Civil Engineering Reference
In-Depth Information
P
j
A
B
P
j
,F
C
U
D
F
IGURE
15.14
Load-deflection curve for a
linearly elastic member
j
0
j
,F
derived and published it in 1879. One of its primary uses is in the analysis of non-linearly
elastic structures, which is outside the scope of this topic.
Now writing Eq. (15.30) in expanded form we have
P
1,F
P
2,F
P
j
,F
P
n
,F
C
=
1
d
P
1
+
2
d
P
2
+···+
j
d
P
j
+···+
n
d
P
n
(15.33)
0
0
0
0
The partial derivative of Eq. (15.33) with respect to one of the loads, say
P
j
, is then
∂
C
∂
P
j
=
j
(15.34)
Equation (15.34) states that the partial derivative of the complementary energy of an
elastic structure with respect to an applied load,
P
j
, gives the displacement of that load
in its own line of action;
C
in this case is expressed as a function of the loads. Equation
(15.34) is sometimes called the
Crotti-Engesser theorem
after the two engineers, one
Italian, one German, who derived the relationship independently, Crotti in 1879 and
Engesser in 1889.
Now consider the situation that arises when the load-deflection curve is linear, as
shown in Fig. 15.14. In this case the areas OBD and OAB are equal so that the strain
and complementary energies are equal. Thus we may replace the complementary
energy,
C
, in Eq. (15.34) by the strain energy,
U
. Hence
∂
U
∂
P
j
=
j
(15.35)
Equation (15.35) states that, for a linearly elastic structure, the partial derivative of
the strain energy of a structure with respect to a load gives the displacement of the
load in its own line of action. This is generally known as
Castigliano's first theorem (Part
II).
Its direct use is limited in that it enables the displacement at a particular point
in a structure to be determined
only
if there is a load applied at the point and
only
in the direction of the load. It could not therefore be used to solve for the required
displacements at B and D in the truss in Ex. 15.6.
