Contents
• Introduction(绪论)
• Center of Gravity of a 2D Body(两维物体的重心)
• Centroids and First Moments of Areas(形心与面积的一次矩)
• Centroids of Composite Areas(组合面积的形心)
• Determination of Centroids by Integration(积分法求形心)
• Centroids of Common Shapes of Areas(常见平面图形的形心)
2
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.
3
Center of Gravity of a 2D Body
• Center of gravity of a plate
y i i
x i i
M xW x W
xdW
M yW y W
ydW
4
Centroids and First Moments of Areas
first moment with respect to
first moment with respect to
y
x
xW x dW
x gAt x gt dA
xA x dA S
y
yA y dA S
x
• Centroid of an area (assuming
uniform thickness and density)
5
Centroids and First Moments of Areas
• 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.
6
Centroids of Composite Areas
• Composite plates
i i
i i
X W xW
Y W yW
• Composite area
i i
i i
X A x A
Y A y A
7
1 1 1
;n n n
x xi i i y i i
i i
S S A y S A x
1
1
10 2000 20 70 2800
2000 280056.66
50
n
i iy i
n
i i
x i
A xS
xA A
A yS
yA A
x yA A
S ydA Ay S xdA Ax
O
x
y100
20
20
140
C
C1
C2
Ⅰ
Ⅱ
Sample Problem
8
For the plane area shown, determine
the first moments with respect to the
x and y axes and the location of the
centroid.
SOLUTION:
• Divide the area into a triangle, rectangle,
and semicircle with a circular cutout.
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
• Find the total area and first moments of
the triangle, rectangle, and semicircle.
Subtract the area and first moment of the
circular cutout.
• Calculate the first moments of each area
with respect to the axes.
Sample Problem
9
3 3
3 3
506.2 10 mm
757.7 10 mm
x
y
S
S
• Find the total area and first moments of the
triangle, rectangle, and semicircle. Subtract the
area and first moment of the circular cutout.
10
23
33
mm1013.828
mm107.757
A
AxX
mm 8.54X
23
33
mm1013.828
mm102.506
A
AyY
mm 6.36Y
• Compute the coordinates of the area
centroid by dividing the first moments by
the total area.
11
Determination of Centroids by Integration
ydxy
dAyAy
ydxx
dAxAx
el
el
2
2
el
el
xA x dA
a xa x dy
yA y dA
y a x dy
drr
dAyAy
drr
dAxAx
el
el
2
2
2
1sin
3
2
2
1cos
3
2
dAydydxydAyAy
dAxdydxxdAxAx
el
el• Double integration to find the first moment
may be avoided by defining dA as a thin
rectangle or strip.
12
Determine by direct integration the
location of the centroid of a parabolic
spandrel.
SOLUTION:
• Determine the constant k.
• Evaluate the total area.
• Using either vertical or horizontal
strips, perform a single integration to
find the first moments.
• Evaluate the centroid coordinates.
Sample Problem
13
SOLUTION:
• Determine the constant k.
21
21
2
2
2
2
2
yb
axorx
a
by
a
bkakb
xky
• Evaluate the total area.
3
30
3
20
2
2
ab
x
a
bdxx
a
bdxy
dAA
aa
14
• Using vertical strips, perform a single integration
to find the first moments.
2
2
0
4 2
2
0
2
2
2
0
2 5 2
4
0
4 4
1
2 2
2 5 10
a
y el
a
a
x el
a
bS x dA xydx x x dx
a
b x a b
a
y bS y dA ydx x dx
a
b x ab
a
15
• Or, using horizontal strips, perform a single
integration to find the first moments.
2 2
0
2 22
0
1 2
1 2
23 2
1 2
0
2 2
1
2 4
10
b
y el
b
x el
b
a x a xS x dA a x dy dy
a a ba y dy
b
aS y dA y a x dy y a y dy
b
a abay y dy
b
16