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where
k
i
k
i
B
+
k
i
w
k
i
B
and
k
i
=
is the stiffness matrix for element
i
.
are given in Example 4.4.
w
v
i
is the vector of displacements on the ends of the element
i
.
p
i
0
is the nodal force vector due to distributed loading along the element.
P
∗
is the assembled vector of concentrated forces and moments applied to the system
nodes.
P
0
is the assembled vector representing the applied distributed loads
p
i
0
.
K
is the assembled system stiffness matrix.
V
is the assembled vector of system nodal displacements.
That is,
=
V
1
V
N
T
V
V
2
···
(6.9a)
where
N
is the number of nodes. Also, for this beam
=
U
Zk
Yk
T
V
k
(6.9b)
EXAMPLE 6.1 Numerical Example of a Beam on Elastic Foundation
Consider the beam on an elastic foundation, loaded with concentrated forces, as shown in
Fig. 6.3. Let
E
320 MN/m
2
.
With the approximate stiffness matrix for the beam element on an elastic foundation given
by Eq. (17) of Example 4.4, the problem is solved using Eq. (6.8), following the displacement
method. Here we choose to study the effect of refining the mesh (increasing the number
of elements) on the accuracy of the solution. Recall that with the displac
e
me
nt
method the
displacement boundary conditions are applied to the expression
KV
m
2
,
I
08 m
4
,
k
w
=
=
21 GN
/
=
1
.
P
0
. However,
for a beam element resting on an elastic foundation, the boundary conditions are on the
forces (M and V) at the ends of the beam. That is, no displacement boundary conditions
occur and Eq. (6.8) is solved without reduction. Since, the elastic foundation prevents rigid
body motion, the system stiffness matrix K is not singular. At each node, there are two
unknown displacements (degrees of freedom). These are the deflection
P
∗
=
+
w
and slope
θ
.
The vector
V
is established for a straight beam that assures that the deflection
w
and
θ
=−
w
are continuous (the same as in the case of a simple beam) at a node for each
element connected at the node. However, nothing in the interpolation based finite element
method of analysis assures that the moment
M
and shear force
V
are continuous at a node.
slope
FIGURE 6.3
Beam on elastic foundation.
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