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Moments of Inertia

- ????????????????????????????
- Centroid ?????????????????????????????????????????

?????????????????? - ??????????????????????????????????????????????????

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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

- 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

- 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

- 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,

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

- 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

- 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)

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.

- Solution
- Part (a)
- Differential element chosen, distance y from x

axis - Since dA b dy

- Solution
- Part (b)
- Moment of inertia about an axis passing through

the base of the rectangle obtained by applying

parallel axis theorem

- 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.

- Solution
- Composite Parts
- Composite area obtained by subtracting the circle

form the rectangle - Centroid of each area is located in the figure

- Solution
- Parallel Axis Theorem
- Circle
- Rectangle

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

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

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

- Solution
- Parallel Axis Theorem
- Rectangle A

- Solution
- Parallel Axis Theorem
- Rectangle B

- Solution
- Parallel Axis Theorem
- Rectangle D

- Solution
- Summation
- For moment of inertia for the entire

cross-sectional area,

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