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Centroids - · PDF fileCentroid Location Complex Shapes 7. Sum the products of each simple...

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  • Centroids

  • 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.

    Object is in state of equilibrium if

    balanced along its centroid.

  • Centroid Principles

    What is an objects 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

    objects 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 circle or semi-circle is

    located at a distance of 4*R/3 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

    r2 (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

    x3.00in.x3.00in. = 4.50in.2

    4.5in.2

    (3.00in.)2 = 9.00in.2

    9in.2

    side2

    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 shapes

    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 shapes 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 shapes 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 shapes 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

    -x3.0in.x3.0in. = -4.5 in.2

    - 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 shapes

    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 shapes 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 shapes 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 shapes 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)

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