ENDS 231 Note Set 12 F2007abn
1
y
x
el
xdx
dA = y⋅dx
θ
pole
o
r
x
x x x
Moments of Inertia
• The cross section shape and how it resists bending and twisting is important to understanding
beam and column behavior.
• Definition: Moment of Inertia; the second area moment
∫
=dAxIy2
∫
=dAyIx2
We can define a single integral using a narrow strip:
for Ix,, strip is parallel to x for Iy, strip is parallel to y
*I can be negative if the area is negative (a hole or subtraction).
• A shape that has area at a greater distance away from an axis through its centroid will have a
larger value of I.
• Just like for center of gravity of an area, the moment of inertia can be determined with
respect to any reference axis.
• Definition: Polar Moment of Inertia; the second area moment using polar coordinate axes
∫∫∫ +== dAydAxdArJo222
yxo IIJ +=
• Definition: Radius of Gyration; the distance from the moment of
inertia axis for an area at which the entire area could be considered as
being concentrated at.
⇒= ArI xx
2
A
I
rx
x= radius of gyration in x
A
I
ry
y= radius of gyration in y
A
J
ro
o= polar radius of gyration, and ro2 = rx2 + ry2
