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. Structural Calculators Properties of Areas Centroid.

Moment of inertia of a circle