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FIGURE 4.14
Beam with ramp loading. Examples 4.5, 4.9, and 4.10.
The boundary conditions are
w
0
=
θ
0
=
w
L
=
M
L
=
0
.
Insert the displacement conditions
(w
0
=
θ
0
=
w
L
=
0
)
in the right-hand side of
=
−
V
0
M
0
V
L
M
L
w
0
θ
0
w
k
i
p
i
0
L
θ
L
(2)
p
i
k
i
v
i
p
i
0
=
−
Solve for
θ
L
from the final row of this relationship with
M
L
=
0
.
Substitute this value of
θ
L
back into (2) to compute the left end reactions (Sign Convention 2)
p
0
L
2
V
0
=−
(
2
/
5
)
p
0
L,
M
0
=
(
1
/
15
)
(3)
This example is provided to illustrate that stiffness equations in the form developed in
this chapter can be used for the solution of simple pr
ob
lems. However, the most important
use of the stiffness relations is in the form of
k
i
v
i
p
i
0
,
w
herein the distributed loadings
on an element are applied as end loads and inserted in
p
i
−
Then this stiffness relationship
is combined (assembled) with those of the other elements of the structure to form a set of
global stiffness equations. See Chapter 5.
.
4.4.5 Conversion of a Stiffness Matrix to a Transfer Matrix
The transformation of a transfer matrix into a stiffness matrix is a useful technique for
deriving stiffness matrices. Of course, the process can be reversed to convert a stiffness
matrix into a transfer matrix. To see this, begin with Eq. (4.12) which was established using
Sign Convention 2.
p
a
=
k
aa
v
a
+
k
ab
v
b
(4.76)
p
b
=
k
ba
v
a
+
k
bb
v
b
We wish to reorganize these equations so that they are in transfer matrix form. The first
relationship is readily changed to
k
−
1
k
−
1
v
b
=
ab
p
a
−
ab
k
aa
v
a
(4.77)
Substitute
v
b
of Eq. (4.77) into the second of Eqs. (4.76) and find
=
k
ba
ab
k
aa
v
a
k
bb
k
−
1
k
bb
k
−
1
−
+
p
b
ab
p
a
(4.78)
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