1. Engineering Mechanics: Statics
Appendix A: Area Moments of Inertia

2. Moment of Inertia
When forces are distributed continuously over an area, it is often necessary to calculate moment of these forces about some axis (in or perpendicular to the plane of area)
Frequently, intensity of the distributed force is proportional to the distance of the line of action from the moment axis, p = ky
dM = y(pdA) = ky2dA
I is a function of geometry only!
Hydrostatic pressure
Bending moment in beam
Moment of inertia of area/
Second moment of area (I )
Torsion in shaft

3. Definitions Rectangular moment of inertia Polar moment of inertia
-- Moment of inertia about x-axis
Notice that Ix, Iy, Iz involve the square of the distance from the inertia axis
-- always positive!
dimensions = L4 (ex. m4 or mm4)

4. Sample Problem A/1 -- Must remember!: for a rectangular area,
Determine the moments of inertia of the rectangular area about the centroidal x0- and y0-axes, the centroidal polar axis z0 through C, the x- axis, and the polar axis z through O.
-- Must remember!: for a rectangular area,
: for a circular area, see sample problem A/3

5. Radius of Gyration For an area A with moment of inertia Ix and Iy
Visualize it as concentrated into a long narrow strip of area A a distance kx from the x-axis. The moment of inertia about x-axis is Ix. Therefore,
The distance kx = radius of gyration of the area about x-axis

6. Radius of Gyration
Similarly,
Do not confused with centroid C!

7. Transfer of Axes
Moment of inertia of an area about a noncentroidal axis
The axis between which the transfer is made must be parallel
One of the axes must pass through the centroid of the area
and with
the centroid on x0-axis
Parallel-axis theorems

8. Composite Areas Centroid of composite areas: Part Area, A Sum SA S S
100 mm
400 mm
Part Area, A
Sum SA S S

9. Composite Areas I = SI + SAd2
The moment of inertia of a composite area about a particular axis is the sum of the moments of inertia of its component parts about the same axis.
I = SI + SAd2
The radius of gyration for the composite area cannot be added, k = I/A
Part Area, A dx dy Adx2 Ady2
Sum SA SAdx2 SAdy2 S S

10. Example A/7
Calculate the moment of inertia and radius of gyration about the x-axis for the shaded area shown

11. Products of Inertia Unsymmetrical cross section Ixy = xydA
may be positive, negative or zero
Ixy = 0 when either the reference axes is an axis of symmetry
because x(-y)dA cancel x(+y)dA
Transfer of Axes
Ixy = (x0+dy)(y0+dx)dA
Ixy = Ixy + Adx dy

12. Sample Problem A/8 & A/10
Determine the product of inertia of the area shown with respect to the x-y axes. 

13. Rotation of Axes To calculate the moment of inertia of an area
about an inclined axes
Ix’ =  y’2 dA =  (ycos q – xsin q )2 dA
Iy’ =  x’2 dA =  (ysin q – xcos q )2 dA
-- expand & substitute sin2q = (1- cos 2q)/2
cos2q = (1+ cos 2q)/2

14. Rotation of Axes called “Principal Axes of Inertia”
The angle which makes Ix’ and Iy’ either max or min
dIx’/dq = (Iy - Ix)sin 2q - 2Ixycos 2q = 0
The critical angle a:
This equation gives two value of 2a [tan 2a = tan (2a+p) ]
obtain two values for a (differ by p/2)
axis of minimum moment of inertia
axis of maximum moment of inertia
called “Principal Axes of Inertia”

15. Mohr’s Circle of Inertia
Draw x-axis as I and y-axis as Ixy
Plot point A at (Ix, Ixy) and B at (Iy, -Ixy)
Find the center of the circle at O R S
4. Radius of the circle is OA or OB
Imax
Imin
Imax = O + R and Imin = O - R
5. Angle 2a is found from AS and OS as

16. Sample Problem A/11
Determine the orientation of the principle axes of inertia through the centroid of the angle section and determine the corresponding maximum and minimum moments of inertia.