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FIGURE 11.10
Midbeam deflections as a function of P .
is plotted in Fig. 11.10 as the second order sol uti on. Also illustrated is a first order solu-
tion, which is the deflection due to moments Pe applied to the en ds of the beam, with
no consideration given to the change in influence of the axial load P on the response as
the deflection changes, i.e., the terms involving H are ignored in Eq. (11.23) or (11.25).
Note that the first order solution is linear, while that for the second order case is non-
linear.
If the critical or unstable situation is associated with inordinately large deformations, then
(3) and (4) can be used to define the axial load leading to bifurcation. Note from (3) that
if sin
ε =
0 or from (4) if cos
ε/
2
=
0 , the deflection grows without limit. These equalities
occur if
ε = π
. Then
= π
2 EI
L 2
=
P
P cr
(5)
This critical value of P defines the horizontal line of Fig. 11.10. Other values of
ε
can also
lead to critical responses. In the case of sin
ε =
0 ,
ε
can assume the values n
π
, where
n
0 , the axial force P must also be zero and the me-
chanical problem has been altered (a “trivial” solution). Under normal circumstances, for
n
=
0 , 1 , 2 , 3 ,
....
However, for n
=
, the physical problem is of little interest, since the structure has already been
subjected to a critical deformation (at n
=
2 , 3 ,
...
=
1).
11.2.4
Element Matrices Using Analytical Solutions
The closed form solutions with analytical functions of Eqs. (11.32) and (11.33) are for a bar
with constant bending stiffness EI , for which second order effects are taken into account.
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