Moments of inertia for areas Previously determined centroid for an area by considering first moment of area about an axis, Previously determined centroid for an area by considering first moment of area about an axis, Moment of inertia → second moment of an area (“moment of inertia” is a misnomer) Moment of inertia → second moment of an area (“moment of inertia” is a misnomer) Moments of inertia of a differential planar area dA about the x and y axes are dI x = y 2 dA and dI y = x 2 dA, respectively SHOW Moments of inertia of a differential planar area dA about the x and y axes are dI x = y 2 dA and dI y = x 2 dA, respectively SHOW For the entire area, For the entire area, Second moment of differential area dA about the pole O or z axis is called the polar moment of inertia, dJ o = r 2 dA Second moment of differential area dA about the pole O or z axis is called the polar moment of inertia, dJ o = r 2 dA For the entire area, For the entire area,

Parallel-Axis Theorem SHOW SHOW However, x’ passes through the centroid of the area However, x’ passes through the centroid of the area Similarly, Similarly,

Radius of Gyration of an Area A quantity often used in the design of columns A quantity often used in the design of columns (units are length) (units are length)

Moments of inertia for an area by integration (Case 1) Specify the differential element dA (usually a rectangle with finite length and differential width) Specify the differential element dA (usually a rectangle with finite length and differential width) Differential element located so it intersects the boundary of the area at an arbitrary point (x,y) Differential element located so it intersects the boundary of the area at an arbitrary point (x,y) Case 1: Case 1: –Length of element is oriented parallel to axis, SHOW –I y determined by a direct application of since element has an infinitesimal thickness dx (all parts of the element lie at the same moment-arm distance x from the y-axis)

Moments of inertia for an area by integration (Case 2) Case 2: Case 2: –Length of element is oriented perpendicular to axis, SHOW – does not apply, all parts of the element will not lie at the same moment-arm distance from axis –First calculate moment of inertia of the element about a horizontal axis passing through the element’s centroid –Determine moment of inertia of the element about the x-axis by using parallel-axis theorem –Integrate to yield I x EXAMPLES (pg 523) EXAMPLES (pg 523)

Moments of inertia for composite areas Divide the area into its “simpler” composite parts Divide the area into its “simpler” composite parts Determine the perpendicular distance from the centroid of each part to the reference axis (often centroid of entire area) Determine the perpendicular distance from the centroid of each part to the reference axis (often centroid of entire area) Determine the moment of inertia of each part about its centroidal axis (parallel to the reference axis) Determine the moment of inertia of each part about its centroidal axis (parallel to the reference axis) If the centroidal axis does not coincide with the reference axis, apply the parallel-axis theorem to determine the moment of inertia of the part about the reference axis If the centroidal axis does not coincide with the reference axis, apply the parallel-axis theorem to determine the moment of inertia of the part about the reference axis The moment of inertia of the entire area about the reference axis is obtained by summing the results of its parts (“holes” are subtracted) The moment of inertia of the entire area about the reference axis is obtained by summing the results of its parts (“holes” are subtracted) EXAMPLES (pg 530) EXAMPLES (pg 530)