Civil Engineering Reference
In-Depth Information
is given by
P
j
,F
c
j
=
j
d
P
j
(15.27)
0
It may be seen from Fig. 15.13(b) that the area OABD represents the work done by a
constant force
P
j
,F
moving through the displacement
j
,F
. Thus from Eqs (15.26) and
(15.27)
u
j
+
c
j
=
P
j
,F
j
,F
(15.28)
It follows that since
u
j
has the dimensions of work,
c
j
also has the dimensions of
work but otherwise
c
j
has no physical meaning. It can, however, be regarded as the
complement of the work done by
P
j
in producing the displacement
j
and is therefore
called the
complementary energy
.
The total strain energy,
U
, of the structure is the sum of the individual strain energies
of the members. Thus
n
U
=
u
j
j
=
1
which becomes, when substituting for
u
j
from Eq. (15.26)
j
,F
n
U
=
P
j
d
j
(15.29)
0
j
=
1
Similarly, the total complementary energy,
C
, of the structure is given by
n
C
=
c
j
j
=
1
whence, from Eq. (15.27)
P
j
,F
n
C
=
j
d
P
j
(15.30)
0
j
=
1
Equation (15.29) may be written in expanded form as
1,F
2,F
j
,F
n
,F
U
=
P
1
d
1
+
P
2
d
2
+···+
P
j
d
j
+···+
P
n
d
n
(15.31)
0
0
0
0
Partially differentiating Eq. (15.31) with respect to a particular displacement, say
j
,
gives
∂
U
∂
j
=
P
j
(15.32)
Equation (15.32) states that the partial derivative of the strain energy in an elastic
structure with respect to a displacement
j
is equal to the corresponding force
P
j
;
clearly
U
must be expressed as a function of the displacements. This equation is
generally known as
Castigliano's first theorem (Part I)
after the Italian engineer who
