Moments of Inertia

Definition of Moments of Inertia for Areas: Measures the beam ability to resist bending. The larger the M.I the less the beam will bend. M.I is a geometrical property of a section and depends on a reference axis. The smallest M.I about any axis passes through the centroid.

Moments of Inertia for Composite Areas Composite area consist of a series of connected simpler parts or shapes Moment of inertia of the composite area = algebraic sum of the moments of inertia of all its parts Procedure for Analysis Composite Parts Divide area into its composite parts and indicate the centroid of each part to the reference axis Parallel Axis Theorem Moment of inertia of each part is determined about its centroidal axis

Moments of Inertia for Composite Areas Procedure for Analysis Parallel Axis Theorem When centroidal axis does not coincide with the reference axis, the parallel axis theorem is used Summation Moment of inertia of the entire area about the reference axis is determined by summing the results of its composite parts

Example 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 below.

Solution

Example Compute the moment of inertia of the composite area about the x axis & y-axia.

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Example Compute the moment of inertia of the composite area about the y-axis.

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Example Compute the moment of inertia of the composite area about the x/-axia.

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Example Determine the moment of inertia of the beam’s cross-sectional area about the x axis.

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Example Determine the moment of inertia of the composite area about the x axis.

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Example Determine the moments of inertia IXo and IYo of the L-secion shown below with respect to the centroidal axes shown in the same figure. All dimensions are in mm.

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Example Determine the moment of inertia of the cross-sectional area about x & y/ axes.

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Example Determine the moment of inertia of the composite area about the x &y-axes. 7.5 mm

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Determine the H-beam,s moment of inertia about the x &y-axes. Example Determine the H-beam,s moment of inertia about the x &y-axes.

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1)For the T-section below , determine the moment of inertia about centroidal axes.

2) Determine the moment of Inertia about x/ & y-axis

Assignment 3) Determine the moment of Inertia about y-axis