2012 Project Lead The Way Inc Principles of Engineering Centroid Principles Objects center of gravity or center of mass Graphically labeled as Centroid Principles Point of applied force caused by acceleration due to gravity ID: 591267
Download Presentation The PPT/PDF document "Centroids" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Centroids
© 2012 Project Lead The Way, Inc.
Principles
of
EngineeringSlide2
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 LocationSymmetrical 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 LocationThe 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
23
2. Determine a reference axis.Slide12
Centroid Location Complex ShapesReview: Calculating area of simple shapes
Side2Width * Height
πr2
½ (base)(height)
Area of a square =
Area of a rectangle =
Area of
a circle =
Area of a triangle =Slide13
Centroid Location Complex Shapes3. Calculate the area of each simple shape.Assume measurements have 3 digits.2Area of shape #1 =Area of shape #2 =
Area of shape #3 =3.00in. x 6.00in. = 18.0in.
218in.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 Shapes4. 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 Shapes5. 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 Shapes6. 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 Shapes7. 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 Shapes8. Sum the individual simple shape’s area to determine total shape area.Shape
Ai
1
18in.
2
2
4.5in.
2
3
9in.
2
31.5in.
2
18in.
2
4.5in.
2
9in.
2
Slide19
Centroid Location Complex Shapes9. Divide the summed product of areas and distances by the summed object total area.
31.5in.2
85.5in.385.5in.
3
2.71in.
2.7in.
2.7in.
2.71in.
Does this shape have any lines of symmetry?
Slide20
Alternative SolutionThe same problem solved a different wayPrevious method added smaller, more manageable areas to make a more complex part.Alternative Method = Subtractive MethodUses the exact same equationsUses nearly the exact same processStart with a bigger and simpler shapeTreat shapes that need to be removed as “negative” areasSlide21
Centroid Location – Subtractive MethodDetermine reference axis and start with an area that is bigger than what is givenSquare = Shape 1Remove an area to get the centroid of the complex shapeTriangle = Shape 2
6 in.
6 in.
3 in.
3 in.Slide22
Centroid Location Complex Shapes3. 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 heightArea 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 Shapes4. Determine the centroid of each simple shape.Shape #1 Centroid LocationCentroid is located at the intersection of the lines of symmetry.Middle of the squareCentroid 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 Shapes5. 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 Shapes6. Multiply each simple shape’s area by its distance from centroid to reference axis.Shape
Area (Ai
)
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 Shapes7. 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 Shapes8. Sum the individual simple shape’s area to determine total shape area.Shape
Ai
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 MemberCross Section View
Neutral Plane(Axes of symmetry)Slide36
Neutral Plane
Tension
Compression
Neutral Plane
(Axes of symmetry)