Civil Engineering Reference
In-Depth Information
of statical equilibrium. Another form of the indeterminacy of a structure is expressed
in terms of its
degrees of freedom
; this is known as the
kinematic indeterminacy, n
k
,of
a structure and is of particular relevance in the stiffness method of analysis where the
unknowns are the displacements.
A simple approach to calculating the kinematic indeterminacy of a structure is to
sum the degrees of freedom of the nodes and then subtract those degrees of freedom
that are prevented by constraints such as support points. It is therefore important
to remember that in three-dimensional structures each node possesses 6 degrees of
freedom while in plane structures each node possess three degrees of freedom.
E
XAMPLE
16.1
Determine the degrees of statical and kinematic indeterminacy of
the beam ABC shown in Fig. 16.10(a).
F
IGURE
16.10
Determination of the
statical and kinematic
indeterminacies of the
beam of Ex. 16.1
r
1
r
2
r
2
A
B
C
A
C
B
(a)
(b)
The completely stiff structure is shown in Fig. 16.10(b) where we see that
M
=
4 and
N
3. The number of releases,
r
, required to return the completely stiff structure to
its original state is 5, as indicated in Fig. 16.10(b); these comprise a moment release
at each of the three supports and a translational release at each of the supports B and
C. Therefore, from Eq. (16.3)
=
n
s
=
3(4
−
3
+
1)
−
5
=
1
so that the degree of statical indeterminacy of the beam is 1.
Each of the three nodes possesses 3 degrees of freedom, a total of nine. There are
four constraints so that the degree of kinematic indeterminacy is given by
n
k
=
9
−
4
=
5
E
XAMPLE
16.2
Determine the degree of statical and kinematic indeterminacy of
the truss shown in Fig. 16.11(a).
F
IGURE
16.11
Determinacy of the
truss of Ex. 16.2
(a)
(b)
