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the displacement form of the principle of virtual work, specialized for beams, takes the
form
x
δ
−
δ
s
L
0
u
T
u
D
T
ED
u
u
dx
u
T
p
dx
u
T
−
δ
W
=
x
δ
−
u
T
u
D
T
ED
u
u
p
dx
−
δ
s
L
0
u
T
=
x
δ
−
=
0
(1)
For engineering beam theory [Chapter 1, Eq. (1.110)],
EA
0
0
E
=
0
k
s
GA
0
(2)
0
0
EI
Then, with
D
u
from Eq. (8) of Example 2.6,
.
.
x
dEAd
x
0
0
.
.
.
.
...........
...........
...........
.
.
0
x
dk
s
GA d
x
x
dk
s
GA
k
D
=
u
D
T
ED
u
=
(3)
.
.
.
.
...........
...........
...........
.
.
0
k
s
GA d
x
x
dEId
x
.
.
+
k
s
GA
where
d
x
=
dx
and
x
d
is the ordinary derivative on the preceding variable. If shear defor-
mation effects are not included, matrix
E
reduces to [Chapter 1, Eq. (1.107)]
d
/
EA
0
E
=
(4)
0
EI
and
D
u
is given by [Chapter 1, Eq. (1.101)]
d
x
0
=
D
u
(5)
0
−
d
x
Then
k
D
becomes
x
dEAd
x
0
k
D
u
D
T
ED
u
=
=
(6)
x
d
2
EI d
x
0
u
T
u
T
It is important to remember that the operator
u
D
acts on
δ
in (1). Hence,
δ
h
as not
(
u
D
T
ED
u
u
been factored out of the integrand of the integral of (1) and the quantity
does
not represent Euler's equations for the beam. The operator
k
D
is most useful in establishing
a discrete model on which a computational solution can be based. In order to reduce (1) to a
form from which the local equations can be extracted, it is necessary to first apply integration
by parts (or the divergence theorem) to transfer the derivatives away from
−
p
)
u
T
u
T
δ
so that
δ
can be correctly factored out. This immediately provides the local beam equations
d
dx
EA
du
0
dx
−
p
x
=
0
(7)
dx
2
EI
d
2
dx
2
d
2
w
−
p
z
=
0
along with appropriate boundary conditions.
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