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. Alternatively, one can solve for the required cross-sectional area, Areq bh (in 2) as the basis for designing or analyzing a rectangular beam for shear, corresponding to an allowable shear stress, allow (psi or ksi) for maximum shear force, V (lb or kips). 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.
In other words, the sum of 𝑎 + 𝑏 × 𝘩, all divided by two.Consider a rectangular cross-section having ABCD as its vertices. So, to get the area for just the one, we divide that total by two. That gives us the area for two trapeziums. Labelling the diagram to show each distinct edge demonstrates that an easy way to calculate the area of a trapezium is to calculate the area of the resulting parallelogram. The area of the parallelogram is the same as the area of the rectangle.īack to the trapezium, or trapeziums, since we have now created two by splitting the parallelogram in half.
When a triangle is chopped off a rectangle and moved to the other end, a parallelogram is formed. The area of a parallelogram can be measured by multiplying its length by its perpendicular height.īecause if one end were chopped off, it would fit perfectly at the other end. But a trapezium can be looked at as half a parallelogram. Calculating the area of a trapezium is a little different to calculating the area of a rectangle, because a trapezium does not have two pairs of equal sides.
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