Polymorf - Knowhere - Cantilever beam

Like columns, the degree of bending the beam exhibits will also depend on the modulus of

elasticity, E, of the beam's material. The higher the material's E the stiffer the beam is and

the less it will tend to bend. If the elastic limit of the material is not exceeded the beam

will return to its unloaded shape after the load has been removed. However, if the elastic

limit has been exceeded its shape will remain distorted or bent even after the load has

been removed. If the stresses generated by the load are significantly larger than the tensile

or compressive strength of the beam it may fracture or break.

And like columns, the higher the moment of inertia, Ι, of its cross-sectional area the stiffer

the beam will be and the less it will tend to bend. Unlike a column though, in which I is

determined by its narrower dimension, for a beam with an unsymmetrical cross-section, Ι

depends on the orientation of the beam. Take the solid wood 2 x 4 rectangular board

shown previously for example. As before two moments of inertia can be calculated for it



I = B ( H )³
12



	I = 4 in. ( 2 in. )³	Ι = 2 in. ( 4 in. )³
	12	12
	= 2.67 in.⁴	= 10.67 in.⁴
Fig. Moment of inertia for identical beams with different orientations (visualization models)

depending on which dimension is designated as H, the height (thickness), or B, the width.

For a beam the dimension of the horizontally oriented side upon which the load is applied

is called B and the vertical side is H. Notice that simply orienting the 2 x 4 so its longest

cross sectional dimension is oriented vertically increases its load bearing capacity by a

factor of four. This is because more of the mass of the vertically oriented beam is placed

farther above and below the axis of the beam than the horizontal beam. Recall that the

bending moment experienced by a beam is greater the farther the load is applied from its

root. So too the farther away the material of a beam is situated from its longitudinal axis

the greater is the moment that the tensile and compressive forces of that material can exert

in resisting the stresses induced by the load. Thus the beam is stiffer. That is why a plank

oriented horizontally makes a good diving board. But oriented vertically can be used as a

supporting beam for an overhanging porch.

Exercise: Construct a rubber band tensioned cantilever beam with Polymorf. Measure

the bending moment for loads placed at different distances from the root. Graph

and compare your results. (details)

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Page 92 - Building stability - Cantilever beams