# Centroids Moment of Inertia by swenthomasovelil

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```									                       Lecture Notes

Module 2: Centroids & Moment of Inertia
Centroid
• Centroid or center of gravity is the point within an object
from which the force of gravity appears to act.
• Centroid of 3D objects often (but not always) lies
somewhere along the lines of symmetry.
Hollowed pipes, L shaped section have centroid
located outside of the material of the section

• The Centroid of any area can be found by taking Centroidal axis
or Neutral
moments of identifiable areas (such as rectangles or
triangles) about any axis. i.e A.y or A.x

• The moment of an area about any axis is equal to the
algebraic sum of the moments of its component areas.
• The moment of any area is defined as the product of
the area and the perpendicular distance from the
Centroid of the area to the moment axis.
First Moment of Area-Centroid
• Composite areas
–   Subdivide
–   Select coordinate axes
–   Find the overall area
–   Apply the composite
centroid formula
Moment of Inertia (I)
• Also known as the Second Moment of the Area is a term
used to describe the capacity of a cross-section to resist
bending.
• It is a mathematical property of a section concerned with a
surface area and how that area is distributed about the
reference axis. The reference axis is usually a centroidal
axis.

where
Cross Section of Beam

Y

Y
Internal Forces

• Naming those internal forces

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Perpendicular Axis Theorem

• For flat objects the
rotational moment of inertia
Iy = (1/12) b3d
of the axes in the plane is
related to the moment of
inertia perpendicular to the
Ix = (1/12) bd3                     d     plane.
Iz  Ix  Iy

b
Iz  Ix  Iy
Iz = (1/12) bd(b2 + d2)
Izz  Polar Moment of Inertia
Parallel-Axis Theorem for an Area
• Used to calculate the moment of inertia about other axis.
• The moment of inertia of an area about an axis is equal to the
moment of inertia of the area about a parallel axis passing through
the area’s Centroid plus the product of the area and the square of
the perpendicular distance between the axes.

- Ix’: Moment of Inertia about new axis
- Ix: Moment of Inertia about original
axis
- A: Area of shape                               A
- d: Perpendicular distance from new to
original axis                           x                          x

d
I X '  I X  Ad 2                x'                         x'

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Moment of Inertia - Parallel
Axis Theorem

Parallel axis theorem:
Consider the moment
of inertia Ix of an area
A with respect to an
axis AA’. Denote by y
the distance from an
element of area dA to
I x   y dA
2

AA’.
Moment of Inertia - Parallel
Axis Theorem
Consider an axis BB’
parallel to AA’ through the
Centroid C of the area,
known as the centroidal
axis. The equation of the
moment inertia becomes

I x   y dA    y  d  dA
2               2

  y dA  2 ydA  d
2                   2
 dA
Moment of Inertia - Parallel
Axis Theorem
The first integral is the
centroid.
I x   y dA
2

The second component is the first moment area
about the centroid i.e if the area is located at the C.G
yA   ydA  y  0
  ydA  0
Moment of Inertia - Parallel
Axis Theorem
Modify the equation obtain
the parallel axis theorem.

Ix   y 2 dA  2  ydA  d 2  dA

 Ix  d A
2
Moments of Inertia and Radius of
Gyration
• In general case                                   Polar Moment of
Inertia
IX
kX 
A
Iy
ky 
A
JO
kO 
A

• It is a mathematical property of a section concerned with a surface
area and how that area is distributed about the reference axis. Look
at IX of following shapes:

x               x              x             x

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Moment of Inertia example
simple rectangular shape

dA  bdy

Centroid
or Neutral axis
Rectangular Area
•      Moment of Inertia about centroidal axes for a rectangle          To
Remember
1         1
IX    bh 3     (300mm )(150mm ) 3  562500 mm 4
12        12                                                            1
IX       bh 3
1         1                                                           12
IY  hb 3  hb 3  (150mm )(300mm ) 3  1125000 mm           4
1
12        12                                                    IY       hb 3
– Which one is bigger?                                                   12
New Unit!
y                     1. Standard sections (C-shapes,
Wide flange beams, Hollow steel
sections)

150mm       x                               x     2. Rectangular shapes:
h

Calculate yourself, memorize
equations.
y                     3. Other shapes (circles, triangles,
300mm
b                       etc..): The equation or value will
be given to you.
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Moments of Inertia for
Composite Areas
• If the moment of inertia of each simpler area (a part of the
composite area) is known or can be determined about a common
axis, then the moment of inertia of the composite area equals the
algebraic sum of the moments of inertia of all its parts.

• Steps to follow:
– Divide the composite area into smaller and simpler parts
– Find the Centroid of each part
– Calculate the moment of inertia (I) for each part about its centroidal axis
– Use parallel-axis theorem to calculate the moment of inertia of each
part about the given axis (normally the centroidal axis of composite area,
if this is the case, very likely you need to find the Centroid of that
composite area first)
– Take the algebraic sum of the moments of inertia of all parts to get the
moment of inertia of the composite area
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Find the M.I of the I - section                    Y
24 mm

6mm           X3                                        X3

8 mm                      y3                               h3y
h2y
48 mm
X                                         X
X2
X2
y2                y      h1y
6mm
X1                                       X1

48 mm
y1                   Y

Ix = Ix1 + A1h1y2 + Ix2 + A2h2y2 + Ix3 + A3h3y2   Symmetric about Y axis hence
h1x, h2x, h3x = 0
C.G  x = 0 ,
y = (A1y1 + A2y2 + A3y3)/ (A1 + A2 + A3)

Ix1 , Ix2, Ix3= (b1d13 )/12, ……….

h1y = y-y1       h2y = y – y2       h3y = y – y3

Similarly find Iy.

Iy1 = (d1b13 )/12

Radius of Gyration along X & Y axis

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