Moment of Inertia of a Beam
Moment of Inertia is the cross sectional property of a beam. If we cut a simple beam in a plane perpendicular to its length we see a rectangle with a base b and height h. This beam is not really strong. We want to make a beam as strong as possible so we usually use I-Beam which belongs to the structural beam family. The horizontal part of the I-Beam is called a Flange and the vertical part is called the Web. The flange is moving the centroid of that section away from the neutral axis so it makes it less vulnerable to bending. So some flanges are made in triangular shapes where the centroid is 1/3 from the base . When we bend a beam we can see the top of the beam is either compressed or stretched and consequently the bottom of the beam is either stretched or compressed. Somewhere in between it switches over so it is called the neutral axis or centroidal axis. Every beam has a neutral axis. For a simple symmetrical I-Beam the neutral axis lies along a line parallel to the top and bottom rectangular flanges and is at the center of the beam. However, for non-symmetrical beams we have to find the centroid first and then the moment of inertia. So if the moment of inertia of the section about its horizontal (XX) axis was required then the vertical (y) centroid would be needed first by using:
Where x-axis is along the base of the lower flange.
So when we consider bending of a beam around an axis, first imagine the beam is along the y-axis therefore we see the beam is bending around the x-x axis so we have to find the moment of inertia about the x-x axis. If the beam is along the x-axis then the beam is bending around the y-axis. You have to pay attention to the position of the base and height of the beam when we bend the beam around an axis.
For Ixx =1/12*b*h3 and for Iyy =1/12*b3*h. The above formulas are for rectangular shaped beams i.e a beam that is made of rectangular sections. The only time we can use the above equations by themselves is when centroids of different pieces of the beam i.e rectangles are on the same line of neutral axis. Otherwise we have to use Parallel axis theorem which is really a transform of each centroid to the neutral axis by using the following equation: Ixx =1/12*b*h3 + A*d2 where A is the area of the rectangular piece and d is the distance between the centroid of that rectangle to the neutral axis. Observe the unit for the moment of Inertia is the fourth power of whichever unit you are using like m4 and the like. If the beam is made of an irregular shape we can find Ixx =∫(y2*dA) and Iyy = ∫(x2*dA). Observe the above integrals can be considered as double integrals but most of the time we can solve the integral by finding one variable in terms of the other so we will get a simple integral.