Mechanics of Materials
CIVL 3322 / MECH 3322
Centroids
and
Moment of Inertia
Calculations
Centroids
x=
∑x A
i
i
i=1
n
∑A
i
i=1
2
n
n
n
y=
∑y A
i =1
n
i
i
∑A
i =1
i
Centroid and Moment of Inertia Calculations
z=
∑z A
i
i
i=1
n
∑A
i
i=1
Parallel Axis Theorem
¢  If
you know the moment of inertia about a
centroidal axis of a figure, you can
calculate the moment of inertia about any
parallel axis to the centroidal axis using a
simple formula
I z = I z + Ay 2
3
Centroid and Moment of Inertia Calculations
P07_045
4
Centroid and Moment of Inertia Calculations
P07_045
5
Centroid and Moment of Inertia Calculations
An Example
¢  Lets
start with an example problem and
see how this develops
x=
1 in
n
∑x A
i =1
n
1 in
1in
3 in
6
Centroid and Moment of Inertia Calculations
i
∑A
i =1
1 in
i
i
An Example
¢  We
want to locate both the x and y
centroids
n
x=
1 in
∑x A
i =1
n
i
i
∑A
i
i =1
1 in
1 in
1in
3 in
7
Centroid and Moment of Inertia Calculations
An Example
¢  There
isn’t much of a chance of
developing a function that is easy to
integrate in this case
n
x=
1 in
∑x A
i =1
n
1 in
1in
8
Centroid and Moment of Inertia Calculations
3 in
i
∑A
i =1
1 in
i
i
An Example
¢  We
can break this shape up into a series
of shapes that we can find the centroid of
n
x=
1 in
∑x A
i =1
n
i
i
∑A
i
i =1
1 in
1 in
1in
Centroid and Moment of Inertia3Calculations
in
9
An Example
¢  There
are multiple ways to do this as long
as you are consistent
n
x=
1 in
∑x A
i =1
n
1 in
1in
3 in
10
Centroid and Moment of Inertia Calculations
i
∑A
i =1
1 in
i
i
An Example
¢  First,
we can develop a rectangle on the
left side of the diagram, we will label that
n
as area 1, A1
x=
∑x A
1 in
i
i =1
n
∑A
i =1
1 in
A1
1 in
1in
11
Centroid and Moment of Inertia Calculations
3 in
An Example
¢  A
second rectangle will be placed in the
bottom of the figure, we will label it A2
n
x=
1 in
1 in
A1
1in
3 in
Centroid and Moment of Inertia Calculations
1 in
i
i
∑A
i =1
A2
12
∑x A
i =1
n
i
i
i
An Example
¢  A
right triangle will complete the upper
right side of the figure, label it A3
n
x=
1 in
1 in
A1
i =1
n
i
i
∑A
i
i =1
A3
A2
∑x A
1 in
1in
3 in
13
Centroid and Moment of Inertia Calculations
An Example
¢  Finally,
we will develop a negative area to
remove the quarter circle in the lower left
hand corner, label it A4
n
x=
1 in
1 in
A1
A4
1in
14
3 Calculations
in
Centroid and Moment of Inertia
i =1
n
1 in
i
i
∑A
i =1
A3
A2
∑x A
i
An Example
¢  We
will begin to build a table so that
keeping up with things will be easier
¢  The first column will be the areas
1 in
ID
A1
Area
(in2)
2
A2
3
A3
1.5
A4
-0.7854
1 in
A1
x=
A3
n
∑x A
i
∑A
i
i =1
A2
A4
i
i =1
n
1 in
1in
3 in
15
Centroid and Moment of Inertia Calculations
An Example
¢  Now
we will calculate the distance to the
local centroids from the y-axis (we are
calculating an x-centroid)
n
ID
Area
2
xi
A1
(in )
2
(in)
0.5
A2
3
2.5
1.5
2
A3
A4
1 in
1 in
-0.7854 0.42441
A1
x=
A3
A4
A2
3 in
Centroid and Moment of Inertia Calculations
i =1
n
i
i
∑A
i =1
1in
16
∑x A
i
1 in
An Example
¢  To
calculate the top term in the expression
we need to multiply the entries in the last
two columns by one another
n
ID
Area
xi
xi*Area
A1
(in )
2
(in)
0.5
(in )
1
A2
3
2.5
7.5
A3
1.5
2
3
2
A4
x=
1 in
3
1 in
A1
i =1
n
i
i
∑A
i
i =1
A3
-0.7854 0.42441 -0.33333
∑x A
A2
A4
1 in
1in
17
Centroid and Moment of Inertia Calculations
3 in
An Example
¢  If
we sum the second column, we have the
bottom term in the division, the total area
n
ID
Area
xi
xi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
A4
-0.7854 0.42441 -0.33333
5.714602
x=
1 in
1 in
A1
A2
1in
18
Centroid and Moment of Inertia Calculations
i =1
n
3 in
i
i
∑A
i =1
A3
A4
∑x A
i
1 in
An Example
¢  And
if we sum the fourth column, we have
the top term, the area moment
ID
Area
xi
xi*Area
A1
(in )
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
2
A4
n
x=
1 in
-0.7854 0.42441 -0.33333
5.714602
11.16667 1 in
A1
∑x A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
19
Centroid and Moment of Inertia Calculations
3 in
An Example
¢  Dividing
the sum of the area moments by
the total area we calculate the x-centroid
n
ID
Area
xi
xi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
A4
-0.7854 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
x=
1 in
1 in
A1
A2
1in
20
Centroid and Moment of Inertia Calculations
i =1
n
3 in
i
i
∑A
i =1
A3
A4
∑x A
i
1 in
An Example
¢  You
can always remember which to divide
by if you look at the final units, remember
that a centroid is a distance
n
ID
Area
xi
xi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
A4
-0.7854 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
x=
1 in
1 in
A1
∑x A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
21
Centroid and Moment of Inertia Calculations
3 in
An Example
¢  We
can do the same process with the y
centroid
n
ID
Area
xi
xi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
A4
-0.7854 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
y=
1 in
1 in
A1
A2
1in
22
Centroid and Moment of Inertia Calculations
i =1
n
3 in
i
i
∑A
i =1
A3
A4
∑y A
i
1 in
An Example
¢  Notice
that the bottom term doesn’t
change, the area of the figure hasn’t
changed
n
ID
Area
xi
xi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
A2
3
2.5
7.5
A3
1.5
2
3
A4
-0.7854 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
y=
1 in
1 in
A1
∑y A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
23
3 in
Centroid and Moment of Inertia Calculations
An Example
¢  We
only need to add a column of yi’s
ID
Area
xi
xi*Area
yi
A1
(in2)
2
(in)
0.5
(in3)
1
(in)
1
A2
3
2.5
7.5
0.5
A3
1.5
2
3
1.333333
A4
n
-0.7854 0.42441 -0.33333 0.42441
5.714602
11.16667
x bar
1.9541
1 in
y=
∑y A
i =1
n
i =1
i
1 in
A1
A4
Centroid and Moment of Inertia Calculations
i
∑A
A3
A2
1in
24
i
3 in
1 in
An Example
¢  Calculate
the area moments about the x-
axis
ID
Area
xi
xi*Area
yi
yi*Area
A1
(in2)
2
(in)
0.5
(in3)
1
(in)
1
(in3)
2
A2
3
2.5
7.5
0.5
1.5
1.5
2
3
1.333333
2
A3
A4
n
y=
-0.7854 0.42441 -0.33333 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
i =1
n
i
i
∑A
i
i =1
1 in
A1
1 in
∑y A
A3
A2
A4
1 in
1in
25
3 in
Centroid and Moment of Inertia Calculations
An Example
¢  Sum
ID
the area moments
Area
2
xi
xi*Area
3
yi
yi*Area
(in)
1
(in3)
2
(in )
2
(in)
0.5
(in )
1
A2
3
2.5
7.5
0.5
1.5
A3
1.5
2
3
1.333333
2
A1
A4
n
-0.7854 0.42441 -0.33333 0.42441 -0.33333
5.714602
11.16667
5.166667
x bar
1.9541
y=
∑y A
i =1
n
i
i
∑A
i =1
i
1 in
1 in
A1
A4
A3
A2
1in
26
Centroid and Moment of Inertia Calculations
3 in
1 in
An Example
¢  And
make the division of the area
moments by the total area
ID
Area
2
xi
xi*Area
yi
yi*Area
3
A1
(in )
2
(in)
0.5
(in )
1
(in)
1
(in3)
2
A2
3
2.5
7.5
0.5
1.5
1.5
2
3
1.333333
2
A3
A4
n
y=
-0.7854 0.42441 -0.33333 0.42441 -0.33333
5.714602
11.16667
5.166667
x bar
y bar
1.9541
0.904117
1 in
A1
1 in
A4
∑y A
i =1
n
i
∑A
i =1
A2
3 in
Centroid and Moment of Inertia Calculations
Parallel Axis Theorem
¢  If
you know the moment of inertia about a
centroidal axis of a figure, you can
calculate the moment of inertia about any
parallel axis to the centroidal axis using a
simple formula
I y = I y + Ax
I x = I x + Ay
28
Centroid and Moment of Inertia Calculations
2
2
i
A3
1in
27
i
1 in
Parallel Axis Theorem
¢  Since
we usually use the bar over the
centroidal axis, the moment of inertia
about a centroidal axis also uses the bar
over the axis designation
I y = I y + Ax 2
I x = I x + Ay
29
2
Centroid and Moment of Inertia Calculations
Parallel Axis Theorem
¢  If
you look carefully at the expression, you
should notice that the moment of inertia
about a centroidal axis will always be the
minimum moment of inertia about any axis
that is parallel to the centroidal axis.
I y = I y + Ax
I x = I x + Ay
30
Centroid and Moment of Inertia Calculations
2
2
Another Example
¢  We
can use the parallel axis theorem to
find the moment of inertia of a composite
figure
31
Centroid and Moment of Inertia Calculations
Another Example
y
6"
3"
6"
x
6"
32
Centroid and Moment of Inertia Calculations
Another Example
¢  We
can divide up the area into smaller
areas with shapes from the table
y
6"
6"
I
3"
II
x
III
6"
33
Centroid and Moment of Inertia Calculations
Another Example
Since the parallel axis theorem will require the area for
each section, that is a reasonable place to start
ID
I
II
III
Area
(in2)
36
9
27
y
6"
6"
I
3"
II
x
III
6"
34
Centroid and Moment of Inertia Calculations
Another Example
We can locate the centroid of each area with respect
the y axis.
ID
Area
xbari
(in2)
(in)
I
36
3
II
9
7
III
27
6
y
6"
6"
I
3"
II
x
III
6"
35
Centroid and Moment of Inertia Calculations
Another Example
From the table in the back of the book we find that the
moment of inertia of a rectangle about its y-centroid
axis is
3
y
6"
3"
y
1
I = bh
12
ID
36
Area
xbari
(in2)
(in)
I
36
3
II
9
7
III
27
6"
6
Centroid and Moment of Inertia Calculations
I
II
x
III
6"
Another Example
In this example, for Area I, b=6” and h=6”
1
3
( 6in )( 6in )
12
I y = 108in 4
Iy =
ID
37
Area
xbari
(in2)
(in)
I
36
3
II
9
7
III
27
y
6"
6"
I
3"
II
x
III
6"
6
Centroid and Moment of Inertia Calculations
Another Example
For the first triangle, the moment of inertia calculation
isn’t as obvious
y
6"
6"
I
3"
II
x
III
6"
38
Centroid and Moment of Inertia Calculations
Another Example
The way it is presented in the text, we can only find
the Ix about the centroid
h
y
6"
x 6"
I
3"
II
x
III
b
39
6"
Centroid and Moment of Inertia Calculations
Another Example
The change may not seem obvious but it is
just in how we orient our axis. Remember an
axis is our decision.
h
x
y
6"
6"
I
3"
b
x
II
b
x
III
6"
h
40
Centroid and Moment of Inertia Calculations
Another Example
So the moment of inertia of the II triangle can
be calculated using the formula with the
correct orientation.
1 3
bh
36
1
3
I y = ( 6in )( 3in )
36
I y = 4.5in 4
Iy =
41
y
6"
6"
I
3"
II
x
III
6"
Centroid and Moment of Inertia Calculations
Another Example
The same is true for the III triangle
1 3
bh
36
1
3
I y = ( 6in )( 9in )
36
I y = 121.5in 4
Iy =
42
y
6"
6"
I
Centroid and Moment of Inertia Calculations
3"
II
x
III
6"
Another Example
Now we can enter the Iybar for each sub-area into the
table
SubArea
Area
xbari
Iybar
(in2)
(in)
(in4)
I
36
3
108
II
9
7
4.5
III
27
6
121.5
43
y
6"
6"
I
3"
II
x
III
6"
Centroid and Moment of Inertia Calculations
Another Example
We can then sum the Iy and the A(dx)2 to get
the moment of inertia for eachy sub-area
6"
3"
6"
I
II
x
III
SubArea
44
Area
xbari
Iybar
A(dx)2
Iybar +
A(dx)2
(in2)
(in)
(in4)
(in4)
(in4)
I
36
3
108
324
432
II
9
7
4.5
441
445.5
III
27
6
121.5
972
1093.5
Centroid and Moment of Inertia Calculations
6"
(in2)
36
9
27
I
II
III
(in4)
108
4.5
121.5
(in)
3
7
6
(in4)
324
441
972
(in4)
432
445.5
1093.5
1971
Another Example
And if we sum that last column, we have the
Iy for the composite figure
y
6"
6"
I
3"
II
x
III
SubArea
Area
xbari
Iybar
A(dx)2
Iybar +
A(dx)2
(in2)
(in)
(in4)
(in4)
(in4)
I
36
3
108
324
432
II
9
7
4.5
441
445.5
III
27
6
121.5
972
1093.5
6"
1971
45
Centroid and Moment of Inertia Calculations
Another Example
We perform the same type analysis for the Ix
ID
I
II
III
46
Area
(in2)
36
9
27
Centroid and Moment of Inertia Calculations
y
6"
6"
I
3"
II
x
III
6"
Another Example
Locating the y-centroids from the x-axis
Sub-Area
Area
(in2)
36
9
27
I
II
III
47
ybari
(in)
3
2
-2
y
6"
6"
I
3"
II
x
III
6"
Centroid and Moment of Inertia Calculations
Another Example
Determining the Ix for each sub-area
Sub-Area
48
Area
ybari
Ixbar
(in2)
(in)
(in4)
I
36
3
108
II
9
2
18
III
27
-2
54
Centroid and Moment of Inertia Calculations
y
6"
6"
I
3"
II
x
III
6"
Another Example
Making the A(dy)2 multiplications
SubArea
y
6"
6"
I
3"
II
Area
ybari
Ixbar
(in2)
(in)
(in4)
(in4)
I
36
3
108
324
II
9
2
18
36
III
27
-2
54
108
49
x
III
)2
A(dy
6"
Centroid and Moment of Inertia Calculations
Another Example
Summing and calculating Ix
SubArea
Area
)2
A(dy
Ixbar +
A(dy)2
ybari
Ixbar
(in2)
(in)
(in4)
(in4)
(in4)
I
36
3
108
324
432
II
9
2
18
36
54
III
27
-2
54
108
162
648
50
Centroid and Moment of Inertia Calculations
y
6"
6"
I
3"
II
x
III
6"
Homework Problem 1
Calculate the centroid of the composite shape in reference to the bottom of the
shape and calculate the moment of inertia about the centroid z-axis.
51
Centroid and Moment of Inertia Calculations
Homework Problem 2
Calculate the centroid of the composite shape in reference to the bottom of the
shape and calculate the moment of inertia about the centroid z-axis.
52
Centroid and Moment of Inertia Calculations
Homework Problem 3
Calculate the centroid of the composite shape in reference to the bottom of the
shape and calculate the moment of inertia about the centroid z-axis.
53
Centroid and Moment of Inertia Calculations