Centroid and Moment of Inertia Notes

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
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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
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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 ]
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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
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