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

                    equation but are not.


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,

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Page 81 - Building stability - Forces and reactions

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