Centroids
Method of Composite Areas
A small boy swallowed some coins and was taken to a hospital. When his grandmother telephoned to ask how he was a nurse said 'No change yet'.
Centroids
¢  Previously,
we developed a general
formulation for finding the centroid for a
series of n areas
n
x=
∑x A
i =1
n
i
∑A
i =1
2
i
i
Centroids by Composite Areas
Monday, November 12, 2012
1
Centroids
¢  xi
was the distance from the y-axis to the
local centroid of the area Ai
n
x=
∑x A
i =1
n
i
∑A
i =1
3
i
i
Centroids by Composite Areas
Monday, November 12, 2012
Centroids
¢  If
we can break up a shape into a series of
smaller shapes that have predefined local
centroid locations, we can use this formula
to locate the centroid of the composite
shape
n
x=
4
Centroids by Composite Areas
∑x A
i =1
n
i
i
∑A
i =1
i
Monday, November 12, 2012
2
Centroid by Composite Bodies
¢  There
is a table in the back cover of your
book that gives you the location of local
centroids for a select group of shapes
¢  The point labeled C is the location of the
centroid of that shape.
5
Centroids by Composite Areas
Monday, November 12, 2012
Centroid by Composite Bodies
¢  Please
note that these are local centroids,
they are given in reference to the x and y
axes as shown in the table.
6
Centroids by Composite Areas
Monday, November 12, 2012
3
Centroid by Composite Bodies
¢  For
example, the centroid location of the
semicircular area has the y-axis through
the center of the area and the x-axis at the
bottom of the area
¢  The x-centroid would be located at 0 and
the y-centroid would be located at
4r
3π
7
Centroids by Composite Areas
Monday, November 12, 2012
Centroid by Composite Bodies
¢  If
we wanted the centroid with respect to
another axis, say along the top of the
semicircle and along the left edge, the
values in the table couldn’t be used exactly
y
x
C
8
Centroids by Composite Areas
Monday, November 12, 2012
4
Centroid by Composite Bodies
¢  The
table would give you the distance of C
above the base of the semicircle, but that
isn’t the distance from the centroid to the
x-axis
9
Centroids by Composite Areas
Monday, November 12, 2012
Centroid by Composite Bodies
y
x
4r/3π
10
C
Centroids by Composite Areas
Monday, November 12, 2012
5
Centroid by Composite Bodies
¢  Since
the radius of the semicircle, in this
case the distance to the y-centroid would
be
4r ⎞
⎛
y = −⎜r −
⎟
3
π
⎝
⎠
11
Centroids by Composite Areas
Monday, November 12, 2012
Centroid by Composite Bodies
y
x
4r ⎞
⎛
y = −⎜r −
⎟
3π ⎠
⎝
4r/3π
12
C
Centroids by Composite Areas
Monday, November 12, 2012
6
Centroid by Composite Bodies
¢  By
the same logic, the distance to the xcentroid would be
x=r
13
Centroids by Composite Areas
Monday, November 12, 2012
Centroid by Composite Bodies
y
x
r
r-(4r/3π)
4r/3π
14
C
Centroids by Composite Areas
Monday, November 12, 2012
7
An Example
¢  Lets
start with an example problem and
see how this develops
x=
1 in
n
∑x A
i =1
n
i
i
∑A
i
i =1
1 in
1 in
1in
3 in
15
Centroids by Composite Areas
Monday, November 12, 2012
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 =1
i
1 in
1 in
1in
3 in
16
Centroids by Composite Areas
Monday, November 12, 2012
8
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
i
i
∑A
i
i =1
1 in
1 in
1in
17
Centroids by Composite Areas
Monday, November 12, 2012
3 in
An Example
¢  We
can break this figure up into a series of
shapes and find the location of the local
n
centroid of each
xA
x=
1 in
∑
i =1
n
i
i
∑A
i =1
i
1 in
1 in
1in
18
Centroids by Composite 3
Areas
in
Monday, November 12, 2012
9
An Example
¢  There
are multiple ways to do this as long
as you are consistent
n
x=
∑x A
i
i =1
n
i
∑A
1 in
i
i =1
1 in
1 in
1in
3 in
19
Centroids by Composite Areas
Monday, November 12, 2012
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=
1 in
∑x A
i =1
n
i
∑A
i =1
1 in
i
i
A1
1 in
1in
20
Centroids by Composite Areas
3 in
Monday, November 12, 2012
10
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
∑x A
i =1
n
i
∑A
i
i =1
A2
i
1 in
1in
3 in
21
Centroids by Composite Areas
Monday, November 12, 2012
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 =1
A3
A2
∑x A
i
1 in
1in
3 in
22
Centroids by Composite Areas
Monday, November 12, 2012
11
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
∑x A
i =1
n
i
∑A
i
i =1
A3
A4
A2
i
1 in
1in
in
Centroids by Composite3Areas
23
Monday, November 12, 2012
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
A4
n
x=
A3
∑x A
i =1
n
i
∑A
i =1
A2
i
i
1 in
1in
3 in
24
Centroids by Composite Areas
Monday, November 12, 2012
12
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
xi
2
A1
(in )
2
(in)
0.5
A2
3
2.5
1.5
2
A3
A4
1 in
x=
A1
1 in
-0.7854 0.42441
A3
∑x A
i =1
n
i
∑A
i
i =1
A2
A4
i
1 in
1in
3 in
25
Centroids by Composite Areas
Monday, November 12, 2012
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
-0.7854 0.42441 -0.33333
A4
∑x A
i =1
n
A2
i
∑A
i =1
A3
i
i
1 in
1in
26
Centroids by Composite Areas
3 in Monday, November 12, 2012
13
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
∑x A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
27
3 in
Centroids by Composite Areas
Monday, November 12, 2012
An Example
¢  And
if we sum the fourth column, we have
the top term, the area moment
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
n
x=
1 in
-0.7854 0.42441 -0.33333
5.714602
11.16667 1 in
A1
A4
∑x A
i =1
n
A2
i
∑A
i =1
A3
i
i
1 in
1in
28
Centroids by Composite Areas
3 in
Monday, November 12, 2012
14
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
∑x A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
29
3 in
Centroids by Composite Areas
Monday, November 12, 2012
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
A4
∑x A
i =1
n
A2
i
∑A
i =1
A3
i
i
1 in
1in
30
Centroids by Composite Areas
3 in
Monday, November 12, 2012
15
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
∑y A
i =1
n
i
∑A
i
i =1
A3
A2
A4
i
1 in
1in
31
3 in
Centroids by Composite Areas
Monday, November 12, 2012
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
A4
∑y A
i =1
n
A2
i
∑A
i =1
A3
i
i
1 in
1in
32
Centroids by Composite Areas
3 in
Monday, November 12, 2012
16
An Example
¢  We
only need to add a column of yi’s
ID
Area
xi
2
xi*Area
yi
3
A1
(in )
2
(in)
0.5
(in )
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
i
∑A
i
i =1
1 in
A1
A3
A2
A4
1 in
1in
33
3 in
Monday,
November 12, 2012
Centroids by Composite Areas
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
A3
1.5
2
3
1.333333
2
A4
-0.7854 0.42441 -0.33333 0.42441 -0.33333
5.714602
11.16667
x bar
1.9541
1 in
n
y=
∑y A
i =1
n
i
i
∑A
i =1
i
1 in
A1
A4
A3
A2
1 in
1in
34
Centroids by Composite Areas
3 in
Monday,
November 12, 2012
17
An Example
¢  Sum
the area moments
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
A3
1.5
2
3
1.333333
2
A4
n
y=
-0.7854 0.42441 -0.33333 0.42441 -0.33333
5.714602
11.16667
5.166667
x bar
1.9541
∑y A
i =1
n
i
i
∑A
i
i =1
1 in
1 in
A1
A3
A2
A4
1 in
1in
35
3 in
Monday,
November 12, 2012
Centroids by Composite Areas
An Example
¢  And
make the division of the area
moments by the total area
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
A3
1.5
2
3
1.333333
2
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
1 in
A1
A4
∑y A
i =1
n
i
i
∑A
i =1
i
A3
A2
1 in
1in
36
Centroids by Composite Areas
3 in
Monday,
November 12, 2012
18
Example Problem 9-60
•  Problem 9-60
37
Centroids by Composite Areas
Monday, November 12, 2012
Example Problem 9-61
38
Centroids by Composite Areas
Monday, November 12, 2012
19
Homework
¢  Problem
9-64
¢  Problem 9-65
¢  Problem 9-71
39
Centroids by Composite Areas
Monday, November 12, 2012
20