Engineering Statics

F-22


Two channels and two plates are used to form the column section shown. Determine the moments of inertia and radii of gyration of the combined section with respect to the centroidal axis.

The body shown is separated into four sections.  The channels are 
standard shapes, hence their areas and centroidal moments of inertia are 
gotten from tables.
The moment of inertia of the top plate is found about its centroidal axis.
The moment of inertia of A<sub>3</sub> about the y axis is found by 
use of the transfer of axis theorem.
Note that d=100+16.1.  100 is the distance from the center to the edge 
of the channel and 16.1 is the distance from the edge of the channel to 
the centroid of the channel given in tables.
The total moment of inertia is found by adding the moments of inertia 
of the four sections.  Note that each moment of inertia has been found 
about the same axis.  The moments of inertia of body 1 and 2 are equal as 
are those of body 3 and 4.
Numbers are substituted.

The moment of inertia of body 1 about its centroidal axis is found.
The transfer of axis theorem is used.
The total moment of inertia is found by adding the three moments of inertia.
Numbers are substituted.
Arithmetic is performed.
The radius of gyration is defined as the square root of the moment of 
inertia divided by the area.  Numbers are substituted for k<sub>x</sub>.
Arithmetic is performed.
The radius of gyration, k<sub>y</sub> is found.
Arithmetic gives the answer.