Polymorf - Knowhere - Cantilever beams

[ If Newtons and meters are used for the units of measurement then the bending moment is

given in Newton-meters (N-m)]


.
M₁ = P L <	M₂ = P ( 2L ) <	M₃ = 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