The following figures analyze the stability of the previous polygons using Euler's equation.

 a) hexagon b) pentagon c) square d) triangle       6<2(6)- 3 9=2(6)- 3 5<2(5)- 3 7=2(5)- 3 4<2(4)- 3 5=2(4)- 3 3=2(3)- 3 unstable stable unstable stable unstable stable stable (need + 3 (need + 2 (need + 1 members) members) members) Fig. 123 - Analyzing the stability of polygons with Euler's equation M = 2 J - 3

Notice that the equation only refers to how many structural elements (members and joints)

each polygon has.  That is, it describes the topology of the polygonal structure regardless

of the length of its members or the shape of the polygon (regular or irregular, segmented

or not).  This is the same type of topological analysis that was used in the previous lesson

to describe the structure of crystals.

 EXERCISE: Derive Euler's equation for stable polygons using simple
 algebra or a computer graphing program.

Euler's equation gives a fairly reliable indication of the minimum number of members

required for stable polygons.  Obviously stable polygons have more members than the

required minimum.  For example the square can be stabilized with seven or eight

members instead of the required minimum of five.  And eleven or twelve members can be    7 > 2 (4) - 3 8 > 2 (4) - 3 11 > 2 (6) - 3 12 > 2 (6) - 3 redundant redundant redundant redundant Fig. 124 - Stable but redundant polygons  (demonstration models)

used to stabilize the hexagon instead of a minimum of nine.  Such structures are said to be

stable but redundant.

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 Page 80 - Building stability - Two-dimensional stability