General Relations Involved in Deformable SolidsDisplacements1KinematicsExternalforceStrainsandmomentsTMaterial modelsEquilibrium4[2StaticInternalforcesequivalencyStressesandmoments311
General Relations Involved in Deformable Solids 11
Kinematics. From observation, the angle of twist of theshaft is proportional to the applied torqueand to the shaft length.aΦ LΦαT,: When subjected to torsion, every cross-section of a circular shaft remains planeand undistorted.h. Cross-sections for hollow and solidcircular shafts remain plain andundistorted because a circular shaft isaxisymmetric.: Since every cross section of the bar is identical, and since everycross section is subjected to the same internal torque, we say thatthe bar is in pure torsion.12
• From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length. T, L • When subjected to torsion, every crosssection of a circular shaft remains plane and undistorted. • Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric. Kinematics 12 • Since every cross section of the bar is identical, and since every cross section is subjected to the same internal torque, we say that the bar is in pure torsion
Kinematics. Consider an interior section of the shaftAs a torsional load is applied, an elementon the interior cylinder deforms into arhombus.: Since the ends of the element remain(a)planar, the shearing strain may be relatedto the angle of twist.It follows thatLy= pp or = P(b)L: Shearing strain is proportional to theangle of twist and radiuscoPand(c)maxmaxLC13
• Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus. • Shearing strain is proportional to the angle of twist and radius max max and c L c L L or • It follows that • Since the ends of the element remain planar, the shearing strain may be related to the angle of twist. 13 Kinematics
Hooke's Law for Shearing Deformation: A cubic element subjected to a shearing stresswill deform into a rhomboid. Thecorresponding shear strain is quantified interms of the change in angle between the sidesTxy = f(yxy): A plot of shearing stress vs. shear strain issimilar the previous plots of normal stress vsU晋+Ynormal strain except that the strength valuesare approximately half. For small strainsTxy=GYxy Tyz =GYyz Tzx =Gzxwhere G is the modulus of rigidity or shearmodulus.14
• A cubic element subjected to a shearing stress will deform into a rhomboid. The corresponding shear strain is quantified in terms of the change in angle between the sides, xy xy f • A plot of shearing stress vs. shear strain is similar the previous plots of normal stress vs. normal strain except that the strength values are approximately half. For small strains x y G x y yz G yz zx G zx where G is the modulus of rigidity or shear modulus. Hooke’s Law for Shearing Deformation 14
A Simple Example of Shearing Deformation2.5inSOLUTION:Sin: Determine the average2inangular deformation orshearing strain of the block: Apply Hooke's law for shearingstress and strain to find theA rectangular block of material withmodulus of rigidityG=90ksi iscorresponding shearing stressbonded to two rigid horizontal plates.Thelowerplateisfixed,whilethe: Use the definition of shearingupper plate is subjected to a horizontalstress to find the force Pforce P. Knowing that the upper platemoves through 0.04 in. under the actionof the force, determine a) the averageshearing strain in the material, and b)the force P exerted on the plate.15
A Simple Example of Shearing Deformation A rectangular block of material with modulus of rigidity G = 90 ksi is bonded to two rigid horizontal plates. The lower plate is fixed, while the upper plate is subjected to a horizontal force P. Knowing that the upper plate moves through 0.04 in. under the action of the force, determine a) the average shearing strain in the material, and b) the force P exerted on the plate. SOLUTION: • Determine the average angular deformation or shearing strain of the block. • Use the definition of shearing stress to find the force P. • Apply Hooke’s law for shearing stress and strain to find the corresponding shearing stress. 15