The
following figures analyze the stability of the previous polygons using
Euler's equation. |
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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 |
|
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members) |
|
members) |
|
members) |
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Fig. 123 - Analyzing the stability of polygons
with Euler's equation M = 2 J - 3 |
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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. |
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EXERCISE: Derive Euler's equation for stable polygons using simple |
|
algebra or a computer graphing program. |
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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 |
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Fig. 124 - Stable but redundant polygons
(demonstration models) |
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|
used to stabilize the
hexagon instead of a minimum of nine. Such structures are said to be |
stable but redundant. |
Back to
Knowhere |
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Page 80 -
Building stability - Two-dimensional stability |
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