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CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Introduction
• The earth exerts a gravitational force on each of the particles
forming a body. These forces can be replaced by a single
equivalent force equal to the weight of the body and applied
at the center of gravity for the body.
• The centroid of an area is analogous to the center of
gravity of a body. The concept of the first moment of an
area is used to locate the centroid.
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Center of Gravity of a 2D Body
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Center of Gravity of a 2D Body
• Center of gravity of a plate
My
xW
x W
x dW
Mx
yW
y W
y dW
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Centroids and First Moments of Areas
• Centroid of an area
xW
x At
x dW
x t dA
xA
x dA Q y
first moment wit h respect to y
yA
y dA Qx
first moment wit h respect to x
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
First Moments of Areas and Lines
• An area is symmetric with respect to an axis BB’
if for every point P there exists a point P’ such
that PP’ is perpendicular to BB’ and is divided
into two equal parts by BB’.
• The first moment of an area with respect to a
line of symmetry is zero.
• If an area possesses a line of symmetry, its
centroid lies on that axis
• If an area possesses two lines of symmetry, its
centroid lies at their intersection.
• An area is symmetric with respect to a center O
if for every element dA at (x,y) there exists an
area dA’ of equal area at (-x,-y).
• The centroid of the area coincides with the
center of symmetry.
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Centroids of Common Shapes of Areas
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
SOLUTION:
• Divide the area into a triangle, rectangle,
and semicircle with a circular cutout.
• Calculate the first moments of each area
with respect to the axes.
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
• Find the total area and first moments of
the triangle, rectangle, and semicircle.
Subtract the area and first moment of the
circular cutout.
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
• Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.
Qx
506.2 103 mm 3
Qy
757.7 103 mm 3
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
xA
A
X
X
Y
yA
A
Y
757.7 103 mm 3
13.828 103 mm 2
54.8 mm
506.2 10 3 mm 3
13.828 10 3 mm 2
36.6 mm
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Distributed Loads on Beams
L
W
wdx
dA
A
0
OP W
xdW
L
OP A
xdA
0
xA
• A distributed load is represented by plotting the load
per unit length, w (N/m) . The total load is equal to
the area under the load curve.
• A distributed load can be replaced by a concentrated
load with a magnitude equal to the area under the
load curve and a line of action passing through the
area centroid.
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
SOLUTION:
• The magnitude of the concentrated load
is equal to the total load or the area under
the curve.
• The line of action of the concentrated
load passes through the centroid of the
area under the curve.
• Determine the support reactions by
A beam supports a distributed load as
summing moments about the beam
shown. Determine the equivalent
ends.
concentrated load and the reactions at
the supports.
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
SOLUTION:
• The magnitude of the concentrated load is equal to
the total load or the area under the curve.
F 18.0 kN
• The line of action of the concentrated load passes
through the centroid of the area under the curve.
X
63 kN m
18 kN
X
3.5 m
CHAPTER 4: CENTER OF GRAVITY AND
CENTROIDS
Sample Problem
• Determine the support reactions by summing
moments about the beam ends.
MA
0 : By 6 m
18 kN 3.5 m
By
MB
0:
Ay 6 m
0
10.5 kN
18 kN 6 m 3.5 m
Ay
7.5 kN
0