Civil Engineering Reference
In-Depth Information
THE PRINCIPLE OF THE STATIONARY VALUE OF THE
TOTAL COMPLEMENTARY ENERGY
Suppose that an elastic structure comprising
n
members is in equilibrium under
the action of a number of forces,
P
1
,
P
2
,
...
,
P
k
,
...
,
P
r
, which produce corres-
ponding actual displacements,
1
,
2
,
...
,
k
,
...
,
r
, and actual internal forces,
F
1
,
F
2
,
...
,
F
j
,
...
,
F
n
. Now let us suppose that a system of elemental virtual forces,
δ
P
r
, are imposed on the structure and act through the actual
displacements. The external virtual work,
P
1
,
δ
P
2
,
...
,
δ
P
k
,
...
,
δ
δ
W
e
, done by these elemental virtual forces
is, from Section 15.2,
δ
W
e
= δ
P
1
1
+ δ
P
2
2
+···+δ
P
k
k
+···+δ
P
r
r
or
r
δ
W
e
=
k
δ
P
k
(15.36)
k
=
1
At the same time the elemental external virtual forces are in equilibriumwith an elem-
ental internal virtual force system,
δ
F
1
,
δ
F
2
,
...
,
δ
F
j
,
...
,
δ
F
n
, which moves through
actual internal deformations,
δ
1
,
δ
2
,
...
,
δ
j
,
...
,
δ
n
. Hence the internal elemental
virtual work done is
n
δ
W
i
=
δ
j
δ
F
j
(15.37)
j
=
1
From Eq. (15.23)
r
n
k
δ
P
k
=
δ
j
δ
F
j
k
=
1
j
=
1
so that
n
r
δ
j
δ
F
j
−
k
δ
P
k
=
0
(15.38)
j
=
1
k
=
1
Equation (15.38) may be written as
&
(
F
j
n
r
'
)
=
δ
δ
j
d
F
j
−
k
P
k
0
(15.39)
0
j
=
1
k
=
1
FromEq. (15.30) we see that the first term inEq. (15.39) represents the complementary
energy,
C
i
, of the actual internal force system, while the second term represents the
complementary energy,
C
e
, of the external force system.
C
i
and
C
e
are opposite in sign
since
C
e
is the complement of the work done
by
the external force system while
C
i
is
the complement of the work done
on
the structure. Rewriting Eq. (15.39), we have
δ
(
C
i
+
C
e
)
=
0
(15.40)
