Chapter Objectives Chapter Outline Method for determining the moment of inertia for an area Definition of Radius of Gyration Chapter Outline Definitions of Moments of Inertia for Areas Parallel-Axis Theorem for an Area Radius of Gyration of an Area Moments of Inertia for Composite Areas

10.1 Definition of Moments of Inertia for Areas The centroid of area is first order of moment of an area, Moment of inertial is the second order of moment of an area Moment of inertia is used to calculate, for example, when calculating a moment about an axis for distributed load, where the force is proportion to distance Mainly applied in Mechanics of Materials, Structural Mechanics, Fluid Mechanics, Mechanical Design

10.1 Definition of Moments of Inertia for Areas For example the pressure of fluid p is proportional to depth y, where is Specific Weight of fluid The force on a small area is dF = pdA = Moment of a force about the x axis is The value is called the moment of the area about x-axis

10.1 Definition of Moments of Inertia for Areas Moment of Inertia Consider the area A on the play x-y The moment of inertial of the small are about x and y is The moment of inertia about x and y axis is

10.1 Definition of Moments of Inertia for Areas Polar Moment of Inertia The second order of moment of the area dA about O or about the z axis is (this axis is called polar axis) r is the perpendicular distance from pole (z axis) to the area dA Polar moment of inertia

10.2 Parallel Axis Theorem for an Area The theorem is used to find the moment of inertia of an area about an arbitrary axis parallel to the axis passing the centroid of the area (the moment of inertia about the centroid must be known) Consider the area, in which x’ is passing through the centroid The area dA has the distance from x to x’ equal to y’ The moment of inertial of the area dA about x with the distance from the centroid x’ axis x’ equal to dy

10.2 Parallel Axis Theorem for an Area The first term is the moment of inertial about the axis passing the centroid The second term is zeros since x’ passes the centroid The third term is Hence and And where and

10.3 Radius of Gyration of an Area Radius of gyration is used in column design is This is similar to an equivalent radius when finding the moment of inertia The equation is similar to the moment of inertia

Procedure for Analysis Consider Integration of dA Example Find Ix Case 1 Consider dA parallel the x axis Ix Case 2 Consider dA perpendicular to the x axis Ix Find find Ix of small area from parallel axis theorem Case 1 Case 2

Example 10.1 Determine the moment of inertia for the rectangular area with respect to the centroidal x’ axis, (b) the axis xb passing through the base of the rectangle, (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’, Part (b) By applying parallel axis theorem,

Solution Part (c) For polar moment of inertia about point C, First determine, Then:

Example 10.2 Determine the moment of inertia for the area about x axis. Case 1 Case 2

Solution Solution I (case 1) Differential element area; Moment of Inertia

Solution Solution II (case 2) To determine the moment of inertia of the differential element area dA, by parallel-axis theorem. For a small rectangle element Then; by parallel-axis theorem

Example 10.3 Determine the moment of inertia for the area about x axis. Case 1 Case 2

Solution Solution I (case 1) Differential element area; Moment of Inertia

Solution Solution II (case 2) To determine the moment of inertia of the differential element area dA, For a small rectangle element Then; Integrating with respect to x

10.4 Moments of Inertia for Composite Areas The composite area is composed of areas with many simple geometries The moment of inertia of composite areas can be found by the sum of individual moment of inertia with sign (holes are negative) Procedure for Analysis Composite areas Divide a composite area into simple geometrical areas and locate the centroids of each of the areas and find the moment of inertias about each of their own centroids Parallel Axis Theorem Apply the parallel axis theorem to calculate the moment of inertia about the required axis Summation Sum all of the moment of inertia, beware of signs

Example 10.4 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 Parallel Axis Theorem Circle Rectangle Summation For moment of inertia for the composite area,