WHAT IS RADIUS OF GYRATION?

We were discussing “Area moment of inertia” and “The difference between centroid and centre of gravity”,” and also we have seen the “Basic principle of complementary shear stresses” with the help of our previous posts.

Now we are going further to start our discussion to understand the basic concept of radius of gyration with the help of this post.

Let us first see here the basic concept of radius of gyration

Radius of gyration of a body or a given lamina is basically defined as the distance from the given axis up to a point at which the entire area of the lamina will be considered to be concentrated.

We can also explain the radius of gyration about an axis as a distance that if square of distance will be multiplied with the area of lamina then we will have area moment of inertia of lamina about that given axis.

Let us see the following figure which indicates one lamina with area A. Let us assume that lamina, displayed here, is made with number of small elemental areas a₁, a₂, a₃, a₄ …etc. As we have considered above that the total area of the lamina is A, now we need to determine here the radius of gyration about an axis for the given lamina.

Where,

x₁= Distance of the C.G of the area a₁ from OY axis

x₂= Distance of the C.G of the area a₂ from OY axis

x₃= Distance of the C.G of the area a₃ from OY axis

x₄= Distance of the C.G of the area a₄ from OY axis

Similarly, we will have

y₁= Distance of the C.G of the area a₁ from OX axis

y₂= Distance of the C.G of the area a₂ from OX axis

y₃= Distance of the C.G of the area a₃ from OX axis

y₄= Distance of the C.G of the area a₄ from OX axis

Area moment of inertia about the OY axis = a₁.x₁²+ a₂.x₂² + a₃.x₃²+ a₄.x₄²+…….

I_yy = a₁.x₁²+ a₂.x₂² + a₃.x₃²+ a₄.x₄²+…….

I_yy = Σ a.x²

Let us assume that entire area of the given lamina is concentrated at a distance k from the reference axis i.e. OY axis and therefore area moment of inertia for the entire area about the reference axis i.e. OY axis will be A.k².

I_yy = A.k²

Where k is the radius of gyration

Similarly, Area moment of inertia about the OX axis

I_xx = a₁.y₁²+ a₂.y₂² + a₃.y₃²+ a₄.y₄²+…….

I_xx = Σ a.y²

Let us assume that entire area of the given lamina is concentrated at a distance k from the reference axis i.e. OX axis and therefore area moment of inertia for the entire area about the reference axis i.e. OX axis will be A.k².

I_xx = A.k²

Where k is the radius of gyration

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