Objects 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 ID: 661133
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Slide1
CentroidsSlide2
Centroid Principles
Object’s center of gravity or center of mass
Graphically labeled as Slide3
Centroid Principles
Point of applied force caused by acceleration due to gravity
Object is in state of equilibrium if balanced along its centroidSlide4
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.
Slide5
Centroid Principles
One can determine a centroid location by utilizing the cross-section view of a three-dimensional object.Slide6
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.Slide7
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.Slide8
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.Slide9
Centroid Location
The centroid of a ½ circle or semi-circle is located at a distance of away from the axis on its line of symmetry
.849in.Slide10
Centroid Location Equations
Complex ShapesSlide11
Centroid Location
Complex Shapes
1. Divide the shape into simple shapes.
1
2
3
2. Determine a reference axis.Slide12
Centroid Location
Complex Shapes
Review:
Calculating area of simple shapes
Side
2
Width * Height
π
r
2
½ (base)(height)
Area of a square =
Area of a rectangle =
Area of
a circle =
Area of a triangle =Slide13
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
side
2
½ base x height
width x heightSlide14
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.Slide15
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.
1.5in.
4in.Slide16
Centroid Location
Complex Shapes
6. Multiply each simple shape’s area by its distance from centroid to reference axis.
Shape
Area (A
i
)
1
x
2
x
3
x
Shape
Area (A
i
)
1
x
2
x
3
x
Shape
Area (A
i
)
1
18.0in.
2
x
2
4.50in.
2
x
3
9.00in.
2
x
Shape
Area (A
i
)
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.Slide17
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
1
54.0in.
3
2
18.0in.
3
3
13.5in.
3
Shape
1
54.0in.
3
2
18.0in.
3
3
13.5in.
3
Shape
1
27.0in.
3
2
18.0in.
3
3
40.5in.
3
Shape
1
27.0in.
3
2
18.0in.
3
3
40.5in.
3
85.5in.
3
85.5in.
3
Slide18
Centroid Location
Complex Shapes
8. Sum the individual simple shape’s area to determine total shape area.
Shape
A
i
1
18in.
2
2
4.5in.
2
3
9in.
2
31.5in.
2
18in.
2
4.5in.
2
9in.
2
Slide19
Centroid Location
Complex Shapes
9. Divide the summed product of areas and distances by the summed object total area.
31.5in.
2
85.5in.
3
85.5in.
3
2.71in.
2.7in.
2.7in.
2.71in.
Does this shape have any lines of symmetry?
Slide20
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 MethodUses the exact same equationsUses nearly the exact same process
Start with a bigger and simpler shapeTreat shapes that need to be removed as “negative” areasSlide21
Centroid Location – Subtractive Method
Determine reference
axis and start with an area that is bigger than what is given
Square = Shape 1
Remove an area to get the centroid of the complex shape
Triangle = Shape 2
6 in.
6 in.
3 in.
3 in.Slide22
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
.Slide23
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 LocationSlide24
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.Slide25
Centroid Location
Complex Shapes
6. Multiply each simple shape’s area by its distance from centroid to reference axis.
Shape
Area (A
i
)
1
x
2
x
Shape
Area (A
i
)
1
x
2
x
Shape
Area (A
i
)
1
36in.
2
x
2
-4.5in.
2
x
Shape
Area (A
i
)
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.Slide26
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
1
108in.
3
2
22.5in.
3
Shape
1
108in.
3
2
22.5in.
3
Shape
1
108in.
3
2
22.5in.
3
Shape
1
108in.
3
2
22.5in.
3
85.5in.
3
85.5in.
3
Slide27
Centroid Location
Complex Shapes
8. Sum the individual simple shape’s area to determine total shape area.
Shape
A
i
1
36 in.
2
2
-4.5 in.
2
31.5in.
2
3 in.
6 in.
6 in.
3 in.
Slide28
3 in.
3 in.
Centroid Location
Complex Shapes
9. Divide the summed product of areas and distances by the summed object total area.
31.5in.
2
85.5in.
3
85.5in.
3
2.71in.
2.71in.
Does this shape have any lines of symmetry?
2.7in.
2.7in.
6 in.
6 in.
Slide29
Centroid Location Equations
Complex ShapesSlide30
Common Structural ElementsSlide31
Angle Shape (L-Shape)Slide32
Channel Shape (C-Shape)Slide33
Box ShapeSlide34
I-BeamSlide35
Centroid of Structural Member
Cross Section View
Neutral Plane
(Axes of symmetry)Slide36
Neutral Plane
Tension
Compression
Neutral Plane
(Axes of symmetry)