[ If
Newtons and meters are used for the units of measurement then the bending
moment is 
given in
Newtonmeters (Nm)] 
. 



. 
M_{1} = P L
< 
M_{2} = P ( 2L )
< 
M_{3} = P ( 3L ) 
(demonstration models) 
Fig. 138  Bending moments of fixed
cantilever beams vs. distance of load from root 

. 
For
example, in the above figure an identical load is applied one length (L),
two lengths 
(2L),
and three lengths (3L) from the root resulting in bending moments that are
one, two, 
and
three times greater respectively. In this respect the beam acts like
the arm of a lever. 
Of
course the bending moment can also be increased by simply keeping the
length of the 
beam
constant and increasing the load. 

The
bending moment will induce stresses in the beam that will be resisted by
the internal 
forces
of the beam's material. If the reactive forces are weaker than the
stresses, the beam 
will
become unstable, or deflect, in the same direction as the force of the
load that is being 
applied
to it. The following is an idealized model of a fixed cantilever
beam showing the 

stresses induced in it by a load applied to its unsupported end.


Notice that the upper rubber band, which represents the upper 
edge of the beam, is stretched in tension when the beam deflects 
down. And the lower rubber band, which represents the bottom 
edge of the beam, is collapsed in compression. The model shows 
that the tensile forces of the upper edge of the beam react to and 
resist the tensile stresses exerted by a load, while the compressive 
forces of the lower edge resist the compressive stresses. 
. 

Fig.
139  Fixed cantilever beam bending in reaction to a load
(training aid model) 

In a
real, solid, fixed cantilever beam there is a gradual shift in internal
stresses and 
reactive
forces from being mainly tensile in the upper part of the beam to
compressive 
mainly
in the lower section in reaction to an applied load. The beam will
continue to bend 
until
the internal tensile and compressive stresses induced by the load are
balanced by the 
internal
tensile and compressive forces exerted by the beam's material. At
that point it will 
bend no more and maintain a condition of stable equilibrium. 

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Page 91 
Building stability  Cantilever beams 

