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PSG College of Technology
**Course:
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B.Tech
**Specialization:
**Mechanical Engineering**Downloads:
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2.001 - MECHANICS AND MATERIALS I
Lecture #24
12/4/2006
Prof. Carol Livermore
Torsion (Twisting)
Mxx = Mt
Examples where torsion is important:
- Screwdriver
- Drills
- Propellers
Use superposition
Example: Unifrom along length, has circular symmetry
1

ϕ(x) = Angle of Twist (Total) at Point x (Built Up)
Compatibility:
d = ϕ(x + dx)R − ϕ(x)R
d = γdx
So:
γdx = ϕ(x + dx)R − ϕ(x)R
γdx = dϕR
dγ
γ=R
dx
What is γ?
So:
γ = γrθ Shear Strain
Thus:
γrθ = R
2
dϕ
dx

And:
γ
R dϕ
=
2
2 dx
θx =
rr = θθ = xx = rθ = rx = 0
Constitutive Relations:
σθx = 2Gθx = Gγxθ = Gr
dϕ
, where G is the shear modulus.
dx
E
2(1 + ν)
G=
σxx = σθθ = σrr = σrθ = σrx = 0
σθx = Gr
Recall beam bending σxx =
dϕ
dx
−Ey
ρ .
Equilibrium:
Mx = 0
rdF = 0
−Mt +
A
dF = σθx dA
rσθx dA = 0
−Mt +
A
Mt =
rGr
A
3
dϕ
dA
dx

Mt =
dϕ
dx
Gr2 dA
A
Recall beam bending:
M=
1
ρ
Ey 2 dA
A
Note:
dϕ
happens
dx : what
2
Gr
dA:
eﬀective stiﬀness
A
Mt : what we apply
If G is constant ”Special Case”
Mt =
J≡
dϕ
G
dx
r2 dA
A
r2 dA Polar Moment of Inertia
A
So for ”special case” G is constant.
Mt = GJ
dϕ
dx
Recall beam bending:
M=
EI
ρ
If G is not constant:
(GJ)ef f =
Gr2 dA
A
When G is constant:
Mt = GJ
dϕ
dϕ
and generally σθx = Gr
dx
dx
Mt
σθx
=
GJ
Gr
So:
σθx =
4
Mt r
J

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