![]() 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.There are two different methods for finding the moment of inertia of any object, that is, the parallel axis theorem and the other one is the perpendicular axis theorem. However, when we change the location of the axis of rotation the formula as well as the value of the moment of inertia of a rectangle changes with it. To sum up, the formula for finding the moment of inertia of a rectangle is given by I=bd³ ⁄ 3, when the axis of rotation is at the base of the rectangle. H is the depth and b is the base of the rectangle. In this case, the formula for the moment of inertia is given as, The variables are the same as above, b is the width of the rectangle and d is the depth of it.įormula when the axis is passing through the centroid perpendicular to the base of the rectangle When the axis is passing through the base of the rectangle the formula for finding the MOI is, The formula for finding the MOI of the rectangle isĭ = depth or length of the rectangle Formula when the axis is passing through the base of the rectangle When the axis of rotation of a rectangle is passing through its centroid. ![]() Formula when the axis is passing through the centroid Let us see when we change the axis of rotation, and then how the calculation for the formula changes for it. ![]() Therefore, the equation or moment of inertia of a rectangular section having a cross-section at its lower edge as in the figure above will be, Similar to mathematical derivations, as we found the MOI for the small rectangular strip ‘dy’ we’ll now integrate it to find the same for the whole rectangular section about the axis of rotation CD. If we see the area of a small rectangular strip having width ‘dy’ will beĪnd the moment of inertia of this small area dA about the axis of rotation CD according to a simple moment of inertial formula which is And after finding the moment of inertia of the small strip of the rectangle we’ll find the moment of inertia by integrating the MOI of the small rectangle section having boundaries from D to A. Involvement of this ‘dy’ will make the assumptions and calculations easier. Now, let us find the MOI about this line or the axis of rotation CD.Īlso, consider a small strip of width ‘dy’ in the rectangular section which is at a distance of value y from the axis of rotation. ![]() ![]() Consider the line or the edge CD as the axis of rotation for this section. Where b is the width of the section and d is the depth of the section. Consider a rectangular cross-section having ABCD as its vertices. ![]()
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