M9
Shafts: Torsion of Circular Shafts
Reading: Crandall, Dahl and Lardner 6.2, 6.3
A shaft is a structural member which is long and slender and subject to a torque
(moment) acting about its long axis. We will only consider circular crosssection shafts
in Unified. These have direct relevance to circular crosssection shafts such as drive
shafts for gas turbine engines, propeller driven aircraft and helicopters (rotorcraft).
However, the basic principles are more general and will provide you with a basis for
understanding how structures with arbitrary crosssections carry torsional moments.
Torsional stiffness, and the shear stresses that arise from torsional loading are important
for the design of aerodynamic surfaces such as wings, helicopter rotor blades and turbine
fan blades.
Modelling assumptions
(a)
Geometry (as for beam).
Long slender, L >> r (b,h)
Note: For the time being we will work in tensor notation since this is all about shear
stresses and tensor notation will make the analysis more straightforward. Remember
we can choose the system of notation, coordinates to make life easy for ourselves!
(b)
Loading
Torque about x
1
axis, T (units of Force x length). We may also want to consider
the possibility of distributed torques (Force x length/unit length) (distributed aerodynamic
moment along a wing, torques due to individual stages of a gas turbine)
No axial loads (forces) applied to boundaries (on curved surfaces with radial normal, or
on x
1
face)
=
11
22
=
33
=
0
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Deformation
Cross sections rotate as rigid bodies through twist angle
, varies with x
1
(cf. beams
– plane sections remain plane and perpendicular)

No bending or extensional deformations in x
1
direction
Crosssection:
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 Fall '05
 MarkDrela
 Force, Torsion, Trigraph, Crandall, ) x2

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