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q
x
p z
y
M a
M b
z, w
V b
V a
b
a
x
k T
k w w
FIGURE 4.13
Beam element with Sign Convention 2.
This expression comes from Example 2.6 with the addition of the beam on elastic foundation
term in which k
w w
is the reactive pressure due to the foundation, which is assumed to react
linearly elastic.
Introduction of the material law and the kinematics will permit the virtual work to be
expressed in terms of displacements. As indicated by (3a), if shear deformation is neglected
κ =−
d 2
dx 2 .
w/
Then
δw EI
(w + κ T
δ
W i
=
[
) δw
p z + δw
k w w
] dx
(8)
x
w =
where
d
w/
dx and the contribution of a temperature differential has been included.
Choice of Approximate Trial Displacements
Choose as an approximate deflection a four term polynomial for this element with four
nodal degrees of freedom. The element, with Sign Convention 2, is shown in Fig. 4.13.
The proposed trial function for the displacement
of this beam on elastic foundation
element is the same one utilized in this chapter for a simple beam element. That is, assume
(Eq. 4.40)
w(
x
)
C 3 x 2
C 4 x 3
w(
x
) =
C 1 +
C 2 x
+
+
=
N u
w
(9)
with
xx 2
x 3 ]
N u
(
x
) =
[1
w T
=
[ C 1
C 2
C 3
C 4 ]
where the generalized degrees of freedom are the constants C 1 , C 2 , C 3 , C 4 . This leads to the
deflection expressed in terms of the unknown displacements v i
as
w(
x
) =
N u
(
x
)
w
=
N u
(
x
)
Gv
=
N
(
x
)
v
(10)
where N
, which is defined in Eq. (4.46), contains the shape functions for this polynomial
approximation of
(
x
)
w(
)
x
.
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