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Eq. 1.136)
EA
d
2
u
dx
2
=−
p
x
(2)
with boundary conditions, whe
re
u
is the axial displacement,
E
is Young's modulus,
A
is
the area of the cross section, and
p
x
is the applied axial distributed traction. By comparison,
the correspondence between these problems is
Heat Equation
Extension Equation
Temperature
T
<
−−−−−
>
Displacement
u
K
<
−−−−−
>
EA
Heat Source
Q
<
−−−−−
>
Load
−
p
x
In addition to this correspondence between the terms in the differential equations, the
relationship between the boundary conditions must be established. For the extension bar,
the boundary conditions can be
1. A fixed end or an end with a prescribed displacement,
u
=
0or
u
=
u
2. A free end,
du
/
dx
=
0
For the heat equation, the boundary conditions can be classified as
1. Prescribed temperature at the boundary,
T
=
0or
T
=
T
2. Insulated,
dT
/
dx
=
0
If the bar is attached to a spring with the spring constant
k
at one end, the boundary
condition at this end can be written as
EA du
This is analogical to the convection
boundary condition for the heat equation which has the form
dT
/
dx
=
ku
.
/
dx
=±
hT,
where
h
(>
0
)
is a constant. The sign
is for the right end.
With these equivalent coefficients, the heat equation can be solved by using a computer
program developed for analyzing an extension bar. In doing so, the values of
K
and
Q
are input to the entries for
EA
and
P
. If quantities
E
and
A
are entered separately,
A
can
be assigned a value first and then let
E
+
is for the left end and
−
In setting up the first value, care should
be taken to assure that it is within the range that the computer program can accept. The
boundary conditions are handled similarly. After executing the program, the computed
u
corresponds to the temperature distribution
T
, with appropriate units.
=
K
/
A
.
7.5.2 Stiffness Matrices
As in Example 7.6 for the Ritz method, we can use the field problem of torsion of a bar to
illustrate the development of stiffness matrices for a finite element solution for the warping
functions across the cross section of a bar.
From Eqs. (4) and (5) of Example 7.6, the equation that can be solved for the warping
function
, and that will permit the cross-sectional characteristics of a bar in torsion to be
computed, takes the form
ω
∂
∂
∂
∂
y
dA
y
δω
∂ω
y
+
∂
z
δω
∂ω
−
∂
∂
+
y
δω
z
z
δω
=
0
(7.85)
∂
∂
∂
z
A
or
A
δω
[
(
∂∂
+
∂∂
)ω
+
(
∂
z
−
∂
y
)
]
dA
=
0
(7.86)
y
y
z
z
y
z
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