Finally,
consider a solid wood column that is 2.828 in. x 2.828 in. and has the
same cross 
sectional area of the 2 x 4 shown previously. 
. 
e) model of 2.828 x 2.828 

f) model of 2 x 4 


Area = 8 in.^{2} 
Area = 8 in.^{2} 
I
= 2.828 in. (2.828 in.)^{3} 
Ι
= 4 in. (2 in.)^{3} 
12 
12 
= 5.33 in.^{4} 
= 2.67 in.^{4} 
. 
Fig. 132  Moment of inertia of solid
columns with the same cross sectional area 

. 
Notice
that the moment of inertia of the 2.828 inch square solid column is twice
that of the 
2 x 4
column despite the fact that their cross sectional areas are equal.
Regardless whether 
it is
solid or hollow, a column with a square cross sectional area will have a
larger moment 
of
inertia and load bearing capacity than a rectangular column of equal
cross sectional 
area,
all other factors being equal. For this reason, structural columns
are usually square 
or
circular in cross section and are often hollow because this design affords
the greatest 
load
bearing capacity for the amount of materials used. Of course if the
walls of the hollow 
column get too thin
their tendency to buckle locally, or crinkle, will increase. 

The
other dimensional component of a column is its length. To repeat,
increasing its length 
decreases a column's
load bearing capacity by the square of the increase. For example in 
Fig. 133 as the length of the columns is doubling the loading capacity is
decreasing by a 
factor of four.
Notice that the crosssectional areas of these columns do not change.
Thus 
. 


. 
. 
Fig. 133  The load bearing capacity of

S.R. narrow face = 14.4 in. / 2 in. = 7.07 
columns with doubling lengths 
S.R. wide face = 14.4 in. / 2.828 in. = 5 
(visualization models) 
Fig. 134  Slenderness ratio of a column 

. 
the
longer columns are more slender than the shorter ones, which obviously
accounts for 
their
relative weakness. Engineers have combined the forgoing factors that
determine a 
column's
strength into a single, simplified measure called the slenderness ratio.
The 
slenderness ratio is simply the column's length divided by its width, i.e.
L / D. For columns 
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Page 88 
Building stability  Moment of inertia of columns 

