bme204_supplement lecture on moment of inertia

Geometric Properties of Plane Areas

-From Appendix C of CD- centroids

- second moment of area (i.e. moment of inertia, polar moment of inertia)

-second moment of inertia about inclined axis-parallel axes theorem

- Mohr’s circle method for MOI skipped

First Moment of Area-Centroid

First moments of area:

Centroids: zandy

First Moment of Area-Centroid

• Composite areas– Subdivide– Select coordinate axes– Find the overall area– Apply the composite

centroid formula

Second Moment of an Area

Moment of inertia, appear in the bending of beams

Polar moment of inertia appear in the torsion of beams

Second Moment of and Area• Moment of inertia of a rectangle

Second Moment of an Area• Polar moment of inertia of a circle about its center

Second Moment of an Area• Parallel axis theorem: shift the coordinate axis away

from the centroid, what will be the new MOI?

Second Moment of an Area• Calculate the MOI wrt y’.

Second Moment of an Area

• Composite areas– Subdivide– Establish centroidal

reference axis for each subdivision

– Apply parallel axes theorem for each division

– Sum them up

(Iy)i

Second Moment of an Area

• Product of inertia: required in the bending of asymmetric beams

Parallel axis theorem

Composite areas

For the rectangle about y-z axes:

Second Moment of an Area

• Moment of inertia about an inclined axis

2sin2cos22' yz

zyzyz I

IIIII

Second Moment of an Area

• Principal moments of inertia: the angle at which formulations in the previous slide for primed axes attain a minimum and a maximum.

Second Moment of an Area