geometric arrangement of the members into the triangle. The triangle
then must be an 
inherently stable structure. 

With
that in mind let's take another look at the other polygonal shapes to see
how they 
might be
made stable. For example let's place an additional member in the
parallelogram 
so that
it joins two of its opposite joints together as shown in Fig. 122 a). This action, called 
triangulation since we are dividing the polygon into two triangular shaped
sections, makes 



a) square 
b) pentagon 
c) hexagon 
Fig. 122  Stabilizing polygons by triangulation
(demonstration models) 


the
parallelogram stable and forms it into a square. Let's do the same
for the pentagon. 
First we
insert one extra member so as to form a triangle with any two adjacent
edges of 
the
pentagon as shown in Fig. 122 b)
above. This helps stabilize part of the structure but 
the
other three members can still move. We add another member forming
three triangular 
sections
in all. The pentagon is now stable also. Finally, we
triangulate the hexagon in a 
similar
manner and find that it takes a minimum of three extra members to make it
stable 
(Fig.
122 c). In each of these examples, the rigidity imparted to the polygon
is due only to 
the fact
that it was triangulated since all of its joints are hinged.
Triangulation changed 
them
from being inherently unstable to being inherently stable. 

Using
such hit and miss methods ancient builders eventually discovered the
principle of 
triangulation in their attempts to build stable structures. But
structural engineers do not 
like
being eaten by wolves any more than the third little pig did. So
they wisely use a 
mathematical equation, attributed to Euler, that gives a good indication if
a polygonal 
structure with flexible joints is inherently stable or not. And if
it is not stable the equation 
predicts
how many additional bracing members it takes to stabilize it: 

M = 2 J  3
where M = members 
J = joints 

Back to
Knowhere 

Page 79 
Building stability  Twodimensional stability 

