Product of Inertia and Mohr's Circle

5/3/2011

Lecture 1

LECTURE 1 TOPICS
I. Product of Inertia for An Area
Definition Parallel Axis Theorem on Product of Inertia Moments of Inertia About an Inclined Axes Principal Moments of Inertia Mohr’s Circle for Second Moment of Areas

II. Unsymmetrical Bending II Unsymmetrical Bending
Unsymmetrical Bending about the Horizontal and Vertical Axes of the Cross Section Unsymmetrical Bending about the Principal Axes

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Lecture 1, Part 1

Product of Inertia for an Area
Consider the figure shown below y x A dA y x Product of Inertia of A wrt x and y axis: Product of Inertia of Element dA:

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Product of Inertia for an Area
Consider the figure shown below y x A dA y x Unit: length4 – m4, mm4, ft4, in4 g NOTE: 1. Ixy can be positive, negative or zero. 2. The product of inertia of an area wrt any two orthogonal axes is zero when either of the axes is an axis of symmetry. Product of Inertia of A wrt x and y axis:

Product of Inertia for an Area
Parallel Axis Theorem Parallel‐Axis Theorem y’ y x’ Product of Inertia of A wrt x and y axis: dA C dy x dx y’ x’ ’ Product of Inertia of Element dA:

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Product of Inertia for an Area
Parallel Axis Theorem Parallel‐Axis Theorem y’ y x’

dA C dy

y’ x’ ’

x dx

The product of inertia of an area wrt any two perpendicular axes x and y is equal to the product of inertia of the area wrt a pair of centroidal axes parallel to the x and y axes added to the product of the area and the two centroidal distances from the x and y axes.

Product of Inertia for an Area
Example 1 y b

Determine the following: a) Product of Inertia, Ixy

h

x

b) Product of Inertia, Ix’y’ , with ) , xy respect to a pair of centroidal axes x’ and y’ parallel to the given axes x and y

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Product of Inertia for an Area
Example 1 y x dA h y x dy b

Solution: a) Product of Inertia, Ixy • Consider the strip ⎛ x⎞ dI xy = ⎜ ⎟ ydA ⎝ 2⎠ • The area,