Centroids & Centers of Gravity

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

• Evaluate the centroid coordinates.

2

3 4

yxA S

ab a bx

ax

4

3

2

3 10

xyA S

ab aby

by

10

3

17

Centroids of Common Shapes of Areas

18

Centroids of Common Shapes of Areas

19