MOMENT OF INERTIA

Type of moment of inertia Moment of inertia of Area Moment of inertia of mass

Also known as second moment Why need to calculate the moment of Inertia?

To measures the effect of the cross sectional shape of a beam on the beam resistance to a bending moment

Application Determination of stresses in beams and columns

Symbol I – symbol of area of inertia

Ix, Iy and Iz

Application : Design Steel ( Section properties)

Moment of Inertia of Area

xhxAxIy dd 22

x

y

C

b

h

dy

x

y

C

b

h

dy

ybA dd Area of shaded element,Moment of inertia about x-axis

ybyAyIx dd 22

Integration from h/2 to h/2

2

2

2dh

h

x ybyI

123

32

2

3 bhbyh

h

xhA dd Area of shaded element,Moment of inertia about y-axis

Integration from b/2 to b/2

2

2

2db

b

y xhxI

123

32

2

3 hbhxb

b

dx

36

3bh

36

3hb

4

161

r

4

41

r

4

161

r

4

81

r

4

81

r

xI yI

12

3bh12

3hb 22

121

hbbh

yx

b

h

y

x

y

y

xr

yx

y

x

r

x

yb

h

y

x

Shape J = polar moment of inertia

1. Triangle

2. Semicircle

3. Quarter circle

4. Rectangle

5. Circle

Table 6.2. Moment of inertia of simple shapes

4

81

r

x

y

r 4

41

r 4

41

r 4

21

r

PARALLEL - AXIS THEOREM There is relationship between the moment of inertia about two

parallel axes which is not passes through the centroid of the area.

From Table 6.1; Ix = and Iy =

The centroid is, ( , ) = (b/2, h/2) Moment of inertia about x-x axis, Ixx = Ix + Ady

2 where dy is distance at centroid y

Moment of inertia about y-y axis, Iyy = Iy + Asx2

where sx is distance at centroid x

12

3bh

12

3hb

y

x

x

y

b

h

x y

Example 6.2

PART AREA(mm2) y(mm) x(mm) Ay (103)(mm3) Ax (103) (mm3)

1 60(200) = 12000

200/2 = 100

60/2 = 30 1200 360

2 160(60)=9600 60/2 = 30 60 + [160/2] = 140

288 1344

3 60(200) = 12000

200/2 = 100

220 +60/2= 250

1200 3000

Σ: 33 600 Σ: 2688 x 103 Σ: 4704 x 103

140 mm

60 mm

160 mm

60mm 60 mm

x

y

mmA

Axx 140

mmA

Ayy 80

Determine centroid of composite area

140 mm

60 mm

160 mm

60mm 60 mm

x

y

1

2

3

PART AREA (A)(mm2)

Ix = bh3/12(106) (mm4) dy = |y-y|

(mm)Ady

2(106)(mm4)

1 60(200) = 12000

60(2003)/12 = 40|100– 80| = 20 4.8

2 160(60)=9600 160(603)/12 = 2.88 |30 – 80| = 50 24

3 60(200) = 12000

60(2003)/12 = 40|100 – 80| =20 4.8

Σ: 33 600

Second moment inertia

xxI [Ix + Ady2]1 + [Ix + Ady

2]2 +[Ix + Ady2]3

= [44.8 + 26.88 + 44.8] x106 = 116.48 x 106 mm4

PART AREA(mm2)Iy = b3h/12(106) (mm4) Sx=|x-x| (mm) Asx

2(106)(mm4)

1 60(200) = 12000 603(200)/12=3.6

|30-140|=110145.2

2 160(60)=9600 1603(60)/12=20.48

|140 – 140 |= 00

3 60(200) = 12000 603(200)/12=3.6

|250 – 140|=110 145.2

Σ: 33 600

yyI [Iy + As2]1 + [Iy + As2]2 +[Iy + As2]3

= [148.8 + 20.48 + 148.8] x106

= 318.08 x 106 mm4