Adjacent Equilibrium
Columns
The differential equation governing equilibrium on the secondary path for the column is linear
because the factor
N
happens to be constant.
(
29
(
29
2
2
1
0
2
1
0
2
iv
N
EA
EA u
w
EIw
EA u
w
w
ε
′
′
′
′
′
=
=
+
=
′
′
′′

+
=
(1)
If the primary path is, say, a straight line through the origin, as for a column, the adjacent
equilibrium configurations occur at a bifurcation point.
The linear equations necessary for this
process may be derived from the nonlinear Eqs. (1) by use of a perturbation technique in which
u
is replaced by
0
1
u
u
+
where
u
denotes the displacement field,
0
u
represents an equilibrium
configuration on the primary path, and
1
u
is a small increment.
Now let
0
1
0
1
u
u
u
w
w
w
→
+
→
+
(2)
where the arrows are read “be replaced by.”
The variables (
0
0
,
u
w
) represent a configuration on
the primary equilibrium path, the incremental displacements (
1
1
,
u
w
) are infinitesimally small,
and both (
0
0
,
u
w
) and (
,
u w
) are equilibrium configurations.
Then, in the equations obtained by
introducing Eqs. (2) into Eqs. (1), one sees that
1 In each equation the sum of all terms containing
0
0
,
u
w
alone is equal to zero because
0
0
,
u
w
satisfy Eqs. (1).
1
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2 Second and higherorder terms in
1
1
,
u
w
may be omitted because of the smallness of
the incremental displacements.
The resulting equations are
(
29
(
29
1
0
1
2
1
0
0
1
0
1
0
1
0
1
0
2
iv
u
w w
EIw
EA
u
w
w
w
u
w w
′
′
′
′
+
=
′
′
′′
′′
′
′
′

+
+
+
=
(3)
Equations (3) are linear in the unknowns
1
1
,
u
w
as desired, the variables
0
0
,
u
w
appearing as
constants.
For the column, the primary equilibrium path represents undeflected configurations.
Thus,
0
0
w
=
for all values of
x
, and Eqs. (3) reduces to
1
1
0
1
0
0
iv
u
EIw
EAu w
′′
=
′
′′

=
(4)
For
0
u
the equilibrium equation for the column in the undeflected form yields
0
P
u
x
EA
= 
(5)
Substituting Eq. (5) into the second of Eqs. (4), gives
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 Summer '10
 Staff
 Equilibrium, Quadratic equation, Elementary algebra

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