Lecture 17

ENGR-1100 Introduction to Engineering

Analysis

CENTROID OF COMPOSITE AREAS

In-Class Activities:

• Reading Quiz

• Applications

• Centroid

• Determine Centroid

Location

•Method of Composite

Areas

• Concept Quiz

• Group Problem Solving

• Attention Quiz

Today’s Objective :

Students will:

a) Understand the concept of centroid.

b) Be able to determine the location of

the centroid using the method of

composite areas.

CENTROID OF A BODY

Similarly, the coordinates of the centroid of volume, area, or

length can be obtained by replacing W by V, A, or L,

respectively.

dA

dAy~y

dA

dAx~x

CONCEPT OF CENTROID

The centroid coincides with the center of

mass or the center of gravity only if the

material of the body is homogenous (density

or specific weight is constant throughout the

body).

If an object has an axis of symmetry, then

the centroid of object lies on that axis.

In some cases, the centroid may not be

located on the object.

The centroid, C, is a point defining the

geometric center of an object.

STEPS TO DETERME THE CENTROID OF AN AREA

1. Choose an appropriate differential element dA at a general point (x,y).

Hint: Generally, if y is easily expressed in terms of x

(e.g., y = x2 + 1), use a vertical rectangular element. If the converse

is true, then use a horizontal rectangular element.

2. Express dA in terms of the differentiating element dx (or dy).

4. Express all the variables and integral limits in the formula using

either x or y depending on whether the differential element is in

terms of dx or dy, respectively, and integrate.

3. Determine coordinates (x, y) of the centroid of the rectangular

element in terms of the general point (x,y).

~ ~

EXAMPLE

2. dA = y dx = x3 dx

3. x = x and y = y / 2 = x3 / 2~~

Solution:

1. Since y is given in terms of x, choose

dA as a vertical rectangular strip.

Given: The area as shown.

Find: The centroid location (x , y)

Plan: Follow the steps.

EXAMPLE(continued)

4. x = ( A x dA ) / ( A dA )~

0 x (x3 ) d x 1/5 [ x5 ]1

0 (x3 ) d x 1/4 [ x4 ]1

= ( 1/5) / ( 1/4) = 0.8 m

1= =

1

0

0

1A y dA 0 (x3 / 2) ( x3 ) dx 1/14[x7]1

A dA 0 x3 dx 1/41

=y =

~

=

= (1/14) / (1/4) = 0.2857 m

0

APPLICATIONS

How can we easily determine

the location of the centroid for

different beam shapes?

The I-beam (top) or T-beam

(bottom) shown are commonly

used in building various types

of structures.

When doing a stress or

deflection analysis for a beam,

the location of its centroid is

very important.

STEPS FOR ANALYSIS

1. Divide the body into pieces that are known shapes.

Holes are considered as pieces with negative weight or size.

2. Make a table with the first column for segment number, the second

column for size, the next set of columns for the moment arms, and,

finally, several columns for recording results of simple intermediate

calculations.

3. Fix the coordinate axes, determine the coordinates of centroid of

each piece, and then fill in the table.

4. Sum the columns to get x, y, and z. Use formulas like

x = ( xi Ai ) / ( Ai )

This approach will become straightforward by doing examples!

EXAMPLE

Solution:

1. This body can be divided into the following pieces:

rectangle (a) + triangle (b) + quarter circular (c) –

semicircular area (d). Note the negative sign on the hole!

Given: The part shown.

Find: The centroid of

the part.

Plan: Follow the steps

for analysis.

EXAMPLE (continued)

39.8376.528.0

27

4.5

9

- 2/3

54

31.5

– 9

0

1.5

1

4(3) / (3 )

4(1) / (3 )

3

7

– 4(3) / (3 )

0

18

4.5

9 / 4

– / 2

Rectangle

Triangle

Q. Circle

Semi-Circle

y A

( in3)

x A

( in3)

y

(in)

x

(in)

Area A

(in2)

Segment

Steps 2 & 3: Make up and fill the

table using parts a, b,

c, and d. Note the location

of the axis system.

4. Now use the table data results and the formulas to find the

coordinates of the centroid.

x = ( x A) / ( A ) = 76.5 in3/ 28.0 in2 = 2.73 in

y = ( y A) / (A ) = 39.83 in3 / 28.0 in2 = 1.42 in

C

Area A x A y A

28.0 76.5 39.83

EXAMPLE (continued)

READING QUIZ

1. A composite body in this section refers to a body made of ____.

A) Carbon fibers and an epoxy matrix in a car fender

B) Steel and concrete forming a structure

C) A collection of “simple” shaped parts or holes

D) A collection of “complex” shaped parts or holes

2. The composite method for determining the location of the

center of gravity of a composite body requires _______.

A) Simple arithmetic B) Integration

C) Differentiation D) All of the above.

CONCEPT QUIZ

Based on the typical centroid

information, what are the minimum

number of pieces you will have to

consider for determining the centroid of

the area shown at the right?

A) 4 B) 3 C) 2 D) 1

3cm 1 cm

1 cm

3cm

ATTENTION QUIZ

2. For determining the centroid of the area, two

square segments are considered; square ABCD

and square DEFG. What are the coordinates

(x, y ) of the centroid of square DEFG?

A) (1, 1) m B) (1.25, 1.25) m

C) (0.5, 0.5 ) m D) (1.5, 1.5) m

~ ~

1. A rectangular area has semicircular and

triangular cuts as shown. For determining the

centroid, what is the minimum number of

pieces that you can use?

A) Two B) Three

C) Four D) Five2cm 2cm

2cm

4cm

x

y

A1m

1m

y

E

FG

CB x

1m 1m

D

GROUP PROBLEM SOLVING

Given: A plate as shown.

Find: The location of its centroid

Plan:

Follow the solution steps to

find the centroid by integration.

Solution

1. Choose dA as a vertical

rectangular strip.

GROUP PROBLEM SOLVING (continued)

GROUP PROBLEM SOLVING (continued)

2

2