Extra Problem 1. Hollow cylinder under inner pressure
Consider the problem of an hollow cylinder (with inner radius
a
, outer radius
b
and length
L
)
made of a linearly elastic homogeneous isotropic material (with Young’s modulus
E
and
Poisson’s ratio
ν
) subjected to an internal pressure
p
. The outer surface is assumed to be traction
free and the two ends of the cylinder are assumed to be constrained by the presence of two
(frictionless) rigid walls. Starting from an assumed form of the displacement field (in cylindrical
coordinates), obtain the stress distribution everywhere in the cylinder.
Solution
The problem geometry and loading suggest that, of the three displacement components
u
r
r
,
!
,
z
(
)
,
u
!
r
,
!
,
z
(
)
,
u
z
r
,
!
,
z
(
)
(
)
, only the radial one is nonzero and depends only on
r
. So our
initial guess for the displacement field is
u
r
r
,
!
,
z
(
)
=
f r
( )
u
!
r
,
!
,
z
(
)
=
0
u
z
r
,
!
,
z
(
)
=
0
"
#
$
%
$
.
(1)
Our goal is to find the expression of
f r
( )
which would satisfy the equilibrium equations and the
boundary conditions.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Lambros
 mechanics, Trigraph, hollow cylinder, displacement field

Click to edit the document details