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

Chapter 5 - Geometric properties of an area. The following will be discussed in this chapter: First moment of area, moment of inertia for an area, moment of inertia for some simple areas, parallel - axis theorem.

STRENGTH OF MATERIALS

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?

5.2 First Moment of Area

• 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 thathas the coordinates

1/10/2013 6

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

5.2.3 The centroid of composite area

5.2 First Moment of Area

x A S

y A S

1/10/2013 8

5.3 Moment of Inertia for an Area

2 ( )

or zero

• Dimension: [L4]; Unit: m4, cm4,

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 yaxis 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:

x y

b

x

5.5 Paralell-axis Theorem

• In the xy coordinates, an area

has geometric properties: S x , S y ,

I x , I y , I xy

• In the uv coordinates: O'u//Ox,

O'v//Oy và:

• Geometric properties of an area

in the coordinates O'uv are:

5.5 Paralell-axis Theorem

1/10/2013 14

5.5 Paralell-axis Theorem

Problem 5.6.1. An area with the shape

and the dimension as shown in the

figure Determine the principal moment

of inertia for area

coordinates x0y0 as shows in the figure

Divide the composite area to 2 simple

- Draw the principal coordinates Cxy

- The Principal moment of inertia for an area:

Example 5.1

Problem 5.2.

Example 5.2

1/10/2013 18

Example 5.2

Example 5.3

1/10/2013 20

Example 5.3

THANK YOU FOR

ATTENTION !