Civil Engineering Reference
In-Depth Information
From the principle of the stationary value of the total complementary energy with
respect to the load
P
2
, we have
n
∂
C
∂
P
2
=
∂
F
j
∂
P
2
−
1
δ
j
2
=
0
(ii)
j
=
from which
n
1
δ
j
∂
F
j
2
=
(iii)
∂
P
2
j
=
Note that the partial derivatives with respect to
P
2
of the fixed loads,
P
1
,
P
3
,
...
,
P
k
,
...
,
P
r
, vanish.
To complete the solution we require the load-displacement characteristics of the
structure. For a non-linear system in which, say,
δ
j
)
c
F
j
=
b
(
where
b
and
c
are known, Eq. (iii) becomes
F
j
b
1
/
c
n
∂
F
j
∂
P
2
2
=
(iv)
j
=
1
In Eq. (iv)
F
j
may be obtained from basic equilibrium conditions, e.g. the method of
joints, and expressed in terms of
P
2
; hence
∂
F
j
/∂
P
2
is found. The actual value of
P
2
is
then substituted in the expression for
F
j
and the product (
F
j
/
b
)
1
/
c
∂
F
j
/∂
P
2
calculated
for each member. Summation then gives
2
.
δ
j
is, from Sections 7.4 and 7.7, given by
In the case of a linearly elastic structure
F
j
E
j
A
j
δ
j
=
L
j
in which
E
j
,
A
j
and
L
j
are Young's modulus, the area of cross section and the length
of the
j
th member. Substituting for
δ
j
in Eq. (iii) we obtain
n
F
j
L
j
E
j
A
j
∂
F
j
∂
P
2
2
=
(v)
j
=
1
Equation (v) could have been derived directly fromCastigliano's first theorem (Part II)
which is expressed in Eq. (15.35) since, for a linearly elastic system, the complementary
and strain energies are identical; in this case the strain energy of the
j
th member is
F
j
L
j
/
2
A
j
E
j
from Eq. (7.29). Other aspects of the solution merit discussion.
We note that the support reactions at A and B do not appear in Eq. (i). This convenient
absence derives from the fact that the displacements,
1
,
2
,
...
,
k
,
...
,
r
, are the
actual displacements of the truss and fulfil the conditions of geometrical compatibility
