Integrating curvatures over beam length, the deflection, at some point along x-axis, should also be reversely proportional to I. For standard cross-sections of bars, the moments of. When performing calculations, it is often necessary to calculate the moments of inertia of complex sections about various axes. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the developed curvature is reversely proportional to the moment of inertia I. The area moment of inertia of the section A about any axis is the sum of elementary areas dA, multiplied by the square of their distance to this axis. With an understanding of how a beam will respond to the expected forces in a specific application, it is possible to optimize the design of the beam for support, as well as mass and cost.Making similar considerations, the moment of inertia of the angle, relative to axis y0 is: Using the moments of inertia, the response of the beam to various forces can be evaluated. Sk圜iv also offers a Free Moment of Inertia Calculator for quick calculations or to check you have applied the formula correctly. If the same T beam as above were inverted, its moment of inertia would be calculated as follows: Application Of The Moment Of Inertia Of A T BeamĪ T beam is a common structural element used in engineering designs. In the event that a T beam is used in an inverted orientation, the moment of inertia about the x-axis will be different than that about a traditional T beam orientation.
In fact, since the pieces are identical, it is only necessary to calculate the moment of inertia for one section, and then use the same numerical value and the distance from the centroid on both sides. In the case of a double T beam, n will be two, but the same concept can be applied for calculating the total moment of inertia for objects that are divided into more than two pieces, so long as the individual moment of inertia of each piece is calculated about the same centroid. The above equation may look familiar, as it is a slight variation on the parallel axis theorem. To calculate the moment of inertia of such a beam, it is necessary to split it into two identical pieces and sum the individual moments of inertia about the centroid of the complete structure, using the following equation: Double T Beam Moment Of InertiaĪ double T beam is made up of three plates, as shown in the following figure: Using the same T beam, the moment of inertia about the y-axis is as follows: Extension Of T Beam Moment Of Inertiaīeginning with the moment of inertia of a T beam, there are further extensions, which are introduced below. Therefore, the moment of inertia of section A is calculated as follows:Īnd the moment of inertia of section B is calculated as follows:įinally, adding the two together, the total moment of inertia of the T beam is calculated as follows: Example Calculation To calculate the moment of inertia of a T beam about its y-axis, the beam must again be divided into two sections, as follows:
ȳ c is the centroid of the entire beam, with SI units of mm.ȳ i is the centroid of the i-th part of the T beam, with SI units of mm.In the case of a T beam, the centroidal axis is ȳ c, so the distance between the two axes can be calculated with the following equation: To apply the parallel axis theorem to calculate the moment of inertia of a T beam, the first step is to determine the distance between the two axes, Ix’ and Ix. d is the distance between the two axes, with SI units of mm.A is the cross-sectional area beam, with SI units of mm 2.I is the moment of inertia about the centroid axis, with SI units of mm 4.I’ is the moment of inertia about the non-centroid axis, with SI units of mm 4.The parallel axis theorem specifies that the moment of inertia about a non-centroid axis can be calculated using the moment of inertia about a centroid axis, so long as the two axes are parallel.
In that case, the moment of inertia about the x’-axis, I x’, will have SI units of mm 4.
Note that in common engineering applications, the dimensions of a T beam may be given in mm. t f is the thickness of the flange, with SI units of m.b is the width of the flange, with SI units of m.h is the height of the beam, with SI units of m.t w is the width of the thickness of the web, with SI units of m.