A STUDY OF WELDED BUILT-UP BEAMS MADE FROM TITANIUM AND A TITANIUM ALLOY A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Narendra Babu Poondla May, 2010 A STUDY OF WELDED BUILT-UP BEAMS MADE FROM TITANIUM AND A TITANIUM ALLOY Narendra Babu Poondla Thesis Approved: Accepted: Advisor Dr Anil Patnaik _ Department Chair Dr Wieslaw K Binienda Co-Advisor Dr T.S Srivatsan Dean of the College Dr George K Haritos Committee Member Dr Craig Menzemer Dean of the Graduate School Dr George R Newkome Date ii ABSTRACT Titanium is well recognized as a modern and high performance metal that is much stronger and lighter than the most widely used steels in the industry There is a growing need to reduce the part weight, cost and lead time, while concurrently facilitating enhanced performance of structural parts made from titanium and titanium alloys Structural components made from titanium have the advantage of high strength-to-weight ratio, and high stiffness-to-weight ratio Owing to good resistance to corrosion and superior ballistic properties, titanium is used in several defense applications This thesis presents a summary of the research conducted on welded built-up titanium beams so as to eventually facilitate the design, fabrication, and implementation of titanium in large structural members An alternative to machining a structural component from thick plates or billets is to fabricate beams using the built-up concept Rolled plates and sheets of titanium alloys can be cut to size and welded together to fabricate a built-up structural component The primary objective of this project is to investigate structural performance of built-up welded beams fabricated from commercially pure (Grade 2) titanium and a common alloy (Ti-6Al-4V) under both static and fatigue loading conditions Six welded built-up titanium beams were fabricated and tested to experimentally and theoretically evaluate iii structural performance Analysis and design approaches for static and fatigue performance of built-up beams were also studied and it is clearly demonstrated that it is feasible to fabricate large built-up titanium beams by welding parts together using GMAW-P welding process The welds produced by this method were found to be sound and without any visible cracks The study also revealed that there is no deleterious influence of welding on structural performance of the built-up welded beams of commercially pure titanium and Ti-6Al-4V titanium alloy With suitable modifications to the current AISC steel design specifications a preliminary design methodology was developed for the titanium beams The failure loads, deflections and strains of welded built-up titanium beams are predictable to a reasonably good level of accuracy The test beams also demonstrated significant reserve strength and ductility following yielding The deflection curves and the load versus strain relationships obtained from the test results demonstrate a reasonably close match between the theoretical predictions and experimental test results up until the elastic limit of the material The fatigue tests conducted for this research revealed that the welded built-up beams made from the commercially pure titanium have better life than those made from the Ti-6Al-4V However additional work is required to develop further insight into the fatigue behavior of welded built-up titanium beams Finally the proposed welded built-up beam approach is anticipated to be a cost effective alternative to fabricating large structural elements and members by machining of the parts from thick plates or billets iv ACKNOWLEDGEMENTS I am heartily thankful to my advisor, Dr Anil Patnaik, whose encouragement and guidance has helped me to accomplish an in depth understanding in every step of my research He has been a constant source of great inspiration for me to my research in the field of Welded built-up titanium beams I am also very thankful to my committee members, Dr T.S Srivatsan and Dr Craig Menzemer for their invaluable suggestions and corrections and also for being on my committee Also, I would like to thank my family and friends for their continued love and support v TABLE OF CONTENTS Page LIST OF TABLES .x LIST OF FIGURES xi CHAPTER I INTRODUCTION 1.1 Background 1.2 Research Motivation and Significance 1.3 Concept of Built-Up Welded Beams 1.4 GMAW-P: A New Welding Technology Developed at Picatinny Arsenal (NJ) .5 1.5 Objectives 1.6 Thesis Outline II LITERATURE REVIEW 2.1 Titanium Alloys .12 2.2 Structural Applications of Titanium and Its Alloys 18 2.3 Titanium Applications for Architectural and Other Engineering Structures 23 2.4 Summary of Commercial Available Titanium and its Alloys 26 vi III PROCUREMENT AND TESTING OF MATERIALS 28 3.1 Commercially Pure Titanium 28 3.2 Ti-6Al-4V 30 3.3 Experimental Procedures 31 3.4 Results and Discussion .34 3.5 Hardness Tests 38 3.6 Concluding Comments .43 3.7 Summary 45 IV THEORETICAL ANALYSIS OF BEAMS USING THE MATERIAL PROPERTIES OF Ti ALLOYS……………………………………………………………………… 46 4.1 Design Specifications .46 4.2 Review of Steel Design Specifications for Potential Application in the Design of Titanium Built-Up Beams………………………………………………………….49 4.3 Discussion 74 V DESIGN OF TEST BEAMS USING COMMERCIALLY PURE TITANIUM AND Ti ALLOYS…………………………………………………………………………… 76 5.1 The Design Basis .77 5.2 Material Properties 77 5.3 Section Properties 78 5.4 Design Procedure .81 5.5 Summary 84 VI FABRICATION OF TEST BEAMS 85 6.1 Preparation of Parts for the Test Beams 86 6.2 Machining of Parts for the Test Beams 87 6.3 Fixture for Welding the Test Beams 89 vii 6.4 Welding of Test Beams 90 6.5 Test Beams .94 6.6 Summary 95 VII STATIC BEND TESTS OF TITANIUM ALLOY BEAMS 96 7.1 Test Set-Up 96 7.2 Instrumentation 99 7.3 Test procedure 102 7.4 Test Results 102 7.5 The Load versus Deflection Curves .110 7.6 The Load versus Strain Relation 113 7.7 Summary 119 VIII FATIGUE TESTS OF TITANIUM ALLOY BEAMS 120 8.1 Test Set-Up and Instrumentation 120 8.2 Test Procedure 123 8.3 Stress Ratios and Loads for Fatigue Tests 124 8.4 Summary .135 IX ANALYSIS OF TEST RESULTS .136 9.1 Tensile Deformation, Fracture behavior, Influence of Material Composition on Microstructural Development, and Hardness .137 9.2 Static Bend Tests of welded Built-Up Titanium Beams .137 9.3 Fatigue Tests of welded Built-Up Titanium Beams 139 9.4 Summary .139 X CONCLUSIONS 140 viii 10.1 Tensile Deformation, Fracture behavior, Influence of Material Composition on Microstructural Development, and Hardness 140 10.2 Welded Built-Up Titanium Beams 143 REFERENCES 146 APPENDICES 150 APPENDIX A LIST OF NOTATIONS 151 APPENDIX B IMAGE GALLERY OF ARCHITECTURAL APPLICATIONS OF TITANIUM 169 ix LIST OF TABLES Table Page 1: Details of CP Ti (Gr 2)……………………………………………………………….29 2: Nominal expected chemical composition of Ti-6Al-4V (in weight percent)…………30 3: A compilation of microhardness test data made on the two materials Ti-6Al-4V alloy and commercially pure titanium (Grade 2)…………………………………….39 4: A compilation of macrohardness test data made on the two materialsTi-6Al-4V alloy and commercially pure titanium (Grade 2)…………………………………… 39 5: Plate Thickness for Different Elements of Test Beams (in Inches)………………… 78 6: Summary of Test Beam Design……………………………………………………….84 7: Details of Welds Made for Different Test Beams…………………………………….92 8: Procedure Qualification Record……………………………………………………….93 9: Maximum and minimum loads applied to the test specimens for fatigue tests…… 124 10: Summary of Failure Loads and Predicted Strengths……………………………….138 x http://www.worldenough.net/picture/English/lab/Lab_street/picts%20videos%20own/Guggenhei m%20Bilbao%202.jpg http://www.nsc.co.jp/en/product/titan/pdf/TC029.pdf http://www.george-square-hotels.com/images/city_pics/glasgow_science_centre_ulybugs.jpg www.h6.dion.ne.jp/~furuchan/page002.html www.fukuokatalk.com/2008/11/15/hawkstown/ Design Calculations Four sets of design calculations have been used in this project These design calculations correspond to the two material types and two grooving conditions (one with grooves and the other without grooves) Beams B1 & B2 Sectional Properties: Area of Flange Af = x 0.267 = 0.801 in2 Area of Web Aw = 3.875 x 0.267 = 1.034in2 Moment of Inertia Ixx = 8.1755in4 Sxc = Sxt = 3.708in3 Elastic Modulus E = 18000 Ksi Yield Stress Fy = 137 Ksi 187 a) Yielding : 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 3.708 × 137 × b) = 46.57 𝐾𝑖𝑝 − 𝑓𝑡 12 Lateral torsional buckling: 𝐸 𝐹𝑦 𝐿 𝑝 = 1.76𝑟 𝑦 𝑟𝑦 = 𝐼𝑦 = 𝐴 0.6 = 0.785 0.9732 𝐿 𝑝 = 1.76 × 0.785 × 𝐸 𝐿 𝑟 = 1.95𝑟𝑡𝑠 0.7𝐹𝑦 18000 = 15.83 137 0.7𝐹𝑦 𝑆 𝑥 ℎ 𝑜 + + 6.76 𝐸 𝐽𝑐 𝐽𝑐 𝑆 𝑥ℎ 𝑜 𝑏𝑓 𝑟 𝑡𝑠 = = 0.785 𝐴 12(1 + 6𝐴𝑤 𝑓 𝐽= 𝑏 𝑖 𝑡 𝑖 = 0.0626 ℎ 𝑜 = 3.875 + 0.267 = 4.142 188 Substituting all of the above values, we get Lr =38.74 in The unbraced length of the beam Lb = 24 in Hence, 𝐿 𝑝 < 𝐿 𝑏 < 𝐿 𝑟 the section is in the inelastic zone, so the governing moment equation is: 𝑀𝑛 = 𝐶𝑏 𝑀 𝑝 − (𝑀 𝑝 − 𝑀 𝑟 ) 𝐿 𝑏 −𝐿 𝑝 𝐿 𝑟 −𝐿 𝑝 𝑀 𝑛 = 1.14 596.47 − (596.47 − 417.529) 27 − 15.83 38.74 − 15.83 𝑀 𝑛 = 48.38 𝐾𝑖𝑝 − 𝑓𝑡 c) Flange local buckling: 𝜆 𝑝 = 0.38 𝜆 𝑟 = 1.0 𝜆= 𝐸 = 4.355 𝐹𝑦 𝐸 = 11.46 𝐹𝑦 𝑏 = 5.617 2𝑡 𝑓 Hence 𝜆 𝑝 < 𝜆 < 𝜆 𝑟 the section is in Inelastic zone, the governing moment equation is, 𝑀𝑛 = 𝑀 𝑝 − (𝑀 𝑝 − 𝑀 𝑟 ) 𝜆− 𝜆𝑝 𝜆𝑟 − 𝜆𝑝 𝑀 𝑛 = 596.47 − (596.47 − 417.529) 𝑀 𝑛 = 47.04 𝐾𝑖𝑝 − 𝑓𝑡 d) Web local buckling: 189 5.617 − 4.355 11.46 − 4.355 𝜆 𝑝 = 3.76 𝐸 = 43.1 𝐹𝑦 𝜆 𝑟 = 5.76 𝜆= 𝐸 = 66 𝐹𝑦 ℎ = 14.51 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 3.708 × 137 × = 46.57 𝐾𝑖𝑝 − 𝑓𝑡 12 The least moment governs the design, and hence the moment strength (Mn) = 46.57 Kip-ft 𝑃× 10 = 0.833𝑃 = 46.57 12 𝑃 = 55.91 𝑘𝑖𝑝𝑠 2𝑃 = 111.84 𝑘𝑖𝑝𝑠 The ultimate load carrying capacity of the beams B1 & B2 in bending is 112 Kips Shear: Shear strength of the web is determined using the expression, 𝑉𝑛 = 0.6𝐹𝑦 𝐴 𝑤 𝐶 𝑣 190 5.0 𝐾 𝑣 = 5.0 + 𝑎 = 5.75 ( )2 ℎ 𝜆 𝑝 = 1.10 𝐾𝑣 𝐸 = 30.2 𝐹𝑦𝑤 𝜆 𝑟 = 1.37 𝐾𝑣 𝐸 = 37.64 𝐹𝑦𝑤 𝜆= ℎ = 14.51 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, and hence, the value of Cv =1.0 Therefore, 𝑉𝑛 = 0.6 × 137 × 1.034 × 1.0 = 85.0 𝐾𝑖𝑝𝑠 The total shear strength of the web is 𝑉𝑛 = × 85 = 170 𝐾𝑖𝑝𝑠 which is greater than 112 kips meaning that the beam will fail in bending by excessive deflection and flexure Beam B3: Sectional Properties: Area of Flange Af = x 0.267 = 0.801 in2 Area of Web Aw = x 0.267 = 1.068in2 Moment of Inertia Ixx = 8.7255in4 Sxc = Sxt = 3.958in3 Elastic Modulus E = 18000 Ksi Yield Stress Fy = 137 Ksi a) Yielding : 191 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 = 49.7 𝐾𝑖𝑝 − 𝑓𝑡 12 𝑀 𝑛 = 1.10 × 3.958 × 137 × b) Lateral torsional buckling: 𝐸 𝐹𝑦 𝐿 𝑝 = 1.76𝑟 𝑦 𝑟𝑦 = 𝐼𝑦 = 𝐴 0.6 = 0.785 0.9732 𝐿 𝑝 = 1.76 × 0.785 × 𝐿 𝑟 = 1.95𝑟𝑡𝑠 𝐸 0.7𝐹𝑦 𝐽𝑐 𝑆 𝑥ℎ 𝑜 18000 = 15.83 137 + + 6.76 𝑏𝑓 𝑟 𝑡𝑠 = 0.7𝐹𝑦 𝑆 𝑥 ℎ 𝑜 𝐸 𝐽𝑐 = 0.785 𝐴 12(1 + 6𝐴𝑤 𝑓 𝐽= 𝑏 𝑖 𝑡 𝑖 = 0.0634 ℎ 𝑜 = + 0.267 = 4.267 Substituting all the above values we get Lr =37.44 in The unbraced length of the beam Lb = 24 in Hence 𝐿 𝑝 < 𝐿 𝑏 < 𝐿 𝑟 the section is in the inelastic zone, so the governing moment equation is: 𝑀𝑛 = 𝐶𝑏 𝑀 𝑝 − (𝑀 𝑝 − 𝑀 𝑟 ) 192 𝐿𝑏 − 𝐿𝑝 𝐿𝑟 − 𝐿𝑝 𝑀 𝑛 = 1.14 596.47 − (596.47 − 417.529) 27 − 15.83 38.74 − 15.83 𝑀 𝑛 = 48.38 𝐾𝑖𝑝 − 𝑓𝑡 c) Flange local buckling: 𝜆 𝑝 = 0.38 𝐸 = 11.46 𝐹𝑦 𝜆 𝑟 = 1.0 𝜆= 𝐸 = 4.355 𝐹𝑦 𝑏 = 5.617 2𝑡 𝑓 Hence 𝜆 𝑝 < 𝜆 < 𝜆 𝑟 the section is in Inelastic zone, the governing moment equation is, 𝑀𝑛 = 𝑀 𝑝 − (𝑀 𝑝 − 𝑀 𝑟 ) 𝜆− 𝜆𝑝 𝜆𝑟 − 𝜆𝑝 𝑀 𝑛 = 596.47 − (596.47 − 417.529) 5.617 − 4.355 11.46 − 4.355 𝑀 𝑛 = 47.04 𝐾𝑖𝑝 − 𝑓𝑡 d) Web local buckling: 𝜆 𝑝 = 3.76 𝐸 = 43.1 𝐹𝑦 𝜆 𝑟 = 5.76 𝜆= 𝐸 = 66 𝐹𝑦 ℎ = 14.98 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 193 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 3.958 × 137 × = 49.7 𝐾𝑖𝑝 − 𝑓𝑡 12 The least moment governs our design and thus, the moment of resistance (Mn) = 47.04 Kip-ft 𝑃× 10 = 0.833𝑃 = 47.04 12 𝑃 = 56.47 𝑘𝑖𝑝𝑠 2𝑃 = 113 𝑘𝑖𝑝𝑠 The ultimate load carrying capacity of the beam B3 in bending is 113 Kips Shear: Shear strength of the web is determined by using the expression, 𝑉𝑛 = 0.6𝐹𝑦 𝐴 𝑤 𝐶 𝑣 5.0 𝐾 𝑣 = 5.0 + 𝑎 = 5.8 ( )2 ℎ 𝜆 𝑝 = 1.10 𝐾𝑣 𝐸 = 30.36 𝐹𝑦𝑤 𝜆 𝑟 = 1.37 𝐾𝑣 𝐸 = 37.812 𝐹𝑦𝑤 𝜆= ℎ = 14.98 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, and hence the value of Cv =1.0 Therefore, 𝑉𝑛 = 0.6 × 137 × 1.068 × 1.0 = 87.78 𝐾𝑖𝑝𝑠 The total shear strength of the web is 𝑉𝑛 = × 87.78 = 176 𝐾𝑖𝑝𝑠 194 Beams B4 & B5: Sectional Properties: Area of Flange Af = x 0.395 = 1.185 in2 Area of Web Aw = 3.875 x 0.125 = 0.484 in2 Moment of Inertia Ixx = 11.44 in4 Sxc = Sxt = 4.9 in3 Elastic Modulus E = 15560 Ksi Yield Stress Fy = 62.6 Ksi a) Yielding : 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 4.9 × 62.6 × b) = 28.11 𝐾𝑖𝑝 − 𝑓𝑡 12 Lateral torsional buckling: 𝐿 𝑝 = 1.76𝑟 𝑦 𝑟𝑦 = 𝐼𝑦 = 𝐴 𝐸 𝐹𝑦 0.890 = 0.838 1.265 𝐿 𝑝 = 1.76 × 0.838 × 195 15560 = 29.8 62.6 𝐸 𝐿 𝑟 = 1.95𝑟𝑡𝑠 0.7𝐹𝑦 0.7𝐹𝑦 𝑆 𝑥 ℎ 𝑜 + + 6.76 𝐸 𝐽𝑐 𝐽𝑐 𝑆 𝑥ℎ 𝑜 𝑟 𝑡𝑠 = 𝑏𝑓 = 0.837 𝐴 12(1 + 6𝐴𝑤 𝑓 𝐽= 𝑏 𝑖 𝑡 𝑖 = 0.125 ℎ 𝑜 = 3.875 + 0.395 = 4.27 Substituting all the above values we get Lr =107.86 in The unbraced length of the beam Lb = 24 in Hence 𝐿 𝑏 < 𝐿 𝑝 < 𝐿 𝑟 the section is in the plastic zone, so the governing moment equation, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 4.9 × 62.6 × c) = 28.11 𝐾𝑖𝑝 − 𝑓𝑡 12 Flange local buckling: 𝐸 = 6.0 𝐹𝑦 𝜆 𝑝 = 0.38 𝜆 𝑟 = 1.0 𝜆= 𝐸 = 15.76 𝐹𝑦 𝑏 = 3.797 2𝑡 𝑓 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 4.9 × 62.6 × 196 = 28.11 𝐾𝑖𝑝 − 𝑓𝑡 12 d) Web local buckling: 𝜆 𝑝 = 3.76 𝐸 = 59.57 𝐹𝑦 𝜆 𝑟 = 5.76 𝐸 = 90.77 𝐹𝑦 𝜆= ℎ = 31 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 4.9 × 62.6 × = 28.11 𝐾𝑖𝑝 − 𝑓𝑡 12 The least moment governs our design and hence the moment of resistance (MR) =28.11 Kip-ft 𝑃× 10 = 0.833𝑃 = 28.11 12 𝑃 = 33.74 𝑘𝑖𝑝𝑠 2𝑃 = 67.5 𝑘𝑖𝑝𝑠 The ultimate load carrying capacity of the beams B4 & B5 in bending is 67.5 Kips Shear: Shear strength of the web is determined by 𝑉𝑛 = 0.6𝐹𝑦 𝐴 𝑤 𝐶 𝑣 5.0 𝐾 𝑣 = 5.0 + 𝑎 = 5.75 ( )2 ℎ 197 𝜆 𝑝 = 1.10 𝐾𝑣 𝐸 = 41.58 𝐹𝑦𝑤 𝜆 𝑟 = 1.37 𝐾𝑣 𝐸 = 51.786 𝐹𝑦𝑤 𝜆= ℎ = 31 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, and hence the value of Cv =1.0 Therefore, 𝑉𝑛 = 0.6 × 62.6 × 0.484 × 1.0 = 18.18 𝐾𝑖𝑝𝑠 The total shear strength of the web is 𝑉𝑛 = × 18.18 = 36 𝐾𝑖𝑝𝑠 The shear strength of the beam is 36 kips which is less than the moment strength of the beam Therefore, the beam is predicted to fail in shear by web local buckling Beam B6: Sectional Properties: Area of Flange Af = x 0.395 = 1.185 in2 Area of Web Aw = x 0.125 = 0.5 in2 Moment of Inertia Ixx = 12.137 in4 Sxc = Sxt = 5.067 in3 Elastic Modulus E = 15560 Ksi Yield Stress Fy = 62.6 Ksi a) Yielding : 198 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 5.067 × 62.6 × b) = 29.071 𝐾𝑖𝑝 − 𝑓𝑡 12 Lateral torsional buckling: 𝐸 𝐹𝑦 𝐿 𝑝 = 1.76𝑟 𝑦 𝑟𝑦 = 𝐼𝑦 = 𝐴 0.890 = 0.838 1.265 15560 = 29.8 62.6 𝐿 𝑝 = 1.76 × 0.838 × 𝐿 𝑟 = 1.95𝑟𝑡𝑠 𝐸 0.7𝐹𝑦 𝑟 𝑡𝑠 = 𝐽𝑐 𝑆 𝑥ℎ 𝑜 + + 6.76 𝑏𝑓 0.7𝐹𝑦 𝑆 𝑥 ℎ 𝑜 𝐸 𝐽𝑐 = 0.837 𝐴 12(1 + 6𝐴𝑤 𝑓 𝐽= 𝑏 𝑖 𝑡 𝑖 = 0.125 ℎ 𝑜 = + 0.395 = 4.395 Substituting all the above values we get Lr =105.34 in The unbraced length of the beam Lb = 24 in Hence 𝐿 𝑏 < 𝐿 𝑝 < 𝐿 𝑟 the section is in the plastic zone, so the governing moment equation, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 5.067 × 62.6 × 199 = 29.071 𝐾𝑖𝑝 − 𝑓𝑡 12 c) Flange local buckling: 𝐸 = 6.0 𝐹𝑦 𝜆 𝑝 = 0.38 𝐸 = 15.76 𝐹𝑦 𝜆 𝑟 = 1.0 𝜆= 𝑏 = 3.797 2𝑡 𝑓 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 5.067 × 62.6 × d) = 29.071 𝐾𝑖𝑝 − 𝑓𝑡 12 Web local buckling: 𝜆 𝑝 = 3.76 𝐸 = 59.57 𝐹𝑦 𝜆 𝑟 = 5.76 𝐸 = 90.77 𝐹𝑦 𝜆= ℎ = 31 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, the governing moment equation is, 𝑀 𝑛 = 𝑍 𝑥 × 𝐹𝑦 𝑀 𝑛 = 1.10 × 5.067 × 62.6 × = 29.071 𝐾𝑖𝑝 − 𝑓𝑡 12 The least moment governs our design and hence the moment of resistance (MR) =29.71 Kip-ft 200 𝑃× 10 = 0.833𝑃 = 29.071 12 𝑃 = 34.89𝑘𝑖𝑝𝑠 2𝑃 = 69.79 𝑘𝑖𝑝𝑠 The ultimate load carrying capacity of the beam B6 in bending is 69.79 Kips Shear: Shear strength of the web is determined by 𝑉𝑛 = 0.6𝐹𝑦 𝐴 𝑤 𝐶 𝑣 5.0 𝐾 𝑣 = 5.0 + 𝑎 = 5.8 ( )2 ℎ 𝜆 𝑝 = 1.10 𝜆 𝑟 = 1.37 𝜆= 𝐾𝑣 𝐸 = 41.768 𝐹𝑦𝑤 𝐾𝑣 𝐸 = 52.01 𝐹𝑦𝑤 ℎ = 32 𝑡𝑤 Hence 𝜆 < 𝜆 𝑝 < 𝜆 𝑟 the section is in plastic zone, and hence the value of Cv =1.0 Therefore, 𝑉𝑛 = 0.6 × 62.6 × 0.5 × 1.0 = 18.78 𝐾𝑖𝑝𝑠 The total shear strength of the web is 𝑉𝑛 = × 18.78 = 38 𝐾𝑖𝑝𝑠 The shear strength of the beam is 36 kips which is less than the moment strength of the beam Therefore, the beam is predicted to fail in shear by web local buckling 201