Torsion Part 1

1

http://videos.howstuffworks.com/auto/car-

parts-videos-playlist.htm#video-30400

Machines that rely on torsion to function

Torque Wrench

Smart Car

Maserati MC12

Motor

Steam Turbine

Gearbox

Courtesy of Shawn Sheng NREL GRC

Wind

Power

GJ

TL

J

T

=

=

φ

ρτShear Stress,

Angle of Twist,

OBJECTIVES: You will be able to effectively apply the

torsion formulas to calculate stresses and deformation.

1. You will know what each variable represents

2. You will understand and apply the sign convention

for torque, shear stress, and shear strain

3. You will know the limitations of the formulas so that

you can properly analyze shafts subjected to torque

Torsion Part 1

2

EQUATION DEVELOPMENT PROCEDURE:

1. Geometry of Deformation

2. Internal Loads

3. Material Relations

For a circular shaft, plane sections remain plane

For a rectangular shaft, these cross sectional planes warp

Imagine a shaft composed of a series of rigid disks, each disk

slips relative to its neighbor to produce deformation

circular shaft rectangular shaft rigid disks

LIMITATION

1. Geometry of Deformation

Arc length, BB’ = ρφ

BB’

ρ φ

Torsion Part 1

3

1. Geometry of Deformation

Arc length, BB’ = ρφ

Chord length, BB’ = L sin(γ)

B B’

γ

L

1. Geometry of Deformation

( )1_EQLK

ρφγ =∴

Arc length, BB’ = ρφ

Chord length, BB’ = L sin(γ)

LIMITATION

If γ is small, then sin(γ) →γ

If γ is small, then arc length → chord length

Equating BB’ →ρφ = L γ

1. Geometry of Deformation

( )1_EQLK

ρφγ =∴

Conclusion: shear strain varies

linearly with the radial position ρ

Q: How does angle of twist φφφφ change with radial position ρρρρ?

Torsion Part 1

4

1. Geometry of Deformation

Even though the shear strain is

a function of radial position,

the angle of twist is not.

Sign Convention

Positive

internal

torque

Shear stresses at points A, B, & C

for positive internal torque

Shear strain for positive

internal torque

Q: Draw an FBD with a negative internal torque.

C

C

2. Internal Loads

ΣM = 0; To – T = 0

(Internal)

(Applied)

Torsion Part 1

5

(Applied)

2. Internal Loads

Internal torque must be equivalent

to the resultant moment from all

internal shear forces.

∫= dFT ρ

2. Internal Loads

Internal torque must be equivalent

to the resultant moment from all

internal shear stresses.

∫= dFT ρ

dAdF τ=

( )2_EQdATA

K∫= ρτ

2. Internal Loads

Torsion Part 1

6

3. Material Relations

τ

γ

G

Hooke’s Law

Linear elastic material behavior

( )3_EQG Kγτ =

3. Material Relations

τ

γ

G

Hooke’s Law LIMITATION

Does not apply beyond the yield

strength τY

τY

Combine EQ 1-3

( )2_EQdATA

K∫= ρτ

( )3_using EQdAGTA

K∫= γρ

( )1_using EQdAL

GTA

K∫=φ

ρρ

∫=A

dAL

GT2ρ

φ

Polar moment of inertia, J

JL

Gφ=

GJ

TL=∴ φ

Angle of twist

in radians( )

( ) ( )∫=L

xJxG

dxxT

0φGeneralize to

Torsion Part 1

7

Combine EQ 1-3

( )2_EQdATA

K∫= ρτ

( )3_using EQdAGTA

K∫= γρ

( )1_using EQdAL

GTA

K∫=φ

ρρ

∫=A

dAL

GT2ρ

φJ

L

Gφ=

GJ

TL=∴ φ

Angle of twist

in radians

( )3_EQG Kγτ =

LG

φρτ =

GJL

TLGρτ =

J

Tρτ =∴

Elastic

torsion

formula

Polar Moment of Inertia for Circular Sections

ro

ri

dAJA

2

∫= ρ

ρρπθρρ ddddA 2==

∫=o

i

r

rdJ ρρπ 32

( )444

242

io

r

r

rrJ

o

i

−=

=

πρπ

( )44

32io

DDJ −=π

OR, in terms of diameters…

Q: Would it be valuable in this analysis to determine J for noncircular

sections?

Q: What is J for a solid circular section?

http://www.pci.org/pdf/publications/parking/P_MNL-129-98.pdf

Structures must resist torsion to carry loads

Spandrel beam

cross-section

Spandrel beam

Since the floor beam reaction

does not occur at the shear

center of the spandrel beam

there is torsion

Torsion Part 1

8

GJ

TL

J

T

=

=

φ

ρτShear Stress,

Angle of Twist,

In summary:Internal torque

Radial distance

Polar moment

of interia

Shear modulus

Shaft length

Limitations: circular sections, elastic behavior, small

shear strain, angle of twist formula only valid if T, G, and J

are constant over L

Shaft with a slit animation from E MCH 213D

Combine EQ 1-3

( )2_EQdATA

K∫= ρτ

( )3_using EQdAGTA

K∫= γρ

( )1_using EQdAL

GTA

K∫=φ

ρρ

∫=A

dAL

GT2ρ

φ

Polar moment of inertia, J

JL

Gφ=

GJ

TL=∴ φ

Angle of twist

in radians

( )3_EQG Kγτ =

LG

φρτ =

GJL

TLGρτ =

J

Tρτ =∴

Elastic

torsion

formula

Generalize to( )

( ) ( )∫=L

xJxG

dxxT

0φ