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References
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Int. J.
for Numerical Methods in Engineering
, Vol. 15, pp. 1771-1812.
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conforming and non-conforming solutions,
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, Air Force Institute
of Technology, Wright-Patterson Air Force Base, Dayton, OH.
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,
Vol. 40, pp. 51-88.
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Formulas for Stress, Strain, and Structural Matrices
, Wiley, New York.
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Analysis and Design of Elastic Beams, Computational Methods
, Wiley, New York.
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Problems
In-Plane Deformation
13.1 Given the relationships
x
=
r
cos
φ
,y
=
r
sin
φ
,
use the Jacobian to show that
∂
r
∂
r
∂φ
∂
1
r
∂φ
∂
1
r
x
=
cos
φ
,
y
=
sin
φ
,
x
=−
sin
φ
,
y
=
cos
φ
∂
∂
Hint:
Form
cos
∂
∂
x
r
∂
y
∂
∂
∂
φ
sin
φ
r
∂
∂φ
∂
∂
r
x
∂
∂
x
∂
∂
x
∂
∂
=
=
=
J
−
r
sin
φ
r
cos
φ
∂
y
∂φ
∂
x
∂φ
y
y
y
Then
J
−
1
cos
∂
∂
∂
∂
r
∂
∂
φ
−
sin
φ
x
r
∂
∂φ
r
∂
∂φ
=
=
∂
r
sin
φ
cos
φ
y
13.2 Show that Eq. (13.11a) corresponds to the conditions of equilibrium for the element
of Fig. 13.3.
13.3 Find the radial displacements and internal forces in a circular disk subject to a radial
pressure of magnitude
p
0
(force/length) on the inner periphery at
r
=
a
.
The outer
=
rim at
r
b
is free of loading.
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