"CESE 656– Advanced Mechanics for Structural Engineers Civil Structural Engineering University of Alabama at Birmingham TorsionElastic Torsion of Circular Shafts 2 Torsion CESE 656Review Elastic Torsion ? Observed deformation behavior for circular shaft ? Cross-sections remains circular and radius is unchanged ? Cross-sections remain flat (no warping) ? Radial lines remain straight, meaning that shear strains, g,vary linearly from zero at the center to a maximumvalue at the outer edge ? Under linear-elastic behavior ( t = G g) shear stresses,t, also vary linearly ? ? ? = ? ( ??? ) = ? ( ) ? ? ? ??? ? ? ? ? ??? ??? 2 ? ? = ? = ? ? ? ? ? ? ? = and t = ??? ? ? 3 Torsion CESE 656 ?? ?? ??Angle of Twist ? rd f =gdx ? g =t/G and ? t = T(x) r/J(x) T(x) ? d f ? dx J(x)G L T(x) f ? dx ? J(x)G 0 ? For constant cross-section TL ? f ? JG 4 Torsion CESE 656Polar Moment of Inertia c 2 2 ? J ? r dA ? r ?2 ?rd r ? ? ? A 0 c c 4 r 3 ? J ? 2 ? r d r ? 2 ? ? 4 0 0 ? ? 4 4 ? J ? c ? d 2 32 5 Torsion CESE 656Maximum Elastic Torque ? For elastic behavior ? ? = ??? ? ? Sett =t and solve for T max Y ? ? ? ? ? = ? ? ? 3 ? ? = ? c for a solid shaft ? ? 2 6 Torsion CESE 656 ??Elastic Torsion of Solid, Non- circular Sections 7 Torsion CESE 656Elastic Torsion of Non-circular Sections ? Observed deformation behavior for non-circular shafts ? Cross-sections remains the same shape ? Cross-sections DO NOT remain flat, instead they warp alongthe longitudinal axis? “Radial” lines DO NOT remain straight, meaning that shearstrains,g, DO NOT vary linearly from zero at the center to amaximum value at the outer edge 8 Torsion CESE 656St. Venant’s Semi-inverse Method ? 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 ? Warping is small ? Warping is nearly the same for all cross-sections 9 Torsion CESE 656St. Venant’s Semi-inverse Method ? Assumptions indicate that onlyg , g , t , andt are non- yz xz yz xz zero ? General solution – (See full derivation in the textbook) ? ? ? ? ? ? ?? ? - = -2? ? ? ? ? ? whereq is the angle of twist per unit length or rate of twist ? ? ? ? ?? ?? ? = = 0 ? ? ?? ? ? ? ? ?? ? ? ? + = 0 ? ? ? ? ? t andt are independent of z, and can be written as zx zy ?? ?? ? ? = , and? = - ?? ? ? ? Making the torsion problem a problem of findingf 10 Torsion CESE 656 ?? ??"