1
AE 321 – Homework #9
Due: Not Graded. Do NOT hand in.
Chapters 6, 7, 8: Torsion, Plane Problems and other problems
1.
(a) Show that the stress function for the torsion of a long cylinder of
solid
triangular cross
section (shown below) subjected to a given torque
T
is given by:
±
x
,
y
()
=
Cx
²
y
²
2
3
h
³
´
µ
¶
·
¸
x
+
3
y
²
2
3
h
³
´
µ
¶
·
¸
x
+
1
3
h
³
´
µ
¶
·
¸
(b) Find the stress, strain and displacement state for this bar assuming a homogeneous, isotropic
linear elastic solid (
E
and
±
) and no body forces.
2.
(Old exam question) A long hollow linearly elastic, isotropic and homogeneous thick walled
cylinder is loaded by a uniform axial pressure
P
and a net torque
T
as shown in the figure.
(a) Briefly describe the concepts of (i) Superposition and (ii) Saint Venant's principle.
(b) What are the boundary conditions on the end faces, the exterior lateral surface and the
interior surface of the cylinder?
(c) Find all stress components in the cylinder in terms of the loading
P
and
T
.
Hint:
Investigate using the known solution for torsion of a thick walled solid (as opposed to
hollow) cylinder, i.e.
13
=
²
μ
³
x
2
and
23
=
²
x
1
with all other stresses zero and the angle
of twist
±
related to the net torque T by
=
2
T
b
4
.
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 Spring '08
 Dutton,J
 mechanics, Force, elastic constants

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