Lecture Strength of Materials I: Chapter 5 - PhD. Tran Minh Tu

TRAN MINH TU - University of Civil Engineering, Giai Phong Str... First moment of area.[r]

STRENGTH OF MATERIALS

TRAN MINH TU -University of Civil Engineering, Giai Phong Str 55, Hai Ba Trung Dist Hanoi, Vietnam

CHAPTER

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Geometric Properties of an Area

5.1 Introduction 5.2 First moment of area 5.3 Moment of inertia for an area 5.4 Moment of inertia for some simple areas 5.5 Parallel - axis theorem

5.6 Examples

5.1 Introduction

Dimension, shape?

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5.2 First Moment of Area

5.2.1 Definition

( )

x

A

S   ydA

( )

y

A

S   xdA

• Centroidal axes: are axes, which first moment of a plane A about them

is zero

• The first moment of a plane A about

the x- and y-axes are defined as

• Value: positive, negative or zero

• Dimension: [L3]; Unit: m3, cm3,

5.2.2 The centroid of an area

• The centroid C of the area is defined as the point in the xy-plane that has the coordinates

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y C

S x

A

C

S y

A



xC

5.2 First Moment of Area

• If the origin of the xy-coordinate system

is the centroid of the area then Sx=Sy=0

• Whenever the area has an axis of

symmetry, the centroid of the area will lie

on that axis

1

n i

i



 

1

n i

i



 

• If the area can be subdivided in to simple geometric shapes (rectangles, circles, etc., then

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5.2.3 The centroid of composite area

5.2 First Moment of Area

1

1

n

Ci i

y i

i i

x A S

x

A

A





  



1

1

n

Ci i

x i

i i

y A S

y

A

A





  



x

y

C1

C2

C3

xC1

yC1

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5.3 Moment of Inertia for an Area

2

( )

x

A

( )

y

A

5.3.1 Moment of inertia

5.3.2 Polar moment of inertia

2 ( )

A

5.3.3 Product of inertia

( )

xy

A

• The value of moment of inertia and polar moment of inertia always positive, but the product of inertia can be positive, negative,

or zero

• Dimension: [L4]; Unit: m4, cm4,

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5.3 Moment of Inertia for an Area

- The product of inertia Ixy for an area will be zero if either the x or the y axis is an axis of symmetry for the area

- The area with hole, then the hole’s

area is given by minus sign

- The composite areas:

1

n

i

i



 

1

n i

i



 

1

n i

i



 

1

n i

i



 

10

5.4 Moment of Inertia for some simple areas

• Rectangular

• Circle

• Triangular

3 12

x

bh

12

y

hb

I 

4

0,1

2 32

p



4

0,05

x y

I  I     D



3

12

x

bh

I 

b

x y

D

x y

b

x