,rd V Ed V pl,rd V wp V axial compression design plastic resistance to normal forces of te net cross-section design plastic resistance to normal forces of te gross cross-section design values of te resistance to normal forces caracteristic resistance to normal force of te critical cross section caracteristic value of resistance to compression compression or tension force acting on Plane ɵ te increase in compression or tension force in te plane ɵ radius of root fillet design value of resistance centre to centre spacing of web openings edge to edge distance of web openings tickness flange tickness web tickness design value of internal warping torsional moment average sear force at web opening design sear resistance design sear force at te web openings plastic design sear resistance orizontal web-post sear force orizontal sear force across te web-post XXXIII

35 v-v w b y e α α LT α cr γ M ε ε u ε y η η η cr λ λ p,b λ LT λ 1 μ ν ρ σ σ com,ed σ cr σ x,ed σ z,ed τ Ed minor principal axis (were tis does not coincide wit te z-z axis) bending deflection of te unperforated beam subject to uniform loading dept of te elastic section portion of a part of a cross section in compression imperfection factor minimum force amplifier to reac te elastic critical buckling load general partial factor strain ultimate strain yield strain factor for sear area conversion factor sape of elastic critical buckling mode slenderness of strut slenderness ratio of compression element non dimensional slenderness for lateral torsional buckling slenderness value to determine te relative slenderness efficiency factor Poisson s ratio in elastic stage reduction factor to determine reduced design values of te resistance to bending moment making allowance for te presence of sear forces stress maximum design compressive stress in an element Elastic critical buckling stress design value of te local longitudinal stress design value of te local transverse stress design value of te local sear stress XXXIV

36 τ τ v φ φ φ 0 orizontal sear stresses at te mid of te web-post vertical sear stresses at te mid of te web-post factor representing te stress ratio at te ends of te plate value to determine te reduction factor χ basic value for global initial sway imperfection φ LT χ χ χ LT χ y χ z ψ ψ value to determine te reduction factor χ LT reduction factor for te relevant buckling curve reduction factor for relevant buckling mode reduction factor for lateral-torsional buckling reduction factor due to flexural buckling (y-y axis) reduction factor due to flexural buckling (z-z axis) ratio of moments in segment stress or strain ratio XXXV

37 CHAPER 1 INTRODUCTION TO THIN STEEL SECTIONS WITH WEB OPENINGS 1.1 Introduction Tin walled steel members using cold formed galvanised steel ave been widely used in building construction, bot as load-bearing frames and as secondary elements, suc as purlins and floor joists. Stainless steel cold formed sections are used were te members are exposed visually or were tere is a concern about corrosion. Tin walled steel sections may be perforated by large circular or elongated openings for visual reasons, to reduce teir weigt or to pass services troug. However, te tin walled sections are relatively tin in order to economise on materials use and te size of te web openings can be up to 75% of te section dept. In te case of beams, te loss of sear resistance due to te web openings is significant. Terefore, te key design issue is te transfer of sear and its effect on te local buckling around te web openings particularly for Class 4 sections (according to Eurocode 3). Tis researc concentrates on te beaviour of perforated cold formed steel beams in bending and sear. Proposed design formulas are derived based on test results to determine te sear and bending resistance of sections wit bot widely and closely spaced web openings. Te additional deflection due to web openings was also considered in tis researc and proposed formulas were derived based on te elastic finite element analysis combined wit a teoretical approac based on te effective openings widt in bending and sear. 1. Design of Beams wit Large Web Openings Te design of ot rolled steel sections and composite beams wit large rectangular web openings is based on te transfer of sear by Vierendeel bending of te web flange Tee sections at te four corners of te opening. Te plastic resistance of te Tees can be developed in Class 1 and steel sections, wose definition is based on te widt to tickness of te elements of te cross-section. Tese limits are presented in BS EN : Eurocode 3. Te early work on ot rolled steel beams wit circular web openings was carried out by Samel (1969) on welded transverse beams and torsion bars wit large openings. An approac for circular openings was introduced, wic was based on equilibrium of te local stresses on radial planes around te opening. However, tis metod does not apply to closely spaced openings wen limited

38 by te stresses in te web-post between te openings. Later, Redwood (1973) carried out work on ot rolled steel beams wit circular web openings and it was sown tat an isolated circular opening could be treated as an equivalent rectangle for Vierendeel bending in wic te effective lengt of te opening is taken as 0.45 opening diameter and its dept as 0.9 diameters. In 1987, te SCI /CIRIA recommendations provided te guidance on composite beams wit web openings, and in 1990, SCI P100 provided guidance on cellular beams wit circular openings (Ward, 1990). Tis was based on detailed finite element studies of te buckling of te web-post in non-composite cellular beams. Design guidance for steel and composite beams wit large web openings is now presented in SCI Publication 355 (Lawson, 011), and was based on te application rules of BS EN (BSI, 003a) and BS EN : Eurocode 4 (BSI, 003b). Altoug considerable test data exists on Class 1 and ot rolled steel sections, tere is little data or design guidance on Class 3 or 4 stainless steel or galvanised steel cold formed members wit large openings, particularly using iger strengt grades. Tese sections are strongly influenced by te local buckling of te web. Furtermore, local buckling of stainless steel sections is influenced by te less well defined yield point of many types of stainless steel (Gardner et al, 014). 1.3 Tests on Galvanised Cold Formed C Sections wit Web Openings Moen and Scafer (008) performed tests on C sections wit elongated openings in compression tat are mainly used in racking systems and studied te buckling beaviour of tin steel. In 009, tey made analytical predictions of te buckling strengt of bot perforated webs and flanges of C sections compared wit finite element analyses. Modifications to te Direct Strengt Metod (DSM) equations and curves in AISI specifications were made. A variety of opening sapes, suc as circular, square, elongated and slotted sapes were considered. Later, Berooz and Rad (010) carried out an investigation on four different types of cold formed beams wit different opening depts. An elastic buckling analysis togeter wit te experimental study was used to evaluate te modification made to te DSM local and distortional curves, but te equations could not be evaluated. Recommendations were made to consider beams wit smaller distortional slenderness to investigate te validity of te transition equations proposed for elastic/inelastic distortional buckling effects. Furter tests were carried out by Moen et al. (01) on 4.8 m span joists of 03 mm dept wit square openings in pure bending and also for comparison, on te same un-perforated joists. Te objective was to examine te effect of te openings on te local buckling resistance of te flanges and Figure 1.1 sows one of te tests in wic, te beam failed by distortional buckling. Te design approac for perforated sections was presented in terms of te DSM, in wic te compression resistance may be expressed directly in terms of te critical buckling strengt and element slenderness.

39 Figure 1.1: Failure modes by distortional buckling for one of te specimens tested by Moen (Moen et al, 01) 1.4 Tests on Stainless Steel in Structural Applications In 199, a wide range of tests on cold formed Lean Duplex square ollow columns was investigated in detail by Young and Hancock. For te cannel sections failing by lateral torsional buckling, te buckling curve in EN was revised to use an imperfection parameter α = 0.35 (buckling curve b ) and a slenderness cut-off of O = 0.4. In 006, studies of ig strengt Austenitic steels by cold working were made by Gardner et al. Later, tests on Lean Duplex stainless steels used in beams and columns were carried out by Teofanous et al. (009). Four structural ollow section sizes from mm to mm tick were tested for wic te average 0.% proof strengt was 633 N/mm, as sown in Figure 1.. Bending tests on sort span beams ad a test resistance to plastic bending resistance ratio of 1.15 to Based on tis researc, it was concluded tat te flange widt: tickness ratio corresponding to te Class and 3 limits in BS EN could be increased. Te column tests ad a slenderness ratio of 0.57 to.0. It was concluded tat te best prediction using te design metod in BS EN was obtained by an imperfection parameter α = 0.49 (buckling curve c ) and a slenderness cut-off of O = 0.4. Tis is now incorporated into te revision of BS EN Real et al. (014) reviewed te use of te Ramberg-Osgood expression for representation of te stress strain curves for all types of stainless steel, wic builds on previous work. It as led to te 3

40 expressions given in BS EN Te effective widts of Class 4 stainless steel elements currently in BS EN are less tan for carbon steel, but te revision to BS EN as led to a general armonisation wit te equivalent carbon steel limits in BS EN Figure 1.: Failure modes of stub columns in stainless steel (Teofanous et al., 009) Te sear buckling resistance of stainless steel beams was ten investigated by Saliba et al. (014) by carrying out tirty four tests on stainless steel plate girders to determine te effect of tension field action wit a rigid end post. Proposals were made for an improved sear buckling formula in BS EN A review of te uses of stainless steel in construction identified te important callenges and researc needs. In te context of Eurocodes, BS EN Design of steel structures supplementary rules for stainless steels extends te application of BS EN (covering general rules for te structural design of building-type structures made from ot rolled and welded carbon steel sections) and BS EN (covering design of cold-formed ligt gauge carbon steel sections) to ot rolled welded and cold-formed stainless steels. BS EN as completed its first revision and uses recent researc information tat as elped to improve its design provisions. Oter information on te structural design of stainless steel is given in a Euro Inox /SCI publication and in a recent AISC Guide. Te stress strain caracteristics of stainless steels are non-linear and also stainless steels possess a ig ultimate tensile strengt in comparison to carbon steels. Terefore, te post-elastic caracteristics of stainless steels are important in understanding teir structural beaviour (Gardner et al, 015). Lately, Yousefi et al. (016) investigated te effect of circular openings on te web crippling strengt of cold formed stainless steel lipped cannel sections wit various dimensions and ticknesses for 4

41 tree different stainless steel grades: Lean Duplex, Austenitic and Ferretic. New web crippling strengt reduction factor equations were proposed and were found to be in good agreement wit te Finite element analysis results. 1.5 Beaviour of Beams wit Web Openings in Sear and Bending Te effect of web openings on te beam sear beaviour is significant due to te loss of te web area and ence its sear resistance but its relatively small on bending resistance were te web openings are located at te centre of te beam. However, te sape, size and location of te web openings ave a direct effect on te sear resistance. Few investigations were undertaken to study te effect of web openings on te sear beaviour. In Canada, Scuster (1995) tested 13 beams wit different web opening sizes and sapes subject to point load. Later, furter investigations were carried out by San et al. (1994) and ten by Eiler (1997) to investigate te beaviour of beams wit web openings subject to uniform load. It was found tat te dept of te web opening to clear web eigt ad te greatest effect on te sear resistance of te beam and te ratio of te eigt of te web to its tickness ad a little effect. A proposal was made to calculate te sear resistance of a beam wit web opening by using a reduction factor wic can be applied to te nominal sear resistance of a solid beam. Kankanamge and Maendran (011) carried out various testing and numerical studies, and design equations were proposed to calculate te sear resistance of cold formed beams by applying a reduction factor to te sear resistance of a beam wit solid web. Tis was later found to be unconservative in use for beams wit large web openings. Anoter investigation was carried out recently by Keertan and Maendran (013) on lipped cannel beams, as sown in Figure 1.3 in wic, a new reduction factor was proposed wic was a function of te ratio between te dept of te web opening and te clear web dept. Figure 1.3: Failure modes by web buckling for lipped cannel beams (Keertan and Maendran, 013) 5

42 1.6 Design Metods for Cold-Formed Steel Members Cold-formed steel structural members are affected by buckling under compressive stresses due to teir slenderness. Currently, two design metods are provided by BS EN and te American Iron and Steel Institute (AISI) for te design of cold-formed steel members. Te effective widt metod relies on te reduced area provided by eac element in te cross-section subject to local buckling. Te Direct Strengt Metod (DSM) considers te stability of te wole cross-section including te interaction between elements and te stress tat it can resist Effective widt metod Cold formed steel members are slender members and often exibit local buckling before te steel reaces its yield strengt. However, cold-formed steel members ave a considerable amount of postbuckling reserve were eac plate element in te cross-section provides a continuous support for te adjacent elements and restraints te plate buckling wic in turn increases te resistance of te member, as illustrated in Figure 1.4 (Berooz and Rad, 010). Figure 1.4: Illustration of transverse restraints in te buckling of te stiffened elements (Berooz and Rad, 010) Te effective widt metod was first introduced in 1930 s to calculate te strengt of tin plates in compression and was later extended to te design of cold-formed steel members (Von Karman and Secler, 193) and (Winter, 1947). At early loading stages, te stress distribution is uniform across te widt of te element. By increasing te applied load, non-uniform stress distributions wit iger stresses applied at te stiffened edges occurs until failure occurs due to te yielding of te regions of maximum stress at te stiffened edges. Te effective widt metod can be simplified by assuming tat a uniform stress equal to f max (= f y) is applied on te reduced widt b eff of te element (Berooz et al, 013). 6

43 For a lipped C-section in bending, te effective section modulus is calculated based on te effective widt of eac of te compressive elements in te cross-section considering te modified position of te neutral axis, as sown in Figure 1.5. Figure 1.5: Effective widt concept for cold-formed steel beams (Barooz and Rad, 010) 1.6. AISI Direct Strengt Metod (DSM) Te AISI Direct Strengt Metod (DSM) was introduced in 004 and uses te elastic buckling properties of te entire cross-section to calculate te cross-section resistance (Scafer, 00). Te elastic buckling mode sapes related to te design of cold formed sections can be obtained from an elastic buckling curve generated wit a finite strip analysis (Ceung, 1998). Te elastic buckling values corresponding to te buckling mode sapes are used in combination wit te DSM equations to estimate te resistance of te cold-formed steel members. Te formulas used to calculate te resistance of flexural members, as well as te DSM design curves for lateraltorsional, local and distortional buckling modes, are provided in AISI specifications, wic are only applicable to un-perforated members. In te DSM, te flexural capacity of a beam is calculated considering tree limit states global buckling, local-global buckling interaction, and distortional buckling (AISI 007, Appendix 1). Modifications to te DSM equations for cold-formed steel beams wit web-openings were proposed by Moen in (008) and ten by Moen and Scafer (010) in wic, te flexural resistance of a beam is calculated considering tree limit states global buckling, local-global buckling interaction, and distortional buckling. Te minimum strengt from te tree limit states is taken as te beam s flexural capacity, i.e. M n is te minimum of M ne, M nl and M nd. 7

44 Were: M ne M nl M nd is te nominal flexural strengt for lateral-torsional buckling is te nominal flexural strengt for local-global buckling is te nominal flexural strengt for distortional buckling Te modified equations were found to be very sensitive to te size of te web opening, and an interaction between buckling and lateral-torsional buckling was made by Moen and Scafer in 008 and confirmed by te finite element investigation tat was carried out by Rad and Moen in 010. Te modified equation proposed by Moen is sown below: For lateral-torsional buckling, Moen stated tat te nominal section bending strengt M ne can be calculated using te following equations: a) For M cre < 0.56M y M ne = M cre 10 10M y b) For.78M y M cre 0.56M y M ne M y (1 ) 9 36M c) For M cre >.78M y M ne = M y Were: M cre = critical elastic lateral-torsional buckling load cre M y = S f f y S f = section modulus of te outer fiber In tis researc, te effective widt metod is considered and a simplified comparison between te proposed tangential stress metod and te finite element analysis results is presented in Capters 10 to Significance, Novelty and Knowledge Gap Significance Tin walled steel sections wit large web openings are increasingly used and te significance of te researc can be summarised in te following: 1. Tin steel sections are generally Class 4 according to Eurocode 3 and are susceptible to local buckling.. No specific guidance is given on web openings in tin galvanized steel or in stainless steel sections. 3. Te size of te opening is a large proportion of te beam dept and te openings are often closely spaced, wic leads to failure of te web-post between te openings. 8

45 4. Circular and exagonal openings are preferred for structural reasons and for service distribution and so guidance sould be applicable to tese cases. 5. Optimising te size and te sape of openings is required wit respect to te applied loads. 6. Te failure modes of stainless steel sections may differ in comparison to te equivalent galvanized steel sections because of te non-linear stress-strain caracteristics of stainless steel Novelty Currently, tere are no recommendations or studies on te beaviour of tin perforated steel sections in bending and sear and ow to treat te local buckling around te openings were te section dept: tickness, d/t 100 to 150 is ig in comparison to ot rolled sections were generally, d/t 50. A study on cold formed steel sections wit web openings was carried out by Moen (008) at Jon Hopkins University, USA, but it concentrated on te compression resistance of perforated sections. A study was carried out on te beaviour of composite beams wit web openings (SCI-P355, 011), but tis applies to ot rolled steel or plated sections, and not to tin steel sections were te beaviour is dominated by local buckling Knowledge gap To understand te stress regime around openings and te effect of opening sape on local buckling, te following knowledge gaps are considered in tis researc: 1. Understanding te development of local buckling around and between large web openings.. Influence of te material caracteristics of stainless steel verses galvanized cold formed steel. 3. Influence of different sape of openings; circular, exagonal and diamond sape. 4. Influence of edge distance between te openings on bending and sear resistance due to web-post buckling. 5. Effect of web slenderness (d/t w) ratio on web buckling next to circular openings. 6. Effect of proportionate opening dept ( o/), te sape and spacing of web openings on additional deflections. 1.8 Material Caracteristics Stainless steel sections An important application tat as been identified for beams in stainless steel is te use of tin steel cantilever beams to support balconies. Te beams may pass troug te cladding and attac to te primary structure. Te large perforations and te low termal conductivity of stainless steel greatly reduce cold bridging troug te section, wic is important in modern energy efficient design. Stainless steel is also used in exposed and visually important applications. A new grade of Lean 9

46 Duplex stainless steel as been developed wic as iger strengt wit low nickel content (refer to Capter 3) Manufacture of cold formed steel sections Cold-formed sections, also called Ligt Gauge or Cold Rolled steel sections are manufactured in a process tat involves forming steel sections in a cold state from strip steel. Te tickness of steel seet used in cold formed construction is usually 1 to 3 mm wit minimum yield strengt of 80 N/mm. Steel coils of 1.0 to 1.5 m widt are usually used and can be cut longitudinally to te correct widt appropriate to te section required before feeding tem into a series of roll forms. Tese rolls, containing male and female dies and are arranged in pairs, moving in opposite direction so tat, as te seet is fed troug tem, its sape is gradually formed to te required profile. Te number of pairs of rolls depends on te complexity of te cross sectional sape and varies typically from 5 to 15. At te end of te rolling stage, a searing macine cuts te member into te desired lengts Difference between cold formed and stainless steel Te stress-strain beaviour of stainless steel differs from tat of carbon steels in a number of respects. Carbon steel exibits linear elastic beaviour up to te yield stress and a plateau before strain ardening occurs wile stainless steel as a more rounded response wit no well- defined yield stress, as sown in Figure 1.6. Terefore, stainless steel yield strengts are generally quoted in terms of a proof strengt defined for a particular offset permanent strain. Figure 1.6: Carbon steel and stainless steel stress-strain beaviour ( 10

47 1.9 Opening Configurations Common sape of openings Various common sapes of web openings are illustrated in Figure 1.7. Circular and triangular openings as in (a) and (b) are relatively simple sapes tat can be fabricated by puncing. Circular openings are efficient structurally. Diamond saped openings in (c) are more complex but could offer advantages. Multiple slots in (d) ave been sown to reduce te eat flux troug external walls in ligt steel framing. t o so 60 o s o (a) Circular openings (b) Triangular openings s o b o so o o sv (c) Diamond openings (d) Multiple slots Figure 1.7: Various forms of web openings in cold formed sections (ttp:// Design cases Steel beams wit large rectangular openings are subject to local moment at eac corner of te openings due to te applied sear force; tis is generally known as Vierendeel bending. Te webflange Tee sections ave to be sufficiently strong to resist tis local moment. For Vierendeel bending due to transfer of sear forces, a circular opening may be treated as an equivalent rectangular opening of a lengt equal to 0.45 times its diameter (Redwood, 1973) and so circular openings are teoretically more efficient in terms of transferring sear forces tan square openings of te same area. For cantilever beams, ig sear forces act togeter wit ig bending moments. Tis is potentially a complex penomenon resulting in axial forces in te Tees as well as local bending moments due to transfer of sear around openings. Sear transfer across closely spaced openings can also result in web-post buckling between te openings. 11

48 In tis researc, a series of tests was performed on tin steel and stainless steel C sections as simply supported beams to simulate teir use as cantilever beams in balconies and to evaluate te bending - sear interactions around openings. Tis structural system is illustrated in Figure 1.8. Following on from te tests and teir analysis, design recommendations are prepared for tin walled members wit large openings, wic are based on te BS EN and BS EN metodology for cold formed and stainless steel sections respectively. Te metod covers Class 3 or 4 sections. Te tests were compared to te results of finite element analyses using te measured structural properties of stainless and cold formed steels (a) Cantilever balcony (b) Equivalent beam wit centre point load Figure 1.8: Tests on an equivalent beam to simulate te beaviour of a cantilever subject to a point load by applying te load at mid-span 1.10 Scope of Work Te scope of work for te researc was divided into two pases, as follows: Pase 1: Literature review on previous researc and proposed tests 1. Collection of data on existing tests on stainless steel and cold formed steel members wit large openings for members in bending and sear.. Review of existing design guidance to BS EN 1993 for Class 3 and 4 members wit openings. 3. A test series aimed at evaluating te combined bending-sear interactions at large openings. Te range of tested beams was as follows: Cold rolled galvanized C sections of 50 mm dept wit nominal flange widt of 63 mm and S390 grade. C sections of 10 mm dept and 70 mm widt, Austenitic and Lean Duplex stainless steel. Steel tickness of 1., 1.5, 1.8 and mm for te galvanized steel beams and and 3 mm for stainless steel beams. Circular openings of 150 and 180 mm dept at a variable spacing of 50 to 50 mm. Diamond and exagonal openings tested as bot isolated and closely spaced. Loading applied as a single point load in mid-span via steel blocks to simulate te combined moment and sear effects in a cantilever beam. 1

49 Beam span of 1.4 to 1.5 m using C sections tested in pairs to reduce instability effects and to cause failure in sear. Pase : Testing and Analysis Te test and analysis covered te following stages:.1 A test series on beams wit openings sizes and spacing to establis te sear resistances at te various openings and also web-post buckling between te openings.. Comparison of te test results wit elastic analyses using effective section properties for Class 3 and 4 sections..3 Finite element analyses of te tested beams concentrating on te areas around te openings and using te elastic and elasto-plastic measured material properties..4 Deriving a design procedure for cold formed and stainless steel beams wit large openings in te form of application rules to BS EN and BS EN respectively. For cold formed galvanized steel, ticknesses of 1. to 1.8 mm were selected so tat failure would occur by local buckling of te web-post. For stainless steel, and 3 mm tick steel were used and for te ticker steel, local buckling was expected to occur only at ig elasto-plastic strains. Beams wit web openings at close centres were expected to fail in web-post buckling wile beams wit web openings at wider spacing were expected to fail in Vierendeel bending and local buckling Outline of te Tesis Te outline of te following capters is presented, as follows: In Capter, a literature review of cold formed steel in structural applications is made togeter wit a review of te relevant clauses of BS EN , and also a review of previous researc on C sections is presented to obtain a general understanding of te design of C sections. In Capter 3, a literature review of stainless steel in structural applications is made. A review of te relevant clauses of BS EN , and also a review of previous researc on C sections are presented to obtain a general understanding of stainless steel and its design. In Capter 4, a general review of te design metodology for cold formed beams wit large web openings is presented to obtain a general understanding of te beaviour of stainless steel and cold formed steel sections wit large circular web openings. In Capter 5, a literature review of beams wit exagonal openings and a review of previous researc and te design of tese beams is presented. 13

50 In Capter 6, a teoretical metodology for predicting te additional deflection of beams wit web openings is presented based on loss of sear and bending stiffness at te openings. In Capter 7, tests on cold formed steel beams wit circular, exagonal and diamond-saped web openings and tests on stainless steel beams wit circular web openings are presented and assessed to obtain te sear and te bending resistance at te openings. In Capter8, a general review of te linear and non-linear finite element analysis using Abaqus was reviewed wit an explanation on ow te modelling of te stainless steel and galvanized steel C sections wit web openings was carried out. In Capter 9, te finite element analysis and results for galvanized cold formed steel beams wit circular, exagonal and diamond web openings are presented and compared to te test results and proposed mecanical metod as presented in Capter 4 and 5. In Capter 10, te finite element results for stainless steel beams wit circular web openings are presented and compared to te test results and te proposed mecanical metod as presented in Capter 4. In Capter 11, conclusions obtained for te researc are presented, togeter wit future researc ideas. 14

51 CHAPTER LITERATURE REVIEW ON COLD FORMED AS A STRUCTURAL MATERIAL.1 Cold Formed Steel Structural Applications.1.1 Introduction Over te past 15-0 years, ligt steel framing construction as become a competitor to te wood frame construction and as become popular in Europe, USA and Australia. Ligt steel frames use cold formed steel sections, usually C sections, as te main structural elements, as sown in Figure.1 to Figure.3. Cold formed sections are sections produced by bending and cold rolling of flat seet (Hancock, 003). Te steel used in cold formed sections is relatively tin, typically 0.9 to 3. mm and is galvanized (zinc coated) for corrosion protection. Steel strengts range from S80 to S450 (yield strengt in N/mm ). Cold-formed steel in coiled strip form is te primary raw material. Te steel seets can be formed into many sapes and forms by a variety of manufacturing process, te different processes available allow great flexibility in te use of te cold-steel sections. Figure.1: Typical ligt gauge steel frame (ttp:// 15

52 Figure.: Infill wall in a structural steel frame (ttp:// Figure.3: Worksop building, Mongolia was built using cold formed steel sections as its main structural components (ttp:// 16

53 .1. Type of sections Tere are various types of ligt steel sections typically in C and Z sapes, as sown in Figure.4. A wide variety of oter sections can be produced by cold forming using a static press, but te majority is produced by continuous cold rolling. C-sections are usually used in ligt steel framing. Tey are typically of 75 to 00 mm dept wit a flange widt of 40 to 75 mm. Te edge of te sections is stiffened by a lip of 10 to 15 mm dept in order to increase te resistance of te section to te local buckling. t t t x x D x x D x x D t t t x x D x x D x x D Figure.4: Different sapes of C and Z sections (ttp:// Design of Cold Formed Steel Sections to BS and BS EN Te design of ligt steel structures as become very efficient bot from te structural design and detailing for manufacture points of view. Te former design standard, BS , was in operation since Te introduction of BS EN : Eurocode 3: "Design of Steel Structures" Part 1.3 and its National Annex are more complex and cross-refer to oter parts of BS EN BS EN replaces BS 5950 Part 5 and design of seeting in BS 5950 Part 6. Limit state design uses partial factors, buckling parameters etc., taken from BS EN and BS EN Resistances are determined by te relevant modes of failure wic include: Effective section properties for bending. Lateral torsional buckling (in bending). Effective section properties in compression. 17

54 Lateral buckling (in compression). Web crusing at points of support or applied loads...1 Design of tin walled sections according to BS EN : (Eurocode 3) Cold formed cross-sections are classified as slender (or Class 4 in Eurocode terminology) because based on te amount of material in te cross-section, tey cannot reac teir full compression resistance because tey tend to buckle locally under compression. Terefore, effective section properties sould be used in structural calculations. Te cold forming can increase te strengt of te material by 3 to 10%, depending on te number of bends in te section. For S80 and S350 steel grades, te design strengt of te steel, f y is taken as its yield strengt, and te partial factor for steel is set to 1.0 (BS EN , 006)... Elastic buckling Te full compression resistance of a perfectly flat plate supported on two longitudinal edges can be developed for a widt-to-tickness ratio of about 40. At greater widts, buckles form elastically causing a loss in te overall compressive resistance of te plate. Tis is due to te inability of te more flexible central portion of te plate to resist as muc compression as te outer portions, wic are partly stabilised by te edge supports (BS EN ). Te critical compression stress at wic elastic buckling of te plate occurs is given by te expression: Were: P cr = 1 K E t = K (t/b) N/mm (Equation.1) (1 v ) b b t E v is te plate widt is te steel tickness Young s Modulus of steel Poisson s ratio Te term K, referred to as te buckling coefficient, represents te influence of te boundary conditions and te stress pattern on plate buckling. Plates are normally considered to be infinitely long but ave various support conditions along teir longitudinal edges. Te two common cases are, firstly, simple supports along bot edges were K can be taken as 4, and secondly, one simple support and te oter free edge were K reduces dramatically to Tis indicates tat plates wit free edges do not perform well under local buckling. Tese cases are illustrated in Figure.5. 18

55 Buckled sape pcr Buckled sape pcr Supported edge Supported edge pcr Junction remains straigt pcr Edge is free to displace Adequate lip No edge lip Figure.5: Local buckling of plates wit different boundary conditions (BS EN , 006)..3 Post-critical beaviour Plates are not perfectly flat and terefore, by increasing te applied load, tey deform out-of-plane gradually rater tan buckle instantaneously at te critical buckling stress. Tis means tat te non-uniform stress state exists trougout te loading regime, and tends to cause te plate element to fail at loads less tan te critical buckling value. Tis is a dominant effect in te b/t range from 30 to 60 for plates simply supported on bot edges (BS EN ). Tere are opposing effects for plate elements wit iger b/t ratios were firstly, a membrane or in-plane tensions are generated caused by te development of te buckling wave wic leads to transfer of in-plane stresses and provides restraint resisting any furter buckling, and secondly, te zone of compression yielding extends from te longitudinal supports to encompass a greater widt of te plate elements. Tese post-critical effects cause an increase in te load-carrying capacity of wide plate elements (b/t > 60) relative to tat given by Equation.1. Te parameter, wic is used to express te beaviour of plate elements in compression, is te effective widt. Tis is te notional widt, wic is assumed to act at te yield strengt of te steel. Te remaining portion of te plate element is assumed not to contribute to te compression resistance, as illustrated in Figure.6. Actual stress distribution Simplified equivalent stresses b b eff / b eff / Y s b b Ys Ys Figure.6: Illustration of effective widt of compression plate (BS EN , 006) (Y s= f y = yield stress) 19

56 ..4 Effective widt metod in BS EN Te load buckling of tin plate elements in BS EN cross-refers entirely to BS EN , wic is te relevant standard for plated steel sections. Te effective widt concept is described in Clause 4.4 of tis standard, and te reduction factor is given by te empirical formulae: b eff = b 1. 0 for Were: ( 3 ) for (Equations.) Te slenderness ratio of te flat plate is defined by: Were: 0. 5 b / t ( f y / cr ) (Equation.3) 8. 4 k = (35/f y) k is buckling factor, wic is conservatively equal to 4 for flanges in pure compression restrained along bot longitudinal edges. is a factor representing te stress ratio at te ends of te plate (= 1 for pure compression) σ cr is a critical buckling stress of te plate is a reduction factor Te effective widt of te plate in compression is terefore b eff = b. Altoug te formulae are different, te effective widts to BS and BS EN give similar results. BS EN is less conservative for b/t >60. For an unstiffened element, te same basic approac is used to determine teir effective widt except tat: 1.0 for (3 ) for (Equations.4) Were: k = buckling factor, wic is equal to 0.45 for a plate supported along one longitudinal edge 0

57 ..5 Influence of stiffeners Tere are two types of stiffeners: tose at te edge of a plate element, and tose internally witin a plate element and are known respectively as edge and intermediate stiffeners, in te form of lips and fold as sown in Figure.7. Unstiffened element Internal element Simple lip Compound lip Intermediate stiffener a) Section wit unstiffened elements b) Sections wit elements stiffened by lips c) Section wit intermediately stiffened element Figure.7: Types of element and stiffeners (BS EN , 006) Te edge stiffeners comprising a simple lip or rigt angle bend and to be fully effective in providing longitudinal support to prevent buckling of te edge of te plate, tey sould not be less in dept tan one-fift of te widt of adjacent plate element (BS EN , 006). If te stiffener is adequate, te plate element may ten be treated as simply supported along bot longitudinal edges, wit a K value of 4. BS EN presents a metod wic includes te influence of partially effective stiffeners on plate buckling. Intermediate stiffeners are intended to reduce te flat widt of te plate elements so tat te section operates more effectively...6 Effectiveness of edge stiffeners in BS EN In BS EN , te treatment of edge stiffeners to te flanges acting in compression or major axis bending is by a beam on elastic foundations analogy. In tis metod, te edge stiffener is supported by a spring equivalent to te transverse bending stiffness of te adjacent flange. Te effectiveness of te spring determines te effective area of te stiffener, as illustrated in Figure.8. In BS EN , tis dept of stiffener is not fully effective, wic means tat te section properties will be less to BS EN in comparison to BS

58 b b e1 b e b e1 b e b 1 c c eff A s k k (a) Effective widt of compression elements (b) Spring stiffeners model for edge stiffeners Figure.8: Spring stiffness of edge stiffener (BS EN , 006) Te performance of compression flanges stiffened by edge lips is affected by te flexibility of te attacment of te flange to te web, wic causes te flange tends to move towards te elastic neutral axis. Te spring stiffness of te flange and web is given by BS EN , as follows: 3 Et 1 K 3 4(1 ) b b 0.5b b 1 w 1 1 w k f (Equation.5) Were: b 1 & b are te distance from te web-to-flange junction to te centroid of te effective stiffener for flange 1 and respectively w k f is te web dept = 0 for sections in major axis bending k f E = 1.0 for symmetric sections in compression = N/mm for galvanized steel = 0.3 Te critical buckling stress cr of te stiffened flange is given by: σ (Equation.6) cr kei s /As Were: and A s is te cross-sectional area of te stiffener and its adjacent connected flange I s is te second moment of area of te stiffener and its adjacent connected flange. Te slenderness ratio for te edge stiffener acting in compression is given by: 0.5 d ( f y / cr ) (Equation.7) Te effective area of te compression flange is multiplied by a reduction factor for distortional d buckling wic depends on te slenderness d and is given by a tri-linear relationsip of te form:

59 For 0.65, 1. 0 d d d , d (Equations.7) 1.38, 0.66/ d d d..7 Beaviour of webs in sear and bending Webs of cross-sections are subject to sear, bending and local compression at teir supports. Tese local effects dominate te design of cold-formed sections. d..7.1 Web sear Slender webs normally fail in sear by sear buckling. Te buckling coefficient K for a simply supported plate in pure sear tends to a value of BS EN refers to BS EN for plate girders, and te maximum web dept before web buckling may control is given by w 7t w, were t w is te web tickness and = (35/f y) 0.5. Te web slenderness ratio used to determine te sear buckling resistance is defined by: w w (Equation.8) 86.4t w For w 0.83, te sear strengt of te web is given by 0.83/ w multiplied by te sear strengt of te web, wic is given in BS EN as equal to f / 3, were f y is te steel yield strengt. Te sear buckling strengt is ten multiplied by te web area to determine te sear resistance of te section. y..7. Web bending Webs of sections in bending are subject to varying compressive stress, reducing from a maximum at te junction wit te flange to zero at te elastic neutral axis position. Very deep webs can be influenced by local buckling in compression. However, te varying stress in te web leads to a deeper plate element before buckling tan compared to a plate element under pure compression. Tis is reflected in te teoretical value of te buckling coefficient K of 3.9 rater tan 4. Te effective widt concept is also used to determine te post-buckling bending resistance of deep webs by considering two separate zones adjacent to te neutral axis and to te compression flange Effective widt formula for webs In BS EN , for webs, te same effective widt formulae apply as for flanges, except webs are not in pure compression wen subject to in-plane bending. In tis case, te buckling factor k is modified to take account of te stress ratio, at te ends of te web, defined by: 3

60 = / 1 (Equation.9) Were 0-1.0, wic is te case for a web in bending, and 1 is te maximum compression stress acting on te web. Te buckling factor for te web in tis stress range becomes: k = (Equation.10) In BS EN , tis effective dept of te web in compression, b eff is divided into two zones given by 0.4 b eff next to te compression flange and 0.6 b eff next to te elastic neutral axis, as illustrated in Figure b 4 t eff + b c + Ineffective web 0.6beff 36 t + b Elastic neutral axis _ Were: = / 1 b c= b 1 - b = eff b c _ = (35 / fy) 0.5 (a) Effective widt of web to BS EN (b) Simplified effective widt for symmetric C-section Figure.9: Effective dept of webs in bending (BS EN , 006) In using te effective widt formulae, it is apparent tat depts of web in compression less tan approximately 60 t w are not subject to local buckling. For greater web depts in compression, te effective widt of te web may be treated for simplicity as being represented by a zone of 4 t w next to te compression flange and 36 t w next to te elastic neutral axis. Te effective widt factors for webs in compression are presented in Table.1 for two stress ratios corresponding to webs in bending (in tis case for S80 steel wit = 0.91). 4

61 Table.1: Effective widt of webs to BS EN using S80 steel (BS EN , 006) w/t w Stress ratio, = -1.0 Stress ratio, = -0.7 b eff/t b eff/t Were: w = web dept b eff = effective web dept in compression..8 Beaviour of Members in Bending to BS EN Bending resistance of section Te effective bending resistance of sections in bending can be obtained by multiplying te elastic section modulus considering te effective widts of te compression elements by te design strengt of te steel once te elastic neutral axis position is known. Bot te neutral axis position and te section modulus are terefore functions of te operating stress of te compression flange. Tis calculation procedure is more complex in BS EN because te elastic neutral axis is determined by iteration depending on te effectiveness of te web. For symmetric sections, te effective section modulus of te compression plate is not greater tan tat in tension and terefore compression yielding occurs first. However, for some non-symmetric sections, tension yielding may occur first causing plastification in te tension flange...8. Lateral torsional buckling Simply supported members are generally attaced to floors, so tat te compression flange is prevented from displacing laterally. Were te lateral restraints are sufficiently wide apart, lateral torsional buckling (LTB) may occur, as illustrated in Figure.10. Te elastic lateral buckling resistance moment of an equal flange I-section or a symmetrical C-section bent in te plane of te web is given by te formula: 5

62 M cr = AED 1 1 / 0 LE t C L b ry ry E 0.5 (Equation.11) Were: L E r y A D C b is te distance between points of lateral restraint is te radius of gyration of te section in te lateral direction is te section dept is te gross cross-section area is te overall dept of te cross- section is te factor representing te sape of te bending moment diagram (unity for constant moment and 1.15 for uniformly distributed loading). In BS EN , te critical buckling moment, M cr may be expressed in terms of slenderness ratio for LTB, wic is given by: M M / λ cr y LT λ LT (λlt / λ ) (Equations.1) π(e/f y ) λ Figure.10: Lateral-torsional buckling of a simply supported beam ( Account may be taken of te support conditions in modifying te effective lengt L E of te beam. Te ratio L E/r y defines te slenderness, of te member. BS EN 1993 uses te slenderness ratio, LT. As te slenderness ratio reduces, so M cr increases, and eventually te bending resistance, M y of te section is reaced. 6

63 Te effective slenderness ratio, LT of te beam may be given by te expression: Were: LT = C b -0.5 UV (Equation.13) U V is approximately equal to 0.9 for C or I section is te torsional parameter, as follows: V = t 0.5 (Equation.14) For tin steel sections, V is in te range of 0.9 to 1.0. For a beam subject to uniform loading, te effective slenderness ratio for LTB tends to te approximation of = 0.8. Tis reflects te LT beneficial effects of non-uniform stress and torsional stiffness on lateral torsional buckling of te section in comparison to a strut of slenderness ratio,. For C sections, buckling curve b is used for lateral torsional buckling. Te buckling resistance moment is determined from te same formulae as for columns, as follows: Were: M χ M for λ 0. (Equation.15) b, Rd LT y,rd LT χ LT 1 φlt φlt λ LT (Equation.16) and φ λ 0. LT LT LT λ (Equation.17) Te full bending resistance of te section is reaced wen te slenderness ratio LT is less tan 0., as opposed to 0.4 in BS Torsional flexural buckling to BS EN and BS For tin omonosymmetric cross-sections suc as C sections, account sould be taken of te possibility tat te resistance of te member to torsional-flexural buckling migt be less tan its resistance to flexural buckling (BS EN , Clause 6..3). Tis is because of te separation of te centroid and sear centre. Te approac in BS to analyse te torsional flexural buckling is to modify te effective lengt for lateral buckling to take into account te possibility of a lower torsional flexural failure mode. Tis is acieved by te use of te effective lengt multiplication factor. Appropriate values for a range of common sections are presented in Appendix C of BS (BS , Clause 6.3). 7

64 .3 Previous Researc on Cold Formed Steel Beams wit Web Openings.3.1 Tests of Tin-Webbed Beams wit Unreinforced Holes by Redwood, Baranda, and Daly and Buckling of Webs wit Openings by Redwood and Uenoya Tests and analysis on tin-walled flexural members wit web openings ave been performed by Redwood et al. (1978 and 1979) using tecniques for bot ot rolled and welded steel plates wit circular and rectangular openings. Tese investigations on tin-walled elements were concerned wit beams of open web section and te analysis of te sear, moment, and teir interaction. Typically, te location of te concentrated load(s) was far from te web opening and terefore precluded web crippling in te vicinity of te web opening. Te loads were used to generate desired sear or moment regions in te member in te vicinity of te web opening. In te portion of te member located in te vicinity of te web opening, te compression region of te cross section beaved like a tee section under compression because of te free edge along te web opening. Terefore, te compression region of te web near te web opening was igly susceptible to buckling. Te study stated tat te most critical factors influencing te beaviour of te sections wit web openings are: 1. Sear force at te openings.. Moment at te opening centreline. 3. Web slenderness. 4. Slenderness of te web of tee section formed by te part of te beam above or below te opening. 5. Lengt of te opening. 6. Sape of te opening. 7. Presence of transverse stiffeners near te opening. Tey found tat te bending resistance is reduced by only to 5%, but te sear resistance is significantly reduced by te presence of te openings..3. Investigation on te Beaviour of Web Elements wit Openings Subjected to Bending, Sear and te Combination of Bending and Sear by Yang, La Boube and Yu Te objective of tis investigation by Yang et al. (1994) was to study te structural beaviour of cold-formed steel members wit perforated webs. A total of 68 tests was carried out; 30 in combined bending and sear, 0 in pure bending and 18 in pure sear. Te a/ and /t ratios (were a is te opening dept, is web dept and t is te steel tickness) ad te most significant effect on te 8

65 beam resistance. As an example, reducing te a/ ratio by 31%, increased te beam sear and bending resistance by approximately 39% wile increasing te /t ratio by 4% reduced te beam resistance by approximately 7%. Modifications to te current AISI specification were made to allow for te reduction in sear and bending capacities due to te presence of web openings..3.3 Bending and Torsion of Cold-Formed Cannel Beams by Put, Pi, and Traair In tis paper, Put et al. (1999) reported te results of 34 bending and torsion tests on simply supported cold formed steel cannel beams loaded eccentrically at mid-span. Tese results and tose of 10 concentrically loaded beams were compared wit analytical predictions and wit simple design formulas. Te test results clearly sowed tat te strengts of te un-braced cold formed cannels reduces as te eccentricities of te concentrated loads increase. Te reduced strengts were greater wen te sense of te load eccentricity was positive (above te sear centre), in wic case te stress in te compression flange lip was reduced, and final failure occurred by local buckling of te compression flange-web junction. Te strengts were lower wen te eccentricity was negative, so tat te stress in te compression flange lip was increased, and final failure was by local buckling of te compression lip. Te senses and magnitudes of te initial eccentricity and twist ad teir greatest effects on point loaded beams. Analytical predictions were made by a finite-element metod tat allowed for initial out of strengten and twist and residual stresses, as well as for te beam material, geometry, loading, and restraint conditions. Te analytical predictions sowed close agreement wit te test results. An extended series of analytical predictions were made and used to develop interaction equations to approximate te strengts of eccentrically loaded un-braced 100 mm deep x 1.9 mm tick C sections. Te simple interaction equations developed provided approximate lower bounds to te test and analytical strengts and were suitable for use in design..3.4 Distortional Buckling Tests on Cold-Formed Steel Beams by Yu and Scafer Tis paper by Yu and Scafer (00) covered te set-up of te distortional buckling tests, te test results, finite element analysis and discussion of te current design metods for laterally braced coldformed steel beams failing due to local or distortional buckling wen te compression flange is not restrained by attacment to seating boards. Tests were carried out by on a wide variety of standard laterally braced C and Z beams in wic te compression flange was unrestrained over a distance of 64 inces (1.6 m) and wic indicated tat distortional buckling is te most likely failure mode. Distortional failures occurred even wen local buckling was at a lower critical elastic moment tan distortional buckling. 9

66 .3.5 Local Buckling Tests on Cold-Formed Steel Beams by Yu and Scafer In tis paper, Yu and Scafer (00) described a series of flexural tests on cold-formed C and Z section, wit details selected specifically to ensure tat local buckling was free to form, but distortional buckling and lateral-torsional buckling were restricted. Te members selected for te tests provided systematic variation in te web slenderness (/t w) wile varying oter relevant nondimensional parameters (/b, b/t, d/t w, d/b). Initial analysis of te completed testing indicates tat overall test-to-predicted ratios for AISI, S136, NAS, and te Direct Strengt Metod are all adequate, but systematic differences were observed. Overall, te test results indicated tat te design metods in AISI 1996, S , and te NAS- 001 provided adequate strengt predictions. However, te overall agreement was sligtly skewed by a number of quite conservative predictions for non-slender members tat ad observable inelastic reserve capacity (M test /M y). Among te considered metods, te Direct Strengt Metod was found to provide te best test-to-predicted ratio for bot slender and non-slender specimens..3.6 Direct Strengt Design of Cold-Formed Steel Members wit Perforations by Moen Te current design metods available to predict te strengt of cold formed sections wit openings are prescriptive and limited to specific perforation locations, spacing, and sizes. Te Direct Strengt Metod (DSM), is a relatively new design metod for cold formed steel members validated for members witout openings. It predicts te ultimate strengt of cold formed steel columns or beams using te elastic buckling properties of te member cross-section (plate buckling) and te Euler buckling load (flexural buckling). Tis researc project by Moen (006) extended te use of DSM to cold formed steel beams and columns wit web perforations. Buckled mode sapes unique to members wit openings were categorized. Parameter studies demonstrated tat critical elastic buckling loads eiter reduce or increase due to te presence of openings, depending on te member geometry and opening size, spacing and location..3.7 Numerical Study of Cold-Formed Steel Beams Subject to Lateral-Torsional Buckling by Kankanamge, Nirosa, Maendran and Maen Tis study by Kankanamge et al. (011) focused on te use of cold-formed steel sections as flexural members subject to lateral torsional buckling. For tis purpose, a finite element model of a simply supported lipped cannel beam under uniform bending was developed and was validated using available numerical and experimental results and used in a detailed parametric study. Te moment capacities results were ten compared wit te predictions from te current cold-formed steel codes of Australia, New Zealand, Nort America and Europe. Te results sowed tat, te current design 30

67 rules in AS/NZS 4600 are un-conservative for lipped cannel beams subject to lateral-torsional buckling and terefore new design equations were proposed. Te new design equations were sown to predict te moment capacities accurately. Te BS 5950 Part 5 design equations were also found to be un-conservative. However, te BS EN design equations were found to be accurate for cold formed steel beams wit ig slenderness values wile teir predictions for beams wit intermediate slenderness were over-conservative. Terefore, a new buckling curve was proposed using BS EN equations for cold-formed steel beams..3.8 Experiments on Cold-Formed Steel C-Section Joists wit Unstiffened Web Openings by Moen, Scudlic, and Heyden In tis investigation by Moen et al. (01), tests were conducted on cold-formed steel C-section joists wit rectangular un-stiffened web openings. Te conclusion of te tests stated tat, un-stiffened web openings decreased te joist capacity and amplified distortional buckling deformation for te cold-formed steel C-section joists considered in tis study. Te distortional buckling deformation was accompanied by un-stiffened strip buckling of te compressed web wen opening dept was approximately two-tirds te web-dept. Wen te opening depts approaced te web-dept, unstiffened strip buckling was suppressed, and sudden local-strut buckling of te compressed flange above te opening occurred. Tis mode was also identified by finite strip elastic buckling analysis of te net section..3.9 Tests on Cold-Formed Steel Beams wit Holes by Rad, Moen, Wollmann and Cousins Te objective of tis study by Rad et al. (013) was to investigate te validity of te proposed modified Direct Strengt Metod (DSM) for cold-formed steel beams wit web openings. Four types of beams were evaluated in tis study. Te opening depts were varied trougout te study. In order to acieve te researc objective, a tin sell finite element eigen-buckling study of te beams was performed using ABAQUS. Te critical buckling modes were identified and te load corresponding to eac mode sape was recorded. Te study investigated te influence of te web openings on te beaviour as well as te failure mecanism of te beams. Load-displacement plots were generated and te tested capacities of te beams were evaluated and compared. Results from te elastic buckling and te tests were used to evaluate te modifications made to te DSM local and distortional curves. Te conclusions of tis researc are summarized below: Distortional buckling mode was determined as te dominant buckling mode causing failure of te 8 and 10 inc (00 to 50 mm) deep beams wile te 1 inc (300 mm) beams failed due to an interaction of local and lateral-torsional buckling modes. A mixing of te local and te distortional buckling modes was observed in te beaviour of te beams incorporating web openings. Crosssection dimensions induced global imperfections in te specimen reduced te tested capacity wit 31

68 an increasing eigt of te beams. Evaluation of te test data demonstrated tat an increase in opening dept results in a decrease in te tested capacity as well as te post-peak ductility. Te modified DSM equations could not be evaluated wit te test data acieved troug tis researc. Additionally, it was observed tat te DSM equations do not take into account te influence of member global imperfections created by cross-section distortion during manufacturing Experimental Studies of te Sear Beaviour and Strengt of Lipped Cannel Beams wit Web Openings by Keertan, Maendran A study was made by Keertan and Maendran (013) on te sear beaviour of lipped cannel beams wit web openings. A total of 40 sear tests was carried out to investigate te sear beaviour and strengt of cannel beams wit web openings. Simply supported test sections wit aspect ratios of 1.0 and 1.5 were loaded at mid-span until failure. Tis paper presents te details of tis experimental study and te results of teir sear capacities and beavioural caracteristics. Test specimens were selected to fail in sear and all tree sear failures, namely elastic sear buckling, inelastic sear buckling and sear yielding, were simulated in tis experimental study. 3 sear tests were conducted wit straps to eliminate te flange distortion due to te presence of unbalanced sear flows and 8 sear tests were conducted witout straps to investigate te effect of flange distortion on te section resistance. Te test results sowed tat te current design rules were very conservative for te sear design of lipped cannel beams wit web openings. Improved design equations were proposed to predict te sear strengt lipped beams wit web openings based on te test results from tis study. A reduction factor metod was developed wic is applied to te sear capacity of te same section witout web openings Summary of te general review on previous papers and researc Te effect of web openings on te resistance of cold formed beams as been researced but tere is a lack in te available information on members subject to local buckling around te openings. Terefore, furter investigation and testing are required to develop a relationsip between dept, tickness, lengt, diameter of te opening, grade of te steel and te spacing of te openings as well as preparing design recommendations for beams wit large openings. Application rules are required to BS EN and BS EN for galvanized steel and stainless steel respectively. 3

69 CHAPTER 3 STAINLESS STEEL AS A STRUCTURAL MATERIAL 3.1 Stainless Steel Structural Applications Introduction Stainless steel is te name given to a family of corrosion and good termal conductivity steels containing a minimum of 10.5% cromium. A wide range of stainless steels wit progressively iger levels of corrosion resistance and strengt wit a variety of grades as a result of te controlled addition of alloying elements is available, eac offering specific attributes in respect of strengt and ability to resist different environments. Te cost of stainless steel sections can be controlled by selecting te adequate steel grade for te required application witout being unnecessarily igly alloyed (Baddoo and Burgan, 001). Te designation systems adopted for stainless steel in BS EN are te European steel number and steel name, for example, te former grade 304L as a steel number , were: Denotes steel Denotes one group of Individual grade stainless steel identification Standard steel grades of and are now extended by te greater use of (Duplex) grades, wic ave iger strengt and are available in relatively tin strip form. Duplex grades are advantageous in general building applications. Wit cromium contents above 10.5% and in te presence of air or any oter oxidising environment, a transparent and tigtly aderent layer of cromium-ric oxide forms on te surface of te steel. Altoug te film is very tin (about mm), it is bot stable and non-porous, tus preventing te steel from reacting furter wit te atmospere, wic is called a passive layer. Te stability of tis passive layer depends on te composition of te steel, its surface treatment and te corrosive nature of its environment. Its stability increases as te cromium content increases and is furter enanced by alloy additions of nickel and molybdenum. Te grades of stainless steel based on teir nickel and cromium content can be classified into te following five basic groups: Austenitic stainless steels Te most widely used types of stainless steel and based on 17 to 18% cromium and 8 to 11% nickel additions wit a good corrosion resistance and ig ductility. Austenitic steels are amenable to cold forming and are readily weldable. Tey ave significantly better tougness over a wide range of temperatures, compared wit standard structural grades and 33

70 can be strengtened by cold working, but cannot be strengtened by eat treatment. Te corrosion performance can be enanced by additions of molybdenum (Baddoo and Burgan, 001: 1). Ferritic stainless steels Te cromium content of te most widely used Ferritic stainless steels is between 10.5% and 18%. Ferritic stainless steels contain less nickel tan Austenitic grades, terefore, are less ductile, less formable and less weldable tan Austenitic stainless. Tey can be strengtened by cold working, but to a more limited degree tan te Austenitic grades and tey cannot be strengtened by eat treatment. Ferritic stainless steel sections are not as corrosion resistant as te Austenitic stainless steels and terefore, teir applications are limited to indoor components (Baddoo and Burgan, 001: 1). Duplex stainless steels Duplex stainless steels ave a mixed microstructure of Austenite and Ferrite, and so are sometimes called Austenitic Ferritic steels. Tey contain 1 to 6% cromium, 4 to 8% nickel and 0.1 to 4.5% molybdenum additions offering te combination of relatively ig strengt and good corrosion performance compared to te Austenitic and Ferritic steels. Altoug Duplex stainless steels ave good ductility, teir iger strengt results in more restricted formability, compared to te Austenitic grades. Cold working can be used to strengten Duplex steel sections, but like te Austenitic and Ferritic on wic tey are based, tey cannot be strengtened by eat treatment. Te modern compositions of Duplex stainless steels ave good weldability and good resistance to stress corrosion cracking and ig fatigue strengt. Tey are currently used in te cemical and offsore industries for tubing, safts and valves as well as for components in desalination plants and ave also been used for tension bars and pins in te construction industry (Baddoo and Burgan, 001: ). Martensitic stainless steels Martensitic stainless steels ave a similar microstructure to Ferritic and structural Carbon steels but, due to teir iger carbon content, can be strengtened by eat treatment unlike any oter stainless steel type. Martensitic stainless steels ave a similar corrosion resistance to tat of Ferritic grades but teir ductility is more limited tan oter grades. Altoug most Martensitic stainless steels can be welded, tis may require pre-eat and post weld eat treatments, wic can limit teir use in welded components. Te low corrosion resistance of Martensitic stainless steels limits te range of suitable applications to components suc as valves and knife blades (Baddoo and Burgan, 001: ). 34

71 Precipitation ardened steels Precipitation ardened steels can be strengtened by eat treatment to very ig strengt. Te strengtening mecanism is different from tat in te Martensitic grades; due to te lower carbon levels, te strengt after eat treatment of precipitation ardened steels is generally not as ig as in te Martensitic grades, but te tensile strengt and tougness can be expected to be better. Tese steels are not normally used in welded fabrication. Te corrosion resistance of tese steels is generally better tan te Martensitic or Ferritic grade and is similar to te 18% cromium, 8% nickel Austenitic grades. Altoug tey are mostly used in te aerospace industry, proprietary grades suc as FV.50B ave been used for certain eavy duty connections in buildings as well as for tie-bolts and reinforcing bars (Baddoo and Burgan, 001: ). Classification of stainless steel types is sown in Figure 3.1, wile te various types of stainless steel covered by BS EN are presented in Table 3.1 in wic te former grade is sown in te second column and te corresponding steel number is sown in te final column. Te most commonly used are te Austenitic grades and % Ni 0 15 Austenitic steels 10 Ferriticaustenitic steels 5 Precipitation ardening steels Figure 3.1: 0 10 Martensitic steels % Cr Ferritic steels Classification of stainless steel according to nickel and cromium content (Design Manual for Structural Stainless Steel) 30 35

72 Table 3.1: Grades of stainless steel included in BS EN (Design Manual for Structural Stainless Steel, 010) Formal Grade Designated grades to BS EN using te European steel number and name 301LN Cromium-nickel Austenitic L , LN L , 1.443, Cromium-nickel-molybdenum Austenitic 316LN Ti LN L Super Austenitic % molybdenum Duplex weldable Ferritic Refer to Clasue for te definition of te European steel numbers and names 3.1. Applications of stainless steel Stainless steel is used for many purposes worldwide, for arcitectural and structural engineering applications, suc as cladding, andrails, structural sections, reinforcement bars, beams, columns, lintels and masonry supports, as sown in Figure 3. (Gardner, 005). Stainless steel usage as increased since te 1990 s, for example in te facade of te Petronas Twin Towers in Kuala Lumpur, Malaysia, structural elements of te Sanomatalo in Helsinki, Finland, te upper facade of te Crysler Building in New York (Figure 3.3), te Gateway Arc in St. Louis and te Tames Barrier in London. Stainless steel as also been used in te construction of pedestrian bridges, suc as, te BP Bridge in Cicago, USA and te Double Helix Bridge, as sown in Figure 3.4 and

73 Figure 3.: Stainless steel beams and columns (ttp:// Figure 3.3: Crysler Building in New York and Petronas Twin Towers sows stainless steel cladding (ttp:// 37

74 Figure 3.4: Bp pedestrian bridge in Cicago, USA (ttp:// Figure 3.5: Double Helix Bridge was built using stainless steel tubes (ttp:// Basic stress-strain beaviour Stainless steel yield strengts are generally quoted in terms of a proof strengt defined for a particular offset of permanent strain as indicated in Figure 3.6 ( 38

75 N/mm² 600 E , 400 0, / Carbon steel (grade S355) 00 E 0 0,00 0,005 0,010 0,015 Figure 3.6: Stress strain beaviour of stainless steel and carbon steel ( For te design purposes, te equivalent yield point is usually defined at 0.% plastic strain (0.% proof stress). Ramberg-Osgood proposed te most commonly used relationsip between stress n σ σ ε E 0 σ (Equation 3.1) 0. Were: ε E 0 σ σ 0. Steel strain, Young s modulus Steel stress Material 0.% proof stress n Strain ardening component. Te value of n may be obtained from te ratio of te stress at te limit of proportionality (conventionally te 0,01% proof strengt, σ 0,01 ) to te 0,% proof strengt, σ 0,, as follows: log(0.05) n (Equation 3.) logσ 0.01 σ 0. Tus te ratio σ 0, 01 /σ 0, may also be used as an indicator of te degree of non-linearity. Table 3. sows te averaged stress-strain caracteristics obtained from te test programme carried out for te First Edition of te Design Manual of Stainless Steel. 39

76 Table 3.: Representative values of stress-strain caracteristics for materials in te annealed condition ( 3. Design of Stainless Steel Beams to BS EN Classification of cross-sections Stainless steel sections are generally classified for local buckling as Class 3 or 4 sections to BS EN 1993, particularly using iger strengt grades, and so are susceptible to local buckling around te openings, wic as to be taken into account. 3.. Effective widts of elements in class 4 cross-sections Te properties of Class 4 cross-sections may be establised by calculation te effective widts of te component parts in full or partial compression. Te effective area of eac element is te effective breadt b eff calculated below multiplied by te element tickness. Te effective widts of elements in full compression (ψ =1) or partial compression (ψ < 0) may be obtained from Table 3.3 for internal elements. Te effective widts of flange elements in compression may be based on te stress ratio ψ determined for te gross cross-section. Te reduction factor ρ may be calculated as follows for stainless steel sections: ρ but 1.0 (Equation 3.3) λ λ p p For cold formed outstand element ρ but 1.0 (Equation 3.4) λ λ p For welded outstand element ρ but 1.0 (Equation 3.5 λ λ p p p 40

77 Were λ p is te element slenderness defined as: b t λp (Equation 3.6) 8.4ε k σ in wic: t is te relevant tickness k σ b is te buckling factor corresponding to te stress ratio ψ is te relevant widt as follows: b = d for webs except Rectangular Hollow Section (RHS) b = b for internal flange elements (except RHS) b = c for outstand flanges ε is te material factor. Generally, te neutral axis of te effective section will sift by a dimension e compared to te neutral axis of te gross section. Tis sould be taken into account wen calculating te properties of te effective cross-section Flange curling Te effect on te load bearing resistance of curling (i.e. inward curvature towards te neutral plane) of a very wide flange in a profile subjected to flexure, or of a flange in an arced profile subjected to flexure in wic te concave side is in compression, sould be taken into account unless suc curling is less tan 5% of te dept of te profile cross-section. If te curling is larger, ten te reduction in load bearing resistance, for instance due to a reduction in te lengt of te lever arm for parts of te wide flanges, and te possible effect of te bending of te webs sould be taken into account. Widt-to-tickness ratios of flanges in typical stainless steel beams are unlikely to be susceptible to flange curling. 41

78 Table 3.3: Internal compression elements for uniform and non-uniform stress (BS EN , 006) 3..4 Resistances of cross-sections subject to sear Te plastic sear resistance of a cross-section, V pl,rd may generally be taken as: Were: V pl, Rd = A v(f y/ 3)/γ M0 (Equation 3.7) A v A v A b r t f t w is te sear area wic may be taken as follows: = A bt f + (t w + r) t f for rolled cannel sections wit loads parallel to web is te cross-sectional area is te overall section breadt is te root radius is te flange tickness is te web tickness (if te web tickness is not constant, t w sould be taken as te minimum tickness) 4

79 3.3 Stainless Steel Design Manual s Recommendation for te Design of Stainless Steel Beams (SCI Publication P91) Te following recommendations for te design of stainless steel beams are based on previous tests and researc results and were presented in SCI Publication 91, as follows: Lateral-torsional buckling For an idealised perfectly straigt elastic beam, tere are no out-of-plane deformations until te applied moment reaces te critical moment M cr wen te beam buckles by deflecting laterally and twisting. Te failure of an initially straigt slender beam is initiated wen te additional stress induced by elastic buckling reaces yield. An initially straigt beam of intermediate slenderness may yield before te critical load is reaced, because of te combined effects of in-plane bending stresses and residual stresses, and may subsequently buckle in-elastically. Real beams differ from te idealised beams in muc te same way as real compression members differ from idealised struts. In a strut, te compression is generally constant trougout its lengt, but in a beam te bending moment and terefore te force in te compression flange usually varies along its lengt. Te variation of te flange compression along te beam affects te buckling load of te member. Tis is taken account wen calculating te slenderness LT. Likewise, te effect of various restraint conditions and weter te load is destabilising or not are also accounted for in te calculation of LT. Te design line proposed in te First Edition of te Stainless Steel Design Manual for cold formed sections was based on an imperfection coefficient of = 0.34 and a limiting slenderness 0, LT = 0. (as compared to = 0.1 and λ 0, LT = 0. for cold formed carbon steel members in BS EN ). However, carbon steel data suggested tat te plateau region is muc longer and in BS EN , no allowance needs to be made for lateral torsional buckling wen λ For stainless steel, tere were insufficient data to support tis and a more conservative cut-off requirement wen λ 0.3 was introduced. Since te buckling curve recommended for stainless steel cold formed LT sections ( = 0.34) was te next lower curve to tat for carbon steel cold formed sections ( = 0.1), it was suggested tat = 0.76 may be suitable for welded stainless steel sections (compared to = 0.49 for welded carbon steel sections). LT 3.3. Sear resistance Te general approac for establising te sear resistance of webs is based on te simple post-critical metod of EN In common wit oter forms of plate buckling, slender plates under sear 43

80 are able to reac ultimate strengts iger tan te elastic critical stress values. Te metod takes advantage of tis in te design line for carbon steel. Tis enancement is also to be expected for slender stainless steel webs, as te stresses are low. However, were te web slenderness is suc tat te elastic critical stress is approximately equal to te yield stress (at λ w = 1.0), a relatively large reduction in sear strengt occurs Web crusing A test programme was carried out to measure te web crusing and crippling resistance of nine grades welded I-section beams wic were subjected to concentrated point loads (Sali et al. 011). On five beams, te load was applied far from te girder end (patc loading) and on te remaining four beams te load was applied near an unstiffened end (end patc loading). For te patc loading, te beams were doubly symmetric, wit w/t w varying from 50 to 110 and te lengts of te beams varying from 996 mm to 168 mm. Bot ends of te beams were stiffened wit vertical steel plates. Loading plates of widt 40 mm and 80 mm were used. Te load was applied on te upper flange, centrally over te web at te mid-span of te simply supported beam. Te results indicated tat te design procedure given in BS EN , gives te best agreement between test and predicted values for bot patc loading and end patc loading Determination of deflection of stainless steel beams Te load-deflection curve for stainless steel is affected by te non-linear material stress-strain relationsip and may be influenced by local buckling effects in te compression flange, terefore, te calculation of te deflection of stainless steel members is a complex matter. In te case of carbon steel members, te modulus is constant (i.e. equal to Young s modulus), and for stainless steel members, te (tangent) modulus may vary trougout te beam according to te value of stress at eac section. 3.4 Previous Researc on Stainless Steel Beams Te Lateral Torsional Buckling Strengt of Cold-Formed Stainless Steel Beams by Breden amp' and Den Berg In tis study, Breden and Den Berg (1994) investigated te lateral torsional buckling strengt of stainless steel beams. Te sections under consideration were stainless steel lipped cannel sections tat were spot-welded back to back to form doubly-symmetric lipped I-beams. Te purpose of tis study was to compare te experimental lateral torsional buckling strengts of doubly-symmetric beams to te teoretical predictions proposed by te ASCE Specification for te Design of Cold- Formed Stainless Steel Structural Members (1991) in wic, te tangent modulus E (defined as te slope of te tangent to te stress-strain curve at eac value of stress) was used to determine te lateral torsional buckling strengt of stainless steel beams. 44

81 It was concluded from te test results on cold-formed stainless steel doubly symmetric lipped I-beam tat, te ASCE metod using te tangent modulus was very conservative compared to test results. However, as a result of previous work on lateral torsional buckling of stainless steel beams, it was suggested tat te tangent modulus approac sould be used for predicting beam strengt of sections using type 3CR1 corrosion resisting steel Distortional Buckling of Cold-Formed Stainless Steel Sections: Experimental Investigation by Lecce and Rasmussen In tis paper by Lecce and Rasmussen (006), a total of 19 distortional buckling tests on simple lipped cannels and lipped cannels wit intermediate stiffeners was carried out. Austenitic 304, Ferritic 430, and Ferritic-like 3Cr1 cromium weldable steel stainless steel alloys were brakepressed from steel strip into simple-lipped cannels and lipped cannels wit intermediate stiffeners were considered for testing. Te material investigation of te different stainless steel alloys used was essential for te design of suitable sections and to assess te distortional buckling beaviour of te sections tested. Data of interest were te strengts in tension and compression as well as te strengt enancement due to cold working of te brake-pressed corners wic were determined by carrying out tension and compression coupon tests on te flat seet material used in te sections. All tests failed by distortional buckling except for one test in wic, signs of local and distortional buckling interaction, wit local buckling occurred at a lower load tan distortional buckling. All distortional buckling tests wit twin sections acieved repeatability of ultimate load witin %, confirming reliability of testing procedures. Te post-ultimate beaviour, owever, was not necessarily identical, even for nominally identical test sections. For example, two sections wit te same nominal cross-section dimensions and lengt obtained te same ultimate load but exibited different post-ultimate beaviour. Tests on bot sections produced te same number of distortional buckling alf-waves but te flanges of one section moved away from te geometric centroid of te section wereas te flanges of te oter section moved in towards te centroid. Terefore, it was suggested tat, te inward or outward movement of te flanges did not ave significant effect on te post-buckling resistance or ultimate load capacity and it was concluded tat te sear deformation effects were not significant, particularly since te ends were fixed. An observation was made by comparing te test results of four sections wit te same flange and web dimensions but different lip depts. Two sections ad a lip dept of 10 mm wereas te oter two ad a lip dept of 14 mm. For te two sections wit 10 mm deep lips, an increase in lengt of approximately 80 mm resulted in a decrease in te ultimate load capacity by approximately 3.% wile for te oter two wit 14 mm deep lips, an increase in lengt of approximately 80 mm resulted 45

82 in a decrease in ultimate load of 1.3%. For column sections wit te same lengt of 600 mm, a relatively small increase of 3% in te gross cross-sectional area increased te section resistance by approximately 16%. It was concluded tat, a more effective lip provided greater stability of te flange and tus greater strengt Distortional Buckling of Cold-Formed Stainless Steel Sections: Finite-Element Modelling and Design by Lecce and Rasmussen Tis paper by Lecce and Rasmussen (006) described te finite-element model calibration for a recent experimental study on te distortional buckling of stainless steel sections. Results sowed tat material nonlinearity and enanced corner properties govern te ultimate load of te section and material anisotropy as little effect. Finite element analyses (FE) were conducted for simple lipped cannels wit r / t ratios of 1 and.5 for Austenitic 304, Ferritic 430, and Ferritic-like 3Cr1 cromium weldable steel stainless steel alloys. FE results sowed tat te effect of enanced corner properties can be significant for relatively stocky sections but ave little to no effect on slender sections. A total of 81 experimental and numerical tests were compared wit te current design guidelines available for cold-formed stainless steel and also cold-formed carbon steel. Te evaluation sowed tat, for simple lipped cannels, te current codes were un-conservative for Austenitic stainless steels and only te currently proposed BS EN 1993 Part 1-4 (BSI 004b) in conjunction wit BS EN 1993 Part 1-3 (BSI 004a), provide reasonably conservative design strengts for ferritic alloys, provided enanced corner properties are ignored in te design Experimental and Numerical Studies of Lean Duplex Stainless Steel Beams by Teofanous and Gardner In tis paper, Teofanous and Gardner (009) carried out a series of material tests on tensile, compressive and corner coupons extracted from cold-formed Lean Duplex stainless steel Square Hollow Sections (SHS) and Rectangular Hollow Sections (RHS) and eigt major axis tree-point bending tests. Te obtained test data were used to develop FE models, upon te validation of wic, parametric studies were conducted. Te analysis of bot experimental and numerical results allowed te effect of local slenderness, aspect ratio and moment gradient on bot te load bearing and deformation capacity of Lean Duplex stainless steel SHS and RHS to be investigated and te suitability of te codified American, Australian/New Zealand and European provisions for Lean Duplex stainless steel flexural members to be assessed. Te current European slenderness limits seem overly conservative for Lean Duplex stainless steel elements and te adoption of te more relaxed slenderness limits proposed by te autors as been sown to be suitable for oter stainless steel grades. Te current American and Australian/New Zealand design procedures ave been sown to more accurately predict te ultimate moment 46

83 resistance of SHS and RHS in Lean Duplex stainless steel. Te recently proposed deformation-based design metod for structural stainless steel cross-sections, termed te Continuous Strengt Metod (CSM), as been found to provide better estimates of te ultimate moment resistance of Lean Duplex stainless steel tan te American, Australian/New Zealand and European specifications Sear Design Recommendations for Stainless Steel Plate Girders by Saliba, Real and Gardner In tis paper, Saliba et al. (014) investigated te beaviour and design of stainless steel plate girders loaded in sear. A review of te existing metods for te design of stainless steel plate girders, and codified provisions was presented. Tirty four tests were carried out on Austenitic, Duplex and Lean Duplex stainless steel plate girders and were used to assess te current sear resistance design equations obtained from BS EN and BS EN Te comparisons indicated tat te design provisions of BS EN were conservative and tat improved results could be acieved by applying BS EN Based on te structural performance data, revised design expressions for te calculation of te ultimate sear resistance of stainless steel plate girders suitable for incorporation into future revisions of BS EN were proposed based on revised expressions for te sear buckling reduction factor tat account for end post rigidity Web Crippling Design of Cold-Formed Duplex Stainless Steel Lipped Cannel Sections wit Web Openings under End One Flange Loading Condition By Yousefi, Lim, Uzzaman, Lian, Clifton and Young In tis paper, Yousefi et al. (016) investigated te effect of circular web openings on te web crippling strengt of cold-formed stainless steel lipped cannel-sections. A total of 74 finite element models of lipped cannel sections wit various dimensions and ticknesses were analysed for te tree stainless steel grades: Duplex EN 1.446, Austenitic EN and Ferritic EN Te web crippling strengts for sections wit circular web openings were divided by tat for solid web sections wic led to a strengt reduction factor (R) due to te openings. Te effects of parameters, suc as te web opening diameter (a), lengt of bearing plates (N) and location of web openings in te web (x) on web crippling strengt and on te reduction factor were reported. Increasing te a/ ratio from 0. to 0.6 reduced te strengt of te Ferritic grade by 9% (reduction factor increased by 9%). It was also observed tat te reduction in strengt is more sensitive to te orizontal distance of te web opening to te bearing plate and decreasing te x/ ratio from 0.6 to 0. increased te strengt by 7%. It was also observed tat, te failure loads obtained for te sections wit flanges fastened to bearing plates were on average 30% iger tan te failure load for sections wit unfastened flanges. 47

84 From te results of te finite element parametric study, four new web crippling strengt reduction factor equations were proposed for te cases of bot flange unfastened and flange fastened to te bearing plates. A reliability analysis was undertaken of te proposed reduction factor equations and it was sown tat te proposed strengt reduction factors are generally conservative and agree well wit te finite element results Summary of te general review on previous papers and researc From te previous researc and investigations on stainless steel sections, it can be concluded tat, te current codes are un-conservative for Austenitic stainless steels and only te currently proposed BS EN 1993 Part 1-4 (BSI 004b) in conjunction wit BS EN 1993 Part 1-3 (BSI 004a) provide reasonably conservative design strengts for Ferritic alloys, provided tat enanced corner properties are ignored in te design. 48

85 CHAPTER 4 DESIGN OF BEAMS WITH CIRCULAR WEB OPENINGS 4.1 Introduction Te sape, size, location, spacing and number of openings ave a great influence on te beaviour of structural members containing openings. Te beaviour of elements wit large web openings results in a reduction of te member s load carrying capacity. Flexural strengt is only marginally affected by te web openings wereas te sear strengt is greatly reduced. Te web buckling beaviour of beams wit web openings depends on te relative location of tese opening to te supports. At failure, te elements around te web openings are subjected to ig combined stresses generated by axial forces from te global bending action, sear forces and local moments arising from te transfer of sear known as Vierendeel bending action. Te magnitude of eac of tese forces depends on te location of te openings in te beam span. For web openings, failure occurs eiter wen plastic inges are generated around te openings in zones of ig sear and low moment (at te edge of te beam), or wen tensile yield exists in te lower web- flange in zones of lower sear and ig moments (centre of te beam). Te metods of analysis of large web openings were derived firstly for ot rolled steel sections were local buckling does not occur. In a cold formed section, te failure mode is different depends more on buckling. Te following section presents information on members acting in bending and sear considering local buckling effect. 4. Literature Review of te Design of Beams wit Web Openings to BS EN Pure bending of beams wit Web Openings Te bending resistance of flexural members is differentiated according to weter or not te member is laterally braced. Bending resistance may be governed by lateral buckling. Te lateral buckling resistance of a flexural member is sligtly reduced by te presence of web openings. Te crosssection bending resistance (M c) can be evaluated by its plastic resistance based on te reduced dept of te section, as follows: M pl =f y A Tee eff (Equation 4.1) A Tee eff f y is te cross-sectioned area of te Tee is te effective dept of te elastic centroids of te Tees is te steel yield stress 49

86 4.. Vierendeel bending of beams wit web openings Vierendeel bending occurs due to te transfer of sear forces across an opening wic requires te development of local moments in te Tees. Te flexural capacity of te upper and lower Tees under Vierendeel bending is critical. Global bending action results in compressive and tensile forces in te top and bottom Tees, wic are at maximum at mid-span for a simply supported beam. Beams wit web openings ave two basic failure models, depending on te geometry of te openings. Tey are: Plastic tension and compression stress blocks in te lower and upper Tees in te regions of ig overall bending. Failure occurs wen tese forces exceed te axial capacity of te Tee (Figure 4.1). Vierendeel action due to te formation of plastic inges at te four corners of te openings in regions of ig sear. Te sear force across a cell induces secondary bending moments in te top and bottom Tees known as Vierendeel bending. Failure occurs wen te bending resistance of te Tees are exceeded (Figure 4.). Yield in compression eff Yield in tension Figure 4.1: Yielding due to bending of te Tee sections above and below openings Plastic inges Figure 4.: Generation of plastic inges around openings due to local Vierendeel bending subject to a sear force 4..3 Effective of lengt of web openings Te effective lengt te openings for Vierendeel bending may be determined from te following simplified relationsips. For Vierendeel bending, te effective lengt o of te opening may be taken 50

87 as te actual lengt for rectangular openings, 0.45 o for circular openings as proposed by Redwood (1967) and te centre to centre spacing of openings, s increased by 0.5 o for elongated openings. Te effective lengts are terefore: eff = 0.45 o for circular openings eff = o 0.5 o eff = o for elongated openings for rectangular openings Te effective dept of circular openings (but not elongated openings) may be reduced to 0.9 o for te calculation of te Vierendeel bending resistance of te Tees. Te local bending moment (M v) due to Vierendeel action is given by: M (Equation 4.) v V Ed eff Were: V Ed is sear force at te openings eff is effective lengt of te opening eff =0.45 o eff.o = 0.9 o Figure 4.3: Equivalent rectangular opening for te circular openings (Redwood, 1967) 4..4 Sear beaviour of beams wit web openings Te sear strengt of te beam webs is governed by eiter yielding or buckling of te web element, depending on te dept-to-tickness ratio /t w and te mecanical properties of te steel. For beam webs aving small /t w ratios, te nominal sear strengt is governed by sear yielding. Wen te /t w ratio is large, te nominal sear strengt is controlled by elastic sear buckling. For beam webs aving moderate /t w ratios, te sear strengt is based on inelastic sear buckling. Te sear strengt of a cold formed steel web element will decrease due to te presence of web opening and te amount of web dept displaced by te openings. Te transfer of sear becomes critical in te 51

88 web of te top and bottom Tees were web openings are located close to supports or wen te beam is subjected to point load. Two modes of beam sear failure sould be cecked. Te vertical and te orizontal sear capacities of te upper and lower Tees. Te factored sear forces in te beam sould not exceed P vy were: P vy = f y (- o) t w (Equation 4.3) In addition, te orizontal sear in te web-post sould not exceed P v were: P v = 0.6f y s o t w (Equation 4.4) Te factor is due to non-uniform sear flow on an elastic condition. Were: s o t w o is edge to edge spacing is web tickness is te dept of te opening is te dept of te beam 4..5 Horizontal sear force in beams wit web openings Te cange in bending moment across a pair of openings generates a orizontal sear force across eac web-post. Tis effect is illustrated in Figure 4.4 sows te top alf of te beam between te centre lines of adjacent openings. Figure 4.4: Horizontal sear at te mid web-post between two openings 5

89 Te orizontal sear force and stress are obtained as follows: s V V (Equation 4.5) eff τ V (s )t (Equation 4.6) o w Were: V τ s is te orizontal sear force across te web-post is te orizontal sear stress at te mid of te web-post is te centre to centre spacing of te opening eff is te effective dept= (-y e ) y e is te neutral axis dept of te Tee section 4..6 Deflection of beams wit web openings Te bending deflection is obtained by calculating te effective second moment of area of te section (I c). Te sear deflection can be estimated by considering te deformation due to local Vierendeel bending action in te bottom cord (Tee section) as follows: 3 ff V e v 6EI (Equation 4.7) T Were: V is average sear force at te opening eff is effective lengt of te opening I c is in plane second moment of area of a Tee section Terefore, te total mid-span deflection is a combination of sear and bending deflection as influenced by te openings. Te effect of te openings on deflection is explained in more detail in Capter 6 were predicted formulas are presented to obtain te additional deflections for beams depend on te number and te sape of te openings as on te applied load. 4.3 Proposed Tangential Stress Metod Equilibrium of forces around closely and widely spaced circular openings Te design of steel beams wit closely spaced circular openings depends on te local stress distribution around te openings. Te model for determining te stresses around an opening is based on equilibrium of forces acting on a particular plane at an angle relative to te vertical axis at te centre-line of te opening and it was presented by Lawson (011). It may be assumed initially tat te normal stresses acting on tis plane increase linearly from te junction wit te flange to a 53

90 maximum at te edge of te opening. For stocky webs, some local plasticity may be developed near te edge of te opening, wic will modify te stress distribution. Te variation of compressive stress around te edge of te opening determines te tendency for local buckling. It is expected tat tese stresses will be at teir igest for 5 to 30 o to te vertical and will reduce to zero for = 90 o. Two cases for equilibrium of forces on any plane may be considered. Te first case is wen te planes for adjacent openings do not over-lap and te second case is were tey overlap. Te critical plane is were te normal stresses acting on te edge of te opening are maximised. Te local compression or tension stresses may be given as a function of V Ed and te spacing: diameter ratio of te opening, s/ o Equilibrium case 1: tan -1 (s/ o) (widely spaced web openings) Tis case is illustrated in Figure 4.5 sows an idealised stress pattern in te web. Tis is assured to be a linear variation from a maximum at te edge of te opening to zero at te flange. Equilibrium of orizontal forces acting on te inclined plane from te centre-line of te opening is defined by: N N cos V sin (Equation 4.8) f w Were: N w V N f is te compression or tension force acting on plane measured from te centre of te opening is te sear force on te same plane is te increase in compression or tension force in te flange of te section at te projection of tis plane is te angle of te plane to vertical Equilibrium of vertical forces on a plane, is defined by: 0.5V Ed N sin θ V cos θ (Equation 4.9) w θ Were: V Ed is te applied sear force on te section (assumed to be constant between te pair of openings). Equilibrium of moments acting about point A is defined geometrically by: Were: 0.5VEd 0.5tan θ Nwx (Equation4.10) 3 is te opening diameter and x is te lengt of te sear plane, defined by: o 54

91 x 0.5 o (Equation 4.11) cosθ S N f CL N f + N f CL N A f + V Ed (s/) 0.5 V Ed V V 0.5 V Ed N w x Nw 0.5 CL Case 1: V wp 0.5o 0.5o -1 tan (s/) Figure 4.5: Equilibrium of forces acting on plane wen tan -1 (s/) Te factor of /3 is based on an assumed elastic distribution of compression or tension stresses acting on te plane. Solving Equations 4.10 and 4.11 leads to te following expression for te normal force, N w on te plane: N w 1.5sin θ ( 0.5VEd ) (Equation 4.1) 1(o /)cosθ 1.5sin θ By re-arranging Equation 4.9 and substitute N w for ( 0.5VEd ) as per Equation 1(o /)cosθ 4.1, te sear force V on tis plane is given by: V θ 1 1.5sin θ tan θ (0.5VEd ) (Equation 4.13) cos θ 1(o /)cos θ It may be sown tat V 0 for = 30 o, and tat V canges in direction for > 30 o. By substituting Equation 4.1 and 4.13 into Equation 4.8, te increase in axial force in te flanges is given by: ΔN f 0.5 ( /) cos θ o 0.5VEd tan θ (Equation 4.14) 1(o /) cos θ 55

92 Te tension or compression stress acting on te plane based on a linear stress variation, is given by: = N w /(x t w) (Equation 4.15) By substituting Equation 4.11 and 4.1 into Equation 4.15, te edge tension or compression stress acting on plane is given by: w Ed ( 1.5sin θ 1 ( /)cosθ V σ (Equation 4.16) t o For cases controlled by local buckling, it follows tat f y. Terefore, wen = f y, Equation 4.16 tends to te limit of: V Ed 1(o /) cos θ t 1.5sin θ f w y (Equation 4.17) By differentiating wit respect to in Equation 4.16, reaces its maximum value wen is given by te following approximate equation: 1 θ cos [0.5(o /) 0.71] (Equation 4.18) Te pure sear resistance of te perforated web at te opening given by: V Rd t ( )f / 3 (Equation 4.19) w o y It follows tat te maximum compression stress around te opening is = 1.48f y, wen te limiting sear resistance of te perforated section, V Rd is reaced. Tis sows tat plasticity develops around te edge of te opening before te pure sear resistance of te perforated section is reaced. For idealised plastic stresses over lengt x, te coefficient of 1.5 in Equation 4.16 becomes 1.0, wic means tat te edge stresses are reduced by 33% Equilibrium case : > tan -1 (s/ o) (closely spaced web openings) Te case illustrated in Figure 4.6 sows idealised stress pattern from a maximum at te edge of te opening to zero at a notional point A, wic is te intercept of te plane, at mid-way between te openings and below te top flange. Te compression force in te top flange is given from overall equilibrium as: N f = 0.5 V Ed (s/) (Equation 4.0) 56

93 S N f N f V Ed(s/) N f + V Ed (s/) V x v y A 0.5 V Ed 0.5 V Ed N w V V Nw 0.5 C L Case : V wp 0.5 o 0.5o -1 > tan (s/) Figure 4.6: Equilibrium of forces acting on plane wen > tan -1 (s/) Equilibrium of orizontal forces acting on plane is defined by: 0.5V ( s / ) N cos V sin (Equation 4.1) Equilibrium of vertical forces is defined by: Ed w 0.5V V N sin V cos (Equation 4.) Ed v w Were: V v is te remaining vertical sear force acting on te web above point A. Equilibrium of moments about point A is defined by: s 0. 5V Ed (s/) y 0. 5V Ed ( ) Nw x (Equation 4.3) 3 Were: y s cot and s x o sin Solving te above equations leads to: 1.5scosθ N w 0.5V o Ed 1 ( /s)sin θ 0.75cos θ V (Equation 4.4) wp 1 ( /s)sin θ o Were: V wp = orizontal web-post sear force = V Ed (s/). 57

94 It is a requirement tat V wp V wp,rd, were V wp,rd is te sear resistance of te web-post between te openings given by: V s o fy / 3 wp, Rd tw (Equation 4.5) Te sear force V acting on te plane is given by: V θ 0.75cos θ 0.5 Vwp (Equation 4.6) (1(o /s) sin θ sin θ Te vertical sear force acting above point A is given by: V (0.5( /s) sin θ (Equation 4.7) o v 1 (s/)cot θ 0.5VEd (1(o /s) sin θ As for Case 1, te compression or tension stress acting on plane may be given by: Tis Equation becomes: σ Nw /(xtw) (Equation 4.8) σ Ed (Equation 4.9) t w V 1.5sin θ 1 ( o / s) sin Terefore, wen = f y, Equation 4.7 tends to: V Ed 1 ( o / s) sin tw f y (Equation 4.30) 1.5sinθ By differentiating wit respect to in Equation 4.6, reaces its maximum value wen te following approximate equation is satisfied: 1 θ sin [0.5(o /s) 0.71] (Equation 4.31) Te maximum compression stress at te edge of te opening wen > 45 0 as a multiple of te vertical sear stress at te opening or te orizontal sear stress at te web post, were tis stress is iger is presented in Table 4.1. In general, te stresses are igest at = 60 0, wen te opening spacing s is less tan te beam eigt,. For o / ratio = 0.7 and s/ = 1, it follows tat is a maximum at = 6 0 to te vertical. In tis case, =.6 f y at te edge of te opening. Tis stress is similar to te =.57f y at = 5 0 for te same o/ ratio. It follows tat te critical zone for local compression stresses around te opening is an arc 58

95 of between approximately 5 0 and 65 0 to te vertical, wic corresponds to a strut lengt of 0.35 o. Te equivalent slenderness of te edge of te opening in compression is 1. 0 / t w. For a limiting slenderness of 30 wen = f y it follows tat o / t w 5 before local buckling may limit te stresses around te opening. Te variation of te local stresses around te openings for an applied load of 5 kn (sear force of 1.5 kn) are illustrated in Figures 4.7 (a) and (b). Table 4.1: Maximum compression stress for Ө > 45 o as influenced by te spacing of te opening cases controlled by orizontal sear in te web-post are saded o/ s/ o =.0 = 1.7 = 1.5 = 1.3 = 57 o = 59 o = 61 o = 64 o v.1 H.47 H.86 H v.15 v.47 H.86 H v 1.61 v. v.86 H v 1.07 v 1.48 v.48 v Were: H=V E (s/)/[(s- o)t w] Te tangential compression or tension stresses, around te edge of te opening may be expressed as a multiple of te vertical sear stress of te opening, or te orizontal sear stress between te openings wicever is te larger. Tese sear stresses are given by: V = VEd /(tw(-o)) fy 3 (Equation 4.3) H = VEd (s /eff ) /(tw(s-o)) fy 3 (Equation 4.33) For s = 00 mm, H = 1.13 V and terefore H controls. For s = 50 mm, H = 0.71 V, and terefore V controls. Te results of te previous equations for edge stress are presented in Table 4. for te above data. For an opening spacing of 00 mm (edge distance = 50 mm), te maximum edge stress is given by.80 H at = 65 o. Wen H = f y/ 3, = 1.61f y, wic implies tat significant plasticity occurs around te edge of te opening. In order for = f y, it is necessary to reduce te applied sear force by 38%. 59

96 For an opening spacing of 50 mm (edge distance = 100 mm), te maximum stress is given by.6 V at = 5 o. Wen V reaces its maximum value of f y/ 3, ten = 1.51 f y, wic sows tat plasticity develops around te opening. Case a: Widely spaced openings, < tan -1 (s/ o) Case b: Closely spaced openings, > tan -1 (s/ o) Figure 4.7: Variation of tangential stress ( N/mm ) around te openings for widely and closely spaced openings for various opening sizes and spacings subject to sear force of 1.5 kn (Sanmugalingam, 010) 60

97 Table 4.: Opening proportions Variation of tangential stresses around te opening as ratio of sear stress V or H Edge stresses at angle to te vertical around te opening s/ o = H = V 1.16 H 1.79 H 1.86 H 1.80 H 1.46 H.04 H.64 H.80 H.76 H 1.81 H H s/ o = 1.67 V 1.64 V.5 V.6 V.54 V 1.77 V 1.13 V 1.58 V 1.63 V 1.47 V 0.89 V 0.88 V = H 4.3. Comparison wit vierendeel bending for equivalent rectangular opening wit te Tangential Stress Metod Te model for Vierendeel bending around circular openings was developed by Redwood, wo defined te dimensions of an equivalent rectangular opening wose corner is at an angle of 6.5 o to te vertical as widt, eff = 0.45 o and dept, o,eff = 0.9 o. Equilibrium of te transfer of sear force V Ed across te opening is defined by: Were: V Ed eff 4 e M (Equation 4.34) M e is te elastic bending resistance web-flange of a Tee section in te presence of tension or compression due to global bending. Assuming tat te elastic neutral axis of te web-flange Tee section is close to te top flange, and te flanges resist te global bending action: M e ( o) t w f y/1 (Equation 4.35) Using te above equations, it follows tat te bending stress and sear force is linked according to: V Ed = 0.74 (-0.9 o) t w b / o (Equation 4.36) Were b is te bending stress in te Tee at te edge of te opening f y. Re-arranging tis formula as a function of te sear stress v in te reduced web section (Equation 4.3), gives a bending stress of: b = 1.35 o( o ) τ v (0.9 ) o (Equation 4.37) 61

98 However, te bending and sear stresses sould be combined to determine te principal bending stresses 1 according to: 1 = ( b + 3 v ) 0.5 (Equation 4.38) Te results for b and 1 are presented in Table 4.3 in comparison to te tangential stresses obtained from equilibrium around te circular opening. Table 4.3: Comparison of Vierendeel bending stresses and tangential stresses for an equivalent rectangular opening o/ Vierendeel bending model Tangential stress b 1 max v.06 v.03 v v.31 v.4 v v.70 v.57 v v 3.5 v 3.13 v σ 1= Principal stress in Vierendeel bending model and max = tangential stress as obtained from te teory It is apparent tat te agreement between te principal stress according to te Vierendeel bending model and te tangential stresses from te model for circular openings is very good (maximum difference = 5%). 4.4 Treatment of Web-Post Buckling in Class 3 or 4 Sections Local buckling around a circular web opening is influenced by: Magnitude and distribution of te igest edge stress around te opening. Opening diameter: steel tickness ratio. Steel strengt and elastic modulus (and for stainless steel, te variation of elastic modulus at iger stresses. For an isolated opening, te beam tests and finite element analyses, as explained later in Capter 7 and 8, sowed tat te buckling wave is symmetric about an angle of 5 o to te vertical, wic is te point of maximum tangential stress around te opening. Te buckling wave extends from approximately 10 o to 40 o in an arc around te opening, wic is equivalent to 0.5 o, were o is te opening diameter (0.5 o corresponds to alf of te radius). 6

99 To allow for te partial fixity of te ends of te wave and te elastic stress variation along te critical plane, te buckling lengt may be taken as: w w 0. Te slenderness of te equivalent strut may be obtained by dividing byt w/ 1 steel tickness, and so: o, were t w is te 1 w / t w 0. 7o / tw (Equation 4.39) Te strut buckling curve b in BS EN may be used for cold formed steel sections, wic corresponds to an imperfection parameter = Te non-dimensional slenderness ratio is defined by: 1 / 1 were E / f y (Equation 4.40) For f y = 350 N/mm and E = 10 kn/ mm, it follows tat 1 = 77. Te strut buckling curve as a cut-off in slenderness ratio of 0., wic corresponds to te steel strengt, f y. Te limiting ratio of o / t w for no local buckling is given by: 0.7 o /t w = 0. x 77 or o / t w Tis cut-off slenderness ratio of 0. is relatively severe for local plate buckling and it is reasonable to increase te limit to 0.4, wic corresponds to o / t w 40 for S80 steel, or 35 for S350 steel, and 30 for S450 steel. Tis equates to a nominal 5% reduction in buckling strengt, wic is considered negligible considering te effects of strain ardening at iger deformations. Te use of buckling curve b leads to te strengt ratios / f y for various o / t w ratios as presented in Table 4.4. Te test beam corresponds to o / t w = 75, and so te reduction in buckling strengt due to local buckling is about % (for S350 steel). In te normal range of 40 o /t w 100, te reduction in compression strengt due to local buckling is given approximately by: /f y = [ ( o/t w)] (35/f y) 0.5 (Equation 4.41) Were: is te maximum edge stress limited by local buckling. Tis same strengt reduction may also be used for stainless steel. 63

100 Table 4.4: o/t w Maximum tangential stress around circular openings to prevent local buckling as a function of o / t w and steel grade S80 steel S350 steel S450 steel / 1 /f y / 1 /f y / 1 /f y V Ed= v (- o)t w 4.5 Conclusion In tis Capter, a metod of calculating te compression and tension stresses around te circular openings is presented. Two cases for equilibrium of forces on any plane were considered. Te first case was wen te planes for adjacent openings do not over-lap and te second case was were tey overlap. Te critical plane is were te normal stresses acting on te edge of te opening are maximised and it was found to be between 5 and 30 degrees to te vertical in beams wit isolated web openings openings. For closely spaced openings, te critical angle was found to be at 65 degrees to te vertical. Te maximum local compression or tension stresses at te critical plane were given as a function of V Ed and te spacing to diameter ratio of te opening, s/ o. Web-post buckling was found to ave a significant effect on te beam resistance, terefore, an approximate metod of calculating te reduced material resistance taking into account te buckling of te web-post between web openings was considered in tis Capter. A simplified Formula to calculate te reduction in te compression resistance was presented (Equation 4.41). 64

101 CHAPTER 5 LITERATURE REVIEW AND DESIGN OF CASTELLATED BEAMS 5.1 Introduction Castellated beams were invented by Geoffrey Murray Boyd, wo was an engineer working in Argentina in te mid-1930 s (Wakcaure et al, 01). Teir use increased in te 1960 s mainly for ligtly loaded beams in roofs and floors Castellated beams profile Castellated beams are formed by a exagonal cut so tat te steel sections are re-welded at teir webposts to create a deeper and ence a stiffer beam. Te precise dimensions of a castellated beam are not fixed, but te commonly adopted profile for te UK is sown in Figure 5.1(a). Te opening eigt is two-tirds of te beam dept and te web-post widt is one quarter of te opening eigt. Te completed section is 50% deeper tan te parent section. A wide castellation in Figure 5.1(b) is also used on te Continent and its sape more closely approximates to a circular opening. Figure 5.1: Typical narrow and wide profiled castellated beam section 65

102 5.1. Applications of castellated beams Castellated beams ave been traditionally used in relatively ligtly loaded secondary beams were sear is not critical, suc as car-park, roofs and some office buildings, as sown in Figure 5.. However, castellated beams ave also been used in oter more eavily loaded structures, suc as road bridges for spans of 0-30 m, as sown in Figure 5.3. Figure 5.: Use of castellated beams in enclosures and car parks (Scerer Steel Structures) Figure 5.3: Use of castellated beams used in a igway bridge Manufacture of castellated beams Castellated beams are created by cutting a saw toot pattern or castellation in te web of a rolled I section. Te tips of te long cuts are ten welded togeter to join te two pieces, as illustrated in Figure 5.4, wic produces a beam of greater dept wit exagonal openings. Te resulting beam as a greater bending stiffness tan te original section witout an increase in steel weigt. However, te presence of te openings in te web affects te structural beaviour of te beam, mainly in sear. Various opening sapes may be created using te same principles. 66

103 Figure 5.4: Cutting of castellated beams (ttp:// & (ttp:// Failure modes of castellated beams Castellated beams are relatively slender and te web openings control teir sear resistance. Kerdal and Netercot (1984) identified six failure modes for castellated beams for wic web-post buckling and lateral-torsional buckling were te major failure modes. Te six failure modes are summarised as follows: a) Formation of flexure mecanism Te section can fail in pure bending in wic te Tee-sections above and below te openings yield and become completely plastic in compression and in tension, as sown in Figure 5.5. Figure 5.5: Flexural failure of castellated beams (Halleux, 1967) b) Lateral-torsional buckling An investigation by Netercot and Kerdal (1984) on castellated beams sowed tat te lateraltorsional buckling beaviour of castellated beams is similar to tat of solid web beams and tat te 67

104 castellations ave no significant influence on te lateral-torsional buckling beaviour, altoug te torsional stiffness is reduced due to loss of te web, as sown in Figure 5.6. Figure 5.6: Lateral-torsional buckling of castellated beams (Netercot and Kerdal, 1984) c) Formation of Vierendeel mecanism In te absence of local or overall instability, beams wit exagonal openings ave two basic modes of plastic collapse, depending on te opening geometry. Failure often occurs by te development of te local bending resistance of te web-flange Tee sections (Vierendeel bending), particularly for wide castellations, as sown in Figure 5.7. Figure 5.7: Vierendeel mecanism caused by sear transfer troug perforated web zone (Halleux, 1967) d) Rupture of te welded joint in a web-post Rupture of a welded joint in a web-post can result wen te widt of te web-post or lengt of welded joint is small. Tis mode of failure is caused by te orizontal sear force in te web-post, wic is needed for te equilibrium of te vertical sear force acting at te centre of te adjacent openings. Tis mode is sown in Figure

105 Figure 5.8: Rupture of a welded joint in orizontal sear (Halleux, 1967) e) Sear bucking of a web-post Te orizontal sear force in te web-post is associated wit double curvature bending over te eigt of te web-post. In a castellated beam, one inclined edge of te opening will be stressed in tension, and te opposite edge in compression and buckling will occur as an equivalent inclined strut along te lengt of te web-post. Tis is influenced by te slenderness of te web-post, (i.e. opening eigt/ steel tickness), and by te presence of local loads, as sown in Figure 5.9. Figure 5.9: Web-post buckling below a concentrated load (Hosain and Spiers, 1973) 5. Literature Review of Castellated Beams Te literature review on non-composite castellated beams includes previous test work wit a brief description of te main features of eac investigation. Tis is summarised below Experimental Investigation of Open Web Beams by Toprac and Cooke Te objective of te early test series by Toprac and Cooke (1959) was to study te elastic and plastic structural beaviour of castellated beams by testing nine beams. Test results and modes of failure were compared wit te teoretical calculations to determine an optimum expansion ratio (new 69

106 eigt: original eigt) for castellated beams. Loads were applied at four concentrated points. Te failure modes for tese beam included lateral buckling, pure bending, and web-post buckling. In one of te test specimens, te failure was due to yielding and buckling of te compression flange in pure bending region. Beams wit Class flanges failed by buckling of te compression flange in te constant moment region wile te beam wit a Class 1 web Tee section failed troug a Vierendeel mecanism in te ig sear region. 5.. Plastic Beaviour of Castellated Beams by Serbourne In tis test series, Serbourne (1966) tested seven castellated beams to investigate te interaction of sear and moment forces on teir beaviour. Beams were simply supported and ad full dept bearing stiffeners under load and reaction points. Beams tat were subjected to a concentrated load at mid-span failed troug extensive yielding of te troat at mid-dept of te post between te first and second openings. Te second beam was subjected to two concentrated point loads designed to investigate te effect of pure moment. Te failure of tis beam was outside te central control section and was associated wit extensive yielding in te end zones subject to bot sear and moment forces. Te tird beam failed by web buckling in te zone of maximum sear, under te two-point loading system. Te fourt beam specimen failed by web-buckling. Te rest of te beams were tested under pure bending, in wic, two beams failed by flexural mecanisms wile one beam failed by lateral torsional buckling Limit Analysis of Castellated Steel Beams by Halleux In tis study, Halleux (1967) tested five types of beams wit different geometrical properties fabricated from IPE300 rolled steel sections under two equal loads applied at te tird-span points. Calculations in te reference were based on te nominal yield stress of te S35 steel, wic was likely to be lower tan te unreported tensile tests Tests on Castellated Beams by Bazile and Texier Four HEA360 and tree IPE70 simply supported beams wit full dept stiffeners at te supports were tested by Bazile and Texier (1968) to failure under eigt uniformly distributed load points. Te objective of tese tests was to develop a furter understanding of different beam caracteristics, properties, and geometry of castellated beams. Tree beams failed by web buckling in te zone of maximum sear and two beams failed by lateral torsional buckling. Te oter two beams failed by web-post buckling under te concentrated loads applied directly above te unstiffened web posts. Te buckling metodology was based on te column strengt formula obtained from te Canadian standards CSA (1994). 70

107 5..5 Failure of Castellated Beams due to Rupture of Welded Joints by Husain and Speirs Te aim of tese tests by Husain and Speirs (1971) was to study te yielding and rupture of welded joints of castellated beams. Six simply supported beams were tested under various load systems were a single concentrated point load vas applied to four beams and two point loads were applied to two beams. Full dept-bearing stiffeners and sufficient lateral bracings were provided to prevent premature buckling failure. Te measured sear stresses were found to be significantly iger tan te expected values obtained from tensile coupon tests wic were assumed to be a result of strain ardening. Te prediction of' ultimate local capacity based on web-post yield is terefore conservative. Sudden weld rupture was te common mode of failure for all beams Experiments on Castellated Beams by Husain and Speirs In tis later paper, Husain and Speirs (1973) tested twelve simply supported castellated beams wit full dept bearing stiffeners, except for beams C and D wit partial dept stiffeners in order to investigate te effect of opening geometry on teir mode of failure and resistance. Te beams were subject to a single point load at mid-span but beams A-, B-1, C and D were subject to two concentrated point loads. Beams A-1, A-, and B-3 failed by te formation of plastic inges in te Tees at te corners of te openings subject to bot sear and moment forces. For beam G-1and G-, yielding of te flanges in te region of ig bending moment led to flexural failure. Beams 8-, C, and D failed prematurely due to web buckling directly under te point of load application. Beam B- 1 failed by web buckling under te concentrated load before te Vierendeel mecanism ad formed. Te conclusions of te study were as follows: If local and lateral buckling are prevented in a castellated beam, failure is eiter by te formation of a Vierendeel mecanism or by te yielding and fracture of te web weld in sear. Reduction in te lengt of te weld troat, wic makes te design less susceptible to secondary bending effects, reduces te possibility of failure due to a panel mecanism but increases te cance of failure due to rupture of te web weld. Te test results revealed tat beam wit more castellations in a given span ad little effect on te failure load and on elastic stiffness but affected te deformation of te beam at failure significantly. Te test results indicate tat te optimum opening geometry requires a minimum lengt of te weld troat wic makes te design less susceptible to secondary bending effects. It was also found tat full dept stiffeners at concentrated load points are essential to avoid expressive failure of te web-post. 71

108 5..7 Optimum Expansion ratio of Castellated Steel Beams by Galambos, Husain and Speirs Four castellated beams were tested by Galambos et al. (1975) as simply supported beams subject to a point load at mid-span. A numerical analysis approac was used to determine te optimum expansion ratio based on bot elastic and plastic metods of analysis. Te span and weld lengts for te four beams were kept constant, but te beam depts were varied. Ultimate loads were recorded but no furter discussion about te modes of failure was given Design of Castellated Beams by Knowles Knowles s paper (1991) was publised by ICE and by Constrado and summarized te elastic design of castellated beams tat were considered to be in te form of a Vierendeel girder wit points of contra-flexure at te mid-line of te openings and at mid-eigt of te web-posts. Te sear force at te centre-line of te opening is divided equally between top and bottom Tees for symmetric sections. A formula was developed to calculate te deflection of castellated beams based on a correction factor to te deflection calculated using te minimum second moment of area at te opening positions based on a previous series of tests. It was stated tat plastic design of continuous castellated beams is not possible because of insufficient rotation capacity due to premature web-post buckling in a torsional mode. However, plastic design of simply supported beams was found to be possible depending on te section class. Te application of BS 5950 to castellated beams was also reviewed and was summarized as follows: A consideration of widt to tickness ratios is required so tat te section can be correctly classified for local buckling. Compression flange outstands are uncanged by castellation, and terefore fall into te same class as te parent section wit te exception of castellated beams wit semi-compact flanges (381 x 146 x 31 and 305 x 133 x 5 UBs). Te increase of dept in Universal Beams (UB) as an important effect on te web dept to tickness ratio in two respects: (a) Between openings, te web slenderness d/t is increased by a factor of about 1:5. (b) At an opening, te beam consists of two Tee sections. Te webs of all web-posts are at least semi-compact, but many ave d/t w > 63(75/f y) 0.5 (BS , clause 3.6.) and terefore sould be cecked for sear buckling. From te above, it was found tat true plastic design of castellated beams was possible only for a very small number of sections; te remainder must be designed on an elastic basis using te net 7

109 section properties wit due allowance for secondary Vierendeel effects of sear at te openings and for te local effects of any point loads Web Buckling in Tin Webbed Castellated Beams by Zaarour In tis researc, Zaarour (1995) investigated te buckling of te web-post between openings under point loads. Fourteen castellated beams fabricated from 8, 10, 1, and 14 inces (00- to 350 mm) beams wit stiffeners at supports and at load points were tested. Six of tese ad inc (50 mm) ig plates welded between te two beam alves at te web-post mid-dept. Web-post buckling was observed in te failure of 10 beams and two beams failed by local buckling of te Tee-section above te openings. Two beams failed by lateral torsional buckling and were omitted from furter consideration, since interest was in web buckling only. FEM analysis was also used to predict webpost buckling load. Te observations from tese tests were: Beams wit intermediate plates exibited a lower failure load tan beams witout tese intermediate plates. In many cases, tere was a significant reduction in te failure load, and te ratios of like pairs varied from 1.16 to Beams witout intermediate plates ad a more favourable beaviour after buckling and sowed additional load resistance after buckling deformations. Te metod suggested by Redwood (1978) to predict te mecanism mode of failure was found to be reasonably conservative and accurate. For beams tested by Redwood and Demirdjian, conservative results varied by 53 % to 71 % of te buckling predictions by Blodgett (1963). For beams tested by Bazile and Texier (1968), Blodgett's metod sowed an improved estimation of te buckling load for tese te predictions and comparisons varied from +1.5 % to -30 %. Blodgett's metod was found to be not valid for beams wit very tin webs as was te case for te beams tested in tis series wit a ratio of web dept: tickness (d/t) varying from 89 to Castellated Beam Web Buckling in Sear by Redwood and Demirdjian In tese tests on castellated beams, Redwood and Demirdjian (1998) investigated te buckling of te web-posts and te effects of moment-to-sear ratio on te mode of failure. Four simply supported castellated beams wit identical cross-sectional and bearing stiffeners provided at te supports and at load points were tested subject to a mid-span concentrated load. Two beams 10-5(a) and 10-5(b) were identical and ad four openings. A tird beam (10-6) ad six web openings and a fourt (10-7) ad eigt openings. Te mean flange and web yield stress values of te steel sections were obtained 73

110 from coupon tests. Buckling of te web post was te observed mode of failure of all tese beams, except beam 10-7, wic failed prematurely by lateral torsional buckling. Te buckling mode involved twisting of te web-post in opposite directions above and below te mid-dept. Te ultimate load values were given as te peak test loads. Test conditions were ten simulated by modelling using elastic finite element analysis. Te buckling loads were witin 4 to 14% of te predicted loads. Te sear buckling resistance of te web-post of a castellated beam was expressed as a function of te opening dept and te orizontal widt of te web-post as follows: k E b tw V (Equation 5.1),cr /tw Were: b = widt of web-post o o = opening dept t w = tickness of web k = buckling coefficient in sear Te orizontal sear resistance V,cr can be converted into a vertical sear resistance V v,cr as follows: - ye Vv, cr V,cr (Equation 5.) s Were s = centre-centre spacing of openings = section dept y e = dept of elastic neutral axis of Tee from te outside of section For te geometry of a typical castellated section, V v,cr 1.5 V,cr Te sear buckling coefficient varies wit ratio, o/b and may be taken as: k /b For b/t < 30 (Equation 5.3) o For a typical castellated section, o/b = 4, and so k 4. Terefore, for a normal castellated section, te sear buckling resistance of te web-post is given by: 5E btw V (Equation 5.4) v,cr /tw o 74

111 It follows tat te effective buckling lengt of te web-post is approximately 0.4 o. Tis sows tat te buckling wave is constrained of te mid-eigt of te web-post and partial fixity exists at te upper part of te web-post Stability of Castellated Beam Webs by Sevak Demirdjian Te objective of te researc by Demirdjian (1999) was to study te failure of castellated beams wit particular empasis on web-post buckling using te available elastic and plastic analysis metods and to derive expressions to predict critical sear force causing web-post buckling. Te researc used previous test results to provide comparisons wit teoretical approximations and tus validation of te suggested metods described. Demirdjian used te available teoretical metods of analysis to predict failure loads of castellated beams including plastic analysis of te Vierendeel mecanism and yielding of te mid-post weld. Finite element analysis was used to perform elastic buckling analysis and predict critical loads of test beams. Elastic buckling modes were investigated under different moment to sear (M:N) ratios. Welldefined relationsips based on pure sear and pure bending forces to cause web buckling were developed to predict elastic buckling loads under various moment to sear ratios. Results of elastic buckling and mecanism yielding loads were ten combined, and fitted curves were derived to predict ultimate sear forces tat cause web-post bucking. To apply tese expressions in a more general fasion, a parametric study of te beaviour of a wide range of castellated beam geometries was carried out and buckling coefficients under pure sear and bending forces were derived. 5.3 Design of Non-Composite Beams wit Large Web Openings to SCI P355 Te design of non-composite beams wit large web openings may be carried out using essentially te same model and design procedures as are set out in SCI P355: Design of Composite Beams wit Large Web Openings (011), but ignoring te contribution of te slab. In a non-composite beam, te Tee resist tension and compression due to global bending and tis means tat tey are less effective in resisting Vierendeel bending due to N-M interaction Sear resistance of perforated steel section Te sear resistance is establised from te sear area of te perforated steel section. According to BS EN , te design plastic sear resistance is given as: V pl, Rd = (A v f y/ 3 )/ ɣ M0 (Equation 5.5) Were: A v is te sear area. 75

112 For an un-perforated I-section beam, te sear area corresponds to te area of te web. However, te perforated cross section is effectively two Tee sections, te effective sear area of te perforated section is calculated from te following Equation: A v = (A- ot w - b f t f + (r + t w) 0.5t f) (Equation 5.6) For a welded Tee section: A v = t w ( w,t - 0.5t f ) (Equation 5.7) Were: A is te cross-sectional area of te beam b f t f r o w,t is te overall widt of te flange is te flange tickness of te flange is te root radius of te section is te opening dept is te overall dept of te Tee. Te sear areas are sown in Figure 5.10 and te contribution of te flange can be significant for sallow Tees. Te plastic sear resistance of te perforated section is tus: V pl,rd = (A v,bt + A v,tt ) f y/ 3 )/ ɣ M0 (Equation 5.8) Were: A v,tt and A v,bt are te sear areas of te two Tees. Te plastic sear resistance of te web may be used unless it is susceptible to sear buckling. Te nature of sear transfer across large openings in a composite beam means tat Vierendeel bending effects rater tan pure sear resistance normally govern te design. Figure 5.10: Definition of te sear area of rolled and welded sections to EN Vierendeel bending resistances Vierendeel bending is te means by wic sear force is transferred across a large opening due to local buckling. Te sum of te Vierendeel bending resistances at te four corners of te opening 76

113 must terefore not be less tan te design value of te difference in bending moment from one side of te opening to te oter due to te sear force. Tis may be expressed as: M bt,nv,rd + M tt,nv,rd V Ed e (Equation 5.9) Were: M bt,nv,rd is te bending resistance of te bottom Tee, reduced for coexisting axial tension and sear M tt, NV,Rd is te bending resistance of te top Tee, reduced for coexisting axial compression and sear V Ed is te design value of te vertical sear force (taken as te value at te lower moment side of te opening) e is te effective lengt of te opening for Vierendeel bending. Te effective lengt of a circular opening was defined by Redwood as e = 0.45 o Plastic bending resistance of Tees Plastic stress blocks can be considered wen te cross-section of te Tees satisfies te section limits for a Class section Plastic bending resistance in te absence of axial force Te plastic bending resistance of a top or bottom Tee section in te absence of axial force (and in te absence of ig sear) is given by te following expression, assuming tat te plastic neutral axis is in te flange of te Tee: M pl,rd = A w,t f y (0.5 w,t + t f z pl) + A f f y ( 0.5 f z pl + z pl /t f) (Equation 5.10) Were: z pl is te distance between te plastic neutral axis and te extreme fibre of te steel flange = (A f + A w,t)/(b f) A w,t is te cross sectional area of web of te Tee (= w,t t w) A f w,t is te cross sectional area of te flange is te dept of web of te Tee ( wb or wt respectively for te bottom and top Tees) Reduction of bending resistance due to sear Te utilization of te web of a Tee in sear may reduce its effective bending and axial resistances. Tis is treated by defining an effective tickness t w,eff wic is dependent on te utilization factor in sear μ and is given by te following equation: t w,eff = t w (1 - (μ - 1) ) for μ > 0.5 (Equation 5.11) Were: = V Ed /V Rd 77

114 No reduction in web tickness is required for μ 0.5 and V Ed = V Rd. Te reduced cross-sectional area of web of te Tee ( w,t t w,eff) is ten used to determine te plastic bending resistance of te Tee in te presence of axial force and sear, M pl,nv,rd, Were: M pl,n,rd = M pl,rd (1 (N Ed/N pl,rd) ) for Class 1 and sections (Equation 5.1) However, te sear force wic may be resisted by a Tee is limited by Vierendeel bending resistance of te Tee over te lengt of te opening. Terefore, a process of iteration is required to determine te sear force distribution between te top and bottom Tees tat is compatible wit te Vierendeel bending resistance Distribution of sear between top and bottom Tees Conservatively V b,rd can be first set to a lower value tan te top Tee to calculate μ and te associated effective web tickness t w,eff of te top Tee. Te plastic bending resistance of te top Tee, M tt,nv,rd can ten be determined. Te value of plastic bending resistance for te bottom Tee, M bt,nv,rd may ten be calculated for te same utilization factor and te associated sear force in te bottom Tee may be evaluated as: V b,ed = M bt,nv,rd /e (Equation 5.13) After tis, te sear force in te top Tee can ten be evaluated as: V t,ed = V Ed V b,ed Te utilization of te Tees may be determined from te calculated values of V t,ed and V b,ed and te bending resistances re-evaluated Elastic bending resistance in te absence of axial force Te elastic bending resistance of te top or bottom Tee section in te absence of axial force is given by: Μ el, Rd A W, T f yd 0.5 W, T tf - z el Af W, T fyd (zel 0.5t) f tf - zel A W, T f yd W, T /1 (Equation 5.14) z el is te distance from te centroid of te Tee to te extreme fibre of te flange, given by: z el A W,T 0. 5 W, T tf (Af A W, T 0. 5tf Af ) Web-posts between openings Te web-post between adjacent openings is subject to ig stresses as follows: Horizontal sear acts at its narrowest widt Compression exists due to transfer of vertical sear force 78 (Equation 5.15)

115 Local bending is developed in asymmetric sections due to Vierendeel bending action. Te interaction is more complex because of te possibility of buckling due to combinations of tese effects. Compression in te web-post (i.e. due to inclined forces) potentially leads to web-post buckling. For verification of adequacy against buckling in te design model, it is necessary to distinguis between closely spaced openings, were te compression force is resisted by te full widt of te end post, and widely spaced openings, were forces are resisted by effective widts of web adjacent to eac of te openings. Te transition from widely spaced to closely spaced openings may be taken to occur at an edge-toedge spacing equal to te opening lengt, corresponding to: s o = o for circular openings s o = o for rectangular openings Were: o and o are te dept and lengt of te openings respectively. Tis transition is not exact but is used in order to simplify te analysis Design compression force for widely spaced openings For widely spaced or discrete openings, web-post buckling is independent of te spacing of te openings. In tis case, it is considered tat a compression force acts at te edge of te opening over an effective widt of o/. Te magnitude of te compressive force may be taken as equal to te larger of te vertical sear forces in te top and bottom Tees. Te use of te larger of te sear forces in te Tees takes account of any asymmetry in te opening position. Te compressive force in te post at te edge of te opening is tus given by: N wp, Ed = V T,Ed (Equation 5.16) Were: V T,Ed is te larger of te sear forces in te two Tees Design compression force for closely spaced openings For closely spaced openings, te longitudinal sear force on te web-post V wp,ed is used to determine te design compression force on te web-post, rater tan te vertical sear force. Tis takes account of te iger forces acting in te web-post between te openings. For openings placed centrally in te beam dept, te compression stress acting on te web-post is taken as equal to te longitudinal sear stress acting on te web-post. Te compression force acting on te web-post is tus given by: N wp,ed = V wp,ed+ M wp,ed /( o/) (Equation 5.17) were: V wp, Ed M wp,ed is te longitudinal sear force acting on web post. is te value of te web-post moment at te mid-eigt of te opening. 79

116 5.4 Design of Hexagonal Web Openings in Beams to BS EN Sear resistance Te sear resistance of te reduced web is obtained as for a rolled section and is given by: V Rd f y o tw (r t) tf (Equation 5.18) 3 Te orizontal sear force acting in te web-post is: V wp, Ed V Ed b ocot θ eff f y tb 3 (Equation 5.19) Were: b eff = widt of web-post and also te top of castellation = slope of side of opening to te orizontal (60 o for narrow castellation) = effective dept of section to te centroid of te web-flange Tees ( eff 0.9) Te transfer of sear across te opening is also controlled by Vierendeel bending of te web-flange Tee sections. Te local bending resistance of te Tees is reduced linearly depending on te axial stress utilisation in te Tees. Equilibrium in Vierendeel bending is defined by: V Ed 4 M v, e b (1 N N Rd ) (Equation 5.0) Were: M v, e is te elastic bending resistance of te Tee N is te axial force in te Tee = M Ed / eff N Rd is te compression resistance of te Tee = ((b f t )t f + t w(- o)/) f y 5.4. Bending resistance of Tee section Te bending resistance of te Tees is required to determine te resistance to Vierendeel bending. Te elastic neutral axis dept of te Tee from te outer part of section is given by: y e o tw b tw f o tw b f twt f f t (Equation 5.1) Te second moment of area of Tee section is given by: I yy o o t f 3 tw tw ye t f b f tw y e o 4 (Equation 5.) 96 Te elastic bending resistance of Tee section is: 80

117 I yy f y M v, e (Equation 5.3) 0. 5 y o e Web-post buckling Compression stress acting on an equivalent strut between openings as per SCI P355 If te d /t w of te parent section is about 60, te d /t w ratio of te castellated beam is about 90, wic is in te region were web-buckling could occur. Te web-post buckling metod developed in SCI P-355 may be adapted to te analysis of castellated beams by consideration of an equivalent strut acting in compression and an opposite tension tie in te web-post between adjacent exagonal web openings. For te compression strut, te buckling of te unsupported free edge next to te opening is partially restrained by te connection of te buckled wave to te adjacent plate tat is not so igly stressed. Tis action is illustrated in Figure Figure 5.11: Compression and tension action in web-post between exagonal openings Te compression stress, c, in te equivalent strut is determined from te orizontal sear force in te web-post, as follows: σ t V wp,ed c y w sin θ f (Equation 5.4) Te orizontal sear force acting in te web-post is given by equation For a tension or compression stress,, acting in te strut, equilibrium of orizontal forces on te webpost, V, is given by: V = t w b cos 30 0 sin 30 0 = b t w (Equation 5.5) 81

118 Te orizontal sear force, V in te web-post is related to te vertical sear force V Ed at te centreline of adjacent openings according to: o b V = V Ed eff (Equation 5.6) Were: eff = distance between centroids of te upper and lower web-flange Tee sections = 0.9 and o = dept of castellation (normally two-tirds of te beam dept) It follows tat te compression stress acting on te equivalent strut is given by: V Ed o o o σ τv 1 1 t w b b (Equation 5.7) Were: v = vertical sear stress acting on te reduced web at te centre-line of te adjacent opening Proposed metod to calculate te compression resistance of te strut between openings based on te buckling resistance of te strut For te compression strut, te buckling of te unsupported free edge next to te opening is partially restrained by te connection of te buckled wave to te adjacent plate tat is not so igly stressed. Te boundary conditions at te ends of te plate may be taken as simply supported but te adjacent longitudinal edge is partially fixed by its connection to te adjacent plate. Tis action is illustrated in Figure 5.1.Te critical buckling stress of te plate is given by: Ek 11 tw cr (Equation 5.8) bp Were: b p is te plate widt in te direction of te applied stress is Poisson s ratio =0.3 Te buckling coefficient, k is a function of: Te boundary condition on te long edge parallel to te free edge. Te aspect ratio (lengt : widt) of te plate Values are presented in te Teory of Elastic Stability. Te two buckling cases of simply supported or fixed long edges are illustrated in Figure 5.1. For a exagonal opening, te aspect ratio is defined for a free edge at 30 0 to te vertical, wic is as follows: Plate widt: b p = 0.87b Plate lengt: a p = 0.58 o 8

119 Were: b p = widt of web-post of exagonal opening (a) Simply supported edges (b) Fixed long edge Figure 5.1: Buckled waves and boundary conditions next to a exagonal opening For simplicity, te buckling coefficients for te two cases are: Free long edge/simply supported long edge: k = 0.45 (a p/b p) Free long edge/fixed long edge: k = 1.7 (a p/b p) Te critical buckling stress may be converted to an effective lengt factor of te free edge, wic is given by: Were eff 1 k 0.5 o = 0.58 o for exagonal opening o (Equation 5.9) For te two boundary conditions, te effective lengt factor becomes: a. Free long side and tree simply supported sides: eff b 0. 58o o = 1.7 b = 0.4 o b. Free long side and fixed long side and two simply supported ends: b eff o = 0.74 b = 0.4 o o Te slenderness of te equivalent strut reduces to te simple formula: 83

120 λ 1( l eff / tw) (Equation 5.30) λ 1 = π (E/f y) 0.5 (Equation 5.31) λ = λ λ 1 Te buckling curve is defined by te following formula: (Equation 5.3) φ = 0.5 [1 + α ( λ 0.) + λ ] (Equation 5.33) Were: α = 0.34 for cold formed sections = 0.49 for welded castellated section Te reduction factor due to buckling of te web-post as a strut is: 1 χ (Equation 5.34) 0.5 φ φ Te compression stress tat may be resisted by te web-post is: c = f y (Equation 5.35) A comparison between te compression stress resistance obtained from te derived teory in Equation 5.35 and Redwood teory is presented later in section Deflection of perforated beams according to SCI P355 Te deflection of castellated beams may be determined for an effective inertia of: I eff o t eff ( b tw) t (Equation 5.36) f f Te additional sear deflection for a uniformly load beam due to Vierendeel bending across te opening is: w v 3 nob VEd,ser (Equation 5.37) 96EIyy Were: n o is te number of openings in te beam span V Ed,ser is te maximum sear force on te beam at te serviceability limit state. b is te widt of te top of te exagonal opening. Te influence of te openings on sear deflections is small for beams wit a span: dept ratio of more tan 0, wic is generally te case for slender beams. 5.5 Proposed Design of Beams wit Diamond Saped Openings Te sear and bending resistance of a beam wit diamond saped web openings can be calculated as above for beams wit castellated web openings, but web-post buckling mode is different as follows: 84

121 5.5.1 Web-post buckling of closely spaced openings Te compression and tension action in web-post between diamond openings is sown in Figure o s o Figure 5.13: Compression and tension action in web-post between diamond openings Te widt of te equivalent strut is, s cos 45 0 = 0.71 s o, were s o is te edge to edge spacing of te diamond sapes. For a tension or compression stress, acting in te strut, equilibrium of orizontal forces on te web-post, V, is given by: V = t w s cos 45 0 sin 45 0 = s t w (Equation 5.38) Were: s o = spacing of te edges of te openings Te orizontal sear force, V in te web-post is related to te vertical sear force V Ed at te centreline of adjacent diamond saped openings according to: o s V = V Ed eff (Equation 5.39) Were: eff = distance between centroids of te upper and lower web-flange Tee sections = 0.9, were is te section dept and o = dept of diamond opening It follows tat te compression stress acting on te strut for equilibrium is given by: V o o o 1.1 Ed v 1 t s s (Equation 5.40) Were: v = vertical sear stress acting on te reduced web at te centre- line of te adjacent opening From te non-linear finite element analysis, te effective lengt of te equivalent strut, may be taken from te geometry of te diamond sape as a tird of te lengt of te sloping side, as follows: eff 85

122 0.33x(0.5 / cos 45 ) 0. 4 (Equation 5.41) eff o 0 Te slenderness of te equivalent strut reduces to te simple formula: o ( o / tw) 0. 8o / tw (Equation 5.4) Te compression strengt is calculated from buckling curve b to Eurocode 3 for cold rolled sections and tis sould exceed te applied compression stress obtained as above Web-post buckling of widely spaced openings Openings may be considered to be widely spaced wen s > - o. In tis case, te widt of te equivalent strut may be taken as 0.5(- o)/cos 45 0 = 0.7(- o). For a tension or compression stress, acting in te strut, equilibrium of vertical sear forces in te web and axial forces on te web -post is given by: 0.5V Ed = 0.7 (- o) cos 45 0 t w = 0.5 (- o) t w (Equation 5.43) It follows tat te compression stress acting on te strut for equilibrium is given by: Were: V (Equation 5.44) Ed v ot w v is defined as previously. Te effective lengt of te equivalent strut, is taken as for closely spaced openings. eff 5.6 Comparison between te Section Resistances Obtained from te Proposed Teory and te Failure Loads for Four Castellated Beams Tested by Redwood and Demirdjian Four castellated beams, as sown in Figure 5.14 were tested by Redwood and Demirdjian (1998) and were analysed using te proposed design metod. Failure loads for beams as Redwood as obtained from testing and finite element analysis are presented in Table 5.1. Section properties were calculated for sear, orizontal sear, Vierendeel sear and te web-post sear capacities for eac section were determined considering te following tree different effective lengts of te equivalent compression strut: l eff = 0.5 l slope ( effective lengt taken as 0.5 times te lengt of te slope) l eff = 0.5 o ( for te proposed metod to calculate te effective lengt as in section ) l eff = 0.4 o ( as proposed by Redwood in 1998 and explained in section 5..) 86

123 Figure 5.14: Configuration of te tested beams (Redwood and Demirdjian, 1998) Table 5.1: Failure loads for te four beams tested by Redwood and Demirdjian Beam Test failure load Failure load obtained from te FEA 10-5 (a) 9.7 kn 88.6 kn 10-5 (b) kn 88.6 kn kn 84. kn kn 81.3 kn Results sown in te table were obtained from Redwood s paper Te comparison between te test failure loads and te sear resistances is sown in Table 5.. Te proposed teory and te teory proposed by Redwood (1998) based on an effective lengt of te equivalent compression strut l eff = 0.4 o, gave a reasonable agreement wit te test results. Te ratio of te failure load to te section resistance was in te range of 1. to 1.36, as sown in Table

124 Table 5.: Castellated beam sear resistances based on section properties and proposed teory Test Failure loads Beam resistances Beam reference Test failure load (kn) Horizont al sear in webpost at failure (kn) Sear resistance V Rd Horizontal sear resistance V, Rd Vierendeel sear resistance V v, Rd Web-post sear resistance l eff = 0.5l slope V Rd l eff = 1.7b (buckling Curve a ) l eff = 0.4 o (Redwood 1998) Imperfection = 0.34 for buckling curve b and b is te web-post widt (defined as e in Redwood study wic was equal to 0.9 o for te test sections) Table 5.3: Relationsip between te failure loads and te section resistance for te tested beams Ratio V Ed/V Rd Test beam reference Sear Horizontal sear Vierendeel bending Web-post effective lengt (l eff = 0.5l slope ) l eff = 1.7b (buckling Curve a ) l eff = 0.4 o (Redwood 98) Were: V Ed = test sear force at failure and V Rd = sear resistance 5.7 Conclusion Te proposed metod to calculate te resistance of castellated beams based on calculating te compression resistance of te strut between openings sowed a reasonable agreement wit te test results of te tree castellated beams tested by Redwood and Demirdjian (1998) if te effective lengt of te strut is taken as l eff = 1.7b, as in boundary condition a in section wit 3 simply supported sides. Later in Capter 7, a comparison between te teory and te two beams tested at te University of Surrey, sowed also a reasonable agreement if te same boundary condition is considered. 88

125 CHAPTER 6 ADDITIONAL DEFLECTION OF BEAMS DUE TO RECTANGULAR AND CIRCULAR WEB OPENINGS 6.1 Introduction Introducing a web opening in a steel beam reduces te flexural and sear stiffness of te beam locally and results in increased deflections and a difference in te deflection across te opening. In most cases, te influence of a single web opening is small but te influence of many web openings can be significant and sould be considered in te design. Te loss of material in te web also as a direct effect on te beam sear deflection and as to be considered for beams wit sorter span: dept ratios. Te additional deflection for beams wit web openings was addressed in SCI P355 but only for beams acting compositely. In tis tesis, te elastic finite element analysis was carried out for beams wit different opening eigt o/, number of openings n o, and beam span: dept ratio, L/. Calculations of te additional deflection for beams wit rectangular and circular web openings are presented. A comparison between te derived teory and te elastic finite element analysis is also presented in tis capter. For beams wit circular web openings, te calculations of te additional deflection are based on te equivalent rectangular openings in wic te effective widt of te opening is establised. Terefore, a teory to calculate te effective lengt of te opening was developed in te tecnical paper presented in Appendix E. A summary of te findings is provided in tis Capter. 6. Additional Deflection of C Section Beams due to Rectangular Web Openings 6..1 Additional deflection due to bending curvature at an opening For a beam subject to uniform loading, te additional deflection is due to loss of te flexural stiffness at te opening as sown in Figure 6.1. Te ratio of te additional mid-span deflection to te pure bending deflection of te beam for an opening at any position, x from one support according to SCI P355 (Equation 6) is: w w add b x x o, eff EI L L L EIeff, o (Equation 6.1) Were: w b = bending deflection of te unperforated beam subject to uniform loading EI = bending stiffness of non-perforated beam EI eff,o = effective bending stiffness of beam at web opening 89

126 o, eff = effective opening lengt L = beam span x = position of te opening in te span from te nearer support. Figure 6.1: Additional deflection due to bending at a large web opening (SCI P355) For a symmetric steel beam, te effective bending stiffness at te opening position is: I eff, o tw. o Is and 1 3 I s Af Aw (Equation 6.) 1 For a cold formed C section, te effective flange area A f is 30% to 40% of te web area, A w, and so: A f 0.3 A w conservatively. Terefore: I s A w w. = t (Equation 6.3) 1 A For a steel beam, te reduced stiffness of a steel beam due to an opening is terefore: Ieff I, o s o (Equation 6.4) Te factor 1 EI eff, o 1 EI is approximately given by o EI Substitute EI EI eff, o o 3 1 in Equation 6.1 wit 0. 4, te additional mid-span deflection of a beam wit a single opening at position x from te support (were x < 0.5L) and subject to uniform loading becomes: w w add b x x o, 19x L L L eff o 3 90

127 Re-arranging te equation, te additional deflection become: w w add b x L x o, L eff 3 o L (Equation 6.5) However, te effective lengt of te opening is increased due to te dispersion of bending stresses away from te web. Simple linear finite element analysis was carried out on a 5 m long beam wit 150 mm deep x 500 mm long single rectangular web opening subjected to uniform loading. Bending stresses were noticed to occur over a distance of 0.5 o on eac side of te web, as sown in Figure 6.. Te effective lengt of te rectangular opening for pure bending is terefore: o, eff = o o. l eff= l o+0.5 o Figure 6.: Distribution of bending stresses around a rectangular web opening For a beam wit a series of n o rectangular openings and according to SCI P355, te increase in bending deflection relative to tat of te unperforated beam is given by: w add w b 0. 4n o o, eff 3 o L Were o, eff = o o and tus te equation becomes: w w add b 3 o 0. 5o o 0. 4no (Equation 6.6) L Were: n o is te number of regular openings in te span. 6.. Additional sear deflection Te additional deflection of a beam due to te effects of sear on te circular openings is a combination of: 91

128 Pure sear due to te loss of te web at te opening Vierendeel bending of te web-flange Tees Bending of web -post in closely spaced openings Te sear displacement across an opening is sown in Figure 6.3. Figure 6.3: Sear deflection due to a web opening (SCI P355) Te additional sear deflection at mid-span is in all cases alf of te sear deflection across a single opening. Tese tree sear effects on mid-span deflection are considered separately as follows: Pure sear deflection due to rectangular openings From first principles, te additional mid-span sear deflection due to a single opening of lengt is given by: o w v,add,pure 0.5q ( L/ - x) Gt o o o (Equation 6.7) Were: And q = load per unit lengt of te beam G = sear modulus= E/.6 Te additional sear deflection may be compared to te pure bending deflection of an unperforated C section beam driven from first principles, wic is given by: w b 4 5 q L, were I s 0.3 A w., as previously. (Equation 6.8) 384 EIs For a beam wit a single rectangular opening and subject to uniform loading, te additional pure sear deflection relative to tat of te un-perforated beam in bending is given by: w x o o L o L v,pure, add w b 3 (Equation 6.9) For a beam wit n o rectangular openings and subject to uniform loading, it follows tat te additional sear deflection at mid-span is given by: 9

129 w v,pure, add w b o o 5.8n o o L 3 (Equation 6.10) 6... Sear deflection due to Vierendeel bending As noted previously, te mid-span sear deflection due to a single opening is alf of te sear displacement across te opening. Te mid-span sear displacement due to a single opening subject to Vierendeel bending according to SCI P355 is given by: w V,Vier 3 o,eff V (Equation 6.11) 48E I Tee Te dispersion of local Vierendeel bending stresses is assumed to occur over a distance on eac side of te opening. Tis is taken as a distance of 0.5 o and te effective opening lengt is taken as = o o o, eff Te bending stiffness of an unstiffened web-flange Tee section is given approximately by: 3 t I o (Equation 6.1) Tee 4 Inserting te approximate formula for I Tee in Equation 6.11 leads to a formula for te additional deflection at mid-span due to Vierendeel bending across a single opening tat is given by: w q. ( L x) o.eff 3 t V,Vier 3 o 3 (Equation 6.13) Te additional mid-span sear deflection due to Vierendeel bending at a single rectangular opening is terefore: w v,vier, add o w b x L o o 3 3 L (Equation 6.14) For a beam wit n o rectangular openings and subject to uniform loading, it follows tat te additional sear deflection at mid-span due to Vierendeel bending is given by: w v,vier, add 3 no w o b 0.5 o o 3 L 3 (Equation 6.15) 93

130 6...3 Web-post bending deflection Web-post bending is caused by orizontal sear tat leads to an additional sear deflection. Te orizontal sear displacement of te web-post between adjacent openings as obtained from te first principles is given by: w, 3 o, eff V (Equation 6.16) 1 E I WP Te effective eigt of te opening also allows for te dispersion of web-post bending stresses into te web above and below te openings. For simplicity, te effective opening eigt is taken as: o,eff = o +0.5 ( - o) = 0.5 ( + o) (Equation 6.17) V is te orizontal sear force in te web-post, wic is: V = V s / (Equation 6.18) Were s is te centre to centre spacing of te openings 3 tw Te second moment of area of web-post is: I WP so, were s o is te minimum widt of te webpost = s- o. Tis orizontal displacement of te web-post may be converted into an equivalent 1 vertical displacement between te centre-line of te openings according to: w v = s w (Equation 6.19) As noted earlier, te mid-span deflection is alf of te sear deflection across a single opening. For a uniformly loaded beam, te additional mid-span displacement due to web-post bending for an opening at position x is given by: w V, wp, mid 3 q L x o eff s, (Equation 6.0) E t w L s o Tis may be compared to te pure bending deflection of an unperforated C section beam wic leads to a deflection ratio of: w x o s L s o L v,wp, add w b 3 3 (Equation 6.1) For a beam wit n o openings, tis additional deflection due to web-post bending becomes: w v,wp, add w b o s 0.8n o s o L 3 3 (Equation 6.) 94

131 Combined additional deflections for a C section wit rectangular openings subject to uniform loading For a uniformly loaded beam, te combined additional mid-span deflection due to a series of rectangular openings is given by: L s s n L n L n L n w w o o o o o o o o o o o o o o o o o b add (Equation 6.3) 6..4 Combined additional deflections for a C section beam wit a single rectangular opening subject to uniform loading For a uniformly loaded beam wit a single rectangular opening at position x in te span (x < L/), te additional deflection at mid-span is obtained as follows: L s s L x L L x L L x L L x L x w w o o o o o o o o o o o o o b add (Equation 6.4) Te final term in s is zero for a single opening, as web-post bending does not occur Proposed simplified formula for te additional deflection of a C section beam wit rectangular openings and subject to uniform loading A proposed simple formula for te additional deflection of C section beam wit a series of rectangular openings is given by: b add w w = n o L o o o 3 (Equation 6.5) Tis applies for beams wose beaviour is dominated by bending. Te simplified formula is affected by web-post sear and bending and so may be modified to:

132 w add w b = n o o o o 3 o so L wen s o < 0.5 o (Equation 6.6) Te results of te equations in terms of te additional deflection for various openings sizes and beam span: dept ratios are presented in Table Elastic Finite Element Analysis on Beams wit Rectangular Web Openings Subject to Uniform Loading Two beams wit two different profiles were modelled using ABAQUS to determine te deflection of te beam subjected to unifrom loadings. Te additional deflection from te elastic finite element anlaysis for te different beam profiles is presented in Appendix E. Te two beams are described as follows : Beam 1: L=3000 mm, =00 mm (L/=15), 0=0.7, s=1.3 0, n o=16 Te beam was subject to 0 kn distributed uniformly wic was applied as a surface load of 0.33 N/mm to a 10 mm wide zone at te top and bottom flanges. Tis was done to avoid distortion of te cross-section. Figure 6.4: Layout of Beam 1 wit rectangular web openings equivelent to circular openings (L / =15) Figure 6.5: Deflection of alf a beam wit eigt rectangular web openings subject to a uniform load of 6.6 kn/m as obtained from te elastic FEA 96

133 Beam : L=5000 mm, = 50 mm (L/=0), o = 0.7, s=1.05, n 0=18. Again a load of 0 kn was applied uniformly to a 10 mm wide zone at te top and bottom flanges. Figure 6.6: Layout of Beam wit rectangular web openings equivelent to circular openings (L / =0) Te deflection of te beam wit pairs of web openings placed near to te supports and near midspan was also analyzed based on te derived equations. Table 6. presents te output from te finite element analysis and a comparison between te results and te additional deflections calculated using te derived equations for C sections wit circular and recatngular web openings. Figure 6.7: Deflection of alf a beam wit eigt rectangular web opening subject to a uniform load of 4 kn/m as obtained from te elastic FEA 97

134 Table 6.1: Additional deflection of C sections due to rectangular openings based on proposed metod Proportionate dept of opening, o/ Opening lengt = o o: s = Number of openings, n o= L / Accurate formula for beam span: dept ratio L / =15 L / =0 L/ =5 Simple formula (equation 6.6) % 9% 7% 1% % % 17% 3% % 61% 44% 80% Proportionate dept of opening, o/ Opening lengt = 1.5 o o: s = 1.5 Number of openings, n o= 0.67L / (s = 1.5) Accurate formula for beam span: dept ratio L / =15 L / =0 L/ =5 Simple formula (equation 6.6) % 10% 8% 9% % 5% 19% % % 73% 5% 45% Proposed limits of application of simple formula are: 1. o o, o/ 0.7 and L/

135 Table 6.: Comparison of additional deflection for rectangular openings as obtained from te FEA and as calculated from te proposed teory a) Data: L / = 0, o / = 0.7, o / = 0.5, s / = 1.05: Rectangular openings 16 no. uniformly distributed along beam no. at x = 0.43L/0.57L no. at x = 0.38L/0.6L no. at x = 0.07L/0.93L no. at x = 0.1L/0.88L Bending Additional deflection obtained from teory Sear Vierendeel bending Webpost sear Total Simple formula (equation 6.) Additional deflection from FEA 9.% 1.3% 13.% 1.7% 5.4% 6.7% 0.7% 4.4% 0.1% 1.3% 0.1% 5.9% - 5.% 0.4% 0.5% 5.% 0.4% 6.5% - 5.% b) Data: L/ = 15, o/ = 0.7, s/ = 0.9: Rectangular openings Additional deflection obtained from teory Bending Sear Vierendeel bending Webpost sear Total Simple formula (equation 6.) Additional deflection from FEA 16 no. uniformly distributed along beam 1.4% 3.% 31.7% 7.4% 54.7% 47% 56% no. at x = 0.4L/0.58L no. at x = 0.47L/0.53L 5.8% 0.% 1.6% 0.% 6.6% - 6.1% no. at x = 0.03L/0.97L no. at x = 0.06L/0.94L 0.% 1.4% 14.% 3.5% 19.3% - 19.% 6.4 Additional Deflection of C Sections due to Circular Web Openings A simple teory was developed (see Appendix E) to calculate te additional deflection due to single circular openings and due to a series of openings by a single algoritm tat takes into account te additional pure bending, pure sear, Vierendeel bending and web-post bending deflections. Te main focus is on uniformly loaded beams but a similar approac is also adopted for point loaded beams. Te beam cross-section is taken as a C section in wic te flange area is expressed as a proportion of te web area. Te teory was compared to finite element analysis (FEA) of te followings cases: 50 mm deep cold formed C section of 5 m span and 00 mm deep C section of 3 m span subject to uniform loading in wic openings are placed in tree configurations: at te ends of te span, close to te middle of te span, and at a uniform spacing along te span. Te span to dept ratios of 15 and 0 cover te normal range of application and sow te relative contribution of bending and sear deflections. 99

136 mm deep stainless steel C sections wit multiple 150 mm diameter openings, in wic te increase in mid-span deflection is mainly due to sear deformation at te openings. A pair of beams of approximately 1.5 m span was tested by loading troug steel blocks tat were bolted to te beam webs. Te tests were performed on mm and 3 mm tick steel in two grades of stainless steel as explained later in Capter mm deep cold formed C sections wit single or pairs of 150 mm and 180 mm diameter openings. Te tests were performed on 1.4 mm and 1.8 mm tick steel wit relatively closely spaced openings so tat web-post deformation is significant (see Capter 7) Combined deflections for a C section beam wit a single circular opening subject to uniform loading For a uniformly loaded beam wit a single circular opening at position x in te span (x< L/), te additional deflection at mid-span is obtained as follows: s s s L L x L L x L x w w o o o o o o o b add (Equation 6.7) For a uniformly loaded beam, te combined additional mid-span deflection due to a series of n o circular openings is given by: s s s L n L n w w o o o o o o o o o b add (Equation 6.8) 6.4. Combined deflections for a C section beam wit circular openings subject to point loading For a beam wit a central point load, te same approac may be adopted but w b is now te deflection of te unperforated beam for tis load case. For a beam wit a single circular opening at position x in te span (x< L/), te additional deflection at mid-span is obtained as follows: s s s L L L x w w o o o o o o o b add (Equation 6.9) Te combined additional mid-span deflection due to a series of circular openings is given by:

137 101 s s s L n L n w w o o o o o o o o o b add (Equation 6.30) A proposed simple formula for te additional deflection in a C-Section beam wit a series of circular openings is given by: For s <1.5 o,, b add w w = n o s L o o. 4 (Equation 6.31) For s 1.5 o,, b add w w = 0.67 n o L o 4 (Equation 6.3) Comparison wit additional deflection for circular openings from FEA A series of elastic finite element analyses using ABAQUS software was performed on typical perforated beams wit span: dept ratios of 15 and 0 and opening eigt of 70% of te beam dept. Tese beams ad pairs of openings in eiter te ig sear of ig bending parts of te span or ad openings uniformly distributed along te beam. Tis was done in order to isolate te components of deflections in comparison wit te above formulas. Te loading was applied as a line loading to te top of te web and alf of te beam span was modelled wit suitable boundary conditions. Te line load was consistent wit an end sear of 10 kn and te additional deflection was obtained by comparing wit te deflection of te same unperforated beam Beams wit span to dept ratio of 15 and 0 Beam 1: L=3000 mm, =00 mm (L/=15), 0=0.7, s=1.3 0, n o=16 Te beam was loaded by 0kN distributed uniformaly in te same way as for rectangular openings. Figure 6.8: Layout of Beam 1 wit circular web openings

138 Figure 6.9: Deflection of alf of a beam wit eigt circular openings subject to a uniform load of 6.6 kn/m as obtained from te elastic FEA Table 6.3: Comparison of additional deflection obtained from te FEA and te teory for a beam wit span: dept ratio of 15 Circular openings 50 x 75 x mm C wit 140 mm dia. openings 16 no. uniformly along beam no. at x = 0.38L and 0.44L no. at x = 0.536L and 0.6L no. at x = 0.06L and 0.1L no. at x = 0.88L and 0.94L Additional deflection obtained from teory Bending Pure sear Sear Vierendeel bending Webpost sear Total Simple formula (equation 6.31) Additional deflection from FEA 7.7% 3.1% 5.4% 11.% 7.4% 19.7% 8.5% 3.7% 0.3% 0.5% 1.0% 5.5% - 3.5% 0.3% 1.3%.3% 4.5% 8.4% - 8.3% Beam : L=5000 mm, = 50 mm (L/ = 0), o = 0.7, s=1.05, n 0=18 Tis was loaded by 0kN distributed uniformly in te same way as for rectangular openings. Figure 6.10: Layout of Beam wit circular web openings 10

139 Figure 6.11: Deflection of alf of a beam wit 9 circular openings subject to a uniform load of 5 kn/m as obtained from te elastic FEA Table 6.4: Comparison of additional deflection obtained from te FEA and te teory for a beam wit span: dept ratio of 0 Circular openings Data: 50 x 75 x mm C wit 175mm dia. openings 18 no. uniformly along beam no. at x = 0.4L and 0.47L no. at x = 0.53L and 0.58L no. at x = 0.05L and 0.1L no. at x = 0.9L and 0.95L Additional deflection obtained from teory Bending Pure sear Sear Vierendeel bending Webpost sear Total Simple formula (equation 6.31) Additional deflection from FEA 6.5% 1.5%.6%.9% 13.5% 14.4% 15% 3.1% 0.1% 0.1% 0.% 3.5% - 3.8% 0.% 0.5% 1.0% 1.1%.8% -.6% 103

140 CHAPTER 7 TESTS ON C-SECTION BEAMS WITH DIFFERENT SHAPES OF WEB OPENINGS 7.1 Introduction A series of beam tests was set up to investigate te structural performance of cold formed and stainless steel C-section wit large web openings. Te specimens were tested to failure and all tests were designed to lead to web-post buckling defines flexural as pure sear failure. Te beams were tested in pairs wit a single point load applied at te mid span via a 100 mm x 100 mm steel block and 0 mm diameter bolts in sear rater tan to te flange of te section, so tat web crusing and local flange buckling were avoided. Te beams were cecked to ensure tat lateral buckling did not occur over alf of teir span at te expected failure load. For te cold formed C sections, te steel ticknesses of 1., 1.5, 1.8 and 1.9 mm were considered to investigate te effect of te web tickness on te web-post buckling resistance. For te stainless steel C section beams, two grades were considered, a standard Austenitic grade , and a Lean Duplex grade LDX101. Tests were carried out wit circular openings at different spacing, steel tickness and grades. Four tests were carried out by Mr Sanmugalingam and were presented as MSc tesis submitted in 010. Later tests were carried out as part of tis researc. 7. Scedule of Tests Table 7.1 sows te steel tickness of te cold formed C-section wit circular web openings wic were tested in te University laboratory wit openings of 150 mm and 180 mm diameter. Te C- section size was 50 mm deep x 63 mm wide nominal flanges wit 1 mm edge stiffeners. Details of circular web openings are sown on Figure 7.1. Table 7. sows a scedule for cold formed C-sections wit diamond and exagonal web openings tat were tested wit isolated and closely spaced web openings. Te measured steel tickness varied between 1. mm for C-sections wit diamond web openings and 1.5 to 1.9 mm for sections wit exagonal web openings. Details of diamond and exagonal web openings are sown in Figure 7.. Te dept of te diamond saped openings was cosen so tat a 180 mm circular web opening would enclose it. Te dept of te exagonal web openings was taken as two tirds te beam dept based on te standard dimensions for castellated beams. 104

141 Table 7.3 sows te steel grades, tickness and lengt of stainless steel C-sections tat were tested wit different spacing for openings of 150 mm diameter. Te C section size was 10 mm deep x 70 mm wide wit 8 mm edge stiffeners. Beams were provided by Outokumpu Foundation wit a series of opening configurations. As sown in Figure 7.3. Table 7.1: Scedule of cold formed steel C-sections wit circular web openings (50 mm deep x 63 mm wide sections) Beam Beam wit single 150 mm diameter circular web opening Beam wit single 180 mm diameter circular web opening Beam wit 180 mm diameter circular web opening at 60 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Beams wit stiffened single web opening Beams wit stiffened web openings at 80 mm edge distances Measured steel tickness Number of tests 1.80 mm mm 1.50 mm mm mm mm mm 1 See Figure 7.1 for openings To manufacturer s specification Table 7.: Scedule of cold-formed C-sections wit diamond and exagonal web openings (50 mm deep x 63 mm wide sections) Beam Measured steel tickness Details Beam wit single 180mm diamond saped web opening 1. mm Beam wit 180 mm diamond saped openings at 9 mm edge distance Beam wit two 167 mm deep exagonal web openings at 45 mm edge distance 1. mm 1.53 mm See Figure 7. for openings Beam wit two 167 mm deep exagonal web openings at 45 mm edge distance 1.93 mm 105

142 Table 7.3: Scedule of specimens for beams tested as pairs of C sections Stainless steel designation Steel tickness Edge to edge spacing of openings 50 mm 100 mm 50 mm Notes mm LDX 101 mm mm 1550 mm x 4 No mm x 4 No mm x No mm x No mm x No mm x No mm x No. See Figure mm for openings x No. Comparative - tests wit no local buckling Figure 7.1: Details of circular opening positions in te tests on cold formed steel C sections 106

143 Figure 7.: Details of diamond saped and exagonal opening positions in te tests on cold formed steel C-sections (all dimensions in mm) Figure 7.3: Details of circular opening positions in te tests on stainless steel C section 107

144 7.3 Testing Arrangement Te beams were tested in pairs and a single point load was applied at te mid-span via a 100 mm x 100 mm steel block and 0 mm diameter bolts in sear rater tan to te flange of te section, so tat web crusing and local flange buckling was avoided. Te loading arrangement is sown in Figure 7.4. Te failure mode was eiter by orizontal sear in te web-post between te openings or Vierendeel bending around te openings. 7.4 Testing Procedure Loading sequence For te testing of tin steel C-sections, an Instron testing macine wit a capacity of 1000 kn was used wit Blueill static software. Tis macine was displacement-controlled. Te test procedure was as follows: Two exactly similar beams (C-sections) were fixed back to back wit tree 100 mm x 100 mm steel blocks by using two 0 mm bolts at 130 mm spacing, as sown in Figures 7.5 and 7.6. Steel blocks were placed between te C-sections at te load point and supports. Te jack applied te point load to te steel block at te mid-span of te beam. DC Voltage LVDTs were used after being collaborated ( were used at eac end and in te middle). Before te test to failure, te specimen was preloaded to eliminate bolt slip. Te tests were carried out under displacement control (at speed of 1 mm/min) so tat tey could sow te effects of local buckling witout leading to rapid failure. Te initial loading was continued until bolt slip occurred and ten te load was removed. Te second cycle was continued to failure. 108

145 Jack load Bolt 100 Block mm span typically Support 100 (a) Side view of test arrangement x 100 block wit 0 mm bolts troug 65 (b) Plan view of test arrangement Variable diameter Variable M0 bolt 10 Side view of load application via steel block Cross-section at jack Figure 7.4: Details of test arrangement on pairs of C-sections 109

146 Figure 7.5: Loading arrangement via a central jack and steel block Figure 7.6: Test arrangement sowing pair of beams connected using steel blocks 7.4. Local deformation of flanges Te beam C-sections ad a 63 mm wide flanges wic deformed towards te neutral axis due to te presence of te edge stiffener acting as a restraint to local buckling, so tat curling of te beam flange 110

147 occurred, as sown in te Figure 7.7. Te amount of te flange deformation is te difference between jack deflection of te web and flange deflection after initial slip in te bolt. Flange deformation is not dependent on te opening configuration and so is present in all cold formed beams wit edge stiffeners. Figure 7.7: Deformation of te stiffened flanges in bending due to te deep edge stiffeners and flexible flanges 7. 5 Tests on Cold Formed C Sections wit Circular Web Openings Test 1: Isolated 150 mm diameter web opening In Test 1, a pair of 1.8 mm tick C sections wit isolated 150 mm diameter openings was tested. Te arrangement is sown in Figure 7.8. Te failure load was 8 kn wic was applied to two C sections, and terefore, te vertical sear force is equal to test load/4. Figure 7.8: Test arrangement for Test 1 wit isolated 150 mm circular web openings 111

148 Load (kn) At failure, local buckling of te compression flange was observed integrated wit buckling of te web at te load application point and around te web opening, as sown in Figure 7.9. Figure 7.9: Failure mode of Test 1 by bending of top flange and local failure at connection between beams and steel blocks A maximum deflection of 1 mm was recorded at failure and te maximum resistance was followed by a rapid drop off in load, as sown in te relationsip between te load and te deflection in Figure mm tick C section wit isolated 150 mm web opening Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.10: Load-deflection curve for Test 1 wit isolated 150 mm circular web opening 11

149 7.5. Test : Isolated 180 mm diameter web opening In Test, a pair of 1.8 mm tick C-sections wit isolated 180 mm diameter web openings was tested, as sown in Figure Te failure load was 68.5 kn, wic was 83% less tan te failure load of Test 1 wit 150 mm diameter opening. Increasing te ratio between te dept of te opening and te dept of te beam ( o/) from 0.6 to 0.7, reduced te beam resistance by 16%. Buckling around te large diameter opening was more apparent compared to te beam wit 150 mm diameter opening. Failure of te beams was due to te local buckling and Vierendeel bending, as sown in Figure 7.1. Te maximum deflection at failure was 11 mm and te relationsip between te load and te deflection is sown in Figure Figure 7.11: Arrangement for Test wit isolated 180 mm web opening Figure 7.1: Failure mode of Test by local buckling due to te Vierendeel bending 113

150 Load (kn) mm tick C section wit isolated 180 mm web opening Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.13: Load-deflection curve for Test wit isolated 180 mm web opening Test 3: Pairs of 180 mm diameter openings at 90 mm edge distance A pair of 1.5mm tick C-sections wit 180 mm web openings at 90 mm edge distance was tested, as sown in Figure Te failure load was 31.8 kn. Te maximum deflection at failure was 9 mm. Failure was due to web-post buckling, as sown in Figure Te relationsip between te load and te deflection is sown in Figure Figure 7.14: Test arrangement for Test 3 wit 180 mm circular web openings at 90 mm edge distance 114

151 Load (kn) Figure 7.15: Failure mode of Test 3 by web-post buckling between web openings mm tick C section wit 180 mm web openings at 90 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.16: Load-deflection curve for Test 3 wit 180 mm web opening at 90 mm spacing Test 4: Pair of 180 mm diameter openings at 60 mm edge distance A pair of 1.5 mm tick C-sections wit 180 mm web openings at 60 mm edge distance was tested, as sown in Figure Te failure load was 8.8 kn wic is about 10% less tan te failure load in Test 3 wit web openings at 90 mm edge distance. Failure was due to web-post buckling due to te increase in te orizontal sear as a result of reducing te web opening edge distances (V =V o/s o), as sown in Figure Te maximum deflection at failure was 8 mm as sown in te relationsip between te deflection and te applied load in Figure

152 Figure 7.17: Test arrangement for Test 4 wit 180 mm web openings at 60mm spacing Figure 7.18: Failure mode of Test 4 by web-post buckling between web openings 116

153 Load (kn) mm tick C section wit 180 mm web openings at 60 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Displacement(mm) Figure 7.19: Load-deflection curve for Test 4 wit 180 mm web openings at 60mm spacing Test 5: Pair of 180 mm diameter web openings at 90 mm edge distance A pair of 1.8 mm tick C-sections wit 180 mm web openings at 90 mm edge distance was tested, as sown in Figure 7.0. Figure 7.0: Test arrangement for Test 5 wit 180 mm web openings at 60 mm edge distance 117

154 Load (kn) Te failure load was 55 kn wic is 4% more tan te failure load of te 1.5 mm tick C sections wit te same web opening configurations as in Test 3. Increasing o/t w ratio by 0 % reduced te beam resistance by 4%. Failure was due to web-post buckling and bending of te top flange, as sown in Figure 7.1. Te relationsip between te load and te deflection is sown in Figure 7.. Te maximum deflection at failure was 1 mm. Figure 7.1: Failure mode of Test 5 by web-post buckling between web openings mm tick C section wit 180 mm web openings at 90 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.: Load-deflection curve for Test 5 wit 180 mm web opening at 90 mm spacing 118

155 7.5.6 Test 6: Isolated 180 mm diameter web opening A pair of 1.8 mm tick C-sections wit 180 mm web openings was tested, as sown in Figure 7.3 and te failure load was 66.5 kn. Te maximum deflection at failure was 11 mm. Te failure was due to local buckling and Vierendeel bending. Buckling of te un-supported web next to te web openings also occurred wit a buckling lengt of about 60 mm, as sown in Figure 7.4. Te relationsip between te load and te deflection is sown in Figure 7.5. Figure 7.3: Test arrangement for Test 6 wit isolated 180 mm web opening Figure 7.4: Failure mode of Test 6 by local buckling and Vierendeel bending 119

156 Load (kn) mm tick C section wit isolated 180 mm web openings Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.5: Load-deflection curve for Test 6 wit isolated 180 mm web opening Test 7: Isolated stiffened elongated openings A pair of 1.5 mm tick C-sections wit elongated stiffened web openings was tested, as sown in Figure 7.6. Te elongated openings were expected to fail in Vierendeel bending but failure occurred locally at te load point. Figure 7.6: Arrangement for Test 7 wit single elongated stiffened web opening 10

157 Load (kn) Te failure load was 4 kn. Te maximum deflection at failure was 9 mm. Te bending capacity was reaced and te top flange and web buckled locally at te load position, as sown in Figure 7.7. Te relationsip between te load and te deflection is sown in Figure 7.8. It is considered tat failure at te load point was at iger load tan predicted for Vierendeel bending and so all tests were expected to fail at te same way, as was te case. Figure 7.7: Failure mode of Test 7 by local bending of te top flange at load application point mm tick C section wit elongated stiffened web openings Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.8: Load-deflection curve for Test 7 wit single elongated stiffened web opening 11

158 7.5.8 Test 8: Pairs of stiffened elongated openings A pair of 1.5 mm tick C-sections wit elongated stiffened web openings at 80 mm edge distance, as sown in Figure 7.9 was tested and te failure load was 43 kn wic was sligtly iger tan te failure load of beams wit isolated openings. Similar to Test 7, te bending capacity was reaced and te top flange and web buckled locally at te load point and te presence of te pair of web openings did not as an effect on te section resistance. Te maximum deflection at failure was 1.5 mm. Figure 7.9: Arrangement for Test 8 wit pairs of elongated stiffened web opening Te failure load was expected to be lower tan tat of te same beam section wit single web opening (Test 7) but for tese beams, te flange and te web buckled locally at te load point wit no buckling observed for te beam web, as sown in Figure Te relationsip between te load and te deflection is sown in Figure

159 Load (kn) Figure 7.30: Failure mode of te beam in Test 8 by bending of te top flanges at te connection point mm tick C section wit elongated stiffened web openings at 80 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.31: Load-deflection curve for Test 8 wit pairs of elongated stiffened web opening 13

160 7.6 Tests on Cold Formed C Sections wit Diamond and Hexagonal Openings Test 1: Isolated diamond saped openings A pair of 1. mm tick C-sections wit a single diamond saped opening was tested, as sown in Figure 7.3. Te failure load was 3.3 kn. Te failure was due to web buckling troug te diamond opening and deformation of te top flange, as sown in Figure Te relationsip between te load and te deflection is sown in Figure Te maximum deflection at failure was 7.8 mm. Figure 7.3: Test arrangement for Test 1 wit isolated diamond saped opening Figure 7.33: Failure mode of Test 1 by web buckling at te diamond saped opening 14

161 Load (kn) 1. mm tick C sections wit isolated diamond web opening Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.34: Load-deflection curve for Test 1 wit an isolated diamond saped opening 7.6. Test : Pairs of diamond saped opening A pair of 1. mm tick C-sections wit double diamond saped openings at 9 mm edge distance was tested, as sown in Figure Te failure load was 8.4 kn, wic is 88% of te failure load of te same beam wit isolated diamond web openings. Te reduction of te beam resistance can be explained by te presence of te web-post wic was subjected to a ig orizontal sear force and ence te failure due to web-post buckling and buckling of te unsupported edge of te web opening, as sown in Figures Figure 7.35: Arrangement for Test wit pair of diamond saped openings at 9 mm edge distance 15

162 Load (kn) Figure 7.36: Failure mode of Test by web-post buckling Te maximum deflection at failure was 7.6 mm. Te relationsip between te load and deflection is sown in Figure mm tick C section wit diamond web openings at 9 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Figure 7.37: Displacement (mm) Load-deflection curve for Test wit diamond saped openings at 9 mm edge distance 16

163 7.6.3 Test 3: Pairs of exagonal web openings (1.53 mm tick ) A pair of 1.53 mm tick C-sections wit 167 mm deep exagonal web openings at 45 mm edge distance was tested, as sown in Figure Figure 7.38: Test arrangement for Test 3 wit pairs of exagonal web openings Te failure mode was due to web-post buckling due to te increase in te orizontal sear stresses witin te web-post as expected, as sown in Figure Buckling of te unsupported sloping side of te exagonal was apparent. Te failure load was 4.15 kn. Te maximum deflection at failure was 9. mm. Te relationsip between te load and deflection is sown in Figure Figure 7.39: Failure mode of Test 3 by web-post buckling 17

164 Load (kn) mm tick C sections wit exagonal web openings at 45 mm edge distance Flange (1) deflection Flange () deflection Jack deflection Displacement (mm) Figure 7.40: Load-deflection curve for Test 3 wit 167 mm deep exagonal web openings Test 4: Pairs of exagonal web openings (1.93 mm tick ) A pair of 1.93 mm tick C sections wit 167 mm deep exagonal web openings at 45mm edge distance was tested, as sown in Figure Te failure load was 66.3 kn, wic is 57% more tan te failure load of Test 11 wit 1.53 mm tick C section. Te maximum deflection at failure was 6.5 mm. Figure 7.41: Test arrangement for Test 4 wit pairs of exagonal web openings 18

165 Load (kn) Te failure mode was due to web-post buckling but as te tickness of te web ad increased from 1.53 mm as in Test 11 to 1.93 mm, te sear resistance of te web-post ad increased. Bending of te flanges was also observed during testing and te web continued to buckle until te beams were unloaded, as sown in Figure 7.4. Te relationsip between te load and te displacement of te beam is sown in Figure Figure 7.4: Failure mode for Test 4 by web-post buckling and bending of te top flange mm tick C sections wit exagonal web openings at 45 mm edge distance Flange (1) defelction Flange () deflection Jack deflection Displacement (mm) Figure 7.43: Load-deflection curve for Test 4 wit pairs of exagonal web openings 7.7 Tensile Test Results for te Steel Used in te Beam Tests Tensile tests were carried out on te galvanised steel used in te previous tests using te Instron 1341 macine. Te test arrangement and failure of te specimen at te end of te test are sown in Figure Stress-strain curves for te steel wit different ticknesses are sown in Table

166 Stress (N/mm ) Stress (N/mm ) Stress (N/mm ) Figure 7.44: Table 7.4: Tensile test arrangement and failure of specimen in te tensile test Stress-strain curves for te steel used in beam tests Strain Strain 1. mm tick galvanized steel 1.48 mm tick galvanized steel Stress (N/mm ) Strain Strain 1.5 mm tick galvanized steel 1.95 mm tick galvanized steel 130

167 7.8 Tests on Stainless Steel C section wit Circular Web Openings Tests 1 & 9: mm Austenitic steel wit isolated web openings For Tests 1 and 9, a pair of mm tick Austenitic grade stainless steel C sections wit web openings at 50 mm edge distance was tested as duplicate tests to ensure te sensitivity of te results. Te test arrangement is sown in Figure Figure 7.45: Test arrangement for Tests 1 and 9 wit openings at 50 mm edge distance Te failure loads were 58. kn and 61.5 kn for te first and second tests respectively. Te mode of failure was by Vierendeel bending and at failure, bending of te top flange was apparent, as sown in Figure Te displacement at failure was 19.5 mm at te mid-span of te beam, as sown in Figure Figure 7.46: Failure mode of te mm tick Austenitic stainless steel wit circular web opening at 50 mm edge distance by Vierendeel bending 131

168 Load (kn) 70 mm tick Austenitic stainless steel C section wit 150 mm web openings at 50 mm edge distance Flange 1 deflection Flange deflection Jack deflection Displacement (mm) Figure 7.47: Variation of load wit displacement for Test 1 wit isolated circular web opening 7.8. Test : 3 mm Austenitic steel wit openings at 50 mm edge spacing In Test, a pair of 3 mm tick Austenitic stainless steel C sections wit web openings at 50 mm edge distance was tested, as sown in Figure Te failure load was 84 kn and te failure mode was by orizontal sear wit sligt web-post buckling between te openings, as sown in Figure Figure 7.48: Test arrangement of Test wit circular web openings at 50 mm edge distance 13

169 Load ( kn) Figure 7.49: Failure mode of te 3 mm tick Austenitic beam wit circular openings at 50 mm edge distance by orizontal sear and buckling of te web-post Te displacement at failure was 16 mm at mid-span, as sown in Figure In tis Test, strain gauges were fixed to obtain te stresses around te openings and in te flanges as sown in Figure Stresses were recorded at 10 points around te openings in order to correlate te performance wit te teory developed in Capter mm tick Austenitic stainless steel C section wit 150 mm web openings at 50 mm edge distance Jack deflection Flange 1 Flange Displacement ( mm) Figure 7.50: Variation of load wit displacement for Test wit circular web openings at 50 mm edge distance 133

170 MID SPAN STRAIN GAUGE POSITIONS All dimensions are in mm S G 1 S G S tres s (N/mm) S G 3 S G 4 S G 5 S G 6 S G 7 S G 8 S G 9 S G L oa d(kn) Figure 7.51: Stresses around te openings in Test (Sanmugalingam, 010) Test 3: mm Austenitic steel wit openings at 100 mm edge spacing In Test 3, a pair of mm tick Austenitic stainless steel C sections wit web openings at 100 mm edge distance was tested, as sown in Figure 7.5. Te failure load was 44.5 kn wic is 5% less tan te failure load of Tests 1 and 9 due to te increase of te orizontal sear as a result of increasing te o/s ratio. Te displacement at failure was 10 mm at mid-span, as sown in Figure Te failure mode was Vierendeel bending associated wit buckling across te web opening and local buckling at mid-span of top flange as sown in Figure

171 Load (kn) Figure 7.5: Arrangement of Test 3 wit circular web openings at 100 mm edge spacing 50 mm tick Austenitic stainless steel beam wit 150 mm web openings at 100 mm edge distance Jack deflection Flange Flange Displacement (mm) Figure 7.53: Variation of load wit displacement for Test 3 wit circular web openings at 100 mm edge distance 135

172 Figure 7.54: Failure mode of te mm tick Austenitic stainless steel beam wit circular web openings at 100 mm edge distance due to Vierendeel bending and local buckling of top flange Test 4: mm Lean Duplex steel wit openings at 100 mm edge spacing Test 4 was carried out on mm tick Lean Duplex stainless steel C sections wit web openings at 100 mm edge distance, as sown in Figure Te failure load was 71.5 kn wic is 60 % more tan te failure load of te same beam configuration in Austenitic stainless steel as tested in Test 3. Te displacement at failure was 1 mm at mid-span, as sown in Figure Te failure was due to te buckling at top bolt and Vierendeel bending as sown in Figure Bending of te top flange was also apparent at failure. Figure 7.55: Test arrangement of Test 4 wit circular web openings at 100 mm edge distance (Sanmugalingam, 010) 136

173 Figure 7.56: Variation of load wit displacement for Test 4 wit circular web openings at 100 mm edge distance (Sanmugalingam, 010) Figure 7.57: Failure mode of te mm tick Lean Duplex stainless steel beam wit web openings at 100 mm edge distance by buckling at te load point (Sanmugalingam, 010) Test 5: 3 mm Austenitic steel wit openings at 100 mm edge distance Test 5 was carried out on 3 mm tick Austenitic stainless steel C sections wit web openings at 100 mm edge distance, as sown in Figure Te failure load was 86 kn wic is only % less tan te failure load of te same beam wit web openings at closer edge distance as in Test but 94% more tan te failure load of te mm tick beam in Test 3. Tis sows tat te edge distances of te web openings as a little effect on te beam resistance if compared to te effect of o/t ratio. Te failure mode was by orizontal sear between te openings, as sown in Figure Bending of te 137

174 top flange was also apparent at failure. Te displacement at failure was 14 mm at mid-span as sown in Figure Figure 7.58: Arrangement of Test 5 wit circular web openings at 100 mm edge distance (Sanmugalingam, 010) Figure 7.59: Variation of load wit displacement for Test 5 wit circular web openings at 100 mm edge distance (Sanmugalingam, 010) 138

175 Figure 7.60: Failure mode of te 3 mm tick Austenitic stainless steel C section wit circular web openings at 100 mm edge distance (Sanmugalingam, 010) Test 6 & 8: mm Lean Duplex steel and openings at 50 mm edge distance Tests 6 and 8 were repeated tests wit mm tick Lean Duplex stainless steel C sections and wit web openings at 50 mm edge distance, as sown in Figure For te two tests, te failure loads were 54. kn and 55.4 kn respectively. Te displacement at failure was 1 mm at mid-span, as sown in Figure 7.6. As expected, te beam failure was due to web-post buckling due to te increase in te orizontal sear stress as sown in Figure Figure 7.61: Test arrangement of Test 6 and 8 wit circular web openings at 50 mm edge distance 139

176 Load (kn) 60 mm tick Lean Duplex beam wit 150 mm web openings at 50 mm edge distance Flange 1 Flange Jack deflection Displacement (mm) Figure 7.6: Variation of load wit displacement for Test 6 wit circular web openings at 50 mm edge distance Figure 7.63: Failure mode of te mm tick Lean Duplex stainless steel C section wit web openings at 50 mm edge distance by web-post buckling Test 7: mm Austenitic steel wit openings at 50mm edge spacing In Test 7, a pair of mm tick Austenitic stainless steel C sections wit web openings at 50 mm edge spacing was tested as sown in Figure Te failure load was 39.4 kn wic is 35% less tan te failure load of te same beam wit web openings at 50 mm edge distance (Tests 1and 9) 140

177 Load (kn) due to te increase of te o/s ratio from 0.6 to 3. Te failure load was also 7% less tan te failure load for te same beam configuration in Lean Duplex (Test 6 and 8) and so it sows te effect of te material on te beam resistance. Te displacement at failure was 15 mm at mid-span, as sown in Figure Te failure of te beam was due to web-post buckling due to te increase in te orizontal sear stresses witin te web-post, as sown in Figure Bending of te top flange was also apparent. Figure 7.64: Test arrangement of Test 7 wit circular web openings at 50 mm edge distance 45 mm tick Austenitic stainless steel C section wit web openings at 50 mm edge disatnce Flange 1 Flange Jack movement Displacement (mm) Figure 7.65: Variation of load wit displacement for Test 7 wit circular web openings at 50 mm edge distance 141

178 Figure 7.66: Failure mode of te mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance by web-post buckling 7.9 Discussion of Results of Tests on Cold Formed Beams Tests on beams wit circular web openings As expected, te distance between te openings (web-post widt) ad a direct effect on te beam resistance and te mode of failure. Beam wit single web openings failed by Vierendeel bending wit local buckling around te openings wile te failure for beams wit web openings at close edge distances was due to te buckling of te web-post between openings. Te tickness of te beam as a great effect on te beam resistance. For example, Test 3 and 5 ave te same configurations in 1.5 mm tick steel (Test 3) and 1.8 mm tick steel (Test 5). Te failure load of Test 5 was 55 kn, wic is 75% iger tan te failure load of 31.8 kn in Test 3. A summary of test results is sown in Table Comparison of te tangential stress metod wit te test results Te tangential stresses metod presented in Capter 4 may be calculated at te failure loads in te tests and compared to te measured steel strengts, f y (at 0.% strain). Tese results are presented in Table 7.6. Tis ratio varies between 0.5 and 0.8, te lower ratios occurred for Test 3 and 4 wic failed by web-post buckling. Tis sows tat te tangential stress metod is reasonably accurate, wen not affected by web-post buckling. Te test results are expressed as a ratio of te various calculated resistances as in Table 7.7 (based on gross section properties). Te ratio of te applied sear force to pure sear resistance at te opening was in te range of 0.34 to 0.61, wic sows tat te sear forces were relatively ig especially for beams wit single 180 mm web openings. Te ratio of te applied bending moment to 14

179 pure bending resistance at te opening next to te load point was in te range of 0. to 0.43, wic sows tat te bending moment was relatively low. Te ratio of te orizontal sear stress to te sear strengt V /V,Rd was in te range of 0.1 to 0.8 depending on te spacing of te openings. Te ratio of te applied sear force to te Vierendeel bending resistance V/V vier,rd of te Tees was in te range of 0.34 to 0.61, wic sows tat Vierendeel bending was relatively ig in many of te tests. It was concluded from Table 7.7 tat, te Vierendeel resistance of beams wit web openings is te dominant mode of failure wen using te Eurocode metod in calculating te section resistance Tests on beams wit elongated stiffened openings (Test 7 and 8) Te tests failed by web buckling at te load point and te sear force transferred at te openings was over 10 kn at failure. Tere was no difference between te failure load for te single opening result and te case wit two openings placed at 80 mm apart. Te stiffeners around te web openings appeared to be doing all te ard work in preventing buckling around te large openings but furter testing and investigations are required to verify te assumption. Table 7.5: Summary of test results for cold formed C sections wit circular web openings Beam Measured steel tickness Failure load for two beams Mode of failure in test Beam wit single 150 mm diameter circular web opening Beam wit single 180 mm diameter circular web opening Beam wit 180 mm diameter circular web opening at 60 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Beams wit stiffened single web opening Beams wit stiffened web openings at 80 mm edge distances Steel tickness = measured steel (galvanizing) 1.8 mm 8 kn 1.8 mm 66/68 kn 1.5 mm 8.8 kn 1.5 mm 31.8 kn 1.8 mm 55 kn 1.5 mm 4 kn 1.5 mm 41 kn Bending of top flange wit failure at connections Local buckling and Vierendeel bending Post buckling failure of te web Post buckling failure of te web Local buckling and Vierendeel bending Buckling of web (local failure at connection) Buckling of web (local failure at connection) 143

180 Table 7.6: Tangential stress at edge of openings calculated for te test failure loads Test Opening spacing (edge to edge) Failure sear force per C section Local tangential stress, σ around openings (N/mm ) Ratio of tangential stress to steel strengt σ/f y kn 56 (=9.5 0 ) kn 361 (=6 0 ) mm 7.9 kn 01 (=6 0 ) mm 7. kn 183(=65 0 ) mm kn 311 (=6 0 ) kn 351 (=6 0 ) 0.8 Were: σ Tangential stress around opening see teory Angle of maximum stress to te vertical f y Yield strengt of steel (measured as 400 N/mm for 1.5 mm tick steel and 435 N/mm for 1.8 mm tick steel) Table 7.7: Summary of te test results compared to section capacity obtained from te gross section properties Test M/M el,rd V/V vier,rd V/V V,Rd V /V,Rd Were: M M el V V vier,rd V V,Rd V V,Rd Bending moment at opening Elastic bending resistance of perforated section Sear force at failure Vierendeel sear resistance Vertical sear resistance at opening Horizontal sear force Horizontal sear resistance 144

181 7.9.3 Tests on Beams wit Diamond and Hexagonal Web Openings Te mode of failure for beams wit diamond and exagonal web openings was buckling across te opening. Te failure mode of beams wit isolated diamond web openings was buckling around te openings. For te web openings at close edge distances, te failure was due to te web-post buckling of te web as expected. Te tickness of te section as a great effect on te beam failure load; Test 11 and Test 1 ad te same beam configurations wit 1.5 mm tick steel for Test 11 and 1.93 mm tick steel for Test 1. Te failure load of Test 1 was 66.3 kn wic is 57% greater tan te failure load of Test 11 being 4.15 kn. Te test results are expressed as a ratio of te various calculated resistances as in Table 7.8 (based on gross section properties). Te ratio of te applied sear force to pure sear resistance at te opening was in te range of 0.3 to 0.77 wic sows tat te sear forces were relatively ig especially for beams wit exagonal web openings. Te ratio of te applied bending moment to pure bending resistance at te opening next to te load point was in te range of 0.3 to 0.44, wic sows tat te bending moment was relatively low. Te ratio of te orizontal sear stress to te sear strengt V /V,,Rd was in te range of 0.13 to 0.9 depending on te tickness of te steel and te sape and te spacing of te openings. Te ratio of te applied sear force to te Vierendeel bending resistance V/V vier,rd of te Tees was in te range of 0.44 to 0.57, wic sows tat Vierendeel bending was relatively ig in many of te tests. Te ratio of te applied sear force to sear resistance V/V Rd was in te range of 0.3 to 0.77, wic sows tat sear was ig in tests of beams wit exagonal web openings Comparison between te Teory as Presented in Capter 5 and Test Results for Two Castellated Beams Tested at University of Surrey A comparison between te applied stresses wic were obtained from te test results and te compressive strengt of te web based on te compression resistance of te strut between openings (as explained in Capter 5 ) is sown in Table 7.9. Te best correlation wit te test data is obtained for an effective strut lengt, were 0. 4 wit 3 simply supported sides tan to case b wit a fixed long edge. eff o, wic corresponds more accurately to case a 145

182 Table 7.8: Summary of test results for cold formed C-sections wit diamond and exagonal web openings Beam Measured steel tickness Failure load for two beams Mode of failure in test Beam wit single 180 mm deep diamond saped web opening Beam wit 180 mm deep diamond saped openings at 9 mm edge distance Beam wit pairs of exagonal web openings at 45 mm edge distance Beam wit pairs of exagonal web openings at 45 mm edge distance 1. mm 3.3 kn Buckling around web openings 1.5 mm 8.4 kn Web-post buckling 1.5 mm 4.15 kn Web-post buckling 1.93 mm kn Web-post bucking and bending of top flanges Table 7.9: Summary of te test results compared to section resistance obtained from te teory in tis tesis Test M/M el,rd V/V vier,rd V/V V,Rd V /V,Rd σ/ σ c Table 7.10: Comparison between applied stresses and compressive strengt of te tested beams Beam tickness Applied stresses Compression strengt (imperfection=0.34, curve b ) Compression strengt (imperfection =0.1,curve a ) l eff = 1.7b l eff = 0.74b l eff = 1.7b l eff = 0.74b N/mm 101 N/mm 7 N/mm 108 N/mm 53 N/mm N/mm 151 N/mm 93 N/mm 164 N/mm 34 N/mm 146

183 7.10 Discussion of Results of Tests on Stainless Steel Beams Te tests on beams wit closely spaced openings failed by web-post buckling wile tests on wide spaced web openings failed by Vierendeel bending associated wit local buckling around te web openings. Te test results sowed tat openings at 100 mm spacing led to a 5% reduction in sear resistance and openings at 50 mm spacing led to a 34% reduction in sear resistance relative to te case of widely spaced openings. Tis occurred due to te effect of web-post buckling. Increasing te steel tickness from to 3 mm increased te sear resistance by 91 to 113%, wic is equivalent to te steel tickness to te power of 1.7 as a result of te local buckling resistance of ticker steel. Using Lean Duplex rater tan Austenitic steel increased te sear resistance by 41 to 58%, wic is less tan te 8% increase in te steel proof strengt because of te greater effect of local buckling at iger stresses. Summary of te test results is summarised in Table Te sensitivity of opening spacing for two steel ticknesses was also investigated. Local buckling was critical for te mm tick sections wit closely spaced openings. Te Vierendeel bending resistance of te web-flange Tees witout considering local buckling was calculated as in Appendix C using te measured strengts of te steel and using gross properties. Tis is based on an equivalent circular opening of lengt = 0.45x diameter for Vierendeel bending (Redwood, 1973). Te test results are expressed as a ratio of te calculated resistances as in Table 7.1 (based on gross section properties). Te ratio of te applied sear force to pure sear resistance at te opening, V/V V,Rd, was in te range of 0.4 to 0.9, wic sows tat te sear forces were relatively ig. Te ratio of te applied bending moment to pure bending resistance at te opening next to te load point was in te range of 0.9 to 0.64, wic sows tat te bending moment was relatively low. Te ratio of te orizontal sear stress to te sear strengt V /V,Rd was in te range of 0.36 to 0.83 depending on te spacing of te openings. Te ratio of te applied sear force to te Vierendeel bending resistance, V/V vier,rd, of te Tees was in te range of 0.48 to 0.94, wic sows tat Vierendeel bending was critical in many of te tests. Te direct stresses σ around te openings (see te following section) are in te range of 0.7 to 1.7 x measured steel strengt, f y. Tis sows tat te tangential stress metod is more accurate wit respect to te test results. Also Test 4 and 6 were affected by local buckling around te opening, and Test 3 was affected by te bearing strengt of te bolts and local buckling of te web at te load application point. It was concluded tat, Eurocode BS EN 1993 does give conservative section resistance wen Vierendeel bending is critical especially for stainless steel beams wit large web openings. 147

184 Table 7.11: Test series, failure loads and mode of failure of stainless steel C sections wit 150 mm diameter web openings Test Stainless steel type/ tickness Opening edgeedge spacing Failure load for two beams Mode of failure in test Comments / mm / 3 mm / mm LDX 101/ mm / 3 mm LDX 101/ mm / mm 50 mm 58 / 61.5 kn 50 mm 84 kn 100 mm 100 mm 100 mm 50 mm 44.5 kn 71.5 kn Vierendeel bending wit local buckling Horizontal sear wit sligt local buckling of web-post Buckling at top flange above load application point Buckling at top bolt of load application before web-post failure 86 kn Horizontal sear causing web-post buckling 54. / 55.4 kn 50 mm 39.4 kn Horizontal sear causing web-post buckling Horizontal sear causing web-post buckling (5% less tan Tests 1&9) (60% Increase on Test 3) (93%Increase on Test 3) (35% less tan Test 4) (35% less tan Tests 1&9) Table 7.1: Failure loads compared to bending and sear resistances calculated using gross properties of te section Test Ratio between applied loads and beam resistance Ratio between tangential stress and yield stress M/M el,rd V/V vier,rd V/V V,Rd V /V,Rd σ/f y 1& & Were: σ f y Tangential stress around opening Yield strengt of steel (measured as a 0.% proof strengt of 85/310 N/mm for Austenitic steel and 50 N/mm for Lean Duplex steel) 148

185 CHAPTER 8 FINITE ELEMENT ANALYSIS ON THIN WALLED STEEL BEAMS WITH WEB OPENINGS 8.1 Introduction to Finite Element Analysis A finite element analysis package ABAQUS/CAE (version 6.11-) was used for te simulations in tis study. Te model geometry is entered in terms of features wic are sub-divided into finite elements in order to perform te analysis. Te software contains predefined material property database for virtually all materials of interest to structural engineers. Linear, non-linear and buckling analysis was considered in tis study, difference between different analysis and wy eac analysis was used (as explained in ABAQUS Manual) is summarised below. 8. Linear and Nonlinear Analysis Linear finite element analysis assumes tat all materials are linear elastic and tat deformations are small enoug to not significantly affect te overall beaviour of te structure. Te following factors are indicating wy nonlinear finite element analysis is required: Gross canges in geometry Permanent deformations Buckling- load on lateral buckling Stresses in te post-elastic stage. ABAQUS as tree types of non-linear analysis wic are summarised in ABAQUS User s Manual, as follows: 8..1 Geometric nonlinearity Tis source of nonlinearity is related to canges in te geometry of te structure during te analysis. Geometric nonlinearity occurs wenever te magnitude of te displacements affects te response of te structure. Tis may be caused by: Large deflections or rotations Initial stresses or load stiffening. 8.. Boundary nonlinearity Boundary nonlinearity occurs if te boundary conditions cange during te analysis. As an example, consider te cantilever beam sown in Figure 8.1 tat deflects under an applied load until it its a stop Te vertical deflection of te tip is linearly related to te load (if te deflection is small) until 149

186 it contacts te stop, and ten tere is a cange in te boundary condition, preventing any furter vertical deflection at te tip, and so te beam response is stiffened. Figure 8.1: A cantilever beam taken as an example to sow te boundary nonlinearity (ABAQUS User s Manual) 8..3 Material nonlinearity Most metals ave a linear stress/strain relationsip at low strain values but at iger strains te material yields, at wic point te response becomes nonlinear and irreversible, as sown in Figure 8.. Material nonlinearity may be related to factors oter tan strain. Strain-rate-dependent material data and material failure are bot forms of material nonlinearity. Figure 8.: Curve sowing te linear and nonlinear stress/strain relationsip (ABAQUS User s Manual) 8.3 Buckling and Post Buckling Analysis Linear buckling analysis Buckling is wen a flexible structure loses its stability, wic may lead to a sudden and catastropic failure, suc as te complete collapse or breakage of te structure. [Ugural, 1987] Linear buckling is te most common type of analysis and is easy to execute, but it is limited in te results it can provide. Linear-buckling analysis calculates te buckling load magnitudes tat cause buckling in its response modes. FEA programs provide calculations of a large number of buckling modes and te associated buckling-load factors (BLF). Te BLF is expressed by a number by wic 150

187 te applied load must be multiplied (or divided - depending on te particular FEA package) to obtain te buckling load Non-Linear buckling analysis As wit any oter nonlinear analysis, nonlinear-buckling analysis requires tat a load is applied gradually in multiple steps rater tan in one step, as in a linear analysis. Eac load increment canges te structure s sape, and tis, in turn, canges te structure s stiffness. Terefore, te structure stiffness must be updated at eac increment. In tis approac, wic is called te load control metod, load steps are defined eiter by te user or automatically so te difference in displacement between te two consecutive steps is not too large (ABAQUS User s Manual &Tecnical Gazette, 01). Altoug te load-control metod is used in most types of nonlinear analyses, it would be difficult to implement in a buckling analysis. Wen buckling occurs, te structure undergoes a momentary loss of stiffness and te load control metod would result in numerical instabilities. Nonlinear buckling analysis requires anoter way of controlling load application, te arc lengt control metod. Here, points corresponding to consecutive load increments are evenly spaced along te loaddisplacement curve, wic itself is constructed during load application. In contrast to linear-buckling analysis, wic only calculates te potential buckling sape wit no quantitative values of importance, nonlinear analysis calculates actual displacements and stresses. 8.4 Eigenvalue and Riks Metods Eigenvalue buckling Eigenvalue buckling is generally used to estimate te critical buckling loads of stiff structures (classical eigenvalue buckling). Stiff structures support teir design loads primarily by axial or membrane action, rater tan by bending action. Teir response usually involves little deformation prior to buckling. Wen te response of a structure is nonlinear before collapse, a general eigenvalue buckling analysis can provide useful estimates of collapse mode sapes. For te base state, te buckling loads are calculated relative to te base state of te structure. If te eigenvalue buckling procedure is te first step in an analysis, te initial conditions form te base state; oterwise, te base state is te current state of te model at te end of te last general analysis step. If geometric nonlinearity is included in te general analysis steps prior to te eigenvalue buckling analysis, te base state geometry is te deformed geometry at te end of te last general analysis step. If geometric nonlinearity was omitted, te base state geometry is te original configuration of te body (ABAQUS User s Manual & Tecnical Gazette, 01). In simple cases, linear eigenvalue analysis will be sufficient for design evaluation; but if tere is concern about material nonlinearity, geometric nonlinearity prior to buckling, or unstable post- 151

188 buckling response, a load-deflection (Riks) analysis must be performed to investigate te problem furter. Te Riks metod uses te load magnitude as an additional unknown; it solves simultaneously for loads and displacements. Terefore, anoter quantity must be used to measure te progress of te solution; ABAQUS/Standard uses te arc lengt along te static equilibrium pat in loaddisplacement space Te Riks metod In nonlinear static analysis for buckling, post-buckling, or collapse beaviour, te tangent stiffness from te load-displacement response curve could cange signs wen system canges its stability status, as sown in Figure 8.3. Te classical Newton s metod will not work in tis situation because te corrections for approacing equilibrium solutions during iterations may become difficult to determine wen te tangent stiffness is close to null. Terefore, te static equilibrium states during te unstable pase of te response can be found by using te modified Riks metod [Abaqus, 007]. Te basic Riks algoritm is essentially Newton s metod wit load magnitude as an additional unknown to solve simultaneously for loads and displacements, tus, can provide solutions even in cases of complex and unstable response (Zao 008). Te Riks metod is generally used to predict unstable, geometrically nonlinear collapse of a structure can include nonlinear materials and boundary conditions and often follows an eigenvalue buckling analysis to provide complete information about a structure s collapse and can also be used to speed convergence of ill-conditioned or snap-troug problems tat do not exibit instability (Tecnical Gazette, 01). Te Eigenvalue linear analysis was te first step in te non-linear analysis of te C sections wit web openings to determine te critical buckling modes of eac beam. After te eigenvalue linear analysis, te nonlinear Riks static analysis was carried out for elastic-linear and nonlinear elastoplastic material definition to determine te maximum beam sear resistance and te corresponding stresses around te web openings. Te Maximum load was obtained so tat te maximum principal stresses at failure are less tan or equal te ultimate strengt of te material obtained previously from tensile tests. 15

189 Figure 8.3: Proportional loading wit unstable response (Tecnical Gazette, 01) 8.5 Effect of Imperfections and Effect of Plastification of Deformable Elements Actual members always ave imperfections, bot in te way te load is applied (eccentricity wit respect to te centroid of te cross-section or inclination wit respect to te bar axis) and wit respect to te geometry of te section (residual curvature, non-constant cross-section, etc.). As a consequence of tese unavoidable perturbations, te axial force causes bending even wen it takes a value wic is smaller tan te critical load (Silva, 006). Te post buckling curve of an initially perfect system does not by itself give sufficient information. To obtain correct information about post buckling beaviour, it is needed to consider imperfections of sape or/and eccentricities of loading wic are present in all real structures (Brubak and Hellesland, 007). Wen ideal load is applied, until te buckling load is reaced, no internal forces are necessary in te elastic element to balance te applied load. Te bending deformation introduces additional stresses, wic become larger wen te load gets close to te critical value. As a consequence, te critical load predicted by te Euler s formula is usually not reaced, since plastic deformations or material failure take place before tis point (Tecnical Gazette, 01). Unavoidable imperfections of te structures may influence teir stability beaviour considerably, wit respect to te value of te critical load, and even in terms of te caracteristics of te deformation (Silva, 006). However, in actual structures, te large deformations caused by te imperfection, wen te load gets close to te critical value, do limit te loading capacity, even in te case of stable post-critical beaviour. Wen te deformable elements of a compressed structure enter te elasto-plastic regime, te corresponding loss of stiffness usually causes a considerable reduction in te maximum load of te structure. Yielding transforms te stable post buckling beaviour into unstable, since, after yielding, an increase in te deformation causes a decrease of te corresponding 153

190 permissible load (Silva, 006). Since te post buckling beaviour may become unstable wen elastoplastic deformations take place, it is very important to investigate te influence of imperfections on te loading capacity of te structure. 8.6 Modelling Tin Steel C-Sections wit Circular Web Openings using ABAQUS In te FE analysis, C sections wit circular, diamond and exagonal web openings at different edge to edge distance were modelled to investigate te beaviour of te beams. Te purpose was to:- Understand te stress flow around te openings. Predict were local buckling occurs. Investigate non-linear effect wic is not part of te elastic metod based. For te finite element analysis (FEA), one of te beams C sections was considered, and by using te symmetry of te beam member, alf of te member was modelled wit correct boundary conditions in order to minimise te number of nodes. Tis minimises te solving time and te memory required for te execution. Te analysed models, are sown in Figure 8.4. Figure 8.4: Finite element model of alf beam as modelled using ABAQUS Mes generation Te model geometry is defined in terms of features (points, lines, surfaces and volumes). Tese features are mesed to generate te finite element model ready for solution. Te member is defined as a combination of tree dimensional sell elements. Figure 8.5 illustrates te mes used for te analysis of beam. A ig mes density will increase te accuracy of te results obtained at te expense of computation time, wile low mes density can lead to inaccuracies. Irregular mes uses an element size equal to 1 mm and 5 mm was used for stainless steel and cold formed C-sections respectively. Mes sensitivity study was carried out on bot te stainless steel and galvanized steel beams as explained in te finite element results in Capters 9 and

191 Figure 8.5: Typical finite element model sowing its mes in ABAQUS 8.6. Material caracteristics Stainless steel properties Te stainless steel material beaviour was assumed to be omogeneous and isotropic. Typical graps for Austenitic and Lean Duplex steel obtained from te tensile tests wic were carried out by Sanmugalingam in 010, are sown in Figure 8.6, 8.7, and 8.8. Te effective yield strengt was taken at te limit of elastic beaviour. Te proof strengt at 0.% strain and for comparison, te proof strengt at 1% strain are given in Table 8.1. Te ultimate tensile strengt occurred at very ig strains (over 0% for steel) so te strengt at 6% strain is also presented. Table 8.1: Tensile test results for stainless steel C sections Steel Tickness Effective yield strengt N/mm Proof strengt at 0.% strain N/mm Proof strengt at 1% strain N/mm Strengt at 6% strain N/mm Ultimate strengt N/mm mm mm LDX 101 mm

192 Stress (N/mm ) Stress (N/mm ) Strain (%) Figure 8.6: Stress-strain curve for mm tick Austenitic stainless steel Strain (%) Figure 8.7: Stress-strain curve for mm tick Lean Duplex stainless steel 156

193 Stress (N/mm ) Strain (%) Figure 8.8: Stress-strain curve for 3 mm tick Austenitic stainless steel Galvanized steel properties Similar to stainless steel, te galvanized material beaviour was assumed to be omogeneous and isotropic. Te effective yield strengt is taken at te limit of elastic beaviour. Te yield strengt at 0.% strain and for comparison at 1% strain and te ultimate strengt are given in Table 8.. Stressstrain grap obtained from te tensile tests are presented in Capter 7 in Table 7.4. Table 8.: Tensile test results for galvanized steel C sections Tickness Effective yield strengt N/mm Yield strengt at 0.% strain N/mm Yield strengt at 1% strain N/mm strengt at % strain N/mm Ultimate strengt N/mm 1.0 mm mm mm mm Boundary conditions Te beam was analysed as a simply supported beam. Terefore bot supports are treated as pinned. By using te symmetric beaviour, alf of te beam lengt is modelled for tis analysis wit correct boundary conditions. 157

194 Tose boundary conditions of te supports are given in Table 8.3 and sown in Figure 8.9. Figure 8.9: Model wit support arrangement in ABAQUS Table 8.3: Boundary Conditions in FE Model Direction Support at left Support at rigt (mid-span) X-along te beam Free Fixed Y-vertical Fixed Free Z-transverse Fixed Fixed Te model was fixed in Y, Z direction at one end to present te support and it was free in te Y direction at mid span to allow te beam mid-span to deflect vertically under te applied load Loading ABAQUS as different type of loadings and in tis analysis, sell edge load was applied to te web of te beam section to represent te point load applied during testing in wic te sear force applied to te model is divided by te dept of te web, as sown in Figure Finite Element Analysis Results Linear and non-linear analyses were considered in tis researc. Te linear analysis was carried out on all beam sections assuming tat all materials are linear elastic in beaviour to obtain te maximum tangential stresses around te web openings and to compare te results wit te proposed tangential metod proposed in Capter 4. Te results from te linear analysis were conservative wen compared wit te results from te proposed teory. Te non-linear analysis was carried out to obtain te sections resistance. Te Riks analysis metod was considered and te section resistances obtained were in te range of 70-90% of te test failure loads for cold formed and stainless steel C sections. 158

195 CHAPTER 9 FINITE ELEMENT RESULTS FOR COLD FORMED BEAMS WITH DIFFERENT SHAPED WEB OPENINGS 9.1 Introduction Te objective of te finite element analyses was to evaluate te stresses around te openings of te beams and to compare te failure loads predicted from te finite element analysis wit te test results in Capter 7. Te performance of cold formed C section beams wit circular, diamond and exagonal web openings was investigated by using linear and non-linear finite element analysis. Te variable parameters were te distance between te openings, te sape of te openings, te tickness of te steel and te relative eigt of te openings, as sown in Table 9.1 and 9.. For all models, te C sections were 50 mm deep x 63 mm nominal flange widt. For all beams, te web dept: tickness ratios ( w /t w) > 7, terefore, web buckling was expected to ave significant effect on te sections resistance. Te beam clear span was 1.4 m for te 1.5 m long beams. Table 9.1: Beam Model 1 Beam wit single opening Model Beam wit single opening Model 3 Beam wit pair of openings Model 4 Beam wit pair of openings Model 5 Beam wit pair of openings Model 6 Beam wit single elongated stiffened opening Model 7 Beam wit pair of elongated stiffened opening Cases analysed in finite element models for cold formed steel C sections wit circular web openings Diameter of opening Edge distance of openings 159 Nominal steel tickness Steel tickness (less galvanizing) Web dept: tickness ratio w /t w 150 mm mm 1.76 mm mm mm 1.76 mm mm 60 mm 1.5 mm 1.46 mm mm 90 mm 1.5 mm 1.40 mm mm 90 mm 1.8 mm 1.76 mm mm mm 1.46 mm mm 80 mm 1.5 mm 1.46 mm 169

196 Failure load ( kn) Table 9.: Beam and openings Single diamond saped opening Cases of finite element models of cold formed steel C sections wit diamond and exagonal web openings Heigt of openings Edge distance Nominal steel tickness Measured steel tickness w /t w 180 mm - 1. mm 1. mm 03 Pair of diamond saped openings Pair of exagonal web openings 180 mm 9 mm 1. mm 1. mm mm 45 mm 1.5 mm 1.53 mm 161 Pair of exagonal web openings 167 mm 45 mm 1.9 mm 1.93 mm 17 Steel tickness less galvanizing is measured tickness mm Edge distance is between te outermost edges of te openings 9. Mes Sensitivity Study on Cold Formed C Sections A sensitivity study for te mes global size was carried out in ABAQUS on one of te tested beams (50 mm deep x 1.5 mm tick C section wit 180 mm circular web openings at 60 mm spacing) in order to establis te average mes global size tat can be considered in te finite element analysis. Different mes sizes were considered and te failure load was calculated for eac case. A global size of 5 mm was found to be a reasonable mes size and gives accurate average results Failure load for different mes size Test failure load Mes global size (mm) Figure 9.1: Failure load for different mes size comparing to te test failure load for 1.5 mm tick C section wit 180 mm web opening at 60 mm edge distance 160

197 9.3 Linear Analysis of Cold Formed C section wit Circular Web Openings Linear finite element analysis (FEA) was carried out on all of te tested beams. For all models, alf a beam section wit te web openings was modelled and analysed in ABAQUS subject to a sell edge load applied to te beam web equal to te sear failure load (quarter te failure load for two beams) as sown in Figure 9.. Maximum tangential stresses were recorded and plotted as a function of te angle to te vertical around te openings. Principal stresses around web openings were also recorded. Te maximum principal stresses and deflection at mid-span obtained from te linear FE analysis are sown in Table 9.3. For Model and Model 5, te maximum principal stresses were iger tan te material yield stress wic can be explained by te presence of residual stresses in cold formed sections in wic, te yield stress can be increased due to tis (Scafer 1998). Table 9.3: Maximum principal and mid-span deflection for eac model obtained from te linear elastic finite element analysis FE results FE Model Applied sear force at failure Maximum principal stresses at failure load from FEA (N/mm ) Angle to vertical of maximum stress Deflection at mid-span at failure (mm) Material yield stress (N/mm ) Model kn o Model 16.6/17.1 kn 490/ o Model 3 7. kn o Model kn o Model kn o Model kn o Model kn o

198 Figure 9.: Sear force applied to te web of te beam as a sell load in ABAQUS Model 1: 1.8 mm tick beam wit 150 mm diameter isolated web opening Te corresponding principal stress around te web opening of te linear analysis is sown in Figure 9.3. Te maximum principal stress was 430 N/mm. Local buckling of te bottom flange is apparent. Principal stresses around te opening were plotted against te angle to te vertical around te openings, as sown in Figure 9.4. Te maximum principal stress was at an approximate angle of 30 o to te vertical. Figure 9.3: Maximum principal stresses of 430 N/mm around web opening for Model 1 subject to a sear force of 0.5 kn 16

199 Principal Stresses (N/mm ) Angle to vertical Stresses around opening Figure 9.4: Principal stresses around circular opening for model 1 wit isolated 150 mm diameter web opening 9.3. Model : 1.8 mm tick beams wit 180 mm diameter web openings Te corresponding principal stresses around te web opening of te linear analysis are sown in Figure 9.5. Te maximum principal stress was 507 N/mm. Te buckling of te beam flange and te buckling across te web opening are apparent. Principal stresses around te opening were plotted, as sown in Figure 9.6. As for te smaller opening, te maximum principal stress was at an angle of 30 o to te vertical. Figure 9.5: Maximum principal stresses of 507 N/mm around te web opening for Model subject to a sear force of 17.1 kn 163

200 Principal Stresses (N/mm ) Stresses around opening Angle to vertical Figure 9.6: Principal stresses around circular opening for Model wit isolated 180 mm diameter web opening Model 3: 1.5 mm tick beams wit 180 mm web openings at 60 mm edge distance Te corresponding principal stresses around te web opening are sown in Figure 9.7. Concentrated stresses occurred around te web openings due to te local buckling across te web openings. Te maximum stress was 310 N/mm. Te deformation of te beam flange is apparent. Principal stresses around te openings were plotted, as sown in Figure 9.8. Despite te relatively narrow web-post, te maximum principal stresses were still at an angle of 30 o to te vertical. Figure 9.7: Maximum principal stresses of 310 N/mm around te opening for Model 3 subject to a sear force of 7. kn 164

201 Principal Stresses (N/mm ) Stresses around opening near support Stresses around opening near load Angle to vertical Figure 9.8: Principal stresses around circular opening for Model 3 wit 180 mm diameter web openings at 90 mm edge disatnce Model 4: 1.5 mm tick beams wit 180 mm web openings at 90 mm edge distance Te deformation of te beam and te corresponding principal stresses of te linear analysis are sown in Figure 9.9. Te maximum stress was 340 N/mm wic ad occurred around te opening near te load point. Te orizontal sear stress witin te web-post equals te orizontal sear force divided by te area of te web-post wic explains te ig stresses witin te narrow web-post. Figure 9.9: Maximum principal stresses of 340 N/mm around te opening for Model 4 subject to a sear force of 7.9 kn Principal stresses around te openings were plotted, as sown in Figure Te maximum principal stresses around te opening near te support were between angles of 30 0 and For te opening near te load point, te maximum principal stress occurred at an angle of 60 0 to te vertical. 165

202 Principal Stresses (N/mm ) Stresses around opening near support Stresses around opening near load Angle to vertical Figure 9.10: Principal stresses around circular opening for Model 4 wit 180 mm diameter web openings at 60 mm edge disatnce Model 5: 1.8 mm tick beam wit 180 mm web openings at 90 mm edge distance Te corresponding principal stresses of te linear analysis are sown in Figure Te maximum stress was 450 N/mm and it occurred around te opening nearer to te load point. Deformation of te top flange and buckling around te web openings are apparent. Figure 9.11: Maximum principal stress of 450 N/mm around te web opening for Model 5 subject to a sear force of kn Principal stresses around te opening were plotted, as sown in Figure 9.1. Te maximum principal stresses around te opening occurred at an angle of 30 0 to te vertical for bot web openings similar to te beam wit web openings at 60 mm edge distance. Tis corresponded to an effective buckling lengt of l eff =0.5 o. Despite te wider web-post, te stresses witin te web-post were greater tan te stresses witin te narrower web-post in Model

203 Principal Stresses (N/mm ) Angle to vertical Stresses around opening near support Stressess around opening near load point Figure 9.1: Principal stresses around circular opening for Model 5 wit 180 mm diameter web openings at 90 mm edge disatnce Model 6 &7: 1.5 mm tick beams wit 180 mm isolated elongated stiffened web openings and pairs of openings at 80 mm edge distance For te beam wit single web opening, te maximum principal stress was 367 N/mm and te buckling of te beam bottom flange is also apparent. Buckling around te stiffened web opening occurred wic can be explained by te Vierendeel bending of te top and bottom Tee sections above and below te web opening wic caused twisting of te top flange, as sown in Figure Figure 9.13: Maximum principal stresses of 367 N/mm around web opening for Model 6 subject to a sear force of kn 167

204 For te beam wit pairs of elongated stiffened web openings at 80 mm edge distance, te deformation of te beam for linear analysis is sown in Figure Deformation of te top and bottom flanges occurred at failure and te bending across te stiffened web opening is apparent. Te maximum principal stress was recorded as 376 N/mm around te opening near te load point and 50 N/mm around te opening near te support. Figure 9.14: Maximum principal and bending stresses of 376 N/mm around te openings for Model 7 subject to a sear force of kn 9.4 Stresses from FEA Models Compared wit Tangential Stress Metod for Cold Formed Steel Beams wit Web Openings at Different Edge Distances Te principal stresses against te angle to te vertical around te web openings obtained from te FE models subject to sear force corresponding to test failure loads for beams wit closely and widely spaced web openings were compared wit te Tangential Stress Metod proposed in Capter 4. Te comparisons are discussed below and sown in Figure 9.15 to Figure 9.19 for te same load. Two curves are presented for te proposed metod, as follows: Stress for isolated openings wic will control for te stresses between 10 0 and 40 0 to te vertical. Stress for closely spaced openings wic will control for te stresses between 50 0 and 80 0 to te vertical Beams wit isolated web openings (Model 1 and ) For Model 1 wit isolated 150 mm diameter web openings, te maximum principal stress around te web openings as obtained from te FEA was 40 N/mm wic occurred at an angle of 40 0 to te vertical compared to a maximum stress of 6 N/mm at an angle of 30 0 based on te proposed tangential metod. For Model wit 180 mm diameter web opening, te maximum principal stress was 478 N/mm as obtained from te FEA and occurred an angle of 30 0 compared to a maximum stress of 360 N/mm at an angle of 8 0 as obtained from te proposed metod. Te proposed metod is conservative for beams wit isolated web openings. 168

205 Principal stresses (N/mm ) Principal Stresses (N/mm ) Angle to Vertical Teoretical stressesisolated opening FE stresses around opening Figure 9.15: Comparison between te principal stresses around te openings from te FEA at te test failure load and from te design metod for 1.8 mm tick beam wit 150 mm diameter isolated web opening Teoretical stresses- Isolated opening FE stresses around web opening Angle to vertical Figure 9.16: Comparison between te principal stresses around te openings from te FEA at te test failure load and from te design metod for te 1.8 mm tick beam wit 180 mm diameter isolated web opening Beams wit closely spaced web openings (Models 3, 4 and 5) For beams wit closely spaced web openings, te stresses obtained from te proposed tangential stress metod taking into account te effect of isolated and closely spaced web openings were 169

206 Principal Stresses (N/mm ) conservative in comparison wit te stresses obtained from te FEA around web openings, as sown in Figures 9.17 to Principal Stresses (N/mm ) FE stresses around opening near load point Teoretical stresses -closely spaced openings for so/o= 1.33 FE stresses around opening near support Teoretical stresses-isolated opening Angle to vertical Figure 9.17: Comparison between te principal stresses around te openings from te FEA at te test failure load and from te design metod for Model 3 wit 180 mm web openings at 90 mm edge distance and 1.5 mm tick steel Angle to vertical FE stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 FE stresses around opening near support Figure 9.18: Comparison between te principal stresses around te openings from te FEA at te test failure load and from te design metod for Model 4 wit 180 mm web openings at 60 mm edge distance and 1.5 mm tick steel 170

207 Principal Stresses (N/mm ) FE Stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 FE Stresses around opening near support Angle to vertical Figure 9.19: Comparison between te principal stresses around te openings from te FEA at te test failure load and from te design metod for Model 5 wit 180 mm web openings at 90 mm edge distance and 1.8 mm tick steel 9.5 Non-Linear Analysis of Cold Formed C Sections wit Circular Web Openings Imperfection sensitivity study Different imperfection values were applied to te 50 mm deep x1.5 mm tick beams wit 180 mm web openings at 90 mm edge distance. A comparison between te failure load for eac imperfection and te test failure load, as sown in Figure 9.0 sows tat imperfection in te web of te beams as little effect and te failure occurs once te web ad reaced its buckling capacity. Terefore, Riks analysis was only considered in te following finite element modelling and gave a good agreement wit te test failure loads, as sown in Table 9.4. Table 9.4: Failure load (kn) % of failure load Failure load for different imperfection values compared to te test failure load Imperfection (mm) Test Riks analysis % 88% 91% 87.1% 171

208 Load (kn) Figure 9.0: mm imperfection Test results 0.mm imperfection 0.5mm imperfection Riks Displacement (mm) Comparison between te test failure load and te failure load for different imperfection values for 50 mm x 1.5 mm tick beam wit 180 mm web openings at 90 mm edge distance 9.5. Riks analysis on cold formed C sections wit circular web openings Half te beam section was modelled in Abaqus and Riks analysis was carried out on eac beam to obtain te sear resistance and te maximum deflection at failure. A comparison between te section resistance obtained from te non-linear FE analysis and te sear force at te test failure load is sown in Table 9.5. Table 9.5: Comparison between test failure sear force and te failure sear force obtained from te Riks analysis Beam Beam wit single 150 mm diameter circular web opening Beam wit single 180 mm diameter circular web opening Beam wit 180 mm diameter circular web opening at 60 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Beam wit 180 mm diameter circular web opening at 90 mm edge distance Steel tickness less galvanizing Test failure sear force for one beam Riks analysisfailure sear load for one beam (kn) Riks analysisdeflection 1.76 mm 0.5 kn 19 (9 %) 13.8 mm 1.76 mm 16.6/17.1 kn 13.5 (80 %) 1.5 mm 1.46 mm 7. kn 6.0 (83%) 13.3 mm 1.46 mm 7.9 kn 5.8 (74%) 11.3 mm 1.76 mm 13.7 kn 10.1 (74%) 13.8 mm 17

209 Load (kn) 9.6 Finite Element Analysis Results of Cold Formed C Sections using te Riks Metod Model 1: 1.8 mm tick beams wit isolated 150 mm diameter web openings Te deformation of te C section at failure is sown in Figure 9.1 were te local buckling around te web opening and deformation of te top flange are apparent. Te maximum principal stress around te web openings was 509 N/mm wic is sligtly less te ultimate strengt of te 1.8 mm tick steel. Te beam ad reaced its maximum resistance and te post-buckling failure was due to te local buckling of te web around te web opening Figure 9.1: Maximum principal stresses at failure based on te non-linear FEA of Model 1 Te failure load obtained from te Riks analysis was 76.5 kn, wic is about 93% of te test failure load. Te maximum deflection at failure was 13.8 mm, as sown in Figure Failure Load from te FE analysis compared to te test failure load for two 1.8 mm tick C sections wit isolated 150 mm web openings Flange 1 deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 9.: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x 1.8 mm tick C sections wit isolated 150 mm diameter web opening 173

210 LOAD (kn) 9.6. Model : 1.8 mm tick beams wit isolated 180 mm diameter web openings Te deformation of te C section at failure is sown in Figure 9.4. Te failure was due to local buckling around te web opening and te bending of te top flange due to te Vierendeel bending. Te maximum principal stress was 503 N/mm wic is close to te maximum stress of Model 1 and indicates tat post-buckling failure occurred before te beam reaces its ultimate resistance of 510 N/mm. Te failure load obtained from te FE analysis was 54.3 kn. Tis is about 80% of te test failure load. Te maximum deflection at failure was 1.5 mm. A comparison between te test failure load and te load obtained from te FE analysis is sown in Figure 9.4. Figure 9.3: Maximum principal stresses at failure based on te non-linear FEA of Model 80 Faliure load obtained from te FE analysis compared to test failure load for 1.8 mm tick beams wit isolated 180 mm web opening Flange 1 deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 9.4: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x1.8 mm tick C sections wit isolated 180 mm diameter web opening 174

211 LOAD (kn) Model 3: 1.5 mm tick beams wit 180 mm web openings at 60 mm edge distances Te deformation of te C section at failure is sown in Figure 9.5. Te mode of failure was deformation of te top flange and web-post buckling wit an effective lengt of about alf te opening eigt. Concentrated stresses are apparent between te web openings due to te web-post buckling. Te maximum principal stress around te web opening near te load point and witin te web-post were 470 N/mm and 460 N/mm respectively. Te failure load obtained from te FE analysis was 4 kn. Figure 9.5: Maximum principal stresses at failure based on te non-linear FEA of Model 3 35 Failure load based on FE analysis compared to te test failure load for beams 1.5 mm tick wit 180 mm web openings at 60 mm edge distance Flange 1deflection Flange deflection Riks Displacement(mm) Figure 9.6: Failure load for two beams based on te non-linear FE analysis compared to te test failure load for two 50 mm deep x 1.5 mm tick C section wit 180 mm web openings at 60 mm edge distances 175

212 As sown in te comparison between te failure load obtained from te non-linear FE analysis and te test failure load in Figure 9.6, te failure load obtained from te FE analysis was 83% of te test failure load for te beam wit web openings at 60 mm edge distance wit a maximum deflection of 13.3 mm Model 4: 1.5 mm tick beams wit 180 mm web openings at 90 mm edge distances Te failure of te C section was due to te web-post buckling as sown in te deformation of te section at failure in Figure 9.7. Te web-post buckling was due to te increase in te orizontal sear witin alf te dept of te web-post wit an effective buckling lengt of l eff =0.5 o. Buckling around te web openings at failure is apparent. Te maximum principal stress at failure was 51 N/mm witin te web-post and around te web opening near te load point due to te web-post buckling. Te maximum ultimate strengt was reaced and te failure was due to te plastic deformation of te section. Figure 9.7: Maximum principal stresses at failure based on te non-linear FEA of Model 4 Te failure load obtained from te FE analysis for Model 4 was 74 % of te test failure load wit a maximum deflection of 11.3 mm. A comparison between te test failure load and te load obtained from te FE analysis is sown in Figure

213 LOAD (kn) 35 Failure load based on FE analysis compared to te test failure load of two 1.50 mm tick C sections wit 180 mm web openings at 90 mm edge distance Flange 1deflection Flange deflection Riks 0 Figure 9.8: Displacement(mm) Failure load for two beams based on te non-linear FE analysis compared to te test failure load for 50 mm deep x1.5 mm tick C-section wit 180 mm web openings at 90 mm edge distance Model 5: 1.8 mm tick C section wit 180 mm web openings at 90 mm edge distance Te deformation of te beam at failure is sown in Figure 9.9. Te mode of failure was web-post buckling wit an effective buckling lengt of l eff =0.5 o. Deformation of te top flange is apparent and was due to Vierendeel bending of te top Tees. Te failure load obtained from te FE analysis was 74% of te test failure load for Model 5 wit maximum deflection of 13.8 mm at failure. Corresponding principal stresses at te failure are sown in Figure Te maximum principal stresses were 510 N/mm around te web openings and between web openings due to web-post buckling. Te ultimate strengt of te material is about 510 N/mm wic indicates tat plastic failure ad occurred. 177

214 Load (kn) Figure 9.9: Maximum principal stresses at failure based on te non-linear FEA of Model 5 60 Failure load obtanied from te FE analysis compared to te test failure load for two 1.8 mm tick C sections wit 180 mm web openings at 90 mm edge disatnce Flange 1deflection Flange deflection Riks Displacement (mm) Figure 9.30: Failure load for two beams obtained from te non-linear FE analysis compared to te test failure load for two 50 mm deep x 1.8 mm tick C-sections wit 180 mm web openings at 90 mm edge distance 178

215 9.7 Analysis of Results and Comparison between te Tests and Finite Element Analysis For cold formed C sections wit circular web openings, reasonable agreement between te test failure loads and te failure loads from te finite element analysis was obtained, as sown in Table 9.6. Te failure load was in te range of 74 to 93% of te test failure loads. Te stiffened elongated beams failed at te load point wit no deformation to te web, and tus were not considered in te non-linear analysis. Table 9.6: Comparison between te test failure loads and te failure loads obtained from te FE analysis for two beams Test beam configuration Failure load for two beams (kn) Test result 1.8 mm tick C section wit single 150 mm diameter circular web opening mm tick C section wit single 180 mm diameter circular web opening 66.5/ mm tick C section wit 180 mm diameter circular web opening at 60 mm edge distance mm tick C section wit 180 mm diameter circular web opening at 90 mm edge distance mm tick C section wit 180 mm diameter circular web opening at 90 mm edge distance 55 Figure in brackets are percentage of te test failure load Riks analysis 76 (9%) 54 (81/78%) 4 (83%) 3. (74%) 40.4 (74%) 9.8 Comparison of te Proposed Teory on Additional Deflection Due to Web Openings as Presented in Capter 6 wit Tests on Sort Span Cold Formed C Sections Te test series on 50 mm deep cold formed steel beams were on S390 steel and nominally 1.5 mm and 1.8 mm ticknesses, and te results are presented in Table 9.7. For an opening diameter of 180 mm, te edge spacing was 60 or 90 mm. Isolated openings of 150 mm and 180 mm diameter were also tested. Again a sear force of 10 kn was used in te comparisons, except for one test were te deflection was obtained by extrapolation from te elastic range. Te additional deflection obtained from te proposed teory as presented in Capter 6 is presented in Table 9.7. Te teory and finite element models are in close agreement and are close to te test results, as sown in Table 9.9. Indeed, in some cases, te teory gave iger deflections tan te tests, unlike in te stainless steel tests. 179

216 Te deformation of te cold formed C sections and te corresponding deflection for te applied sear force from te FEA are sown in Figures 9.31 to 9.35 for te1.5 and 1.8 mm tick sections. Figure 9.31: Deflection of 1.8 mm tick C section wit isolated 150 mm web opening subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.3: Deflection of 1.8 mm tick C section wit isolated 180 mm web opening subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.33: Deflection of 1.8 mm tick C section mm wit 180 mm web openings at 90mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA 180

217 Figure 9.34: Deflection of 1.5 mm tick C section mm wit 180 mm web openings at 60mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA Figure 9.35: Deflection of 1.5 mm tick C section wit 180mm web opening at 90 mm edge distance subject to sear force of 10 kn as obtained from te elastic FEA Table 9.7: Opening spacing no. single openings 4 no. 90 mm edge distance 4 no. 60 mm edge distance Comparison of test deflections for cold formed steel beams Diameter 150 mm 180 mm 180 mm Steel tickness 1.76 mm net 1.4 mm net Test deflection for sear force of 5 kn Jack Flanges Support Net 1.01 mm 1.3 mm 1.50 mm 0.96 mm 1.00 mm 1.35 /1.01 (1.18 mm av) 1.5 /1.47 (1.49 mm av) 1.80 /1.69 (1.74 mm av) 1.5 /1.11 (1.18 mm av) 1.33 /1.04 (1.19 mm av) Test deflection for sear force of 10 kn (extrapolated) 0.13 mm 0.88 mm 1.76 mm 0.08 mm 1.15 mm.30 mm 0.14 mm 1.36 mm.7 mm 0.07 mm 0.89 mm 3.56 mm 0.07 mm 0.93 mm 3.7 mm 181

218 Table 9.8: Deflections of 50 mm deep cold formed steel sections according to te proposed teory in Capter 6 Opening spacing no. single openings 4no. 90 mm edge distance 4no. 60 mm edge distance Diameter 150 mm 180 mm 180 mm Steel tickness 1.76 mm net 1.4 mm net Deflection of solid web beam Span of 1.4 m and sear force of 10 kn in all cases Bending Sear Bending Additional deflection due to openings Sear & Vier. bending Webpost sear Total deflection 1.0 mm 0. mm 0.05 mm 0.5 mm mm 1.0 mm 0.0 mm 0.05 mm 0.30 mm 0.05 mm 1.6 mm 1.0 mm 0.0 mm 0.10 mm 0.60 mm 0.50 mm.4 mm 1.35 mm 0.5 mm 0.10 mm 0.80 mm 0.70 mm 3. mm 1.35 mm 0.5 mm 0.10 mm 0.80 mm 1.10 mm 3.6 mm Table 9.9: Comparison of test deflections, FEA and teory for 50 mm deep cold formed C-section Opening spacing Diameter Steel tickness Deflection for sear force of 10 kn Test FEA Teory Single opening 90 mm edge distance 150 mm 180 mm 1.7 mm net 1.4 mm 180 mm 60 mm edge net distance Span of 1.4 m and sear force of 10 kn in all cases 1.8 mm 1.6 mm 1.5 mm.3 mm mm 1.6 mm.7 mm.6 mm.4 mm 3.5 mm 3.6 mm 3. mm 3.7 mm 3.7 mm 3.6 mm 18

219 9.9 Linear Analysis of C Sections wit Diamond and Hexagonal Web Openings In te linear analysis, alf beam sections were modelled and subject to a sear force at mid-span equal to te test failure sear force. Sear forces were applied as sell loads to te web of te beams, as explained previously. Table 9.10 summarises te results of te linear elastic analysis. Table 9.10: Summary of te linear analysis of cold formed C sections wit diamond and exagonal web openings FE Model and opening arrangement Test failure load for beams Applied sear force Maximum principal stress (N/mm ) Deflection at mid-span at failure Model 1: 1. mm tick wit single diamond web opening 3.3 kn 8.1 kn mm Model : 1. mm tick wit diamond web openings at kn 7. kn mm mm edge distance Model 3: 1.53 mm tick wit exagonal web openings at 4.1 kn 10.5 kn mm 45 mm edge distance Model 4: 1.93 mm tick wit exagonal web openings at 45 mm edge distance 66 kn 16.5 kn mm Te maximum principal stresses were te largest value recorded around te web openings at te failure load of te beam Model 1: 50 mm deep x 1. mm tick C section wit isolated diamond saped opening Te corresponding principal stresses around te web opening based on te linear FE analysis of te beam subjected to te test sear failure load are sown in Figure Concentrated stresses occurred around te sarp corner of te opening. Te maximum principal stress was 87 N/mm. Figure 9.36: Maximum principal stresses of 87 N/mm around te opening based on te linear FE analysis of Model 1subject to a sear force of 8.1 kn 183

220 9.9. Model : 50 mm deep x 1. mm tick C section wit pairs of diamond saped openings at 9 mm edge distance Te corresponding principal stresses around te web openings based on te linear analysis of te beam are sown in Figure Stresses around te openings were due to te local buckling around te web openings. Sligt buckling of te web-post and buckling of te top and bottom flange are apparent. Te maximum principal stress around te opening near te load point was 77 N/mm. Figure 9.37: Maximum principal stress of 77 N/mm around te opening near te load point based on te linear FE analysis of Model subject to a sear force of 7. kn Model 3: 50 mm deep x 1.53 mm tick C section wit 167 mm deep exagonal opening at 45 mm edge distance Te deformation of te C section based on te linear analysis is sown in Figure For w/t w ratio of 03, te web is expected to fail by web-post buckling before reacing its material resistance. Elastic buckling can be considered te mode of failure. Te maximum principal stress was 400 N/mm at te corner of te web opening near te load point. Figure 9.38: Maximum principal stress of 400 N/mm around te opening for linear analysis of Model 3 subject to sear force of 10.5 kn 184

221 9.9.4 Model 4: 50 mm deep x 1.93 mm tick C section wit 167 mm deep exagonal opening at 45 mm edge spacing Te mode of failure of Model 4 is similar to Model 3 and altoug te beam tickness was greater tan tat of Model 3, te applied sear force was 57% more tan te applied sear force for Model 3 and buckling of te web-post occurred. Te corresponding principal stresses around te web opening based on te linear analysis are sown in Figure Concentrated stresses occurred around te opening are apparent. Te maximum principal stress was 479 N/mm around te corners of te web openings near te web-post wic can be explained by development of Vierendeel bending over te flat web of te exagonal openings. Figure 9.39: Maximum principal stress of 485 N/mm around te web opening based on te linear analysis of Model 4 subject to a sear force of 16.5 kn 9.10 Imperfection Sensitivity Study on C Sections wit Diamond Saped Openings A comparison between te failure load for eac imperfection value and te test failure load was carried out for C sections wit diamond and exagonal web openings to investigate te effect of imperfection on te load resistance. Riks analysis was considered in te comparison and a summary of te results is presented as follows: mm deep x 1. mm tick C section wit diamond saped openings at 9 mm edge distance Te comparisons in Table 9.11 and Figure 9.40 sows tat Riks analysis gave conservative failure loads compared to te failure load obtained from te buckling analysis taking into account te web imperfection. 185

222 Load (kn) Table 9.11: Comparison between te test failure load and te failure loads obtained from te FE Riks and buckling analysis wit different imperfection values for a 50 mm x 1. mm tick beam wit diamond saped openings at 9 mm edge distance Failure load for two beams (kn) % of test failure load Test failure load (kn) Imperfection (mm) Riks analysis % 106.% 106% 98% Flange 1 Flange Riks 0.1mm Imperfection 0.mm Imperfection 0.5mm Imperfection Displacement (mm) Figure 9.40: Comparison between test failure load and failure loads obtained from te FE analysis for different imperfection values for beam wit diamond saped openings at 9 mm edge distance mm deep x 1.53 mm tick C section wit exagonal openings at 45 mm edge distance Te comparison of te analysis and test sown in Figure 9.41 sows tat te failure load obtained from te buckling analysis wit an imperfection value of 0.5, (equal to te eigt /10), was te closest to te test failure load. Te Riks analysis gave te lowest failure load. Buckling analysis wit different imperfection values gave different failure loads, as sown in Table

223 Load (kn) Table 9.1: Comparison between te test failure load and te failure loads obtained from te finite element Riks and buckling analysis wit different imperfection values for 50 mm deep x 1.53 mm tick beam wit exagonal openings at 45 mm edge distance Failure load for two beams (kn) Test Imperfection (mm) Riks analysis % of test failure load % 98% 98% 87.5% Flange 1 Flange Riks 0.1mm Imperfection 0.mm Imperfection 0.5mm Imperfection Displacement (mm) Figure 9.41: Comparison between test failure load and failure loads obtained from te FE analysis for different imperfection values for 1.53 mm tick beam wit exagonal openings at 45 mm edge distance Te comparison between te test mode of failure, Riks analysis mode of failure and te mode of failure obtained from te buckling analysis wit an imperfection value of 0.5 mm sowed tat te mode of failure obtained from te Riks analysis was similar to te test mode of failure as sown in Figure 9.4 and tus, Riks analysis was considered for te non-linear analysis to obtain te sear resistance of te C section due to web-post buckling. 187

224 Deformation of te beam web Failure mode at 0.5 mm Failure mode at Riks web during testing imperfection analysis Figure 9.4: Comparison between te test mode of failure and two different mode of failure based on te FE analysis for a 50 mm deep x 1.53 mm tick beam wit exagonal openings at 45 mm edge distance 9.11 Non-linear Finite Element Results of C Sections wit Diamond Saped and Hexagonal Web Openings Non-linear finite element analysis, as explained in Capter 8, was carried out on te cold formed steel C sections wit diamond and exagonal web openings to obtain te failure loads for eac C section. Table 9.13 summarises te sear resistance of eac beam section compared to te sear force applied to eac beam at failure. Table 9.13: Comparison between testing sear forces at failure and te sear resistance for eac C section as obtained from te FE analysis and te corresponding deflection at failure FE Model Steel tickness Opening edge distance Test failure sear force for one beam FE failure sear force for one beam Test deflection at failure FE non-linear deflection at failure Model mm kn 6.65 kn 7.8 mm 5.7 mm Model 1.18 mm 9 mm 7. kn 5.65 kn 7.6 mm 7.46 mm Model mm 45 mm 10.6 kn 8.5 kn 9. mm 5.1 mm Model mm 45 mm 16.6 kn 1.55 kn 6.5 mm 5.7 mm Steel tickness excludes zinc tickness 188

225 Load (kn) Model 1: 50 mm deep x 63 mm wide x 1. mm tick C section wit diamond saped openings Te sear resistance for two beams was 6.6 kn, wic is 8% of te test failure load being 3.4 kn for te beam wit isolated web openings. Te maximum stress around te web opening due to te buckling across te opening was 501 N/mm wic is close to te ultimate resistance of te steel being f u= 510 N/mm. Web buckling troug te web openings and buckling of te top flange ad occurred at failure, as sown in Figure Figure 9.43: Principal stresses based on te non-linear FE analysis of Model 1 A comparison between te test failure load, te beam resistance obtained from FE analysis and te corresponding deflections is sown in Figure Te maximum deflection at maximum load resistance based on te FE analysis was 5.7 mm compared to a deflection of 7.8 mm at te test failure load. 35 Failure load obtained from te FEA compared to te test failure load of two 1. mm tick C sections wit isolated diamond opening Flange 1 deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 9.44: Failure load from FE analysis for two beams compared to te test failure load of two 50 mm deep C sections wit isolated diamond saped opening 189

226 9.11. Model : 50 mm deep x 1. mm tick C section wit diamond saped openings at 9 mm edge distance Te sear resistance for two beams was.6 kn for te beam wit diamond saped openings at 9 mm edge distance based on te FE analysis of te beam section, wic is 78 % of te test failure load of 8.8 kn. Te failure was due to te web-post buckling due to te orizontal sear at middept of te web-post. Buckling of te top flange also occurred at failure. Te maximum principal stresses at failure were 480 N/mm and 460 N/mm around te web openings and witin te web-post respectively due to te orizontal sear in te upper part of te web-post were interaction between te stresses around te web opening ad occurred. Web-post buckling was te mode of te failure and ence te concentrated stresses witin te web-post between web openings. Figure 9.45: Principal stresses based on te non-linear FE analysis of Model Te maximum deflection at failure based on te FE analysis was 7.46 mm compared to te deflection of 7.6 mm at te test failure wic can be explained by te plastic deformation of te beam section at failure. A comparison between te test failure load, te beam resistance obtained from FE analysis and te corresponding deflections is sown in Figure

227 Load (kn) 30 Failure load obtained from te FEA compared to te test failure load of two 1. mm tick C sections wit pair of diamond openings Flange 1deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 9.46: Failure load from te FE analysis for two beams compared to te test failure load of two 50 mm deep x 63 mm wide C sections wit a diamond saped web openings at 9 mm edge distance Model 3: 50 mm deep x 1.53 mm tick beam wit exagonal web openings at 45 mm edge distance Te sear resistance for two beams in wic te material resistance does not exceed te ultimate resistance was 33 kn, wic is 78% of te test failure load of 4.1 kn. Te failure was due to webpost buckling wit an effective eigt of about 50 % of te opening eigt. Twist of te web-post ad occurred due to te increase of te applied load and ence, te increase of te orizontal sear force at mid-dept of te web-post. Te deformation of te beam section at failure is sown in Figure Figure 9.47: Principal stresses around te openings based on te non-linear FEA of Model 3 191

228 Load (kn) Te corresponding maximum principal stress around te web openings at failure was 450 N/mm wic is very close to te ultimate strengt f u = 465 N/mm and indicates a post-buckling failure of te web. Interaction between te stresses around te web openings and te ig stresses between te web openings due to te web post buckling are apparent. Te maximum deflection at te maximum load capacity was 5.1 mm wic is less tan te actual deflection at testing of 9. mm. Te relationsip between te test failure load and te failure load obtained from te non-linear analysis is plotted in Figure Failure load obtained from te FEA comapred to test failure load of two 1.53mm tick C section wit exagonal web openings at 45mm edge distance Figure 9.48: Flange (1) deflection Flange () deflection Failure load based on te FEA compared to te test failure load of two 50 mm deep x 1.53 mm tick C section wit exagonal openings at 45 mm edge distance Riks Jack deflection Displacement (mm) Model 4: 50 mm deep x 1.93 mm tick beam wit exagonal openings at 45mm edge distance Te sear resistance for te 1.9 mm tick beams in wic te material properties does not exceed te ultimate resistance was 50. kn, wic is 76% of te test failure load of 66 kn. Te deformation of te sections at failure was due to web-post buckling, as sown in Figure Buckling of te unsupported free edge next to te web opening is apparent and te effective buckling lengt is about alf te sloped side of te exagonal. 19

229 Load (kn) Figure 9.49: Principal stresses around te openings based on te non-linear FEA of Model 4 Te maximum principal stress around te web openings at failure was 460 N/mm. Interaction between te stresses around te web openings is apparent and te ig stresses witin te web-post were as expected due to te web post buckling of te beam and te orizontal sear at te web-post mid-dept. Stresses at failure were close to te ultimate strengt of te beam (f u= 50 N/mm ) and indicates a plastic deformation followed by plastic failure. Te maximum deflection at failure of te non-linear FE analysis was 5.75 mm compared to te test failure deflection of 6.5 mm, wic is due to te local plastic deformation of te beam. Te relationsip between te test failure load and te beam resistance obtained from te non-linear FE analysis is sown in Figure Failure load obtained from te FEA compared to te test failure load of two 1.93 mm tick C sections wit exagonal web openings at 45mm edge distance Flange (1) defelction Flange () deflection Riks Jack defelction Displacement (mm) Figure 9.50: Failure load based on te FE analysis compared to te test failure load of two 50 mm deep x1.9 mm tick beam wit castellated web openings at 45 mm edge distance 193

230 9.1 Discussion of Results of FE Analyses Te failure loads obtained from te FE analysis were in reasonable agreement wit te test failure loads for all te beam sections except te 1. mm tick C section wit isolated web openings, as sown in Table Increasing te steel tickness from 1.53 to 1.93 mm increased te sear resistance by 5 % for C sections wit exagonal web openings. Te sear resistance of te beams wit isolated diamond saped web openings was 18% more tan te sear resistance of te same beam sections wit diamond saped openings at 9 mm spacing due to te effect of web post buckling. Table 9.14: Comparison between test failure loads and FE failure loads FE model Edge distance Steel tickness Test Failure load for two FE failure load for two beams beams (kn) Model 1-1. mm 3.3 kn 6.6 (8%) Model 9 mm 1. mm 8.4 kn.6 (80%) Model 3 45 mm 1.50 mm 37 kn 33 (89%) Model 4 45 mm 1.90 mm 66 kn 50. (76%) Figures in brackets are te percentage of te test failure load 194

231 CHAPTER 10 FINITE ELEMENT ANALYSIS RESULTS FOR STAINLESS STEEL BEAMS WITH CIRCULAR WEB OPENINGS 10.1 Introduction Te objective of te finite element analyses was to evaluate te stresses around te openings of te beams and to compare te predicted failure load obtained from te finite element analysis wit tose obtained from te test results presented in Capter 7 and te teoretical analysis presented in Capter 4. Te performance of stainless steel C section beams wit circular web openings was investigated by using linear and non-linear finite element analysis considering te models sown in Table Te variable parameters are distance between te openings, sape of openings, and tickness of te steel. For all models, te web opening diameter was 150 mm. Table 10.1: Cases analysed in Finite element models of stainless steel beams FE model Stainless Steel type Stainless steel tickness Distance between edge of openings 1 Austenitic mm 50 mm Austenitic 3 mm 50 mm 3 Austenitic mm 100 mm 4 Lean Duplex mm 100 mm 5 Austenitic 3 mm 100 mm 6 Lean Duplex mm 50 mm 7 Austenitic mm 50 mm All beams were 10 mm deep x 70 mm nominal flange widt 10. Mes Sensitivity Study on Stainless Steel C Sections A sensitivity study for te mes global size was carried out on one of te tested beams (10 mm deep x mm tick) Lean Duplex stainless steel C section wit 150 mm web openings at 50 mm edge distance in order to establis te average mes global size tat can be considered in te optimised finite element analysis. Different mes sizes were considered and te failure load was obtained for eac case and altoug a mes size of 38 mm was found to give a very close failure load compared 195

232 Load (kn) Failure load (kn) to te actual test failure load, a global size of 1 mm was found to be a reasonable mes size giving reasonably accurate average results, as sown in Figures 10.1 and Test failure load Failure load for different mes size Mes global size Figure 10.1: Comparison between te test failure load and te failure loads for different mes sizes for test on mm tick Lean Duplex stainless steel C section wit 150 mm web openings at 50 mm edge distance Flange 1 Flange Jack Movement Mes size 38 Mes size 30 Mes size 5 Mes size 0 Mes Size 15 Mes size 1 Mes size 10 Figure Displacement (mm) Mes comparison of mm tick Lean Duplex stainless steel C section wit 150 mm web openings at 50 mm edge distance 196

233 10.3 Linear Finite Element Analysis Linear finite element (FE) analysis was carried out on all te tested stainless steel C sections were alf te beam section was modelled and subject to a sell load applied to te web area equal to te test failure sear forces to obtain te maximum principal stresses around te web openings Principal Stresses Te transfer of sear around te openings caused local Vierendeel bending stresses, wic are combined wit te overall bending and sear stresses acting on te beams. Tis FE analysis sows ig principal stresses because beam fails wen te wole section becomes effectively plastic around te openings, wic leads to iger failure loads. Te maximum principal stress and deflection corresponding to te applied sear forces are summarised in Table 10.. Table 10. Applied sear force at failure and te corresponding maximum principal stress of te linear finite element analysis of te stainless steel models FE model Stainless steel type/ tickness/ opening edge distance Web openings Applied sear force Maximum principal stress (N/mm ) Yield stress at 0. % strain N/mm Maximum deflection at failure / mm 50 mm 15.4 kn mm / 3 mm 50 mm 1 kn mm / mm 100 mm 11.1 kn mm 4 LDX 101/ mm 100 mm 17.9 kn mm / 3 mm 100 mm 1.5 kn mm 6 LDX 101/ mm 50 mm 13.7 kn mm / mm 50 mm 9.8 kn mm 197

234 Principal Stresses (N/mm ) 10.5 Linear Finite Element Analysis Results of Stainless Steel C Sections Model 1: mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Te corresponding principal stresses of te linear elastic analysis are sown in Figure Te maximum principal stress was 35 N/mm and it occurred around te opening near te load point. Deformation of te top flange due to te Vierendeel bending is also apparent. Figure 10.3: Maximum principal stress of 35 N/mm based on te linear FE analysis of Model 1 subject to sear force of 15.4 kn Principal stresses around te opening were plotted against te angle to te vertical around te openings as sown in Figure Maximum principal stresses were at an angle of 30 0 to te vertical Stresses around opening near support Stresses around opening near load Angle to vertical Figure 10.4: Principal stresses around te web openings of FE Model 1 subject to a sear force of 15.4 kn 198

235 10.5. Model : 3 mm tick Austenitic stainless steel C section wit web openings at 50mm edge distance Te corresponding principal stresses based on te linear FE analysis are sown in Figure Te maximum stress was 359 N/mm due to web-post buckling. Figure 10.5: Maximum principal stress of 359 N/mm around web openings based on te linear FE analysis of Model subjected to a sear force of 1 kn Principal stresses around te opening were plotted around te openings as sown in Figure Te maximum principal stress was at an angle of 5 0 for bot openings near te support and te load point. Te stresses were constant between angles of 30 0 and 60 0 were te web-post is subjected to ig stresses due to te orizontal sear. Principal Stresses (N/mm ) Stresses around opening near support Stresses around opening near load point Angle to vertical Figure 10.6: Principal stresses around te web openings of FE Model subject to a sear force of 1 kn 199

236 Principal Stresses (N/mm ) Model 3: mm tick Austenitic stainless steel C section wit web openings at 100 mm edge distance Te maximum stresses around te web openings are sown in Figure Te maximum principal stress was 35 N/mm around te web openings. Sligt buckling of top flange and web-post is apparent. Figure 10.7: Maximum principal stress of 35 N/mm around web openings based on te linear FE analysis of Model 3 subject to a sear force of 11.1 kn Principal stresses around te opening were plotted around te openings as sown in Figure Te maximum principal stresses were at an angle of 30 0 for bot openings near te support and te load point Stresses around opening near support Stresses around opening near load point Angle to vertical Figure 10.8: Principal stresses around te web openings of FE Model 3 subject to a sear force of 11.1 kn 00

237 Principal Stresses (N/mm ) Model 4: mm tick Lean Duplex stainless steel C section wit web opening at 100 mm edge distance Te deformation of te C section based on te linear FE analysis is sown in Figure Bending of te top flange and web-post buckling are apparent. Te maximum principal stress was 580 N/mm around te web openings. Figure 10.9: Maximum principal stress of 580 N/mm around te web opening based on te linear FE analysis of Model 4 subject to a sear force of 17.8 kn Principal stresses around te opening were plotted around te openings as sown in Figure Te maximum principal stresses were at angles of 30 0 and 35 0 around te opening near te load point and te support respectively Stresses around opening near support Stresses around opening near load point Angle to vertical Figure 10.10: Principal stresses around te web openings of FE Model 4 subject to a sear force of 17.8 kn 01

238 Model 5: 3 mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Te deformation of te C section based on te linear FE analysis is sown in Figure Buckling of te top and bottom flanges is apparent. Sligt web-post buckling and buckling around te web openings are also apparent. Maximum principal stresses occurred around te web openings. Te maximum principal stress was 349 N/mm, as sown in Figure Figure 10.11: Maximum principal stress of 349 N/mm based on te linear FE analysis of Model 5 subject to a sear force of 1.5 kn Principal stresses around te opening were plotted around te openings, as sown in Figure Te maximum principal stresses around openings near te support and te load point were at angle of 5 0 to te vertical. Hig tangential stresses between te web openings occurred at angles of 30 0 and 60 0 to te vertical. Principal Stresses (N/mm ) Stresses around opening near support Stresses around opening near load point Angle to vertical Figure 10.1: Principal stresses around te web openings of FE Model 5 subject to a sear force of 1.5 kn 0

239 Principal Stresses (N/mm ) Model 6: mm tick Lean Duplex stainless steel C section wit web openings at 50 mm edge distance Buckling of te top flange and web-post buckling were te mode of failure for Model 6. Te deformation of te beam is sown in Figure Maximum principal stresses occurred around te web openings and witin te web-post. Te maximum principal stress was 500 N/mm. Tangential stresses around te opening were plotted against te angle to te vertical around te openings as sown in Figure Te maximum principal stress around te opening near te load point was at an angle of 35 0 and te maximum stresses around te opening near te support were at angles between 35 0 and Figure 10.13: Maximum principal stress of 500 N/mm around te opening based on te linear analysis of Model 6 subject to a sear force of 13.7 kn Stresses around opening near load Stresses around opening near support Angle to vertical Figure 10.14: Principal stresses around web openings of FE Model 6 subject to a sear force of 13.7 kn 03

240 Principal Stresses (N/mm ) Model 7: mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Te linear deformation of Model 7 is sown in Figure Buckling of te top and bottom flanges occurred. Te ig stresses witin te web-post are due to web-post buckling and orizontal sear at mid-dept of te web-post. Te maximum principal stress around te web openings was 35 N/mm. Figure 10.15: Maximum principal stresses of 35 N/mm around te web openings based on te linear analysis of Model 7 subject to a sear force of 9.8 kn Principal stresses around te opening were plotted around te openings as sown in Figure Te maximum principal stress around te opening near te load point was at angle of Te maximum principal stresses around te opening near te support witin te ig sear zone were at angles of 30 0 and 75 0 to te vertical Stresses around opening near load Stresses around opening near support Angle to vertical Figure 10.16: Principal stresses around te web openings of FE Model 7 subject to a sear force of 9.8 kn 04

241 Principal Stressess ( N/mm ) 10.6 Stresses from FEA Models Compared wit Tangential Stress Metod for Stainless Steel Beams Te results of te FEA models were obtained at te sear force corresponding to te failure load in te tests. Te FEA results of te principal stresses around te openings are presented in Figures to Te results are compared wit te Tangential Stress Metod for te same load, and for te tests for openings at 50 mm, 100 mm and 50 mm edge distances. Two curves are presented for te proposed metod: Stress for isolated openings wic will control for te stresses between 0 and 40 0 to te vertical Stress for closely spaced openings wic will control for te stresses between 50 0 and 90 0 to te vertical Beams wit isolated web openings For Model 1 wit isolated web openings, te tangential stresses obtained from te proposed metod around web openings were in good agreement wit te stresses obtained from te FE analysis, as sown in Figure Stresses around opening near load point Teoretical stresses-isolated opening Stresses around opening near support Angle to vertical Figure 10.17: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model Beams wit closely spaced web openings Te total stresses obtained from te proposed metod for isolated and closely spaced web openings were in good agreement wit tose obtained from te FE analysis for beams in Models 3,5,6 and 7 and conservative for Models wit 3 mm tick Austenitic stainless steel and Model 4 wit mm 05

242 Principal Stresses (N/mm ) Principal Stresses (N/mm ) tick Lean Duplex stainless steel C section. Te metod is very conservative for all Models except for Model, as sown in Figures to Angle to vertical Stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 Stresses around opening near support Figure 10.18: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model wit web openings at 50 mm edge distance Opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 Stresses around opening near support Angle to vertical Figure 10.19: Comparison between te principal stresses around te openings from te FE analysis at failure and te stresses obtained from te design metod for Model 3 wit openings at 100 mm edge distance 06

243 Principal Stresses (N/mm ) Principal Stresses ( N/mm ) Stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 Stresses around opening near support Angle to vertical Figure 10.0: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model 4 wit openings at 100 mm edge distance Angle to vertical Stresses around opening near load point Teoretical stresses-isolated opening Stresses around opening near support Teoretical stresses-closely spaced openings for so/o=1.33 Figure 10.1: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model 5 wit openings at 100 mm edge distance 07

244 Principal Stresses ( N/mm ) Principal Stresses (N/mm ) Angle to vertical Stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= 1.33 Stresses around opening near support Figure 10.: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model 6 wit openings at 50 mm edge distance Stresses around opening near support Stresses around opening near load point Teoretical stresses-isolated opening Teoretical stresses -closely spaced openings for so/o= Angle to vertical Figure 10.3: Comparison between te principal stresses around te openings from te FE analysis at te test failure load and from te design metod for Model 7 wit openings at 50 mm edge distance 08

245 10.7 Non-Linear Finite Element Analysis on Stainless Steel Beams wit Circular Web openings Non-linear finite element analysis was carried out on te tested stainless steel C sections to predict te failure loads for eac C section. Failure occurred at different arc lengts and te corresponding failure sear forces for beam sections are summarised in Table Te failure sear force obtained from te FE analysis was ten multiplied by 4 to obtain te failure load for two beams wic was ten compared to te test failure load for eac test. Table 10.3: Comparison between te test failure sear forces and te sear force obtained from te Riks analysis and te corresponding deflection FE Model Stainless Steel type/ tickness Web opening edge distance Test failure sear force for one beams Riks failure sear force (kn) one beam Maximum deflection at failure (mm) / mm 50 mm 15.3 kn 1.63(88%) / 3 mm 50 mm 1.0 kn 16.8 (80%) / mm 100 mm 11.1 kn 10. (9%) LDX 101/ mm 100 mm 17.9 kn 1. (68%) / 3 mm 100 mm 1.5 kn.5 (105%) LDX 101/ mm 50 mm 13.8 kn 9.8 (70%) / mm 50 mm 9.8 kn 7.1 (7%) 8.0 (Te figure in bracket is % of te test failure load) Imperfection sensitivity study To investigate te effect of web imperfections on te beam resistance, a non-failure finite element analysis was carried out on Model 7, were 10 mm deep x mm tick Austenitic stainless steel C section was considered. A comparison between te failure load obtained from te FE analysis for eac imperfection value and te test failure load is sown in Figure and Table Te imperfection in te web of te beam ad little effect on te beam resistance and te failure occurred once te web reaced its buckling capacity. Terefore, Riks analysis was only considered in te nonlinear finite element analysis. 09

246 Load (kn) Table 10.4: Failure load for two beams (kn) % Test failure load Comparison between te test failure load and te failure loads obtained from te finite element Riks and buckling analysis wit different imperfection values Test Imperfection (mm) Riks analysis % 73% 73% 73% 7% Test failure load KN Riks failure load =8.3 KN Flange 1 Flange Riks 0.1mm Imperfection 1mm Imperfection 0.mm Imperfection 0 Imperfection Displacement (mm) Figure 10.4: Comparison between test failure load and failure loads from te FE analysis for different imperfection values 10

247 10.8 Finite Element Analysis Results of Stainless Steel C sections using te Riks Metod Model 1: mm tick Austenitic stainless steel C section wit web openings at 50 mm edge distance Te deformation of te beam at failure is sown in Figure Te beam continued to resist additional load until te Vierendeel failure occurred due to te yielding of te top Tee. Local buckling around te web openings is also apparent. Te failure load for one beam was 5. kn. At failure, ig stresses witin te top and bottom Tee sections were due to Vierendeel bending at failure. Te maximum principal stress around te web opening at failure was 34 N/mm wic indicates tat post-yielding ig deformation occurred (f y= 80 N/mm ) and failure occurred at 3% strain locally related to stress-strain curve. Figure 10.5: Deformation and maximum principal stresses at failure load based on te FE analysis of Model 1 Te failure load from te non-linear finite element analysis was found to be 83% of te test failure load as sown in Figure Te deflection at failure was mm. 11

248 Load (kn) Flange 1 deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 10.6: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 10 mm x mm tick Austenitic stainless steel C-section wit 150 mm web openings at 50 mm edge distance Model : 3 mm tick Austenitic stainless steel C section wit150 mm web openings at 50 mm edge distance Te failure load for one beam was kn and te failure mode was due to te web-post buckling and buckling of te top flange. Te deformation at failure is sown in Figure Te maximum principal stress around te web opening was 35 N/mm and was due to te local buckling and te web-post buckling between openings. Te ig stresses above te openings were due to te Vierendeel bending of te top Tees. Post-yielding Failure occurred as in Model 1 Figure 10.7: Deformation and maximum principal stresses at failure load based on te FE analysis of Model 1

249 Load (kn) Te failure load from te non-linear finite element analysis was 67.1 kn, wic is 80% of te test failure load as sown in Figure Te maximum deflection at failure was 8.6 mm Riks Displacement (mm) Figure 10.8: Failure load as obtained from te non-linear FE analysis for two 10 mm deep x 3 mm tick Austenitic stainless steel C sections wit 150 mm web openings at 50 mm edge distance Model 3: mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Te failure load for one beam was for 0.5 kn and failure mode was due to te buckling of te top flange and te local buckling between and around te web openings, as sown in Figure Webpost buckling is apparent. Te maximum principal stress at failure was 34 N/mm wic again indicates tat failure occurred post-yielding. Figure 10.9: Deformation and maximum principal stresses at failure based on te non-linear FE analysis of Model 3 13

250 Load (kn) Te failure load from te non-linear finite element analysis using te Riks metod was found to be 9% of te test failure load wit a corresponding deflection of mm, as sown in Figure Jack Load Flange Flange 1 Riks Displacement (mm) Figure 10.30: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick Austenitic stainless steel C sections wit 150 mm web openings at 100 mm edge distance Model 4: mm tick Lean Duplex stainless steel C section wit 150 mm web opening at 100 mm edge distance Te failure mode of te beam was due to te increase of te orizontal sear forces causing webpost buckling wit an effective lengt equal to approximately tird of te opening eigt. Buckling of te top flange ad also occurred and te principal stresses around te web openings at failure are sown in Figure Te maximum principal stresses occurred around and between te web openings were due to te web-post buckling and te plastic deformation of te beam. Te maximum principal stress was 773N/mm wic is sligtly less tan te ultimate strengt of te material f u= 790 N/mm. Te failure load for one beam as obtained from te non-linear finite element analysis was 4.4 kn, wic is 68% of te test failure load. Te relationsip between te load and te corresponding deflection is sown in Figure Te maximum deflection at failure was 30.3 mm. 14

251 Load (kn) Figure 10.31: Deformation and maximum principal stresses at failure based on te non-linear FE analysis of Model Riks Displacement (mm) Figure 10.3: Failure load as obtained from te non-linear FE analysis for two 10 mm deep x mm tick Lean Duplex stainless steel C section wit 150 mm web openings at 100 mm edge distance Model 5: 3 mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Te failure load for one beam was 45. kn and te failure mode was due to te orizontal sear causing buckling of te web-post. Vierendeel bending at te top Tee also occurred above te opening near te load point. Te corresponding principal stresses around te opening at failure are sown in Figure Te maximum principal stress around te web openings was 400 N/mm. Te maximum principal stress witin te top Tee was 395 N/mm and was due to te Vierendeel bending. Te stress at failure 15

252 Load (kn) equals to te proof strengt at 3.% strain wic indicates tat post-yielding failure occurred (f y= 90 N/mm ).Te failure load from te non-linear finite element analysis using te Riks metod was 105% of te test failure load. Te maximum deflection at failure was 17.6 mm, as sown in Figure Figure 10.33: Deformation and maximum principal stresses at failure load based on te nonlinear FE analysis of Model Riks Displacement (mm) Figure 10.34: Failure load as obtained from te non-linear FE analysis for two 10 mm deep x 3 mm tick Austenitic stainless steel C section wit 150 mm web openings at 100 mm edge distance Model 6: mm tick Lean Duplex stainless steel C section wit 150 mm web opening at 50 mm edge distance Te failure load for one beam was 19.7 kn and te failure was due to orizontal sear causing webpost buckling. Te twisting of te web-post in te buckling mode caused buckling of te top flange. Te principal stresses at failure are sown in Figure

253 Load (kn) Maximum stresses occurred around te web openings and witin te web-post between openings. Te maximum principal stresses around te openings and witin te web-post at te plastic failure were 735 and 750 N/mm respectively, wic were sligtly less tan te ultimate strengt of te material f u=790 N/mm. Te ig stresses can be explained by te combined effect of te web-post buckling, pure bending of te beam and local buckling around te web openings. Figure 10.35: Deformation and maximum principal stresses at failure load based on te non-linear FE analysis of Model 6 Te failure load from te FE analysis was 70% of te test failure load. Te maximum deflection at failure was 36.7 mm wic was due to te local deformation around te opening. A comparison between te test failure load and te load obtained from te non-linear FE analysis is sown in Figure Flange 1 deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 10.36: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick Lean Duplex stainless steel C sections wit 150 mm web openings at 50 mm edge distance 17

254 Load (KN) Model 7: mm tick Austenitic stainless steel C section wit 150 mm web opening at 50 mm edge distance Te failure load for Model 7 was 8.3 kn and te failure mode was due to te orizontal sear causing web-post buckling between te web openings, as sown in te deformation sape at failure in in Figure Te maximum principal stresses around te web openings and witin te webpost were 340 N/mm and 357 N/mm respectively (corresponding to 3.% strain). Hig stresses witin te web-post were due to te increase of te orizontal sear causing te web to twist. Te failure load from te finite element analysis using Riks metod was found to be 7% of te test failure load, as sown in te comparison in Figure Figure 10.37: Deformation and maximum principal stresses at failure load as per te FE non-linear analysis of Model Flange 1deflection Flange deflection Riks Jack deflection Displacement (mm) Figure 10.38: Failure load as obtained from te non-linear FE analysis compared to te test failure load for two 10 mm deep x mm tick Austenitic stainless steel C section wit 150 mm web openings at 50 mm edge distance 18

255 10.9 Analysis of te Results and Comparison between te Test and Finite Element Analysis for te Stainless Steel Tests Reasonable agreement between te test failure loads and te failure loads from te finite element analysis using Riks metod was obtained, as sown in Table10.5. Te failure load for te FE analysis was in te range of % of te test failure loads. Te resistance of te 3 mm tick beam was 33% more tan te resistance of te mm tick beams wit te same web opening configuration. Te load resistance of te Lean Duplex beams was 37% more tan te resistance of te same Austenitic stainless steel beam (Model 6 and 7) Table 10.5: Comparison between te test failure loads and te failure loads obtained from te FE analysis for te stainless steel tests Model Stainless Steel type/ tickness Test failure load for beams Riks failure load (kn) for beams FE mode of failure 1 & / mm 58/61.5 kn 50.5 (88%) Vierendeel bending / 3 mm 84 kn 67.1(80%) / mm 44.5 kn 41 (9%) 4 LDX 101/ mm 71.5 kn 48.8 (68%) Horizontal sear and web-post buckling Buckling of top flange and web-post buckling Buckling of top flange and web-post buckling / 3 mm 86 kn 90.4 (105%) Web-post buckling 6 & 8 LDX 101/ mm 54./55.4 kn 38.8 (70%) Web-post buckling / mm 39.4 kn 8.3 (7%) (Value in brackets is percentage of te test failure load) Horizontal sear causing web-post buckling Comparison of te Proposed Teory on Additional Deflection Due to Web Openings as Presented in Capter 6 wit Tests on Stainless Steel C Sections Te test series on 10 mm deep stainless steel beams were on Austenitic and Lean Duplex grades and in mm and 3 mm ticknesses and te results are presented in Table 10.6 and Te elastic range of te load-deflection curve corresponded to a sear force of 10 kn wic was used in te comparisons, except for one test wic failed at a lower load and so te deflection in te elastic range was obtained by extrapolation. For stainless steel, te elastic modulus is non-linear and a load of 10 kn corresponds to a bending stress of about 150 N/mm for te mm tick steel. Using tese elastic modulus in te teory and 19

256 finite element models give close agreement but are lower tan te test results, as sown in Table Tis may be attributed to te iger local stresses at te bolts at te load application points wic migt lead to iger deflections due to te reduced elastic modulus. Te elastic deformation of te stainless steel beams and te corresponding deflection subject to te applied load from te FEA are sown in Figures to Figure 10.39: Deflection of mm tick Austenitic stainless steel beam wit web openings at 50 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA Figure 10.40: Deflection of mm tick Austenitic stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA Figure 10.41: Deflection of mm tick Austenitic stainless steel beams wit web openings at 50 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA 0

257 Figure 10.4: Deflection of 3 mm tick Austenitic stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA Figure 10.43: Deflection of mm tick Lean Duplex stainless steel beams wit web openings at 100 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA Figure 10.44: Deflection of mm tick Lean Duplex stainless steel beams wit web openings at 50 mm edge distance subject to a sear force of 10 kn as obtained from te elastic FEA 1