Civil Engineering Reference
In-Depth Information
T
ABLE
16.3
F
1,
j
L
j
Member
L
j
(mm)
F
1,
j
F
a,
j
(N)
AB
4000
1.33
7111.1
−
700
BC
3000
1.0
3000.0
−
525
−
CD
4000
1.33
7111.1
700
DA
3000
1.0
3000.0
−
525
AC
5000
−
1.67
13 888.9
875
DB
5000
−
1.67
13 888.9
875
=
48 000.0
The forces,
F
a,
j
, in the members of the complete truss are given in the final column of
Table 16.3.
An alternative approach to the solution of statically indeterminate trusses, both self-
straining and otherwise, is to use the principle of the stationary value of the total
complementary energy. Thus, for the truss of Ex. 16.8, the total complementary energy,
C
, is, from Eq. (15.39), given by
F
j
n
C
=
δ
j
d
F
j
−
P
C
0
j
=
1
in which
C
is the displacement of the joint C in the direction of
P
. Let us suppose
that the member BD is short by an amount
λ
BD
(i.e. the lack of fit of BD), then
F
j
n
C
=
δ
j
d
F
j
−
P
C
−
X
1
λ
BD
0
j
=
1
From the principle of the stationary value of the total complementary energy we have
n
1
δ
j
∂
F
j
∂
C
∂
X
1
=
∂
X
1
−
λ
BD
=
0
(16.8)
j
=
Assuming that the truss is linearly elastic, Eq. (16.8) may be written
n
F
j
L
j
A
j
E
j
∂
F
j
∂
X
1
−
∂
C
∂
X
1
=
λ
BD
=
0
(16.9)
j
=
1
or since, for linearly elastic systems, the complementary energy,
C
, and the strain
energy,
U
, are interchangeable,
n
F
j
L
j
A
j
E
j
∂
F
j
∂
X
1
=
∂
U
∂
X
1
=
λ
BD
(16.10)
j
=
1
Equation (16.10) expresses mathematically what
is generally referred to as
Castigliano's second theorem which states that
