Torsion. Torsion is a moment that twists/deforms a member about its longitudinal axis
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1 Mehanis of Solids I Torsion Torsional loads on Cirular Shafts Torsion is a moment that twists/deforms a member about its longitudinal axis 1
2 Shearing Stresses due to Torque o Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque, ( ) T ρdf ρ da o Torque applied to shaft produes shearing stresses on the faes perpendiular to the axis. o Conditions of equilibrium require the existene of equal stresses on the faes of the two planes ontaining the axis of the shaft Shaft Deformations o From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length. φ T φ L o Cross-setions for hollow and solid irular shafts remain plain and undistorted beause a irular shaft is axisymmetri. o Cross-setions of nonirular (nonaxisymmetri) shafts are distorted when subjeted to torsion.
3 Shearing Strain Consider an interior setion of the shaft. As a torsional load is applied, an element on the interior ylinder deforms into a rhombus. Sine the ends of the element remain planar, the shear strain is equal to angle of twist. It follows that Lγ ρφ or ρφ γ L Shear strain is proportional to twist and radius φ ρ γ and γ γ L Stresses in Elasti Range da πρ dρ 4 1 π 4 4 ( ) 1 π 1 o Multiplying the previous equation by the shear modulus, ρ Gγ Gγ ρ From Hooke s Law, Gγ, so The shearing stress varies linearly with the radial position in the setion. o The sum of the moments from the internal stress distribution is equal to the torque on the shaft at the setion, T da da ρ ρ polar moment of inertia T 3
4 Torsion formula o The elasti torsion formula, T and Tρρ where imum shear stress, shear stress, T internal torque at the setion outer radius of the shaft polar moment of inertia ρ distane from enter of the ross setion o For solid shaft o For tubular shaft 1 4 π 4 4 π ( 1 ) 1 Shear Stress Distribution 4
5 Normal Stresses o Elements with faes parallel and perpendiular to the shaft axis are subjeted to shear stresses only. Normal stresses, shearing stresses or a ombination of both may be found for other orientations. o Consider an element at 45 o to the shaft axis, ( ) F A0 os45 A0 F A0 σ o 45 A A 0 o Element a is in pure shear. o Element is subjeted to a tensile stress on two faes and ompressive stress on the other two. Note that all stresses for elements a and have the same magnitude Torsional Failure Modes o Dutile materials generally fail in shear. Brittle materials are weaker in tension than shear. o When subjeted to torsion, a dutile speimen breaks along a plane of imum shear, i.e., a plane perpendiular to the shaft axis. o When subjeted to torsion, a brittle speimen breaks along planes perpendiular to the diretion in whih tension is a imum, i.e., along surfaes at 45 o to the shaft axis. 5
6 Proedure for Analysis Find absolute imum torsional stress o Need to find loation where ratio T/ is imum o Draw a torque diagram (internal torque vs. x along shaft) o Sign Convention: T is positive, by right-hand rule, is direted outward from the shaft o One internal torque throughout shaft is determined, imum ratio of T/ an be identified Sample Problem 3.1 Shaft BC is hollow with inner and outer diameters of 90 mm and 10 mm, respetively. Shafts AB and CD are solid of diameter d. For the loading shown, determine (a) the minimum and imum 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. 6
7 Angle of Twist in Elasti Range Reall that the angle of twist and imum shearing strain are related, φ γ L In the elasti range, the shearing strain and shear are related by Hooke s Law, T γ G G Equating the expressions for shearing strain and solving for the angle of twist, φ G If the torsional loading or shaft ross-setion hanges along the length, the angle of rotation is found as the sum of segment rotations i i φ i G i i Angle of Twist in Elasti Range Reall that the angle of twist and imum shearing strain are related, φ γ L In the elasti range, the shearing strain and shear are related by Hooke s Law, T γ G G Equating the expressions for shearing strain and solving for the angle of twist, φ G If the torsional loading or shaft ross-setion hanges along the length, the angle of rotation is found as the sum of segment rotations i i φ i G i i 7
8 Sign Conventions for T and φ Problem 3. Shaft BC is hollow with inner and outer diameters of 90 mm and 10 mm, respetively. Shafts AB and CD are solid of diameter 80 mm. Find angle of twist of B relative to A (φ B/A ), φ C/B and φ D if point A is fixed. Given shear modulus G77Gpa. 8
9 Statially Indeterminate Shafts o Given the shaft dimensions and the applied torque, we would like to find the torque reations at A and B. o From a free-body analysis of the shaft, T + T 90lb ft A B whih is not suffiient to find the end torques. The problem is statially indeterminate. o Divide the shaft into two omponents whih must have ompatible deformations, A 1 B L 1 TB TA G 1 G L1 o Substitute into the original equilibrium equation, L T T 90lb ft 1 A + A L 1 Sample Problem 3.4 Two solid steel shafts are onneted by gears. Knowing that for eah shaft G 11. x 10 6 psi and that the allowable shearing stress is 8 ksi, determine (a) the largest torque T 0 that may be applied to the end of shaft AB, (b) the orresponding angle through whih end A of shaft AB rotates. 9
10 Design of Transmission Shafts o Prinipal transmission shaft performane speifiations are: Power,P Speed,ω o Designer must selet shaft material and ross-setion to meet performane speifiations without exeeding allowable shearing stress. o Determine torque applied to shaft at speified power and speed, P Tω π ft P P T ω π f o Find shaft ross-setion whih will not exeed the imum allowable shearing stress, T π T 3 π T ( solid shafts) 4 4 ( 1 ) ( hollow shafts) Stress Conentrations o The derivation of the torsion formula, T assumed a irular shaft with uniform ross-setion loaded through rigid end plates o The use of flange ouplings, gears and pulleys attahed to shafts by keys in keyways, and ross-setion setion disontinuities an ause stress onentrations o Experimental or numerially determined onentration fators are applied as T K 10
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