Concept

Second moment of area

The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an (for an axis that lies in the plane of the area) or with a (for an axis perpendicular to the plane). In both cases, it is calculated with a multiple integral over the object in question. Its dimension is L (length) to the fourth power. Its unit of dimension, when working with the International System of Units, is meters to the fourth power, m4, or inches to the fourth power, in4, when working in the Imperial System of Units or the US customary system. In structural engineering, the second moment of area of a beam is an important property used in the calculation of the beam's deflection and the calculation of stress caused by a moment applied to the beam. In order to maximize the second moment of area, a large fraction of the cross-sectional area of an I-beam is located at the maximum possible distance from the centroid of the I-beam's cross-section. The planar second moment of area provides insight into a beam's resistance to bending due to an applied moment, force, or distributed load perpendicular to its neutral axis, as a function of its shape. The polar second moment of area provides insight into a beam's resistance to torsional deflection, due to an applied moment parallel to its cross-section, as a function of its shape. Different disciplines use the term moment of inertia (MOI) to refer to different moments. It may refer to either of the planar second moments of area (often or with respect to some reference plane), or the polar second moment of area (, where r is the distance to some reference axis). In each case the integral is over all the infinitesimal elements of area, dA, in some two-dimensional cross-section.

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https://en.wikipedia.org/wiki/Second_moment_of_area

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Official source

https://en.wikipedia.org/wiki/Second_moment_of_area

About this result

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xa0 Calculate the moment of inertia of a hollow cylinder of width e and mass M with respect to its revolution axis. We remind the moment of inertia of a full cylinder of mass M₁ :

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. xa0 Density of material:

\rho \Rightarrow \rho = \frac{M}{V}

V = \underbrace{h\pi R^2}_{V_1} - \underbrace{h\pi (R - e)^2}_{V_2}

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This approach is general. We can also reach this result more directly :

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xa0 The head of Mickey is composed of three disks of constant density ρ and width e. The radius of the figure is R and the radius of his ears is

\frac{R}{2}

. We note (Oz) the axis going through he middle of the figure of Mickey ( see sketch hereby). Calculate the moment of inertia of the head of Mickey around the (Oz), as function of R and of its mass M. Reminder : Moment of inertia of a homogeneous disk of radius R and of mass M around its axis of revolution :

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