If we consider the diameter of a circle D, then we must also take ‘r’ the radius as D/2. Moment of Inertia of a Circle about its Diameter In the case of a quarter circle the expression is given as: In case of a semi-circle the formula is expressed as: In the case of a circle, the polar moment of inertia is given as: Similarly, the moment of inertia of a circle about an axis tangent to the perimeter(circumference) is denoted as: The moment of a circle area or the moment of inertia of a circle is frequently governed by applying the given equation: The moment of Inertia formula can be coined as:
Mathematically, it is the sum of the product of the mass of each particle in the body with the square of its length from the axis of rotation. Yes, the proper definition of the moment of inertia is that a body tends to fight the angular acceleration. When a body starts to move in rotational motion about a constant axis, every element in the body travels in a loop with linear velocity, which signifies, every particle travels with angular acceleration. It can be inferred that inertia is related to the mass of a body. Rod Smith, P.E.First of all, let us discuss the basic concept of moment of inertia, in simple terms. For that application of load, you'll have bolts in tension, but I believe the compression will typically be carried through contact of the plates (unless there are standoff sleeves around the bolts carrying the compression, of course).
For out-of-plane moments (producing tension or prying on the bolts), I think you would typically calculate the moment of inertia about an axis closer to the compression edge of the plate. In my design work, I've never had occasion to need Ix or Iy for a bolt group individually only combined for the polar I to calculate forces due to an in-plane moment (producing shear on the bolts, for web splice plates on an I-beam, e.g.). I wonder if we may have skipped over a more important aspect about how and when to apply the moments of inertia once you have calculated them.
Centroidal polar moment of inertia of a circle how to#
With all of the posts, you should have a pretty good handle on how to calculate Ix and Iy, and the polar I (Ix + Iy, or the equation I posted). Yes Amar-Dj, you've got a handle on the units. Rod Smith, P.E., The artist formerly known as HotRod10 RE: Moment of Inertia of a bolt group You can add them up individually, or combine and reduce the terms to simplify it for larger groups. I don't have time right now to recreate the derivation, but it should just be a matter of rearranging and combining the equations for Ix and Iy given in the thread I linked to above.Įdit: In it's most basic sense, the polar moment of inertia of a bolt group is the summation of d 2, where d is the distance from the centroid of the group to the center of each bolt. The formula I use for the total polar I of a bolt group (that I either derived or found a long time ago) is:
The formula and derivation can be found in this thread.
The moment of inertia of the bolts themselves about their individual centroids is ignored as being inconsequential.įor the polar moment of inertia, which is what you would use to calculate the force for a bolt group where a moment is about the centroid of the bolt group, is Ix + Iy. The reason the units are mm 2 is that the "I" of the bolt group only considers the "Ad 2" portion of the moment of Inertia calculation (Io + Ad 2), where "A" is set equal to 1 for convenience of the calculations (so you don't have to multiply by the area of the bolt to get stress and divide it back out to get force).