# Centroids & Moment of Inertia

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"Centroids & Moment of Inertia"

```					Centroids & Moment of Inertia

EGCE201 Strength of Materials I
Instructor: ดร.วรรณสิ ร ิ พันธอุไร (อ.ปู)
์
ห้องทำงำน: 6391 ภำควิชำวิศวกรรมโยธำ
E-mail: egwpr@mahidol.ac.th
โทรศัพท: 66(02) 889-2138
์                        ตอ 6391
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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
Centroidal axis
• The centroid of any area can be found by taking or Neutral
moments of identifiable areas (such as rectangles or
Sum MAtotal = MA1 + MA2 + MA3+ ...

• 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.
centroid example
simple rectangular shape
y

h/2                                      Sum MAtotal = MA1 + MA2 + MA3+ ...
centroid
area x distance | from the centroid
h/2                                       of the area to the moment axis
ZZ’
b
h h     h     h h
(bh)Y  (b )        (b ) 
2 2     4     2 4
Take ZZ’ as the reference axis                   h  3h h 
and take moment w.r.t ZZ’ axis
 (b )  
2  4 4

 (b )h 
h
2
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
Moment of Inertia example
simple rectangular shape

I z   y dA
h
2                    2
I z   y bdy
2

h
2

dA  bdy                         y3
h
2

y                      b
3       
h
2

h/2                                   b  h 3  h 3 
z             
h/2
dy              3  8  8 
bh 3
Centroid

or Neutral axis
b                        12
“I” is an important value!
• It is used to determine the state of stress in a section.
• It is used to calculate the resistance to bending.
• It can be used to determine the amount of deflection in a
beam.       y                                h

h/2
b/2
z                            z
h/2                               b/2

b                                 y

bh 3             >                hb 3
Iz                               Iz 
12        Stronger section        12
Built-up sections
• It is often advantageous to combine a
number of smaller members in order to
create a beam or column of greater
strength.
• The moment of inertia of such a built-up
section is found by adding the moments of
inertia of the component parts
Transfer formula
• There are many built-up sections in which the
component parts are not symmetrically distributed about
the centroidal axis.
• To determine the moment of inertia of such a section is
to find the moment of inertia of the component parts
about their own centroidal axis and then apply the
transfer formula.
• The transfer formula transfers the moment of inertia of
a section or area from its own centroidal axis to another
parallel axis. It is known from calculus to be:

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