MECHANICS OF MATERIALS

Third Edition

CHAPTER

MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf

Torsion

Lecture Notes: J. Walt Oler Texas Tech University

© 2002 The McGraw-Hill Companies, Inc. All rights reserved.

Third Edition

MECHANICS OF MATERIALS

Beer • Johnston • DeWolf

Contents Introduction

Statically Indeterminate Shafts

Torsional Loads on Circular Shafts

Sample Problem 3.4

Net Torque Due to Internal Stresses

Design of Transmission Shafts

Axial Shear Components

Stress Concentrations

Shaft Deformations

Plastic Deformations

Shearing Strain

Elastoplastic Materials

Stresses in Elastic Range

Residual Stresses

Normal Stresses

Example 3.08/3.09

Torsional Failure Modes

Torsion of Noncircular Members

Sample Problem 3.1

Thin-Walled Hollow Shafts

Angle of Twist in Elastic Range

Example 3.10


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Torsional Loads on Circular Shafts • Interested in stresses and strains of circular shafts subjected to twisting couples or torques • Turbine exerts torque T on the shaft • Shaft transmits the torque to the generator • Generator creates an equal and opposite torque T’


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Net Torque Due to Internal Stresses • Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque, T = ∫ ρ dF = ∫ ρ (τ 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. © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Axial Shear Components • 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 components 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.


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Shaft Deformations • From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length. φ ∝T φ∝L

• When subjected to torsion, every cross-section 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. • Cross-sections of noncircular (nonaxisymmetric) shafts are distorted when subjected to torsion. © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Shearing Strain • 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 shear strain is equal to angle of twist. • It follows that Lγ = ρφ or γ =

ρφ L

• Shear strain is proportional to twist and radius γ max =


cφ ρ and γ = γ max L c

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Stresses in Elastic Range • Multiplying the previous equation by the shear modulus, Gγ =

ρ c

Gγ max

From Hooke’s Law, τ = Gγ , so τ=

ρ c

τ max

The shearing stress varies linearly with the radial position in the section.

J = 12 π c 4

• Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, τ τ T = ∫ ρτ dA = max ∫ ρ 2 dA = max J c c

(

J = 12 π c24 − c14

)

• The results are known as the elastic torsion formulas, τ max =


Tc Tρ and τ = J J 3-8

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Normal Stresses • Elements with faces parallel and perpendicular to the shaft axis are subjected to shear stresses only. Normal stresses, shearing stresses or a combination of both may be found for other orientations. • Consider an element at 45o to the shaft axis, F = 2(τ max A0 )cos 45 = τ max A0 2

σ

45o

=

F τ max A0 2 = = τ max A A0 2

• Element a is in pure shear. • 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 © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Torsional Failure Modes • 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 45o to the shaft axis. © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Sample Problem 3.1 SOLUTION: • Cut sections through shafts AB and BC and perform static equilibrium analysis 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 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. © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

• 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

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Sample SOLUTION:Problem 3.1 • Cut sections through shafts AB and BC and perform static equilibrium analysis to find torque loadings

∑ M x = 0 = (6 kN ⋅ m ) − TAB

∑ M x = 0 = (6 kN ⋅ m ) + (14 kN ⋅ m ) − TBC

TAB = 6 kN ⋅ m = TCD

TBC = 20 kN ⋅ m


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Sample Problem 3.1 • Apply elastic torsion formulas to find minimum and maximum stress on shaft BC

J=

• Given allowable shearing stress and applied torque, invert the elastic torsion formula to find the required diameter

π ( c24 − c14 ) = [(0.060 )4 − (0.045)4 ] 2 2

π

= 13.92 × 10− 6 m 4

τ max = τ 2 =

TBC c2 (20 kN ⋅ m )(0.060 m ) = J 13.92 × 10 − 6 m 4

τ max =

Tc Tc = J π c4 2

65MPa =

6 kN ⋅ m π c3 2

c = 38.9 × 10 −3 m d = 2c = 77.8 mm

= 86.2 MPa

τ min c1 = τ max c2

τ min 86.2 MPa

τ min = 64.7 MPa

=

45 mm 60 mm

τ max = 86.2 MPa τ min = 64.7 MPa


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Angle of Twist in Elastic Range • Recall that the angle of twist and maximum shearing strain are related, γ max =

cφ L

• In the elastic range, the shearing strain and shear are related by Hooke’s Law, γ max =

τ max G

=

Tc JG

• Equating the expressions for shearing strain and solving for the angle of twist, φ=

TL JG

• If the torsional loading or shaft cross-section changes along the length, the angle of rotation is found as the sum of segment rotations Ti Li i J i Gi

φ =∑


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Statically Indeterminate Shafts • 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, TA + TB = 90 lb ⋅ 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, φ = φ1 + φ2 =

TA L1 TB L2 − =0 J1G J 2G

LJ TB = 1 2 TA L2 J1

• Substitute into the original equilibrium equation, LJ TA + 1 2 TA = 90 lb ⋅ ft L2 J1


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Sample Problem 3.4 SOLUTION: • Apply a static equilibrium analysis on the two shafts to find a relationship between TCD and T0 • Apply a kinematic analysis to relate the angular rotations of the gears • Find the maximum allowable torque on each shaft – choose the smallest

Two solid steel shafts are connected by gears. Knowing that for each shaft • Find the corresponding angle of twist G = 11.2 x 106 psi and that the for each shaft and the net angular allowable shearing stress is 8 ksi, rotation of end A determine (a) the largest torque T0 that may be applied to the end of shaft AB, (b) the corresponding angle through which end A of shaft AB rotates. © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Sample Problem 3.4 SOLUTION: • Apply a static equilibrium analysis on the two shafts to find a relationship between TCD and T0

∑ M B = 0 = F (0.875 in.) − T0

• Apply a kinematic analysis to relate the angular rotations of the gears

rBφ B = rCφC rC 2.45 in. φC = φC rB 0.875 in.

∑ M C = 0 = F (2.45 in.) − TCD

φB =

TCD = 2.8 T0

φ B = 2 .8 φ C


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Sample Problem 3.4 • Find the T0 for the maximum • Find the corresponding angle of twist for each allowable torque on each shaft – shaft and the net angular rotation of end A choose the smallest

φA / B =

τ max =

TAB c T (0.375 in.) 8000 psi = 0 π (0.375 in.)4 J AB 2 TCD c 2.8 T0 (0.5 in.) 8000 psi = π (0.5 in.)4 J CD 2

T0 = 561lb ⋅ in.

T0 = 561lb ⋅ in

(

)

= 0.387 rad = 2.22o

φC / D =

T0 = 663 lb ⋅ in.

τ max =

(561lb ⋅ in.)(24in.) TAB L = J ABG π (0.375 in.)4 11.2 × 106 psi 2 2.8 (561lb ⋅ in.)(24in.) TCD L = J CDG π (0.5 in.)4 11.2 × 106 psi 2

(

= 0.514 rad = 2.95o

(

)

)

φ B = 2.8φC = 2.8 2.95o = 8.26o φ A = φ B + φ A / B = 8.26o + 2.22o


φ A = 10.48o 3 - 18

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Design of Transmission Shafts • Principal transmission shaft performance specifications are: - power - speed • Designer must select shaft material and cross-section to meet performance specifications without exceeding allowable shearing stress.

• Determine torque applied to shaft at specified power and speed, P = Tω = 2πfT T=

P

ω

=

P 2πf

• Find shaft cross-section which will not exceed the maximum allowable shearing stress, τ max =

Tc J

T J π 3 = c = τ max c 2

(

(solid shafts )

)

T π 4 4 J c2 − c1 = = τ max c2 2c2


(hollow shafts )

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Stress Concentrations • The derivation of the torsion formula, τ max =

Tc J

assumed a circular shaft with uniform cross-section 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 or numerically determined concentration factors are applied as τ max = K


Tc J

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Plastic Deformations • With the assumption of a linearly elastic material, τ max =

Tc J

• If the yield strength is exceeded or the material has a nonlinear shearing-stress-strain curve, this expression does not hold. • Shearing strain varies linearly regardless of material properties. Application of shearing-stress-strain curve allows determination of stress distribution. • The integral of the moments from the internal stress distribution is equal to the torque on the shaft at the section, c

c

0

0

T = ∫ ρτ (2πρ dρ ) = 2π ∫ ρ 2τ dρ


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Elastoplastic Materials • At the maximum elastic torque, TY =

J τ Y = 12 πc3τ Y c

φY =

Lγ Y c

• As the torque is increased, a plastic region ρ τ = τY ) τ = τ ( Y ) develops around an elastic core ( ρY =

ρY

Lγ Y

φ

⎛

ρY3 ⎞⎟

T=

2 πc 3τ ⎜1 − 1 Y⎜ 3 4

T=

3⎞ ⎛ 4 T ⎜1 − 1 φY ⎟ 3 Y⎜ 4 3⎟

⎝

⎝

c ⎟⎠ 3

=

⎛

4 T ⎜1 − 1 3 Y⎜ 4

⎝

ρY3 ⎞⎟ c3 ⎟⎠

φ ⎠

• As ρY → 0, the torque approaches a limiting value, TP = 43 TY = plastic torque


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Residual Stresses • Plastic region develops in a shaft when subjected to a large enough torque • When the torque is removed, the reduction of stress and strain at each point takes place along a straight line to a generally non-zero residual stress • On a T-φ curve, the shaft unloads along a straight line to an angle greater than zero • Residual stresses found from principle of superposition

Tc ′ = τm J © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

∫ ρ (τ dA) = 0 3 - 23

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Example 3.08/3.09 SOLUTION: • Solve Eq. (3.32) for ρY/c and evaluate the elastic core radius • Solve Eq. (3.36) for the angle of twist A solid circular shaft is subjected to a torque T = 4.6 kN ⋅ m at each end. Assuming that the shaft is made of an elastoplastic material with τ Y = 150 MPa and G = 77 GPa determine (a) the radius of the elastic core, (b) the angle of twist of the shaft. When the torque is removed, determine (c) the permanent twist, (d) the distribution of residual stresses.


• Evaluate Eq. (3.16) for the angle which the shaft untwists when the torque is removed. The permanent twist is the difference between the angles of twist and untwist • Find the residual stress distribution by a superposition of the stress due to twisting and untwisting the shaft

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Example SOLUTION: 3.08/3.09 • Solve Eq. (3.32) for ρY/c and evaluate the elastic core radius ⎛ 1 ρY3 ⎞ 4 T = 3 TY ⎜1 − 4 3 ⎟ ⇒ ⎜ c ⎟⎠ ⎝ J=

1 πc 4 2

=

1π 2

ρY

⎛ T = ⎜⎜ 4 − 3 c TY ⎝

(25 ×10 m)

• Solve Eq. (3.36) for the angle of twist ⎞ ⎟⎟ ⎠

1

3

−3

= 614 × 10−9 m 4

τY =

TY c J

τ J ⇒ TY = Y c

( 150 × 106 Pa )(614 × 10−9 m 4 ) TY = 25 × 10

−3

m

φ ρ = Y c φY

⇒ φ=

φY ρY c

(

)

TY L 3.68 × 103 N (1.2 m ) = φY = JG 614 × 10-9 m 4 (77 × 10 Pa )

(

)

φY = 93.4 ×10−3 rad 93.4 × 10−3 rad = 148.3 × 10−3 rad = 8.50o φ= 0.630

φ = 8.50o

= 3.68 kN ⋅ m

ρY

4.6 ⎞ ⎛ = ⎜4 −3 ⎟ c 3.68 ⎠ ⎝

1

3

= 0.630

ρY = 15.8 mm © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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Example 3.08/3.09 • Evaluate Eq. (3.16) for the angle which the shaft untwists when the torque is removed. The permanent twist is the difference between the angles of twist and untwist φ′ =

• Find the residual stress distribution by a superposition of the stress due to twisting and untwisting the shaft

(

)(

Tc 4.6 × 103 N ⋅ m 25 × 10−3 m ′ = = τ max J 614 × 10-9 m 4

)

= 187.3 MPa

TL JG

( 4.6 × 103 N ⋅ m )(1.2 m ) = (6.14 ×109 m4 )(77 ×109 Pa ) = 116.8 × 10 −3 rad φp = φ − φ′

(

)

= 116.8 × 10−3 − 116.8 × 10−3 rad = 1.81o

φ p = 1.81o


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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 • For uniform rectangular cross-sections, τ max =

T c1ab 2

φ=

TL c2 ab3G

• At large values of a/b, the maximum shear stress and angle of twist for other open sections are the same as a rectangular bar.


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Thin-Walled Hollow Shafts • Summing forces in the x-direction on AB, ∑ Fx = 0 = τ A (t A∆x ) − τ B (t B ∆x ) τ At A= τ Bt B = τ t = q = shear flow

shear stress varies inversely with thickness • Compute the shaft torque from the integral of the moments due to shear stress dM 0 = p dF = pτ (t ds ) = q( pds ) = 2q dA

T = ∫ dM 0 = ∫ 2q dA = 2qA

τ=

T 2tA

• Angle of twist (from Chapt 11) φ=


TL

ds

∫ 4 A 2G t

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Example 3.10 Extruded aluminum tubing with a rectangular cross-section has a torque loading of 24 kipin. Determine the shearing stress in each of the four walls with (a) uniform wall thickness of 0.160 in. and wall thicknesses of (b) 0.120 in. on AB and CD and 0.200 in. on CD and BD. SOLUTION: • Determine the shear flow through the tubing walls • Find the corresponding shearing stress with each wall thickness


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Example 3.10 SOLUTION: • Determine the shear flow through the tubing walls

• Find the corresponding shearing stress with each wall thickness

with a uniform wall thickness, τ=

q 1.335 kip in. = 0.160 in. t

τ = 8.34 ksi

with a variable wall thickness A = (3.84 in.)(2.34 in.) = 8.986 in.2 q=

T kip 24 kip - in. 1 . 335 = = in. 2 A 2 8.986 in.2

(

)

τ AB = τ AC =

1.335 kip in. 0.120 in.

τ AB = τ BC = 11.13 ksi τ BD = τ CD =

1.335 kip in. 0.200 in.

τ BC = τ CD = 6.68 ksi © 2002 The McGraw-Hill Companies, Inc. All rights reserved.

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