Civil Engineering Reference
In-Depth Information
where is the slope of the curve for
a
1
at the point c
1
in
Suppose the line
a-b-c
in Figure 10.17a is parallel to line
a
1
-c
1
and the
point
a
is at a distance
e
Ó to the left of the origin 0, where
e
Ó<
e
Ó
1
. It is clear
from Figure 10.17a that the difference
D
between the values of ¦ at
c
1
on the
curve for a
1
and that at
c
on the line
a-b-c
is greater than zero. That is,
where
is the value of ¦ on the
curve for the value of
a
1
at
c
1
;
is
the value of ¦ on the line
a-c
at c.
Following the arguments in Section 10.4.1, it is clear that the column is in
stable equilibrium at
b
(Figure 10.17a). That is,
a
1
<
a
crit
.
In Figure 10.17b, the lines
a
2
-
c
2
and
a-c
are parallel and have slopes equal
to
a
2
.
The line
a
2
-c
2
touches the moment-deflection curve for
a
2
at the point
c
2
; hence Eqn 10.27b holds at
c
2
. Since the line
a-c
is above the line
a
2
-c
2
(i.e.
e
Ó>
e
Ó
2
) in Figure 10.17b, the difference
D
between the value of ¦ at
c
2
on the
curve for
a
2
and that at
c
on the line
a-b-c
is less than zero. That is
where and are as shown in Figure 10.17b.
In this case, the external moment
M
t
(i.e. ¦
t
value on the line
a-c
) always
exceeds the internal moment
M
(i.e. the ¦ value on the curve for
a
2
in
Figure 10.17b) and equilibrium is impossible. That is,
a
2
>
a
crit.
In Figure 10.17c, the line
a-c
having a slope equal to
a
3
touches the
curve for
a
3
at
c
3
, (i.e. at c). The column is in unstable equilibrium, that is,
a
=
a
crit
. In this case,
where are shown in Figure 10.17c.
Before further discussion of the implication of Eqns 10.30, 10.31 and
10.32, it is helpful to define the general expression for
D
:
