CE 201 - Statics

Chapter 10 – Lecture 1

MOMENTS OF INERTIA

Objective

Develop a method for determining the moment of inertia for an area

Moment of inertia is an important parameter to be determined when designing a structure or a mechanical part

Definition of Moments of Inertia for an Area

• To find the centroid of an area by the first moment of the area about an axis was determined ( x dA )

• Integral of the second moment is called moment of inertia ( x2 dA)

• Consider the area ( A ) • By definition, the moment of inertia of the differential area about

the x and y axes are dIx and dIy• dIx = y2dA Ix = ( y2 dA)• dIy = x2dA Iy = ( x2 dA)• to find the moment of inertia of the differential area about the

pole (point of origin) or z-axis, ( r ) is used

( r ) is the perpendicular distance from the pole to dA for the entire area

Jo = r2 dA = Ix + Iy (since r2 = x2 + y2 )

A

dA

y

x

O

y

x

r

Definition of Moments of Inertia for an Area

Parallel-Axis Theorem for an Area

If moment of inertia of an area about an axis passing through its centroid is known, then it will be convenient to determine the moment of inertia about any parallel axis by the parallel-axis theorem.

How to derive the theorem?

• find the moment of inertia of the area about the axis

• Ix = (y + dy)2 dA• = y2 dA + 2dy y dA + d2y

dA• The first integral is the moment of

inertia about the centroid axis Ix• The second integral is zero since the x

axis passes through the area's centroid C

y dA = y dA = 0 (since y = 0)• Ix = Ix + A d2y• Iy = Iy + A d2x• Jo = Jc + A d2

A

dA

dy

dx

O

y

x

d

xy

y

xC

MOMENTS OF INERTIA FOR COMPOSITE AREAS

Composite areas consist of simpler parts

If moment of inertia of each part is known or can be determined, then the moment of inertia of the composite area is the algebraic sum of moments of inertia of all parts

Procedure for Analysis

Divide the area into simpler parts Indicate the perpendicular distance from the centroid of

each part to the reference axis Find moment of inertia of each part about its centroidal

axis Use parallel-axis theorem to find moments of inertia about

the reference axis (I = ¯I + A d2 ) Find moment of inertia of the entire area by summing

moments of inertia of all parts algebraically If a composite body has a hole, then its moment of inertia

is found by subtracting the moment of inertia for the hole from the moment of inertia of the entire part including the hole.