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Substitution of the Assumed Displacement in the Fundamental Beam Equations
The beam element state variables
w
,
θ
,
M
, and
V
can be expressed in terms of the shape
functions
N
=
N
u
G
. Thus, introduce
w(
x
)
of (10) into the expressions of (1), (2), and (3)
Gv
i
w(
)
=
(
)
x
N
u
x
)
=−
w
(
N
u
(
Gv
i
θ(
x
x
)
=−
x
)
)
=−
w
(
N
u
(
Gv
i
κ(
x
x
)
=−
x
)
(11)
Gv
i
EI
N
u
(
M
(
x
)
=
EI
(κ
−
κ
T
)
=−
x
)
−
EI
κ
T
M
(
EI
N
u
(
Gv
i
V
(
x
)
=
x
)
=−
x
)
where
N
u
(
x
)
=
[0026
x
]
N
u
(
x
)
=
[0006]
Since the matrices
G
and
v
i
contain only discrete values (and are not functions of
x
), the
variables
along the element can be expressed as functions of the
assumed displacement and its derivatives.
w(
x
)
,
θ(
x
)
,
M
(
x
)
, and
V
(
x
)
Substitution of the Assumed Displacement into the Principle of Virtual Work
Substitute
Gv
i
w
=
N
u
(
x
)
into the virtual work of (8) for a single element
i
.
v
iT
G
T
b
a
v
iT
G
T
b
a
W
i
N
T
u
EI
N
u
(
v
i
N
u
(
v
i
δ
=
δ
(
x
)
x
)
dx
G
+
δ
x
)
k
N
u
(
x
)
dx
G
w
k
B
k
w
v
iT
G
T
b
a
v
iT
G
T
b
a
N
T
u
N
u
(
+
δ
(
x
)
EI
κ
T
dx
−
δ
x
)
p
z
(
x
)
dx
p
p
z
(12)
where
k
B
is the stiffness matrix for bending of the element. This was developed in Section
4.4.2.
p
0
T
k
w
is the element stiffness matrix to account for the elastic foundation.
p
0
is the loading vector for thermal effects.
T
p
p
z
is the loading vector for the line loading along the element.
N
and
B
u
N
u
. Then
A more common notation is to use
N
u
G
=
N
. Also define
B
=
=
v
iT
b
a
v
iT
b
a
W
i
B
T
EI
B
dx
v
i
N
T
k
v
i
δ
=
δ
+
δ
N
dx
w
k
B
k
w
v
iT
b
a
v
iT
b
a
B
u
EI
B
T
+
δ
κ
T
dx
−
δ
p
z
dx
(13)
p
0
p
p
z
T
Calculation of the Element Stiffness Matrix
k
w
for the Elastic Foundation
Most of the matrices in (12) were developed previously in this chapter. An exception is
k
w
for the elastic foundation, which will be treated here. From (12)
G
T
o
N
u
k
w
N
u
dx
G
k
w
=
(14)
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