Torsion

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Torsion
There are many torsionally loaded structural
elements in life in airplanes, automobiles,
drill equipments, screw drivers, .etc.
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Kinematics of Circular Shafts
? Circular shafts much simpler, more common,
very efficient ((as will be proven later))
? Cylindrical coordinates system
? Assumptions
1) Circular cross sections are rigid.
? ?r ?? ?r? 0
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2) The shaft remains straight and cross-section
- its geometric axis before deformation remains
- after deformation.
? 90? angles remain 90?
? no shear
? ?rz 0
3) The distance between cross sections does not
change.
? ?z 0
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The assumptions above are geometric they do not
depend on the material behavior (elastic or
inelastic). However, these assumptions are
limited to small deformations.
Summary ?r ?? ?z ?r? ?rz 0
Thus, from Hookes Law
?r ?? ?z ?r? ?rz 0
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Thus, we have only ??z ? ??z
Since we have ??z and ??z (?z?), we may drop
the subscripts.
? ? and ?
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Kinematics
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very small deformation Is assumed. ?
er ? straight line ec ? straight line
? ? ABC ? ? CBO
Note that er is common for the two triangles
ABC and CBO.
er ? r?? ? ?r?z
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• r?? ?r?z ?

Let ? ? d ?
The equation above expresses the relative
rotation of the cross section at (zdz) wrt the
section at z in terms of the shear strain at a
distance r from the center.
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Now, we need to find the angle of twist ? and the
shear stress/strain ?/? .
We can not use Statics alone to derive the
equations. (TRY!)
Thus, this can be achieved by utilizing
(1) Equilibrium (2) Geometric Compatibility (3) Ma
terial Behavior
The problem is internally SI.
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Elastic Twisting of Circular Shafts
(1) Equil.
dT (? da) r
? dT ? (?da) r
? internal
((T ? Mz))
? external
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? Written before (2) G. Comp. as it is short
easy. ? go to (2).
(2) Material Behavior
T ? ? r da
? G ? r da
(3) Geometric Compatibility
T ? G ? r da
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Recall from Statics that ? r2 da J
J polar moment of inertia wrt the z-axis
Thus
for solid section
for hollow section
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For uniform shafts (constant T, J, G) as shown
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Thus, for a uniform shaft
? total angle of twist
For nonuniform shafts,
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The three methods of analysis, namely (1) direct
integration (2) discrete element (3)
superposition discussed earlier in axially-loaded
members problems can be used here.
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Shear Stress (?)
? G?
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? r is any radius in the shaft
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Torsion Power Transmission
Power Transmission (HP)
Angular speed (Hz cycle/sec or rpm)
? rev./min
T is needed to design the shaft.
From Physics P T?
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SI Units
? 2 ? f
P Power (watt)
? angular velocity (rad/sec)
f frequency of the rotating shaft (Hz
/sec)
T Torque (N.m)
? P 2? f T (N.m/s watt)
1 Hp 745.7 (N.m/s)
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U.S./English Units
? 2 ? n
n (rpm rev/min) T (in-lb)

1 Hp 550 (ft-lb/sec)
Do NOT forget the units
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Gears
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Gears
See the Example Next.
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