Engineering Mechanics: Statics

Appendix A: Area Moments of Inertia

Moment of Inertia

When forces are distributed continuously over an area, it is often necessary to calculate moment of these forces about some axis (in or perpendicular to the plane of area)

Frequently, intensity of the distributed force is proportional to

the distance of the line of action from the moment axis, p = ky

dM = y(pdA) = ky2dA

I is a function of geometry only!

Hydrostatic pressure

Bending

moment in beam

Torsion in shaft

2M k y dA

Moment of inertia of area/ Second moment of area (I )

Rectangular moment of inertia

Polar moment of inertia

Definitions

2

2

x

y

I y dA

I x dA

-- Moment of inertia about x-axis

2 x yzI r dA I I

• Notice that Ix, Iy, Iz involve the square of the distance from the inertia axis -- always positive!

• dimensions = L4 (ex. m4 or mm4)

Sample Problem A/1

Determine the moments of inertia of the rectangular area about the centroidal x0- and y0-axes, the centroidal polar axis z0 through C, the x-axis, and the polar axis z through O.

-- Must remember!: for a rectangular area,

: for a circular area, - see sample problem A/3

3 3

, 12 12x ybh hbI I

4

4x yrI I

Table D/3

For an area A with moment of inertia Ix and Iy

Visualize it as concentrated into a long narrow strip of area A a distance kx from the x-axis. The moment of inertia about x-axis is Ix. Therefore,

The distance kx = radius of gyration of the area about x-axis

Radius of Gyration

2x xk A I

x xk I A

Similarly,

Do not confused with centroid C!

Radius of Gyration

y yk I A

z zk I A 2 2 2z x yk k k

Transfer of Axes

Moment of inertia of an area about a noncentroidal axis

The axis between which the transfer is made must be parallel

One of the axes must pass through the centroid of the area

2 20

( )x xdAdI y y d dA

2 2

0 0 2x x x xd dA d dA dI I y y dA

2x x xI I Ad

0Ay 0 0y and with the centroid on x0-axis

2

2

2

x x x

y y y

z z

I I Ad

I I Ad

I I Ad

Parallel-axis theorems

Composite Areas

Centroid of composite areas:

i i

i

A xx

A

Part Area, A

Sum SA S S

x y AyAx

Ax Ay

i i

i

Ayy

A

100 mm

400 mm

400 mm

100 mm

Composite Areas The moment of inertia of a composite area about a particular

axis is the sum of the moments of inertia of its component parts

about the same axis.

I = SI + SAd2

o Find the moment of inertia of the T-section

o The radius of gyration for the composite area cannot be added, k = I/A

xIPart Area, A dx dy Adx2 Ady

2 yI

Sum SA SAdx2 SAdy

2 S S xI yI

Products of Inertia

Unsymmetrical cross section

Ixy = xydA

may be positive, negative or zero

Ixy = 0 when either the reference axes is an axis of symmetry

because x(-y)dA cancel x(+y)dA

Transfer of Axes

Ixy = (x0+dy)(y0+dx)dA

xy xy x yI I Ad d

Sample Problem A/8 & A/10

Determine the product of inertia of the area shown with respect to the x-y axes.

Rotation of Axes

To calculate the moment of inertia of an area

about an inclined axes

Ix’ = y’2 dA = (ycos q – xsin q )

2 dA

Iy’ = x’2 dA = (ysin q – xcos q )

2 dA

-- expand & substitute sin2q = (1- cos 2q)/2

cos2q = (1+ cos 2q)/2

q q

q q

q q

'

'

' '

cos2 sin22 2

cos2 sin22 2

sin2 cos22

x y x y

x xy

x y x y

y xy

x y

x y xy

I I I II I

I I I II I

I II I

222 2

' ' ' 2 4

x yx y

x x y xy

I II II I I

Mohr’s Circle of Inertia

1. Draw x-axis as I and y-axis as Ixy

2. Plot point A at (Ix, Ixy) and B at (Iy, -Ixy)

3. Find the center of the circle at O

2 2R OS AS

R

S Imax Imin

6. Imax = O + R and Imin = O - R

tan2AS

OS

5. Angle 2 is found from AS and OS as

4. Radius of the circle is OA or OB

Rotation of Axes

The critical angle :

This equation gives two value of 2

[tan 2 = tan (2+) ]

2tan2

xy

y x

I

I I

obtain two values for (differ by /2)

axis of minimum moment of inertia

axis of maximum moment of inertia

called “Principal Axes of Inertia”

max

min

ave

ave

I I R

I I R

Sample Problem A/11

Determine the orientation of the principal axes of

inertia through the centroid of the angle section and

determine the corresponding maximum and minimum

moments of inertia.