Moment of Inertia
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Transcript Moment of Inertia
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
y
y
dx
dy
h
C
dy
x
h
C
b
x
b
Area of shaded element, dA bdy
Area of shaded element, dA hdx
Moment of inertia about x-axis
Moment of inertia about y-axis
I y x 2dA hx 2dx
I x y 2dA by 2dy
Integration from h/2 to h/2
Integration from b/2 to b/2
h2
Ix
by
b2
2
dy
h 2
Iy
hx dx
2
b 2
h2
by 3
bh3
3 h 2 12
b2
hx 3
b3h
3 b 2 12
Table 6.2. Moment of inertia of simple shapes
Shape
bh 3
36
b3h
36
1 4
r
8
1 4
r
8
1 4
r
4
1 4
r
16
1 4
r
16
1 4
r
8
y
bh3
12
b3h
12
1
bhb2 h2
12
x
1 4
r
4
1 4
r
4
1 4
r
2
x
h
y
x
b
y
2. Semicircle
r
3. Quarter circle
Iy
y
1. Triangle
y
x
y
r
y
x
yx
4. Rectangle
x
h
5. Circle
y
b
x
J = polar moment
Ix
r
of inertia
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.
y
h
x
y
x
b
From Table 6.1; Ix =
The centroid is, ( x ,
bh3
12
y
and Iy =
b 3h
12
) = (b/2, h/2)
Moment of inertia about x-x axis, Ixx = Ix + Ady2
where dy is distance at centroid y
Moment of inertia about y-y axis, Iyy = Iy + Asx2
where
sx is distance at centroid x
Example 6.2
y
y
140 mm
140 mm
60 mm
60 mm
1
3
2
x
x
60mm 160 mm 60 mm
60mm 160 mm 60 mm
Determine centroid of composite area
PART
AREA(mm2)
y(mm)
x(mm)
1
60(200) =
12000
160(60)=9600
200/2 =
100
60/2 = 30
60(200) =
12000
200/2 =
100
2
3
Σ: 33 600
Ay (103)(mm3)
Ax (103) (mm3)
60/2 = 30
1200
360
60 +
[160/2] =
140
220 +60/2=
250
288
1344
1200
3000
Σ: 2688 x 103
Σ: 4704 x 103
Ax
140 mm
A
Ay
y
80 mm
A
x
Second moment inertia
PART
1
2
3
AREA (A)(mm2)
60(200) = 12000
160(60)=9600
60(200) = 12000
Ix = bh3/12
(106) (mm4)
dy = |y-y|(mm)
Ady2(106)(mm4)
60(2003)/12 = 40
|100– 80| = 20
4.8
|30 – 80| = 50
24
|100 – 80| =20
4.8
160(603)/12 = 2.88
60(2003)/12 = 40
Σ: 33 600
I xx
=
=
PART
1
2
3
[Ix + Ady2]1 + [Ix + Ady2]2 +[Ix + Ady2]3
[44.8 + 26.88 + 44.8] x106
116.48 x 106 mm4
AREA(mm2)
60(200) = 12000
160(60)=9600
60(200) = 12000
Iy = b3h/12
(106) (mm4)
603(200)/12
=3.6
1603(60)/12
=20.48
603(200)/12
=3.6
Σ: 33 600
I yy
[Iy + As2]1 + [Iy + As2]2 +[Iy + As2]3
= [148.8 + 20.48 + 148.8] x106
= 318.08 x 106 mm4
Sx=|x-x| (mm)
|30-140|=110
Asx2(106)(mm4)
145.2
|140 – 140 |= 0
0
|250 – 140|=110
145.2