AAE 221 – Aerospace Structures II – Spring 2004
Chapter 2.2 – Beam Torsion
1.
An uniform beam with elliptical cross-section is submitted to a twisting moment
of 400 Nm.
The length of the beam is 4m. The beam is made of aluminum (E =
70,000 N/mm
2
),
ν
= 0.30).
The dimensions of the cross-section are illustrated
below.
Determine the total twisting angle between the two ends of the beam, the
location and value of the maximum warping, and of the maximum shear stress.
2.
Refer to the thin-walled circular tube illustrated below.
a)
Using the exact solution seen in class for the solid (i.e., not hollow) cylinder,
derive the expression of the torsion constant
J
for the hollow cross-section as a
function of the radii
a
i
and
a
e
.
b)
Compute an approximate expression for
J
using the fact that the thickness
h
is
very small compared with the radii and that we can therefore assume a linear
variation of the stress function
Ψ
through the thickness
(
)
⎪
⎩
⎪
⎨
⎧
≤
≤
≤
≤
−
=
Ψ
i
e
i
e
a
r
0
k
a
r
a
r
a
h
k
for
for
as illustrated below.

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