445.204

Introduction to Mechanics of Materials

(재료역학개론)

Chapter 6: Torsion

Myoung-Gyu Lee, 이명규

Tel. 880-1711; Email: myounglee@snu.ac.kr

TA: Chanmi Moon, 문찬미

Lab: Materials Mechanics lab.(Office: 30-521)

Email: chanmi0705@snu.ac.kr

Contents

- Torsion- Torsion of circular shafts- Torsion of thin walled tubes- Elastic-perfectly plastic torsion

2

Torsion of circular shafts Observation and assumptions- Plane section remains plane during

deformation; no warpage- Radius R does not change, length

does not change; i.e., normal strains in the radial and length are zeros; also no strain in the plane of cross section

- That is, cross-section simply rotates as a rigid about the axis of the shaft (or x-direction in the left figure)

- Only single strain component

3

Torsion of circular shafts

' 'x

B C rd d rdx dx dxθ

φ φγ = = =

Angle of twist per unit length

x xdG G rdxθ θφτ γ= =

Constitutive equation

Equilibrium

2 2x x

A A A

d d dM r dA r G dA G r dA G Jdx dx dxθφ φ φτ= = = =∫∫ ∫∫ ∫∫

xMddx GJφ=

Polar moment

4

Torsion of circular shafts

Angle of twist for constant torque

xMddx GJφ=

0 0L L x xM M Ld dx

GJ GJφ φ= = =∫ ∫

xx

M rJθτ =

PLEA

δ =c.f.

5

Torsion of thin walled, non-circular closed shafts Assumptions- Tube is prismatic (or constant cross section)- Wall thickness can be variable- Define middle surface or midline (along this coordinate s is defined)

6

Torsion of thin walled, non-circular closed shafts

Applying equilibrium into the infinitesimal element

0d dtt x s t s xds dsττ τ − ∆ + + ∆ + ∆ ∆ =

0dt dt sds ds

ττ + ∆ =

( ) 0d tds

τ = tant cons tτ =

“Shear flow” is constant

Moment (torque) equilibrium

( ) ( ) ( )( )2xM t d t d t Aτ τ τ= × = × =∫ ∫i r s r s i

2xM tAτ=2

xMtA

τ =

A: total plane area enclosed by the midline s7

Torsion of thin walled, non-circular closed shafts

Applying conservation of energy law,For the net rotation

01 12 2

LxM tdsdx

Gτφ τ= ∫

d Ldxφφ =

22

2 2

1 14

xx

MdM tds tdsdx G G t Aφ τ = =∫ ∫

24xMd ds

dx GA tφ = ∫

24

xM Sddx GA tφ =

Constant thickness 8

Elastic-perfectly plastic torsion

From elastic circular shaft xx

M rJθτ = x

xJM

rθτ

=or

Elastic

R

a

plastic

τs

a

a/r*τs

4 3max, 2 2

s sel s

JT R RR Rτ τ π π τ = = =

( )3

20 0 2

3R

hinge s sRT r rdrdπ τ θ πτ= =∫ ∫

max,

43

hinge

el

TT

=

( )

( )( )

20 0

2 3 30

123 2

as

R ssa

rT r rdrda

r rdrd R a

π

π

τ θ

πττ θ

= +∫ ∫

= −∫ ∫

Elastic-perfectly plastic torsion

Elastic

R

a

plastic

τs

a

( )

( )( )

20 0

2 3 30

123 2

as

R ssa

rT r rdrda

r rdrd R a

π

π

τ θ

πττ θ

= +∫ ∫

= −∫ ∫

1/33 64

s

Ta Rπτ

= −

Radius of elastic core

xd rdxθφγ = xd

dx rθγφ

=

At r=a, sx Gθ

τγ = sddx Ga

τφ=Thus,

Elastic-perfectly plastic torsion

xMddx GJφ=

Residual rate of twist

From,

res plastic

d d Tdx dx GJφ φ = −

Questions ?