MOMENTS OF INERTIA

1
MOMENTS OF INERTIA

• Distributed forces act on volume or surface
  elements.
• Gravity
• Hydrostatic pressure in a fluid
• Bending stress in beams

2
Center of Gravity for Area

• Center of gravity ?W gt?A
• First moments Qx ?ydA Qy ?xdA

3
Bending Stresses in Beams

• Distributed forces proportional to area element
  and distance from an axis.

?F ky?A y?F ky2?A
4
Area Properties

• Area (magnitude) A
• First moment (geometrical center)
• Second moment (distribution) Ix, Iy

Areas, geometrical centers the same.
5
Second Moments of Area

• Second moments about x (Ix), y (Iy) axes.

6
Properties

• The dimensions of Ix, Iy are length4.
• Ix, Iy are always positive.
• Ix, Iy measure how area is distributed with
  respect to a line, i.e., how much and how far.
• The origin of the x,y axes need not be at the
  centroid. It can be any point.

7
Ix By Integration

• Element area dA (a-x)dy
• Element 2nd moment
• dIx y2dA y2(a-x)dy

8
Iy By Integration

• Element area dA ydx
• Element 2nd moment
• dIy x2dA x2ydx

9
The Rectangle

• Compute Ix, Iy .
• dA bdy
• dIx y2bdy

10
Ix by Vertical Strip

• b dx , h y

11
Polar Moment of Inertia

• Polar moment J0

12
Radius of Gyration

• A, Ix

kx radius of gyration