6 Bars subjected to Torsion Let us now consider a straight bar supported at one end and acted upon by two pairs of equal and opposite forces. P1 P2 Then each pair of forces and form a couple that tend to twist the bar about its longitudinal axis, thus producing surface tractions and moments. Then we can write the moments as T 1= Pd 1 1 T2= Pd 2 2
7 Torsion of Circular Shafts In this chapter, we will examine uniaxial bars subject to torque. Where does this occur? Transmission Shaft Force Couples
8 Recall: External loads (T) produce internal loads which produce deformation, strain and stress. Torsion
9 Torsion of Circular Shafts cont d We assume Bar is in pure torsion Small rotations (the length and radius will not change) How does the bar deform? Cross-section of the bar remains the same shape, bar is simply rotating. Cross-section remains perpendicular to axis of cylinder (cylinder does not warp). Not true for most non-circular bars
10 Torsional Deformation of a Circular Shaft Torque is a moment that twists a member about its longitudinal axis. If the angle of rotation is small, the length of the shaft and its radius will remain unchanged. Chapter 5: Torsion
11 Angle of Twist Deformation of a circular shaft subjected to pure torsion Fix left end of shaft A moves to A φ = angle of twist (in radians) What are the boundary conditions on φ? x φ (x) = 0 at x = 0 φ (x) = φ at x = L For pure torsion, φ is linear.
12 Shearing Strain Calculate the surface shear strain in the cylinder. Consider an element of length dx. Recall we assume small φ & small γ. dx This equation applies to any function φ (x). For pure torsion φ (x) = φ x / L, so C C
13 Shearing Strain cont d Maximum shear strain on surface The maximum shear strain on the surface of the cylinder occurs when ρ=c. We can express the shearing strain at any distance from the axis of the shaft as
14 The Torsion Formula When material is linear-elastic, Hooke s law applies. A linear variation in shear strain leads to a corresponding linear variation in shear stress along any radial line on the cross section.
15 The Torsion Formula If the shaft has a solid circular cross section, If a shaft has a tubular cross section,
16 Stress Profiles: Shear stress profile YOU MUST UNDERSTAND THIS!!!! Where is shear stress max? zero? How does it vary along the length and circumference?
17 Stress Profiles:
18 Shearing Strain cont d We can also apply the equation for maximum surface shear strain to a hollow circular tube. c c This applies for all types of materials: elastic, linear, nonlinear, plastic, etc.
19 Elastic Shearing Stress Calculate shear stress in a bar made of linearly elastic material. Recall Hooke s Law for shearing stress: τ=gγ
20 Torque We still need to relate τ to the applied torque T, which is generally the known, applied load. First, find the resultant moment acting on a cross-section and set this equal to T. c
21 Torque cont d Continuing from previous slide: Where J is the polar moment of inertia of the cross section of the bar Plug this into the equation for τ max.
22 Torque cont d For a non-uniform bar n n φ = φ = T i L i i i=1 i =1 G i J i For a continuously varying bar
23 Inclined Plane Cut a rectangular element along the plane at an angle θ.
24 Inclined Plane cont d y x
25 Inclined Plane cont d τ max occurs at θ = 0º, ±90º σ max occurs at θ = ±45º τ max = σ max When σ θ is max, τ θ = 0, and when τ θ is max, σ θ =0.
26 Transmission Shafts In a transmission, a circular shaft transmits mechanical power from one device to another. Generator Turbine ω = angular speed of rotation of the shaft The shaft applies a torque T to another device To satisfy equilibrium the other device applies torque T to the shaft. The power transmitted by the shaft is
27 Transmission Shafts cont d Units for P=Tω ω = rad/s T = N m (SI) T = ft lb (English) P = Watts (1 W = 1 N m/s) (SI) P = ft lb/s (1 horsepower = hp = 550 ft lb/s) (English) We can also express power in terms of frequency.
28 Stress Concentrations in Circular Shafts Up to now, we assumed that transmission shafts are loaded at the ends through solidly attached, rigid end plates. In practice, torques are applied through flange couplings and fitted keyways, which produce high stress concentrations. Fitted keyway Flange coupling One way to reduce stress concentrations is through the use of a fillet.
29 Stress Concentrations in Circular Shafts cont d Maximum shear stress at the fillet τ max = K Tc J Fillet Tc/J is calculated for the smaller-diameter shaft K = stress concentration factor
30 Στρέψη περιορισμοί για άξονες κυκλικής διατομής Το μήκος του μέλους που υπόκειται σε στρέψη είναι σημαντικά μεγαλύτερο από τη μεγαλύτερη διάσταση της διατομής. Η περιοχή που εξετάζεται δεν περιέχει συγκεντρώσεις τάσεων. Η μεταβολή της εξωτερικής στρέψης ή η μεταβολή των εμβαδών των διατομών είναι βαθμιαία, με εξαίρεση τις περιοχές συγκέντρωσης τάσεων. ΟΙ εξωτερικές στρεπτικές ροπές δεν εξαρτώνται από το χρόνο, το πρόβλημα είναι στατικό. Η διατομή είναι κυκλική, επιτρέποντας τη χρήση της συμμετρίας ως προς τον άξονα για τη μείωση της παραμόρφωσης.
31 Στρέψη - Παραδοχές 1. Τα επίπεδα τμήματα που είναι κάθετα στον άξονα περιστροφής παραμένουν επίπεδα κατά τη διάρκεια της παραμόρφωσης. 2. Σε μια διατομή, όλες οι ακτινικές γραμμές περιστρέφονται κατά ίσες γωνίες κατά τη διάρκεια της παραμόρφωσης. 3. Οι ακτινικές γραμμές παραμένουν ευθείες κατά τη διάρκεια της παραμόρφωσης. 4. Οι παραμορφώσεις είναι μικρές. 5. Το υλικό είναι γραμμικά ελαστικό. 6. Το υλικό είναι ισότροπο. 7. Το υλικό είναι ομογενές κατά μήκος της περιοχής της διατομής.
35 ΠΑΡΑΔΟΧΕΣ Το υλικό είναι ομογενές μεταξύ x 1 και x 2 (G=σταθερό) Ο άξονας δεν λαμβάνει κωνοειδές σχήμα (J =σταθερή) Η εξωτερική (και επομένως και η εσωτερική) στρέψη δεν μεταβάλλεται με το x μεταξύ x 1 και x 2 (T = σταθερό)
36 NON-UNIFORM TORSION Uniform/Pure torsion torsion of prismatic bar subjected to torques acting only at the ends Non-uniform torsion the bar need not be prismatic and the applied torque may act anywhere along the axis of bar Non-uniform torsion can be analysed by Applying formula of pure torsion to finite segments of the bar then adding the results Applying formula to differential elements of the bar and then integrating
37 NON-UNIFORM TORSION CASE 1: Bar consisting of prismatic segments with constant torque throughout each segment φ = n n = i i φi i= 1 i= 1 Gi ( I p ) i T L CASE 2: Bar with continuously varying cross sections and constant torque φ L φ L = d = Tdx GI p ( x) 0 0
38 NON-UNIFORM TORSION CASE 3: Bar with continuously varying cross sections and continuously varying torque φ L L = dφ = T ( x) dx GI ( x) 0 0 p
39 NON-UNIFORM TORSION Limitations Analyses described valid for bar made of linearly elastic materials Circular cross sections (Solid /hollow) Stresses determined from the torsion formula valid in region of the bar away from stress concentrations (diameter changes abruptly/concentrated torque applied For the case above, Angle of twist still valid Changes in diameter is are small and gradually (angle of taper max 10o)
40 SOLID NON-CIRCULAR SHAFTS
41 Solid Noncircular Shafts The maximum shear stress and the angle of twist for solid noncircular shafts are tabulated as below:
42 Comparison of Axial and Torsion formulae. AE =Axial rigidity Axial Stiffness AE k A = L Axial Flexibilty: Axial displacement δ = i Axial stress σ = P A PL i i A E i i 1 f A = k A GJ =Torsional rigidity Torsional Stiffness GJ k T = L Torsional Flexibilty: Torsional displacement φ = i Ti Li J G Torsional stress τ = Tρ J i i 1 f T = k T
43 Sample Problem 1 SOLUTION: Cut sections through shafts AB and BC and perform static equilibrium analyses to find torque loadings. Shaft BC is hollow with inner and outer diameters of 90 mm and 120 mm, respectively. Shafts AB and CD are solid and 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. 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.
44 Sample Problem 1 SOLUTION: Cut sections through shafts AB and BC and perform static equilibrium analysis to find torque loadings. Fig. 1 Free-body diagram for section between A and B. Fig. 2 Free-body diagram for section between B and C. T M AB x = = 0 = ( 6kN m) T M = 0 = ( 6kN m) + ( 14kN m) 6kN m = T CD AB T BC x = 20kN m T BC
45 Sample Problem 1 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. J τ τ τ τ Fig. 3 Shearing stress distribution on cross section. π = π 4 4 ( c c ) = [( 0.060) ( 0.045) ] 2 1 = max min max min = τ 2 = 6 = 86.2MPa = = c c m T BC c J 64.7 MPa 2 4 = τmin 86.2MPa ( 20kN m)( m) = 45mm 60mm 6 m 4 τ τ max min = 86.2MPa = Fig. 4 Free-body diagram of shaft portion AB. τ max 64.7 MPa = Tc J = c = π 2 3 Tc c m 4 65MPa d 6kN m = 3 c π 2 = 2 c = 77.8mm
46 Statically Indeterminate Shafts (a) Shaft with central applied torque and fixed ends. (b) free-body diagram of shaft AB. (c) Free-body diagrams for solid and hollow segments. Given the shaft dimensions and the applied torque, we would like to find the torque reactions at A and B. From a free-body analysis of the shaft, T A + T B = 90lb ft which is not sufficient to find the end torques. The problem is statically indeterminate. Divide the shaft into two components which must have compatible deformations, TAL TBL φ = φ φ2 = = 0 TB = J G J G Substitute into the original equilibrium equation, T L 1 J2 T = 90lb ft L J A + A L1 J L J T A
47 Example 2 The tapered shaft is made of a material having a shear modulus G. Determine the angle of twist of its end B when subjected to the torque. Solution: From free body diagram, the internal torque is T. Thus, at x, For angle of twist,
48 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS Procedure for analysis: use both equilibrium and compatibility equations Equilibrium Draw a free-body diagram of the shaft in order to identify all the torques that act on it. Then write the equations of moment equilibrium about the axis of the shaft. Compatibility To write the compatibility equation, investigate the way the shaft will twist when subjected to the external loads, and give consideration as to how the supports constrain the shaft when it is twisted.
49 STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS (cont) Express the compatibility condition in terms of the rotational displacements caused by the reactive torques, and then use a torque-displacement relation, such as Φ = TL/JG, to relate the unknown torques to the unknown displacements. Solve the equilibrium and compatibility equations for the unknown reactive torques. If any of the magnitudes have a negative numerical value, it indicates that this torque acts in the opposite sense of direction to that indicated on the free-body diagram.
50 EXAMPLE 3 The solid steel shaft shown in Fig. has a diameter of 20 mm. If it is subjected to the two torques, determine the reactions at the fixed supports A and B.
51 EXAMPLE 3 (cont) Solution It is seen that the problem is statically indeterminate since there is only one available equation of equilibrium and there are 2 unknowns M T b x = T A = 0 (1) Since the ends of the shaft are fixed, the angle of twist of one end of the shaft with respect to the other must be zero. φ A / B = 0
52 EXAMPLE 3 (cont) Solution Using the sign convention established, Using Eq. 1, ( 0.2) ( 800 T )( 1.5) ( 300 T )( 0.3) TB + JG T = 645 N m B T A JG (Ans) The negative sign indicates that acts in the opposite direction of that shown in Fig. (b) B + = 345 N m B JG = 0
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