"CESE 656 – Advanced Mechanicsfor Structural EngineersCivil Structural EngineeringUniversity of Alabama at BirminghamTorsion Inelastic Torsion2 Torsion CESE 656 Inelastic Torsion for Circular Sections? Strains still vary linearly? Stresses vary according to the t-?relationship? Total torque is still? ? = ? (? )? ? Define dA as a differential ring2 ? ? = 2 ? ? ( ? )? ? ?? 3 Torsion CESE 656??Elastic-Perfectly Plastic Behavior? 2 ? ? = 2? ? ? ? ?? ? 0 ? For a solid shaft? ? ? ? 2 2 ? ? = 2? ? ? ?? + 2? ? ? ??? ? ? ? 0 ? ? ? ? Elastic Core Plastic Annulus3 3 3 2 ??? 2 ??? 2 ??? ? ? ? ? ? ? ? = + -4 3 3 ?? ? 3 3 ? ? = 4 ? - ? for solid shaft only? 6 4 Torsion CESE 656?? ?? ??Fully Plastic Torque (perfectly plastic)? 2 ? ? = 2? ? ( ? )?? ?? 0 ? For a solid shaft? 2 ? ? = 2? ??? ?? ? ? 0 2 ? 4 3 ? ? = ? ? = ? for solid shaft only? ? ? 3 3 5 Torsion CESE 656 Elastic-Strain Hardening Behavior? Ultimate torque instead of fully plastic torque? Strain at the outer edge is set to ? u? Stress distribution is found from t-? diagram as t(?)? 2 ? ? = 2? ? (?) ? ??For a solid shaft? 0 ? 2 ? ? = 2? ? (?) ? ??For a hollow shaft? ? ? 6 Torsion CESE 656 Fully Plastic Torque For Solid Non-CircularSections? From St. Venant’s semi-inverse method?? ?? ? ? = , and? = -? ? ?? ? ? 2 2 ? ? = ? + ? = ? for the fully plastic torque? ? ? 2 2 ?? ?? 2 ? + = ?? ?? ? ? ? The maximum slope of the stress function, f, everywherein the cross-section is equal to t Y7 Torsion CESE 656?? ??Examine a Square Cross-section? If the slope of f is t , everywhere, then f looks like aY pyramid when plotted? The total torque is given by? ? = 2 ? ? ? ? ?? ? T = 2 x volume under f1 2 ? ? = 2 2? ? ?? ? 3 8 3 ? ? = ? ?? ? 3 8 Torsion CESE 656 Other Cross-sections (from textbook)9 Torsion CESE 656 Fully Plastic Torque For Hollow Section? T = T – T P PS PH? T = Fully Plastic Torque for the hollow sectionP ? T = Fully Plastic T orque for a solid section having aPS boundary at the outer cross-section of the hollowmember? T = Fully Plastic Torque for a solid section having aPH boundary at the inner cross-section of the hollowmember= –10 Torsion CESE 656 "