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In-Depth Information
Transfer Matrix Method
Use the transfer matrix of Eq. (11.47), along with the methods of Chapter 5 for incorporating
the in-span support, to develop a global transfer matrix. The boundary conditions applied
to the global transfer matrix equations lead to the characteristic equation and the critical
axial load of
064
EI
P
cr
=
7
.
(1)
2
This is the exact critical load as no approximation, other than approximations involved in
engineering beam theory, is made.
Displacement Method Related Techniques
The goal here is to set up the eigenvalue problem of Eq. (11.67). First, establish the linear
stiffness matrix
K
lin
by assembling the element stiffness matrices
k
lin
i
=
1
,
2
,
3 of Eq. (11.63)
to obtain
36
−
18
−
36
−
18
0
0
0
0
V
a
M
a
/
V
b
M
b
/
V
c
M
c
/
V
d
M
d
w
a
θ
a
w
b
θ
b
w
c
θ
c
w
−
18
12
18
6
0
0
0
0
−
36
18
60
6
−
24
−
12
0
0
−
18
6
6
20
12
4
0
0
EI
=
(2)
−
−
−
0
0
24
12
36
6
12
6
3
0
0
−
12
4
6
12
6
2
0
0
0
0
−
2 6 26
d
θ
0
0
0
0
−
6
2
6
4
/
d
Next, the global geometric stiffness matrix
K
geo
should be assembled using the element
geometric stiffness matrices
k
i
geo
.
Use the consistent geometric stiffness matrix of Eq. (11.64). Assemble the global matrix
for the beam. Apply the displacement boundary conditions. Then the eigenvalue problem
appears as
#
$
%
12
18
6
0
0
2
/
15
1
/
10
−
1
/
30
0
0
18
60
6
−
12
0
1
/
10
12
/
5
0
−
1
/
10
0
6
6
20
4
0
−
λ
−
1
/
30
0
4
/
15
−
1
/
30
0
V
=
0
(3)
#
&
0
−
12
4
12
2
0
−
1
/
10
−
1
/
30
4
/
15
−
1
/
30
0
0
0
2
4
0
0
0
−
1
/
30
2
/
15
K
lin
+
K
geo
V
=
0
λ
=
ε
2
=
2
/
where
P
EI
. This provides a critical load of magnitude
EI
P
cr
=
7
.
298
(4)
2
which is above the “exact” value of (1).
Varying Axial Force
Suppose this stepped col
um
n is subject to axial forces applied at
b
and
c
, as shown in
Fig. 11.42b, in addition to
P
applied at the right
en
d
. T
he
so
lution procedure
re
m
ain
s the
same, except n
ow
the axial force in element 1 is
P
P
c
, and
in element 3 is
P
. In order to find the buckling load, normal
ly
it is
ass
ume
d
that
th
e axial
forces remain in fi
xe
d proportion to each other, e.g., suppose
P
b
+
P
c
+
P
b
,
in element 2 is
P
+
=
α
P
and
P
c
=
β
P
. Then,
the buckling load
P
is calculated.
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