Bedford, Fowler: Statics. Chapter 8: Moments of Inertia, Examples via TK Solver
Moments of Inertia
This chapter and the previous one are mainly reviews of geometric properties covered in
introductory courses in calculus. The area centroid and area moment of inertia topics are
important in a later course on beam stress analysis. For example, in a beam with an internal
moment (see next chapter) of M(x) the bending normal stress, , at a height of y, above the
centroid of the area, is given by
where is the area moment of inertia of the cross‐section of the beam. Likewise, the mass
moment of inertial of a rigid body will be extremely important in this book’s later chapters on
Here, I simply want to point out that there are two common choices for representing the
integral geometric properties of an object. The first scalar definition is used in this chapter and
the previous chapter on centroid calculations. The second matrix (or more correctly tensor)
form is used later in chapter 20. They differ in the sign of the product inertia definition, and
thereby how they transform under coordinate rotations. The tensor form is mainly employed
for three‐dimensional dynamics. Having the wrong sign on half the terms can lead to serious
errors in solid body dynamic results.
Tensor Form (see appendix chapter 20)
Measure (mass) where is the mass density, or unity for a geometric property.
Centroid (first moment) with 1, 2, 3 , ,
Inertia (second moment) ∑ , 1 , 0.
Parallel axis theorem: + (∑ ‐
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