a)
model of 2 x 4 |
|
b)
model of 4 x 4 |
|
|
I
= 4 in. ( 2 in.)3 |
I
= 4 in. ( 4 in.)3 |
12 |
12 |
=
2.67 in.4 |
=
21.3 in.4 |
. |
Fig. 130 - Moments of inertia for columns with different
cross sections (visualization
models) |
|
. |
Notice
that doubling the thickness of the 2 x 4 to 4 x 4 inches increases the
moment of |
inertia
by a factor of eight. The 4 x 4 is much stiffer than the 2 x 4
because more of its mass |
is
concentrated farther away from the center of its cross section.
Since the critical buckling |
factor
of a column is directly proportional to its moment of inertia, doubling
the thickness |
of a
column increases its load bearing capacity, not by a factor of two as
might be thought, |
but by a
factor of eight! |
|
Consider
another example. Column c) is a solid steel column 2 in. x 2 in.
square. Column |
d) is a
hollow steel column with the same cross sectional area of mass as column
c). |
. |
c) model of a |
|
d) 2.828 x 2.828 with 2 x 2 hole |
solid 2 x 2 |
|
Area = 4 in.2 |
Area = 4 in.2 |
I
= 2 in. ( 2 in.)3 |
I
= I
(large square) - Ι
(small square) |
12 |
I
= 2.828 in. ( 2.828 in.)3_ 2 in. (2 in.)3 |
= 1.33 in.4 |
12
12 |
|
|
= 4 in.4 |
. |
Fig. 131 - Moments of inertia for columns
with equal cross sectional areas of mass |
|
. |
Despite
the fact that both columns have the same cross sectional area of mass, and
thus the |
same
weight of material per linear foot of column, the moment of inertia of the
hollow |
column is three times that of the solid column. Therefore the hollow
column will support |
three
times the weight for the same cost of materials. |
Back to
Knowhere |
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Page 87 -
Building stability - Moment of inertia of columns |
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