centroids and moment of inertia calculation

7/31/2019 Centroids and Moment of Inertia Calculation

1/27

Mechanics of MaterialsCIVL 3322 / MECH 3322

Centroids

and

Moment of Inertia

Calculations

Centroid and Moment of Inertia Calculations2

Centroids

x =

xiA

i

i=1

n

Ai

i=1

n

1

1

n

i i

i

n

i

i

y A

y

A

=

=

=

z=

ziA

i

i=1

n

Ai

i=1

n


2/27


Parallel Axis Theorem

If you know the moment of inertia about acentroidal axis of a figure, you can

calculate the moment of inertia about any

parallel axis to the centroidal axis using a

simple formula

Iz=I

z+Ay2

P07_045



3/27

P07_045


An Example

Lets start with an example problem andsee how this develops

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=

1in

1 in

1 in

3 in

1 in

6 Centroid and Moment of Inertia Calculations


4/27

An Example

We want to locate both the x and ycentroids

1in

1 in

1 in

3 in

1 in

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=


An Example

There isnt much of a chance ofdeveloping a function that is easy to

integrate in this case

1in

1 in

1 in

3 in

1 in

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=



5/27

An Example

We can break this shape up into a seriesof shapes that we can find the centroid of

1in

1 in

1 in

3 in

1 in

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=


An Example

There are multiple ways to do this as longas you are consistent

1in

1 in

1 in

3 in

1 in

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=



6/27

An Example

First, we can develop a rectangle on theleft side of the diagram, we will label thatas area 1, A1

1in

1 in

1 in

3 in

1 in

A1

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=


An Example

A second rectangle will be placed in thebottom of the figure, we will label it A2

1in

1 in

1 in

3 in

1 in

A1

A2

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=



7/27

An Example

A right triangle will complete the upperright side of the figure, label it A3

1in

1 in

1 in

3 in

1 in

A1

A2

A3

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=


An Example

Finally, we will develop a negative area toremove the quarter circle in the lower left

hand corner, label it A4

1in

1 in

1 in

3 in

1 in

A1

A2

A3

A4

1

1

n

i i

i

n

ii

x A

x

A

=

=

=



8/27

An Example

We will begin to build a table so thatkeeping up with things will be easier

The first column will be the areas

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=

ID Area

(in2)

A1 2

A2 3

A3 1.5

A4 -0.7854


An Example

Now we will calculate the distance to thelocal centroids from the y-axis (we are

calculating an x-centroid)

1

1

n

i i

i

n

ii

x A

x

A

=

=

=

ID Area xi

(in2) (in)

A1 2 0.5

A2 3 2.5

A3 1.5 2

A4 -0.7854 0.42441

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



9/27

An Example

To calculate the top term in the expressionwe need to multiply the entries in the lasttwo columns by one another

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4


An Example

If we sum the second column, we have thebottom term in the division, the total area

1

1

n

i i

i

n

ii

x A

x

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



10/27

An Example

And if we sum the fourth column, we havethe top term, the area moment

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602 11.16667

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4


An Example

Dividing the sum of the area moments bythe total area we calculate the x-centroid

1

1

n

i i

i

n

ii

x A

x

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



11/27

An Example

You can always remember which to divideby if you look at the final units, rememberthat a centroid is a distance

1

1

n

i i

i

n

i

i

x A

x

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4


An Example

We can do the same process with the ycentroid

1

1

n

i i

i

n

ii

y A

y

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



12/27

An Example

Notice that the bottom term doesntchange, the area of the figure hasntchanged

1

1

n

i i

i

n

i

i

y A

y

A

=

=

=

ID Area xi xi*Area

(in2) (in) (in

3)

A1 2 0.5 1

A2 3 2.5 7.5

A3 1.5 2 3

A4 -0.7854 0.42441 -0.33333

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4


An Example

We only need to add a column of yis

1

1

n

i i

i

n

ii

y A

y

A

=

=

=

ID Area xi xi*Area yi

(in2) (in) (in

3) (in)

A1 2 0.5 1 1

A2 3 2.5 7.5 0.5

A3 1.5 2 3 1.333333

A4 -0.7854 0.42441 -0.33333 0.42441

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



13/27

An Example

Calculate the area moments about the x-axis

1

1

n

i i

i

n

i

i

y A

y

A

=

=

=

ID Area xi xi*Area yi yi*Area

(in2) (in) (in

3) (in) (in

3)

A1 2 0.5 1 1 2

A2 3 2.5 7.5 0.5 1.5

A3 1.5 2 3 1.333333 2

A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333

5.714602 11.16667

x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4


An Example

Sum the area moments

1

1

n

i i

i

n

ii

y A

y

A

=

=

=


(in2) (in) (in

3) (in) (in

3)

A1 2 0.5 1 1 2

A2 3 2.5 7.5 0.5 1.5

A3 1.5 2 3 1.333333 2

A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333

5.714602 11.16667 5.166667x bar 1.9541

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4



14/27

An Example

And make the division of the areamoments by the total area

1

1

n

i i

i

n

i

i

y A

y

A

=

=

=


(in2) (in) (in

3) (in) (in

3)

A1 2 0.5 1 1 2

A2 3 2.5 7.5 0.5 1.5

A3 1.5 2 3 1.333333 2

A4 -0.7854 0.42441 -0.33333 0.42441 -0.33333

5.714602 11.16667 5.166667

x bar 1.9541 y bar 0.904117

1in

1 in

1 in

3 in

1 in

A2

A3

A1

A4




If you know the moment of inertia about acentroidal axis of a figure, you can

calculate the moment of inertia about any

parallel axis to the centroidal axis using a

simple formula

2

2

= +

= +

y y

x x

I I Ax

I I Ay


15/27



Since we usually use the bar over thecentroidal axis, the moment of inertia

about a centroidal axis also uses the bar

over the axis designation

2

2

= +

= +

y y

x x

I I Ax

I I Ay



If you look carefully at the expression, youshould notice that the moment of inertia

about a centroidal axis will always be the

minimum moment of inertia about any axis

that is parallel to the centroidal axis.

2

2

= +

= +

y y

x x

I I Ax

I I Ay


16/27


Another Example

We can use the parallel axis theorem tofind the moment of inertia of a composite

figure


Another Example

y

x

6" 3"

6"

6"


17/27


Another Example

y

x

6" 3"

6"

6"

I

II

III

We can divide up the area into smallerareas with shapes from the table


Another Example

Since the parallel axis theorem will require the area foreach section, that is a reasonable place to start

y

x

6" 3"

6"

6"

III

III

ID Area

(in2)

I 36

II 9

III 27


18/27


Another ExampleWe can locate the centroid of each area with respect

the y axis.

y 6" 3"

6"

6"

I

II

III

ID Area xbari

(in2) (in)

I 36 3

II 9 7

III 27 6


Another Example

From the table in the back of the book we find that themoment of inertia of a rectangle about its y-centroid

axis is31

12=

yI b h y

x

6" 3"

6"

6"

III

III

ID Area xbari

(in2) (in)

I 36 3

II 9 7

III 27 6


19/27


Another ExampleIn this example, for Area I, b=6 and h=6

( )( )3

4

1 6 612

108

=

=

y

y

I in in

I iny

x

6" 3"

6"

6"

I

II

III

ID Area xbari

(in2) (in)

I 36 3

II 9 7III 27 6


Another Example

For the first triangle, the moment of inertia calculationisnt as obvious

y

x

6" 3"

6"

6"

III

III


20/27


Another Example

The way it is presented in the text, we can only findthe Ix about the centroid

x

b

h

y

x

6" 3"

6"

6"

I

II

III


Another ExampleThe change may not seem obvious but it is

just in how we orient our axis. Remember anaxis is our decision.

x

b

h

x

h

b

y

x

6" 3"

6"

6"

III

III


21/27


Another ExampleSo the moment of inertia of the II triangle can

be calculated using the formula with thecorrect orientation

.

( )( )

3

3

4

1

36

16 3

36

4.5

=

=

=

y

y

y

I bh

I in in

I in

y

x

6" 3"

6"

6"

I

II

III


Another ExampleThe same is true for the III triangle

( )( )

3

3

4

1

36

1

6 936

121.5

=

=

=

y

y

y

I bh

I in in

I in

y

x

6" 3"

6"

6"

III

III


22/27


Another ExampleNow we can enter the Iybar for each sub-area into the

table

y

x

6" 3"

6"

6"

I

II

III

Sub-Area Area xbari Iybar

(in2) (in) (in4)

I 36 3 108

II 9 7 4.5

III 27 6 121.5


Another ExampleWe can then sum the Iy and the A(dx)

2 to getthe moment of inertia for each sub-area

y

x

6" 3"

6"

6"

I

II

III

Sub-

Area Area xbari Iybar A(dx)2

Iybar+

A(dx)2

(in2) (in) (in4) (in4) (in4)

I 36 3 108 324 432

II 9 7 4.5 441 445.5

III 27 6 121.5 972 1093.5


23/27


Another ExampleAnd if we sum that last column, we have the

Iy for the composite figure

y 6" 3"

6"

6"

I

II

III

Sub-

Area Area xbari Iybar A(dx)2

Iybar+

A(dx)2

(in2) (in) (in4) (in4) (in4)

I 36 3 108 324 432

II 9 7 4.5 441 445.5

III 27 6 121.5 972 1093.5

1971


Another Example

We perform the same type analysis for the Ix

ID Area

(in2)

I 36

II 9

III 27

y 6" 3"

6"

6"

I

II

III


24/27


Another ExampleLocating the y-centroids from the x-axis

y 6" 3"

6"

6"

I

II

III

Sub-Area Area ybari

(in2) (in)

I 36 3

II 9 2

III 27 -2


Another ExampleDetermining the Ix for each sub-area

y 6" 3"

6"

6"

I

II

IIISub-Area Area ybari Ixbar

(in2) (in) (in4)

I 36 3 108

II 9 2 18

III 27 -2 54


25/27


Another ExampleMaking the A(dy)

2 multiplications

y 6" 3"

6"

6"

I

II

IIISub-

Area Area ybari Ixbar A(dy)2

(in2) (in) (in4) (in4)

I 36 3 108 324

II 9 2 18 36

III 27 -2 54 108


Another ExampleSumming and calculating Ix

y 6" 3"

6"

6"

I

II

III

Sub-Area Area ybari Ixbar A(dy)

2

Ixbar+A(dy)

2

(in2) (in) (in4) (in4) (in4)

I 36 3 108 324 432

II 9 2 18 36 54

III 27 -2 54 108 162

648


26/27

Homework Problem 1


Calculate the centroid of the composite shape in reference to the bottom of theshape and calculate the moment of inertia about the centroid z-axis.

Homework Problem 2




27/27

Homework Problem 3



centroids and moment of inertia calculation

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