Name: how area shape affects beam stiffness
Uploaded: 2025-05-22T09:06:05.000Z
Duration: 1 min 12 s
Channel: JoVE

The second moment of an area of an object about any axis can be estimated by concentrating its area into a thin strip parallel to the same axis.
The distance between this strip and the axis is defined as the radius of gyration. It is expressed as the square root of the ratio of the second moment of area about that axis and the area.
Similarly, the polar radius of gyration is expressed as the square root of the ratio between the polar moment and area.
Consider a rectangular beam with a cross-sectional area equal to the product of its height and width.
The second moment of area about a centroidal axis is obtained by multiplying the rectangle&#39;s width by the cube of its height and dividing it by twelve.
The radius of gyration is calculated using the area and second moment of area. The result equals the beam&#39;s height divided by the square root of twelve.
A higher radius of gyration indicates greater resistance against bending deformation due to an increased moment of inertia.

radius of gyration of an area

Radius of Gyration of an Area

The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object's resistance against bending deformation, or stiffness. To accurately estimate the second moment of area along any axis, one needs to concentrate all areas associated with that object into a thin strip, which should be placed parallel to that particular axis.
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As a result, the distance between this strip and the concerned axis can be determined by calculating its radius of gyration. The radius of gyration is represented as the square root of the ratio between the second moment of area and total area.
To illustrate this concept, consider a rectangular beam for which the area is equal to the product of the beam's height and width. In such a case, we can calculate its second moment of area about the centroidal axis through a simple formula: multiplying its width with the cube of its height and dividing the result by twelve.
<img alt="Area moment of inertia formula, \(I_x=\frac{bh^3}{12}\), structural analysis equation." src="/files/ftp_upload/14345/14345_PT_Equation_1.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium formula Ix=bh³/12, equation for moment of inertia calculation.">
The radius of gyration resulting from the product of area and the second moment of area gives a numerical value equal to the beam's height divided by the square root of twelve.
<img alt="Equilibrium formula kx=h/√12, static balance, educational equation illustration." src="/files/ftp_upload/14345/14345_PT_Equation_2.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium formula, kx=h/√12, educational diagram for physics analysis.">
A higher radius of gyration directly equates to increased resistance against deformations due to greater moments of inertia occurring therein. As such, understanding and manipulating the radius of gyration is one way for mechanical engineers to create durable structures that can safely bear significant load capacities.

The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object&#39;s resistance against bending ...

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Moment of Inertia

KEY TERMS AND CONCEPTS

how area shape affects beam stiffness

The second moment of area helps explain how a shape resists bending. It is also called the moment of inertia of area, and it is a key measure of stiffness. Engineers use it to understand how an object may deform under load.

To estimate the second moment of area along a chosen axis, the area is concentrated into a thin strip. That strip is placed parallel to the axis being studied. The distance from the strip to that axis is found using the radius of gyration.

The radius of gyration is the square root of the second moment of area divided by the total area. A rectangular beam shows this idea clearly. Its area equals height times width, and its second moment of area about the centroidal axis equals width times height cubed divided by twelve.

For that rectangle, the radius of gyration becomes the height divided by the square root of twelve. A larger radius of gyration means a larger moment of inertia and greater resistance to deformation. That is why understanding radius of gyration helps mechanical engineers design durable structures that can carry significant loads.

How Area Shape Affects Beam Stiffness

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The second moment of area helps explain how a shape resists bending. It is also called the moment of inertia of area, and it is a key measure of stiffness. Engineers use it to understand how an object may deform under load. To estimate the second moment of area along a chosen axis, the area is concentrated into a thin strip. That strip is placed parallel to the axis being studied. The distance from the strip to that axis is found using the radius of gyration. The radius of gyration is the...

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Moment of Inertia for Triangles

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key terms and concepts

Learn how the second moment of area and radius of gyration describe beam stiffness, using a rectangular beam example and the link to load resistance.

How Area Shape Affects Beam Stiffness | JoVE K12

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How Area Shape Affects Beam Stiffness

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