Centroids

Centroid Principles

Object’s center of gravity or center of mass.

Graphically labeled as

Centroid Principles

Point of applied force caused by

acceleration due to gravity.

Object is in state of equilibrium if

balanced along its centroid.

Centroid Principles

What is an object’s centroid location used for in statics?

Theoretical calculations regarding the

interaction of forces and members are

derived from the centroid location.

Centroid Principles

One can determine a centroid location

by utilizing the cross-section view of a

three-dimensional object.

Centroid Location

Symmetrical Objects

Centroid location is determined by an

object’s line of symmetry.

Centroid is located on

the line of symmetry.

When an object has multiple lines of symmetry,

its centroid is located at the intersection of the

lines of symmetry.

H

B

Centroid Location

The centroid of a square or rectangle is

located at a distance of 1/2 its height and 1/2

its base.

2

B

2

H

H

B

Centroid Location

The centroid of a right triangle is located at a

distance of 1/3 its height and 1/3 its base.

Centroid Location

The centroid of a ½ circle or semi-circle is

located at a distance of 4*R/3π away from the

axis on its line of symmetry

4

3

R

4 2 .

3

in

0.849 in. = 0.8in.

.849in.

Centroid Location Equations Complex Shapes

i i

i

y Ay=

A

i i

i

x Ax=

A

i i

i

z Az=

A

Centroid Location Complex Shapes

1. Divide the shape into simple shapes.

1

2

3

2. Determine a reference axis.

Centroid Location Complex Shapes

Review: Calculating area of simple shapes

Side2 Width * Height

πr2 ½ (base)(height)

Area of a

square = Area of a rectangle =

Area of

a circle =

Area of a triangle =

Centroid Location Complex Shapes

3. Calculate the area of each simple shape. Assume measurements have 3 digits.

2

Area of shape #1 =

Area of shape #2 =

Area of shape #3 =

3.00in. x 6.00in. = 18.0in.2

18in.2

½x3.00in.x3.00in. = 4.50in.2

4.5in.2

(3.00in.)2 = 9.00in.2

9in.2

side2

½ base x height

width x height

Centroid Location Complex Shapes

4. Determine the centroid of each simple shape.

1/3 b

1/3 h

Shape #1 Centroid Location

Shape #2 Centroid Location

Shape #3 Centroid Location

Centroid is located at the

intersection of the lines

of symmetry.

Centroid is located at the

intersection of the lines

of symmetry.

Centroid is located at the

intersection of 1/3 its

height and 1/3 its base.

Centroid Location Complex Shapes

5. Determine the distance from each simple shape’s

centroid to the reference axis (x and y).

4in.

4.5in.

1.5in.

3in

.

1.5

in.

4in

.

Centroid Location Complex Shapes

6. Multiply each simple shape’s area by its distance

from centroid to reference axis.

Shape Area (A) xi Axi

1 x

2 x

3 x

Shape Area (A) yi Ayi

1 18.0in.2 x

2 4.50in.2 x

3 9.00in.2 x

18.0in.2

4.50in.2

9.00in.2

1.50in.

4.00in.

4.50in.

27.0in.3

18.0in.3

40.5in.3

54.0in.3

18.0in.3

13.5in.3 1.50in.

4.00in.

3.00in.

Centroid Location Complex Shapes

7. Sum the products of each simple shape’s area and

their distances from the centroid to the reference axis.

Shape Ayi

1 54.0in.3

2 18.0in.3

3 13.5in.3

Shape Axi

1 27.0in.3

2 18.0in.3

3 40.5in.3

3

3

3

27.0in.

+ 18.0in.

+ 40.5in.

85.5in.3

Ax=

i

3

3

3

54.0in.

+ 18.0in.

+ 13.5in.

Ay=

i

85.5in.3

Centroid Location Complex Shapes

8. Sum the individual simple shape’s area to determine total shape area.

Shape A

1 18in.2

2 4.5in.2

3 9in.2

2

2

2

18.0in.

+ 4.5in.

+ 9.0in.

31.5in.2

A=

18in.2

4.5in.2

9in.2

Centroid Location Complex Shapes

9. Divide the summed product of areas and distances

by the summed object total area.

3

231.5

85.5

in.

i .=

n =

31.5in.2 A=

85.5in.3 Ax

=i

Ay=

i 85.5in.3

3

231.5

85.5

in.

i .=

n = 2.71in.

2.7in.

2.7

in.

2.71in. Does this shape have any lines

of symmetry?

Alternative Solution

• The same problem solved a different way.

– Previous method added smaller, more

manageable areas to make a more complex

part.

– Alternative Method = Subtractive Method

• Uses the exact same equations

• Uses nearly the exact same process

– Start with a bigger and simpler shape

– Treat shapes that need to be removed as

“negative” areas

Centroid Location – Subtractive Method

1. Determine reference axis and start with an area that is bigger than what is given

Square = Shape 1

2. Remove an area to get the centroid of the complex shape

Triangle = Shape 2

6 in.

6 in.

3 in.

3 in.

Centroid Location Complex Shapes

3. Calculate the area of each simple shape. Assume measurements have 3 digits.

Area of shape #1 =

6.0in. x 6.0in. = 36 in.2

-½x3.0in.x3.0in. = -4.5 in.2

-½ base x height

width x height

Area of shape #2 =

6 in.

6 in.

3 in.

3 in.

Note: Since the area is being

removed, we are going to call it

a negative area.

Centroid Location Complex Shapes

4. Determine the centroid of each simple shape.

Shape #1 Centroid Location

Centroid is located at the

intersection of the lines

of symmetry.

Middle of the square

Centroid is located at the

intersection of 1/3 its

height and 1/3 its base.

6 in.

6 in.

3 in.

3 in.

1/3 b

1/3 h

Shape #2 Centroid Location

Centroid Location Complex Shapes

5. Determine the distance from each simple shape’s

centroid to the reference axis (x and y).

6 in.

6 in.

3 in.

3 in.

5in.

3in.

3in

.

5in.

Centroid Location Complex Shapes

6. Multiply each simple shape’s area by its distance

from centroid to reference axis.

Shape Area (A) xi Axi

1 x

2 x

Shape Area (A) yi Ayi

1 36in.2 x

2 -4.5in.2 x

36in.2

-4.5in.2

3.0in.

5.0in.

108in.3

-22.5in.3

108in.3

-22.5in.3 5.0in.

3.0in.

6 in.

6 in.

3 in.

3 in.

5 in.

3 in.

3 in

.

5 in.

Centroid Location Complex Shapes

7. Sum the products of each simple shape’s area and

their distances from the centroid to the reference axis.

Shape Ayi

1 108in.3

2 22.5in.3

Shape Axi

1 108in.3

2 22.5in.3

3

3

108.0in.

+ -22.5in.

85.5in.3

Ax=

i

Ay=

i

85.5in.3

3

3

108.0in.

+ -22.5in.

Centroid Location Complex Shapes

8. Sum the individual simple shape’s area to determine total shape area.

Shape A

1 36 in.2

2 -4.5 in.2

2

2

36.0in.

+ -4.5in.

31.5in.2

A=

3 in.

6 in.

6 in.

3 in.

3 in.

3 in.

Centroid Location Complex Shapes

9. Divide the summed product of areas and distances

by the summed object total area.

3

231.5

85.5

in.

i .=

n =

31.5in.2 A=

85.5in.3 Ax

=i

Ay=

i 85.5in.3

3

231.5

85.5

in.

i .=

n = 2.71in.

2.71in. Does this shape have any lines

of symmetry? 2.7

in.

2.7in.

6 in.

6 in.

Centroid Location Equations Complex Shapes

i i

i

y Ay=

A

i i

i

x Ax=

A

i i

i

z Az=

A

Common Structural Elements

Angle Shape (L-Shape)

Channel Shape (C-Shape)

Box Shape

I-Beam

Centroid of Structural Member

Cross Section View

Neutral Plane

(Axes of symmetry)

Neutral Plane

Tension

Compression

Neutral Plane (Axes of symmetry)