Centroid and Moment of Inertia Notes

MEC 244 – Strengths
Lesson 8
Chapter 8
Centroid Review and
Moment of Inertia
Center of Gravity – Centroids
• Centroids are CG in 2 dimensions
• Table 7-2 shows standard formulae
• Note that x-bar and y-bar are based
on origin of x-y co-ordinates at
bottom left
1
2
Moment of an Area
Moment of Area Example
• Imagine the area of a surface
equals pounds of force at the
centroid
• Then create a moment = “area”
times distance from centroid to
axis
• M x = A x (y-bar)
• M y = A x (x-bar)
• End result = in 3
(but not cubic in.)
y
x-bar
y-bar
3
x
4
Calculating Centroids of Objects
• If area x centroidal dist. = moment
• And if an area is made up of
various shapes of known areas, we
can determine individual moments,
add them together and divide by
area to get centroid of entire object.
Sample Spreadsheet
Area Dimension Area x-bar A x (x-bar) y-bar A x (y-bar)
1 150x12 1800 6 10800 75 135000
2 88x12 1056 56 59136 6 6336
Total Area= 2856 Total A*x= 69936 Total A*y= 141336
Total x-bar= 24.5 Total y-bar= 49.5
5
6
1

Areas of Volumes a la Centroid
• Area = length x width
• Volume = area x dist. Centroid travels
• If centroid rotates about an axis, a
circular volume is formed
–Rotate rectangle = a cylinder
–Rotate a circle = a toroid (a donut)
Volume by Rotation
creates a cone
7
8
Moment of Inertia
• A mathematical term that
determines the “stiffness to
bending” that a beam would have,
based solely on its cross-sectional
shape.
• Units are in 4 – Table 8-1
9
Moment of Inertia Theory
• Important: I c is based around
centroidal axis, NOT like moment
of area.
• Moment of Inertia is the Sum of all
small areas x (centroidal distance) 2
(above and below the axis)
• I c = Σ[(areas) x (y-bars) 2 ]
10
Sample Problem 9
Transfer Formula
• Note that calculations are based
around the centroidal axis, both
about x and y.
• Note how beam is “flimsier” when
bent about the Y-Y axis
• Material does not matter in
• Composite beams or beams of various
shapes (like I-beams) are calculated
separately, then added.
• I a-a = I x + Ad 2
• Rules: If new axis = part axis, then just
calculate I x
• Rule: If new axis is NOT on part axis,
still figure I x but then add Ad 2 for each
figuring Moment of Inertia
11
12
2

Moments for Composite Areas
Example:
• Separate the beam into simple
areas. Use the Transfer Formula if
needed to calculate I for areas not
located at the centroidal axis.
• All individual moments of inertia
are always added (cumulative)
• Subtract any areas representing an
• Find Moment of Inertia for I-beam
air space 13
14
x
1
2
2
6
1
x
Reserved for board work
15
3