Civil Engineering Reference
In-Depth Information
F
IGURE
16.8
Determination of
the degree of statical
indeterminacy of a
structure
(a)
(b)
(c)
equivalent is shown in Fig. 16.8(b) where we observe by inspection that it consists of
three rings, each of which is six times statically indeterminate so that the completely
stiff structure is 3
18 times statically indeterminate. Although the number of
rings in simple cases such as this is easily found by inspection, more complex cases
require a more methodical approach.
×
6
=
Suppose that the members are disconnected until the structure becomes singly con-
nected as shown in Fig. 16.8(c). (A singly connected structure is defined in the same
way as a singly connected foundation.) Each time a member is disconnected, the num-
ber of nodes increases by one, while the number of rings is reduced by one; the number
of members remains the same. The final number of nodes,
N
, in the singly connected
structure is therefore given by
N
=
M
+
1
M
=
number of members)
Suppose now that the members are reconnected to form the original completely stiff
structure. Each reconnection forms a ring, i.e. each time a node disappears a ring is
formed so that the number of rings,
R
, is equal to the number of nodes lost during the
reconnection. Thus
N
−
R
=
N
where
N
is the number of nodes in the completely stiff structure. Substituting for
N
from the above we have
R
=
M
−
N
+
1
In Fig. 16.8(b),
M
3 as deduced by inspection. There-
fore, since each ring is six times statically indeterminate, the degree of statical
indeterminacy,
n
s
, of the completely stiff structure is given by
=
10 and
N
=
8 so that
R
=
n
s
=
6(
M
−
N
+
1)
(16.1)
For an actual entire structure, releases must be inserted to return the completely stiff
structure to its original state. Each release will reduce the statical indeterminacy by 1,
