centroid of area and moment of inertia calculation

8/11/2019 Centroid of Area and Moment of Inertia Calculation

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McMaster Faculty of Engineering

Hamilton Ontario Canada

Materials 2H04 Measurement and Communications2005-2006 Calculation Assignment(draft )

Centroid of Area and Moment of Inertia CalculationThe ability to calculate the moment of inertia (second moment) of a given cross sectionis important for beam analysis and design. The moment of inertia of a complexgeometric object can be calculated by first dividing the larger region up into smaller,simpler objects. The effect of each of the smaller objects on the overall moment of inertiacan than be summed together.

In order to calculate the moment of inertia of an object, the location of the centroid(centre of mass) must be known. The centroid can be thought of as the location on anobject that experiences only translation and no rotation when a force is applied to theobject. For a sample of uniform density, the centroid of an object is independent of the

density, and only the distribution of area need be known.

Centroid of Area

The centroid of area for a symmetrical object such as a square or circle is located at thecentre of the object, however this is not necessarily the case for more complex objects.For a complex object the overall centroid is calculated by breaking it into smaller objectsusing a weighted average (by area) as shown in Equation 1.

In the equation yiand xiform an ordered pair that represents the location of the centreof mass for the smaller objects. A iis the area of the smaller object and ATis the totalarea of the cross section. The products yiAiand xiAiare commonly termed the firstmoments of area relative to the x axis and y axis.

T

i

i

i

A

Ax

X

=

T

i

i

i

A

Ay

Y

= (1)

Example1

Calculate the centroid of area for the object shown in Figure 1

Step 1 Chose a frame of reference and a point of or igin .

1The example is adopted from: Beer et al. Mechanics of Materials 3

rded. New York: McGraw Hill,

2004.


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Materials 2H03 Measurement and Communications 2005-2006

x

y

Figure 1: T Shape (all dimensions in mm)

Step 2 Find the areas of the sub-regions

2

1

2400

6040

mm

mmmmA

=

=

2

2

1600

8020

mm

mmmmA

=

=

2

21

4000mm

AAAT

=

+=

Step 3 Find the firs t moment of area relative to the x axis

34

2

111,

1020.7

240030

mm

mmmm

AyQx

=

=

=

35

2

222,

1012.1

160070

mm

mmm

AyQx

=

=

=

35

2,1,,

1084.1 mm

QQQ xxTx

=

+=

Step 4 Find the Y coordinate of the centroid

mm

mm

mm

A

QY

T

Tx

46

4000

1084.12

35

,

=

=

=

Student Exercise Find the X coordinate for the centro id

Prepared by Chris Harris and Allison Eppel

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Calculating the Moment of Inertia

The moment of inertia depends on the dimensions and distribution of mass within anobject; it also depends on the objects orientation, therefore the moment of inertia aboutthe x axis is different than the moment of inertia about the y axis. For a complex object

the moment of inertia about the x axis can be calculated using Equation 2, due to theparallel axis theorem. The parallel axis theorem compensates for each smaller objectscentroids distance from the actual centroid. The moment of inertia about the y axis canbe calculated in an identical fashion.

In Equation 2 1,'xI is the moment of inertia of the smaller object about the x axis, located

at iy . The variable d is the distance between iy andY .

2

222,'

2

111,', dAIdAII xxTx +++= (2)

Example Determine the moment of inertia for Figure 1relative to the

x-axis

Step 1 Determine the moment of inertia for area 1

46

223

2

111

3

11

2

111,'1,

1033.1

)4630(2400)60(4012

1

)(12

1

mm

mmmmmmmmmm

Yyhbhb

dAII xx

=

+=

+=

+=

Step 2 Determine the moment of inertia for area 2

45

223

2

222

3

22

2

222,'2,

1075.9

)4670(1600)20(8012

1

)(12

1

mm

mmmmmmmmmm

Yyhbhb

dAII xx

=

+=

+=

+=

Step 3 - Find the total moment of inertia about the x axis

46

4346

46

2,1,,

103.2

)/100.1(103.2

103.2

m

mmmmm

mm

III xxTx

=

=

=

+=


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Student Exercise Find the moment of inertia about the y axis

In Class Calculations

For the object assigned to you calculate the following:

1) The centroid2) The moment of inertia about the x axis3) The moment of inertia about the y axis

Ensure that all of your work is recorded in a legible and logical fashion.

Hint: Think about how you can use the location of the origin to simplif y your calculations .

Figure 2: I Beam Figure 3: L Shape

x

y

Figure 4: C Shape

Note: all dimensions are in mm


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centroid of area and moment of inertia calculation

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