Civil Engineering Reference
In-Depth Information
P
P
B
C
B
C
B
C
X
BD
X
BD
1
1
Cut
45
°
A
D
A
D
A
D
F
IGURE
16.18
Analysis of a
statically
indeterminate truss
L
(a)
(b)
(c)
We are assuming that the truss is linearly elastic so that the relative displacement of
the cut ends of the member BD (in effect the movement of B and D away from or
towards each other along the diagonal BD) may be found using, say, the unit load
method as illustrated in Exs 15.6 and 15.7. Thus we determine the forces
F
a,
j
, in the
members produced by the actual loads. We then apply equal and opposite unit loads
to the cut ends of the member BD as shown in Fig. 16.18(c) and calculate the forces,
F
1,
j
in the members. The displacement of B relative to D,
BD
, is then given by
n
F
a,
j
F
1,
j
L
j
AE
BD
=
(see Eq. (viii) in Ex. 15.7)
j
=
1
The forces,
F
a,
j
, are the forces in the members of the released truss due to the actual
loads and are not, therefore, the actual forces in the members of the complete truss.
We shall therefore redesignate the forces in the members of the released truss as
F
0,
j
.
The expression for
BD
then becomes
n
F
0,
j
F
1,
j
L
j
AE
BD
=
(i)
j
=
1
In the actual structure this displacement is prevented by the force,
X
BD
, in the redun-
dant member BD. If, therefore, we calculate the displacement,
a
BD
, in the direction
of BD produced by a unit value of
X
BD
, the displacement due to
X
BD
will be
X
BD
a
BD
.
Clearly, from compatibility
BD
+
X
BD
a
BD
=
0
(ii)
from which
X
BD
is found. Again, as in the case of statically indeterminate beams,
a
BD
is a flexibility coefficient. Having determined
X
BD
, the actual forces in the members
of the complete truss may be calculated by, say, the method of joints or the method of
sections.
InEq. (ii),
a
BD
is the displacement of the released truss in the direction of BDproduced
by a unit load. Thus, in using the unit load method to calculate this displacement, the
