"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamTorsion Thin-wall Torsion Members withRestrained Ends2 Torsion CESE 656 Effect of Restrained Ends? The restrained end is no longer free to warp? Normal stresses develop at the restrained end tocounteract the tendency to warp3 Torsion CESE 656 I-Sections? Near the wall (a), Warping Torsiondominates. ? Torque is transmitted mainly by a lateralshear force in each flange ? The shear force creates a lateral bendingmoment in the flange? The lateral bending moment producesnormal stresses in the flange? Away from the wall (c), Saint VenantTorsion dominates.? Here torque is transmitted mainly bytorsional shear stresses? Between (a) and (c), the torque istransmitted by a combination ofwarping torsion and Saint Venanttorsion4 Torsion CESE 656 Cantilever I-Section? Warping torsion = ? = ? ??1 ? St. Venant torsion = ? = ?? ?2 ? ? + ? = ?1 2 ? ? ?? + ?? ? = ?(1)? (1 eqn, 2 unknowns)5 Torsion CESE 656 Cantilever I-Section? A second equation comes from lateral bending of the topflange2 ?? ? ? ? ?? ? = -? (Assuming? ? )??? 2 2 ? ? 2 2 2 h ? ? h ? ? ? ? = ? ? =2 2 2 ? ? 2 ? ? ?? ? = ??? ?? h ?? ? = -?4 ?? ?? ? = ? ??? 2 ?? h ? ? ? = -? ? (2)2 4 ? ? 6 Torsion CESE 656 Cantilever I-Section? ? ?? + ?? ? = ? (1)2 ?? h ? ? ? = -? ? (2)2 4 ? ? 2 2 ?? h ? ? ? ? - + ? =2 4 ? ? h ?? ? Let ? =2 2 ? ? ? 2 ? -? + ? =2 ? ? 7 Torsion CESE 656?? ?? ?? ??Cantilever I-Section - Solution? Angle of Twist of Beam Length L? ? ? ? = ? - ?? ? ??? ? ? ? ? - ? ? ? ? ? , > 2.5? ? ? ? Maximum Lateral Bending Moment in Flange? ? ? ? = ?? ? ??? ? ? h ? ? ? ? ? ? ? , > 2.5? ? ? h ? ? ? ? ? ? ? 6 ? 2 2 ? ? = = ? ? =? ? ? 3 ? ? ? 2 ? ? ? ? ? ? ? ? ? ? 12 8 Torsion CESE 656?? ??Other SolutionsNote that ? in these tablesshould be ß to be consistentwith the derivation in thetextbook and presented onthe previous slides9 Torsion CESE 656 Example – Cantilever I-beam? The I-beam shown is made of steel for which E = 200 GPa and G = 77.5 GPa).? The beam is 8.00 m long, fixed at one end, and free at the other.? T = 7.00 kN·m? Determinea. The maximum normal stress in the beamb. The maximum shear stress in the beamc. The total angle of twist, ß, at the free end? Section Properties3 3 20 270 2 60 156 4 ? ? = 2 + = 65.7 10 ? ?? ? 12 12 1 3 3 6 4 ? ? = 2 270 ?? 20 ?? + 260 ?? 15 ?? = 1.73 10 ? ?3 280 200? ? ? ( 65. 7) ? ? = = 1,390 ??2 77. 5 ? ? ? ( 1. 73) 10 Torsion CESE 656?? ?? ?? ?? ??"