Moments of Inertia

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Moments of Inertia

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• Stress within the beam varies linearly with the
  distance from an axis passing through the
  centroid C of the beams cross-sectional area
• s kz
• For magnitude of the force acting
• on the area element dA
• dF s dA kz dA

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• Definition of Moments of Inertia for Areas
• Since this force is located a distance z from the
  y axis, the moment of dF about the y axis
• dM dF kz2 dA
• Resulting moment of the entire stress
  distribution applied moment M
• Integral represent the moment of inertia of area
  about the y axis

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• Moment of Inertia
• Consider area A lying in the x-y plane
• Be definition, moments of inertia of the
  differential plane area dA about the x and y axes
  
• For entire area, moments of
• inertia are given by

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• Moment of Inertia
• Formulate the second moment of dA about the pole
  O or z axis
• This is known as the polar axis
• where r is perpendicular from the pole (z axis)
  to the element dA
• Polar moment of inertia for entire area,

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Parallel Axis Theorem for an Area

• For moment of inertia of an area known about an
  axis passing through its centroid, determine the
  moment of inertia of area about a corresponding
  parallel axis using the parallel axis theorem
• Consider moment of inertia
• of the shaded area
• A differential element dA is
• located at an arbitrary distance
• y from the centroidal x axis

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• The fixed distance between the parallel x and x
  axes is defined as dy
• For moment of inertia of dA about x axis
• For entire area
• First integral represent the moment of inertia of
  the area about the centroidal axis

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• Second integral 0 since x passes through the
  areas centroid C
• Third integral represents the total area A
• Similarly
• For polar moment of inertia about an axis
  perpendicular to the x-y plane and passing
  through pole O (z axis)

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Moments of Inertia for an Area by Integration

• Example 10.1
• Determine the moment of
• inertia for the rectangular area
• with respect to (a) the centroidal
• x axis, (b) the axis xb passing
• through the base of the
• rectangular, and (c) the pole or
• z axis perpendicular to the x-y
• plane and passing through the
• centroid C.

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• Solution
• Part (a)
• Differential element chosen, distance y from x
  axis
• Since dA b dy

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• Solution
• Part (b)
• Moment of inertia about an axis passing through
  the base of the rectangle obtained by applying
  parallel axis theorem

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• Solution
• Part (c)
• For polar moment of inertia about point C

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• Example 10.5
• Compute the moment of
• inertia of the composite
• area about the x axis.

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• Solution
• Composite Parts
• Composite area obtained by subtracting the circle
  form the rectangle
• Centroid of each area is located in the figure

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• Solution
• Parallel Axis Theorem
• Circle
• Rectangle

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• Solution
• Summation
• For moment of inertia for the composite area,

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• Example 10.6
• Determine the moments
• of inertia of the beams
• cross-sectional area
• about the x and y
• centroidal axes.

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• Solution
• Composite Parts
• Considered as 3 composite areas A, B, and D
• Centroid of each area is located in the figure

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• Solution
• Parallel Axis Theorem
• Rectangle A

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• Solution
• Parallel Axis Theorem
• Rectangle B

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• Solution
• Parallel Axis Theorem
• Rectangle D

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• Solution
• Summation
• For moment of inertia for the entire
  cross-sectional area,