"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamTorsion Elastic Torsion of Hollow, Non- circular Sections2 Torsion CESE 656 Closed, Thin-walled Sections? Assumptions? Isotropic materials? Cross-section does not change due to torsional deformation? Each cross-section rotates approximately as a rigid body aboutan axis of twist? Warping occurs and is unrestrained? Walls are thin compared to cross-section dimensions? Shear stress is uniform across the wall thickness 3 Torsion CESE 656 Shear Flow? Examine equilibrium of element AB, applying theassumption of uniform shear stress? Sum forces along the axis of the member? ? ( ? ? ? ) - ? ( ? ? ? ) = 0? ? ? ? ? ? ? = ? ? ? ? = ?? ? ? ? ? ? ? Shear flow, q, is constant around the cross-section? t occurs where t is smallestmax 4 Torsion CESE 656 Shear Stress? Examine equilibrium of the member, applying a constantshear flow? The effect of the shear flow must equal the internaltorque? ? = h? ? = h ? ? ? = ? h ? ?? ? ? ? h ds = 2 times the area of blue triangle? h ? ? = 2 ?? ? ? A = Area enclosed by the mean perimeterm ? T = 2A q = 2A ttm m ? ? ? ? = or ? =2? ? 2 ? ? ? 5 Torsion CESE 656 Rate of Twist? Using Energy Methods? Strain Energy 2 ? ? ? ? = ? ? , with ? =? 2? 2 ? ? 2 ? ? ? = ? ?? 2 2 8 ? ? ? ? 2 ? ? ? ? ? = ? ? , with ? ? = ? ? ?? ? 2 2 8 ? ? ? ? 2 ? ?? ? ? = ? ?? ? 2 8 ? ? ? ? ? External Work1 ? ? = ?? ? ?? 2 ? External Work = Internal Strain Energy2 1 ? ?? ? ?? ? ? = ? ?? ? ? 2 2 8 ? ? ? ? ? ?? ? ? = ,where ? = the rate of twist and ? 2 4 ? ? ? ? ? the angle of twist, is? = ? ? ? ? 6 Torsion CESE 656 Example – Hollow Section with Constant t? A hollow elliptical section is used to resist a torque of 250 ft·lb? The mean semi-major axis is b = 1.5 in? The mean semi-minor axis is 0.5b = 0.75 in? The thickness of the tube is t = 0.125 in? The tube is steel for which G = 11,200 ksi? Determinea. The maximum shear stress in the tubeb. The rate of twist for the tube7 Torsion CESE 656 a) Shear Stress? Thickness is uniform, therefore t is constant? ? ? =2 ? ? ? ? ? 1? ? ? 250 ?? · ?? 12 1,0 00? ? =2? 1. 5? ? 0. 75? ?0. 125 ? ? ? t = 3.40 ksi8 Torsion CESE 656?? ??b) Rate of Twist? Thickness is uniform? ? ? =? 2 4 ? ? ? ? ? ? ? = ??? 2 4 ? ?? ? ? ?? = mean perimeter of the ellipse ? 2 2 ? + ? ? This is actually difficult to calculate, but can be approximated as2?2 ? The “exact” solution can only be expressed as an infinite series? ? 1 ? ? ? 250 ? ? · ?? 12 2 2 1. 5? ? + 0. 75 ? ? 1,0 00? ? = 2 ?2 4 ? 1. 5 ? ? 0. 75? ? 11, 20 0 ?0. 1 25 ? ? 2 -6 ? ? = 320(10 ) rad/in9 Torsion CESE 656?? ?? ?? ??Example – Hollow Section with Variable t? A hollow rectangular section is used to resist a torque of 2.5 kN·m? The height is a = 40 mm? The width is b = 3a = 120 mm? The wall thicknesses are ? t = 10 mm in the flanges and 1 ? t = 6 mm in the webs2 ? The tube is steel for which G = 77.5 GPa? Determinea. The maximum shear stress in the tubeb. The rate of twist for the tube10 Torsion CESE 656 "