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) |
|
. |
Back to
Knowhere |
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Page 92 -
Building stability - Cantilever beams |
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