Civil Engineering Reference
In-Depth Information
X
X
(a)
(b)
u
x
y
u
z
u
y
u
z
z
x
z
x
u
y
X
F
IGURE
16.2
Statical
indeterminacy
of a ring
y
u
x
(c)
RINGS
The simplest approach is to insert constraints in a structure until it becomes a series
of completely stiff
rings
. The statical indeterminacy of a ring is known and hence that
of the completely stiff structure. Then by inserting the number of releases required to
return the completely stiff structure to its original state, the degree of indeterminacy
of the actual structure is found.
Consider the single ring shown in Fig. 16.2(a); the ring is in equilibrium in space
under the action of a number of forces that are not coplanar. If, say, the ring is cut
at some point, X, the cut ends of the ring will be displaced relative to each other as
shown in Fig. 16.2(b) since, in effect, the internal forces equilibrating the external
forces have been removed. The cut ends of the ring will move relative to each other
in up to six possible ways until a new equilibrium position is found, i.e. translationally
along the
x
,
y
and
z
axes and rotationally about the
x
,
y
and
z
axes, as shown in Fig.
16.2(c). The ring is now statically determinate and the internal force system at any
section may be obtained from simple equilibrium considerations. To rejoin the ends of
the ring we require forces and moments proportional to the displacements, i.e. three
forces and three moments. Therefore at any section in a complete ring subjected to
an arbitrary external loading system there are three internal forces and three internal
moments, none of which may be obtained by statics. A ring is then six times statically
indeterminate. For a two-dimensional system in which the forces are applied in the
plane of the ring, the internal force system is reduced to an axial force, a shear force
and a moment, so that a two-dimensional ring is three times statically indeterminate.
