Torsion.

Transcription

1 Torsion

2 Contents Introduction to Torsion( 扭转简介 ) Examples of Torsion Shafts( 扭转轴示例 ) Sign Convention of Torque( 扭矩符号规则 ) Torque Diagram( 扭矩图 ) Power & Torque( 功率与扭矩 ) Internal Torque & Stress - Static Indeterminacy( 内力扭矩和应力 - 超静定概念的引入 ) General Relations Involved in Deformable Solids( 分析可变形固体的几大基本关系 ) Kinematics( 几何关系 ) Hooke s Law for Shearing Deformation( 剪切变形物理关系 ) Static Equivalency( 静力等效关系 ) Torsional Stress & Angle of Twist( 圆轴扭转的应力与扭转角 ) 2

3 Contents Nonuniform Torsion of Circular Shafts( 圆轴的非均匀扭转 ) Strength & Stiffness Analysis( 强度和刚度条件 ) Strain Energy and its Density( 扭转应变能与应变能密度 ) Polar Moments of Inertia & Section Modulus( 圆截面的极惯性矩与扭转截面系数 ) Theorem of Conjugate Shearing Stress( 切应力互等定理 ) Stresses on Oblique Cross Sections( 扭转轴斜截面上的应力 ) Failure Modes of Torsional Shafts( 扭转轴的失效模式 ) Stress Concentrations( 扭转轴的应力集中 ) Torsion of Noncircular Members( 非圆截面杆的扭转 ) Membrane Analogy( 薄膜比拟 ) Torsion of Thin-walled Open Shafts( 薄壁开口截面杆的扭转 ) Torsion of Thin-walled Hollow Shafts( 薄壁空心截面杆的扭转 ) 3

4 Introduction to Torsion Turbine exerts torque T on the shaft Shaft transmits the torque to the generator Generator creates an equal and opposite torque T Cross section remains planar Interested in stresses and strains of circular shafts subjected to twisting couples or torques 4

5 Introduction to Torsion 5

6 Examples of Torsion Shafts Power generation shaft Automotive power train shaft Ship drive shaft Complex crank shaft Tire shift drive Screwdriver 6

7 Sign Convention of Torque Sign (Positive) convention (right-hand rule): thumb crosssection normal; rest fingers torque 7

8 Torque Diagram Abscissa: cross-section position Ordinate: torque 8

9 Power & Torque Transformation from electric power to mechanical power motor power: P (kw); generated torque: M e (N m): n (rpm, revolutions per minute): P M e2πn P M e π n n (rps, revolutions per second): 1000 P P M e2πn P1000 M e 159 2π n n ω (rad/s, radian per second): P M e P1000 M e 1000 P n 9

10 Internal Torque & Stress - Static Indeterminacy Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque, T 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 statically indeterminate must consider shaft deformations. Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform. 10

11 General Relations Involved in Deformable Solids 11

12 Kinematics 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. 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. 12

13 Kinematics Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus. Since the ends of the element remain planar, the shearing strain may be related to the angle of twist. It follows that L or L Shearing strain is proportional to the angle of twist and radius c max and L c max 13

14 Hooke s Law for Shearing Deformation 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 f xy 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 xy G xy yz G yz zx G where G is the modulus of rigidity or shear modulus. zx 14

15 A Simple Example of Shearing Deformation SOLUTION: Determine the average angular deformation or shearing strain of the block. 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. Apply Hooke s law for shearing stress and strain to find the corresponding shearing stress. Use the definition of shearing stress to find the force P. 15

16 Determine the average angular deformation or shearing strain of the block. xy tan xy 0.04in. 2in. xy 0.020rad Apply Hooke s law for shearing stress and strain to find the corresponding shearing stress. 3 xy G 9010 psi 0.020rad 1800psi xy Use the definition of shearing stress to find the force P. P A xy psi8in. 2.5in lb P 36.0kips 16

17 Static Equivalency c Multiplying the previous equation by the shear modulus, G G max c From Hooke s Law, G, so max c The shearing stress varies linearly with the radial position in the section. Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, c c max 2 max T da da I p The results are known as the elastic torsion formulas, (I p Polar moment of inertia) Tc I max and p T I p 17

18 Torsional Stress & Angle of Twist Torsional section modulus: [m 3 ]: Torsional stress W I c. Tc T T T max, I W I W c Angle of twist per unit length p p p p p T T TL L L G G I GI GI GI p : Torsional rigidity [N m 2 ] P p p p The equations are limited to bars of circular cross section (either solid or hollow) that behave in a linearly elastic manner. 18

19 Nonuniform Torsion of Circular Shafts Pure torsion formula refers to torsion of a prismatic bar subjected to torques acting only at the ends. Nonuniform torsion differs from pure torsion in that the bar need not to be prismatic and the applied torques may act anywhere along the axis of the bar. Bars in nonuniform torsion can be analyzed by applying the formulas of pure torsion to finite segments of the bar and then adding the results, or by applying the formulas to differential elements of the bar and then integrating. TL i i i i i Gi Ip i L d 0 0 L x T x dx GI p 19

20 Strength Analysis Strength check: max τ TW τ P max Cross-section design: W T p max [] Maximum allowable load: Tmax W p [ ] 20

21 Stiffness Condition Angle of twist per unit length: Stiffness condition: max L T GI p []: allowable angle of twist per unit length [radian/m] or [degrees/m] NC machine: [] = degrees/m Ordinary shaft: [] = degrees/m Shaft of drilling machine: [] = degrees/m Stiffness check; Cross-section design; Allowable load. 21

22 Strain Energy Strain energy and work in pure torsion 2 1 T L GI p 2 TL U W T 2 2GI p 2L GI p Nonuniform torsion Prismatic bar in pure shear U 2 i i Ui i i 2Gi Ip i T L U L du L T x GI 2 x p dx Torque-rotation diagram 22

23 Strain Energy Density Consider a differential cube of side h subjected to shearing stresses on its sides U W V h h h U u G 3 h 2 2 2G Total strain energy in torsion U udv V udadx dadx L A L A 2G T T 2 T dx dadx da dx L A 2 2G I L 2 A L p GI p 2GI p 23

24 Polar Moment of Inertia & Section Modulus Solid circular shafts da 2πd I p d 2 A 2 da 2 2πd 0 πd 32 4 dρ ρ d τ W p I p πd 2 πd r 32 d

25 Hollow circular shafts da I p A π 1 16 p p D I W D π D d d 4 4 π 32 D d 4 4 π 1 32 D d D d ρ dρ D τ 25 Polar Moment of Inertia & Section Modulus

26 Polar Moment of Inertia & Section Modulus Thin-walled tubes: (δ << r 0 ) D 2 r, d 2 r, A 2πr δ r 0 D D 2 2 I da r da 2πr p W p d d 2 2 I r p 0 2πr 2 0 d D Torsional stress in thin-walled tubes can be simplified to T W A p T 2 r 2 0 T 2 r d 0 d r0 2 r0 2 T r A r A A 0 26

27 Sample Problem SOLUTION: Cut sections through shafts AB and BC and perform static equilibrium analysis to find torque loadings Apply elastic torsion formulas to find minimum and maximum stress on shaft BC Shaft 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. Given allowable shearing stress and applied torque, invert the elastic torsion formula to find the required diameter 27

28 SOLUTION: Cut sections through shafts AB and BC and perform static equilibrium analysis to find torque loadings T AB T 0 6kN m T 6kN m T CD AB T BC T 0 6kN m 14kN m T 20kN m BC 28

29 Apply elastic torsion formulas to find minimum and maximum stress on shaft BC Given allowable shearing stress and applied torque, invert the elastic torsion formula to find the required diameter I p c c m kN m0.060m T c BC 2 max I p m min max min 86.2 MPa c c MPa min 86.2MPa 45mm 60mm Tc Tc 65MPa max 4 3 I p 2c 2c c m d 2c 77.8 mm 6kN m 29

30 Sample Problem The horizontal shaft AD is attached to a fixed base at D and is subjected to the torques shown. A 44-mmdiameter hole has been drilled into portion CD of the shaft. Knowing that the entire shaft is made of steel for which G = 77 GPa, determine the angle of twist at end A. SOLUTION: 1 TABLAB T L T L G I I I BC BC CD CD i pab pbc pcd G i rad

31 Sample Problem SOLUTION: Apply a static equilibrium analysis on the two shafts to find a relationship between T CD and T 0 Apply a kinematic analysis to relate the angular rotations of the gears Two solid steel shafts are connected by gears. Knowing that for each shaft G = 11.2 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 corresponding angle through which end A of shaft AB rotates. Find the maximum allowable torque on each shaft choose the smallest Find the corresponding angle of twist for each shaft and the net angular rotation of end A 31

32 Apply a static equilibrium analysis on the two shafts to find a relationship between T CD and T 0 Apply a kinematic analysis to relate the angular rotations of the gears T M M CD B C T 0 F F 0.875in. 2.45in. T T 0 CD r B B B B r r C B r C C 2.8 C C 2.45in in. C 32

33 Find the T 0 for the maximum allowable torque on each shaft Find the corresponding angle of twist for each shaft and the net angular rotation of end A T T AB 0 max 4 I pab in. 0 CD 0 max 4 I pcd 2 0.5in. 0 T c 663lb in. T c 561lb in. T 8000psi 8000psi 0.375in. 2.8T 0.5in. 561lb in. 24 in. AB AB / 4 6 GI pab rad in psi in CD C/ D 4 6 GI pcd 2 B T T L L rad in psi o o 2.8C A B A/ B o lb in o o o o 33

34 Sample Problem Determine the angle of twist at end A of the shaft. Solution x dx d A db d A L d x I d d d L L Tdx A 0 GI 4 x px A B A A px L Tdx 0 x G d A db d A 32 L 32T L G 3dB d A d A db 32TL G db d A d A db

35 Sample Problem Determine the angle of twist at cross-section B of the shaft. Solution L L T tx tl 2 dx dx B GI GI 2GI 0 p 0 p p 35

36 Sample Problem Given: n = 500 rpm, input gear A (P 1 = 400 kw), output gears C and B (P 2 = 160 kw, P 3 = 240 kw), [τ] = 70 MPa, [ ] = 1 degree/m, G = 80 GPa. Find (1) d 1 and d 2 ; (2) d if d AC = d BC is required; (3) How to arrange the input and output gears to make the transmission more reasonable? A M e1 d1 d2 C B M e2 M e 3 36

37 Solution: 1. Torque diagram M M M e1 P n N m 160 M 3060 N m 400 e2 e1 240 M 4580 N m 400 e3 e1 A M e1 d1 d2 7640N m C 4580N m B M e2 M e 3 37

38 2. Find d 1 From strength condition: T max 3 πd T d1 6 π[ ] π7010 From stiffness condition: max 4 πd1 T 180 G π T m 82.2 mm d Gπ [ ] 8010 π 1 Take d 1 = 86.4 mm m 86.4 mm 38

39 3. Find d 2 From strength condition: d T π[ ] π m 69.3mm From stiffness condition: d T Gπ [ ] 8010 π 1 Take d 2 = 76mm m 76mm 4. If d = d 1 = d 2 is required, take the maximum value: d d mm 39

40 A 5. Rearrangement of gears to make the transmission more efficient. Put the input gear in between two output gears. M e1 d1 d2 C B M e2 M e N m 4580N m C M e2 d1 d2 A B M e1 M e N m 3060 N m Smaller maximum load carried by the system (4580/7640=60%) 40

41 Theorem of Conjugate Shearing Stress Torque applied to shaft produces shearing stresses on the faces perpendicular to the axis. Conditions of equilibrium require the existence of equal stresses on the faces of the two planes containing the axis of the shaft. The existence of the axial shear is demonstrated by considering a shaft made up of axial slats. The slats slide with respect to each other when equal and opposite torques are applied to the ends of the shaft. 41

42 Stresses on Oblique Cross Sections Elements with faces parallel and perpendicular to the shaft axis are subjected to shearing stresses only. Normal stresses, shearing stresses or a combination of both may be found for other orientations. Consider an element at 45 0 to the shaft axis, F 2 A cos 45 A 2 max 0 max 0 F A 2 o 45 A A0 2 Element a is in pure shear. max 0 Element c is subjected to a tensile stress on two faces and compressive stress on the other two. Note that all stresses for elements a and c have the same magnitude What about the normal and shearing stress for an arbitrarily oriented element? max 42

43 Failure Modes of Torsional Shafts 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 45 o to the shaft axis. 43

44 Stress Concentrations The derivation of the torsion formula assumed a circular shaft with uniform crosssection loaded through rigid end plates. The use of flange couplings, gears and pulleys attached to shafts by keys in keyways, and cross-section discontinuities can cause stress concentrations Experimental determined concentration factors are applied as max K Tc I p 44

45 Torsion of Noncircular Members Previous torsion formulas are valid for axisymmetric or circular shafts Planar cross-sections of noncircular shafts do not remain planar and stress and strain distribution do not vary linearly The mathematical theory of elasticity for straight bars with a uniform rectangular cross section predicts that [S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd ed., McGraw- Hill, New York, 1969, Article 109]: T TL c ab G max 2 3 c1ab 2 45

46 Membrane Analogy The determination of the deformation of the membrane depends upon the solution of the same partial differential equation as the determination of the shearing stresses in the bar. The shearing stress at Q will have the same direction as the horizontal tangent to the membrane at Q, and its magnitude will be proportional to the maximum slope of the membrane at Q. The applied torque will be proportional to the volume between the membrane and the plane of the fixed frame. 46

47 Torsion of Thin-Walled Open Shafts Using the membrane analogy to help us visualize the shearing stresses, we note that, if the same torque is applied to each member, the same volume will be located under each membrane, and the maximum slope will be about the same in each case. Thus, for a thin-walled member of uniform thickness and arbitrary shape, the shearing stress is the same as for a rectangular bar with a very large value of a/b and may be determined from T TL max ab 0.333ab G 47

48 Torsion of Thin-Walled Hollow Shafts Summing forces in the x-direction on AB F x 0 A t x t x At A Bt B t q shear flow Shearing stress varies inversely with t. A B B da A Compute the shaft torque from the integral of the moments due to shearing stress 0 0 dm p df p t ds q pds 2q da T dm 2q da 2 qa T 2tA where A is the area bounded by the center line of the wall cross section. 48

49 Torsion of Thin-Walled Hollow Shafts Strain energy density 2 2 T T 2 ta u 2G 8Gt A 2 2 A Strain energy and work done by the torque 2 L L T U u tdsdx 2 2 tdsdx 0 0 8Gt A 0 L 2 2 T ds T L ds dx 2 2 GA t GA t 8 8 Angle of twist by energy method U 1 TL 2 4GA T 2 ds t 49

50 Sample Problem Extruded aluminum tubing with a rectangular cross-section has a torque loading of 24 kip-in. Determine the shearing stress in each of the four walls with (a) uniform wall thickness of 0.16 in. and wall thicknesses of (b) 0.12 in. on AB and CD and 0.2 in. on CD and BD. SOLUTION: Determine the shear flow through the tubing walls Find the corresponding shearing stress with each wall thickness 50

51 SOLUTION: Determine the shear flow T q 2A 24 kip-in in. 2.34in. kip in. Find the corresponding shearing stress with each wall thickness (a) with a uniform wall thickness, q 1.335kip in ksi t 0.160in. (b) with a variable wall thickness 1.335kip in kip in. AB AC 11.13ksi, BD CD 6.68ksi 0.120in in. 51

52 Contents Introduction to Torsion( 扭转简介 ) Examples of Torsion Shafts( 扭转轴示例 ) Sign Convention of Torque( 扭矩符号规则 ) Torque Diagram( 扭矩图 ) Power & Torque( 功率与扭矩 ) Internal Torque & Stress - Static Indeterminacy( 内力扭矩和应力 - 超静定概念的引入 ) General Relations Involved in Deformable Solids( 分析可变形固体的几大基本关系 ) Kinematics( 几何关系 ) Hooke s Law for Shearing Deformation( 剪切变形物理关系 ) Static Equivalency( 静力等效关系 ) Torsional Stress & Angle of Twist( 圆轴扭转的应力与扭转角 ) 52

53 Contents Nonuniform Torsion of Circular Shafts( 圆轴的非均匀扭转 ) Strength & Stiffness Analysis( 强度和刚度条件 ) Strain Energy and its Density( 扭转应变能与应变能密度 ) Polar Moments of Inertia & Section Modulus( 圆截面的极惯性矩与扭转截面系数 ) Theorem of Conjugate Shearing Stress( 切应力互等定理 ) Stresses on Oblique Cross Sections( 扭转轴斜截面上的应力 ) Failure Modes of Torsional Shafts( 扭转轴的失效模式 ) Stress Concentrations( 扭转轴的应力集中 ) Torsion of Noncircular Members( 非圆截面杆的扭转 ) Membrane Analogy( 薄膜比拟 ) Torsion of Thin-walled Open Shafts( 薄壁开口截面杆的扭转 ) Torsion of Thin-walled Hollow Shafts( 薄壁空心截面杆的扭转 ) 53