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

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 Page 79 - Building stability - Two-dimensional stability