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In matrix notation, this expression takes the form
x
dEA d
x
.
.
.
u
···
v
···
w
···
φ
p
x
···
p
y
···
p
z
···
m
φ
···
···
···
···
.
x
d
2
EI
yy
d
x
.
.
···
···
···
···
−
.
.
x
d
2
EI
zz
d
x
.
u
T
x
δ
dx
=
0
(12.32)
···
···
···
···
.
.
.
x
d
2
EI
ωω
d
x
.
.
.
+
x
dGJ
t
d
x
k
D
u
p
where the boundary terms have not been included. The symbol
x
d
used in the matrix
k
D
indicates the application of
d
/
δ
u
T
.
dx
to the preceding variable
As explained in Chapter 4,
the differential operator matrix
k
D
forms the basis of the element stiffness matrix for the
bar.
12.1.7
Governing Local Equations
The differential equation form of the governing equations can be obtained from Eq. (12.32)
by utilizing integration by parts. Recall that integration by parts is the one-dimensional
equivalent of the divergence theorem. The equations provided by the principle of virtual
work are the equilibrium relations. We find
EA u
)
+
(
p
x
=
0
EI
yy
v
)
−
(
p
y
=
0
(12.33)
EI
zz
w
)
−
(
p
z
=
0
ωω
φ
)
−
(
GJ
t
φ
)
−
(
EI
m
φ
=
0
with the boundary conditions
EI
yy
v
−
M
z
=
0
EI
zz
w
+
M
y
=
0at
x
=
a
and
x
=
b
ωω
φ
+
EI
M
ω
=
0
and
EA u
−
N
=
0
EI
yy
v
)
+
(
V
y
=
0
at
x
=
a
and
x
=
b
EI
zz
w
)
+
(
V
z
=
0
ωω
φ
)
−
GJ
t
φ
+
(
EI
M
φ
=
0
The first order local governing equations can be obtained from the relationships devel-
oped in this section following the procedures outlined in Chapter 1 or 2.
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