3ContentsTorsional Loads on Circular ShaftsNet Torque Due to Internal StressesAxial Shear ComponentsShaft DeformationsShearing StrainStresses in Elastic RangeNormal StressesTorsional Failure ModesSample Problem 3.1Slide 12Slide 13Angle of Twist in Elastic RangeStatically Indeterminate ShaftsSample Problem 3.4Slide 17MECHANICS OF MATERIALSFourth EditionFerdinand P. BeerE. Russell Johnston, Jr.John T. DeWolfLecture Notes:J. Walt OlerTexas Tech UniversityCHAPTER© 2006 The McGraw-Hill Companies, Inc. All rights reserved. 3Torsion© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 2ContentsIntroductionTorsional Loads on Circular ShaftsNet Torque Due to Internal StressesAxial Shear ComponentsShaft DeformationsShearing StrainStresses in Elastic RangeNormal StressesTorsional Failure ModesSample Problem 3.1Angle of Twist in Elastic RangeStatically Indeterminate ShaftsSample Problem 3.4Design of Transmission ShaftsStress ConcentrationsPlastic DeformationsElastoplastic MaterialsResidual StressesExample 3.08/3.09Torsion of Noncircular MembersThin-Walled Hollow ShaftsExample 3.10© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 3Torsional Loads on Circular Shafts•Interested in stresses and strains of circular shafts subjected to twisting couples or torques•Generator creates an equal and opposite torque T’•Shaft transmits the torque to the generator•Turbine exerts torque T on the shaft© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 4Net Torque Due to Internal Stresses   dAdFT•Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque,•Although the net torque due to the shearing stresses is known, the distribution of the stresses is not.•Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform.•Distribution of shearing stresses is statically indeterminate – must consider shaft deformations.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 5Axial Shear Components•Torque applied to shaft produces shearing stresses on the faces perpendicular to the axis.•The existence of the axial shear components is demonstrated by considering a shaft made up of axial slats.•Conditions of equilibrium require the existence of equal stresses on the faces of the two planes containing the axis of the shaft.•The slats slide with respect to each other when equal and opposite torques are applied to the ends of the shaft.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 6•From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length.LTShaft Deformations•When subjected to torsion, every cross-section of a circular shaft remains plane and undistorted.•Cross-sections of noncircular (non-axisymmetric) shafts are distorted when subjected to torsion.•Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 7Shearing Strain•Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus. •Shear strain is proportional to twist and radiusmaxmax and cLcLL or •It follows that•Since the ends of the element remain planar, the shear strain is equal to angle of twist.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 8Stresses in Elastic RangeJcdAcdATmax2max•Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section,421cJ 414221ccJ  and maxJTJTc•The results are known as the elastic torsion formulas,•Multiplying the previous equation by the shear modulus,maxGcG maxcFrom Hooke’s Law,G, soThe shearing stress varies linearly with the radial position in the section.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 9Normal Stresses•Elements with faces parallel and perpendicular to the shaft axis are subjected to shear stresses only. Normal stresses, shearing stresses or a combination of both may be found for other orientations. max00max450max0max22245cos2oAAAFAAF•Consider an element at 45o to the shaft axis,•Element a is in pure shear. •Note that all stresses for elements a and c have the same magnitude•Element c is subjected to a tensile stress on two faces and compressive stress on the other two.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 10Torsional Failure Modes•Ductile materials generally fail in shear. Brittle materials are weaker in tension than shear. •When subjected to torsion, a ductile specimen breaks along a plane of maximum shear, i.e., a plane perpendicular to the shaft axis.•When subjected to torsion, a brittle specimen breaks along planes perpendicular to the direction in which tension is a maximum, i.e., along surfaces at 45o to the shaft axis.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf3 - 11Shaft BC is hollow with inner and outer diameters of 90 mm and 120 mm, respectively. Shafts AB and CD are solid of diameter d. For the loading shown, determine (a) the minimum and maximum shearing stress in shaft BC, (b) the required diameter d of shafts AB and CD if the allowable shearing stress in these shafts is 65 MPa.Sample Problem 3.1SOLUTION:•Cut sections through shafts AB and BC and perform static equilibrium analyses to find torque loadings.•Given allowable shearing stress and applied