SHAFTS: TORSION LOADING AND DEFORMATION


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1 ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220 Mehanis of Materials Department of Civil and Environmental Engineering University of Maryland, College Park ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 1 orsion oading Introdution Members subjeted to axial loads were disussed previously. he proedure for deriving loaddeformation relationship for axially loaded members was also illustrated. his hapter will present a similar treatment of members subjeted to torsion by loads that to twist the members about their longitudinal entroidal axes.
2 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 2 orsion oading orsional oads on Cirular Shafts Interested in stresses and strains of irular shafts subjeted to twisting ouples or torques urbine exerts torque on the shaft Shaft transmits the torque to the generator Generator reates an equal and opposite torque ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 3 orsion oading Net orque Due to Internal Stresses Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque, ( τ ) df da Although the net torque due to the shearing stresses is known, the distribution of the stresses is not Distribution of shearing stresses is statially indeterminate must onsider shaft deformations Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads an not be assumed uniform.
3 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 4 orsion oading Introdution Cylindrial members Fig. 1 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 5 orsion oading Introdution Retangular members Fig. 2
4 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 6 orsion oading Introdution his hapter deals with members in the form of onentri irular ylinders, solid and hollow, subjeted to torques about their longitudinal geometri axes. Although this may seem like a somewhat a speial ase, it is evident that many torquearrying engineering members are ylindrial in shape. ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 7 orsion oading Deformation of Cirular Shaft Consider the following shaft Fig. 3 a b d l l
5 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 8 orsion oading Deformation of Cirular Shaft In referene to the previous figure, the following observations an be noted: he distane l between the outside irumferential lines does not hange signifiantly as a result of the appliation of the torque. However, the retangles beome parallelograms whose sides have the same length as those of the original retangles. ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 9 orsion oading he irumferential lines do not beome zigzag; that is ; they remain in parallel planes. he original straight parallel longitudinal lines, suh as ab and d, remain parallel to eah other but do not remain parallel to the longitudinal axis of the member. hese lines beome helies*. *A helix is a path of a point that moves longitudinally and irumferentially along a surfae of a ylinder at uniform rate
6 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 10 orsion oading Deformation of Cirular Shaft Subjeted to orque Fig. 4 a b ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 11 orsion oading Deformation of a Bar of Square Cross Setion Subjeted to orque Fig. 5 a b
7 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 12 orsion oading Shaft Deformations From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length. φ φ When subjeted to torsion, every rosssetion of a irular shaft remains plane and undistorted. Crosssetions for hollow and solid irular shafts remain plain and undistorted beause a irular shaft is axisymmetri. Crosssetions of nonirular (nonaxisymmetri) shafts are distorted when subjeted to torsion. ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 13 orsion oading An Important Property of Cirular Shaft When a irular shaft is subjeted to torsion, every ross setion remains plane and disturbed In other words, while the various ross setions along the shaft rotate through different amounts, eah ross setion rotates as a solid rigid slab.
8 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 14 orsion oading An Important Property of Cirular Shaft his illustrated in Fig.4b, whih shows the deformation in rubber model subjeted to torsion. his property applies to irular shafts whether solid or hollow. It does not apply to nonirular ross setion. When a bar of square ross setion is subjeted to torsion, its various setions are warped and do not remain plane (see Fig. 5.b) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 15 orsion oading Fig. 6 Stresses in Cirular Shaft due to orsion Consider the following irular shaft that is subjeted to torsion B C A
9 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 16 orsion oading Stresses in Cirular Shaft due to orsion A setion perpendiular to the axis of the shaft an be passed at an arbitrary point C as shown in Fig. 6. he Freebody diagram of the portion BC of the shaft must inlude the elementary shearing fores df perpendiular to the radius of the shaft. ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 17 orsion oading Stresses in Cirular Shaft due to orsion B C Fig. 7 df τ da
10 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 18 orsion oading Stresses in Cirular Shaft due to orsion But the onditions of equilibrium for BC require that the system of these elementary fores be equivalent to an internal torque. Denoting the perpendiular distane from the fore df to axis of the shaft, and expressing that the sum of moments of ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 19 orsion oading Stresses in Cirular Shaft due to orsion of the shearing fores df about the axis of the shaft is equal in magnitude to the torque, we an write r df τ da area (1) area
11 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 20 orsion oading Stresses in Cirular Shaft due to orsion B C r τ da area (2) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 21 orsional Shearing Strain If a plane transverse setion before twisting remains plane after twisting and a diameter of the the setion remains straight, the distortion of the shaft of Figure 7 will be as shown in the following figures (Figs. 8 and 9):
12 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 22 orsional Shearing Strain 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 φ γ φ or γ Shear strain is proportional to twist and radius φ γ max and γ γ max ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 23 orsional Shearing Strain Shearing Strain φ Fig. 8
13 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 24 orsional Shearing Strain Shearing Strain γ a φ a a a aa tanγ sin beauseγ is very small a Fig. 9 γ a ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 25 orsional Shearing Strain Shearing Strain From Fig. 9, the length aá an be expressed as But herefore, a a tanγ γ aa φ φ γ φ γ (3) (4) (5)
14 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 26 orsional Shearing Strain Shearing Strain For radius, the shearing strain for irular shaft is γ φ For radius, the shearing strain for irular shaft is φ γ (6) (7) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 27 orsional Shearing Strain Shearing Strain Combining Eqs. 6 and 7, gives γ γ φ (8) herefore γ γ (9)
15 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 28 orsional Shearing Stress he Elasti orsion Formula If Hooke s law applies, the shearing stress τ is related to the shearing strain γ by the equation τ Gγ (10) where G modulus of rigidity. Combining Eqs. 9 and 10, results in τ G τ τ τ (11) G ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 29 orsional Shearing Stress he Elasti orsion Formula When Eq. 11 is substituted into Eq. 2, the results will be as follows: r 0 area τ τ da τ da area 2 da area τ da (12)
16 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 30 orsional Shearing Stress Polar Moment of Inertia he integral of equation 12 is alled the polar moment of inertia (polar seond moment of area). It is given the symbol J. For a solid irular shaft, the polar moment of inertia is given by π J da ( 2π d) (13) 2 0 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 31 orsional Shearing Stress Polar Moment of Inertia For a irular annulus as shown, the polar moment of inertia is given by b J 2 da b π 2 4 πb π 2 ( 2π d ) 4 4 ( r r ) r 0 outer radius and r i inner radius o i (14)
17 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 32 orsional Shearing Stress Shearing Stress in erms of orque and Polar Moment of Inertia In terms of the polar seond moment J, Eq. 12 an be written as τ τ J r A 2 d area Solving for shearing stress, τ J (16) (15) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 33 orsional Shearing Stress Shearing Stress in erms of orque and Polar Moment of Inertia τ τ max J J (17a) τ shearing stress, applied torque radius, and J polar moment on inertia (18a)
18 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 34 orsional Shearing Stress Distribution of Shearing Stress within the Cirular Cross Setion τ τ max τ τ τ max τ min J r o ri Fig. 10 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 35 Stresses in Elasti Range 4 J 1 π ( ) J 1 π Multiplying the previous equation by the shear modulus, G γ Gγ max From Hooke s aw, τ Gγ, so τ τ max he shearing stress varies linearly with the radial position in the setion. Reall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the setion, τ 2 da max τ τ da max J he results are known as the elasti torsion formulas, τ max and τ J J
19 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 36 orsional Shearing Stress Example 1 A hollow ylindrial steel shaft is 1.5 m long and has inner and outer diameters equal to 40 mm and 60 mm. (a) What is the largest torque whih may be applied to the shaft if the shearing stress is not to exeed 120 MPa? (b) What is the orresponding minimum value of the shearing stress in the shaft? ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 37 orsional Shearing Stress Example 1 (ont d) 60 mm 40 mm 1.2 m Fig. 11
20 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 38 orsional Shearing Stress Example 1 (ont d) (a) argest Permissible orque Using Eq.17a τ maxj τ max max J Using Eq.14 for laulating J, 4 [ ] 4 4 π 4 ( ) ( 0.03) ( 0.02) π J r o r i m (19) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 39 orsional Shearing Stress Example 1 (ont d) Substituting for J and τ max into Eq. 19, we have Jτ r o max 6 6 ( )( ) 4.05 kn m 0.03 (b) Minimum Shearing Stress r i J (0.02) τ 79.3 MPa
21 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 40 orsional Displaements Angle of wist in the Elasti Range Often, the amount of twist in a shaft is of importane. herefore, determination of angle of twist is a ommon problem for the mahine designer. he fundamental equations that govern the amount of twist were disussed previously ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 41 orsional Displaements Angle of wist in the Elasti Range he basi equations that govern angle of twist are Reall Eqs. 6, φ dθ γ or γ (20) d
22 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 42 orsional Displaements Angle of wist in the Elasti Range τ and J or τ (21) J G τ γ (22) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 43 orsional Displaements Angle of wist in the Elasti Range Reall Eq. 17a and 7 θ τ max γ max J Combining these two equations, gives γ max τ θ max G GJ 1 J G
23 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 44 orsional Displaements Angle of wist in the Elasti Range he angle of twist for a irular uniform shaft subjeted to external torque is given by GJ θ (22) ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 45 orsional Displaements Angle of wist in the Elasti Range Multiple orques/sizes he expression for the angle of twist of the previous equation may be used only if the shaft is homogeneous (onstant G) and has a uniform ross setional area A, and is loaded at its ends. If the shaft is loaded at other points, or if it onsists of several portions of various ross setions, and materials, then
24 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 46 orsional Displaements Angle of wist in the Elasti Range Multiple orques/sizes It needs to be divided into omponents whih satisfy individually the required onditions for appliation of the formula. Denoting respetively by i, i, J i, and G i, the internal torque, length, polar moment of area, and modulus of rigidity orresponding to omponent i,then n n i i θ θi (23) G J i 1 i 1 i i ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 47 orsional Displaements Multiple orques/sizes θ n i 1 θ i n i 1 i i G J i i E 1 E 2 E Fig. 12 Cirular Shafts
25 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 48 orsional Displaements Angle of wist in the Elasti Range he angle of twist of various parts of a shaft of uniform member an be given by n n i i θ θi (24) G J i 1 i 1 i i ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 49 Angle of wist in Elasti Range Reall that the angle of twist and maximum shearing strain are related, φ γ max In the elasti range, the shearing strain and shear are related by Hooke s aw, τ γ max max G JG Equating the expressions for shearing strain and solving for the angle of twist, φ JG If the torsional loading or shaft rosssetion hanges along the length, the angle of rotation is found as the sum of segment rotations i φ i J G i i i
26 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 50 orsional Displaements Angle of wist in the Elasti Range If the properties (, G, or J) of the shaft are funtions of the length of the shaft, then θ dx (25) GJ 0 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 51 orsional Displaements Angle of wist in the Elasti Range Varying Properties θ 0 GJ dx x Fig. 13
27 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 52 orsional Displaements Example 2 What torque should be applied to the end of the shaft of Example 1 to produe a twist of 2 0? Use the value G 80 GPa for the modulus of rigidity of steel. ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 53 orsional Displaements Example 2 (ont d) 60 mm 40 mm 1.2 m Fig. 14
28 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 54 orsional Displaements Example 2 (ont d) Solving Eq. 22 for, we get JG θ (26) Substituting the given values G Pa 0 2π rad θ m 3 rad ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 55 orsional Displaements Example 2 (ont d) From Example 1, J was omputed to give a value of m 4. herefore,using Eq. 26 JG θ ( )( ) ( ) 1.5 N m 1.9 kn m
29 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 56 orsional Displaements Example 3 What angle of twist will reate a shearing stress of 70 MPa on the inner surfae of the hollow steel shaft of Examples 1 and 2? 60 mm 40 mm 1.2 m Fig. 14 ECURE 6. SHAFS: ORSION OADING AND DEFORMAION ( ) Slide No. 57 orsional Displaements Example 3 (ont d) Jτ τ J φ GJ 6 6 ( )(70 10 ) kn m (1.5) ( ) o obtain θ in degrees, we write 360 θ π 0
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