Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
445.204
Introduction to Mechanics of Materials
(재료역학개론)
Chapter 6: Torsion
Myoung-Gyu Lee, 이명규
Tel. 880-1711; Email: [email protected]
TA: Chanmi Moon, 문찬미
Lab: Materials Mechanics lab.(Office: 30-521)
Email: [email protected]
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 ?