Most commonly, the moments of inertia are calculated with respect to the section's centroid. Where x and y are the coordinates of element dA with respect to the axis of interest. (Note 1) I x and I y are the moments of inertia about the x- and y- axes, respectively, and are calculated by: The second moment of area, more commonly known as the moment of inertia, I, of a cross section is an indication of a structural member's ability to resist bending. Therefore, the first moment of the entire area of a cross section with respect to its own centroid is zero. It should be noted that the first moment of an area is either positive or negative depending on the position of the area with respect to the axis of interest. This area extends from the point y 1 to the extreme fiber at the top of the cross section. In this case the point of interest is above the neutral axis, so it is simpler to consider the upper area which is shaded in blue in the figure below. We may calculate the first moment of the area either above or below this location. We are interested in calculating the shear stress at a point located at a distance y 1 from the centroid of the cross section. Where Q is the first moment of the area between the point y 1 and the extreme fiber (top or bottom) of the cross section. Recall that the shear stress at any point located a distance y 1 from the centroid of the cross section is calculated as: The first moment is also used when calculating the value of shear stress at a particular point in the cross section. If you compare the equations for Q above to the equations for calculating the centroid (discussed in a previous section), you will see that we actually use the first moment of area when calculating the centroidal location with respect to an origin of interest. If the area is composed of a collection of basic shapes whose centroidal locations are known with respect to the axis of interest, then the first moment of the composite area can be calculated as: The values x and y indicate the locations with respect to the axis of interest of the infinitesimally small areas, dA, of each element as the integration is performed. Where Q x is the first moment about the x-axis and Q y is the first moment about the y-axis. The first moment of an area with respect to an axis of interest is calculated as: The first moment of area indicates the distribution of an area with respect to some axis. The centroidal distance in the y-direction for a rectangular cross section is shown in the figure below: The centroidal distance, c, is the distance from the centroid of a cross section to the extreme fiber. Where x c,i and y c,i are the rectangular coordinates of the centroidal location of the i th section with respect to the reference point, and A i is the area of the i th section. If a cross section is composed of a collection of basic shapes whose centroidal locations are known with respect to some reference point, then the centroidal location of the composite cross section can be calculated as: The centroidal locations of common cross sections are well documented, so it is typically not necessary to calculate the location with the equations above. Where dA represents the area of an infinitesimally small element, A is the total area of the cross section, and x and y are the coordinates of element dA with respect to the axis of interest. If the exact location of the centroid cannot be determined by inspection, it can be calculated by: If the area is symmetric about only one axis, then the centroid lies somewhere along that axis (the other coordinate will need to be calculated). If the area is doubly symmetric about two orthogonal axes, the centroid lies at the intersection of those axes. The centroid of a shape represents the point about which the area of the section is evenly distributed.
The moment of inertia of a thin disk is Show more detail.ĭetails about the moment of inertia of a sphere. The expression for the moment of inertia of a sphere can be developed by summing the moments of infintesmally thin disks about the z axis. The moment of inertia of a solid sphere isĪnd the moment of inertia of a thin spherical shell is The moment of inertia of a sphere about its central axis and a thin spherical shell are shown. Moment of Inertia, Sphere Moment of Inertia: Sphere
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