Title: Torsion
1Torsion
There are many torsionally loaded structural
elements in life in airplanes, automobiles,
drill equipments, screw drivers, .etc.
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5Kinematics 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
6 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
7The 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
8Thus, we have only ??z ? ??z
Since we have ??z and ??z (?z?), we may drop
the subscripts.
? ? and ?
9Kinematics
10very 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
11Let ? ? 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.
12Now, 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.
13Elastic Twisting of Circular Shafts
(1) Equil.
dT (? da) r
? dT ? (?da) r
? internal
((T ? Mz))
? external
14? 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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16Recall from Statics that ? r2 da J
J polar moment of inertia wrt the z-axis
Thus
for solid section
for hollow section
17For uniform shafts (constant T, J, G) as shown
18Thus, for a uniform shaft
? total angle of twist
For nonuniform shafts,
19The three methods of analysis, namely (1) direct
integration (2) discrete element (3)
superposition discussed earlier in axially-loaded
members problems can be used here.
20Shear Stress (?)
? G?
21? r is any radius in the shaft
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32Torsion 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?
33 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)
34 U.S./English Units
? 2 ? n
n (rpm rev/min) T (in-lb)
1 Hp 550 (ft-lb/sec)
Do NOT forget the units
35Gears
36Gears
See the Example Next.
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