Furthermore, inherent stability in structures depends no only on the
number of structural 

members used but also their placement. For example the 
polygon to the left has the minimum
number of members 
called for by Euler's equation yet it is unstable. Obviously 
then we must seek a deeper understanding of the factors 
responsible for a structure's stability other than just its
topology. 
We
must study the dynamic interplay between the internal and 
external forces a structure experiences when it is stressed. 

21 = 2 (12)  3 
◄
Fig. 125  Unstable octagon that satisfies Euler's
equation 
. 

Exercise: 1) How many more members are needed to stabilize Fig. 125
? Placed where? 

2) Find other polygonal structures that should be stable according
to Euler's 


. 
Forces and reactions 
. 
In the
previous experiment we determined empirically that a polygon is unstable
if its 
shape is
distorted when an outside force acts on it. Newton's first law
states that an object 
that is
at rest will remain at rest provided it is not subjected to an unbalanced
force. When 
you push
on an unstable polygon the structural members move from their at rest
position 
and
deform because the total force pushing on it is greater than the
structure's ability to 
push
back and resist being deformed. If we assign a negative value to the
forces pushing 
on the
structure and a positive value to the forces resisting being pushed, then
the sum of 
the
forces do not equal zero. As a result the structure will be deformed
in the direction that 
the
excess force is pushing it. If, however, the sum of the forces
equals zero then the 
structure is stable and will not deform. 
. 

For example, if you push down on the edge of a Polymorf panel that
is 
resting upright on a table it does not move downward because the 
downward acting force, or load, that you are applying to it (orange
arrow) 
is
balanced by the reaction of the upward force of the table pushing
back 
(green arrow). The harder you push down the harder the table
pushes 
back. The downward (negative) force plus the upward (positive
force) 
equals zero. We say that the forces acting on the panel are in
equilibrium 
and the panel is stable. 
◄ Fig. 126  Pushing down on a panel
induces compressive stresses 

. 
Notice
that the panel is being squeezed between two forces that are external to
it. Yet it is 
not
deformed because the atoms it is made of resist being deformed.
Internal forces, 
Back to
Knowhere 

Page 81 
Building stability  Forces and reactions 

